The Bank of England quarterly model

The Bank of England Quarterly Model Richard Harrison, Kalin Nikolov, Meghan Quinn, Gareth Ramsay, Alasdair Scott and Ryland Thomas Further copies of this publication are available, at £10 plus p&p, from: Publications Group Telephone: 020 7601 4030 email: mapublications@bankofengland.co.uk The Bank of England’s website is at www.bankofengland.co.uk A pdf file of this book and an ASCII file containing the model equations are available at: www.bankofengland.co.uk/publications/beqm/ Bank of England, Threadneedle Street, London, EC2R 8AH c Bank of England 2005 ISBN 1 85730 153 6 Contents Foreword 1 Acknowlegements 3 1 Introduction and overview 5 1.1 The role of models and forecasts at the Bank of England 5 1.2 An overview of BEQM 6 1.3 Some key technical features of BEQM 8 1.4 The structure of this book 9 1.5 Summary 2 Project motivation and model design 10 11 2.1 Motivations and challenges 11 2.2 The design of BEQM 12 2.3 A comparison with other models 15 2.4 Summary 18 3 The core theory 23 3.1 Overview 23 3.2 Characterisation of the agents 26 3.3 Characterisation of the markets 38 3.4 The nominal side of the economy and monetary transmission 44 3.5 Long-run growth 49 3.6 Summary 50 4 The core/non-core hybrid approach 61 4.1 Functional forms 62 4.2 Making the hybrid system work 63 4.3 Summary 67 5 Implementing and solving the model 69 5.1 Setting up the model 69 5.2 Solving the model 74 5.3 Recursive simulations 75 5.4 Applications 79 5.5 Summary 83 i 6 Parameterisation and evaluation 85 6.1 Issues in parameterising the core model 85 6.2 The model-consistent database 86 6.3 Parameterising the structural core model 97 6.4 Parameterising the non-core equations 113 6.5 An evaluation of the model’s forecast performance 121 6.6 Summary 126 7 Model properties 127 7.1 Interpreting the responses 127 7.2 Shock responses 128 7.3 Summary 150 8 Final remarks 151 References 153 Appendices A The core model 165 A.1 Mnemonics 165 A.2 Core model equations 172 B The non-core equations 197 B.1 Mnemonics 197 B.2 Non-core equations 203 C Data transformations and sources 225 D Parameter and exogenous values 243 ii List of figures 2.1 The trade-off between theory and data 12 2.2 A stylised forecast sequence 15 3.1 Key agents in the model macroeconomy 24 3.2 Key flows and assets 25 3.3 Consumption equilibrium in the steady state 27 3.4 Consumption and net foreign asset equilibrium 29 3.5 Production-clearing flows and stocks 39 3.6 The monetary transmission mechanism 49 5.1 Timing conventions for bonds 69 5.2 Timing convention for housing 70 5.3 Timing convention for capital stock 71 5.4 Building a profile under recursive simulations 76 6.1 Ratios of expenditures to private sector output 106 6.2 Ratios of stock values to private sector output 107 6.3 Relative prices 107 6.4 Comparison of growth rate forecasts from BEQM and the MTMM 123 6.5 Theil inequality coefficients 124 6.6 Comparison of growth rate forecasts from the BEQM core and the MTMM 125 7.1 Effects of an interest rate shock 130 7.2 How expectations can affect shock responses 132 7.3 Effects of a productivity shock 135 7.4 Effects of a government spending shock 139 7.5 Effects of a terms of trade shock 142 7.6 Effects of a world demand shock 145 7.7 Effects of a participation shock 148 iii List of tables 6.1 The mapping from data to model concepts 90 6.2 The mapping from labour market data to model concepts 97 6.3 Statistical tests for stationarity of gaps 115 A.1 Endogenous variables 165 A.2 Exogenous variables 168 A.3 Parameters 169 A.4 Working variables 171 B.1 Endogenous variables 197 B.2 Exogenous variables 201 B.3 Parameters 202 C.1 Data sources and transformations for BEQM 225 D.1 Parameter values 243 v List of technical boxes 1 Some recent developments towards hybrid structural models 19 2 The consumer’s maximisation problem 30 3 Private sector output and government output 36 4 Over-discounting and insurance against mortality 42 5 The determination of inflation 46 6 How does the Blanchard-Yaari model make consumption stationary? 51 7 The firm’s maximisation problem 54 8 The union bargaining problem 58 9 The hybrid approach applied to the Ramsey model 64 10 The exogenous variables model 78 11 Conditioning nominal interest rate paths 80 12 Detrending and model units 88 13 The sensitivity of the steady state to changes in parameter values 103 vii Foreword This book contains details of the Bank of England’s new quarterly model which is used to help the Monetary Policy Committee produce its economic projections. The book builds on the previous books on the Bank’s use of economic models. The new quarterly model is a valuable addition to the Bank’s ‘suite of models’. It does not represent a significant shift in the Committee’s view of how the economy functions or of the transmission mechanism of monetary policy. Rather its value lies in the fact that its more consistent and clearly articulated economic structure better captures the MPC’s vision of how the economy functions and so provides the Committee with a more useful and flexible tool to aid its deliberations. The project to develop a new quarterly model has been an important initiative and I am grateful to all the Bank staff who contributed to its success. The Bank has an outstanding group of economists and they are the unsung heroes and heroines of the success of the United Kingdom’s new monetary framework. But all economic models, however good, represent simplifications of reality and, as such, no single model can possibly address the many and varied issues that matter for economic policy. This recognition is central to the Bank’s use of economic models and its approach to economic forecasting. The Bank relies on a plurality of models to help inform the Committee’s projections. And these models are used as tools to help the Committee reach the economic judgements that play a critical role in shaping its projections, rather than simply to generate mechanical forecasts. Economic forecasting is ultimately a matter of judgement. The economy is constantly changing and so too will the quarterly model and the other models used by the Bank. The model described in this book is part of an evolving process and the Bank will continue to devote resources to both reaping the benefits from the advances this new model brings and developing it further. Mervyn King, Governor of the Bank of England January 2005 1 Acknowledgements The authors would like to thank Mark Allan, Pedro Alvarez-Lois, Charles Bean, Andy Blake, Fabio Canova, Spencer Dale, Rebecca Driver, Karen Dury, Philip Evans, Guillermo Felices, George Kapetanios, Hashmat Khan, Lavan Mahadeva, Stephen Millard, Andrew Moniz, Adrian Pagan, Laura Piscitelli, Simon Price, James Proudman, Peter Sinclair, Jan Vlieghe, Peter Westaway, Simon Wren-Lewis and Tony Yates for comments on drafts of this material, and Andrew Holder for his work as editor. We would also like to thank our successors as forecasters and model users – in particular, James Bell, Alex Brazier, Michael Grady, and Iain de Weymarn – for their work during the transition to the new model. The views expressed in this book are those of the authors and should not be thought to represent those of the Monetary Policy Committee. 3 Chapter 1 Introduction and overview The Bank of England has developed a new macroeconomic model for use in preparing the Monetary Policy Committee’s quarterly economic projections. The new Bank of England Quarterly Model (BEQM) was used to an increasing extent during 2003 and is the main tool in the suite of models employed by the staff and the Monetary Policy Committee (MPC) in the construction of the projections contained in the quarterly Inflation Report. This book explains the motivation for BEQM and the economic and modelling approaches underlying it; it also includes a full technical account of the model and its quantitative properties. This chapter (1) describes the role of models at the Bank of England in helping to produce the MPC’s quarterly projections, explains the motivation for the new model, and provides an overview of BEQM and the modelling approaches underlying it. It also includes a guide to the remaining chapters, which describe BEQM in greater detail. 1.1 The role of models and forecasts at the Bank of England The Bank of England is mandated by the Chancellor of the Exchequer to aim at an inflation target – at the time of writing, a 2% annual inflation rate of the Consumer Prices Index (CPI) – and uses a very short-term nominal interest rate as its instrument to pursue this target. Because of the lags between changes to interest rates and the associated effects on inflation, setting monetary policy is inherently a forward-looking exercise. Hence the quarterly Inflation Report, in addition to assessing the current state of the economy, contains projections for output growth and inflation for up to three years out, based on assumptions of both constant and market-based interest rates. These projections represent the Committee’s best judgement of both the most likely central outcome and the range of possible alternative outcomes around that central case. A key element of the analysis contained in the Inflation Report is to consider the major risks and uncertainties surrounding the central projection, rather than to focus simply on the central point predictions for GDP growth and inflation. The Bank uses numerous economic models to help produce these projections. (2) No model can do everything – all models are imperfect, precisely because they are simplifications of reality. And each projection is a judgement of the MPC rather than a mechanical output from any model. Nonetheless the Bank has found, like many other policy institutions, that, when producing its economic projections, it is helpful to use a macroeconomic model as the primary organisational framework to process the various judgements and assumptions made by the Committee. This is the role now played by BEQM. The forecast process at the Bank involves a high degree of interaction between the Bank’s staff and the members of the Monetary Policy Committee. In particular, a key element of the forecast process is for Committee members to assess the extent to which different economic judgements and assumptions concerning the major issues affecting the economy could influence their view of future prospects. This process is critical to understanding the nature of the risks and uncertainties surrounding the central projection. In order to be able to carry out this sort of analysis, the main forecast model ideally needs a relatively explicit economic structure that identifies the key behavioural parameters and channels within the economy. (1) This chapter is based on the article on the new model that was published in the Summer 2004 Quarterly Bulletin, see Bank of England (2004). (2) The Bank’s use of economic models is discussed in more detail in Chapter 1 of Bank of England (1999a). 5 The Bank of England Quarterly Model The importance of having a model suitable for analysing the implications of different economic judgements and assumptions is not new. This role was also central to the design of the previous macro model used by the Bank, the Medium-Term Macro Model (MTMM). (3) Indeed, the basic economic structure of BEQM is very similar to that of the MTMM. The aim of BEQM is not to incorporate a different view of how the economy works or of the role of monetary policy. Rather, the decision to develop a new model reflected the view that recent advances in both economic understanding and, importantly, in computational power meant that it was possible to improve upon the articulation of the economic structure within the MTMM. As Pagan (2003) noted in his report on modelling and forecasting at the Bank of England, the MTMM was no longer ‘state of the art’. In particular, Pagan concluded that ‘It seems highly likely that [a new model] could achieve the same empirical coherence [as the MTMM] with a stronger theoretical perspective’. In doing so, this would provide the Committee with a more flexible and coherent framework to aid its economic deliberations. That, in short, is what the new model tries to achieve through a clearer articulation of the underlying structure of the economy and a more explicit identification of the role expectations play. 1.2 An overview of BEQM BEQM describes the behaviour of the UK economy at a relatively aggregated level that is closely related to the incomes and expenditures recorded in the UK National Accounts. To do this, the model contains formal descriptions of the behaviour of private domestic agents, policymakers and the rest of the world, and their interactions in markets for capital and financial assets, goods, and labour. Households consume imported and domestically produced goods. When deciding on their current level of consumption, and hence their level of saving or borrowing, households are assumed to want to keep their lifetime consumption as smooth as possible. To do this, households can borrow and save using a range of financial assets, including domestic equities, corporate debt, government debt, money, and foreign assets. In addition, in the short run, households’ levels of consumption can be influenced by a variety of other factors, such as short-term fluctuations in their income and their level of confidence about the future. Firms seek to maximise profits by hiring labour and buying capital in order to produce output. Firms and workers bargain over wages and, given the outcome, firms are assumed to choose the labour they wish to employ so that the costs of any extra workers are compensated for by the higher revenues they generate. Similarly, firms’ desired level of capital is determined by the cost of capital and the return to extra investment. The output that firms produce is sold in markets for domestic consumption, investment and government procurement, as well as in housing and export markets. Firms are assumed to face varying degrees of competition in these markets, which implies that firms may receive a different profit margin from the sale of their goods in each market. The composition of total sales will therefore affect revenue and profits, so that relative demand conditions will matter as well as overall demand conditions. Firms face competition from importers for consumption and investment goods, and have to price their products in export markets so as to achieve maximum profits. In addition, various short-run factors can influence firms’ behaviour, such as the short-run prospects for demand affecting the speed with which they invest. The government buys output from domestic firms and labour from households, financed by raising taxes and selling debt, in addition to a small amount of revenue that accrues from seigniorage. Total revenue also has to be sufficient to pay the cost of servicing the existing level of government debt and any government transfers. For long-run solvency, the fiscal authority may at some stage have to adjust a policy instrument – such as a tax rate – to ensure that the fiscal budget constraint is met. A variety of (3) The Medium-Term Macro Model is described in more detail in Bank of England (2000). 6 Introduction and overview fiscal policy ‘rules’ can be considered. In general, these rules assume that any required fiscal adjustment occurs only gradually. The monetary policy maker has the job of anchoring the nominal side of the economy. The nominal target could, in principle, be specified in terms of any nominal aggregate, such as the nominal exchange rate, the growth rate of nominal output, or the growth rate of the money stock. The default assumption is that the central bank targets an annual inflation rate of the CPI of 2%, using the short nominal interest rate as its instrument. An assumption about the policy rule used by the central bank – the monetary policy reaction function – is required for inflation to be anchored in the long run. The structure allows a variety of different reaction functions to be incorporated. BEQM assumes that UK capital markets are ‘small’, in the sense that the demand for and supply of financial assets in the United Kingdom do not affect the level of interest rates prevailing in the rest of the world. Since all claims on domestic firms’ assets and government debt must ultimately be held either by domestic households or the rest of the world, it follows that the United Kingdom’s net foreign asset position is determined jointly by the decisions of firms and the government about how many financial liabilities to issue and by domestic households about how many of these assets to hold. The rest of the world affects these decisions through assumptions about the level of foreign real interest rates and world demand. These decisions also have implications for the United Kingdom’s trade balance. Suppose, for example, UK households were assumed to want to hold only some of the domestic financial assets on offer, such that the United Kingdom maintained a net debt with the rest of the world. This would imply that, in the long run, the United Kingdom would need to have a trade surplus sufficient to meet the costs of servicing this debt. The equilibrium real exchange rate moves so as to ensure that exports and imports achieve this long-run balance. This story is further complicated by the assumption that UK producers have some market power in the prices they set in world markets, so the long-run trade balance will, in general, depend on assumptions made about conditions in both financial and goods markets. The main channels through which changes in monetary policy are transmitted to the rest of the economy are similar to those previously described by the Monetary Policy Committee. (4) The fact that prices and nominal wages move only slowly means that the central bank, by changing the nominal interest rate, has the ability to influence real interest rates. Lower real rates tend to encourage consumers to spend more now. Lower real rates also encourage investment and spending on housing by lowering financing costs, and they make it less costly to hold inventories. The combined effect is to push up domestic demand. To meet that demand, firms will demand more of the factors used in the production of goods and services, namely capital and labour. This in turn is likely to increase the costs of these factors of production. The fact that the UK economy is a small open economy adds an important channel through which monetary policy operates. In particular, a lower domestic real interest rate may tend to encourage a depreciation in the real exchange rate. This will lead to both a direct price effect – the prices of imported goods will rise – and a number of possible indirect (or ‘second-round’) effects, reflecting both any pass-through from higher import prices onto domestic prices and costs, and the impact of any change in competitiveness associated with the change in the real exchange rate on the United Kingdom’s trade balance. The impact of changes in aggregate demand on prices and inflation will depend on the way in which agents – households, firms, policymakers and the rest of the world – interact with each other. Other (4) See Bank of England (1999b). 7 The Bank of England Quarterly Model things being equal, increased demand for workers leads to higher wage costs, which firms will typically attempt to pass on to some degree in the form of higher prices. Similarly, increases in world prices or an exchange rate depreciation create pressure on import prices. And increased demand for domestically produced goods will also create incentives for firms to raise prices. Inflationary pressures reflect the degree of imbalance between the level of demand and the capacity of firms to meet that demand. The level of demand and potential supply will depend on both the current stance of monetary policy and the stance expected in the future. Likewise, firms’ responses to these pressures on capacity will depend on the extent to which they are likely to persist, and hence on the expected stance of monetary policy in the future. The importance of future expectations in determining current inflationary pressures underlines the central importance of monetary policy anchoring private sector expectations of the long-term inflation rate. 1.3 Some key technical features of BEQM The improved economic structure of BEQM is reflected in a number of specific features. First, it has a well defined steady state. This means that, in the long run, all variables in the model settle on paths that are growing consistently with each other in a sustainable equilibrium. This aids analysis of economic issues, since an understanding of the medium term requires an understanding not just of short-run forces, but also of where the economy is heading to in the long run. For example, a stable steady-state solution would not be compatible with a situation in which household debt was increasing without bound. In characterising this steady state, careful attention has been paid to ‘stock-flow’ and ‘flow-flow’ accounting. This is designed to ensure that all economic flows within the economy are accounted for – all income is spent or saved, for example – and that all expenditures have implications for physical and financial stocks. This again aids the understanding of medium-term issues. For example, stock-flow consistency implies that monetary policy cannot stimulate consumption indefinitely, since this would imply an erosion of households’ net wealth, which they could not ignore forever. Another important feature of the new model is that it contains more explicit forward-looking representations of agents’ expectations about the future. These include expectations about future labour income, aggregate demand, the exchange rate, and so on. Models with fully forward-looking agents can sometimes exhibit unrealistic dynamic properties; in particular, if households and firms are assumed to have perfect foresight, they might adjust their behaviour immediately in response to future anticipated events. But in reality the economy does not ‘jump’ about in this fashion. That partly reflects the fact that it is often costly for households and firms to change their behaviour very rapidly. In addition, firms and households do not have perfect foresight. Instead, they have to form expectations on the basis of limited information. BEQM incorporates both of these features. In particular, it is structured in such a way that assumptions about the speed of adjustment and the amount of information available to agents can be changed in order to help the Committee to assess how these assumptions could affect the future path of the economy. These features are not new: some or all of them are present in many other models currently used by policy institutions, such as the Bank of Canada’s Quarterly Projection Model, the FRB/US model at the US Federal Reserve Board of Governors, and the Reserve Bank of New Zealand’s FPS model. Indeed, these features were often an explicit aim of pioneering work on macro modelling in the United Kingdom over the past 25 years, such as the Liverpool model, the London Business School model, the COMPACT model, and various models at the Cambridge Economic Policy Group and the National Institute of Economic and Social Research. The implementation in BEQM may differ in technical details, reflecting 8 Introduction and overview decisions made on how to satisfy the particular demands of forecasting at the Bank, but the basic ideas and motivations are the same. 1.4 The structure of this book This book explains the factors which led the Bank to develop a new model and the way it went about doing this. In doing so, it provides a thorough technical description of the new model, including details of its theory, construction and use. The main text of the chapters is written with the intention of avoiding heavily technical expositions. In some parts, where there are issues that may need more detailed explanation, use is made of boxes, which can be read or passed by as the reader chooses. Technical details are contained in appendices. The following provides a guide to the remaining chapters. Chapter 2 contains a fuller discussion of the particular requirements made upon forecasting models at the Bank of England and how that is reflected in the design of BEQM. The aim of the project was that the new model should provide a richer, more explicit, theoretical structure, while matching the data at least as well as the previous macro model. It would also need to be flexible and reliable under different forecasting assumptions and conventions. This led us to the concept of building the model with two parts – a layer that provides the theoretical core of the model, and a layer of extra dynamics designed, in part, to facilitate judgemental adjustments. The idea of adding ad hoc or ‘data-driven’ dynamics to theoretical structure is not a new one, but the implementation has many variations. The chapter therefore includes comparisons with alternative modelling approaches and some other macro models. Chapter 3 discusses the core theory. The individual building blocks of the theoretical core are largely conventional, as seen in Section 1.2. However, a key focus in the development of BEQM was ensuring that the model works consistently as a system, with close attention paid to the constraints and linkages between agents. Chapter 4 follows with an account of the ad hoc dynamics. These equations take the paths from the core theory and combine them, if needed, with extra persistence and variables that proxy for effects missing in the core theory. These effects might be missing because we choose not to attempt to model them in a fully structural way: the additional structure to do so consistently would make the model much more complicated and potentially difficult to run. Additionally, there are some effects that seem empirically robust, but are very difficult to model formally. Chapter 5 provides technical details of how we solve the model. A key issue here is the treatment of expectations, and in particular how to deal with cases in which agents do not fully anticipate future events. We address this issue by the use of so-called ‘recursive simulations’ that potentially limit the amount of information available to agents from period to period. Chapter 6 discusses the parameterisation of the model. The main problem that we face is that the model is large, in order to be able to handle typical forecast issues with sufficient richness. And in order to treat the theoretical building blocks in the model consistently, the model is highly simultaneous – in other words, one agent’s actions will generally depend on all other agents’ actions at the same time. The model is therefore too large to confront using conventional econometric techniques for estimating simultaneous systems. This issue is not new and confronts all builders of large macro models. Our approach is to separate out parameterisation from evaluation. That is, we select parameter values based on a range of evidence, and then evaluate the whole system against a number of different criteria. The model is parameterised to achieve a plausible long-run relation to observed values for key ratios, and to achieve dynamic properties that are at least as good as those of the previous model, by using econometric evidence and priors about the transmission of shocks. We find the theoretical core does 9 The Bank of England Quarterly Model well at tracking broad movements and does quite well at forecasting at longer horizons (two and three years) over history. But some variables track better than others, and we find econometric evidence that the model’s fit is improved by the inclusion in the non-core equations of proxies for short-run effects such as credit constraints, house price effects, confidence and accelerator effects. Chapter 7 shows how all of the preceding elements come together in terms of model properties. Shock responses are a useful way of illustrating the overall model properties, and we present several, including demand, supply and policy shocks. Finally, Chapter 8 concludes with some remarks on future uses and directions for the model. A number of appendices set out supporting detail to the discussion in these chapters. Appendix A sets out the core model equations and mnemonics, with comments on the economic rationale for the equations. Appendix B presents similar detail for the non-core equations. Appendix C details data transformations and sources and Appendix D sets out parameter values. 1.5 Summary The Bank of England has developed a new macroeconometric model for use in preparing the MPC’s quarterly economic projections. This model uses recent advances in economic understanding and computational power to develop and improve upon existing models used at the Bank. The new model does not represent a change in the Committee’s view of how the economy works or of the role of monetary policy. Indeed, the sensitivity of output and inflation to temporary changes in interest rates is broadly similar to that in existing models used at the Bank. However, the model does provide the Committee with a more flexible and coherent framework to aid its economic deliberations. 10 Chapter 2 Project motivation and model design This chapter describes some of the thinking behind the new model. Section 2.1 sets out the project’s motivation, and how the requirements for the new model relate to the problem of achieving both theoretical and empirical consistency, but in a way that is suitable for practical forecasting. This leads to a description of the design of the new model (Section 2.2). Section 2.3 compares BEQM with some other macroeconomic models, before a summary in Section 2.4. 2.1 Motivations and challenges The main motivation for developing the new model was to improve theoretical consistency and clarity. In particular, one of the key benefits of a formal model is that it can remind us of important implications that are not immediately apparent. The importance attached to understanding the ‘economics’ of the forecast, and to exploring the various risks and uncertainties surrounding the central projection, points to the need for a clear and explicit economic structure. If the forecast were simply a mechanical process – that is, the production of a single, ‘best’ prediction, without alteration or imposition of judgement – then the comparative advantage would lie with atheoretic models such as large-dimensional common factor models. (1) Instead, the MPC wants to understand what is driving the economy. This focuses attention on forces at work in the economy – asking what economic shocks are affecting the economy, how will they work their way through the economy, and what implications do they have for monetary policy. A central problem here is that there are often several possible explanations for observed inflation. For example, a fall in inflation could be the result of an increase in productive potential; a fall in wage growth; an increase in domestic competition; pass-through of lower world prices; or an exchange rate appreciation. Without theoretical consistency and clarity, a model would lack the structure and linkages needed to discriminate between these different hypotheses. The model should be consistent at a general level with the MPC’s view of how the economy works (especially the monetary transmission mechanism). Moreover, to be used as a forecasting device, the model should produce realistic responses, which implies that it has to be matched to the data. At the same time, the final, published forecast is a conditional projection, based on policymakers’ judgements about risks, influenced by other models and information. This, in turn, implies that theoretical strictness should not preclude the ability to apply a wide range of judgement to the model’s ‘naive’ projections. To summarise, the project had three clear challenges: • to incorporate theory that is rich enough to be able to analyse a wide range of economic issues, while remaining tractable, internally consistent, coherent and easily understood; • to make this theoretically tight model match the data at least as well as the previous model; and • to make the model reliable and efficient under different forecasting assumptions, and amenable to the imposition of judgemental adjustments and conditioning paths. (1) Indeed, the Bank maintains a number of such models for comparison with the conditional forecast. 11 The Bank of England Quarterly Model 2.2 The design of BEQM The key design issue was how to meet these three challenges. Models with a high degree of theoretical coherence are helpful for analysing economic issues but are unlikely to match the data as well as purely statistical models that have been designed to maximise coherence with the data. Such atheoretical models might have many parameters but these would be chosen purely on the basis of statistical fit and would be hard to relate to the underlying economics of how agents and markets behave. So macroeconomic modellers face an inherent trade-off, even among ‘state of the art’ models, between achieving theoretical consistency and coherence with the data. (2) Figure 2.1 shows a stylised version of this trade-off, such that the current state of the art describes a ‘frontier’ between the axes. Figure 2.1: The trade-off between theory and data theoretical consistency data coherence In terms of the first challenge, our overall approach was to start, at a relatively general level, from a view of the required theoretical building blocks: which economic agents to include; how they interact; and in which markets. To ensure the desired level of internal consistency, we started with clearly defined optimisation problems for households, firms, and unions that bargain on behalf of workers. Explicit assumptions were also laid out about the behaviour of the government, the monetary authority, and the rest of the world. These basic ingredients were present in the previous macroeconomic model. The new model aims to fill in the gaps by deriving decision rules from first principles, so that the resulting equations are internally consistent. (3) However, all models are abstractions: no model can capture all of the behaviour of the economy. Moreover, some elements of theory were deliberately omitted in order to keep the optimisation problems tractable and the resulting equations clear. In terms of Figure 2.1, we attempted to see how far we could improve theoretical consistency before the complications of additional theoretical richness were felt to outweigh the benefits, while at least maintaining the level of data coherence of the previous model. (2) See, for example, the typology in Pagan (2003). (3) The maximands for the optimisation problems in the core theory can be quite elaborate. We did not start with these but began with relatively basic prototype models, incorporating new features by adding to the optimisation problems. At some points this revealed that certain theoretical building blocks were not compatible with each other, and so a decision had to be made as to which ones would be used. 12 Project motivation and model design The sort of tightly specified structural model that this process delivers is a useful device for thinking about the transmission of shocks and policy, and the identification of different economic stories. However, such a model would probably have some difficulty in matching the data fully, because it would always be missing some potentially important economic elements. Typically, in much of the academic literature over the past 25 years, models have been designed to answer a specific question and can therefore be focused on specific issues. In our case, however, the model is intended to be used as a general-purpose vehicle for policy analysis and forecasting, and we cannot neatly restrict the range of economic issues that the model will face. Moreover, the ultimate users – policymakers on the MPC – must have confidence that the model is able to match general features of the UK economy and its response to shocks, even if some of this behaviour is difficult to model structurally. One approach to matching movements in the data, commonly used for macroeconomic models, is to treat the theory as a guide to the economic variables that appear in econometric regressions. (4) If a strict approach is taken to deriving a reduced-form equation from theory, then the equation parameters will be combinations of the deep parameters from the structural decision rules. If cross-equation restrictions were not strictly enforced, the linkages implicit in the original specifications would be weakened. This would be problematic for our purposes, because so many of the forecast issues and risks revolve around competing structural stories, and we need to be able to trace their different effects through the model. These are not just variations in exogenous effects, such as assumptions for future world trade, but also in how the economy responds to those effects. (5) An alternative approach could be to retain the structural specifications derived from the optimisation problems and to add extra, ad hoc components. But it is not clear where to place ad hoc elements in a micro-founded simultaneous system. Indeed, experiments along these lines confirmed that it would be easy to create a system that generated unpredictable results and that might not even solve. To reap full benefit from a structural system, any additional elements should be worked through consistently from the original optimisation problem, which could risk making the model large and intractable. To secure the benefits of the new model, our approach was to build the model in two distinct parts: a theoretical ‘core’ model, and ‘non-core’ equations that include additional variables and dynamics not modelled formally in the core. When used together, these two parts form the full model that is the actual platform used for producing forecast paths and allows the direct application of judgement. The theoretical core is a structural model, containing a set of decision rules derived from first principles and an associated set of consistency conditions, such as accounting constraints and stock-flow identities. It could be thought of as a dynamic general equilibrium model in which adjustment costs and other frictions are modelled explicitly. The decision rules are dynamic and are derived from the assumptions that agents act to maximise forward-looking objective functions according to dynamic constraints. (6) It describes how we would expect changes in exogenous forces to work their way through the model economy. A change to a given structural parameter would usually feed into many different decision rules and would therefore affect how the system responds to shocks. The paths from this core model are treated as starting points for the final forecast paths. (4) See, for example, the programme laid out in Fair (1993) and (1994). (5) For example, the responses of the economy if goods markets were more or less competitive are well defined where parameters for demand elasticities are identified throughout the system. But with reduced forms, the answer is often buried in the constants and (quasi-) elasticities of the system. An expert user could make use of intercept adjustments, but this would take some skill and time, which is not ideal when a large number of economic issues and uncertainties must be processed quickly. (6) For example, consumers are assumed to maximise expected lifetime utility subject to the constraint that their assets evolve according to a period-by-period budget constraint. 13 The Bank of England Quarterly Model The full forecast model supplements the paths from the theoretical core with a statistical model of the discrepancy between historical outturns and the paths generated by the core model. We supplement the core theory for two reasons. First we might allow for different dynamics, such as more persistence than the theory implies. Second, we might allow for influences from variables that proxy for missing effects, such as credit channel effects and confidence effects through the business cycle. The only restriction on the structure of ad hoc non-core equations is that the projected path for a given variable should always converge to the long-run equilibrium imposed by the core theory. This forecasting model is strictly autoregressive, so that judgement (7) can be used to modify paths in a predictable way, which would be more difficult within the structural core model. This is an important feature, allowing us to impose the Committee’s judgements, using off-model information. Actual forecast paths are thus combinations of three types of information: • theoretical insight from the structural core model; • data-driven evidence on historical correlations of endogenous variables with other factors, especially those that are not formally accounted for in the structural core; and • a direct application of judgement, informed by other models and staff expertise. A key forecast question is how much weight these different contributions should carry in deriving the final forecast path. The profile for a given endogenous variable is built up from the core model and supplemented in the full model by additional variables, dynamics or judgement, as illustrated in the stylised sequence below and in Figure 2.2: (8) • given values for the exogenous variables, a steady-state version of the core model determines a sustainable long-run equilibrium value; • the core model indicates a path that converges from the current starting point to the long-run level that is consistent with the decision rules and constraints in the core theory; and • in the full model, the path might have additional lags or proxy variables added, or judgement applied. (9) While this process allows a relatively free modification of the original path for a given endogenous variable, the system still preserves accounting identities and stock-flow relations, so there are no ‘free lunches’ allowed by the application of judgement (the details of the application of this approach are discussed in Chapter 4). (7) These ‘addfactors’ are additional exogenous elements on the right-hand sides of equations. They are set to zero in the long run, but can take various values to affect the path of a left-hand side variable. For example, suppose we have a system of the form: y1t = a1 · y1t−1 + e1t y2t = a2 · y1t + e2t A ‘type 1’ fix on y2t would be where we use e2t to achieve a desired path for y2t . For a ‘type 2’ fix, we would use e1t to affect y1t and therefore y2t . (8) For simplicity, the illustration is as if stationary and abstracts from trend growth. (9) This representation is very stylised; in practice, judgement might be applied to the long run (for instance, by changing core model parameters) as well as the short run. 14 Project motivation and model design Figure 2.2: A stylised forecast sequence steady-state equilibrium history projection core path history projection full model path (including judgement, extra lags & proxy variables) history projection If some correlations appear to be particularly important and robust, then it would be logical to ask whether some future effort should be made to find out whether we could account for them structurally through an extension to the core theory. For instance, ad hoc dynamics can capture what is missing and suggest modifications to the theory that would make the dynamic responses of the model closer to those observed in the data. This emphasises that model development is a continuous process, as demands and economic knowledge evolve. 2.3 A comparison with other models The motivation behind our two-tier approach is a model that is theoretically consistent and data coherent, but also sufficiently flexible and tractable for forecasting applications. But the basic idea of supplementing theory with ad hoc, data-driven dynamics is not new. So it is worthwhile to put our approach in context with those underlying other models, and to see how economic modelling has evolved in response to advances in dynamic macro theory and the changing demands on models in policymaking institutions. 15 The Bank of England Quarterly Model There has been a long UK tradition associated with cointegration-based econometrics, which has been very influential in macromodelling. This approach uses theory to posit the existence of long-run relations, which would be incorporated into the model if validated on statistical grounds. Examples of this approach include Hall and Henry (1987) and, to some extent, the Bank of England’s earlier MTMM model, which in broad terms is a large, restricted Vector Error-Correction Model (VECM). This approach puts less emphasis on some aspects of theory, insofar as short-run dynamics are largely ‘data driven’, and long-run relations implied by theory have to be confirmed by empirical work. For example, the modeller would not insist that the model has a balanced-growth equilibrium, but instead would test whether the cointegrating relation implied by this was present in the data. (10) Similarly, the existence of forward-looking expectations and susceptibility to the Lucas critique are hypotheses to be tested. (11) In the 1980s, a generation of UK macro models emerged that attempted to respond to the Lucas critique directly with the use of rational expectations econometrics. (12), (13) These models typically placed a strong emphasis on consistency conditions such as stock-flow relations. (14) A typical approach would be to take an equilibrium condition derived under the assumption of rational expectations and estimate the parameters using some form of instrumental variables. For example, a consumption function might be used as the starting point for a regression of current consumption on leads of consumption, lags, and wealth terms. A good example of such an approach is the NIDEM model produced by the National Institute of Economic and Social Research. (15) Such an approach effectively allows the model-builder to test directly the coherence of the theoretical relation against the data. The equations are nonetheless still reduced form, with cross-equation restrictions enforced to varying degrees. (16) Because of this, some ad hoc measures would be necessary if some technical features, such as a long-run balanced-growth solution, were required. (17), (18) In terms of models actively used by policy institutions to support forecasts and policy analysis, there has been a steady shift towards models that place greater emphasis on theoretical consistency. (19), (20) For example, the Bank of Canada shifted from the RDXF model to the QPM in the early 1990s, and the Board of Governors of the Federal Reserve moved from the MPS to the FRB/US model. The QPM model placed more weight on theoretically plausible parameter values than on direct econometric (10) See Doornik and Hendry (1994) for an example of this approach. (11) See, for example, Favero and Hendry (1992). (12) An outstanding early example was the work on the Liverpool model (Minford (1980)). Subsequent work included that at the National Institute for Economic and Social Research (Hall and Henry (1985) and (1987)), the LBS model (Budd et al (1984) and Dinenis et al (1989)), the City University CUBS model, and, more recently, COMPACT (Darby et al (1999)). See Wallis and Whitley (1991) for a commentary. (13) Wallis (1980) was an early investigation of the consequences of rational expectations for macroeconometric specifications. (14) The importance of consistency conditions was emphasised in pioneering work on optimal control in forward-looking models – see, for example, Holly and Zarrop (1983) and Holly (1986). (15) This has several vintages; see Wren-Lewis (1989) for an example. (16) In this sense, these models can be interpreted as overidentified VARs; typical identifying assumptions that would be maintained are homogeneity restrictions. (17) For example, in the case of the NIESR model referred to here, a stable net foreign asset position was ensured by configuring the fiscal rule so that the government effectively took the role of ensuring an economy-wide savings equilibrium. (18) Extensive comparisons of UK macro models have been conducted, especially by the ESRC Macroeconomic Modelling Bureau from 1983 to 1999. See Andrews et al (1984), (1985), (1986); Fisher et al (1987), (1989), (1990); and Church et al (1991), (1993), (1995), (1997), (2000). (19) This was facilitated by advances in solution algorithms that made simulations with large-scale non-linear models with model-consistent expectations feasible; see, for example, Fisher, Holly and Hughes-Hallett (1986). (20) Some, however, would regard existing macroeconometric forecasting models as well within the possible frontier, paying less attention to issues of simultaneity and cross-equation restrictions than would have been the case in macroeconomic models of the 1960s; see Sims (2002). Indeed, the Liverpool model of the late 1970s was theoretically very strict by today’s standards. 16 Project motivation and model design estimates. (21) It uses a calibrated theoretical model to pin down a set of steady-state attractors for error-correcting relationships. (22) Dynamics are driven by assuming, on a partial equilibrium basis, that there are adjustment costs between current and long-run target levels for a variable. (23) In the FRB/US model, theory is used to inform long-run relationships, and some of these are forward looking, such as human wealth. (24) Dynamics are assumed to be driven by generalised adjustment costs, and the existence of higher orders of adjustment costs introduces a role for forward expectations. (25) The full model is a mixture of structural relations implied by a partial equilibrium treatment of theory (such as the decision rule for aggregate consumption) and some reduced-form relations (such as the trade block, which employs error-correcting relationships.) A similar transition was made by the International Monetary Fund (IMF) with the shift to the Mark III vintage of the MULTIMOD multi-country model. Since MULTIMOD was intended to be used more as a simulation model rather than a direct forecasting tool, several of the changes which were implemented in the Mark III version arose from the need to enrich its theoretical structure, so that it could deal with new macroeconomic issues such as current account imbalances. (26) As with the QPM, a steady-state model enforced necessary terminal conditions. (27) Other central banks have followed with variations of their own. In 1997, the Reserve Bank of New Zealand moved away from spreadsheet-based forecasting to a formal model, FPS, that drew on the experience with QPM. (28) That model can be viewed as a forward-looking IS/LM system with a disaggregated IS curve. The difference between ad hoc calibrated dynamics and ‘equilibrium’ dynamics for real variables defines an output gap, which drives the nominal side through a Phillips curve. Work at the Bank of Japan has pursued and extended this approach, (29) and a variant of the QPM model has been used by the Sveriges Riksbank in the form of the RIKSMOD model. Recent work at several institutions indicates that this process may go several steps further. Projects at the Board of Governors of the Federal Reserve and the Bank of Canada are now under way to explore models with theory-based dynamics as well as long-run properties. Work on the GEM model at the IMF and the EDGE and AINO models at Bank of Finland can also be seen in this way. In the case of BEQM, the model is split into two tiers – the structural model is kept intact, with no attempt to introduce ad hoc components directly; forecast paths are constructed as a weighted average of paths from the structural model and paths driven by statistically robust correlations, together with application of policymakers’ judgements. This approach reflects the nature of the forecast process at the Bank of England. As much as possible, an attempt is made to understand the forces at work on the macroeconomy in terms of fundamental economic drivers and constraints, which are articulated in the (21) For example, Coletti et al (1996) comment that ‘there had been a systematic tendency towards over-fitting equations and too little attention to capturing the underlying economics. It was concluded that the model should focus on capturing the fundamental economics necessary to describe how the macro economy functions, and, in particular, how policy works’ (page 14). (22) See Black et al (1994). (23) See Coletti et al (1996). (24) See Brayton and Tinsley (1996). (25) See Kozicki and Tinsley (1999). (26) See Laxton et al (1998). (27) This was to compute the terminal conditions for the forward-looking variables in the model. It also enabled users to relax the assumption of earlier vintages that current accounts have to balance in the long run, which enabled better investigation of sustainability issues. (28) See Black et al (1997) and Hunt et al (2000). (29) See Fujiwara et al (2004). 17 The Bank of England Quarterly Model core model. But our understanding of the macroeconomy is imperfect, so it is logical to ask whether some weight should be given to correlations that are robust in the data but might be difficult to explain or model in a fully structural way. Other, more specialised models can provide insight on specific issues or variables, as can sectoral expertise available within the Bank. These suggest instances when judgement can usefully be applied. 2.4 Summary This chapter discusses some of the factors behind the design of BEQM. The main motivation for developing a new model was to improve theoretical consistency and clarity. A number of specific requirements stem from its role in helping to produce the MPC’s quarterly economic forecast and in analysing the economic issues underlying the forecasts, together with associated risks and uncertainties. This meant that we want a model that is rich enough to be able to analyse a wide range of economic issues; that can match the observed data; and that is reliable and robust under the pressures of a real-time forecasting round – including the ability to impose judgement and conditioning paths. Our approach was to build a model with two distinct parts. We start with a tightly specified theoretical core model, containing dynamic decision rules derived from the solution of dynamic optimisation problems. We supplement this with non-core equations that include additional lags and variables to match dynamics that are not modelled formally in the core. These equations also allow the imposition of judgement based on ‘off-model’ information. The final forecast path can be thought of as a combination of theoretical insight from the structural core model; additional variables and dynamics from the non-core; and direct application of judgement. Finally, we put BEQM into historical context by discussing advances in macroeconomic model building over the past 25 years. Over time, greater emphasis has been placed on theoretical consistency, and advances in computing power have allowed more complex models to be employed. 18 Project motivation and model design Box 1: Some recent developments towards hybrid structural models Substantial effort in recent years has been directed towards ‘hybrid’ models, which preserve relatively strong, theoretically derived identification structures but nonetheless fit the data according to some well defined statistical metric. Some of this work can be thought of as coming from a relatively atheoretic perspective, such as the Vector Auto Regression (VAR) literature; other work takes theoretically tight models, such as from the Dynamic Stochastic General Equilibrium (DSGE) literature, as a starting point, and asks what has to be done to make such models fit the data. The attempt at convergence is logical, because both approaches yield a compact autoregressive form that can be assessed against the data. In the VAR literature more and more use has been made of long-run (and even short-run) identifying restrictions; for example, Leeper and Zha (2001) aimed to produce a VAR model that is ‘useful’ for monetary policy analysis. A small number of papers have attempted to exploit the data-matching properties of VARs together with the story-telling advantages of structural models. McKibbin, Pagan and Robertson (1998), for example, start with a VAR to produce a hybrid model that retains the very short-run properties of the VAR, but is designed to match some of the features of a calibrated structural model. There is now a substantial literature that assesses DSGE models against their corresponding VARs. (a) In a conventional DSGE approach, first-order approximations to the decision rules derived from dynamic optimisation problems are evaluated at a deterministic steady state. The useful ‘trick’ of the DSGE approach that makes the solution of these models tractable is to assume that the exogenous variables follow a simple autoregressive process. Given this assumption, the rational expectations of future variables can be derived as functions of current states of the world, leading to a backward-looking representation of the dynamic solution to the model. The generic state space representation of these models will have the form st yt = Ast−1 + Bu t = Cst (1) (2) where s is a vector of states of the world, u is a vector of shocks and y is a vector of endogenous variables. A, B, and C are conformable matrices, where the elements are combinations of structural parameters. Usually y will be larger-dimensioned than s: given knowledge about the evolution of a relatively limited number of states of the world (eg capital stock, previous levels of consumption), we make inferences about a wide range of variables (such as output, wage rates, employment, and asset prices). In its state-space form, the model can be run recursively against the historical data and prediction errors can be extracted from the difference between predicted and actual y. Notionally at least, these errors could be used to evaluate a likelihood function. A problem arises when applying this to the canonical Ramsey mode, which has a single stochastic process (technology), in that the covariance matrix is singular. But if we augment the range of extrinsic dynamics so that there is a stochastic process for each endogenous variable, then exact maximum likelihood is possible. (a) See, for example, Canova, Finn, and Pagan (1994). 19 The Bank of England Quarterly Model Hence, in recent years we have seen papers (eg Hansen (1985)) in which parameters that would previously have been held fixed are allowed to vary over time. For example, instead of a conventional household maximisation problem with fixed time preference and utility weights on consumption (c) and leisure (the proportion of available time not spent working, 1 − h): max E t ∞ i=0 β i {log ct+i + A log (1 − h t+i )} we could now specify the problem as max E t ∞ i=0 ϕ t+i At+i ϕ t+i {log ct+i + At log (1 − h t+i )} = (1 − ρ ϕ )ϕ̄ + ρ ϕ ϕ t+i−1 + εϕt+i 0 ≤ ρ ϕ < 1 A = (1 − ρ A ) Ā + ρ A At+i −1 + εt+i 0 ≤ ρA < 1 Other extensions include capital-specific and labour-specific effectiveness processes, time-varying investment efficiency, and policy shocks. One would add shock processes to the optimisation problem until there is a stochastic process for each endogenous variable. Then estimation of parameters is possible using the Kalman filter to extract prediction errors to be assessed using the likelihood function. A recent example of adding shock processes to the optimisation problem can be found in Smets and Wouters (2003a) A potential disadvantage with this type of model is that its projections are driven by a large set of unobservable shocks, some of which might be regarded as arbitrary and difficult to interpret. A Bayesian rather than classical approach to this problem can also be taken. One implementation is to use the recursive state-space form of the theoretical model to generate artificial data. The parameters of the system can then be estimated using a pseudo-sample that combines actual with artificial data. The higher the proportion of artificial data used in the pseudo-sample, the higher the weight on theoretical priors. Bayesian ‘shrinkage’ procedures can be used to determine the optimal weight on theory and data. For example, Ingram and Whiteman (1994) showed that using priors and cross-equation restrictions from a Real Business Cycle (RBC) model allowed for a considerable improvement in the performance of a VAR, compared with the unrestricted VAR form and a VAR using the Minnesota (random walk) prior. Recent implementations include Del Negro and Schorfheide (2004). In different contexts, it is conventional to refer to (2) as a measurement equation, reflecting the assumption that the linear transforms of state values in Cs will only be imperfect approximations to observed data in y. For example, we do not observe ‘output’ or ‘marginal product of labour’ directly, but have constructed measures such as ‘private sector value added at current prices’ and ‘unit labour costs’. Sargent (1989) exploited this structure to deal with issues about data mismeasurement. In his schema, we would have (1) as before, but (2) would be augmented to include error terms: yt et 20 = Cst + et = Det−1 + ξ t Project motivation and model design As the error terms are assumed to represent measurement error, they are orthogonal: D and cov(et ) are assumed to be diagonal. However, Ireland (2004) suggests that we interpret the errors as ad hoc processes to make up for the misspecification of the model in fitting the data to y, by allowing D and cov(et ) to have non-zero off-diagonal elements. In this case, the errors are not orthogonal and will be correlated with the elements in s in almost all cases. In other words, the theoretical model embodied in the relations (1) and (2) is assumed to be wrong, and so the question, in the spirit of Watson (1993), is what needs to be done in e in order to match the data in y. While much progress has been made in terms of numerical methods, these sorts of techniques have only been applied to relatively compact systems where the interpretation of the corresponding reduced-form VAR representation is relatively straightforward. It is not yet clear how the added unobservable extrinsic dynamics in these models should be interpreted in terms of the demands of a forecast process and the need to understand the underlying economic drivers of the forecast. Nonetheless, this literature holds some promise in terms of reconciling demands for theoretical consistency with coherence with the data. 21 Chapter 3 The core theory This chapter summarises the core theory, describing the main building blocks, with attention to the interactions between agents and the key assumptions. An overview of the core theory (Section 3.1) is followed in Section 3.2 by discussion of the objectives and constraints of the key agents: households, firms, the government, the monetary authority and the rest of the world. Section 3.3 describes how these agents interact in markets for goods, labour and capital. This is followed in Section 3.4 by an account of the nominal side, including money market equilibrium and the price level, nominal wages and prices, inflation, and the monetary transmission mechanism. Section 3.5 deals with real trend growth, followed by a summary of the chapter in Section 3.6. 3.1 Overview We can think of the core theory as an organising framework for analysing the economy. It should help us tease apart competing explanations of what we observe in the data, by reminding us of the different implications of each story. To do this, the theory in the core model needs to be sufficiently rich and general to handle a wide range of issues, while at the same time being compact enough to be tractable, reliable and clear. In order to secure the level of internal consistency that we want from the core model, we take an optimisation-based approach that begins with clear statements about how the key agents act in the face of constraints. To this end, we wanted the theory to describe an economy where: • households receive wage income and transfers from both firms and the government, and returns from assets, while paying for a bundle of domestically produced and imported consumption goods, making investments in housing and financial assets (shares, bonds and money) and paying taxes to the government; • domestic firms receive income from selling final goods in domestic and overseas markets, while paying taxes and (potentially) receiving transfers from the government, paying for factor inputs of domestically sourced capital goods, imported capital goods, and labour. They finance this activity through the issue of equity and debt, and accumulating and decumulating inventories; • the government generates revenue from taxes, new debt issuance and seigniorage, while purchasing goods and services from firms and labour services from households, and servicing existing debt; • the monetary authority sets a short-term nominal interest rate in order to achieve an inflation target and, consequently, provides nominal stability; and • the rest of the world provides capital, goods and services demanded by the domestic economy, and is a potential market for domestic production. The core theory contains explicit decision rules and constraints that specify how these key agents interact with each other in markets for capital, financial assets, goods and labour. The treatment of the agents differs quite substantially: decision rules for households and firms arise out of explicit optimisation problems, while the rest of the world is exogenous. The monetary authority and the government are given simple reaction functions that specify policy targets and an endogenous instrument. This description of the core model economy is illustrated in Figure 3.1. 23 The Bank of England Quarterly Model Figure 3.1: Key agents in the model macroeconomy Households Firms maximise utility subject to budget constraint maximise profits subject to demand and technology reaction function to specify nominal instrument in reaction to deviations from nominal anchor reaction function to specify fiscal instrument to ensure debt sustainability Monetary authority Government Domestic economy assumed to be 'large' with respect to domestic economy Rest of world World economy Likewise, there are some important differences in the assumptions made for various markets. We wanted the theory to describe an economy in which: • goods markets are monopolistically competitive, leading to profits and an ability for firms to charge non-competitive sticky prices, which clear all of domestic production to satisfy demands (net of imports) for consumption, investment, changes in inventories, government spending and exports; • the labour market equilibrium is not perfectly competitive. Firms and unions bargain over wage levels which generate unemployment, given private sector and public sector labour demand, labour supply and wage curves; and • asset market prices reflect standard no-arbitrage conditions, with (net) assets owned by domestic households and overseas residents, and long-run real interest rates are pinned down by world conditions. This broad view of the economy is unchanged from the previous macro model. Indeed, many of the theoretical building blocks of the previous model have been used here, including the paradigm of a single, constant returns to scale production function on the supply side; the union bargaining framework for the labour market; and the same small open economy assumptions. The key difference here lies in the implementation, where we have now been more explicit about what motivates the agents, their constraints, and the characteristics of the markets in which they interact. This consistent approach leads to a number of technical features: a well defined, balanced-growth steady state; consistent stock-flow accounting; binding budget constraints; an explicit treatment of expectations; and the potential for endogenous policy reaction. Figure 3.2 shows how agents are linked by expenditure flows and the accumulation of physical and financial assets. The government is bound by a budget constraint, so that deficits cumulate into government debt. Firms cumulate physical stocks of inventories and capital goods, and must finance their activities, so accumulating liabilities of debt and equity. The key linkage is that these liabilities must ultimately be held as assets by households or the rest of the world. Households’ decisions about how much to consume – reflecting their desired level of total financial wealth – will ultimately be reflected in the level of net foreign assets that the UK economy holds. This net balance can be positive 24 The core theory Figure 3.2: Key flows and assets Households dwellings investment World economy Firms stockbuilding investment saving Housing stock Inventories Financial assets = balance of payments Capital stock Government bonds + Value of the firm (shares and debt) + Net foreign assets deficits Monetary authority Government Domestic economy or negative, and will imply a particular balance of payments current account if it is to be sustainable. Hence flow decisions by households, firms and government have a stock dimension, which is an important ingredient in determining the sustainability of short-term spending decisions. (1) Expectations play a key role in determining the equilibrium depicted in Figure 3.2: households plan for future consumption and firms make pricing, factor and inventory decisions, based on their expectations of future events. The core theory discussed in this chapter does not depend on any particular assumption about how expectations are formed, but we assume that expectations are ‘model-consistent’. The treatment of expectations is discussed in Chapter 5, along with how we can vary the extent to which agents in the model anticipate future events. Little of the theory that is described in the following sections is innovative. Indeed, the aim was to stay as much as possible within the bounds of ‘textbook’ macroeconomics. However, since storytelling depends in large part on proper identification of the interactions between sectors and agents, the challenge has been to ensure consistency between the various theoretical assumptions and components. Some theories or assumptions that are frequently employed in the current academic literature may not be compatible with other theories that we would want to use, and a considerable amount of judgement is required about what would be lost by using one instead of the other. Furthermore, there are many theories that could be incorporated into the core model, but would render it hopelessly intractable and opaque for general use as a forecasting and policy simulation model. Hence, while the new model is intended to be more ‘complete’ in terms of the interactions between key agents in the macroeconomy, it is still very much a general-purpose model that should not be expected to be able to handle every conceivable macroeconomic issue. In other words, the new model emphasises internal consistency and is intended to complement the Bank’s suite of models, not replace it. (1) The monetary authority is assumed not to purchase goods and services or accumulate stocks, so there are no arrows linking it to any of the assets in this ‘real-side’ depiction of the economy. However, it clearly has a role influencing demand conditions and, hence, the pattern of flows in the short run. 25 The Bank of England Quarterly Model 3.2 Characterisation of the agents 3.2.1 Households Households are important because their choice of a desired level of financial assets, given the supply of domestic assets, determines the long-run equilibrium for sustainable consumption, net foreign assets and the trade balance. A key issue for the theory of the household is how this equilibrium is determined. We also want the household theory to account for the demand for real money balances, the demand for housing, and the split of consumption between domestically produced and imported goods. Further, we want some control over the short-run dynamic convergence to long-run equilibrium for these demands. Our approach starts with a standard utility-maximisation problem, but instead of dealing with a single representative agent, we aggregate individual decision rules to derive total demands for consumption, housing and money. We consider first the optimisation problem for individuals: they aim to maximise lifetime utility subject to their expected lifetime resources. The first source of utility is consumption of domestically produced and imported non-durable goods, and services from housing (which we treat as a very durable good). Individuals also gain utility from holding real money balances, which in this context can be thought of as an implicit requirement for cash for transactions. (2) Individuals also form habits and value leisure time. (3) To keep the model as simple as possible, we abstract from public finance issues that might be relevant for a fiscal authority, so public goods do not enter utility and government consumption and investment do not benefit households directly. (4) Turning to income, households rent their labour to firms or the government and receive wages in return (see Section 3.3.2). Households are also assumed to receive certain transfers directly from firms (such as corporate pension contributions), as well as transfers from the government (such as government benefits). To smooth consumption, households can borrow and save using a range of financial assets, including government securities, corporate bonds, shares and foreign bonds. (5) A commonly used assumption is that of a representative agent who lives for ever, which would imply that the level of net foreign assets is not pinned down to a particular long-run equilibrium level. (6) Our assumption that the domestic economy is small in relation to international capital markets implies that an infinitely lived domestic agent would be able to borrow an infinite amount and pay it off in the indefinite future. If such domestic agents were ‘impatient’ relative to the rest of the world, domestic households would accumulate an infinite amount of net foreign debt; if domestic agents were more ‘patient’ than the rest of the world, the domestic economy would acquire all the assets from the rest of world. (7) This would violate the assumption that the domestic economy is small in relation to international capital markets and is incompatible with our basic requirement that the interaction of agents in the model determines a long-run sustainable equilibrium for consumption, net foreign assets and trade. (2) Money is assumed to provide services that mitigate transactions frictions. Real money balances enter the utility function additively. (3) Technically, this is not implemented in the same way as conventional direct leisure in lifetime utility, and does not affect the marginal propensity to consume. ‘Unions’ are assumed to bargain on behalf of workers – see Section 3.3.2. (4) As long as public goods enter utility additively, they would not affect consumption decisions in any case. (5) See Box 2 on page 30 for the consumer’s maximisation problem. (6) This model does have growth – see Section 3.5 – so that in what follows ‘consumption’ refers to the consumption-output ratio. (7) That is, if β (1 + r) > 1, where β is the individual’s discount factor and r a real return on assets, then consumption will steadily grow and foreign assets will fall. Conversely, if β (1 + r) < 1 then consumption will steadily fall and foreign assets will rise. See Chapter 3 of Barro and Sala-i-Martin (1995). 26 The core theory There are several modifications to the infinitely lived representative agent model that render consumption stationary. (8) However, these methods typically imply that one of consumption, net foreign assets or the current account has to return to a predetermined value, rather than each reacting endogenously to shocks, as we would like. In addition, they usually require that the household rate of time preference equals a constant foreign real interest rate, which is an unattractive feature in our case because movements in foreign real interest rates can be a key forecast issue. Further, these models typically preclude discussions of ‘pure’ wealth effects, which would further restrict the ability of the model to analyse forecast issues. (9) Instead, we ensure that consumption is stationary by assuming that households do not live for ever. In particular, following Blanchard and Yaari, (10) we assume that individual households face a constant probability of survival from period to period, with new households being born to replace those that have died. As with a representative agent model, they aim to maximise expected utility, subject to a period-by-period budget constraint with a standard transversality condition imposed. (11) This leads to intertemporal conditions for individuals (such as the consumption Euler condition) that are the same as in the infinitely lived representative agent case. Aggregate consumption, however, is stationary. The intuitive reason is that, because individuals do not expect to live forever, households in aggregate are prevented from borrowing or saving unlimited amounts. When we aggregate over the entire population, the aggregate economy settles on a constant ratio of consumption to income in the long run. (Box 6 on page 51 explains this formally in a simplified model.) This can be represented in Figure 3.3, (12) where the long-run consumption level (abstracting from growth) is defined by the point where ct = ct−1 = c∗ . Figure 3.3: Consumption equilibrium in the steady state ct c* c t = f (ct -1,HW t ) 45° c* c t-1 (8) Methods include making preferences a function of previous consumption, introducing portfolio adjustment costs, and imposing external financing constraints. See the survey by Schmitt-Grohé and Uribe (2003) for a comparison. (9) See Chapter 7 of Frenkel and Razin (1996) for a discussion of ‘pure’ wealth effects. (10) See Blanchard (1985) and Yaari (1965). Box 6 on page 51 describes the framework in more detail, and particular assumptions that are required. This has been an increasingly popular structure in recent years for models designed to analyse macroeconomic policy. Examples include models at the Bank of Canada (Black et al (1994) and Coletti et al (1996)), the IMF (Laxton et al (1998)), the European Commission (Roeger and in’t Veld (1997)) and the Reserve Bank of New Zealand (Hunt et al (2000)). (11) This transversality condition essentially means that other agents will not allow an agent to keep borrowing to repay existing debt, and is used to rule out explosive paths for financial assets. As is standard, we actually impose the stricter transversality condition limT →∞ (1 + r )−T bt+T +1 = 0, where b is any bond, meaning that agents do not intend to leave available resources unused. This also implies that agents do not leave any intended bequests to future generations. (12) This diagram is taken from Frenkel and Razin (1996). 27 The Bank of England Quarterly Model This point in Figure 3.3 is defined by the intersection of the two lines. The 45 degree line simply shows points consistent with steady-state equilibrium. The other line has a positive intercept because aggregate consumption in the Blanchard-Yaari framework depends on human wealth (HW). As explained in Box 6 on page 51, this is because aggregate consumption can be thought of as the sum of the consumption of ‘newborn’ consumers (equal to the marginal propensity to consume multiplied by their wealth, which consists entirely of expected future earnings – human wealth – on the assumption that they do not inherit bequests of financial or physical capital) and older consumers, who follow an Euler equation for consumption. A unique steady-state level of consumption also requires that the slope of the line in Figure 3.3 is less than unity. (13) For total consumption to be stable, the rate at which older consumers increase their consumption cannot outstrip the rate at which they die and are replaced by newborn consumers with no assets. This means that the survival probability cannot be ‘too high’: the precise condition is discussed in Box 6. (14) A permanent income shock will shift the consumption schedule upwards, raising the sustainable steady-state level of consumption. The slope of the line – and hence the equilibrium level of consumption – is a function of individuals’ preferences and interest rates. An increase in individuals’ discount rates would flatten the schedule and lead to lower steady-state consumption, because higher consumption today leads to higher debt repayment in future, and consequently lower steady-state consumption. This stable steady-state consumption position is also consistent with a desired level of total financial assets, which implies a stable position for both domestic and net foreign assets. Two possible cases are shown in Figure 3.4. (15) The line labelled ċ = 0 shows the combination of consumption and net foreign assets compatible with stable consumption and may have a positive or negative slope. If the discount rate of domestic households is lower than the world real interest rate, individual consumers seek to build up their net foreign asset holdings to sustain a higher level of consumption in future. In contrast, ‘impatient’ households (with a discount rate higher than the world real interest rate) prefer to bring consumption forward by borrowing from overseas, which reduces the future sustainable level of consumption. However, the slope of the sustainable net foreign asset schedule (labelled nḟa = 0) is positive, which implies that the price for borrowing from abroad is lower sustainable consumption, because of higher service costs for foreign debt. An increase in world interest rates would tilt this schedule upwards. Households play the key role in determining the equilibrium of the economy. To a first approximation, given the supply of domestic (financial) assets, households’ decision about how much wealth to accumulate determines net foreign assets, with consequences for the equilibrium current account and real exchange rate. We treat households and firms separately: households own domestic resources and effectively determine how much labour is supplied and the demand for financial assets. Although all (13) This is not guaranteed in the Blanchard-Yaari framework and will be satisfied for particular settings of utility function parameters relative to the world real interest rate. (14) In contrast, simple representative agent models of consumption imply that aggregate consumption follows an Euler equation. In the context of Figure 3.3, this would be represented by a line from the origin with its slope determined by preferences and the world real interest rate. In the special case in which the consumer’s discount rate equals the world real interest rate, the slope of the Euler equation is exactly unity and the line describing it would overlay the 45 degree line: there would be an infinite number of potential steady-state equilibria for consumption. Otherwise, there is no steady-state equilibrium (and consumption either grows indefinitely or shrinks to zero). (15) These diagrams are taken from Blanchard (1985) and use the simplifying assumption that the net supply of domestic assets is zero, so that net foreign assets coincide with total asset holdings. In a more general model, the relative ‘patience’ of the domestic economy determines the sign of the desired total asset position. 28 The core theory Figure 3.4: Consumption and net foreign asset equilibrium 'patient' economy 'impatient' economy C C . c=0 . c=0 . nfa = 0 . nfa = 0 NFA NFA returns from production eventually go to households, we find it convenient to make a distinction between wages, transfers, capital gains and profits. (16) If households do not wish to retain all of the returns from domestic production, they can sell their resources to overseas residents to fund higher spending, so achieving the net foreign asset position we see in the ‘impatient’ economy of Figure 3.4. (17) A further implication of the overlapping-generations structure is that some forward-looking terms for aggregate household behaviour – human wealth, transfer wealth and the marginal propensity to consume – will contain the household survival parameter. (18) That is because individuals know that for any particular date in the future there is a probability that they will be dead, which leads to a phenomenon known as ‘over-discounting’. (19) The most important behavioural consequence of over-discounting is that the model does not display Ricardian equivalence, so that private agents are not indifferent between increases in government spending financed by current taxes or by raising debt. (20) Individuals alive today know that they may not have to pay the taxes required to service future debt. This non-equivalence property does not rely on differences in returns between private agents and the government, but instead it depends crucially on the present value to the individual of future tax obligations per se. (21) (16) This disaggregation reflects a level of detail that is useful for analysis of forecast issues. (17) Technically, domestic residents trade in bonds with overseas residents. (18) It is worth emphasising here that the Blanchard-Yaari structure is mainly a device to ensure that net foreign assets, trade and the real exchange rate are all endogenous with well defined steady-state positions. In its standard form, it is insufficiently rich to take literally as a description of actual cohorts in the economy. This implies that we should treat the survival probability as a calibration constant, rather than as a parameter that we could change, say, to estimate the effects of actual changes in mortality rates. (19) Box 6 on page 51 explains this issue in more detail, and the consequences for how we model financial assets. Households will discount future income by the probability of survival, γ , as well as by the discount factor, β. (20) A further, technical consequence of this framework is that, because the probability of death is known with certainty (by virtue of a law of large numbers assumption) and can be perfectly insured against, there are no (unintended) bequests. Analysis of the macroeconomic consequences of inheritance would require a more traditional overlapping-generations setting, where individual cohorts are tracked separately. (21) Technically, it is not mortality itself that generates the non-Ricardian features of the model but rather that there will be households born in the future that will bear some of the burden; see Buiter (1988). 29 The Bank of England Quarterly Model Box 2: The consumer’s maximisation problem At date t, an individual consumer born at date s ≤ t maximises discounted lifetime utility, defined over the volume of non-durable consumption (C), the stock of dwellings (D) and real money balances (M O N ). (a) The s subscript indicates variables that are specific to an individual consumer s (so consumers are indexed by birth date). (b) Each consumer discounts future utility according to a standard discount factor 0 < β < 1 as well as the (constant) probability of survival, 0 < γ < 1, into the next period. We write the problem as: ∞ β i γ i U Cs,t+i , Ds,t+i , M O Ns,t+i i=0 where the period utility function takes the form: Ut = Utcd 1− σ1c 1− 1 σc −1 (ψ mon )− σ c M O Ns,t 1 + c ψ hab 1−σ σd (λt Nt ) 1− σ1c 1− −1 1 σc where σ c denotes the consumer’s intertemporal elasticity of consumption, ψ mon is a parameter defining the weight on real money balances in the utility function, (c) and U cd is a constant elasticity of substitution subutility function of non-durable consumption and dwellings: ⎤ σd d ⎡ d d Utcd ⎢ = ⎣φ c ψ c Cs,t σ −1 σd ψ hab H ABt + 1 − φc 1 − ψc Ds,t ψ habd H AB Dt σ −1 σd ⎥ ⎦ σ −1 The parameters φ c and ψ c determine the shares of expenditure devoted to non-durable consumption and dwellings and σ d is the elasticity of substitution between the two. Habit formation is incorporated by specifying that utility depends on the levels of non-durable consumption and dwellings relative to the habit stocks H AB and H AB D. Since we assume a so-called external form of habit formation, these variables are treated as exogenous by the individual household. The parameters ψ hab and ψ habd control the extent to which habits are important in utility (setting ψ hab = ψ habd = 0 removes the influence of habits). The consumer faces the following period-by-period nominal budget constraint: = 1 B Fs,t B K s,t (k) + Vt (k) ωs,t (k) dk + PG t BG s,t + PCt M O Ns,t + E Rt 0 1 + r f t−1 B Fs,t−1 1 + rgt−1 + PG t−1 BG s,t−1 + PCt M O Ns,t−1 γ E Rt γ 1 + 0 (1 + rkt−1 (k)) B K s,t−1 (k) + (Vt (k) + DVt (k)) ωs,t−1 (k) dk +ϒt − PCt Cs,t − P DVt 1 + τ dt Ds,t − 1 − δ dt Ds,t−1 (1) (a) Chapter 6 explains how we transform BEQM variables into stationary form, which we call detrended model units. However, we define the agents’ maximisation problems in terms of nominal variables and constraints. These are denoted in upper case and correspond to the actual units in Table 6.1. (b) This means that variables carrying the s subscript are choice variables for the consumer and variables without the subscript are taken as given. (c) The term in real money balances is adjusted for labour productivity λt and population Nt in the presence of habit formation (see below) to ensure a balanced growth steady state. 30 The core theory where the consumer’s income adjusted for transfers and taxes is: ϒt = W L t L s,t + PCt T R AN SCt + PCt T R AN S K Ct + PCt T R AN S K Pt +PCt T R AN S F Pt + PCt R F P R E Mt + PCt RG P R E Mtt −PCt T AX LU M PC (2) The budget constraint (1) describes how the individual consumer’s purchases of assets (left-hand side) are financed by the sale of previously accumulated assets plus net income during the current period (right-hand side). Total asset holdings consist of foreign bonds, B F, (evaluated in domestic currency by adjusting for the nominal exchange rate, E R (a) ); government bonds, BG; real money balances, M O N ; and holdings of corporate debt, B K , and equity, ω (at price V ) issued by domestic firms, which are indexed by k ∈ (0, 1). The first two lines of the right-hand side measure the value of previously accumulated assets after the payment of interest and dividend income. The nominal interest on foreign bonds r f and government bonds rg are adjusted for the survival probability, γ , whereas corporate bond interest rk and equity returns (dividends) DV are not adjusted – see Box 4. The last line in (1) represents income less expenditure on consumption (at price PC) and net investment in dwellings. Dwellings carry the price P D and investment is measured net of depreciation of the dwellings stock (at rate δ dt ) and inclusive of taxation costs levied at rate τ dt . Equation (2) shows that net income consists of income from labour market activities (W L · L) plus net transfers from the government (T R AN SC), firms (T R AN S K C and T R AN S K P) and from overseas (T R AN S F P); non-interest income from overseas and government bond holdings (R F P R E M and RG P R E M); less lump-sum taxes (T AX LU M PC). The return from participating in the labour market is a nominal effective wage rate W L discussed in more detail in the discussion of the wage bargain – see Box 8 – multiplied by the participation decision L s,t . (b) (a) An increase in E R represents an appreciation. (b) The individual consumer can choose either to participate in the labour market (L s,t = 1) or not (L s,t = 0). Solving for individuals’ decision rules leads to expressions for desired levels of directly imported, domestically produced and total consumption, housing stock and money demand. We then aggregate over the total population, factoring in the size of each age group (see Box 6 on page 51 for an example), to generate a set of corresponding per capita expressions for total consumption, directly imported consumption, domestically produced consumption, housing stock and money demand; these are shown in Appendix A2.1. (22) For example, although the model has overlapping generations under the surface, we can solve out for a single, per capita consumption function of the form: ct = mpct · wealth t pct (3.1) where c represents per capita demand for total consumption, mpc is the per capita marginal propensity to consume out of wealth, and pc is the relative price of total consumption. (23) All three expressions on the right-hand side of (3.1) are endogenous and all will usually vary as the effects of a shock work their (22) The core model only requires a single per capita consumption demand expression, rather than separate consumption demand expressions for each age cohort. For a general purpose macro model, this is a considerable advantage over overlapping-generations models that specify fixed lifespans. (23) Price variables in BEQM are scaled relative to a numeraire price level, which we have chosen (arbitrarily) to be the consumption price level, as explained in Section 5.1.2. In practice, therefore, pct = 1 for all t. 31 The Bank of England Quarterly Model way through the economy. The marginal propensity to consume depends on present and future post-tax interest rates, present and future consumption prices, habits, and the user cost of housing. Wealth includes the present value of current and future labour income, the present value of current and future taxes and transfers, returns on financial assets, and the current value of housing. The price of the total consumption bundle will reflect the effects of domestic demand and supply conditions on the price of domestically produced consumption goods, and the effects of shifts in world prices and the exchange rate on the price of imported consumption goods. The choice of desired housing stock reflects household preferences over durables and non-durables, and the user cost of housing capital, which depends on depreciation, capital gains, taxes and interest rates. Housing wealth – which is defined in this model as the value of the housing stock brought into the current period – is part of overall wealth that appears in equation (3.1), as housing is a potential store of value from one period to the next. It is tempting to assume from this that housing wealth ‘causes’ consumption, as higher house prices would make more wealth available to be used for consumption in the next period. However, this is not the case under the assumptions of the core theory. Consumers ‘gain’ from an increase in house prices because this increases the value of the housing stock brought forward from the previous period. But consumers also ‘lose’ since it now costs more for households to own houses. On the other hand, if there had been an exogenous decrease in the housing stock (from, say, an unanticipated ‘depreciation shock’ – a natural disaster, for example) some consumption would need to be sacrificed over a period of time to rebuild the housing stock. And the housing stock deteriorates each period, so some expenditure will be needed to maintain a constant level of housing stock. (24) Finally, we include so-called ‘external’ habit formation. Households assess their consumption against a desired level – the reference habit – so that aggregate consumption will be stickier than otherwise. (25) In other words, households are concerned about the rate of change of their consumption, as well as the level over their expected lifetime. This allows us to influence the degree of persistence of consumption following a shock. Because the habit formation is external, however, the introduction of habits does not necessarily generate a ‘hump-shaped’ aggregate consumption response to shocks, which is often a feature of models with habit persistence. (26) 3.2.2 Firms The theory of the firm must provide an account of factor and inventory accumulation, production of output from these factors, the distribution of wealth, and pricing decisions. As with households, we start with a standard optimisation problem for individual agents, before deriving aggregate decision rules. Individual firms are assumed to maximise profits given cash-flow constraints and market conditions. They are monopolistically competitive and each sells a single, slightly differentiated good in different domestic and foreign markets, while renting labour from households and paying for capital goods. Because firms have some market power, they make a profit on each unit of output such that (abstracting from taxes) the value of the firm is greater than the value of its capital stock. (24) This takes no account of the possible effects on consumption of house price rises, such as through an increase in the ability to borrow, which we do not attempt to model in the core theory. (25) In the case of ‘external’ habits, the reference habit is typically an aggregate measure of consumption, as opposed to individual consumption levels, which would be the case of ‘internal’ habits. External habits can be thought of as ‘catching up with the Joneses’; see Abel (1990). We use the external version of habits to facilitate aggregation in the Blanchard-Yaari structure. (26) See, for example, Fuhrer (2000). 32 The core theory In order to provide an account of different expenditures, the goods market is subdivided into private consumption, housing investment, capital investment, ‘other’ investment, inventories, government procurement, imports and exports (although firms only produce using a single production function, see Section 3.3.1). Firms are assumed to sell their production to these different markets, so sales revenue depends on both sales volumes and prices in these markets. An individual firm aims to sell at a mark-up over its marginal cost and is assumed to face costs of adjusting prices, such that they prefer not to change prices quickly. At any given time, all firms will choose to adjust their prices by a certain fraction towards the desired level, depending on current costs, expected inflation, and usually lagged inflation as well (see also Section 3.4.3). The mark-up is time-varying because of this nominal stickiness, but would be constant if prices adjusted instantly. (27) A firm’s cash flow is equal to sales revenue less labour costs (the gross wage bill plus employers’ social security contributions), investment, (net) debt servicing, transfers to overseas residents and domestic households (such as pension contributions), and lump-sum taxes. After paying tax and debt interest payments, the remainder is distributed to shareholders (households) as dividends. In order to account for aggregate imports, we build demands for imported consumption goods into the optimisation problem of the household, and imported capital goods into the optimisation problem for the firm. We assume that firms desire a particular level of a composite capital good and that capital can be sourced from both domestic and overseas production. (28) This implies that the real exchange rate will affect the user cost of directly imported capital, and hence the supply side. (29) The split between domestic and imported capital will depend on their relative prices and on firms’ preferences. (30) Total capital is accumulated to match its marginal product (corrected for the mark-up over real marginal costs) with the effective user cost. This is a conventional Jorgensonian function of depreciation, taxes and price changes. As a default, firms are assumed to face real rigidities in adjusting both the capital stock and investment, which are modelled as quadratic adjustment costs that have no effect on the steady state. These costs are tangible, involving lost output for firms, and therefore affect the marginal returns to capital when there is dynamic adjustment to equilibrium. (31) The presence of adjustment costs ensures that the path to the new steady state is not as jumpy as would be implied by classical investment theory. While there is no explicit variable in the model representing Tobin’s q, the behaviour of investment is compatible with that theory. (32) However, unlike in the simple textbook representation, the steady-state value of Tobin’s q is not unity, because our model assumes monopolistic competition in the goods market and allows for capital taxes. Total capital is predetermined in the production function, as capital used in the current period must be installed in the previous period, but the amount of labour used is decided in each period. Firms bargain with unions over nominal wages (see Section 3.3.2) and incur adjustment costs in changing employment (just as for capital). We impose a ‘right to manage’ assumption, which means that firms unilaterally decide on the level of employment, once the wage bargain is complete. This results in an equilibrium in which workers are paid their marginal product (adjusted for the mark-up and accounting for adjustment costs). (27) In the steady state, the mark-up on a good is solely a function of the (constant) elasticity of demand in the particular market for that good. (28) As we observe from UK trade and National Accounts data. (29) This is different from the basic neoclassical open economy model in which the capital-labour ratio is tied down by the world real interest rate. (30) Formally, we assume that the capital index is a constant elasticity of substitution (CES) aggregate of imported and domestically produced capital goods. (31) These losses do not result in payments that benefit any other agent – they are ‘frictions’ that are lost to the macroeconomy. (32) See, for example, Chapter 8 of Romer (1995). 33 The Bank of England Quarterly Model If capital is predetermined and labour costly to adjust, the marginal cost curve will be steep. Since pricing decisions are a mark-up over marginal costs, this implies that firms respond to demand shocks with large changes in prices and small changes in production. To create more flexibility for these key model properties, we therefore assume that, in the short term, the firm can vary the intensity with which capital is used, thereby increasing or decreasing the effective level of capital entering production. (33) The ability to utilise capital more intensely flattens the short-run marginal cost curve. However, the consequence of higher (lower) utilisation is higher (lesser) depreciation, as the rate of wear and tear is assumed to increase if machines are run more intensively. In the long run, only an increase in physical capital, labour or productivity can generate more output. With capital and labour in place, firms produce value added using a Constant Elasticity of Substitution (CES) technology. As is well known, an infinite elasticity of substitution means that capital and labour are perfect substitutes, while a zero elasticity of substitution implies a Leontief function, where capital and labour must be combined in fixed proportions; and an elasticity of one implies a Cobb-Douglas technology. The motivation for using a CES production function, instead of the simpler Cobb-Douglas form, is that the elasticity of investment to interest rates would be unrealistically high under the assumption of Cobb-Douglas technology. Correspondingly, we assume that capital and labour tend to be less substitutable for each other than in the Cobb-Douglas case. Firms also hold inventories. The assumptions we make about expectations, discussed in Section 5.1.5, mean that firms’ decisions on optimal production levels are not affected by uncertainty. As such, there is no role in the core model for inventories as insurance against demand surprises. Instead, we assume that firms first make their decisions for capital, labour and prices, and then make a decision about the desired level of inventories. This target level balances two costs: the expected cost of foregone sales and the expected opportunity cost of holding stocks. (34) Target inventories will therefore be affected by average sales revenue per unit, real marginal costs and real interest rates. Some production is diverted from (or added to) sales in order to achieve the desired stock level; the change in stocks enters the goods market clearing condition and is part of total expenditure. As with other factors, we usually assume that stock adjustment is costly, so that firms are not always at their desired level. Firms finance their activities by issuing shares and debt. Since we allow firms and consumers to discount the future at different rates, we impose a debt-equity ratio to prevent firms issuing only debt or only equity. Shares are valued using the standard dividend discount model, accounting for taxes, which emerges from the household optimisation problem (see Section 3.3.3 for details). Each firm is assumed to produce a slightly differentiated good, which gives them market power. Our market structure implies that market power in each goods market is inversely related to the elasticities of demand in those markets. (35) As in Blanchard and Kiyotaki (1987), we make the standard assumption that there is a large number of such firms and that they behave symmetrically. (36) This allows us to aggregate across firms to get the decision rules for employment, domestically produced and imported capital, inventory stocks, output, and prices for each domestic market and the export market. (33) We use the implementation in Burnside and Eichenbaum (1996). (34) The expected cost of lost revenue is proxied by a scalar on next period’s average sales price; given factor costs and interest rates for the storage costs, this dictates the stock-sales ratio. This is a version of the model in Kahn (1987). (35) This is a very useful simplification, for example, see Dixit and Stiglitz (1977). (36) That is, firms are assumed to make the same pricing and factor decisions. See pages 235-236 of Walsh (2003) for details. 34 The core theory We assume that prices are sticky because firms want to avoid changing prices. We do this by assuming price adjustment costs in firms’ ‘disutility’, as in Rotemberg (1982), (37) which means that aggregation is simple and direct. Price adjustment costs are intangible in the sense that firms do not incur direct costs, like labour and capital adjustment costs in our model or in ‘menu cost’ approaches such as Mankiw (1985). (38) An alternative pricing assumption would have been that of Calvo (1983), which we use in the union wage bargaining context (see Section 3.3.2). In a Calvo setting, only a (randomly selected) proportion of firms would be allowed to change their price at a given date. The aggregate price level would therefore be a weighted average of newly set prices and prices set in the past. These different prices would create a source of heterogeneity among firms, implying that firms would chose different levels of employment, capital stocks and utilisation rates. In addition, we would have to account for the fact that firms have different labour demand schedules in our wage bargaining framework – so that aggregation problems would ‘spill over’ into other markets. (39) For tractability, therefore, we assume Rotemberg price adjustment costs, which means that firms behave identically in changing their prices, unlike in the Calvo setting or if fixed-term pricing contracts were assumed. (40) This means that we can talk about ‘the firm’, even though a continuum of differentiated firms underlies the decision rules. 3.2.3 The government The main requirement for the government sector is to be able to account for fiscal revenues and expenditures, at a level that is detailed enough to be able to model the effects of government policy decisions on the demands for goods and labour. (41) This flow accounting needs to be matched by stock accounting that draws out the implications for the supply of government debt to financial markets. Importantly, we do not need to use the model to analyse public finance issues, and so we are able to abstract from an analysis of optimal fiscal decisions arising from social preferences. This simplified purpose means that we can assume a fairly simple fiscal reaction function, given policy targets; we do not need to distinguish between central and local government; and the government redistributes resources according to some simple rules. The government gains revenue from taxes, issuing debt and seigniorage, while making expenditures on government purchases of goods and labour, transfers, and debt servicing. As with households, debt is not allowed to follow an explosive path. This solvency condition is ensured by enforcing the period-by-period fiscal budget constraint, along with a fiscal reaction function. We assume that the government has targets for debt, spending (on goods and on labour) and transfers, so that it does not generally follow a balanced budget each period. Moreover, debt and spending are allowed to vary temporarily from their target levels following a shock. (37) See Pesenti (2002) for an example of another recent model that uses the Rotemberg pricing model. (38) Allowing price adjustment to result in direct costs would not have a first-order effect on pricing decisions. However, it would require us to allocate these adjustment costs to an element of the National Accounts data. (39) It becomes difficult to aggregate individual decisions across the population of firms without additional simplifications (such as an economy-wide rental market for capital goods). While considerable progress has been made in this respect (see for example Chapter 5 of Woodford (2003)), the aggregation usually relies on log-linearisation of decision rules around a particular steady state. This is not suitable for our purposes, because we retain the decision rules in levels in order to analyse shocks that change the steady state of the model. In any case, the resulting pricing behaviour is likely to be similar to adopting the Calvo approach: in simple models, the two approaches yield identical (to a first-order approximation) reduced-form pricing equations, as shown by Rotemberg and Woodford (1999) for example. (40) Neither the Rotemberg nor Calvo alternatives are necessarily realistic descriptions of pricing behaviour at the level of individual firms. At that level, it seems more likely that frictions such as menu costs are perhaps more relevant. When choosing a specification for nominal rigidities in a general purpose macroeconomic model, however, tractability is often the deciding factor. (41) Box 3 on page 36 discusses the relationship between private sector and government output. 35 The Bank of England Quarterly Model Box 3: Private sector output and government output In the core theory, firms are profit-maximising agents selling output in (imperfectly) competitive markets. The concept of output most applicable to the model’s production function is the (value-added) output of private sector firms. And the principal inputs into the production function should, correspondingly, be private sector hours worked and the private sector capital stock. In reality, however, the government also produces output using its own capital and labour inputs, which is reflected in the National Accounts measure of GDP. So the relationship between the core model’s measure of private sector output and National Accounts GDP is given by: gdp = y + yg + cir where gdp is National Accounts GDP, y is private sector value added, yg is the value added of the government, and cir represents actual and imputed rentals on dwellings (included in GDP but not in the core theory’s measure of output). The government’s purchases of private sector output (procurement) are included in y, and the private sector production function is given by: y = F(k, e) where k and e are private sector capital and labour inputs. In principle, a production function for the government’s value added could also be written, dependent on the government’s factor inputs of capital, kg and employment, eg: yg = G(kg, eg) The issue for the core theory is the extent to which the government’s output and factor input decisions are assumed to affect private sector behaviour. The key channel is through its demand for factor inputs. The government’s demand for capital goods has a direct effect on private sector output as they are produced by private sector firms. And the government’s demand for labour competes with the demand for labour from private sector firms, and so will affect wage costs in the private sector. In general, therefore, the government both adds to the demand for private sector output and absorbs factor inputs that could be used to produce private sector goods. Section 3.3.2 explains how government employment and government wages are assumed to interact with the private sector wage bargain. There are two other channels through which the level of government output might in principle affect private sector decisions. First, the provision of public goods (eg public health provision) might affect the marginal utility of consuming some types of private sector goods (eg private health insurance), (a) though the overall effects will depend on the extent of substitution between all goods in the consumption basket. Second, government output (or the government’s capital stock) might enhance private sector productivity – for example, increased investment in transport infrastructure. These two channels are not articulated in the core theory of BEQM. This does not mean that they are assumed to be unimportant, but rather that they are unlikely to affect private sector decisions materially over the normal forecast horizon (which is the most relevant for our purposes). Therefore, the government’s production function does not appear in the model and its demands for factor inputs are specified according to simple rules governing the government’s wage bill and capital spending, and the relationship between private and public sector wages. (a) Technically, this requires public goods to enter households’ utility non-additively. 36 The core theory For simplicity, government bonds take the form of single-period nominal bonds. Government employment and purchases are assumed not to add to the productive capital stock or contribute to household utility. (42) Transfers are made to households, firms and the rest of the world, based on target ratios to output. There is a wide range of taxes: on households and firms, directly on income and expenditure, on stocks and flows, and distortionary and lump-sum. In practice, some of these will be set to zero, if we cannot split them out when looking at the aggregate tax data. But they are included to allow a wide range of scenarios to be considered. The fiscal side of the model is closed by specifying an element of the budget constraint to be the endogenous instrument. Virtually any item in the government budget constraint could be used as the fiscal instrument (in principle, more than one element could be chosen), as long as the constraint is not violated. The fiscal reaction function specifies that the government varies tax revenue according to the difference between debt and its target and the rate of change of the outstanding debt stock. This implies that debt initially acts as a buffer following a shock, before settling back to its long-run target level. (43) As a default, a single household tax is normally assumed to be the endogenous instrument, with target ratios for the remaining taxes relative to the relevant flows. 3.2.4 The monetary authority The monetary authority anchors the nominal side of the economy. As with fiscal policy, we assume a simple reaction function for the monetary authority, rather than an optimising strategy. A number of different targets, instruments and reaction functions could be specified. The current default that ensures that the nominal side is anchored in the long run is a simple Taylor rule in which the short-term (one-period) nominal interest rate is used to ensure that annual CPI inflation is ultimately maintained at a target level of 2%. Some smoothing of the nominal interest rate is usually assumed. (44) 3.2.5 The external sector The characterisation of overseas agents is the most simplified of all. Consistent with the specification of a small open economy, the actions of overseas agents are characterised by exogenous paths for the world interest rate, world prices and inflation, and overseas demand. The domestic economy is completely small in capital markets, which means that there are no restrictions on the flow of foreign capital and domestic agents are price-takers in foreign capital markets. Monetary and fiscal policy actions in the domestic economy do not, therefore, affect interest rates in the rest of the world. The domestic economy is not completely small in international goods markets: domestic and foreign goods are differentiated and have different (common currency) prices. Foreign demand for domestic goods depends on the relative (common currency) price of domestic and foreign goods. (45) One of the key features of this model is that it determines an equilibrium real exchange rate. (46) There is no simple reduced-form expression for the real exchange rate, but the equilibrium real exchange rate can (42) Since these goods do not enter the household utility or private sector production functions, and government policy follows simple rules, it is not necessary to specify a production function for public goods. (43) This is how the model is set to work when simulating the response to shocks. Given the purpose of the government sector in BEQM, we use a simple reaction function rather than attempting to model optimal fiscal strategy. When used for forecasts, assumptions from HM Treasury about fiscal variables are conventionally imposed over the forecast horizon. (44) For forecasts, constant or market interest rates are imposed over the forecast horizon. See Chapter 5 for a fuller discussion. (45) If the domestic economy were completely small, domestic exports and foreign goods would sell at a common price, and foreign demand for domestic exports would be perfectly elastic. (46) The real exchange rate is defined as the nominal exchange rate adjusted for the relative consumer price levels between the domestic economy and the rest of the world. This definition is different from the definition of the real exchange rate as the ratio of tradable and non-tradable goods prices that is common in recent work in ‘new open economy’ models. 37 The Bank of England Quarterly Model be thought of as the relative price that ensures that export demand clears the domestically produced goods market and maintains internal balance (see Section 3.3.1). Additionally, the current account equilibrium is the flow equilibrium consistent with the net foreign asset stock equilibrium; the real exchange rate is the relative price consistent with this flow. In the steady state, these all settle to constant levels (abstracting from growth). Hence household choice, firms’ decisions, and domestic policy all contribute to determining the value of the real exchange rate, given conditions in the rest of the world. The current and capital account identities are not explicit in the model but are standard and enforced. The capital account is simply the change in the net foreign asset position, and is equal in magnitude and opposite in sign to the current account. The current account is the sum of net trade, foreign asset returns (which reflect interest payments and real exchange rate changes) and net transfers between domestic and overseas agents. Net foreign asset accumulation can therefore be defined as output less domestic expenditure, foreign debt servicing and net transfers, so that the net foreign asset stock reflects the gap between domestic supply and demand flows. 3.3 Characterisation of the markets Having described the motivations of agents, we now set out the characteristics of the markets in which they interact: goods markets, a domestic labour market and financial markets. 3.3.1 Goods markets The characterisation of goods markets has to be sufficient to describe equilibrium in demand and supply, which is a statement about how production is cleared. At the same time, we have to be able to map model concepts to National Accounts data. We assume that private sector firms add value using capital and labour. The measure this corresponds to is private sector value added. (47) This approach raises three key issues. First, the choice of a single production function on the supply side makes it difficult to say anything about different expenditure deflators. Second, a significant proportion of imported goods are intermediates in production, so we need some extra modelling to account for final private sector output. Finally, we need to match private sector inputs with private sector output. We describe below how we have attempted to tackle these issues – in particular, how we have dealt with the inherent tension between using a single production function and the need to differentiate between expenditures. Consider first production from firms. As we saw in Section 3.2.2, we assume a continuum of firms and that each produces a slightly differentiated output. Product differentiation is the source of market power and the foundation for thinking about sticky prices. We aggregate the decision rules of firms, so that when we talk about output produced by ‘the’ firm, we are referring to a composite index of production. We take this index to be a measure of value-added production. Firms produce finished goods from capital and labour: F (k, e) ≡ y. (48) To reconcile different categories of expenditure with goods from a single production function, we assume that firms can distribute production to different markets for alternative uses – so the composite good is a sort of wondrous substance that is used for domestic non-durable consumption, accumulation of durables (housing), domestic capital expenditure, stockbuilding, government spending on private sector output, and exports. (47) To avoid potential distortions to real output measures arising from differing indirect tax rates across expenditure components, we use real value added at basic prices. (48) In practice, output is defined net of capital and adjustment costs (see equation (A.30) in Appendix A) 38 The core theory Figure 3.5: Production-clearing flows and stocks Capital stock Labour Production chv + id + ikhv + iov + dels + gv xv + = y Housing stock Inventory stocks This implies that production clearing is defined by the following identity: (49) F (k, e) ≡ yt = chv t + idt + ikhv t + iov t + delst + gv t + xv t meaning that everything produced by domestic firms is distributed to markets for home consumption, investment in dwellings, investment in home capital, other investment, (50) inventory accumulation, government purchases and exports. The production-clearing identity does not require relative prices: it is a statement about how volumes of the composite good are distributed. The consequent flows and can be represented as in Figure 3.5. Figure 3.5 illustrates the assumption that the good from the single production function – call it ‘rice’ – can be sold as ‘consumption rice’ for eating, relabelled as ‘government rice’, costlessly and instantly transformed into ‘rice bricks’ for dwellings, replanted as ‘investment rice’ for capital investment, stored as plain rice for inventories (to be converted later), and sold in export markets. Consequently, the accumulations of inventory stocks, (domestically produced) capital stock, and housing stocks are all in ‘rice units’. (51) In order to match nominal (money) expenditure, however, we do need prices. Otherwise, we would be unable to match National Accounts deflators or analyse the impact of relative price changes on the user costs for capital and housing. We therefore assume that the different markets are segmented, so that the domestically produced good can bear a different price in each market. We do this by assuming different degrees of market power in each market (52) – as a special case, we could assume that all markets are (49) The suffix v denotes that this is a value-added measure of output. There is no distinction between value-added and final expenditure on housing investment or stockbuilding, so these do not carry the suffix. See Appendix A.1 for details of the mnemonics. (50) Here we define iov as other investment, which mainly includes transactions costs and fees on transfers of buildings. (51) Of course this ‘rice’ is the composite good. So we might stretch the metaphor and describe the rice as a mixture of grains produced from different plots of land. 39 The Bank of England Quarterly Model equally competitive, so that all goods are effectively sold in one market. Clearly, this is a compromise because we are trying to account for differences in the price elasticities of demand for different goods. This ignores technological factors, but a more fully articulated multisectoral setting would be needed to account for them properly. (53) The analogue to the production clearing condition depicted in Figure 3.5 is a goods market clearing condition that accounts for imported goods – both final and intermediate goods. The total demand for imports is not modelled by a single equation, but comes from the demands for imported final consumption and final capital goods from the maximisation problems for households and firms (as well as intermediate imports, see below). (54) However, we cannot just add imported goods to the picture in Figure 3.5, as we are then adding ‘corn’ to ‘rice’. We can, however, reconcile expenditure on domestic and imported goods by an identity that divides total money expenditure into consumption, investment, stock-building, government procurement, exports and imports. We also need to account for imports of intermediate goods, which form a significant proportion of total imports and enter the production process creating the final output that is divided between expenditure categories. Clearly, one way of dealing with intermediate imports would be to model a two-stage production process, where intermediates were potentially inputs into domestic production and a factor of final production itself, but this would require a more fully structured multisectoral setting. Alternatively, we could model output as a function of capital, labour and imported intermediates, but this would mean having to subtract a proportion from output in order to match real production clearing on a value-added basis. We assume instead that intermediates must be combined with domestically produced goods in fixed proportions, which vary according to expenditure category. (55) This allows us to match National Accounts concepts while ensuring that consistent market clearing conditions hold. It has the added advantage that the demand for intermediates depends on the composition of expenditure, which is also a feature of the data. The focus on private sector firms means that we have to take account of the effects of government value added and government employment. This is chiefly a matter of ensuring that data series are defined appropriately, as described in Box 3 on page 36. 3.3.2 The labour market and unions Labour market interactions have implications for household income (hence financial asset accumulation and the current account), capital investment (hence wealth generation), net government revenues (through labour tax revenues and government expenditure on wages and benefits, hence deficits and government debt), and nominal wages (hence firms’ costs and inflation). The labour market has to be able to provide an account of non-competitive labour market outcomes – to match observed labour market behaviour – and also movements in labour market participation, employment and unemployment. (52) We use a form of monopolistic competition that ensures that elasticities of demand and market power are inversely related and controlled by the same parameter – see Blanchard and Kiyotaki (1987). In general, we would expect market entry and exit to affect mark-ups in monopolistic competition, but that would make aggregation substantially more complicated and would be impractical for a general purpose model such as this. (53) Pesenti (2002) illustrates a multisectoral treatment of relative prices in an open economy. (54) We could include imported goods in government expenditure, but these are not significant in the data. (55) This is as if we have introduced another layer of firms into the model, which sell gross home consumption and capital goods to domestic households and firms by combining imported intermediate goods with value-added goods using a Leontief technology. Unlike the firms in Section 3.2.2, they do not use capital or labour. If we assume that these firms are perfectly competitive, they will make no profit themselves and hence contribute nothing to value added. 40 The core theory In some models, non-competitive wage outcomes are generated by assuming that workers have a degree of market power, because they each offer a differentiated labour service. (56) However, in such models the choice variable of both households and firms is hours worked, so the labour market always clears. This means that everyone is employed, and hours worked and the real wage move to clear the labour market. (57) Formally, this is because labour market equilibrium implies a real wage that clears demand and supply for labour (in hours). We therefore need to introduce an additional relationship – a ‘wage-setting’ curve – that will allow us to model unemployment. We could use several devices: matching models (see, for example, Mortensen and Pissarides (1994)), insider-outside models (see Layard, Nickell and Jackman (1991) and references therein) or union bargaining models (see Manning (1993)). We have retained the same kind of bargaining framework that was used in the MTMM model. This should be thought of as a metaphor to generate the outcomes we want, rather than a literal description of the UK labour market. The implementation here is structural and so needs to take account of general equilibrium issues that are not usually dealt with in other settings. Labour supply is determined by a binary choice made about participation: workers decide to enter the labour market if the expected return from doing so exceeds their ‘reservation’ (participation) wage – the decision here is whether or not to enter the labour market, not about how much labour to supply. The expected return from entering the labour market is a weighted average of private sector (post-tax) wages, public sector (post-tax) wages and unemployment benefits. (58) The reservation wage reflects the marginal disutility of work, relative to not working, which we assume is exogenously distributed across households. The level of employment is chosen by firms to maximise profits. Unions bargain on workers’ behalf. In any given period, a proportion of (randomly chosen) unions engage with firms in a bargain over the nominal wages of the workers they represent. This fraction is constant, so that we have Calvo (1983) nominal wage setting, rather than contracts for fixed terms as in Taylor (1980). (59) Unions aim to maximise the welfare of an average worker, so the value of the ‘outside’ earnings that could be received if employed by the government or unemployed has a role to play. The private sector wage is determined as the Nash equilibrium in which the firms’ and unions’ strategies are both optimal. The wages of government employees are set according to a simple rule linking government and private sector wages. (60) The natural rate of unemployment in this model depends on many factors, including those affecting the wage bargain: the reservation wage, technology, goods and labour market conditions, taxation and benefits, and unions’ preferences and power. The real exchange rate can affect the natural rate both through profitability and through labour supply decisions (because workers care about their wage in terms of their ability to purchase a bundle of consumption goods). The interaction of the factors described above in the labour market defines an outcome for numbers employed. For simplicity, we assume that average hours worked are given exogenously. (56) In those models, workers seek a mark-up of their wages over their own implicit costs of working (such as the opportunity cost of leisure), just as firms seek a mark-up over real marginal costs. See, for example, Erceg, Henderson and Levin (2000). (57) Galí (1996) employs a mapping from hours directly into unemployment in order to get, partially, around this problem. (58) The weights reflect the probabilities of being employed in the private sector or public sector or of being unemployed. See Box 8 on page 58. (59) While contracts for fixed terms may be more realistic, Calvo-style contracts are particularly tractable. Our assumption here contrasts with the discussion of firms’ pricing in Section 3.2.2. We find Calvo pricing assumption more tractable for the labour market than for firms. (60) Box 3 on page 36 discusses the relationship between private sector and government output. 41 The Bank of England Quarterly Model Box 4: Over-discounting and insurance against mortality In a standard textbook setting with infinitely lived representative agents, households are indifferent between owning debt or equity claims on firms. (a) This arises because all assets bear the same rate of return (given assumptions that financial markets are efficient, with no predictable arbitrage opportunities or tax distortions, and certainty equivalence). Hence we can write such models with rental contracts between households, who own capital, and firms, who want to use it; or with a stock market in which firms own capital, but households can buy claims on the wealth that it generates. This equivalence is complicated by the introduction of Blanchard-Yaari households. Individuals in this setting do not expect to live for ever, and so ‘over-discount’ future income streams: a bond bought now yields a future return which will be discounted by the probability of survival, γ . The Blanchard-Yaari model contains an assumption that lenders can insure themselves against the expected probability that the borrower will die before the debt contract can be fulfilled. If the real interest rate is rt the expected real return from purchasing a quantity bt of bonds at date t is γ1 · (1 + rt ) · bt where 0 < γ < 1 is the probability of survival into the period t + 1. The assumption of perfect insurance for bonds ensures that the average real return is (1 + rt ) · bt . The insurance arises because total payments made at the start of period t + 1 to those who die at the end of period t, a value of (1 − γ ) · (1 + rt ) · bt , are transferred to those who have survived into the new period, thus ensuring a market return of (1 + rt ). In other words, the assumption of actuarially fair premia means that the market rate of return is not distorted by the probability of death. The yield of a one-period consumption bond to the individual is greater than the market rate, which is consistent with the notion that households collectively over-discount future outcomes and require compensation for holding assets. We assume that government and foreign bonds are insured as described above, implying that the steady-state market real rate of return will be the same on each asset. We could also assume that claims on the firm – equities and corporate debt – were similarly insured, but do not do so to avoid unintended wealth effects that could arise when wealth generated by firms is distributed back to households in different forms. Suppose, for example, an increase in the flow of transfers from firms to consumers (such as pension contributions): this would decrease dividend payments, if there were no change in firms’ profitability. Transfers are uninsured and are therefore over-discounted using the survival probability, γ . If equities were insured – ie the aggregate return was the same as the market return on government bonds – then households would not be indifferent to this change. Their insured (dividend) wealth would fall and their uninsured (transfer) wealth rise, so net wealth falls. We want to ensure that there are no such wealth effects on consumption. This is not because we believe that changes in the composition of income streams never affect net wealth. Rather, it is unlikely that such effects are accurately captured by the simple Blanchard-Yaari structure. We therefore assume that the dividend stream associated with holding equities is uninsured. Since the stream of transfer payments from firms is also uninsured, consumers would not care whether they receive higher dividends or higher pensions transfers, and there is no wealth effect. (a) See, for example, page 102 of Obstfeld and Rogoff (1996). 42 The core theory We have found that this effect is potentially non-trivial: gross income flows have been constructed to match the National Accounts and can change substantially. Our assumptions ensure that these transfers net out in terms of private sector income flows, but they would not net out in terms of private sector wealth if we assumed that equities were insured. If equities were uninsured and corporate debt insured, the average rate of return on the household financial asset portfolio would depend on the composition of bonds and equities, and firms would not be indifferent to the source of their finance. We want to avoid this too, so we assume that corporate debt is uninsured. We also assume that firms and consumers discount the future at different rates, which allows leverage over the cost of capital without having to change the household over-discounting parameter, γ . Within this structure, an exogenous debt-equity ratio determines the level of outstanding corporate debt. 3.3.3 Financial and asset markets Financial prices equate the demand and supply of financial assets, which in turn reflect underlying equilibrium in the goods and labour markets. As we saw in Figure 3.2, an important relation in the model is that households’ aggregate demand for financial assets is met by the domestic supply of assets (equities, corporate debt and government debt) and overseas supply of (net) assets. A steady-state position is reached when equilibrium flows are at levels that sustain unchanging (adjusted for growth) asset stock positions at their desired levels. For example, the net foreign asset position is stable when the current account generates exactly enough net income to finance the net interest payments on net foreign assets. To reach a steady-state position, flows must adjust over time so that the assets can be accumulated (or decumulated) to the desired long-run level. For example, a rise in desired net foreign assets requires, on average, smaller current account deficits (or larger surpluses) during the transition period. A key simplification of the treatment of asset prices arises from assuming certainty equivalence (see Chapter 5). In practice, this means that a single risk-free market rate of return is used to value assets. A further simplification is that we have not attempted to model a layer of financial intermediation. Instead, we assume that households own claims on firms, government and overseas residents. The model is not, therefore, directly useful for issues where financial intermediation is of first-order importance, such as collateral effects and financial accelerator phenomena. (61) We prefer to look at such issues using separate, more specialised models and keep the balance sheet dimension of the macro model as simple as possible. The most basic asset in the model is a one-period nominal bond. These bonds can be issued by firms, by the government or by households. Technically, we only allow households to trade bonds with overseas residents, but the net effect is equivalent to allowing firms to hold gross investment positions in foreign assets and liabilities. As the economy is completely small and open in capital markets, uncovered interest parity (UIP) is a standard no-arbitrage condition that prices the exchange rate to equalise the return on riskless domestic and foreign bonds. Firms can issue shares, which are priced according to the standard (post-tax) dividend discount model. (62) (61) See, inter alia, Bernanke, Gertler and Gilchrist (1999). (62) A complication in the Blanchard-Yaari setting is that individuals ‘over-discount’ because they do not expect to live for ever, so it is assumed that households can insure themselves. But this means that certain additional assumptions (set out in Box 4 on page 42) are required to prevent undesirable wealth effects. 43 The Bank of England Quarterly Model The difference between the supply of domestic financial assets and households’ demand for financial assets is taken up fully by overseas residents in the form of net foreign assets. (63) These are single-period bonds priced in foreign currency units, so exchange rate changes will affect the value of domestic households’ portfolios. Although the level of total financial assets held does reflect an optimisation decision by households, there is no portfolio choice because there is no risk-return trade-off to balance across different assets. Asset stocks and prices can reveal potentially important information. Figure 3.2 shows how agents are linked by asset markets. Household financial assets are composed of liabilities from government, firms, and overseas residents. The value of firms’ liabilities reflects decisions about pricing, wage payments, employment, capital, utilisation rates and inventory accumulation. The value of government liabilities will reflect policy decisions for taxes, transfers and spending, given demand conditions. Finally, the value of foreign assets will reflect movements in the current account and the exchange rate. 3.4 The nominal side of the economy and monetary transmission The previous discussion focused on the real side of the model economy. But the analysis of inflation requires attention to nominal magnitudes and to the determinants of nominal wages and prices. 3.4.1 Money market equilibrium Money in BEQM is modelled as a non-interest bearing government liability that enters the government budget constraint through seigniorage. We create a demand for money by the introduction of real money balances to the utility function of households, but money has no idiosyncratic features over and above those of any other good. (64) So money is dominated by assets earning a positive nominal rate of return, and households would not choose to hold any of their portfolio in money if it were not included in the utility function. (65) Given households’ demand for money, the actions of the monetary authority determine the money supply. This can be targeted directly or can react to the effects of interest rate changes, as discussed in Section 3.4.4, depending on the assumed monetary policy reaction function. With an inflation target pursued through interest rates, the assumptions about a balanced real growth equilibrium imply that the money stock grows at the same rate as nominal output in the steady state. Without nominal rigidities, the market equilibrium outcomes for real variables and relative prices would be independent of the supply of money, and the model could be thought of as an exchange economy with no role for money or nominal prices. In such a case, equilibrium in the money market would determine the price level to satisfy households’ demand for real money balances. However, when there are nominal rigidities as in BEQM, the determination of nominal prices and real variables cannot be separated in this way. Relative prices are slower to adjust and changes in nominal and real variables are jointly determined, because demand and supply decisions are affected by the relative prices of goods and factors of production. So nominal prices and inflation will depend on the behaviour of households, firms, the rest of the world and the monetary authority. (63) The theoretical structure determines net positions, but does not say anything about gross positions. (64) As discussed in Chapter 2 of Walsh (2003), there are a variety of assumptions that can be used to introduce a role for money in general equilibrium models. We choose the money in utility approach for tractability. (65) Chapter 2 of Woodford (2003) presents analysis of a ‘cashless economy’. This analysis shows that monetary equilibrium can be sustained in a model in which the traditional motivations for money demand (transactions costs, cash in advance constraints) are absent. 44 The core theory While monetary policy can have short-run effects on the real economy, BEQM is, by construction, neutral with respect to the price level in the long run. (66) But the presence of Blanchard-Yaari households means that the model is potentially non-superneutral (that is, the equilibrium level of output will be affected by the inflation rate). (67) The size of such effects depends on the parameters of the model, but they are quantitatively small under the current parameterisation. 3.4.2 Nominal wages and prices As noted in Sections 3.2.2 and 3.3.1, firms operate in imperfectly competitive markets and are therefore able to sell goods at a mark-up on costs, which we assume can differ across markets. This will be reflected in different relative prices in the long run. As described in Section 3.3.1, imports are made up of final consumption and capital goods, and intermediate goods for consumption, capital investment, government purchases and exports. We assume local currency pricing, which implies gradual pass-through of foreign price shocks to domestic prices (when prices are sticky). The model assumes long-run relative purchasing power parity, as a feature of the balanced-growth equilibrium. Export prices are set in foreign currency and are subject to the same form of price adjustment costs as domestic prices. This ‘pricing to market’ behaviour provides a short-term channel through which the exchange rate affects export prices in terms of domestic currency. The inclusion of intermediate goods implies an extra channel for price effects. Given the assumptions of fixed shares of gross domestic goods and intermediate imports, the relative price of, say, final home consumption is a weighted average of the price of home consumption value added and the price of (imported) intermediate goods. So shifts in imported intermediate costs will (eventually) be passed on in full to consumers. Imported intermediate goods are also used in the production of exports, so domestic export prices will depend on foreign prices through two channels: the competition effect described above, and the effect of intermediate import costs. In addition, the production process for each type of good is assumed to be subject to indirect taxes, which are assumed to be passed on fully into final prices. The final price of each good in National Accounts terms is a market price – that is, inclusive of indirect taxes – as opposed to a basic price. Each expenditure category has a different weight of indirect taxes in final prices. Section 3.3.1 describes how value-added output is combined with imported intermediates in fixed proportions, and we employ the same assumption for indirect taxes. (68) A rise in indirect tax rates leads to a change in final prices relative to basic prices. (69) (66) That is, the economy’s real equilibrium is not affected by the general price level. (67) See Orphanides and Solow (1990). The steady-state inflation rate changes the real value of seigniorage in the government budget constraint. As discussed in Section 3.2.1, the Blanchard-Yaari structure is non-Ricardian and so a change in the real tax burden can have effects on net wealth and consumption. (68) As discussed in Chapter 6, the final price is therefore a weighted average of three components: the price of domestic value added; the price of imported intermediates; and indirect taxes. (69) Market prices include indirect taxes (net of subsidies) on both products and production in the National Accounts. Basic prices only include taxes on production. Taxes on products are defined as taxes linked to the sale of a unit of output, such as VAT and duties on fuel and tobacco, and are typically passed on directly in prices paid by consumers. Taxes on production, on the other hand, are taxes on the overall process and are typically based on the use of fixed capital or the right to undertake particular activities; examples include local authority business rates and vehicle excise duty levied on firms. 45 The Bank of England Quarterly Model Box 5: The determination of inflation When looking at the core model equations (see Appendix A.2), there is apparently no ‘pricing equation’ as in traditional forecasting models, or Phillips curve as often seen in DSGE models. Because the model is written in non-linear levels in stationary units, the dynamic pricing behaviour is summarised in the behaviour of a real marginal cost expression. This box presents a simplified version of the core theory, to show that our representation is consistent with other models. (a) Our starting point is to assume a continuum of monopolistically competitive firms. A firm, indexed by k ∈ (0, 1) , solves the following constrained maximisation problem: max ∞ i=0 β i Pt+i (k) Yt+i (k) − Wt+i E t+i (k) − subject to Pt (k) Pt Yt (k) = and χ 2 −η Pt+i (k) −1 Pt+i−1 (k) (1 + ṗ ss ) 2 Pt+i Yt+i Yt (1) (2) Yt (k) = TFPt E t (k) (3) where 0 < β < 1 is a discount factor, which we assume for simplicity is constant, P is the nominal price of output, Y is the (volume of) the firm’s output, W is the nominal (money) wage level, E is employment, TFP is productivity, and ṗss is the steady-state inflation rate. We denote the adjustment cost parameter as χ > 0 , and η > 1. The firm’s optimisation problem can then be represented as: max ∞ ∞ {Pt+i (k)}i=0 i=0 ⎡ βi ⎣ Pt+i (k) − − χ2 Wt+i TFPt+i Pt+i (k)/Pt+i−1 (k) (1+ ṗss ) Pt+i (k) Pt+i 2 −1 −η Yt+i Pt+i Yt+i ⎤ ⎦ which shows that firms maximise profits less price adjustment costs. (b) This delivers the first order condition at date t: Pt (k) (1 − η) Pt Wt Pt (k) + η At Pt = 0 −η −η Pt Yt Pt (k) −1 (1 + ṗss ) Pt−1 (k) (1 + ṗ ss ) Pt−1 (k) Yt Pt+1 (k) Pt+1 (k) Pt+1 Yt+1 + βχ −1 Pt (k) (1 + ṗss ) Pt (k) (1 + ṗss ) Pt2 (k) Yt − χ (a) The key simplifications are: ignoring multiple goods markets; abstracting from the assumption that adjustment costs depend on lagged inflation; and using a simple production function that is linear in labour and excludes adjustment costs. (b) This is the Rotemberg (1982) form of price adjustment costs. 46 The core theory Imposing a symmetric equilibrium (under the assumption that all firms are identical) allows us to set Pt (k) = Pt , E t (k) = E t and Yt (k) = Yt for all periods t, which allows us to write the first order conditions for prices as: (1 − η) − χ Pt Pt −1 ss (1 + ṗ ) Pt−1 (1 + ṗ ss ) Pt−1 2 Yt+1 Pt+1 Pt+1 +ηR MCt + βχ − 1 ss (1 + ṗ ) Pt Yt (1 + ṗ ss ) Pt2 = 0 where R MCt ≡ Wt / (TFPt Pt ) denotes the real marginal cost of production. Using ṗt ≡ (Pt /Pt−1 − 1) to denote the rate of inflation, we can log-linearise the equation around the steady-state point to give: (a) ṗt − ṗss = β ṗt+1 − ṗ ss + η−1 log R MCt − log (η − 1) /η χ This means that we can think of inflation either in the context of a New Keynesian Phillips Curve or the sort of mark-up pricing equations that are common in some traditional forecasting models. If treated consistently, the information content in each is the same. 1+ ṗt ss (a) This uses the fact that in the steady state R MC = (η − 1) /η and the approximation log 1+ ṗss ≈ ṗt − ṗ for inflation rates close to steady state. 3.4.3 Price and wage inflation The specification of the firms’ maximisation problem leads to a set of price decision rules for the different markets in which goods are sold, which balance the costs of adjusting prices against the loss of profitability that arises if firms do not react to market conditions. These are mark-up equations, showing how much rent individual firms should aim to extract from each unit of output, and they provide the key link between nominal values and real activity for an individual firm. Firms make decisions about their own prices given an aggregate price level that reflects the state of aggregate demand, which can be affected by the monetary authority. The importance of distinguishing between price stickiness and inflation stickiness is well known. (70) To generate inflation persistence, we include an assumption that the target path for prices can depend on lagged inflation as well as the steady-state inflation rate. (71) While there is no explicit Phillips curve, the resulting aggregated pricing equations are consistent with a reduced form that would look much like a New Keynesian Phillips Curve derived under Calvo pricing assumptions. (72) In our case, the reduced form for real marginal costs is more elaborate than is usual, as we have to include capital and labour adjustment costs as well, and the technology is CES instead of the usual Cobb-Douglas form. Further, (70) See, for example, the discussion on page 223 of Walsh (2003). (71) This is similar to the ‘indexation’ assumption used in Smets and Wouters (2003a). (72) See Box 5 on page 46. See also Wolman (1999) and Rotemberg and Woodford (1999) for discussions and derivations of New Keynesian Phillips Curves. 47 The Bank of England Quarterly Model firms in our setting are concerned with the average sales price rather than the price of a single good, because they are each assumed to be selling to different markets, as described in Section 3.3.1. As discussed above, we impose nominal wage rigidity through Calvo contracts: in any given period, a fraction of unions renegotiate nominal wage levels with firms. Wage demands depend on current and expected future prices (with an important role for the exchange rate). In turn, wages affect inflationary pressure, which also reflects assumptions about the institutional structures of goods and labour markets. The degree of inflation persistence, however, also depends on the assumption about the behaviour of the monetary authority (including the inflation target and how action would be taken to achieve it). 3.4.4 The monetary transmission mechanism The monetary authority is assumed to have direct control over the short-term (one-period) nominal interest rate. Since prices are (usually) assumed to be sticky, the monetary authority in the core model has the ability to influence real rates. The consequences of changes in real interest rates are largely conventional. Lower real rates discourage consumption because asset returns are lower (the income effect) but encourage current consumption over future consumption (the substitution effect), and we assume the latter effect dominates. (73) Lower real rates also reduce the costs of capital for investment by firms and households, and lower the opportunity cost of holding inventory stock. The combined effect is to encourage final domestic demand. To meet that demand, firms will have to pay more for factors of production. An open economy has a further important transmission channel: assuming uncovered interest parity, lower real domestic rates would encourage a depreciation in the real exchange rate. (74) Given movements in the exchange rate, we can distinguish a direct price effect (the prices of imported goods change) and a real effect (a lower real exchange rate would encourage exports and raise import costs). There is an important forward-looking dimension to this: asset returns will move to reflect the shift in real interest rates. Clearly, the intertemporal substitution effect relies on forward-looking consumers. But firms are also forward looking, so that current prices are affected by current and future costs. We can represent the transmission channels as shown in Figure 3.6. One aspect that is missing from this exposition is a direct role for nominal interest rates via credit and collateral effects. (75) This is not to deny the potential importance of these effects, (76) but the mechanisms needed to embed them in a general equilibrium context are usually quite elaborate. So we have not attempted to incorporate them in the theoretical core model. (73) That is, the elasticity of intertemporal substitution for consumption, σ c , is less than one. (74) When used for forecasting, alternative paths for the nominal exchange rate might be used that deviate from a strict UIP path. (75) It is difficult to embed such a mechanism in a general equilibrium setting. For example, in the financial accelerator model (see Bernanke, Gertler and Gilchrist (1996)), there is necessarily a specific type of heterogeneity that makes aggregation difficult if the agents are already heterogenous in other dimensions. (76) See Aoki, Proudman and Vlieghe (2002) and Hall (2001) for examples of work at the Bank of England on these effects. 48 The core theory Figure 3.6: The monetary transmission mechanism Market rates Domestic demand Domestic pressure Asset prices Net external demand Instrument Inflation Expectations Exchange rate Import prices 3.5 Long-run growth The core model exhibits neoclassical (exogenous) growth. This framework facilitates a consistent treatment of trend growth factors and, in particular, a clear distinction between effects arising from changes in population and productivity. Some restrictions must be observed in order for the model to settle on a balanced-growth equilibrium. First, there are standard inequality conditions between aggregate steady-state growth, on the one hand, and preferences and real interest rates, on the other, that must be observed for stability. Second, we must assume that technological progress is labour augmenting. (77) The two sources of potential supply growth, ẏ, are therefore net population growth, ṅ, and labour productivity growth, λ̇, so that we have 1 + ẏ = (1 + ṅ) · 1 + λ̇ . The assumption of Blanchard-Yaari households requires a distinction between gross population growth (ie the birth rate) and net population growth, ṅ, which factors in mortality. Third, the balanced-growth path assumes that the rest of the world grows at the same rate in the long run. This does not rule out permanent increases in productivity in both the domestic economy and the rest of the world, but it does rule out the possibility that the domestic economy could grow faster than that of the rest of the world for ever, and so ultimately grow to be larger than the rest of the world. This is a standard implication of our assumption that the domestic economy is ‘small’. Productivity level shocks are implemented as temporary labour-augmenting productivity growth shocks, but we also include a multiplicative constant to the constant returns to scale production function to represent temporary shocks to the level of total factor productivity. Some features of the model (such as adjustment costs) have been specified so that they do not depend on trend growth. However, growth is an important consideration for the discussion of steady-state sustainability. For example, if the economy is growing at a rate ẏ, then the ‘effective’ interest rate on (77) Of course, this is effectively imposed if we assume that the elasticity of factor substitution in the CES production function is unity, so that we have the limiting Cobb-Douglas case. The balanced-growth path is also facilitated by the use of isoelastic utility. For more details on the conditions for the existence of a balanced-growth equilibrium, see pages 54-55 of Barro and Sala-i-Martin (1995). 49 The Bank of England Quarterly Model foreign debt is r f − ẏ, in terms of maintaining a constant ratio of net foreign assets to output. We restrict long-run growth and interest rates so that growth cannot allow consumers to escape for ever from debt obligations. Growth assumptions are important for discussions about the current account and, by consequence, consumption behaviour more generally. 3.6 Summary This chapter explains the core theory of the model. The aim of the theory presented here is not to be a literal description of how the economy works, but to illustrate the key mechanisms that we think are useful for describing and distinguishing economic phenomena. The view of the world is quite simple. We started with a description of how agents – households, firms, policymakers and the rest of the world – interact in markets for goods, labour and financial assets. Households make consumption-savings decisions, based on income from working and from transfers. We use the Blanchard-Yaari model (see Blanchard (1985)) of overlapping generations in order to anchor a stable long-run equilibrium for consumption, total assets, and hence net foreign assets and the current account. Firms are conventional profit maximisers, operating in a world of monopolistic competition (as in Blanchard and Kiyotaki (1987)) and so are able to extract supernormal rents from the goods they sell and set sticky prices (as in Rotemberg (1982)). Unions act as intermediaries between households and firms, negotiating a non-competitive real wage and engaging in sticky nominal contracts, as in the Calvo (1983) model. Faced with the costs for its factors of labour and capital, firms make a mark-up when setting prices, which is responsive to demand and monetary conditions. The monetary authority has the ability to manipulate real interest rates, which is the critical link between monetary policy, the real economy and inflation. Inflation is also affected, via imported goods prices, by movements in the exchange rate and world conditions. 50 The core theory Box 6: How does the Blanchard-Yaari model make consumption stationary? This box presents a simplified version of the Blanchard-Yaari framework that underpins the household equilibrium, showing how it leads to a stationary path for the ratio of aggregate consumption to output (and consequently the current account and net foreign asset ratios). Consider the consumption problem of a consumer born at date t − s, indexed by s (the consumer’s age). All consumers of this age will behave identically because they have the same preferences (including a constant probability of death) and face the same constraints. The consumer’s problem ∞ is to choose a consumption path {cs,t+i }i=0 to maximise expected utility, subject to a budget constraint. We assume that utility takes an isoelastic form. This problem can then be written as a Lagrangian: (a) max 1 1− ∞ 1 σ i=t (γ β)i−t cs,i σ + λ(T Ws,t − ∞ 1− 1 γ i−t h t,i cs,i ) (1) i=t Here cs,t denotes the consumption of a consumer of age s, γ is the probability of survival, β is the discount factor and σ is the elasticity of intertemporal substitution. We use a market discount −1 factor: h t,i = i−1 and h t,t = 1, where r is a real (one-period) rate of return. We also j=t (1 + r j ) use T Ws,t to denote the ‘total wealth’ of a consumer of age s. This is calculated in the conventional way (by iterating on the budget constraint) and is loosely defined here so that we can abstract from the supply side. The first order conditions for dates i = t, t + 1, ... (1) are: (γ β)i −t cs,iσ = λγ i−t h t,i −1 (2) We see that the probability of survival (γ ) appears identically on both sides of the equality in (2) and so does not distort individual consumption decisions. This is apparent if we divide the first order condition at i = t by the first-order condition at i = t + 1. This yields: 1 σ cs,t+1 βcs,t 1 σ − (1 + rt ) = 0 ⇒ cs,t+1 = β σ (1 + rt )σ cs,t (3) which reveals that the behaviour of each individual consumer is driven by a set of Euler equations that is identical to that of the representative consumer. To derive a consumption function for the individual, we can substitute the first order condition (2) into the definition of total wealth as the present value of expenditures to obtain: T Ws,t = λ ∞ −σ γ i−t h t,i i=t β i −t h t,i σ Then we can use the first order condition again to show that cs,t = where the law of motion of t T Ws,t is described by: t = [1 + γ β σ (1 + rt )σ −1 −1 −1 t+1 ] (4) (a) See page 315 of Frenkel and Razin (1996). 51 The Bank of England Quarterly Model To derive the behaviour of per capita consumption we need to aggregate across consumers. If the probability of survival is γ , the expected size of a cohort s periods old is γ s , and the total s population is therefore ∞ s=0 γ . If we assume a constant total population normalised to 1, per s capita consumption is given by ct = (1 − γ ) ∞ s=0 γ cs,t . Then it is easy to see that ct = t T Wt (5) γ s T Ws,t ). Now consider the behaviour of where T Wt is per capita wealth (T Wt = (1 − γ ) consumption. We know that the consumption of individuals follows an Euler equation. So in general, when β(1 + rt ) = 1, an individual’s consumption profile will be increasing or decreasing. How can this be consistent with a constant steady-state level of per capita consumption? The short answer is that people die and are replaced by newborns. To see this, note that, assuming that the law of large numbers applies, we can write per capita consumption as ∞ s=0 ct = γ ctsur + (1 − γ )ctnew (6) where csur denotes the average consumption of surviving agents and cnew is the average consumption of new-born consumers. Defining total wealth as the sum of human wealth and a portfolio of assets, then we have T Ws,t = hwt + (1 + rt−1 )as,t−1 (7) Human wealth is independent of the age of the consumer, and consumers are identical in terms of their labour productivity. But newborns are not endowed with a predetermined asset holding: a0,t ≡ atnew = 0. Using equation (5), this means that the consumption of newborns is given by: ctnew = t hwt (8) The consumption of surviving consumers satisfies the Euler equation (3): sur = β σ (1 + rt )σ ctsur ct+1 (9) The law of large numbers means that in any given period, the average consumption of surviving consumers is equal to the average consumption of all consumers, because the probability of death is the same for all consumers. This means that equation (9) becomes sur = β σ (1 + rt )σ ct ct+1 (10) Using equation (8) (led one period), equations (10) and (6) (led one period) give ct+1 = γ β σ (1 + rt )σ ct + (1 − γ ) t+1 hwt+1 (11) In the context of Figure 3.3 we can see that the requirement for the slope of the line relating ct to ct−1 to be less than unity is equivalent to γ β σ (1 + r)σ < 1. The term β σ (1 + r)σ is the slope of the individual’s Euler equation (as seen from equation (3)). So this condition means that the survival probability multiplied by the slope of the Euler equation must be less than unity. In other words, the consumption of survivors cannot be increasing after adjusting for the death rate. 52 The core theory Now consider the steady-state version of (11). Here we set rt = r, hwt+1 = hw, ct+1 = ct = c and t+1 = = 1 − γ β σ (1 + r)σ −1 . This gives us c = (1 − γ ) 1 − γ β σ (1 + r)σ −1 hw 1 − γ β σ (1 + r)σ (12) Equation (12) demonstrates the crucial role of the survival probability, γ , in determining steady-state per capita consumption. In the Blanchard-Yaari framework, the steady state has two important features that cannot both obtain when households are infinitely lived: the steady state is stable (following a temporary perturbation, the model will converge back to the initial steady-state position), and steady-state consumption is strictly positive and finite. The steady state of the Blanchard-Yaari model can thus be summarised as follows. We can think of the economy as populated by two types of consumer: newborns and survivors. We know that in the general case when β(1 + rt ) = 1, the consumption profiles of survivors will not be flat over time. Asymptotically, the weight of a given surviving cohort will approach infinity or zero. But because individuals do not expect to live for ever, average consumption remains bounded. This is because a constant proportion of the population dies and is replaced by newborns who enter the world with no assets, so their consumption is restricted to the annuity value of their income stream, which is finite and well defined. (a) Thus, when we weight together the consumption of newborns and survivors, we have a well defined steady-state level of per capita consumption. (a) Of course, once a consumer has survived for one period, consumption follows the Euler equation (9). 53 The Bank of England Quarterly Model Box 7: The firm’s maximisation problem The supply side of the model is described by a continuum of monopolistically competitive firms, indexed by k ∈ (0, 1); we normalise the population of firms to unity. In this box we consider the decision problem for an individual firm. In what follows, the firm chooses optimal plans for variables denoted with a (k) index, treating other variables as exogenous. However, there are two exceptions. First, stockbuilding (St+i (k)) decisions are not modelled as part of this optimisation problem: as discussed in Section 3.2.2, there is no role for stocks as insurance against unanticipated demand, so we consider a separate optimisation problem to determine inventory holdings. Second, we assume that other investment (I O and I OV ) is taken as given by firms, and is not modelled as part of the optimisation procedure. Prices are set in domestic currency apart from export prices (P X V F), which we assume are set in foreign currency (this captures the idea that nominal rigidities are local to the market in which output is sold). Firm k’s nominal dividend at date t, DVt (k) is defined as follows: DVt (k) = PC H Vt (k) C H Vt (k) + P K H Vt (k) I K H Vt (k) + P I O Vt I O Vt (k) P X V Ft (k) +P DVt (k) I Dt (k) + PGVt (k) GVt (k) + X Vt (k) E Rt +P SVt St (k) − St−1 (k) − (1 + ecostt ) Wt E t (k) − P SVt St (k) − St−1 (k) − P I Ot I Ot (k) −P K Ht K Ht (k) − 1 − δ kh t − χ z z t (k)1+φ z − (z ss )1+φ z 1 + φz −P K Mt K Mt (k) − 1 − δ km t − χ z z t (k)1+φ z − (z ss )1+φ z 1 + φz K Ht−1 (k) K Mt−1 (k) +B K t (k) − (1 + rkt−1 ) B K t−1 (k) − T AX K t − T AX LU M P K t −T R AN S K Ct − T R AN S K Ft + T R AN S K t (1) Equation (1) shows that dividends are a function of cash flow (sales less expenses) and taxation on firms’ income. Nominal sales revenue is represented by the first three lines of equation (1). It depends on sales volumes in each market – home consumption goods (C H V (k)), home investment (I K H V (k)), dwellings investment (I D (k)), government procurement (GV (k)) and exports (X V (k)) – and the nominal prices that the firm sets for these (PC H V (k), P K H V (k), P DV (k), PGV (k), P X V F (k)). (a) It also includes revenue from sales of other investment (I O V ) and stock building (St (k) − St−1 (k)) but the firm is not assumed to set the prices for these goods. The nominal prices of the firm’s sales carry the V suffix which indicates that these prices are basic prices. (a) Nominal revenue from export sales is measured in domestic currency so that the foreign currency price set by the firm is adjusted for the nominal exchange rate E R (where an increase represents an appreciation). 54 The core theory Labour and investment costs are described in the fourth, fifth and sixth lines of (1). The firm chooses employment E (k) at the private sector nominal wage rate W , and also pays social contributions at rate ecost on behalf of workers. The firm’s nominal investment expenditure on domestic investment goods is the nominal price of home capital goods P K H multiplied by the change in the capital stock. The effective rate of depreciation depends on an exogenous component δ kh and an increasing function of the deviation of the firm’s choice of its capital utilisation rate z, relative to the steady-state rate z ss . Nominal expenditure on imported investment is defined analogously. The prices the firm pays for home investment, imported investment and other investment are market prices and so do not carry the V suffix, some of this expenditure is thus on intermediate imports or indirect tax payments. The remaining components are the contributions to dividends of the firms issuance of corporate bonds, B K , net of interest payments on the outstanding stock and the firms payment of net transfers and taxes. T AX K denotes corporation tax; T AX LU M P K lump sum taxes on firms; T R AN S K C profit transfers from to households; T R AN S K F transfers from firms to overseas; and T R AN S K general government transfers to firms. (a) Firm k’s objective is to maximise a discounted flow of current and future dividends, net of intangible adjustment costs: ⎫⎤ ⎡⎧ 2 pch χ pch ⎪ ⎪ ⎪ ⎪ DV ξ PC H V C H V − (k) (k) t+i t+i t+i ⎪ t+i 2 ⎪ ⎥ ⎢⎪ ⎪ ⎪ ⎪ 2 ⎥ ⎢⎪ ⎪ pg pkh 2 ⎪ ⎪ pkh pg χ χ ∞ ⎥ ⎢⎨ − ⎬ ξ PGV GV P K H V I K H V − ξ (k) (k) t+i t+i t+i t+i t+i t+i 2 2 ⎥ ⎢ max ⎥ t,t+i ⎢ 2 pd px 2 px pd P X V Ft+i χ χ ⎥ ⎢⎪ ⎪ ⎪ X V − − ξ ξ P DV I D (k) (k) i =0 t+i t+i t+i ⎪ ⎥ ⎪ ⎢⎪ t+i t+i 2 E Rt+i 2 ⎪ ⎪ ⎪ ⎦ ⎣⎪ ⎪ ⎪ 2 d ⎪ ⎪ Dt+i−1 (k) ⎩ ⎭ P DV I D − χ2 I Dt+i (k)/I − 1 t+i t+i ss (1+ ẏt+i ) (2) where 1 i =0 t,t+i = i γk i ≥1 l=1 1+rgt+l−1 is the nominal discount factor (common to all firms), (b) and ξ t+i (k), ξ t+i (k), ξ t+i (k) , ξ t+i (k) pd and ξ t+i (k) are costs of adjusting the prices of home consumption, home investment, government procurement, exports and dwellings respectively. pch pkh pg px These costs have similar forms and generally measure the percentage difference between the change in the firm’s prices and a reference inflation rate (usually a weighted average of the inflation target ṗ ss and lagged average inflation in the relevant expenditure category, with weights determined by 0 ≤ ≤ 1 parameters). (c) These costs are described in equations (3)-(7). (a) These categories are included to provide a match with National Accounts concepts. Transfers are measured net, so that these flows can be negative. (b) The discount factor depends on the parameter 0 < γ k ≤ 1 which measures the extent to which firms over-discount future dividends. Setting γ k = γ (where γ is the household survival probability) implies that firms and consumers discount the future at the same rate. (c) The exception is the costs of adjusting export prices. For exports, the reference rate is a weighted average of lagged inflation of world export prices (P X F) and steady-state world inflation. 55 The Bank of England Quarterly Model ξ t+i (k) = pch ξ t+i (k) = pkh PC H Vt+i (k) /PC H Vt+i−1 (k) (PC H Vt+i −1 /PC H Vt+i −2 ) (1 + ṗ ss )1− pchdot −1 (3) pkhdot −1 (4) P K H Vt+i (k) /P K H Vt+i−1 (k) (P K H Vt+i−1 /P K H Vt+i−2 ) ξ t+i (k) = pg ξ t+i (k) = px ξ t+i (k) = pd pchdot pkhdot (1 + ṗss )1− PGVt+i (k) /PGVt+i−1 (k) (PGVt+i−1 /PGVt+i−2 ) pgdot (1 + ṗ ss )1− pgdot P X V Ft+i (k) /P X V Ft+i−1 (k) (P X Ft+i−1 /P X Ft+i−2 ) pxdot 1 + ṗ f ss 1− pxdot P DVt+i (k) /P DVt+i−1 (k) (P DVt+i −1 /P DVt+i−2 ) pddot (1 + ṗ ss )1− pddot −1 (5) −1 (6) −1 (7) Maximisation is subject to several types of constraint. First, the firm must supply whatever level of demand is implied by the price it chooses to set. We assume that preferences are such that each firm faces isoelastic demand schedules for each type of good. The elasticities are denoted η and can vary across expenditure categories. This specification implies that the demand for firm k’s good is a function of the price it charges relative to the industry average and of the overall level of demand for that expenditure category. The relevant demand schedules are therefore: C H Vt+i (k) = PC H Vt+i (k) PC H Vt+i I K H Vt+i (k) = P K H Vt+i (k) P K H Vt+i I Dt+i (k) = P DVt+i (k) P DVt+i GVt+i (k) = PGVt+i (k) PGVt+i X Vt+i (k) = P X V Ft+i (k) P X V Ft+i −ηc −ηk −ηd −η g C H Vt+i (8) I K H Vt+i (9) I Dt+i (10) GVt+i (11) −η x X Vt+i (12) The second set of constraints ensures that the output of the firm must be generated from its inputs according to a production function. Equation (13) shows that output is generated by a constant elasticity of substitution (CES) production function net of the (quadratic) costs of adjusting home capital, imported capital and employment. 56 The core theory The CES production function describes how the firm combines effective employment (average hours avh times labour productivity λ times the level of employment E) and effective capital (capital utilisation z times the capital index K , defined below) to produce output. The parameters of the function α and φ affect the optimal mix of inputs and σ y > 0 is the elasticity of substitution between capital and labour. Yt+i (k) = T F Pt+i (1 − α) {(1 − φ) avh t+i λt+i E t+i (k)} ⎫ ⎧ 2 ss ⎪ χ kh 1 + ẏt+i ξ kh ⎨ ⎬ t+i (k) K Ht+i−1 ⎪ 1 2 km ss km − 1 + ẏt+i ξ t+i (k) K Mt+i−1 +χ ⎪ 2⎪ 2 ⎭ ⎩ +χ l ξ lt+i (k) Yt+i σ y −1 σy + α {φz t+i (k) K t+i−1 (k)} σy σ y −1 σ y −1 σy (13) The index of capital entering the production function is given by a CES aggregator over home and imported capital (14). K t+i (k) = ψ k φ K Ht+i (k) k σ k −1 σk + 1−ψ k 1−φ k K Mt+i (k) σ k −1 σk σk σ k −1 (14) The costs of adjusting home capital, imported capital and employment are given by equations (15), (16) and (17) respectively. They have a similar form to the price adjustment costs in equations (3)-(7) and depend on the difference between the change in factor inputs and a reference rate of change For the home and imported capital stocks, the reference rate is a weighted average of steady-state growth, ẏ ss , and the lagged aggregate change in the relevant capital stock (with weights determined by 0 ≤ ≤ 1 parameters). For employment, the reference rate is trend population growth, ṅ ss . ξ kh t+i (k) = ξ km t+i (k) = 1+ K Ht+i (k) /K Ht+i−1 (k) kh ss 1− ẏt+i (K Ht+i−1 /K Ht+i−2 ) K Mt+i (k) /K Mt+i −1 (k) ss 1 + ẏt+i ξ lt+i (k) = 1− km 1+ (K Mt+i−1 /K Mt+i −2 ) E t+i (k) . ss n t+i E t+i−1 (k) kh km −1 (15) −1 (16) −1 (17) The final constraint faced by the firm is that the sum of demand for all expenditure components is met by supply (18). Yt+i (k) = C H Vt+i (k) + I Dt+i (k) + I K H Vt+i (k) +I O Vt+i + St+i (k) − St+i−1 (k) + GVt+i (k) + X Vt+i (k) (18) 57 The Bank of England Quarterly Model Box 8: The union bargaining problem Firms are assumed to hire labour each period from a continuum of unions that provide differentiated types of labour. There is also a continuum of firms indexed by k ∈ (0, 1), as described in Box 7. A given firm’s input of employment, E, is given by E t (k) = 1 0 E t (h, k) ηw −1 ηw dh ηw ηw −1 where h ∈ (0, 1) indexes the population of unions. If the bargained nominal wage for union h is Wt (h), it can be shown that the cost-minimising demand for labour of type h by firm k is given by: Wt (h) E t (h, k) = Wt −ηw E t (k) so that ηw > 1 represents the (constant) elasticity of demand for each labour type and 1 Wt = 0 Wt (h) 1−ηw 1 1−ηw dh is the average wage rate across unions. This implies that each firm’s demand for each type of labour is a function of the relative wage of that type of labour and the firm’s overall demand for labour, E t (k), which is given by a first order condition relating Wt and E t (k). We can therefore analyse the firm’s problem in two stages: first, firms and unions bargain over wages; second, the firm chooses its overall demand for labour (and hence the demand for each labour type). The nature of the demand functions for each labour type implies that the share of employment of each type is a function solely of the wage of that type relative to the average wage. So we can analyse the firm’s overall demand for labour as if it employs one type of labour at the average wage rate Wt (as in Box 7). We assume that the wage bargaining process seeks to maximise the following Nash maximand: (a) ∞ i=0 u φ t,t+i (1 − γ w )i Ut+i (h) ψu ∞ i=0 k φ t,t+i (1 − γ w )i Ut+i 1−ψ u where we assume (see Section 3.3.2) that unions and firms bargain with probability 0 < γ w ≤ 1 each quarter. The wage bargain takes place at date t and the bargaining maximand is defined over u the stream of expected future ‘utilities’: Ut+i (h) represents the period utility function of the union k representing labour type h; Ut+i is the ‘utility’ of the representative firm. (b) (1 − γ w )i is the probability that the wage bargain remains in force at date t + i; and φ t,t+i is the discount factor applied to the flow of surpluses, which is a function of nominal interest rates, rg, adjusted for inflation ṗ: 1 i =0 φ t,t+i = i γ (1+ ṗt+l ) i ≥1 l=1 1+rgt+l−1 (a) A Nash maximand is usually defined in terms of a surplus over the utility gained from a ‘threat point’ (the case in which the parties fail to strike a bargain). For unions, the utility gained from the threat point is specified as the alternative wage in the period utility function. For firms, we normalise the benefit from the threat point to zero. (b) Since all firms behave symmetrically in equilibrium, we assume there is effectively one firm. More generally, the ‘utility function’ represents the average utility of a firm. 58 The core theory The period utility functions for unions depend on real consumption wages relative to what the worker can expect elsewhere and the level of employment of union workers. At date t + i these functions depend on the wage bargained at date t: u Ut+i (h) = ee 1 − τw t+i − τ t+i PCt+i t,t+i Wt where E t+i (h) = t,t+i Wt (h) (h) − −ηw Wt+i E t+i (h)ψ W At+i PCt+i e Ē t+i is the total demand for labour type h (which depends on the aggregate demand for labour across all firms, Ē); W At+i = 1 − u t+i − µt+i eg ee w ee 1 − τw t+i − τ t+i Wt+i +µt+i 1 − τ t+i − τ t+i W G t+i +u t+i B E Nt+i eg is the alternative wage available to workers elsewhere, which is a linear combination of the (post-tax) average wages in private sector firms (W ) and government employment (W G), and of unemployment benefits (B E N ). It assumes that workers are allocated to these labour market states at random, so the weight on each wage rate is equal to the proportion of participating workers in that state: u is the aggregate unemployment rate and µeg is the proportion of participating workers employed by the government; and i t,t+i = k=0 1 + ẇss (1−εw ) (1 + ẇt+k−1 )ε w is an ‘indexation factor’ specifying how wage settlements evolve. It is a function of steady-state nominal wage growth (ẇss ) and expected private sector nominal wage growth (ẇ). The representative firm’s utility is written as the surplus of sales revenue over wage costs: k Ut+i = SU R Pt+i = PC H Vt+i C H Vt+i + P DVt+i I Dt+i + P K H Vt+i I K H Vt+i +P I O Vt+i I OVt+i + PGVt+i GVt+i + P SVt+i D E L St+i +P X Vt+i X Vt+i − (1 + ecostt+i ) 1 0 Wt+i (h) E t+i (h) dh 59 Chapter 4 The core/non-core hybrid approach The previous chapter described the theory behind the structural core model; this chapter explains how we combine that structural core with ad hoc dynamics and additional variables into one model that we can use for forecasting, including the application of judgement. The basic idea behind our approach is simple. The core model embodies a rich economic structure that can be used to analyse a wide range of economic issues. Some features in the theoretical structure are designed to help match dynamic responses in the data, including the potential for consumption habits, labour adjustment costs, capital and investment adjustment costs, inertia in prices and nominal wages, wage and price inflation stickiness, and slow import price pass-through. However, as discussed in Chapter 2, we know that the core model does not fully capture all of the economic channels and dynamic relationships affecting the observed correlations between economic variables. This in part reflects the choice not to include in the core model some features of the economy, such as credit market frictions, which could risk making the core model too large and complex to be tractable. Moreover, the theoretical underpinnings of some aspects of these correlations, for example the degree of persistence of many nominal variables, are not yet well understood. For these reasons, an extra layer of ad hoc ‘non-core’ dynamics is added to reflect additional short-run correlations that are not matched by the core theory. For example, we can think of a neoclassical story for consumption being combined with proxies for credit effects, or a conventional Jorgensonian user-cost story for investment supplemented by terms for firms’ gearing in the short run. The full model is a hybrid combination of core and non-core elements, which matches past movements in the data better than either element on its own, and enables a straightforward application of judgement to the forecast. One interpretation of this hybrid approach is that the final projections are a weighted average of three types of information: a structural story coming from the core model, extra short-run correlations from the non-core model, and judgement applied by the user through the non-core model (the relative weights on these types of information will vary across different parts of the model). Figure 2.2 on page 15 shows a stylised example of the forecast process. A key feature of this approach is the strict separation between the core and non-core elements of the model. This is reflected in the way that the model is structured, estimated and solved. The need for this separation stems from the incompatibility of ad hoc terms with the simultaneous system derived from the core theory. As explained in Chapter 3, the decision rules embodied in the core theory are based on the optimising behaviour of different agents, derived from a set of consistent assumptions about preferences, technology, market conditions etc. As such, it would not be possible to include additional ad hoc terms or to apply judgement to the system without potentially violating the underlying assumptions, and undermining the theoretical micro-foundations of the decision rules derived from optimisation by key agents. (1) This theoretical coherence is central to the ability to use core theory to analyse various economic issues. Moreover, it is also important in ensuring the stability of the system, (2) which is essential in a real-time forecasting context. Hence, BEQM is structured in such a (1) If we add ad hoc terms to the equations derived from the core theory, it is no longer clear what these equations mean. The simultaneity of the core model means that normalisation is arbitrary: adding ad hoc lags to consumption, for example, would not be the same as introducing habit persistence, because habits would be expected to affect the choice of other variables that affect utility, such as money balances. (2) Experiments with a range of small models have shown that seemingly innocuous additions of own lags and proxy variables, which appear appropriate at the level of an individual equation, can easily create an unstable or wildly oscillating system. For example, two sets of ad hoc extensions might ‘work’ by themselves, in the sense that each achieves the desired marginal change to the system responses, but together they would generate explosive properties or a model that will not always solve. 61 The Bank of England Quarterly Model way that the core and non-core elements of the model are kept separate. The remainder of this chapter explains this structure in more detail. 4.1 Functional forms We can think of the core model as providing dynamic paths that form a starting point for thinking about the forces at work in the economy. The final projections from the full model reflect additional short-term correlations and judgement from the non-core model, as well as the dynamic path from the core model. We can think of a general format for a non-core equation where, for a given endogenous variable y, there is a relationship to the core value y core as follows: A (L) yt = B (L) ytcore + C (L) z t + εt (4.1) Here A, B and C are polynomials in the lag operator, L. We use z t to denote a vector of selected endogenous and exogenous variables, though in much of the discussion below we assume (without loss of generality) that z is a single variable. Finally, εt is an error term. This structure is restricted so that the projected path for the variable y will converge to the path generated by the core model in the long run. (3) But this is not just a statement about long-run convergence: if the model fitted the data well, the path for yt would be very close to ytcore at each point in time and there would be no need for additional ad hoc dynamics. In practice, higher-order lags may be needed for richer adjustment patterns than in the core model (4) and there is a potential role for influences from variables that proxy for missing effects, which are captured in the z variables in equation (4.1). Examples of such additional variables include changes in the value of the housing stock as a proxy for a housing collateral mechanism, which we have decided not to model structurally in the core model but is relevant for business cycle dynamics; other credit channel effects; risk premia; and confidence effects through the business cycle. This hybrid approach can be thought about in two different ways, leading to slightly different empirical formulations. In the first, y core has the status of another regressor, along with the elements of z, in trying to explain observed outturns, y. If, for simplicity, we restrict attention to a specification of A and B with only one lag, then equation (4.1) can be written as: core yt = α 1 yt−1 + β 1 ytcore + β 2 yt−1 + ψ 1 zt + εt (4.2) which will ensure that the non-core equation converges on the core model solution under the parameter restriction 1 − α 1 = β 1 + β 2 . (5) Under this restriction we can manipulate equation (4.2) to give: core + β 1 ytcore + ψ 1 z t + εt yt = − (1 − α 1 ) yt−1 − yt−1 (4.3) (3) This requires that z terms converge to zero in the long run, though effects can be quite persistent relative to the forecast horizon. We typically implement this by differencing the z variables or by expressing them as ‘gap’ terms (such as the difference between a variable and its steady-state level). (4) While we have many levers over general inertia in the core model, they are comparatively straightforward devices such as habits, adjustment costs, and so on. Such relatively simple core dynamics may not match the data at all well. (5) We are also implicitly assuming that α 1 < 1. 62 The core/non-core hybrid approach which is often called an equilibrium correction form. (6) This is clearly equivalent to: core + β 1 ytcore + ψ 1 z t + εt yt = α 1 yt−1 + (1 − α 1 ) yt−1 (4.4) which is often referred to as a partial adjustment form. (7) A slightly different form can be derived from the idea of combining forecasts. One interpretation of our hybrid approach is that we assume the forecast from the structural core theory will not ‘encompass’ those from other models. This can be formally evaluated – consider the regression: core SR + γ 2 ŷt+1 yt+1 = γ 0 + γ 1 ŷt+1 core SR where ŷt+1 is the one-step-ahead forecast produced by the core model and ŷt+1 is a one-step-ahead forecast produced by a statistical ‘short-run’ model. If γ 0 , γ 1 , γ 2 = (0, 1, 0) then we could say that the core model forecast-encompasses the ad hoc short-run model; if γ 0 , γ 1 , γ 2 = (0, 0, 1) then we could draw the opposite conclusion. For any other values of γ 0 , γ 1 , γ 2 , neither model encompasses the other, and both have value for forecasting. (8) This assessment can be done for any k period ahead forecast, but if we restrict our interest to one-step-ahead forecasts and restrict γ 0 = 0 and γ 2 = 1 − γ 1 , then we can derive a compact form: yt = γ 1 · ytcore + 1 − γ 1 yt−1 + γ 3 z t + εt (4.5) This requires first estimating a short-run forecast model ŷtS R = γ 3 z t , and secondly estimating the weighting γ 1 , which can be done simply by least squares methods. Equation (4.5) looks similar to (4.4). Whichever form is used, there is a distinction between our approach and the more common use of equilibrium (or error) correction equations. In such applications, y core represents a long-run attractor that comes from some static optimising theory or cointegration analysis (and is often expressed as a linear combination of other variables), and the remaining terms describe the dynamics. This arises from supposing that y core is a long-run target and that some generic, unspecified adjustment costs explain the slow convergence to the target. (9) In our case, however, y core is a fully dynamic path to the steady state rather than a long-run attractor, so the interpretation is different, even if the notional appearance – such as (4.3) – is similar. 4.2 Making the hybrid system work Equations such as (4.1) need to be used carefully when y core is determined by a structural system. This section considers the implications of the hybrid approach for whether the outcomes from non-core equations could feed back into the determination of y core , and also how the system should be operated to ensure accounting consistency. (6) In this case, we potentially face a generated regressor problem (see Pagan (1984) and Oxley and McAleer (1993) for a review of the literature). While this will affect tests of whether α 1 = 0, it will not affect the point estimates of the parameters. (7) Another equation form that ensures convergence on the core path is obtained under the parameter restrictions β 1 = 1, β 2 = −α 1 , which delivers: core + ψ z + ε yt − ytcore = α 1 yt−1 − yt−1 t 1 t We call this the ‘gap form’. (8) These ideas have a long history, going back to at least Nelson (1972), Cooper and Nelson (1975), with formalisations by Chong and Hendry (1986). For a review, see Diebold and Lopez (1996). (9) See Nickell (1985) for a discussion of conventional error correction equations and their motivation. Kozicki and Tinsley (1999) discuss the higher-order adjustment forms used in the QPM and FRB/US models. 63 The Bank of England Quarterly Model Box 9: The hybrid approach applied to the Ramsey model As an illustration of the hybrid approach and some of the issues raised in this chapter, consider the Ramsey model. The perfect foresight social planner problem aims to maximise a stream of (logarithmic) utility from consumption (ccore ) discounted at rate 0 < β < 1, subject to a resource constraint that the capital stock brought forward for use in production next period (ktcore ) is given by the level of previously accumulated capital (net of depreciation at rate δ) plus production less core α consumption. Here production is given by λt kt−1 where λ is the (exogenous) level of productivity and 0 < α < 1 is the production function parameter. max ∞ core β i log ct+i i=0 s.t. ktcore core core = (1 − δ) kt−1 + λt kt−1 α − ctcore The solution yields a consumption Euler condition, core ctcore = ct+1 β αλt+1 ktcore α−1 +1−δ with the economy-wide resource constraint, core core ktcore = (1 − δ) kt−1 + λt kt−1 α −1 − ctcore (1) (2) Equations (1) and (2) form a simultaneous block, from which we can recursively calculate a number of variables of interest, such as investment: core i tcore = ktcore − (1 − δ) kt−1 output: core ytcore = λt kt−1 and a competitive market (net) real interest rate: rtcore = αλt+1 ktcore α α−1 (3) (4) −δ (5) Equations (1) to (5) would be the equivalent of the core model, yielding paths for ccore , i core , y core , k core and r core . Assume now that we want to make ad hoc adjustments to the consumption and investment paths. We could then create equations following a specialised form of equation (4.2) for actual consumption and investment: ct = α c ctcore − ct−1 + u ct i t = α i i tcore − i t−1 + u it We would then carry over the perpetual inventory consistency condition, (3): kt = (1 − δ) kt−1 + i t which will be sufficient to ensure that kt −→ ktcore . 64 The core/non-core hybrid approach We can then consistently define corresponding values for production and interest rates: α yt = λt kt−1 p rt = αλt+1 ktα−1 − δ p These also have long-run limits rt −→ rtcore and yt −→ ytcore . Expenditure is defined as yt = ct + i t p p which also ensures that yt −→ ytcore so that limt→∞ yt = limt→∞ yt = ytcore . However, yt = yt in the short run, which is a consequence of the ad hoc additions that override the saddlepath dynamics of the core model. (a) (a) The hybrid approach of Ireland (2004) has the same implication. 4.2.1 Feedback Consider a variable y1 in a system determining a set of endogenous variables y1 , ..., yn . The most general approach would be to assume that the actual values of y1 can affect the determination of the core levels for other variables: that y1 is a function of y1core , and y1core a function of y2 , ..., yn together with past values of y1 . Put another way, the system would allow predictions from the non-core equations to feed back into the decision rules for other variables in the core model. In this case, we would not really need to use the labels y and y core , but instead have actual y1 , y2 , and so on, each determined by a combination of theory and ad hoc elements. In such a feedback implementation, desired capital stock, for example, would be a function of full model predictions (including ad hoc elements) for final demand, actual cost of capital, and so on, rather than just the core model predictions of these variables. Likewise, actual consumption would affect profit conditions, which in turn would affect dividends and so core consumption, and so on. But we have ruled out this type of ‘full feedback’ option, for the theoretical and practical reasons discussed at the start of this chapter. Instead, we use a ‘non-feedback’ approach, which maintains the important distinction between y and y core . The core model determines levels for y1core , y2core , ..., yncore , but the value for y1 from the ad hoc non-core equation does not affect the determination of y2core , ..., yncore . Instead, there is a one-way causality from the core model into non-core equations. As long as the core model is stable – which is far easier to guarantee when decision variables are derived consistently from theory – it will converge to its steady state. The restrictions discussed above on the polynomials in (4.1) will generally ensure that a non-core variable converges to the level of its core counterpart. (10) And the ‘non-feedback’ approach permits easy application of judgement, because there is a unique mapping from desired to residual paths, which would not be the case if there were leads in the non-core equations (as there are in the core model). (10) Box 9 on page 64 shows how this would work in the context of a simple Ramsey model. 65 The Bank of England Quarterly Model 4.2.2 Accounting consistency The same consistency conditions – stock-flow identities, accounting constraints – that apply in the core also need to be maintained in the full model. These include standard National Accounting identities and conditions to ensure consistency of related non-core variables. Two main sets of consistency issues need to be considered: Flow-flow consistency: all flows in the core model are fully accounted for; and must be too in the non-core model. Fortunately, these are quite straightforward: the three key conditions are the household budget constraint, the firm’s cash-flow condition, and the government budget constraint. These conditions are carried over from the core model and enforced in the non-core, as shown in Appendix B. There is, however, one consistency condition that is not enforced in the non-core equations. Goods markets in the core model are assumed to clear at all times, so that the sum of consumption, investment, stockbuilding, government and export expenditures (on domestic goods) accounts for all (domestic) output produced. Thus, as set out in Section 3.3.1, the core model enforces the condition: (11) chv + id + ikhv + iov + dels + gv + xv = y = F (k, e) We refer to this as the production-clearing condition, and it means that y core is both total private sector production and demand for private sector output. However, if we imposed ad hoc partial adjustment equations in the non-core model for all of the demand expenditures, such as: chv t idt xv t core − chv t−1 = α ch · chv t−1 core − idt−1 = α id · idt−1 .. . core = α x · xv t−1 − xv t−1 then the sum of expenditures would not in general equal the output implied by evaluating the production function using non-core factor inputs. We allow this to happen rather than force production clearing to hold across the non-core equations. (12) Thus, output is determined by demand over the short run. Stock-flow consistency: The core model observes strict stock-flow consistency, in that flows cumulate into stocks, subject to depreciation and revaluations. We want to maintain the same consistency on the non-core side. If we make ad hoc modifications to a flow, such as investment, then it is simple to use the corresponding cumulation condition to define non-core capital stock. A more complicated problem arises with the asset market clearing condition of the core. The equations of the form (4.1) are restricted so that all non-core expenditures hit their steady-state core values in the long run. The flow-flow consistency conditions are enough to ensure that all flows reach their steady-state core values, and the cumulation equations are enough to ensure that physical stocks will reach their steady-state core values. However, the problem that arises is that an ad hoc consumption equation (which we can think of equally as an ad hoc savings equation) will not by itself generate a level (11) See Section 3.3.1 for an explanation of the variables in this equation. (12) We could easily impose a system where one of the ad hoc expenditures was made a residual to the market-clearing condition (for example, inventories). 66 The core/non-core hybrid approach of cumulated financial wealth that is consistent with the sum of all individual wealth components: these flows have to integrate to a certain level for full stock-flow consistency to be achieved. In our case, this inconsistency would be seen in the path of foreign bonds. While the net addition to foreign bonds (a flow) eventually attains the same value as that in the core (as all of the flows in the household budget constraint reach their core values), the path of the stock (net foreign assets) could increase or decrease without bound. (13) Our solution for this effectively amounts to a form of integral control in the ad hoc consumption equation – if non-core consumption is below the core path for a period, then there must be a period of ‘over-consumption’ relative to the core in later periods, so that the integral of the implied savings path matches the supply of assets. We might think of this as agents being potentially slow to react to a permanent income increase, for a variety of reasons that we do not model in the core (for example, learning, signal extraction, credit constraints), but they eventually make up for short-run mistakes or constraints. In order to achieve this, we add a term in the asset ‘gap’ to the consumption equation. A stylised representation would be: (14) core core B (L) ct = ct−1 + ν at−1 − at−1 + ctres (4.6) 4.2.3 Projections from the hybrid system and the Lucas critique The hybrid system combines structural and (quasi-) reduced form elements, with relative weights estimated from the data. As such, the system is vulnerable to the Lucas critique: baseline projections using the system as standard make the implicit assumption that the coefficients on the short-run proxies and these weights have not shifted. Policy analysis in such a system is therefore made on the assumption that interventions are ‘modest’. (15) However, these coefficients are only a guide. In some circumstances, they might be imposed. For example, if there were a large shock that might change behaviour, such as a significant, anticipated change in a distortionary tax, then we might place more weight on the story from the core model for certain variables. 4.3 Summary This chapter discusses how the core and non-core models are combined into the full forecasting model. Equations in the core model are based on the optimising behaviour of forward-looking agents, derived from a set of consistent assumptions on factors such as preferences and technology. Non-core equations include additional ad hoc dynamics and variables that proxy for ‘missing’ effects; they also allow direct application of judgement. The structure of the full model is restricted so that variables must converge to the long-run paths projected by the core model, but short-term behaviour is potentially quite flexible and amenable to the imposition of judgement. (13) In the long run, the household budget constraint accumulates foreign bonds at a rate (1 + r) > 1 (where r > 0 is the world real interest rate), which is explosive. The consumption path from the core theory ensures that foreign bond holdings do not explode. But if we impose an alternative consumption path such that consumers ‘overspend’ relative to the core path, this implies a lower level of household net assets and, in the absence of any other correction, the level of foreign debt will increase without bound. In constrast, physical stocks accumulate at a rate δ < 1, which implies that convergence to a steady-state flow rate ensures a stock level consistent with the steady state of the core model. (14) The size of the coefficient(s) in ν required to ensure that net foreign assets converge to their appropriate steady-state value will be determined by the properties (roots) of the system of non-core equations. There is no guarantee that an estimated value for ν will ensure convergence – if not, we would need to impose a suitable value. (15) See Leeper and Zha (2001). 67 The Bank of England Quarterly Model There are a number of benefits from this separation of core and non-core elements. If we introduced the ad hoc elements into the core, it would risk violating the underlying theoretical assumptions of the core model, and it could also produce an unstable system. One way of viewing this hybrid approach is that the path from the core model is treated as a regressor, along with additional variables and ad hoc dynamics, in the full model equation. We do not allow projections from non-core equations to feed back into the core model, which would bring about similar problems of instability and an undermining of the micro-foundations of the core theory. Instead we use a ‘non-feedback’ approach, which maintains the distinction between the values from the core and the full forecasting models: there is one-way causality from the core model to the non-core equations. This also facilitates the direct application of judgement to the forecast model, so that it is easy to impose desired paths for particular variables. In the full model, we generally impose the same stock-flow and flow-flow consistency conditions as in the core model. An exception is that output is determined by demand in the short term. 68 Chapter 5 Implementing and solving the model This chapter explains how we implement the theory to generate actual numerical outputs. Some of the details are primarily technical, but are important for an understanding of how we produce material in subsequent chapters, such as steady-state analysis and shock responses. There are four sections before the final summary. The first sets out some important detailed assumptions we make to implement the model. The second explains the basic principles of solving the model to produce numerical outputs. The third describes a recursive simulation technique that we use for dealing with assumptions about information and expectations in the core theory, as well as for temporary fixes of endogenous variables and conditioning the responses on particular nominal interest rate paths. And the fourth section deals with historical simulations and producing projections. 5.1 Setting up the model 5.1.1 Timing In a discrete time model, it is important to be clear about the timing of decisions and contracts. In general, values and quantities in BEQM are recorded at the end of each discrete period, which accords with the treatment of stocks in the National Accounts. Payments, such as interest returns and dividends, are also made at the end of the period, and are therefore available to agents at the beginning of the next period. So financial assets appear in period-by-period budget constraints as providing a return of rt−1 · bt−1 in the current period, reflecting the assumption that the debt contract was made in the previous period. Effectively, interest payments are made between periods and available for use in the next period. This process is shown in Figure 5.1. Figure 5.1: Timing conventions for bonds interest payment made t -1 investment made in new bonds t record value for bond stock at end of period record value for bond stock at end of period value (1+r t -1)·b t -1 available for use in period t Revaluations of physical and financial assets occur because of differences between the current price of the asset and the price when the asset was acquired. An important example is net foreign assets, which yield rf t−1 · bf t−1 /ert from foreign bonds bought in the previous period, bf t−1 , valued at this period’s nominal exchange rate of ert (where a rise in er represents an appreciation) and carry a real (own currency) interest rate of rf t−1 . 69 The Bank of England Quarterly Model Physical stocks – housing, imported capital and domestically produced capital – deteriorate at a rate δ during a period. The timing of depreciation depends upon the period during which the stock provided a service. For example, the stock of housing, dt , is assumed to provide a service flow – that is, U (ct , dt ) – to consumers in the current period. Consumption of housing services is assumed to depreciate the asset. So the value of housing brought into the current period is denoted by pdv t · 1 − δ d · dt−1 : the housing stock accumulated in the previous period, dt−1 , will have deteriorated by δ d by the time it is to be accounted for in the current period, and it will be valued at current (relative) market prices, pdv t . The investment flow made in the period is dated at time t, so that in the case of housing investment we have pdv t · idt = pdv t dt − 1 − δ d dt−1 . This is the process shown in in Figure 5.2, where depreciation is modelled as occurring between periods, like interest payments. Figure 5.2: Timing convention for housing depreciation t -1 investment made in new housing t record value for housing stock at end of period value pdv t (1-δd)·d t -1 available for use in period t service flow from stock of housing, dt, and record value for housing stock at end of period Only capital installed during previous periods is useful for current production: production, F (k, e), is in fact F (kt−1 , et ). If capital were not predetermined in this way, we would be producing resources using a capital stock that has not yet been accumulated. It is only after production has taken place that additions to the capital stock are made. Capital dated kt−1 contributes to production at time t, and depreciates by δ kt as a result. The depreciation rate during period t of capital accumulated up to the end of t − 1 is a function of how hard the capital stock is utilised during period t – ie δ kt = δ (z t ), where z is the utilisation rate. (1) Hence, investment is recorded as i t = kt − 1 − δ kt kt−1 , rather than i t = kt+1 − 1 − δ kt kt as in much of the recent academic literature. This is the process shown in Figure 5.3. The model could be rewritten with the alternative timing convention found in much of the academic literature – yt = F (kt , et ) – without any alteration to the model’s fundamental economic properties. But our notation makes it straightforward for the simulation software to recognise lags of capital as predetermined variables. (1) As explained in Section 3.2.2, firms can make short-term changes to the intensity with which they use capital, and this affects the depreciation rate. 70 Implementing and solving the model Figure 5.3: Timing convention for capital stock produce y using k t -1 capital investment: i t depreciation: -δkt·k t -1 t -1 record value for capital stock at end of period t record value for capital stock at end of period: k t = (1-δkt)·k t -1 + i t 5.1.2 Units of account and relative prices All goods, assets and transactions are priced in units of money. This in turn implies a set of relative prices for different expenditure categories, which can be affected by compositional changes and these can have important economic effects. (2) So what matters to households, for example, is the value of money balances and of wages relative to the cost of the consumption bundle. The relative prices of different expenditure components are connected by the goods market clearing condition. The condition, from Section 3.3.1, for private sector production of value added to satisfy demand is F (k, e) ≡ y = chv + id + ikhv + iov + delsv + gv + xv, meaning that the same product from the factory is distributed to markets for domestic consumption goods, dwellings, domestic (and other) investment goods, inventory accumulation, government procurement and exports. This is a statement about the allocation of output volumes, so the identity holds without any relative prices. The aggregate value-added expenditure identity (at market prices), however, needs to include relative prices: (3) pym·ym = pc·c+ pdv ·id + pk·ik+ pio·io+ psv ·dels+ pg·g+ px ·x − pcm·cm− pkm·ikm− pmin·mi The equation above is written in terms of relative prices. This means that we need to choose a numeraire price against which relative prices are measured. Without loss of generality, we assume that the numeraire price is the price of the non-durable consumption bundle (excluding actual and imputed rents). This means that the relative price of non-durable consumption is always unity (that is, pc = 1) by construction. The choice of the numeraire does not affect the behaviour of the model since we derive the core model equations in terms of nominal variables and constraints. (2) For example, total capital includes domestically produced (home) and imported final components, carrying prices pkh and pkm respectively. The aggregate bundle carries the relative price pk, which can be affected by shifts in the relative volumes of home and imported capital as well as by changes in pkh and pkm. (3) The equation for the value-added expenditure deflator, pym, plays no behavioural role in the model: changes in pym reflect changes in individual relative prices and shifts in the composition of aggregate demand. In particular, while firms sell to a variety of markets and are concerned with their average sales price, they do not set the aggregate price level in the sense of targeting the level of the value-added expenditure deflator itself. 71 The Bank of England Quarterly Model 5.1.3 Stationarity We find it convenient to have a stationary model in which the steady state can be expressed as a series of constant values measured in ‘detrended model units’ (see Chapter 6). This means that policy simulations and comparative static experiments are computationally simple, fast and straightforward to run. For projections, this structure means that it is easy to make consistent changes to trend growth assumptions. Because the model settles on a balanced-growth path (meaning that variables grow at constant, common rates in the steady state), we can write the model in a stationary form by scaling any given variable by the appropriate trend growth rate. In general, therefore, a lagged variable will need to be scaled down by the growth rate, and a lead variable will be correspondingly scaled up (see Box 12 on page 88). When the model is solved, rescaling to levels is a simple cumulation conversion, given a starting point for each variable. For most real variables, such as goods and assets, the appropriate scaling is the growth rate of productive potential, 1 + ẏ. For human wealth, transfer wealth, variables associated with the wage schedule and labour values, the appropriate scaling is by labour productivity growth, 1 + λ̇. For nominal variables, scaling is by the rate of inflation in the numeraire price, 1 + ṗ. 5.1.4 Linearity A standard approach in much of the DSGE literature is to linearise around the non-stochastic steady state and then to solve for the rational expectation of the endogenous variables. Instead, we leave the model in levels, which allows us to analyse how permanent shocks affect the steady state. We use a solution method that can handle systems of non-linear difference equations (see Section 5.2). The non-linearity implies that the movement of the model to a new steady-state equilibrium can be influenced by the starting point. (4) There are, however, some potential costs to working with a non-linear model. First, care is needed because model responses are now dependent on the starting point and the size of the shock. Second, non-linearity makes numerical convergence problems more likely. Third, some exercises require a linear model – for example, conventional classical control exercises. Fortunately, numerical calculation of a linearised version for such purposes is relatively straightforward. 5.1.5 Expectations in the core model Equations in the core model have leads as well as lags and are derived under the assumption that agents know how the paths for endogenous variables are determined from the expected evolution of exogenous variables. This, combined with the assumption that agents form point expectations about future exogenous variables, implies that the model can be treated as deterministic. (5) Formally, suppose that y represents endogenous variables and x represents exogenous variables. If all agents know the model and ∞ the past histories of endogenous and exogenous variables yt− j , xt− j j=0 ; and they have point ∞ expectations for the future paths of exogenous variables xt+ j j=1 , then their expectations of future endogenous variables coincide with the core model solutions generated by those paths. Such expectations are often referred to as model-consistent. (4) A good example is the pure wealth effects that arise as a result of the Blanchard-Yaari paradigm for the household: an increase in the supply of government debt will make households rebalance their asset portfolios, including net foreign asset levels, necessitating a current account and real exchange rate response. The extent of these responses will depend on the net foreign asset position at the starting point. See Chapter 7 of Frenkel and Razin (1996). (5) Point expectations mean that agents place full probability on a particular path for exogenous variables. An implication is that certainty equivalence holds. See pages 57-59 of Sargent and Ljungqvist (2000) for a formal definition. 72 Implementing and solving the model This might appear different to the conventional approach usually taken in DSGE models, where agents ∞ know only current and past y and x: yt− j , xt− j j=0 . However, that approach also implies model consistent expectations, as is made clear in King, Plosser and Rebelo (1988). The difference is that, under that approach, simplifying assumptions are made about the nature of the processes determining the exogenous variables, so that an equilibrium can be calculated from knowledge of y and x only up to time t. We do not follow the standard DSGE approach for practical reasons. In the conventional linear DSGE approach, expectations of future exogenous variables are solved out by assuming that these variables follow simple autoregressive processes. Unfortunately, real-world forecasting issues are usually more complex. For example, we would want to condition on exogenous paths for fiscal plans – taxes, transfers, spending – that may have been pre-announced and will almost certainly not follow simple autoregressive processes. Much of forecasting involves changing assumptions about the present and future processes determining exogenous variables to see what happens to the endogenous variables – which means that we need to be able to manipulate future exogenous variables directly. This is possible in the DSGE approach, but would require constructing auxiliary equations for expectations of a variable for each period in the future (and having to change these equations when expectational assumptions change). (6) In our case, assumptions about future paths of exogenous variables are a frequent issue in forecasts (see Section 5.3), which require a more flexible approach. The assumption of model-consistent expectations has been a standard assumption in macroeconomic modelling for some time, often combined with the additional assumption that agents have full information about the future paths of exogenous variables (including shocks hitting the economy). It is not intended as a realistic description, and there are sometimes cases where we would like a tractable means of imposing an alternative assumption. One model of learning is that of Erceg and Levin (2003), in which agents use a filter to extract signals about the true persistence of the shocks hitting the economy. Agents may have to decide, for example, to what extent an observed change in world demand represents a temporary shock or a permanent change. Such a model of learning could easily be implemented using the recursive simulation methodology (described in Section 5.3) which allows us to specify how point expectations of future exogenous variables are falsified by a sequence of unanticipated shocks. 5.1.6 Stability analysis The conditions for stability for linear rational expectations models are laid out in the well-known paper by Blanchard and Kahn (1980). Essentially, the analysis rests on the eigenvalues of the system, so it cannot be applied directly to non-linear systems. However, we can get some information from a system linearised around a point, such as the steady state. Our linearisation procedures rely on the fact that the saddlepath dynamics of the non-linear model are well described by the dynamics of a linear approximation close to the steady state. (7) Subject to this caveat, the Blanchard and Kahn conditions are satisfied for our model. (6) See Cochrane (1993) for an illustration. (7) The extent to which the saddlepath dynamics are well approximated by the linearised model is discussed by Anderson (1999). 73 The Bank of England Quarterly Model 5.2 Solving the model This section explains the basics of solving the model. As discussed in previous chapters, the model is made up of two distinct parts. The core model is forward looking, which raises special issues in terms of producing numerical solutions: much of the discussion below concerns the technicalities of solving the core model. In contrast, the full model takes the core model as given and adds backward-looking (non-core) dynamics. Since the non-core equations are backward looking, the full model can be solved with conventional techniques for systems of simultaneous equations, conditional on a solution for the core model. The basic requirements are easy to describe: we need a set of starting values and a set of internally consistent terminal values, together with a method to solve for paths connecting the two sets of points that are consistent with the structure of the model. Before addressing these three issues, however, we first address how to solve for the steady state. 5.2.1 Solving the steady state The solution to the steady state of the complete model is the long-run equilibrium implied by the core theory. As the core model is designed to follow a balanced-growth path, we can write both the core and non-core models in stationary form. This makes solving the steady state considerably easier, as the solution to the steady state is defined by a set of constant values, given assumptions for world and domestic inflation, domestic labour productivity and population growth. By stripping out leads and lags from the core model, we can create a separate steady-state model. This is solved directly as a straightforward simultaneous equations problem. The results for steady-state ratios will often be of direct interest (such as when looking at comparative statics); using this approach we can solve the steady-state model without any need to solve the full dynamic model forward. 5.2.2 Starting values We need to know where the system is starting from to solve for the dynamics of the model. In particular, we need values for the system’s predetermined (ie lagged endogenous) variables. There are two alternatives: we can shock the model from an initial steady-state equilibrium that we have solved for as described above; or for forecasting, we can supply the model with current and historical values for the predetermined variables. In the latter case, the ‘shocks’ are implicit in the fact that the system is assumed to be away from its long-run equilibrium, and the model’s responses show how the system gets to this equilibrium. 5.2.3 Terminal values The assumption of model-consistent expectations means that some equations contain leads as well as lags, so we need to have a set of terminal conditions: a solution for up to time T requires values for the leads at T + 1. We use the solution to the steady-state model to define the values of these terminal conditions. When a temporary shock is applied to the model starting from a steady-state equilibrium, the terminal conditions will be those same starting conditions. For a permanent shock, we have to calculate new steady-state levels, which are derived by solving the steady-state model under the new assumptions. (8) (8) The decision rules of the core model are expressed in levels, because we are often interested in what happens when there is a new steady state. In contrast, it is common in the conventional DSGE approach to solve for a steady-state equilibrium, and then derive a recursive law of motion for state variables, given linearised decision rules for the endogenous variables as 74 Implementing and solving the model 5.2.4 Solving the model dynamically We need a procedure to reconcile leads and lags for a set of non-linear difference equations, from a known starting point (the predetermined variables), to reach terminal conditions at T + 1 – a problem of the form Yt = f (Yt+1 , Yt−1 ). We use the popular Stacked Time algorithm of Laffargue (1990), Boucekkine (1995) and Juillard (1996); the LBJ algorithm appears in practice to be faster and more reliable than ‘first-order’ methods. (9) Nonetheless, despite the significant recent advances in solution algorithms and computing power, there may still be situations where we fail to find a solution. Typically a valid solution will exist, but it may be difficult or impossible for the solution algorithm to locate it, given the initial conditions. This can arise when the starting point is a long way from the desired steady state (as can happen in a forecast context) or the shock is large or ‘unusual’. In such cases our approach is to solve a linearised model dynamically in order to derive a series of guesses for future values of endogenous variables from the starting point. The linearisation is performed numerically around the appropriate steady state. We find that this approach substantially increases reliability and reduces total simulation times, even though we have to solve another model. 5.3 Recursive simulations In a standard simulation experiment with a model such as BEQM, we would shock the model from an initial starting point and record the solution to the terminal condition, deriving the entire path for each endogenous variable. But many of the exercises we want to conduct with the model do not necessarily suit this approach. In particular, in a standard simulation, any future changes to exogenous variables after the initial shock are assumed to be fully anticipated. In contrast, we may want to be able to simulate a sequence of unanticipated shocks, especially if analysing how the response of the economy is affected by the degree to which agents anticipate future events. But how should we think of implementing an unanticipated shock that takes place some time in the future? We can change the degree to which agents anticipate future events by using the technique of recursive simulations. The basic idea is simple: if we are dealing with an exogenous variable that is potentially subject to shocks in each future period, instead of describing the entire path of all variables by the results from a single, one-shot projection, we build up a profile in which information about exogenous variables is revealed period by period. The paths of endogenous variables will change as more information about exogenous variables is revealed. We start the forecast by solving the model at period t. That result for the endogenous variables at t + 1 is used as the starting point for a new projection at t + 1, which incorporates one more period’s information about the exogenous variables, and the model is solved again with the endogenous variables from that used as the starting point for period t + 2. We continue the process period by period until we have a complete profile. (10) The following subsections illustrate where this approach is useful. deviations from that steady state. In that case, the model would return to the steady-state equilibrium following a shock (as long as implicit transversality conditions are observed in the theory and the appropriate policy rules are stabilising). (9) See, inter alia, Judd (2002). (10) At some arbitrary period t + j < T in the future, we would be content to run the model out to period T . 75 The Bank of England Quarterly Model 5.3.1 Exogenous variables and expectations Expectations in BEQM are model-consistent, as described in Section 5.1.5. This standard assumption has often been combined with the additional assumption that agents have full information about the future paths of exogenous variables (including future shocks). Although this can be a useful benchmark, it can on occasion lead to unrealistic model responses. In particular, agents can ‘see through’ temporary shocks and so alter their behaviour only slightly, if at all. Likewise, shocks imposed at some time t + i in the future will be fully anticipated and so reflected in behaviour immediately (ie at time t). Such responses may be sensible for some shocks, but not for all. Ideally, therefore, we would like a tractable and reliable model of learning and information processing by private agents, but we have not yet attempted to build this into the core theory. Instead, we use the recursive simulation technique and manipulate information sets to vary the way that agents react to future events. Expectations are still model-consistent, but we can control the available information in each period, so that there may be surprises in future periods. In the case of a future shock, for example, we could simulate recursively so that the shock was completely unanticipated: this would be consistent with the case in which agents are completely surprised by events and have no knowledge of the future. Figure 5.4: Building a profile under recursive simulations exogenous variable assumptions t0 t +1 t +2 t +3 t +4 endogenous path given full anticipation endogenous variable endogenous path given incomplete anticipation t0 76 t +1 t +2 t +3 t +4 Implementing and solving the model A stylised representation of the different responses this approach can lead to is given in Figure 5.4. The blue lines show a path for an exogenous variable in the top panel and the associated path for a given endogenous variable under full anticipation in the bottom panel. (11) The dashed lines in the top panel show a sequence of paths such that only partial information is given about the future path of the exogenous variable – the first line in period t0 , the second in period t1 and so on. The dashed line in the bottom panel shows the result of linking the series of jumps each period under the sequence of exogenous paths shown in the top panel. We have an auxiliary model – the exogenous variables model – that allows us to vary the degree of anticipation of each exogenous variable between the two extremes of complete anticipation and complete surprise (see Box 10 on page 78). 5.3.2 Imposing judgement directly on endogenous variables The preceding section showed how recursive simulations can be used to vary the assumptions about the information available to agents at different points in time, and how this affects the path of endogenous variables. Here we consider a different issue, involving direct manipulation of endogenous variables. On some occasions, such as in a forecast, we may wish to apply judgement directly onto an endogenous variable, for example to take account of ‘off-model’ theories or data. The forward-looking nature of the core model implies that adjusting core model solutions by applying residuals (or shocks) to particular equations is not straightforward. In principle, it is possible to ‘exogenise’ an endogenous variable temporarily. (12) The model will solve as before, but it will be conditioned on the path for the temporarily ‘exogenised’ variable. However, we quickly encounter problems with this approach. The first is purely numerical: the model might not solve for a path that is too long or too far away from the stable path (or saddlepath) for the endogenous variable. Secondly, forward-looking behaviour implies that current variables will be stabilised by expected future market clearing conditions, so exogenising variables that have to ensure future stabilisation can generate current instability. Finally, even if the model can solve for the temporary ‘fix’, the path will be fully anticipated and private agents will make decisions at time t in anticipation of the end of the fix. Hence, even if the fix can be executed, the results may not be a good match to the thought experiment in mind. Instead, we simulate recursively so that an endogenous variable may be fixed for a single period, moving forward with each successive simulation. Single-period fixes are computationally much easier for the simulation software to handle, and recursive simulations allow us to trace out virtually any desired path as a series of one-period unanticipated fixes. (13) The story consistent with this technique is that agents are successively surprised with outturns for an endogenous variable, which differ from their own model-consistent forecasts. Box 11 on page 80 provides an example of this approach in terms of fixing a path for the short-term nominal interest rate. (11) In practice, we would expect dynamics to be much richer than shown here; for clarity we assume that the endogenous variable jumps from its starting point immediately when the shock is known. (12) Technically, we apply a shock term to an equation in the core model to deliver a particular path for an endogenous variable, conditioned on the profiles of all other shocks in the model. In other words, we find a sequence of shocks to the model that deliver a particular profile for an endogenous variable as an equilibrium. This is not equivalent to making the variable truly exogenous, in the sense that all agents in the model understand that the variable will follow the particular path independently of the profiles of other shocks. (13) Numerical problems may still arise if we suppress the model’s natural dynamic reaction for so many periods that it moves far away from steady-state equilibrium. 77 The Bank of England Quarterly Model Box 10: The exogenous variables model The exogenous variables model has two important functions: 1. to enforce a number of technical conditions that need to hold in order for the model to exhibit a balanced-growth path; and 2. to allow us to control the information sets that agents are assumed to use to forecast exogenous variables. This allows us to take a view about how past movements in exogenous variables affect agents’ expectations for these variables going forward into the future. Function 1 allows us to ensure that shocks to exogenous variables that are trended (for example, labour productivity, population, world demand) are converted correctly into the stationary ‘model units’ in which the model is written. This is important to ensure model stability. Function 2 is necessary because it is the values of exogenous variables expected to prevail in the indefinite future that pin down the model’s steady state. When simulating the model recursively, we can specify a different information set for each period, reflecting assumptions about what is known about the future. Simulating over the past requires an assumption about the information available to agents at particular dates in the past (as discussed in Section 5.4.1). One possible assumption is that agents observed the actual path of exogenous variables over the whole period; so in 1980 Q1, agents observed the actual path of (say) world demand from 1980 Q1 to 2003 Q4. One alternative to this perfect foresight assumption, which we have termed the ‘random walk’ assumption, is to suppose that expectations of exogenous variables are determined by e = xt xt,t+i (1) e denotes the value of x expected to prevail at date t + i based on for all periods t + i. Here xt,t+i (a) information at date t. In other words, the expectation at t is projected forward from the current value. In the perfect foresight approach, the latest observation is treated as a temporary phenomenon that has no effect on the level that the exogenous variable will settle on (and hence the steady-state values of endogenous variables). Under the ‘random walk’ approach, all of the observed change in the exogenous variable is interpreted as ‘news’ about the long-run level of the variable: all changes in exogenous variables are treated as permanent. Another way of thinking about expectations of exogenous variables is to suppose that agents use available historical data to work out a good way of predicting future changes in exogenous variables. We could imagine that the experiences of agents allow them to behave like econometricians who fit time-series models to the data on exogenous variables that they observe, and then use these models to generate forecasts of these variables. That would allow agents to decompose recent outturns of x into permanent and temporary components (and judge how quickly any temporary element will unwind). (a) Of course, it is possible that the time series properties of some exogenous variables over the past may support the ‘random walk’ assumption. 78 Implementing and solving the model Specifically, we imagine that forecasts for an exogenous variable x based on information available at date t are generated by an autoregressive process: e − µtx = α i xt − µtx xt,t+i (2) Equation (2) expresses expected values of xt as following a simple AR(1) process around a (projected) time-varying mean, µtx , which is the value that x is ultimately expected to settle on. We could generalise the representation in (2) to incorporate time series processes of more complexity (including multivariate treatments). But the idea would remain the same. Notice that the ‘random walk’ assumption (1) is just a special case of (2) under the assumption that α = 1. Applying fixes to the non-core equations is more straightforward, given the absence of leads in these equations. This means that we can easily adjust the value of an equation residual to deliver a particular value for an endogenous variable, y1 . Such a ‘type 1 fix’ is formally achieved by making y1 temporarily exogenous and specifying the residual, y1res , as the endogenous variable for that equation, so that y1res will take whatever value is required to achieve the desired path of y1 . This structure of the non-core equations also permits ‘type 2 fixes’ in which the residual of another equation, say y2res , changes to deliver a particular value for the endogenous variable y1 : when the model solves, y2res will move to produce values of y2 that would deliver the required path for y1 . This will, of course, only work well if changes in y2 have sufficient effect (direct or indirect) on y1 . This approach allows us to impose judgement on a wide range of variables, if required. 5.4 Applications This section explains how these techniques are brought together for two particular applications: solving over the past and running forecast projections. The ability to solve the core model over the past is an essential ingredient for both parameterising the core model and estimating the non-core equations. Here, we concentrate on the techniques and methodology for solving the model over the past, drawing on the discussion in the earlier sections of this chapter. Chapter 6 then presents the results of parameterisation and estimation, making use of the solution to the core model over the past. 5.4.1 Solving over the past We solve the model over the past in order to assess the performance of the core model and to estimate the non-core equations (see Chapter 6). We have to make a number of decisions when simulating the model over the past: these include whether future changes in exogenous variables are fully anticipated; and whether changes in exogenous variables are assumed to be permanent or temporary. We judge how well the model matches historical data by looking at the performance of the model as a full system. Matching the model to past data involves difficult choices before we can judge how well it explains movements in the data. For example, to what extent were consumers able to anticipate future events when making decisions at any given time? Should we assume, say, that consumers in 1987 Q1 correctly anticipated changes to fiscal policy made in 2000? If so, the effect on their behaviour would need to be included in the model for an accurate characterisation of consumption decisions. The degree of anticipation can be controlled using the recursive simulation technique, together with the ‘exogenous variables model’ described in Box 10 on page 78. 79 The Bank of England Quarterly Model Box 11: Conditioning nominal interest rate paths One application of the recursive simulations technique concerns policy variables and, in particular, the nominal interest rate path. The Bank publishes projections conditioned on given paths for the nominal interest rate, whereas equilibrium in the core model requires endogenous monetary reaction in order to anchor the nominal side of the model. So how can we condition on a nominal interest rate path (whether constant or market rates) while still allowing the model to solve for a sustainable equilibrium? The approach used in BEQM is to apply a sequence of ex ante unanticipated shocks to the monetary policy reaction function that, ex post, delivers the required path for the nominal interest rate, without agents in the model actually expecting that path. Specifically, consider a reaction function for the nominal interest rate rg: rgt = h (...) + εt rg (1) which says that the policy rate, rgt , responds to a function of endogenous variables (for example rg deviation of inflation from target, a measure of demand pressure on capacity). The term εt is a shock to the reaction function. Suppose we want to solve the model under the assumption that ex post interest rates follow a particular path for the first J periods of the projection. Then the task is rg to find a sequence of unanticipated shocks εt ; t = 1, ..., J such that the sequence of interest rates matches the desired path. When solving the core model using a stacked-time algorithm we can find the sequence of policy shocks as follows. We begin by specifying monetary policy as: rgt = ϒt rg t + (1 − ϒt ) h (...) (2) where ϒt is a dummy variable taking a value of 1 in the first period of the simulation and 0 thereafter and rg t is the interest rate path to be imposed on the model. Solving the model with this specification of policy at date t = 1 will deliver a solution in which the interest rate takes a value of rg 1 at date 1 and follows the reaction function h (...) in subsequent periods. To hold the interest rate at rg 2 at date t = 2, we solve the model recursively, moving on to the next period (t = 2) and solving again, this time with ϒ2 = 1 and ϒt = 0 for t = 3, 4, .... Continuing in this way for J periods delivers a projection in which the interest rate follows the path rg t (ex post) for the first J rg periods. We can find the sequence of unanticipated shocks εt ; t = 1, ..., J that generates the same result by computing the difference between the reaction function h (...) and the value of rg t over the first J periods of the simulation. (a) The method therefore proceeds as follows. We temporarily ‘exogenise’ the nominal interest rate for one period at a desired level, and solve the model forward from t. We then take outturns for t + 1 as the starting point for a new projection, which will typically include imposing another value for the nominal interest rate. The resulting path that is built up for the nominal interest rate can match virtually any profile. (a) This is equivalent to shocking the monetary reaction function in autoregressive systems, such as has been done in structural VARs (see Leeper and Zha (2001)) and DSGE models (see, for example, Smets and Wouters (2003b)). We go into some detail here to show what we do, given that we do not have an autoregressive representation that we can shock directly. 80 Implementing and solving the model With each period in the recursion, private agents are surprised by the nominal rate but expect the monetary authority to behave according to the properties of the model’s monetary policy reaction function in the next period. To the extent that the nominal rate path is inconsistent with achieving the inflation target, pressures will accumulate. For example, in the stylised example depicted below, the nominal interest rate is held for three periods above the level that would be consistent with achieving the inflation target. This generates deflationary pressure, which is only addressed when the normal monetary reaction function is allowed to direct the path for interest rates. In general, the longer the nominal interest rate is held away from the level that would be consistent with the inflation target, the more ‘work’ the reaction function will have to do in order to bring inflation back to target. Figure A: when interest rates are not consistent with the inflation target nominal interest rate steady-state equilibrium reaction function resumes inflation rate t0 t +1 t +2 t +3 t +4 t +5 t +6 The recursive simulation technique also requires a choice of whether the values for predetermined variables used at any point in time in the historical simulation are actual outturns in the data, or the endogenous solution from the previous period’s projection. (14) Either option is possible, but our standard assumption is that the predetermined variables at each period are actual outturns. Finally, when simulating over the past we may wish to vary values for some parameters over the sample period (for example the parameter controlling unions’ relative bargaining power) which are summary variables for deeper structure that is not formally modelled (for example, trends in the extent of union coverage). Hence, technically speaking, some parameters are treated as exogenous variables for the purpose of historical simulations. Making sensible historical simulations with a model of this sort essentially amounts to conducting a series of retrospective forecasts. At each past period, we have to take a view on the perception for the exogenous variables, the steady state applicable at the time, and the starting points for endogenous variables. Not surprisingly, the assumptions made about information, expectations for exogenous (14) In other words, we need to choose whether the simulation is static or dynamic. 81 The Bank of England Quarterly Model variables, the values of predetermined variables, and the values of parameters over the past can have large effects on the simulated historical paths of endogenous variables. 5.4.2 Forecast projections In a standard simulation, we would assume that the model economy was initially at equilibrium, shock the model at time t, and solve the model to get new values for endogenous variables. But a forecast uses recent data as the initial starting point and solves the model to see how endogenous variables would need to move over the medium term to achieve (eventually) the assumed steady-state equilibrium. The paths from the model are stationary and effectively represent deviations around underlying trends for labour productivity growth, output growth, and inflation. To derive solutions for data in levels, we scale these solution paths up by the appropriate growth rates, given the starting points. Effectively, the outputs from the full model are converted back into units that correspond to original data sources by reversing the transformations that take place when deriving the model-consistent database – this process is explained in detail in Section 6.2.1. We also make use of a set of transformation equations that takes some of the outputs from the model and recombines them to construct useful diagnostic measures such as the savings ratio. Thus, a forecast profile is built up in the following steps: 1. Profiles for expectations of future exogenous variables are derived using the exogenous variables model. 2. Values for forecast assumptions (endogenous fixes) are decided. 3. Terminal conditions for endogenous variables are calculated using the steady-state model. 4. The dynamic model is simulated recursively, taking the (possibly different) steady-state values at each period in the projection, using the first period’s forecast values from the previous period’s projection. A linearised model is simulated first, which the non-linear core model then uses for initial guesses for endogenous values. The first period’s responses for each projection are extracted and form the final core model values. 5. The full model with non-core dynamics is solved, (15) given the complete path for the endogenous values in the core. Residual adjustments can be made to the non-core equations and the model can be solved again. 6. The outputs from the full model, expressed in model units, are transformed into more familiar measures, such as National Accounts units and growth rates, for analysis. Judgement can be added at each step: for example, if it were thought that consumption was too low over the forecast horizon, this could reflect a number of possible issues such as the long-run equilibrium of the model being too low; or medium-term expectations of future labour income, price changes, taxes or real interest rates needing modification; or the short-term forecast profile looking implausible relative to short-term indicators. The perceived source of the problem would determine whether the forecaster should look at parameters driving the steady state, reconsider the profile for exogenous variables over the forecast horizon, or change residual settings on the non-core model. (15) We could in principle use the responses from the full model as the starting points for the core in the next period, but we do not do so and instead use core values as starting points for the core model going forward. 82 Implementing and solving the model 5.5 Summary This chapter discusses how the model is set up and solved. Importantly, the model is a system of non-linear difference equations that contains leads as well as lags, and so requires specialised solution algorithms. We have also developed further routines employed in simulating the model. These are quite elaborate, so here we summarise the key ingredients. The model is designed to settle on a long-run sustainable equilibrium, so we have to specify those values. Hence all simulations require an internally consistent steady-state solution. The core model is used to generate a steady-state model for this task that can be solved in a straightforward way. We always need starting values for those endogenous variables that have lags. In one type of exercise – typically for demonstrating theoretical properties – we can shock the model from an initial steady-state equilibrium, by changing the value of one or more exogenous variables. For a permanent shock, we would need to re-solve the steady-state model to provide a new set of terminal conditions, but not for a temporary shock. Forecasts use recent data outturns as the starting point. The model’s dynamics show how we can get from these starting values to the sustainable equilibrium implied by the values from the steady-state model. The forecast is a more complicated exercise than a simple ‘one shock’ simulation. Our standard approach is to build up the forecast path by recursive simulations, so that we can apply judgement and deal with issues of expectations, information sets, temporary fixes on endogenous variables. 83 Chapter 6 Parameterisation and evaluation This chapter discusses our approach to parameterising BEQM. After a brief discussion of some initial issues (Section 6.1), Section 6.2 describes the construction of the model-consistent database. The parameterisation of the structural core model is discussed in Section 6.3, and that of the ad hoc non-core equations in Section 6.4. The results of a formal evaluation of the model (Section 6.5) are followed by a summary in Section 6.6. 6.1 Issues in parameterising the core model Our basic approach has been to decide, first, which theoretical mechanisms are required for the model to serve as a useful tool for policy analysis, and then to work towards matching the data. (1) This approach puts a great deal of emphasis on viewing the model as a system. (2) It follows that because the model has been built as a system, and will be operated and used as a system, it should be parameterised and evaluated as a system. However, there are several features of the model that raise difficult questions for parameterisation. First, like many other macro models used in policy institutions, BEQM is large, simultaneous, and non-linear. In ideal circumstances, we would estimate the model using maximum likelihood methods, which have the major benefit of consistent estimates and (asymptotic) normality. This would require us to specify correctly the stochastic structure of the model, but its non-linearity means that we would lose these desirable properties if there were any deviation from a correct specification. (3) A linearised model would provide some protection from the effects of misspecification. However, we cannot assume a fixed steady-state around which to linearise the model, because the steady state itself is assumed to change over the past, and possibly over the projection period as well. An alternative would be to use some form of method of moments estimation, which would not require a full specification of the probability spaces (eg for parameters). But that approach is still not immune to misspecification, and would require a large number of moment conditions in a large model such as BEQM. Aside from representing a significant computational challenge, the number of observations would need to be much larger than the number of moment conditions. In general, ‘textbook’ econometric approaches assume that there is a (very) large number of observations in the sample relative to the number of parameters being estimated. This condition does not hold in our case. (4) There is also a question as to how to evaluate the parameterisation of the model. As described in Chapter 2, the main motivation for the new model was to improve theoretical consistency and clarity, so that policymakers can make projections with confidence that they understand the forces generating the (1) In terms of the modelling ‘frontier’ shown in Figure 2.1, we can think of this as moving north, then east, rather than vice versa. For simulation evidence on the merits of this strategy, see Kapetanios et al (2004). (2) This is hardly new: the Cowles Foundation approach to econometrics is famous for its concentration on the estimation of large, simultaneous equation models, going as far back as the Klein (1950) model. (3) And in practice we will be more concerned with the fit of the model in some dimensions than others. So it is not clear that a classical likelihood perspective is the right one, since maximum likelihood weights different moments according to the amount of information the data have about these moments. See pages 1613-14 of Eichenbaum (1995). (4) This situation is not unusual in macromodelling. Indeed, as Sims (2002, page 15) has noted: ‘Taking account of simultaneity ... seemed to require a lot of work to end up with results that were arbitrary (if based on a truncated instrument list), almost OLS (2SLS with all available instruments), or quirky (FIML and LIML).’ However, alternative approaches may become feasible in future. 85 The Bank of England Quarterly Model outputs from the model and how they are affected by policy decisions. Inevitably this means that the new model will not mechanically generate the best possible forecast, nor fit the data as well as a data-based atheoretical model. At the same time, some congruence with the data is needed to make the exercise meaningful. But the best forecasting models can be of limited use for policymakers if they cannot be used to analyse economic issues. (5) Consequently, a measure of statistical fit is not the be-all and end-all for assessing a model such as BEQM. We do not pretend to have solved these problems. Our parameterisation strategy has been simple and pragmatic, especially for the core model. As a first pass, this has been calibrated using a range of information from different sources. As explained in Section 6.3, there is a good deal of structure to the calibration. First, we have broken the parameterisation of the model down into smaller steps, where results still depend on the whole system but there is a smaller set of variables to consider. For example, some variables affect only the numerical steady state, so we approach these first. Second, while the model is highly simultaneous, there are sensible ways of ordering the use of equilibrium relations from the core model to estimate its underlying parameters. Third, since projections using the model are based on model-consistent expectations, we simulate the model over history on the same basis (in effect, we run a series of recursive projections). Advances in numerical and computational methods may make it feasible in the future to estimate the whole model using more formal methods. In any case, the current parameterisation is not set in stone: it is frequently reappraised, and parameter values may be revisited in response to signs of structural breaks or when there are large data revisions (such as the adoption of chain linking in the National Accounts). This applies not only to the parameterisation of the model’s dynamics but also to the steady-state values shown in Section 6.3. Before we proceed to the choice of parameter values, however, we need to address a number of data issues. 6.2 The model-consistent database The first step towards parameterising the model is to create a model-consistent database. This has two key aspects: the data are detrended so that the model can be written in stationary form; and the data are matched to model concepts. When we refer to ‘model units’, we therefore mean data that have been detrended and may have been modified compared with, say, a National Accounts measure, to make a better match with the relevant economic concept. 6.2.1 Detrending and retransforming to levels BEQM uses data that have been transformed into stationary form (as discussed in Section 5.1.3): variables are detrended by potential output growth, labour productivity growth, or inflation, as appropriate (Box 12 on page 88 sets out the process in more detail). Thus, leads and lags in the model equations are scaled by growth rates of productive potential, productivity or inflation. A simple example for a real flow is the housing investment stock-flow condition: idt = dt − 1 − δ dt · dt−1 1 + ẏt (5) The potential trade-off between theoretical consistency and data coherence is discussed in the context of Figure 2.1 in Chapter 2. 86 Parameterisation and evaluation where id is the flow of dwellings investment, d is the stock of dwellings, and ẏ is the growth rate of productive potential. Where nominal scaling is involved, a detrended form requires lags to be scaled down by growth and inflation (leads are similarly scaled up), as in the expression for seigniorage: pct · mon t − pct−1 · mon t−1 (1 + ṗt )(1 + ẏt ) Such detrended data must be re-transformed if data in levels are required. Examples of these transformations are given in Appendix C, but some general principles can be listed here. For most real variables, such as goods and assets, the appropriate scalar is the growth rate of productive potential, (1 + ẏ). This is a function of the population growth rate and the labour productivity growth rate: (1 + ẏ) = (1 + ṅ) · 1 + λ̇ . (6) So real variables (ie volumes) will be transformed to levels by multiplying by the levels of both the population (N ) and labour productivity (λ). For example, the volume of housing investment in levels (analogous to a constant price measure) will be I D = id · N · λ, where I D is a volume measure and id is in detrended model units. The notation we use to distinguish actual and detrended model units is set out in Section 6.2.2 and Table 6.1. For variables associated with the wage schedule, the appropriate scaling is by labour productivity growth, (1 + λ̇). These variables in model units would be transformed back to levels using the level of labour productivity. So the real private sector wage rate in levels would be given by w · λ. As discussed in Chapter 5, prices in BEQM are relative to a numeraire price level, which we have chosen (without loss of generality) to be the consumption price level (excluding actual and imputed rents). The numeraire price level in the model, PC, is used to detrend other nominal prices and hence turn them into relative prices (denoted here in lower case). By implication, the relative price of the numeraire, pc = 1 in every period. Other price levels (ie deflators) are equal to the appropriate relative price multiplied by the consumption price level (PC). For example, the nominal price of exports is P X = px · PC. (7) Changes in the level of PC over time are given by the inflation rate, ṗ. The numeraire price level simply cumulates inflation from an initial starting point, so there is a unique condition for the total consumption deflator: PCt = PCt−1 · (1 + ṗt ). Nominal variables (current price measures) simply apply both price and real levels transforms. For example, nominal export expenditure is P X · X = x · N · λ · px · PC, scaled back up to a nominal level by re-introducing cumulative inflation (including the relative price, px) and growth, N · λ. The nominal wage, on the other hand, only scales by labour productivity and the price level: W = w · λ · PC. (6) Here, we use ṅ and λ̇ to refer to actual growth rates of population and labour productivity. Their product ẏ, however, is the growth rate of productive potential rather than actual output. (7) We calculate relative prices by dividing each price level by the consumption deflator, PC. For instance, the example shown in the text easily rearranges to pxt = P X t /PCt . For the consumption deflator, however, this becomes pct = PCt /PCt = 1. 87 The Bank of England Quarterly Model Box 12: Detrending and model units The model is constructed to converge on a neoclassical balanced growth path in which all similarly measured variables grow at the same rate. Rather than retain the model in variables with non-stationary levels, the model works with variables that are detrended by the relevant growth path. We find it useful, therefore, to deal with variables adjusted for population and productivity. We illustrate this using the following notation: Nt is the population level, and λt is the level of labour productivity. (a) For a given variable X t , we define a level Xt λt Nt which we refer to as the level of x in ‘detrended model units’. The goods market clearing condition from Figure 3.5, which states that all private sector output must be distributed in domestic and world markets, can be written: xt = Yt = C H Vt + I Dt + I K H Vt + I O Vt + D E L St + GVt + X Vt where D E L St = St − St−1 is stockbuilding. The conversion to detrended model units is trivial: Yt C H Vt I Dt I K H Vt I O Vt D E L St GVt X Vt = + + + + + + λt Nt λt Nt λt Nt λt Nt λt Nt λt Nt λt Nt λt Nt ⇒ yt = chv t + idt + ikhv t + iov t + delst + gv t + xv t . Indeed, most equations in the model can be transformed this way. Some extra thought is required for those with leads and lags. Defining 1 + λ̇t = λt /λt−1 and 1 + ṅ t = Nt /Nt−1 , detrending a given variable X t−1 by current productivity and population gives: X t−1 λt−1 Nt−1 xt−1 X t−1 = · = λt Nt λt−1 Nt−1 λt Nt 1 + λ̇t (1 + ṅ t ) while a lead would have the opposite transform: X t+1 λt+1 Nt+1 X t+1 = · = xt+1 1 + λ̇t+1 (1 + ṅ t+1 ) λt Nt λt+1 Nt+1 λt Nt The same approach can be applied to labour market and nominal variables. For example, private sector employment would be detrended by the population level only: Et Nt and et−1 would appear deflated by 1/ (1 + ṅ t ). The marginal product of labour will be growing at the rate of labour productivity growth, and hence so will the real wage rate, so that effective wages are in productivity units, Wt wt = PCt λt and wt−1 would appear deflated by 1/ 1 + λ̇t (1 + ṗt ) . et = (a) Technically, λ measures labour-augmenting technical progress. 88 Parameterisation and evaluation Prices are expressed relative to numeraire price, which is the price of the total (non-durable) consumption bundle. For example, the relative price of export goods in the model is written as pxt = P X t /PCt . In the steady state, all price levels will increase at the rate of inflation, ṗss , so the relative prices are constant: pxt = P Xt 1 + ṗ ss P X t−1 P X t−1 = = = pxt−1 PCt 1 + ṗ ss PCt−1 PCt−1 But if we want to deal correctly with relative prices with lags or leads, we have to be careful to account for inflation: P X t−1 PCt−1 pxt−1 P X t−1 = · = PCt PCt−1 PCt 1 + ṗt 6.2.2 Matching model concepts The model has been designed to observe a number of consistency conditions, especially stock-flow and flow-flow conditions (such as budget constraints), as discussed in Section 4.2.2. These consistency conditions may not always be observed in the raw data, especially if different components come from inconsistent sources. But the model’s theory does not recognise the differences between, say, National Accounts and public finances definitions of government spending. So we need to adjust the data to satisfy these conditions. In addition, official measures may be only approximate matches to underlying economic concepts. The components of particular expenditures, taxes and transfers need to be aligned as closely as possible with the underlying decision-making structure of the core theory. This may imply disaggregating and reaggregating in different ways (Box 3 on page 36 discusses the treatment of government and private sector output). However, there are some cases where data that have no obvious model counterpart have to be aggregated into other model variables to account for all flows and stocks. This section illustrates how the main model concepts map into the National Accounts and other data sources; more detailed data sources are listed in Appendix C. For simplicity in this section, we do not add time subscripts if a data relationship is contemporaneous. They are included where necessary (such as stock-flow relationships), but we abstract from the scaling that would be implied by the detrending procedures described in Section 6.2.1. Nominal spending, volumes, prices and the National Accounts The key measure of activity in BEQM is private sector output. But there is a straightforward mapping between this and GDP as measured in the National Accounts. As far as possible we apply a consistent mapping between model concepts and National Accounts data. Nominal expenditures on different types of goods in the model are linked to the corresponding National Accounts measures. Similarly, the volume of different types of goods and services produced by firms in the model are linked to the equivalent to the National Accounts chained volume measure; and the prices of different goods and services in BEQM are equivalent to the corresponding National Accounts price deflators. We adopt a particular naming convention for National Accounts data in the model: a ‘cp’ suffix is typically used to denote National Accounts data for nominal expenditure (current prices); a ‘kp’ suffix is used for the corresponding chained volume measure (expressed in terms of the current reference year’s prices); and a ‘def ’ suffix is used to denote the implicit expenditure deflator; detrended model units are 89 The Bank of England Quarterly Model shown in lower case. Table 6.1 demonstrates the straightforward mapping from the data to model concepts, using exports as an example. Appendix C gives a full description of data sources and how National Accounts and other data are transformed into detrended model units. The discussion below focuses on variables in detrended model units. Table 6.1: The mapping from data to model concepts Data Description Actual units xkp chained volume X pxdef implicit deflator PX xcp current prices PX · X Detrended model units x= X λ· N px = px · x = PX PC PX · X λ · N · PC Private sector output, GDP and factor incomes Aggregate expenditure on private sector output at nominal market prices (ie including indirect taxes, less subsidies, on products) in detrended model units is written as: pym · ym = pc · c + pdv · id + pk · ik + pio · io + psv · dels + pg · g (6.1) + px · x − pcm · cm − pkm · ikm − pmin · mi Market price expenditure on private sector output is thus the sum of consumption (excluding actual and imputed rents), investment in dwellings, business investment, other investment (largely transactions costs and fees on transfers of buildings), inventory accumulation, government expenditure on private sector output (procurement, including investment goods) and exports; less imports of consumption, capital and intermediate goods. (8) The prefix p denotes the relative price of the relevant good (here, they are market prices), in terms of the numeraire price (this is total consumption, so implicitly pc = 1). To get from private sector output to the National Accounts measure of GDP at market prices, gdpex p, we add on expenditure on actual and imputed rents on dwellings (cirex p) and the National Accounts measure of government value added, (9) which is given by the government’s wage bill ((1 + ecostg) · wg · eg) plus an imputed amount for the government’s return on capital (gosgex p), essentially depreciation: gdpex p = pym · ym + cirex p + (1 + ecostg) · wg · eg + gosgex p (6.2) (8) In the non-core equations (see Appendix B), we also take account of the statistical discrepancy between the expenditure and average measure of GDP in the National Accounts. (9) Strictly speaking our measure of private sector output should be called ‘business sector’ output in National Accounts terms, because it includes public corporations. One advantage of this treatment, however, is that it avoids discrete shifts in time series resulting from the privatisations of nationalised industries. 90 Parameterisation and evaluation We can split total consumption and investment into home-produced and imported components: c = pch · ch + pcm · cm pk · ik = pkh · ikh + pkm · ikm The split of total consumption is not immediately available from the National Accounts, but has been estimated using input-output tables as a guide. Given the National Accounts measure of total imports, imported intermediate goods are defined by residual. The identity (6.1) can also be valued at basic prices, which excludes indirect taxes (net of subsidies) on products. This gives an expression for the (gross) value added of the private sector: py · y = pym · ym − pbpa · bpa (6.3) where the value of net indirect taxes is equivalent to the National Accounts ‘basic price adjustment’, denoted by pbpa · bpa. The National Accounts measure of GDP at basic prices (10) thus combines (6.2) and (6.3): gdpbpex p = py · y + cirex p + (1 + ecostg) · wg · eg + gosgex p (6.4) We can also write this measure of GDP in terms of factor incomes by allocating private sector value added into the profits of private sector firms and the compensation paid to private sector workers (in model units): gdpbpex p = sur p + cirex p + (1 + ecost) · w · e + (1 + ecostg) · wg · eg + gosgex p where sur p is the operating profit of private sector firms and (1 + ecost) · w · e is the compensation of private sector workers (including the wage component of self-employment income). This breakdown of GDP in terms of model concepts is approximately the same as the breakdown of the income measure of GDP in the National Accounts (11) . Real expenditures and volumes: market prices and basic prices; real value added and gross output Expenditure flows in detrended model units are made up of a real volume and a relative price component. To obtain the income actually received by domestic private sector firms, aggregate final expenditure needs to be adjusted for the indirect taxes (net of subsidies) and direct and intermediate imports. For internal consistency, indirect taxes and intermediate imports need to be allocated to the individual components of expenditure, rather than simply taken off aggregate expenditure, as in equation (6.3) above. And for different National Accounts concepts, we need a behavioural and accounting structure that distinguishes at the component level between real value added and final output, and between market price and basic price volumes. (10) This is sometimes called Gross Value Added (GVA) at basic prices, which is the Office of National Statistics’ preferred method of valuing real output. (11) The only difference is that sur p excludes the actual rental income on dwellings (captured in cir ex p) and includes the ‘profit’ component of self-employment income. It is also a measure of income at basic prices rather than factor cost, and so includes taxes on production such as business rates and motor vehicle excise duty. In BEQM, these are treated as part of the lump sum taxes on firms (see the discussion of fiscal data below). 91 The Bank of England Quarterly Model As discussed in Chapter 3, we assume that each type of final good consists of a domestic value-added component (the value added by domestic private sector firms), an intermediate import component and an indirect tax (or ‘basic price adjustment’) component. These different components are combined using a Leontief (fixed-proportion) technology, which allows straightforward aggregation and accounting. (12) The intermediate import and indirect tax shares can vary according to the different type of good sold, to allow for different import and tax intensities across expenditure categories. So, for example, the real final volume of home-produced consumption goods is written using the shares of each component as: ch = κ chv · ch value-added component + ρ ch · ch + intermediate import component (1 − ρ ch − κ chv ) · ch indirect tax component The shares for each component are derived from input-ouput tables consistent with the National Accounts. We can construct the core measure of output on which the model’s production function is based, by adding up the real value added (denoted by v) for each type of good. In National Accounts terms, the sum of these is real private sector value added at basic prices: y = chv + idv + ikhv + iov + dels + gv + xv and the equivalent real volume at market prices is simply given by ym = y + bpa where bpa = (1 − κ chv − ρ ch ) · ch + (1 − κ ikhv − ρ ikh ) · ikh + (1 − κ i ov ) · io +(1 − κ gv − ρ g ) · g + (1 − κ xv − ρ x ) · x Two exceptions to the standard structure are worth noting. First, we assume there is no basic price or intermediate import component of stockbuilding or investment in dwellings. Second, we assume there is no intermediate import component for ‘other investment’, which reflects transfer costs on land and buildings. (13) Again, there is a simple mapping into the National Accounts volume measures of GDP at basic and market prices. If we denote the National Accounts chained volume measure of final government spending on consumption and investment goods (in detrended model units) as gons, then we can write: gdp = ym + cir + gons − g gdpbp = gdp − bpa where gons − g is implicitly the ONS measure of real government value added. Additivity and chain-linking The discussion above assumed additivity of the components of the main aggregates. In practice, National Accounts aggregates are annually chain-linked, using the previous year’s relative prices to weight the different components of ‘chained volume’ expenditure together. The data that are required (12) A story compatible with this approach assumes that production of final output requires inputs of private sector value added, intermediate imports and ‘permits’ purchased from the government for the right to produce that good. These are combined costlessly by a perfectly competitive industry, which therefore produces no value added. (13) The basic price component of this expenditure is largely stamp duty. 92 Parameterisation and evaluation for fully balanced National Accounts are only available after several years, so the ‘base year’ for chained volume measures is typically around three years in the past. This means that the components of chained volume measures only sum up to the aggregate from the latest base year onwards for annual data (currently 2001 onwards), and for quarterly data only from the first quarter of the year following the base year (currently 2002 Q1 onwards). The ‘chaining discrepancy’ between the sum of the volume components and the chain-weighted aggregate raises issues for a model like BEQM, which assumes that all goods in the economy are produced with the same production function and can be added together to form aggregate output. When parameterising the model over the past, therefore, we choose parameters that match the individual components of expenditure with their counterparts in the data. The sum of output components may deviate from the aggregate, especially further back in the past. We do not force additivity by changing any components. Data for the aggregate chained volume measure of private sector output are used to estimate the trend in labour productivity, as it is a better measure of the volume of output over time. Relative prices As with the real flows, we require a separate treatment of imported and home-produced prices of consumption and investment goods. And we need to distinguish between final goods prices (including intermediate import prices and indirect taxes) and value-added prices. Given that each final good in the model requires a fixed proportion of intermediate imports and a fixed indirect tax component, we can write the relative price of each component of final output as a weighted average of intermediate import prices and the ‘relative prices’ of value added and indirect taxes. (14) So we can write the relative price of domestically produced consumption goods: pch = κ chv · pch domestic value-added price component ρ ch · pmin + intermediate import price component + (1 − ρ ch − κ chv ) · pbpa indirect tax component Again, we can create aggregates that map into National Accounts deflators. The private sector value added deflator at basic prices is defined by an identity that implicitly weights all value added deflators by their volume shares: py = pchv · chv + pdv · id + pkhv · ikhv + piov · iov + psv · dels + pgv · gv + pxv · xv chv + id + ikhv + iov + dels + gv + xv And the GDP deflator at basic prices is given by: pgdp = py · y + cirex p + (1 + ecostg) · wg · eg + gosgex p y + cir + gons − g All prices here are written relative to the numeraire price: the deflator for aggregate consumption (excluding rentals on dwellings), pcde f . To map into actual National Accounts deflators, we need to use the transformation discussed earlier. So the actual private sector output deflator is calculated as: pyde f = py · pcde f (14) The relative price of indirect taxes is analogous to the National Accounts deflator for the basic price adjustment (relative to the consumption deflator), pbpa. 93 The Bank of England Quarterly Model While most of the relative prices used in the model are derived from National Accounts deflators, we use some other price series in the model. In particular, the non-core model uses an auxiliary housing price, phse, which is an average of the Halifax and Nationwide house price indices, scaled into model units. The other house price variable in the model, pdv, is the National Accounts housing investment deflator. The key difference is that phse implicitly reflects the value of land as well as that of the dwelling. Inflation The basic inflation rate in BEQM, ṗ, is the rate of change of the numeraire price, the consumption deflator (excluding actual and imputed rents). From this, we can then construct a number of alternative measures of consumer price inflation, which could potentially be used in the monetary reaction function. In particular, to construct the target measure of CPI inflation, we need to add on the rents component of the CPI and take account of the average wedge between the consumption deflator and CPI (reflecting the different methods of constructing the consumption deflator and the CPI index). This relationship can be approximated by cpi xrdot = ṗ − cpiwedge + (seasonal factors) and cpidot = µcpr · cprdot + (1 − µcpr ) · cpi xrdot where cpi xrdot is the quarterly CPI inflation rate excluding rents, cpiwedge is the wedge that reflects the different weighting schemes, cpidot is the quarterly CPI inflation rate, cprdot is the quarterly inflation rate of CPI rents, and µcpr is the weight on rents in the CPI. (15) A similar set of equations can be used to construct RPI-based inflation measures. Physical stocks and flows Real physical stock measures need to be consistent with the real flows in the model and the assumed depreciation rates. This applies to the stocks of dwellings, domestically produced capital, imported capital and inventories: (16) dt = 1 − δ d · dt−1 + idt kh t = 1 − δ kh · kh t−1 + ikh t km t = 1 − δ km · km t−1 + ikm t st = st−1 + delst Money, exchange rates, financial prices and stocks Money is introduced in BEQM through a money-in-utility assumption, which we think of as a simple way of motivating demand for money for transactions (see Section 3.4.1). We assume that this money stock is conceptually the same as narrow money, measured by private sector holdings of notes and coin. (15) In practice, BEQM uses a more complicated chain-linking formula to weight the two CPI components together. (16) In practice, we exclude the National Accounts alignment adjustment from contributing to the level of stocks. 94 Parameterisation and evaluation Real exchange rate measures take the exchange rate index (ERI) (17) and adjust for relative domestic and overseas consumer price levels. We have a variety of financial assets in the model. They must satisfy the model identities a = v + bk + pg · bg + pc · mon + nfa nfa = bf /q such that real household financial asset holdings are equal to the sum of the real values of equities, corporate bonds, government bonds, money and net foreign assets. Financial intermediaries are not modelled explicitly in BEQM, so households are assumed to hold financial assets directly, including those which in practice are held on their behalf by pension funds and other financial companies. We also use a narrow definition of money, so households’ holdings of deposits with monetary financial institutions and household sector debt are not separately identified. Implicitly, the stock of households’ deposit holdings (net of their secured and unsecured debt) is treated as part of households’ net equity holdings, even though deposit and debt instruments may in practice provide specialist services (for example, bank deposits providing transactions services). This means that the conceptual definitions of different assets in the model only correspond to the balance sheet data found in the National Accounts at a highly aggregated level. (18) Asset stocks are measured at market values. So the accumulation identities in the model need to take account of changes in the market values of different asset stocks, such as the effect of exchange rate movements on the domestic currency value of net foreign assets. Savings flows in the model are assumed, where possible, to be equivalent to the concept of net lending in the National Accounts, so the financial surplus measure used in the government budget constraint is equivalent to the National Accounts measure of general government net lending; and the overseas sector flow equates to the current account of the balance payments plus net capital transfers. (19) The assumption of a small set of one-period, risk-free assets pinned down by arbitrage relationships, means that mapping the model’s revaluation of financial assets onto those observed in the data is not straightforward. If asset price revaluations have moved significantly differently from the arbitrage relationships in the model, we may not be able to match market values of the stocks as well as we can the associated flows. (17) The version of BEQM described here was calibrated on the definition of the ERI current in 2004, not the new index proposed in Lynch and Whitaker (2004). (18) We have matched a conceptually to net financial wealth of the household sector (as defined in the National Accounts). The variable bk is a measure of Private Non-Financial Companies’ debt net of short-term liquid financial assets (constructed from National Accounts components); pg · bg is general government gross debt (from the National Accounts); mon is narrow money holdings (as defined above); and nfa is net foreign assets (as defined in the National Accounts). We define v residually as an implicit measure of the value of UK equities, although, as noted above, in practice this measure will contain other assets as well. (19) Although the private sector financial deficit in the model maps into the National Accounts measure of net lending, the breakdown into household sector and coporate sector savings flows does not correspond exactly to the net lending data in the National Accounts. This is partly because we do not model financial intermediaries, and partly due to the treatment of self-employment income and rents, which affects the split between profits and labour income. But the household sector saving ratio and other National Accounts variables, such as the corporate profit share, are straightforward to construct from model concepts using some auxiliary equations. 95 The Bank of England Quarterly Model Fiscal data The model’s theory makes no distinction between government expenditures recorded in the National Accounts and those in the public finance accounts, and the two are not consistent. Our priority is for flows that are consistent with the National Accounts measure of nominal government spending, because we use the National Accounts framework. Government revenues and expenditures have to be captured by the budget constraint: (new debt) + (seigniorage revenue) + (tax revenue) = (transfer payments) + (procurement of private sector goods and services) + (government spending on factor inputs) + (debt servicing) Total tax revenue is accounted for by various revenue streams as follows: tax = taxw + taxee + taxd + taxef + taxk +taxlumpc + taxlumpk + taxf + taxind + gosgex p Total tax receipts comprise: taxes from labour income; employees’ National Insurance Contributions; taxes on dwellings; employers’ National Insurance Contributions; corporation tax; lump-sum taxes on households and firms; tax receipts from overseas residents; and indirect taxes. The imputed return on government capital (depreciation), gosgex p, is also added to match the National Accounts measure of receipts. Tax rates are calculated as effective rates, by dividing the revenue by the appropriate income flow; see the equation listing in Appendix A for details. Transfer payments are also broken up in spending categories: trans = transc + transu + transk + transf + transksubs + rgprem Total transfer payments include transfers to consumers, the unemployed, firms and overseas residents; product subsidies to firms; and a ‘wedge’ on interest rates applying to government bonds. Transfer rates, as with tax rates, are calculated as effective rates relative to the underlying flow. Labour market The labour market requires data for wages, employment, unemployment and labour supply. Labour supply, employment and unemployment are defined in terms of Labour Force Survey (LFS) aggregates, as is the split between government and private sector employment, (20) and also private sector and whole economy average hours worked. Similar to our treatment of National Accounts data, we adopt a particular naming convention to identify LFS quantities in heads and hours in the model: the suffix hds is denotes heads-based LFS quantities (in thousands) and hr s denotes weekly hours. Average hours are indexed in the model to take the value of 1 in 1995, this index is denoted by avh. For transforming into detrended model units, we use the LFS measure of population aged 16+ (nhds) to equate with the model population concept, N , as discussed in Section 6.2.1. Table 6.2 illustrates this: essentially the detrending process for labour market data converts quantities in terms of heads into rates of the population aged 16+. (20) Private sector employment includes both employees and the self-employed, while all government sector workers are treated as employees. 96 Parameterisation and evaluation Table 6.2: The mapping from labour market data to model concepts Data Description Actual units Detrended model units nhds population (16+) N 1= N N ehds employment E e= E N lhds labour supply (participation) L l= L N uhds unemployment U u= U N Private and public sector wages are derived from the National Accounts definition of wages and salaries, plus an adjustment for the wage income of the self-employed (the number of self-employed multiplied by the wage per employee in the private sector). The public and (adjusted) private sector wage bills are then deflated by LFS employment measures to obtain wages per head. Trade and balance of payments Values and volumes for exports and imports are from the National Accounts. The split between direct and intermediate imports is based on input-output tables and disaggregated trade data. The trade balance and current and capital account data are standard National Accounts measures. World variables BEQM has exogenous paths for world conditions. World activity, cf, is conceptually matched by UK-weighted world imports (constant prices, detrended model units). There are two price levels: pxfdef is matched by M6 (21) export prices and pcfdef by M6 consumer prices, (together they define relative pxfdef world export prices, pxf = pcfdef ). Foreign inflation, ṗ f , is matched by M6 consumer price inflation. The world nominal interest rate, rf, is matched by the M6 measure of three-month nominal interest rates. 6.3 Parameterising the structural core model This section describes the approach used to set the parameters in the core model. There are several challenges. First, the model is large and simultaneous, with many parameters. Second, the time series available for estimating the system are relatively short. Finally, the model has to fit in levels, not just in deviations away from trend values, so we need to produce steady-state values too. The first and second issues, in particular, make conventional systems estimation procedures hard to apply. Over time, the ability to deal with these problems may improve, but for now we have chosen to set core model parameters by a process of informed judgement rather than formal estimation methods. Given the need for judgement in setting each parameter, we do not attempt to give details of the decisions behind each and every choice. Rather, this section outlines our general approach and explores some of our choices in more detail. (21) The M6 (major six) economies are Canada, France, Germany, Italy, Japan and the United States, 97 The Bank of England Quarterly Model 6.3.1 General approach We can address some of the problems noted above by splitting the parameters in the core model into three groups: those that affect only the steady state; those that affect only the dynamics; and those that affect both the steady state and the dynamics. This allows us to divide the parameterisation process into two broad steps. First we choose parameters that affect only the steady state of the model, aiming to match broad trends in the levels of the data. Second, we choose the parameters that affect the dynamics of the model. Because some of these parameters will also affect the steady state, some iteration between the two steps will generally be required. We do not assume that the parameters of the core model are constant over time, because we do not view the core model as a literal description of behaviour defined by fixed underlying parameters. This is because the core model is intended to provide a coherent model of decision making for different economic agents, but to do this in a tractable way means that some important mechanisms cannot be incorporated into the core model and are instead proxied by parameters in the model. To the extent that we believe that such effects have changed over time, we allow the core model parameters to vary over the past. An example is the way in which the core model accounts for changes in relative (value-added) prices, given the assumption that goods are produced by a single production function. We know that there have been significant trends in the relative prices of expenditure components (as measured by National Accounts deflators) over our data sample. But aside from imported intermediates prices and taxation, the only way the core model allows for relative price movements is through different demand elasticities (and hence mark-ups) across expenditure components. So we use assumptions about demand elasticities that allow us to proxy trends in relative prices. (22) Thus, we are not simply choosing a fixed number for each parameter in the model, but in some cases a time profile over the past. (23) 6.3.2 Parameterising the steady state Our first step is to parameterise the steady state of the model, (24) which forms the base for simulations and the long run for forecasts. We do not think it always sensible to aim to match sample averages (as has often been used in the DSGE literature), because some steady-state ratios show apparent regime shifts (eg the inventory-output ratio) or persistent trends (eg the relative price of investment goods). In such cases, a value that differs from the historic sample average would seem more plausible for the future long run. Our method was to make initial assumptions for parameter values and solve the steady-state model. (25) Because it is stationary, the model defines the steady state as a set of numbers, such as ratios to output and relative prices, so that evaluating the steady state against historical data is straightforward. Iteration follows to adjust structural parameters, so that the steady state tells a coherent story relative to these data. While this process is judgemental, it is not unconstrained. First, there are more steady-state values to (22) We recognise that this is not perfect, as some of the observed relative price trend is likely to be due to relative productivity movements. A more fully articulated multisector model might be able to pick up some of these movements by modelling differential productivity trends in different sectors (this is not a trivial task, however, as data for multisectoral splits are often unavailable). (23) This procedure is very close, albeit informally, to estimating time series processes for parameters; see, for example, Smets and Wouters (2003a) and Adolfson et al (2004a). (24) In common with most of the DSGE literature, we are dealing with a deterministic steady state: that is, we assume certainty equivalence when solving the model. (25) See Chapter 5 for a discussion of how we solve the steady-state model. 98 Parameterisation and evaluation match than there are free parameters. And, second, those free parameters affect the different steady-state values simultaneously. So parameters cannot be assigned to fit each steady-state property independently. In this way, the exercise of jointly fitting many steady-state values with a smaller set of parameters in a theory-consistent framework imposes a meaningful discipline on our calibration of the model. Exogenous variables Before choosing the parameters that determine the steady state of the model, we decompose historical movements in exogenous variables into transitory and permanent components. The steady state of the core model at any moment in time depends on the expected long-run values of exogenous variables, rather than their current values, so movements in the steady state could in part reflect changing perceptions of the long-run values of exogenous variables. We have three main types of exogenous variables: 1. domestic trend growth assumptions; 2. domestic policy targets or assumptions; and 3. assumptions for the rest of the world. Permanent and transitory components were initially identified by applying low-frequency filters to the data for exogenous variables. But in some cases, we adopted an alternative approach. For example, over the period of inflation targeting in the United Kingdom we set the steady-state inflation rate equal to the announced target. (26) But there was no explicit inflation target for the early part of the sample period. We could in principle apply filtering techniques to the data over that period. But instead, we derived an implicit inflation target from announced policy objectives. (27) Parameter values In many small models with microfoundations, a unique recursive ordering for parameters can be derived and used to estimate the parameters. For example, calibration of the canonical Ramsey model (discussed in Box 9 on page 64) typically starts by setting the discount factor of consumers in the model, based on observations of (or assumptions about) long-run real interest rates. Then the production function parameters can be chosen to match the observed capital-output ratio, and the depreciation rate to match the investment-output ratio. This process is recursive: it starts with an assumption about the discount factor and moves on to other parameters in the model. Unfortunately, such an approach cannot be applied directly to the core model, since the interactions in the model give rise to a greater degree of simultaneity and complexity – for example, changes in the real exchange rate have implications for supply, which creates a link between the level of potential output and parameters that primarily affect demand. However, we can work out an approximately recursive approach that exploits the way the model is used in practice. Some parameters in the core model can be chosen recursively, conditional on assumptions about certain endogenous variables. Given these judgements, we can calibrate separate parts of the model as (relatively) self-contained blocks. The aim is to chose assumptions about endogenous variables such that subsequent parameter settings are not inconsistent with these assumptions. To the (26) Or the mid-point of the range, which is a straightforward assumption for a model such as this. (27) In particular, we followed closely the approach of Batini and Nelson (2000). For example, in periods of money growth targeting, it is possible to infer an inflation target by adjusting for (published) forecasts of velocity and trend growth. 99 The Bank of England Quarterly Model extent that these inconsistencies arise, the process can be repeated, with updated assumptions for the steady-state endogenous variables. (28) Key policy judgements and supply-side conditions We begin by making judgements about the parameters that affect the historical paths of the steady-state real exchange rate, unemployment rate and participation rate. We choose parameters that deliver a particular steady-state path: for example, the real exchange rate profile depends on a scalar, κ x , in the export demand function. (29) The consistency of the implicit assumptions about κ x required to support our initial judgement can be validated at a later stage (for example, by comparison with the results of direct estimation of an export demand relationship and other estimates of steady state exchange rates). Assumptions about all such parameters combine to give us the steady-state exchange rate. We can validate the choices for such parameters and the steady states jointly by reference to other studies and to our priors. The next step is to make a judgement about the permanent components of (both domestic and imported) relative prices in the data. These assumptions pin down important supply-side quantities, to allow us to apply a recursive approach conditional on key assumptions. For example, given the data for expenditure deflators, we make a judgement about the extent to which changes in the data were perceived as temporary or permanent. (30) The perceived permanent components are treated as the historical path for the steady-state relative prices. For domestic prices, we choose the appropriate profile for demand elasticities across expenditure components; for import prices we choose (exogenous) paths for variables representing the relative world prices of different types of goods (intermediates, investment goods and consumption goods). (31) With assumptions about steady-state relative prices in place, it becomes possible to set parameters for particular parts of the model in a more or less recursive manner. For example, the cost of capital is pinned down by assumptions about steady-state relative prices, depreciation rates of home and imported capital and the steady-state world real interest rate. Maintaining these assumptions (by changing the relevant parameters if necessary) means that we can temporarily suppress ‘second round’ demand effects that would otherwise occur. This allows us to proceed with parameterisation of the supply side. The production function On the supply side we can estimate the production function parameters simultaneously with the level of labour productivity, given a judgement on the path of the steady-state capital stock. (32) The iterative estimation process starts with an initial assumption about the parameters, which is used to estimate the coefficients of a log-linear time trend for labour productivity. Given data for private sector output and (28) This approach could also be used for formal econometric estimation: individual equilibrium conditions or systems of equilibrium conditions could be used to identify values for underlying parameters. However, apart from supply-side parameters, experiments with this approach were not very successful, especially for the key household preference parameters. This is well known: see Attanasio (1999) for a critique of estimating equations using aggregate data. (29) This scalar is included because the underlying foreign demand data are measured as an index number and so do not correspond to the units in which domestic volumes are measured. (30) There are clearly judgements to be made about the extent to which agents may have anticipated or extrapolated apparent relative price trends. A first step is usually to apply a simple filter to the data. (31) Data series at this level of disaggregation are not generally available, so we include parameters to allow judgement regarding relative world price movements to be applied. (32) Specifically, we estimate the production function parameters α, φ (the return-to-capital parameters in the CES production function) and σ y (the elasticity of substitution between capital and labour) jointly with a log-linear time trend for labour productivity. 100 Parameterisation and evaluation the capital stock, and our assumptions about the path of steady-state employment and average hours, we use the production function to estimate labour productivity using non-linear least squares. The resulting coefficients determine both the level of labour productivity and its steady-state growth rate. The ss steady-state productivity growth rate (λ̇ ) affects decision making in the core model (through growth-adjusted rates of return) and the steady state of the model is now solved and the implied path of the steady-state capital stock compared to the assumed path. (33) The production function parameters are then adjusted and the estimation process repeated until the steady-state capital stock matches the assumed path as closely as possible. (34) Preference shares We can also exploit the use of constant elasticity of substitution functions to capture preferences (for example, preferences between non-durable consumption and dwellings, between home and imported consumption, and between home and imported capital). These functions imply that relative demands are solely determined by relative prices and the parameters of the aggregator function. Given assumptions about relative price trends, the coefficients of these functions can be estimated using single equation techniques. This provides initial information on how to set elasticities of substitution and share parameters. For example, the CES aggregator defining the preferences between home and imported consumption goods implies that the relative demands for these goods satisfy the simple relationship: ch t = cm t 1 − φm φm σ m −1 ψ m pch t (1 − ψ m ) pcm t −σ m where φ m and ψ m are share parameters and σ m is the elasticity of substitution. Taking logs allows us to estimate the regression log ch t − log cm t = A − σ m log pch t − log pcm t using ordinary least squares. This gives us a direct estimate of the substitution elasticity and the estimate of A (which is a function of φ m and ψ m ) guides the choice of share parameters. In practice, the left-hand and right-hand side series can take various forms. We could take the actual data for expenditures and relative prices or the assumptions about the steady state components, or a combination of both. Different series give different parameter estimates and this range can be used as a guide to the initial parameter settings. Household preference parameters Holding the supply side fixed now, we can establish a link between demand and asset accumulation. This highlights the implications for steady-state asset positions of different values of the household preference parameters. A useful starting point is the marginal propensity to consume. To illustrate the role of the key parameters, if we were to simplify the model so that there were no dwellings, money, or habits, we would have a steady-state marginal propensity to consume out of wealth given by mpc = 1 − γ β σ (1 + r)σ c c −1 (6.5) where γ is the probability of survival, β is the household discount factor, σ c is the elasticity of intertemporal substitution in consumption, and r is the real interest rate. Equation (6.5) implies that increasing β reduces the marginal propensity to consume out of wealth as households become more (33) This solution adjusts the necessary parameters to deliver the judgements on steady-state relative prices mentioned earlier. (34) Given the small number of parameters, this process was efficiently implemented using a Nelder-Mead algorithm, based on the description in Lagarias et al (1998). 101 The Bank of England Quarterly Model ‘patient’ and forego current consumption to raise future consumption Raising γ also reduces the marginal propensity to consume – if households expect to live longer, they are more inclined to save. The effect of increasing σ c on the marginal propensity to consume depends on whether β (1 + r) is greater or less than 1. And the effect on the marginal propensity to consume of an increase in the steady-state real interest rate depends on the elasticity of intertemporal substitution. (35) Steady-state consumption is equal to the (steady-state) marginal propensity to consume multiplied by (steady-state) wealth. However, the effects of parameter changes on the marginal propensity to consume do not necessarily reflect the effects on consumption because wealth is endogenous. (36) This highlights a key simultaneity when thinking about the effects of changes in household parameters (β, γ , σ c ) on consumption: these parameters affect both the marginal propensity to consume out of wealth and the level of wealth itself. As we are holding household income and the supply of domestic assets fixed, we can now see the implications for the economy’s net financial position. The net foreign asset position is determined by the difference between the demand for financial assets and the supply of domestic assets (equities, corporate bonds, and government debt). (37) nfa desir ed savings − domestic assets = y y Given the supply side and the real exchange rate judgements described above, the demand side assumptions uniquely pin down the net foreign asset position. The accumulation identity for net foreign assets is nfat = (1 + rt−1 ) ·nfat−1 + xt − m t + (net transfers), where r is the world real interest rate and x − m is the trade balance. This shows that the sustainable trade balance is simply given by: m − x − (net transfers) r so that a negative net foreign asset position is sustainable with a positive current account flow – the economy has to maintain a positive trade balance in order to service debt owed to the rest of the world – and vice versa. (38) nfa = World interest rates and cross-border transfers are exogenous from the first step of the calibration when dealing with exogenous (world and policy) variables. Preference share estimation pins down the import share of consumption. Through household time preference parameters we can affect aggregate consumption, and therefore the trade balance and the required net foreign asset level. Holding output and domestic assets fixed, increasing β or γ will increase the steady-state levels of net foreign assets and consumption. In general, we can see that consumption choices are linked to the net foreign asset, trade balance, and real exchange rate positions in the steady state, because households accumulate assets in order to sustain desired consumption over their expected lifetime. In order to say which parameters have large leverage in a general equilibrium context, we need to use numerical simulations. A useful exercise is to calculate new steady-state values for marginal changes in a single parameter: see Box 13 on page 103. (35) As equation (6.5) shows, if σ c = 1, then the expression reduces to 1 − γ β, and the savings rate is a constant that is unaffected by the real interest rate. When σ c > 1 the substitution effect dominates – consumers are relatively willing for a given change in interest rates to swap present for future consumption. When σ c < 1, the income effect dominates. (36) This is true even when steady-state supply is held temporarily fixed, because wealth depends on the level of net foreign assets. (37) This reflects our assumption that the UK economy is small in markets for capital: the rest of the world will buy any assets not wanted by UK households, or alternatively sell as many assets as UK households want. In all cases, the world interest rate would remain unchanged by these shifts in the UK net asset position. (38) Note that we abstract from steady-state growth here. 102 Parameterisation and evaluation Box 13: The sensitivity of the steady state to changes in parameter values Changes in parameters may have wide-ranging effects on the steady state of the model. This box demonstrates how incremental changes in three parameters affect the steady-state values of key endogenous variables in the model. In each chart, the x-axis is measured in terms of the parameter we are varying, (a) and its baseline calibration (consistent with the listing in Appendix D) is shown by a vertical line. Figure A shows what happens when the household discount factor, β, is varied. Increasing the discount factor means that domestic consumers become more ‘patient’ and are more willing to reduce current consumption to raise future consumption. As a result, they are more willing to lend overseas at the (exogenous) world real interest rate. The increase in net foreign assets is associated with a real exchange rate appreciation, which reduces domestic supply capacity through higher cost of capital and lower participation (see Section 7.2). Interest income from the higher level of net foreign assets supports higher consumption and lower exports. Figure A: Effects of changing β on steady state Net foreign assets: private sector output Per cent Real exchange rate (change from baseline) Level 1 0.1 0 0 -1 -0.1 -2 -0.2 -3 -0.3 -4 -0.4 -5 0.980 0.982 0.984 0.986 0.988 β 0.990 0.992 0.994 Capital: private sector output 0.996 0.998 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 β Level Consumption: private sector output Level 0.839 3.230 3.229 3.228 0.837 3.227 3.226 0.835 3.225 3.224 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 0.833 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 β β Exports: private sector output Level Imports: private sector output Level 0.485 0.5122 0.484 0.5119 0.483 0.5116 0.482 0.5113 0.481 0.5110 0.480 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 β Marginal propensity to consume β Level Wealth: private sector output Level 0.026 40 38 0.024 36 0.022 34 0.020 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 β 32 0.980 0.982 0.984 0.986 0.988 0.990 0.992 0.994 β 0.996 0.998 (a) The exercise aims to show the variation that we would see from moderate variation in parameter ranges, rather than testing the consequences of extreme values. See the discussion on pages 382-383 of Kim and Pagan (1995). 103 The Bank of England Quarterly Model Figure B shows the effect of varying the share of capital in the production function, α. There is an increase in the capital-output ratio with a corresponding rise in investment as a share of output. Consumption declines as a share of output (though the level of consumption rises), as it is crowded out by the additional investment expenditure. The increase in the share of capital in production reduces the labour share, as the the value of output increases proportionally more than the wage bill. The reduction in the consumption-output ratio as α rises encourages a real depreciation to sustain a higher level of exports as output expands. The value of net trade relative to output rises, but the ratio of net foreign assets to output falls because a lower level of interest income is required to finance the (improved) steady-state trade balance. Figure B: Effects of changing value of α on steady-state ratios Capital: private sector output Non-durable consumption: private sector output Level Level 0.86 3.4 0.85 3.3 0.84 0.83 3.2 0.82 3.1 0.81 0.80 3.0 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.26 0.35 0.27 0.28 0.29 0.30 α Capital investment: private sector output 0.31 α 0.32 0.33 0.34 0.35 Level Private sector labour share Level 0.145 0.580 0.142 0.575 0.139 0.570 0.136 0.565 0.133 0.130 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.560 0.26 0.27 0.28 0.29 α 0.30 0.31 0.32 0.33 0.34 0.35 α Net foreign assets: nominal output Level Per cent change Real exchange rate -0.10 4 -0.11 3 -0.12 2 -0.13 1 -0.14 0 -0.15 -1 -0.16 -2 -3 -0.17 -4 -0.18 0.26 0.27 0.28 0.29 0.30 α 0.31 0.32 0.33 0.34 0.35 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 α Figure C shows the effect of varying the long-run depreciation rate of domestic capital, δ kh . This experiment is designed to demonstrate that there is often a trade-off between matching different data series. In this case, we cannot use our assumption about the long-run depreciation rate of capital to move both steady-state capital and steady-state investment in the same direction. As the depreciation rate on domestically produced capital increases, the cost of capital rises and the capital-output ratio declines. This is associated with lower wealth and a lower consumption-output ratio. Investment rises as a proportion of output and therefore ‘crowds out’ other expenditure components. A higher depreciation rate means that, even though the capital stock is lower, a higher level of replacement investment is required to maintain it. An increase in the depreciation rate of domestic capital is associated with a decline in the value of net foreign assets relative to output, because a lower stock of assets is required to sustain the lower level of consumption. 104 Parameterisation and evaluation A higher level of the real exchange rate is also associated with a higher rate of capital depreciation, as the value of the trade balance must increase to maintain the lower level of net foreign assets and the appreciation acts to reduce the price (and hence value) of imports. Though the real appreciation is associated with a fall in the volume of exports, the decline in output is more marked, so the share of exports in output actually increases. Figure C: Effects of changing value of δ kh on steady-state ratios Capital: private sector output Level Non-durable consumption: private sector output Level 3.30 0.845 3.25 0.840 3.20 0.835 3.15 0.0054 0.0058 0.0062 δ 0.0066 0.830 0.0054 0.0070 0.0058 kh 0.0062 0.0066 δ Capital investment: private sector output Level 0.0070 kh Real exchange rate (change from baseline) Per cent 0.3 0.145 0.2 0.140 0.1 0 0.135 -0.1 -0.2 0.130 0.0054 0.0058 0.0062 0.0066 δ 0.0070 -0.3 0.0054 0.0058 0.0062 kh 0.0066 δ Exports: private sector output Level 0.0070 kh Net foreign assets: nominal output Level 0.483 -0.13 0.482 -0.14 0.481 -0.15 0.480 -0.16 0.479 0.0054 0.0058 0.0062 0.0066 δ kh 0.0070 -0.17 0.0054 0.0058 0.0062 δ 0.0066 0.0070 kh Results for key ratios Here we present a number of charts of data series together with the implied steady-state values from the most recent parameterisation of the model. The variables we plot will take constant values in the balanced-growth steady state of the model, so we plot ratios rather than levels, and relative rather than absolute prices. The solid blue lines show actual data over the period 1978 Q1 to 2003 Q4; the dashed lines depict the corresponding steady-state values from the model (using the parameter values for 2003 Q4). A key point here is that, to the extent that steady-state values have evolved over time, we should not expect their recent values to be close to the average of the past data. 105 The Bank of England Quarterly Model Figure 6.1 plots ratios of various chained-volume expenditure components to private sector output. In general, the current steady-state ratios implied by the model are relatively close to those observed in recent data. The exceptions are the export and import ratios, where the steady-state ratios are significantly higher than the historical data. This is consistent with the view that the trend increase in gross trade volumes reflects a continuing process of specialisation, so that gross trade volumes will continue to rise as a share of output in the future. However, the assumed increases in gross trade shares are broadly offsetting, since steady-state net trade as a proportion of private sector output is not exceptional in terms of recent experience. Figure 6.1: Ratios of expenditures to private sector output Capital investment (ik ) Non-durable consumption (c ) 1 0.4 0.9 0.3 0.8 0.2 0.7 0.1 steady state 0.6 1978 1982 1986 1990 1994 1998 2002 0 1978 Government procurement (g ) 1982 1986 1990 1994 1998 2002 Exports (x ) 0.4 0.6 0.3 0.5 0.2 0.4 0.1 0.3 0.2 0 1978 1982 1986 1990 1994 1998 1978 2002 Imports (m ) 1982 1986 1990 1994 1998 2002 Net trade (x - m ) 0.6 0.2 0.5 0.1 0.4 0 0.3 -0.1 -0.2 0.2 1978 1982 1986 1990 1994 1998 2002 1978 1982 1986 1990 1994 1998 2002 Figure 6.2 shows the ratios of the value of asset stocks relative to the value of (quarterly) private sector output. For net foreign assets and domestic capital, the steady state of the model using parameters for 2003 Q4 is relatively close to the recent data. But the steady-state ratio of nominal imported capital stock to nominal private sector output is somewhat lower than the historical data – mainly because the steady-state price of imported capital is assumed to be lower than the data as shown in Figure 6.3. Figure 6.3 presents charts of (past) actual and (current) steady-state relative prices, defined as the relevant deflator divided by the consumption deflator (excluding rents). The data show persistent trends in relative prices over the past, though in general the steady-state relative prices are close to recent outturns. 106 Parameterisation and evaluation Figure 6.2: Ratios of stock values to private sector output Stock of domestic capital (kh ) Stock of imported capital (km ) 3 11 10 2 9 8 1 steady state 7 6 1978 1982 1986 1990 1994 1998 2002 0 1978 Net foreign assets (nfa ) 1982 1986 1990 1994 1998 2002 Inventory stocks (s ) 2 1.2 1 1 0 0.8 -1 -2 1978 1982 1986 1990 1994 1998 2002 0.6 1978 1982 1986 1990 1994 1998 2002 Figure 6.3: Relative prices Domestic non-durable consumption (pch ) Imported non-durable consumption (pcm ) steady state 2.0 2.0 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 1978 1982 1986 1990 1994 1998 0.6 1978 2002 Domestic capital (pkh ) 1982 1986 1990 1994 1998 2002 Imported capital (pkm ) 2.0 2.0 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 1978 1982 1986 1990 1994 1998 2002 0.6 1978 Exports (px ) 1982 1986 1990 1994 1998 2002 Imported intermediates (pmin ) 2.0 2.0 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 1978 1982 1986 1990 1994 1998 2002 0.6 1978 1982 1986 1990 1994 1998 2002 107 The Bank of England Quarterly Model 6.3.3 Parameterising the core dynamics As noted above, some parameters affect both the steady state and the dynamic properties of the model. For example, depreciation rates affect steady-state stock ratios as well as the amplitude and rate of decay of the investment boost that follows a permanent increase in the desired level of capital. We also have a set of parameters in the core model that affect only the dynamic behaviour of the model. These include: consumers’ habits; capital and investment adjustment costs; inventory stock adjustment costs; the elasticity of capital utilisation; nominal price and wage stickiness; and nominal wage and price indexation parameters. These parameters can be set to generate a model with a fully flexible equilibrium in which quantities and prices are free to adjust immediately in the face of a shock. As with the calibration of the steady state, our approach to setting of these parameters has been largely informal. We have used three main exercises to judge how to set them: first, historical fit against the data; second, responses to marginal shocks (such as those shown in Chapter 7); and third, the estimation results from the non-core equations, which can be viewed as a statistical assessment of the degree to which the core model can capture the properties of the historical data. The process for setting the parameters was iterative. The starting values represented our priors with which we evaluated the model using the three criteria (fit, shock responses and non-core results) before adjusting the parameter values in the light of the results. Clearly this process is very informal, since ‘trial and error’ adjustment of parameters does not guarantee values that best meet our criteria. Moreover, while each criterion can be summarised by some kind of formal metric, (39) the weights applied to each are essentially judgemental. The starting values (or priors) come from two main sources. First, some parameters have plausible values when interpreted literally. For example, if wage stickiness is implemented using Calvo contracts, the adjustment probability determines the expected duration of contracts and a plausible length for the average wage contract implies a specific value of the adjustment probability. Second, we can use estimation results from previous studies. There are potential problems with both of these approaches. Literal interpretation of parameters may not be appropriate, as the core theory is a highly stylised representation of reality. And the direct relevance of existing estimates is often unclear: the precise data sources may not be directly comparable; existing estimates are usually conditional on the assumed structure of a model, which may differ from BEQM; and not all empirical evidence is based on UK data, and parameters may vary across countries. For example, parameters that relate to the costs of adjusting prices could take guidance from estimates of New Keynesian Phillips curves (relating inflation to expected future inflation and real marginal costs). But the results of estimation using, say, CPI inflation may not be of direct use in choosing parameters that affect price setting at a more disaggregated level, as in BEQM. Similarly, some estimated parameters may not be of direct use if based on a different assumption about the production technology to that employed in BEQM. This problem applies even more forcefully to microeconomic studies, which often allow for important roles for factors that are ignored or highly simplified in macroeconomic models (for example heterogeneity and uncertainty). (40) (39) See the discussion in Section 6.3.4. When matching impulse responses, the metric could (in principle) be in terms of squared deviations from specific shock responses. When fitting the data, we could (in principle) use the likelihood function as the metric. (40) Such difficulties are summarised succinctly on page 546 of Browning et al (1999): ‘A parameter that is valid for a model in one economic environment cannot be uncritically applied to a model embedded in a different economic environment’. 108 Parameterisation and evaluation Before discussing the parameter choices in detail, a summary of some of the main conclusions is as follows: • habits and capital adjustment costs are needed to generate inertia in real variables; • nominal wage stickiness and substantial nominal price stickiness are needed to ensure that supply responds elastically to movements in demand; (41) • we need to allow lagged nominal price and wage inflation to influence price and wage setting, so that inflation builds up gradually, rather than jumping immediately after a shock; • the strength of the response of inflation relative to output depends on the slope of the short-run marginal cost curve. Variable capital utilisation is also needed to ensure that supply responds elastically to movements in demand. Otherwise, even with substantial nominal wage and price stickiness, the incentive for firms to change prices rather than quantities is large; and • substantial import price stickiness is required to ensure that exchange rate changes are passed through gradually into consumer price inflation. We also assume a degree of dependence on past import price inflation, which slows down the rate of pass-through. The discussion below is ordered according to generic groups of rigidities affecting demand, supply and nominal prices and wages. A full listing of parameter values can be found in Appendix D. We consider evidence using data from different countries, different samples and employing different modelling assumptions. As noted above, these issues make it difficult to assess the relevance to parameter choices for BEQM, so we do not comment on the implications of all discrepancies between our parameter choices and the range of available empirical estimates. First we look at parameters influencing demand behaviour. The value set for the elasticity of intertemporal substitution for consumption in utility (42) is relatively low (σ c = 0.2) compared with the logarithmic utility assumption (σ c = 1) favoured in many real business cycle calibrations. (43) A low elasticity of intertemporal substitution implies (other things being equal) that consumers are less willing to substitute consumption across time in response to a temporary change in real interest rates. The range of empirical estimates is very wide for this parameter, coming from different countries, data sets and sample periods. (44) Indeed, Smets and Wouters (2004) obtain a estimate of around 0.6 using US data, compared with around 0.35 from Edge et al (2003). The single equation estimate of Nelson and Nikolov (2002) based on UK data suggests a value of at least 0.34, whereas the full model estimate of Bergin (2003) suggests a value very close to zero. Our assumptions about habit formation imply that an individual’s current utility places some weight on past consumption: ψ hab = ψ habd = 0.7. (45) Habit formation means that consumers gain utility from keeping consumption close to previous levels, which can introduce inertia into the response of consumption to temporary shocks. Studies using slightly different formulations of the utility function (41) Some studies, for example Christiano, Eichenbaum and Evans (2001), find that nominal wage stickiness alone is sufficient. These authors note, however, that their result is conditional on the assumption of low marginal costs of changing capital utilisation. Our finding, that nominal price stickiness is also needed, is more in accord with Smets and Wouters (2003a). (42) Note that this parameter does not only affect the dynamic responses of the core model – it will also affect the steady state through its influence on the marginal propensity to consume. (43) The imposition of a balanced growth steady state with endogenous labour supply may restrict the class of preferences that can be used (an example is to assume logarithmic utility in consumption). We are not constrained in our choice because of the way in which we implement participation decisions. (44) See Cromb and Fernandez-Corugedo (2004). (45) When ψ hab = 0, an individual’s current utility does not depend on the habit variable (in BEQM a function of past per capita consumption). When ψ hab = 1, utility in BEQM is defined in terms of the ratio of individual consumption to past (per capita) consumption. Our parameterisation is therefore towards the latter specification of utility. See Box 2 on page 30 for the precise specification of the utility function. 109 The Bank of England Quarterly Model have often found a role for habit formation. (46) For example, Smets and Wouters (2003a) find a value close to 0.6 for euro-area data; Smets and Wouters (2004) find a value slightly higher than 0.6 for US data; and Juillard et al (2004) estimate a value around 0.8 for US data. Banerjee and Batini (2003) find estimates of around 0.8 using UK data, with a similar utility function but a slightly different specification for the habit variable. Turning to supply, our parameterisation imposes significant costs of adjusting capital, which reduce the firm’s incentive to change the capital stock in response to changes in the cost of capital. The particular values of the adjustment cost parameters for home and imported capital (χ kh and χ km ) are not particularly informative, since they are conditional on the precise specification of the costs. To aid comparisons across models, such parameters are often converted into a more useful metric – for example, the proportion of any immediate increase in investment that must be paid in adjustment costs, starting from a steady state. Our parameterisation implies that increasing home capital investment by 1% above the steady-state rate leads to a cost of around 0.005% of steady-state home investment. (47) For imported capital, the cost is around 0.003% of imported investment. Estimates from other studies suggest costs of 0.009% (Edge et al (2003)) and 0.018% (Christiano et al (2001)) based on US data and 0.034% for euro-area data (Smets and Wouters (2003a)). (48) Using a capital adjustment cost formulation similar to the one employed in BEQM, Bergin (2003) estimates a cost of around 0.22% on UK data. The parameterisation for costs of adjusting dwellings investment implies that increasing dwellings investment by 1% from steady state incurs a cost of 0.05% of steady state dwellings investment. (49) The adjustment cost specification in BEQM also allows capital adjustment costs to depend partly on lagged changes in the capital stock (with a weight of kh = km = 0.7) and partly on trend growth (with a weight of 0.3). The above studies using US and euro-area data use an adjustment cost specification better approximated by kh = km = 1. Variable intensity of capital utilisation is an important feature of the core theory, which effectively ‘flattens’ the real marginal cost curve so that meeting changes in demand is possible without requiring large changes in marginal costs that would other things being equal, imply large price changes. (50) Most of the comparator studies do not employ the same type of adjustment cost function for changes in utilisation that we use in BEQM. This makes comparisons less straightforward, but a qualitative feature of the studies is that they tend to find that the elasticity of the cost with respect to utilisation is relatively low (generally much lower than that implied by a quadratic cost function). The BEQM core model assumes that effective depreciation rates for home and imported capital are adjusted by a correction for utilisation (z) with an elasticity φ z = 0.1. Estimates for this parameter by Basu and Kimball (1997), (46) Generally the studies quoted here assume that utility depends on the difference between individual consumption and the habit variable, rather than the ratio as we assume in BEQM. However, the Euler equations implied by these two specifications are broadly similar (to a log-linear approximation as used in those studies) in terms of the coefficients on future and past consumption. (47) We can calculate this cost as follows. The form of the adjustment cost implies that, close to the steady state, the cost as a K = χ K proportion of steady-state investment is χ2 KK δ −1 , where the equality follows from the fact that I K 2 steady-state investment covers capital depreciation. If the rate of investment increases by 1% above the steady-state rate then the capital stock increases by K = 0.01δK . This implies that the cost as a proportion of steady-state investment is χ 2 2 [0.01] δ. (48) These figures are based on the analysis of Edge et al (2003) who report an investment adjustment cost function that 2 specifies net (of adjustment cost) investment as exp − χ2 2 I I 2 I which means that the cost as a proportion of investment is: 1 − exp − χ2 0.012 . They report their estimate of χ and compare this directly with the parameter estimates in other studies. (49) The costs in this case take the form χ2 ratio of investment, is χ2 [0.01]2 . (50) See Christiano et al (2001). 110 I D 2 I D so that the cost of increasing investment by 1% above steady state, as a ID Parameterisation and evaluation based on US data, imply a very large confidence interval centred on a value of around unity. (51) Baxter and Farr (2001) find that a calibrated value of this elasticity of 0.1 improves the ability of an international real business cycle model to match certain properties of the data compared with the benchmark calibration of unity. Our parameterisation implies that labour adjustment costs are absent: χ l = 0. We assume that average hours worked are exogenous to the firm’s decision process and cannot respond to shocks. The other component of total hours worked is employment in heads – making this slow to adjust (χ l > 0) would steepen the real marginal cost schedule, which would tend to offset the beneficial effects of variable capital utilisation described above. The responsiveness of participation to changes in the wage rate that can be expected to be earned from labour market activity is relatively low: the elasticity of participation with respect to the real wage is ηl = 0.1. The estimation results on UK data by Bergin (2003) suggest an elasticity with respect to the wage rate very close to zero. Other estimates suggest higher values: Smets and Wouters (2004) estimate an elasticity around 0.5 for US data and Smets and Wouters (2003a) obtain an estimate close to 0.4 for the euro area. We implement nominal rigidities in two ways: as discussed in Chapter 3, we use Rotemberg (1982) adjustment costs for prices set by domestic firms and a Calvo (1983) specification for nominal wages and import prices. To make the comparison more straightforward we convert our Rotemberg adjustment cost parameter settings into the equivalent Calvo adjustment probabilities. (52) Our parameterisation implies that the Calvo probability of adjusting prices for domestically produced consumption, capital and government procurement is around 0.15. The associated probability for export prices is around 0.4. This reflects the fact that we assume price stickiness in the foreign currency price of exports, which is a relatively volatile data series. We assume that dwellings prices are perfectly flexible (χ d = 0). The adjustment probabilities for import (consumption, investment and intermediates) prices are close to those for domestic prices at 0.15. Other studies estimate a variety of different adjustment probabilities for prices. For the United States, Christiano et al (2001) estimate a probability of 0.5; Edge et al (2003) find 0.33; (53) and Smets and Wouters (2004) estimate a value of 0.1. For the euro area, Smets and Wouters (2003a) estimate a value of 0.1. Also using euro area data, Adolfson et al (2004b) estimate probabilities of 0.12, 0.73, 0.44 and 0.34 for domestic prices, imported consumption prices, imported investment prices and export prices respectively. And Bergin (2003) estimates a probability of 0.06 using UK data. (54) As well as parameters governing the extent of price stickiness (such as χ pch ) the core model also includes parameters (such as pchdot ) that determine the extent to which price adjustment costs are influenced by lagged inflation. For domestic prices (home consumption, home capital and government procurement) and exports, we assume that adjustment costs place a 50% weight on lagged inflation ( pchdot = pkhdot = pgdot = pxdot = 0.5). For import prices, past inflation rates are assumed to play more of a role in determining adjustment costs: we set pcmdot = pkmdot = pmindot = 0.9. The specification of Edge et al (2003) allows for separate identification of a parameter measuring the dependence of adjustment costs on lagged inflation. Their results using US data suggest little role for (51) The 95% confidence interval they report is [-0.2,2]. (52) We do this by noting the well known property that, under the assumptions defining the firm’s maximisation problem, the log-linearised pricing equations have the same reduced form as a Calvo approach. The parameters linked by the equation η−1 (1−βζ )(1−ζ ) , where η is the demand elasticity, χ is the Rotemberg adjustment cost, β is the household discount factor χ = ζ and (1 − ζ ) is the Calvo adjustment probability. In our calculations we assume β = 0.99, which is the standard calibration in the papers we use to compare estimates. (53) Here we use the formula mentioned in the footnote above to convert the adjustment cost estimate into a Calvo probability. (54) Here we use the formula mentioned in the footnote above to convert the adjustment cost estimate into a Calvo probability. 111 The Bank of England Quarterly Model lagged inflation in price adjustment costs. The results of Smets and Wouters (2004) on US data suggest a weight (for domestic price setting) of around 0.47. (55) For the euro area, Smets and Wouters (2003a) also estimate a weight of 0.47 for domestic prices and Adolfson et al (2004b) estimate weights of 0.21 and 0.15 for domestic prices and export prices respectively. Nominal wage stickiness is implemented according to a Calvo scheme. The parameterisation sets the adjustment probability at 0.5. Using US data, Christiano et al (2001) estimate a probability of 0.3.and Smets and Wouters (2004) find an estimate of 0.2. For the euro area, Smets and Wouters (2003a) estimate a probability of 0.26 and Adolfson et al (2004b) find 0.30. The type of Calvo mechanism used in the core model allows the specification of an indexation factor, which specifies the extent to which unadjusted private sector nominal wage contracts are uprated in line with lagged average (private sector) wage inflation. Our parameterisation of wdot = 0.9 implies that a high weight is placed on lagged wage inflation. Smets and Wouters (2004) estimate a weight on lagged inflation of around 0.32 using US data. For the euro area, Smets and Wouters (2003a) estimate a value of 0.76 and Adolfson et al (2004b) estimate 0.51. 6.3.4 More formal approaches The procedure described above is by no means the only possible approach. There is now a substantial literature that attempts to formalise the traditional approaches to parameterisation. For example, the exercise in Box 13 on page 103 could be extended to estimate sampling uncertainty, which is lacking in informal calibration exercises. (56) As several authors have noted, calibration amounts to imposing moment conditions and, in theory, can be formalised in a way analogous to GMM estimators. Several papers have used the state-space form of a model to relate to VARs, (57) or to deduce key model properties. (58) There are problems in applying these approaches to a model as large as BEQM, but it is possible to work out ways by which a large system (the theoretical model) can be formally related to smaller systems, which can then be directly related to the data. (59) There has also been considerable progress in recent years in directly estimating the underlying parameters of micro-founded models by maximum likelihood methods. In theory, the state-space representation of the model can be used to provide prediction errors, and so a likelihood function can be evaluated. In practice, however, the likelihood surface is usually relatively flat and numerical search methods will not converge or will converge on local maxima. Bayesian priors can be used to assist estimation, (60) but Bayesian maximum likelihood has not yet been applied to systems are large as BEQM. (61) Nevertheless, the technology is developing rapidly and it may prove possible to use such techniques in future: they have been applied to increasingly large systems as developments have been made in numerical methods. (62) (55) This study uses a Calvo mechanism for price stickiness. However, the log-linearised pricing equations from both the Rotemberg and Calvo pricing mechanisms imply that Smets and Wouters’ parameter γ p can be interpreted analogously to the parameters reported in the main text. (56) See Canova (1994, 1995); for an application to a large model see Amano et al (2002). (57) See, for example, Ingram and Whiteman (1994). (58) See, for example, Watson (1993). (59) See Kapetanios et al (2004). (60) See, for example, Schorfheide (1999) and Fernández-Villaverde and Rubio-Ramírez (2004). (61) No use is made in the core model of so-called ‘extrinsic dynamics’ (ie persistent shock processes) to bolster the fit of the model. (62) See the pioneering work by Smets and Wouters (2003a). More recently, some relatively large systems have been estimated; see Adolfson et al (2004a). 112 Parameterisation and evaluation 6.4 Parameterising the non-core equations As described in Chapter 4, the non-core equations pick up correlations in the data that are not explained by the theory set out in Chapter 3. In effect, we can think of these equations as modelling the gap between the core model and the historical data. 6.4.1 Estimating the non-core equations Generating core model paths As described in Chapter 5, we generate core model paths by running the core model as a full system. Simulating the core model over the past requires us to make two important choices: first, how to treat values of predetermined variables; and second, what information to assume was available to agents at each date in the past when generating the core solution. Given values for the predetermined variables, the core model generates paths for endogenous variables that converge on the steady state. The path of the predetermined variables forms part of this core model solution: for example, the path of the capital stock is given by accumulated investment flows. One option would be to give the core model the values of predetermined variables at the start of the historical sample and allow the solutions for these variables to be given by the relationships in the core model. So in this case, the core model solution for the capital stock would equal the initial value of the capital stock observed in the data at the start of sample, plus cumulated net investment from the core model. However, we choose to update the values of the predetermined variables from the data: core model solutions in period t depend on the data for the capital stock at period t − 1, as discussed in Section 5.1.1. This means that the core model solutions for the historical data period represent a sequence of one-period projections, conditional on the data up to the previous period. As such, the core model is allowed to react to news in both exogenous variables and changes in the predetermined endogenous variables. (63) The second choice concerns the amount of information about the future that we assume is available in each period of the historical sample. This is important because, in each period, the path of the core model depends on the expected future path of exogenous variables. One assumption would be to allow perfect foresight of exogenous variables over the historical sample: in each past period, we would allow agents in the core model to observe the actual path of exogenous variables in the full data set. This assumption is straightforward, but not realistic for many variables. Instead, we assume that expectations of future exogenous variables are generated by an ‘exogenous variables model’. As described in Box 10 on page 78, this approach assumes that agents in the core model decompose movements in exogenous variables into permanent and transitory components and forecast how the exogenous variable returns to its permanent level. In general, we use a low-frequency filter to identify the permanent component and then estimate a simple time-series process for the transitory component. However, judgement is also applied where appropriate. (64) (63) One way of thinking about this is that the core model is repeatedly ‘surprised’ when the data for predetermined variables turn out differently to what was decided upon last period. These surprises represent shocks to the model and in response agents formulate new optimal plans in each period. (64) For example, when tax changes are pre-announced, we may want to ensure that this news is contained in the permanent component of the tax rate from the announcement date. 113 The Bank of England Quarterly Model Equation form The equations we estimate takes the general form of equation (4.3) in Chapter 4, which we repeat here for convenience. core yt = − (1 − α 1 ) yt−1 − yt−1 + β 1 ytcore + ψ 1 z t + εt In general, the we found the β 1 coefficients to be insignificantly different from zero and so terms in y core do not appear in the estimated non-core equations presented in Section 6.4.2. Our specification has the advantage of smoothing through the initial responses of the core model. Even though the core model contains extensive real and nominal inertia, endogenous variables can still jump in response to disequilibrium between starting points and equilibrium targets. A primary role of the non-core equations is therefore to allow for an extra layer of persistence to forecasts of a sort that is, as yet, difficult to capture by micro-founded dynamic optimising models. Lags and proxy variables One way of approaching the estimation of non-core equations would be to try to specify the best single equation for each variable. We could imagine a process in which we began with a very general specification of each equation (with many lagged dependent variables and proxy variables) and tested down to a preferred equation. Such an approach is likely to lead to a set of non-core equations each containing a relatively large number of lagged dependent variables and proxy variables. Though individually these equations would have sound properties (for example, high measures of fit such as R 2 statistics) they may not perform at all well when combined as a system to produce forecasts. One reason is that the inclusion of a large number of proxy variables means that these too must be forecast and that can affect the overall forecasting performance of the system. Another reason is that the interaction of equations with large numbers of lagged dependent variables can lead to implausible system-wide properties. Section 2.2 discussed the trade-off between theoretical consistency and data coherence, illustrated in a stylised form in Figure 2.1 – our approach to estimating the non-core equations can be seen as reflecting our choice as to which part of the frontier we should aim for. In particular, we follow a strategy that is consistent with the remit of the model development project to improve theoretical coherence and fit the data at least as well as the previous MTMM model. We start with the core model and parsimoniously add non-core dynamics to improve the empirical properties of the full model. (65) At the limit, we could imagine a case in which the core theory was sufficiently rich that it did not require any additional non-core equations. In practice, we do find that lagged dependent variables and proxy variables are useful. Proxy variables are selected, based on a view of what is known to be missing in the core theory, the evidence gained from using the previous MTMM model in forecasting, and specialists’ conjunctural experiences. In general, we find that changes in aggregate activity variables (such as output or unemployment) are necessary to pick up accelerator-like effects. (66) Changes in nominal variables (such as interest rates and prices) are also useful; we cautiously interpret those as picking up credit-related effects. However, in terms of the simulation responses seen in Chapter 7, these effects are not very powerful. (65) An alternative, statistical, argument for parsimony is that while inclusion of a relevant variable may improve the accuracy of the forecast mean (and hence reduce bias) it will also increase parameter uncertainty (and so add to the forecast error variance). It is possible, therefore, that a misspecified but parsimonious model can out-forecast the data generating process. (66) We hesitate to interpret this literally as evidence for accelerator mechanisms, as they can easily arise in the reduced forms of neoclassical rational expectations models, especially when data are subject to mismeasurement: see, for example, Sargent (1989). 114 Parameterisation and evaluation A key implication of our approach is that we place more emphasis on getting the right system properties rather than on single equation results. This does not mean there is no value in single equation diagnostics, but rather that some are more useful than others. For example, tests of residual serial correlation are useful indications of whether our regressors adequately capture the persistence of the data. However, tests for the normality and heteroscedasticity of single equation residuals are less important to the extent that we do not use t-statistics as the sole criterion for including regressors (as might be the case when testing down from a general to specific equation). 6.4.2 The non-core equations To assess whether each core path can be treated as an attractor (in a statistical sense), we performed tests to determine whether the gap between the core path and the data can be treated as stationary. Tests for stationarity are well known to have low power in small samples (67) and Table 6.3 presents tests of the stationarity of the gaps between the core path and the data using a number of well-known tests. (68) Table 6.3: Statistical tests for stationarity of gaps Gap log ct − log ctcore log cm t − log cm core t log et − log etcore log avh t − log avh core t log idt − log idtcore log ikt − log iktcore log ikm t − log ikm core t log lt − log ltcore ṗt − ṗtcore log pxt − log pxtcore log wt − log wtcore log xt − log xtcore Philips-Perron -2.17 -3.43 -2.88 -2.63 -9.39 -4.18 -4.64 -2.89 -10.02 -3.14 -9.21 -3.54 (a) Test statistic ADF (4 lags) (b) -1.05 -3.73 -2.98 -3.61 -2.54 -2.08 -2.39 -3.68 -2.94 -3.40 -3.53 -2.69 KPSS (c) 1.03 0.12 0.26 0.18 0.98 0.22 0.42 0.91 0.32 0.34 0.21 0.73 (a) Sample period: 1978 Q1 to 2003 Q4. Critical values: 5% -1.944; 1% -2.588. (b) Sample period: 1979 Q1 to 2003 Q4. Critical values: 5% -2.89; 1% -3.50. (c) Sample period: 1977 Q4 to 2003 Q4. Critical values: 5% 0.46; 1% 0.74. The Phillips-Perron and Augmented Dickey-Fuller approaches assume a null hypothesis that the gap between the two series is non-stationary. Here, however, we are seeking a strong relationship between y and y core , so the appropriate null hypothesis is that the gap is stationary and we therefore also present results from the KPSS test. The results show that evidence of statistically significant non-stationarity in the gaps depends on the particular test used. For the Phillips-Perron test, there is no evidence of non-stationarity (failure to reject the null hypothesis of a unit root) at the 5% level, and in most cases at the 1% level too. But the (67) See, for example, pages 444-445 of Hamilton (1994) and references within for a discussion. (68) We implement the tests in Eviews 5.0. The Augmented Dickey-Fuller (ADF) test uses four lags and for the Phillips-Perron test, we use a Bartlett kernel with four lags. See Chapter 20 of Davidson and MacKinnon (1993) for a description of the ADF and Phillips-Perron tests and Kwiatkowski et al (1992) for the KPSS test. 115 The Bank of England Quarterly Model Augmented Dickey-Fuller (ADF) test shows some evidence of non-stationarity (failure to reject the null hypothesis of a unit root) for consumption, dwellings investment, total and imported capital investment and exports. And the KPSS test suggests evidence of non-stationarity (here, by rejection of null hypothesis of stationarity) for consumption, dwellings investment, participation and exports. Consumption The non-core equation contains an error correction to non-durable consumption. An extra argument in assets is included to ensure convergence in net foreign assets (see Section 4.2.2). Proxy variables for effects missing from the core include changes in the value of the housing stock, as a proxy for housing collateral effects; changes in household income, as a proxy for the existence of rule-of-thumb individuals; changes in the employment rate, as a proxy for confidence and uncertainty effects; and changes in nominal interest rates, as a proxy for credit and cash-flow effects. log ct = log eaggt − 0.197 (0.699) 1.125 (2.203) rgt + 0.221 (2.316) log lyt + 0.194 (3.872) log( phset dt−1 ) core core ) + 0.006 (log at−1 − log at−1 ) − 0.125 (log ct−1 − log ct−1 (1.509) (3.077) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period = = : : 0.344 0.010 F-statistic = 2.778 [p-value = 0.031] 1978 Q2 – 2003 Q4 Imported consumption This equation includes an error correction term in the ratio of imported to total (non-durable) consumption. Extra variables include changes in total consumption and in the relative price of imported and home-produced consumption goods. These capture short-run income and substitution effects on the demand for imported consumption. log cm t = 1.134 (5.926) log ct + 0.221 (1.170) log ct−1 − 0.187 (1.652) core − 0.080 log (cm t−1 /ct−1 ) − log cm core t−1 /ct−1 (2.761) Adjusted R 2 = 0.227 s.e. of equation = 0.024 LM test for serial correlation : F-statistic = 1.304 [0.274] Estimation period: : 1978 Q2 – 2003 Q4 116 log ( pcm t / pch t ) Parameterisation and evaluation Dwellings investment The non-core equation contains an error correction term for dwellings investment. Extra variables include consumption growth and changes in nominal interest rates (to proxy the role of cash-flow effects on short-term housing demand). log idt = − 6.859 (2.212) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: rgt−1 + = = : : 1.667 (3.077) log ct−2 − 0.153 (1.844) core log idt−1 − log idt−1 0.132 0.069 F-statistic = 4.271 [0.003] 1978 Q3 – 2003 Q4 Total capital investment This equation includes error correction to total capital investment. Extra variables include lagged changes in total investment; a steady-state capital gap term; and an accelerator term in the change in output growth. Lagged investment is included to capture long decision lags that are not fully captured by adjustment costs in the core model. The steady-state capital gap term implies that investment is boosted when there is an increase in the long-run desired capital stock log ikt = log yt−1 + 0.346 (1.355) 0.067 (0.652) log ikt−1 − 0.255 (2.553) ss log kt−1 − log kt−1 core − 0.107 log ikt−1 − log ikt−1 (2.150) Adjusted R 2 = 0.077 s.e. of equation = 0.032 LM test for serial correlation : F-statistic = 1.475 [0.216] Estimation period : 1978 Q3 – 2003 Q4 Imported capital investment The non-core equation contains an error correction term in imported capital investment. Extra variables are lagged changes in imported capital investment, and changes in the relative price of imported and domestically produced capital goods. The lags in imported capital investment capture the sluggishness in the investment data; the relative price term captures substitution effects between domestic and imported capital goods. log ikm t = 0.242 (2.554) log ikm t−1 − 0.526 (4.493) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : log pkm t − 0.056 pkh t (1.893) log ikm t−1 − log ikm core t−1 0.123 0.049 F-statistic = 3.267 [0.0148] 1978 Q2 – 2003 Q4 117 The Bank of England Quarterly Model Exports The non-core equation specifies an error correction term in exports and also includes changes in world demand. log xt = 0.730 (4.89) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : log c f t − core log xt−1 − log xt−1 0.172 (3.176) 0.23 0.024 F-statistic = 5.255 [0.001] 1978 Q2 – 2003 Q4 Private sector real wages This equation includes an error correction term in private sector real wages. Extra variables include lagged real wage growth and changes in RPI inflation, which are likely to influence wage setting; terms in RPI and the change in the consumption deflator, reflecting the wedge between the two measures (in effect the equation is estimated on nominal wages deflated by RPI); and the gap between the unemployment rate and steady-state unemployment, to capture cyclical influences on wage setting. log wt + ṗt − r pidotsat Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : = 0.508 ( log wt−1 + ṗt−1 − r pidotsat−1 ) (3.248) − 0.488 (2.008) r pidotsat − − 0.519 (2.317) r pidotsat−2 − 0.599 (2.835) 0.120 (1.576) r pidotsat−1 u t − u ss t core − 0.420 log wt−1 − log wt−1 (3.761) 0.192 0.008 F-statistic = 0.249 [0.909] 1989 Q1 – 2003 Q4 Public sector real wages The non-core equation includes an error correction term in private sector real wages, adjusted for the wedge between public and private sector wages. An additional term in lagged public sector real wage growth captures sluggishness in real wage adjustment. log wgt = 0.348 (1.423) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: 118 = = : : log wgt−1 − 0.166 (2.541) log wgt−1 − log µwg wt−1 0.196 0.015 F-statistic = 1.519 [0.210] 1989 Q1 – 2003 Q4 Parameterisation and evaluation Private sector employment The non-core equation specifies an error correction to private sector employment. Extra variables include lagged changes in private sector employment and changes in private sector output. These terms proxy sluggish employment adjustment and labour demand effects respectively. A lagged term in factor utilisation is also included, as increases in factor utilisation tend to raise subsequent employment growth. log et = log et−1 + 0.094 (3.570) 0.649 (11.705) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : f u t−1 − 0.031 100 (3.143) log yt + 0.053 (2.983) core log et−1 − log et−1 0.824 0.002 F-statistic = 0.685 [0.604] 1978 Q2 – 2003 Q4 Average hours This equation includes an error correction term for average hours. Additional dynamic terms include lagged growth in average hours and private sector output – reflecting sluggish adjustment and labour demand effects respectively. log avh t = log avh t−1 + 0.560 (7.17) − 0.048 (2.345) Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : 0.037 (1.778) log yt + 0.055 (2.655) log yt−1 log avh t−1 − log avh core t−1 0.426 0.002 F-statistic = 1.554 [0.193] 1978 Q2 – 2003 Q4 Participation The non-core equation includes an error correction term in participation, together with lagged participation growth (reflecting sluggish labour market adjustment) and changes in average real wages (as a proxy for the return to entry into the labour market). log lt = 0.499 (5.456) log lt−1 + 0.249 (2.658) log lt−2 + 0.044 (2.202) log (wt−2 et−2 + wgt−2 egt−2 ) eaggt−2 core − 0.099 log lt−1 − log lt−1 (4.037) Adjusted R 2 = 0.529 s.e. of equation = 0.002 LM test for serial correlation : F-statistic = 1.042 [0.390] Estimation period: : 1978 Q3 – 2003 Q4 119 The Bank of England Quarterly Model (Non-housing) consumption deflator inflation The non-core equation specifies error correction to the inflation rate. Lagged inflation changes reflect sluggish inflation adjustment; there is an activity effect from the deviation from the steady state in aggregate employment; a term in private sector factor utilisation captures cyclical variations in margins and short-run marginal costs not captured by the core theory; and there is an additional effect from imported intermediate price inflation. ṗt = 0.138 ( f u t−1 /100) − 0.421 (3.773) (3.515) ṗt−1 − 0.350 (3.950) ṗt−2 + ss ss − 0.332 + 0.072 log eaggt−1 − log et−1 + egt−1 (2.811) (2.475) 2 Adjusted R = 0.469 s.e. of equation = 0.007 LM test for serial correlation : F-statistic = 1.501 [0.208] Estimation period: : 1978 Q3 – 2003 Q4 0.047 (2.453) min ṗt−1 core ṗt−1 − ṗt−1 Export prices This equation specifies sluggish error correction. log pxt = 0.373 (3.716) log pxt−1 + Adjusted R 2 s.e. of equation LM test for serial correlation Estimation period: = = : : 0.119 (1.216) log pxt−3 − 0.046 (1.394) core log pxt−1 − log pxt−1 0.064 0.015 F-statistic = 0.634 [0.640] 1978 Q4 – 2003 Q4 House prices The non-core equation for the house price index features error correction to the housing investment deflator (with an estimated adjustment for the differences in the levels of these series). Additional variables include lagged house price and interest rate changes, the latter proxying for credit effects. log phset = 0.611 (6.469) log phset−1 + 0.192 (1.739) log phset−2 + 0.071 (0.723) log phset−3 − 1.452 rgt−2 − 0.019 (log phset−1 − log pdv t−1 + 2.029 ) (2.054) (2.125) (25.915) 2 Adjusted R = 0.66 s.e. of equation = 0.014 LM test for serial correlation : F-statistic = 0.130 [0.971] Estimation period: : 1978 Q4 – 2003 Q4 120 Parameterisation and evaluation 6.5 An evaluation of the model’s forecast performance As discussed in Section 6.4.1, the simultaneity of the model means that its empirical fit should be evaluated as a system, rather than on a single-equation basis. In particular, a collection of equations that appear to fit well for known values of the right-hand-side variables may not perform well when combined to produce out-of-sample forecasts. Given that we wanted BEQM to fit the data at least as well as the previous model, a natural test is to compare the properties of forecasts from BEQM and from the MTMM We place less emphasis, therefore, on single equation properties than on how the equations work together as a system. The alternative approach of an equation-by-equation comparison of the non-core equations with corresponding MTMM equations would be of limited use in any case, because the equations are not estimated conditional on the same information. Method We evaluate the forecasting performances of the MTMM and BEQM by running the two models as systems over history and measuring the prediction errors for a set of key endogenous variables. In doing so, we are in effect testing the performance of the models under something approaching actual forecast conditions. The main – and important – difference between what has been done here and actual forecasts is the application of judgement: the projections in this exercise have no judgemental adjustment, other than the routine residual adjustments required to enforce identities in the MTMM. Hence, the size of the forecast errors is likely to be substantially bigger than actual forecast errors. We started the assessment by forecasting with each model at 1992 Q2, and recording the full paths for the model’s projections of key endogenous variables over 13 quarters. We then moved one quarter ahead to 1992 Q3 and repeated the exercise, allowing the models to ‘see’ actual outturns at 1992 Q2, rather than starting the new projection with the previous projection’s forecast values. This prevented errors in early projections from building up over time. We then repeated for each subsequent period to produce a sequence of forecasts for each forecast horizon. Once the series of in-sample forecasts was finished, we analysed the prediction errors of the two models. Some variables do not have well defined behavioural equations in the MTMM, including: nominal interest rates, the nominal exchange rate, nominal government spending and the value of equities. To ensure comparability, we conditioned the MTMM projections on the paths for these variables generated by BEQM. The model versions The exercise described above used the versions of the model that were current in the first half of 2003. This was to ensure a fair comparison, because the MTMM was not fully rebuilt and re-estimated to take account of the shift to chain-weighted data in October 2003, so it would not be expected to perform as well against more recent data. The in-sample period therefore runs from 1992 Q2 to 1999 Q3. It is constrained at the beginning by the number of lags in the MTMM and the lack of comparable data for earlier periods, and at the end because we assessed forecasts up to 13 periods ahead. 121 The Bank of England Quarterly Model Outputs We focus on a subset of key variables: inflation (defined as the four-quarter growth rate of the RPIX index), (69) the level of unemployment (the rate as defined in the Labour Force Survey) and the four-quarter growth rates of GDP (at market prices), consumption, business investment, exports, imports, unit labour costs, and import prices (the import price deflator). Prediction errors for inflation and unemployment rates are the percentage point difference of the actual and projected rates in each individual forecast; other prediction errors are measured as percentage point differences of actual and projected four-quarter growth rates. From these errors, we can calculate a variety of statistical measures, such as Root Mean Squared Errors, Mean Absolute Errors and mean errors. Information on the nature of the errors can be gained by decomposing squared prediction errors into bias, variance and covariance contributions. (70) We used Theil’s inequality coefficient to describe how good the forecasts are on an absolute scale. (71) And we used the Diebold-Mariano (1995) statistic (72) to test whether forecasts were significantly different from each other. Results The bar charts in Figure 6.4 summarise the relative forecasting performance of the two models over history. For each variable at a specified forecast horizon, the bar shows the root mean squared forecast error from the MTMM less that from BEQM. A result of zero therefore indicates that the models have identical forecasting performance. The scale is in percentage point differences, so that a result of +1% means that the MTMM had a 1 percentage point higher root mean squared error than BEQM in forecasting that variable at that horizon. The results were mixed across variables and forecast horizons. But in general, BEQM tended to have a slightly lower forecast error at the two- and three-year horizon. (73) Nevertheless, these differences were not great. Formal tests of whether the forecast errors are significantly different from each other suggest that the difference was not statistically significant for the majority of the variables at the five- and nine-quarter horizons. BEQM was significantly better for unemployment at both horizons, while worse for GDP growth at the nine-quarter horizon; for the remainder, there was no statistically significant difference. Overall, these results suggest that BEQM has a ‘hands free’ forecasting performance that is at least as good as that of the MTMM. (74) (69) The inflation target in the first half of 2003, when the exercise reported here was undertaken, was still set in terms of RPIX. (70) The mean of squared prediction errors can be decomposed into bias, variance and covariance: 2 1 T 2 2 yts − yta = yts − yta + σ s − σ a + 2 (1 − ρ) σ s σ a T t=1 where yts , yta , σ s , and σ a are the means and standard deviations of the series y s (simulated or predicted series) and y a (actual data) respectively, and ρ is their correlation coefficient. (71) See pages 30-37 of Theil (1961). The statistic computed here is Theil’s ‘U1’ measure. Comparisons using this statistic contain more information than comparisons of root mean squared errors, since the U1 measure also accounts for differences in the variability of predicted outcomes. (72) This procedure is designed to test the null of equal predictive ability between two models by considering the mean of the differences of squared prediction errors of the two competing models. We used a small-sample correction of this test statistic, proposed by Harvey, Leybourne and Newbold (1997). (73) A similar picture also holds if we consider the mean absolute and simple mean errors, not shown here. (74) This was an important part of the remit for developing BEQM as discussed in Section 2.1. 122 Parameterisation and evaluation Figure 6.4: Comparison of growth rate forecasts from BEQM and the MTMM 3 -1 -2 -2 -3 -3 pr ic es C R et ai l Im po rts Im po U r tp ne ric m pl es oy m en t( ra U ni te tl ) ab ou rc os ts 0 -1 Ex po rts 1 0 G D P on s Bu um si pt ne io ss n in ve st m en t 2 13 quarters ahead 3 -2 -2 -3 -3 Bu si ne ss Ex po r G D C on su m Im po rts Im po U r tp ne ric m pl es oy m en t( ra U ni te tl ) ab ou rc os ts -1 ts 0 -1 pt io n in ve st m en t 1 0 P 2 1 pr ic es 2 R et ai l Im po rts Im po U r tp ne ric m pl es oy m en t( ra U ni te tl ) ab ou rc os ts ts Ex po r G D P C on s Bu um si pt ne io ss n in ve st m en t pr ic es 3 1 3 9 quarters ahead R et ai l 5 quarters ahead 2 Im po rts Im po U r t ne pr m ic pl es oy m en t( ra U ni te tl ) ab ou rc os ts Ex po rts G D P C on s Bu um si pt ne io ss n in ve st m en t R et ai l pr ic es 1 quarter ahead However, these comparisons do not tell us anything about whether the forecasts are individually ‘good’ or ‘bad’ in absolute terms. It is quite possible that, using statistical measures of forecast performance, relatively simple time-series models would be able to out-perform BEQM (without judgement). Indeed, using Diebold-Mariano tests of predictive ability, we were generally unable to reject the null hypothesis that the RMSEs from five- and nine-quarter ahead forecasts using BEQM were no different to those from a simple ‘no change’ model that predicted each future growth rate over the forecast horizon to be equal to the most recent observation. (75) In light of this evidence, it seems likely that we could specify atheoretical statistical models that would outperform BEQM using this metric. However, as set out in Chapter 2, the remit for the development of BEQM stressed the importance of increasing the theoretical consistency of the new model, as well as maintaining the ability to fit the data at least as well as the previous model. In particular, Chapter 2 discusses the trade-off between theoretical consistency and data coherence. An atheoretical statistical model might be able to outperform BEQM on statistical tests of ‘hands free’ forecasting performance, but could not be used to analyse economic issues or to distinguish between alternative explanations for the observed behaviour of the economy. In terms of the stylised frontier shown in Figure 2.1, the remit for BEQM rules out a position close to either end of the frontier. (75) The exception was for nine quarter ahead forecasts of business investment, where there was evidence that the BEQM RMSEs were significantly smaller. 123 The Bank of England Quarterly Model Another comparison between BEQM and the MTMM is shown in Figure 6.5, which depicts Theil inequality coefficients at the nine-quarter horizon. These are scaled to lie between 0 and 1: 0 would be a perfect fit and 1 would be the worst possible forecast. (76) To see what drives these errors, we can break the Theil statistics down into the proportional contributions made by ‘forecast bias’, ‘forecast variance’ and ‘forecast covariance’. This tells us whether the forecast is consistently biased; unbiased but not getting the variability of actual outturns right; or that errors represent the idiosyncratic or erratic component of the forecast error (in the sense that it accounts for the remaining error over and above the systematic components). A striking feature of these error decompositions is that, without any judgemental adjustment, there is a contribution from the bias component in the MTMM forecasts for inflation. That is not to say that the MPC’s projections have been biased, but that the MTMM required the consistent application of judgement to avoid bias. Figure 6.5: Theil inequality coefficients Inflation 1.0 Covariance Variance Bias GDP 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.2 0.2 0.4 0.2 0.0 MTMM BEQM Investment MTMM 1.0 MTMM MTMM BEQM Exports 0.0 MTMM 1.0 BEQM Imports 1.0 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 MTMM 1.0 BEQM 0.0 0.8 BEQM Import Prices 1.0 Consumption BEQM Unemployment 0.0 MTMM 1.0 BEQM Unit Labour Costs 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 MTMM BEQM 0.0 MTMM BEQM (76) The statistic equals zero for a model with a root mean squared error of zero. The statistic equals unity when the predictions from the model are perfectly negatively correlated with the actual data. 124 Parameterisation and evaluation We can also assess the relative contribution made by the core structural model to BEQM’s forecasts, by repeating the evaluation exercise using only the core model. The results in Figure 6.6 indicate that the non-core equations improve the short-run dynamics of the model, especially for capital investment and exports. For exports, this is not surprising, since there are no significant frictions influencing export demand determination in the core theory. For capital investment, this indicates that additional dynamic terms are useful, despite the presence of capital adjustment costs in the core model. However, the relative role of the core model increases over longer horizons and the relative contribution of the non-core model diminishes. Figure 6.6: Comparison of growth rate forecasts from the BEQM core and the MTMM 3 -1 -2 -2 -3 -3 -4 -4 -5 -5 Ex po r pr ic es R et ai l ts 0 -1 Im po rts Im po U rt ne pr m ic pl es oy m en t( ra U ni te tl ) ab ou rc os ts 1 0 G D P C on s Bu um si pt ne io ss n in ve st m en t 2 13 quarters ahead 3 -1 -2 -2 -3 -3 -4 -4 -5 -5 C Im po r et ai l ts Im po U rt ne pr m ic pl es oy m en t( ra U ni te tl ) ab ou rc os ts 0 -1 Ex po rts 1 0 G D P on s Bu um si pt ne io ss n in ve st m en t 2 1 pr ic es 2 R U tp ne ric m pl es oy m en t( ra U ni te tl ) ab ou rc os ts Im po rts Im po r Ex po rts G D P C on s Bu um si pt ne io ss n in ve st m en t pr ic es 3 1 3 9 quarters ahead R et ai l 5 quarters ahead 2 Im po rts Im po U r t ne pr m ic pl es oy m en t( ra U ni te tl ) ab ou rc os ts Bu si ne ss Ex po rts pt io n in ve st m en t P G D C on su m R et ai l pr ic es 1 quarter ahead Finally, we repeated the evaluation using the latest vintages BEQM and the data. There have been significant changes to the data since mid-2003, most notably the revisions implied by the move to annual chain-linking in the 2003 Blue Book. And the core model parameters were adjusted in response to these changes in the data. Therefore, comparisons with the results reported above will be affected by changes in both the model and the data. Nevertheless, the results suggest that changes in the data and model have not substantially affected the overall picture: the root square mean errors from the newer version of BEQM are generally lower than the MTMM and, as in the results above, the core model performs well at longer horizons. 125 The Bank of England Quarterly Model 6.6 Summary This chapter describes our approach to parameterising and evaluating BEQM against the data. A first step was to create a database with the required characteristics. Variables need to be detrended by the appropriate growth rate so that the model can be written in stationary form. And some modification of variables is necessary to make a better match with the underlying economic concepts. The combination of a large, simultaneous model with the assumption of model-consistent expectations raises difficult issues for parameterisation. These make conventional systems estimation procedures hard to apply, given the relatively short time series that were available. We therefore split the core model parameters into three groups according to whether they affect the steady state, the dynamics, or both. This allows us to calibrate the steady state first, using an approximately recursive approach and calibrating separate parts of the model as relatively self-contained blocks. In some cases we aim to match recent observations rather than full sample averages, where there are apparent trends or regime shift. We adopt a largely informal approach to parameterising the dynamics of the core model: we judge the fit against historical data; in terms of the responses to marginal shocks; and relative to the estimation results for non-core equations. We choose parameters that give desired properties, such as inertia in real variables, gradual pass-through from import price changes, and an appropriate mix of output and inflation responses. The non-core equations are estimated mainly with error correction to the relevant core model paths, which are derived using actual data for predetermined variables and assuming that expectations are generated using the ‘exogenous variables model’ discussed in Chapter 5. We present a number of evaluations of the model’s performance against the data. The design and structure of BEQM mean that it should be evaluated as a system, rather than on a single-equation basis. We compare prediction errors for a key set of endogenous variables from running comparable versions of BEQM and the MTMM over the past. In general, the results suggest that BEQM has slightly lower forecast errors at the two- and three-year horizons, but few of the differences are statistically significant. Further tests suggest that there is a greater contribution from bias to inflation forecasts in the MTMM than in BEQM; and that the non-core equations improve the short-term dynamics of BEQM, although less so at longer horizons. Finally, we repeat the evaluation using the latest versions of both BEQM and the data. This suggests that recent changes to the model and the data have not substantially altered the overall performance of BEQM relative to the MTMM. 126 Chapter 7 Model properties This chapter presents the results of a series of model simulations to illustrate the properties of BEQM. Section 7.1 sets out some introductory remarks on the nature of the simulations and interpretation of the results, which are described in Section 7.2. Finally, Section 7.3 summarises. 7.1 Interpreting the responses The shock responses shown in Section 7.2 are intended to highlight specific mechanisms of the model, and so we focus on a few stylised experiments. Several points are worth noting about the nature of these results. First, we assume that the economy is at its long-run equilibrium when the shock hits. (1) This is a different type of exercise from forecasts, which can be thought of as showing how an economy that is away from a sustainable long-run equilibrium could move towards such a position. Second, we generally assume that the shocks are unanticipated, but once they occur they are recognised immediately and fully understood. In other words, there is no assumption of partial information or learning (discussed in Chapter 5), apart from in the variant of the interest rate shock (shown in Figure 7.2) that we use to illustrate the effect of changing assumptions about agents’ expectations. Third, each shock represents an isolated change to a single exogenous variable, which helps clarity but is a simplification compared with most forecast issues or historical episodes, which often involve simultaneous shocks to a number of exogenous variables. Moreover, in the simulations presented here, the rest of the world is assumed not to react in any way. For example, we shock (exogenous) world prices and world demand (see Sections 7.2.4 and 7.2.5), but no other world variables are assumed to change in the experiments presented here. In many cases, we would expect other variables to change in a well articulated story about economic shocks, depending on the ultimate source of the shock. Fourth, policy assumptions are important. In the first shock – designed to illustrate the impact of interest rate changes – we shock monetary policy directly. But interest rates are an endogenous variable, typically moving in response to some change in the economy. This shock represents an erratic deviation from the normal policy reaction function, which is different from most policy changes. The other simulations are run under the assumption that policy reacts immediately: we do not condition on a particular path for short-term interest rates or fiscal variables. Differences in policy assumptions can have a significant impact on the responses, so it is important to interpret the reactions to shocks in the light of the policy reaction functions used in the simulations. Here, we assume that the monetary authority targets inflation using short-term nominal interest rates and a Taylor-type reaction function; (2) and a lump-sum household tax adjusts to ensure fiscal solvency, given policy targets for government expenditure, debt, transfers and other taxes. (1) The model’s decision rules are written in levels. The curvature of the assumed utility, production and demand functions imparts a mild non-linearity to the whole system. Shock responses therefore always depend on the starting point. Testing has indicated that reasonable differences in the initial steady-state equilibria are unlikely to make a qualitative difference to the overall response. (2) The version we use gives some weight to lagged interest rates. See Appendix A for full details of the reaction function we use for these simulations. 127 The Bank of England Quarterly Model Finally, it matters whether the shocks are permanent or temporary. As explained below, the interest rate shock used here is a temporary shock, so that the model can return to equilibrium. The other shocks are configured as permanent shocks, in order to show how the model moves from one steady-state equilibrium to another. This has implications for behaviour, particularly through the behaviour of agents’ expectations and asset prices. Shocks that are understood to be temporary tend to have smaller effects on asset prices, because there is no long-term movement in asset prices (unlike the results of the permanent shocks discussed in Sections 7.2.2 to 7.2.6). 7.2 Shock responses This section discusses the behaviour of the full model in the face of shocks to monetary policy, productivity, government spending, the terms of trade, world demand, and labour supply. We have chosen these shocks to illustrate the interactions of households, firms and policymakers in goods, labour and financial markets. The temporary monetary policy shock is described by the impact of the initial shock and how the economy returns to equilibrium. However, we find it more intuitive to describe the other, permanent shocks by starting with the determination of the new steady-state equilibrium, and then the short-run dynamics. Where possible, we discuss the responses in the light of the conclusions of directly comparable empirical literature. For each shock, we show a standard set of eight charts (using the same scales for each shock); for the permanent shocks in Section 7.2.2 to Section 7.2.6 we show some additional detail to help illustrate the particular effects. The new steady state in the permanent shocks is shown as a dashed line. All the shocks discussed in this chapter start with the economy at a steady-state equilibrium. Some asset stocks, especially net foreign assets, are quite slow to get to their new steady-state equilibrium positions in the face of most of the permanent shocks. So flows can be very close to their long-run equilibrium positions, but it may take some time before stocks reach equilibrium. In the case of net foreign assets, adjustment occurs through net trade and the short-run trade responses can be quite different from the long-run balance required for a sustainable equilibrium. In general, the persistence of expenditures means that we often see short-term overshooting in net foreign assets. (3) 7.2.1 An interest rate shock Figure 7.1 illustrates the effect of a 1 percentage point rise in nominal interest rates for four quarters. It is designed to show the direct impact of interest rate changes on the model economy. In the other simulations presented in this section, interest rates react endogenously to movements in inflation and output that come about because of some other shock. But here, we start with an interest rate movement and follow the reaction of the rest of the model. In particular, we assume that monetary policy can be described by a simple Taylor-type reaction function in which nominal interest rates react to inflation and output gaps, and we fix the nominal interest rate by shocking the monetary policy reaction function directly. (4) (3) Eigenanalysis of the model confirms that it is stable, but the estimated dynamics imply that convergence is slow. (4) The mechanics of implementing the shock are described in Box 11 on page 80. 128 Model properties Implemented this way, using the recursive simulation technique discussed in Chapter 5, the shock can be thought of as a sequence of unanticipated deviations from the monetary policy that would be expected on the basis of private agents’ understanding of the economy and, in particular, the monetary policy reaction function. This is not the same as exogenising monetary policy by removing the monetary reaction function. In each period, agents see interest rates that do not accord with their understanding about how monetary policy is usually set; (5) they nonetheless expect monetary policy in subsequent periods to be set according to the normal reaction function, and remain confident that inflation will remain anchored at the original target in the long run. The responses to this shock reflect the theoretical structure of the model discussed in Chapter 3. Optimal decisions are made on the basis of relative prices, taking account of the general level of prices. This ensures long-run neutrality of monetary policy so that the only long-run change is that nominal prices all move by the same proportion. Relative prices are unaffected in the steady state, so there are no long-run effects on real expenditures or the rate of inflation, which is brought back to target by the monetary reaction function. But there are potentially significant effects in the short run. The model incorporates a number of rigidities that reflect costly adjustment of prices and quantities. Broadly speaking, the balance of these rigidities determines the extent to which a monetary policy shock will affect inflation or real activity in the short run. (6) For example, we assume that it is costly for firms to adjust their factor inputs and nominal prices. Firms balance the costs of adjusting factors of production and of adjusting prices, so that the overall price/quantity response will depend on the relative costs of adjustment. There is a variety of other rigidities that also affect the overall output/inflation response. And there are additional channels of short-run ‘non-neutrality’ of monetary policy shocks. For example, changes in nominal interest rates may affect real consumption growth in the short run, to the extent that they represent changes in the credit constraints faced by consumers. Because nominal prices and wages are assumed to be sticky and inflation is assumed to be persistent, the increase in the nominal interest rate leads to an increase in the real interest rate, and hence the cost of borrowing to finance consumption and investment. The nominal exchange rate immediately appreciates with the increase in nominal interest rates, in line with uncovered interest parity. (7) This has two effects. First, domestic price stickiness means that there is a real exchange rate appreciation, which reduces the demand for exports. Second, the nominal appreciation puts downward pressure on import prices, which feeds through gradually because of price stickiness. Other things equal, this effect would increase the demand for imports, but the income effect from lower demand dominates in the short run. (5) Leeper and Zha (2003) make the distinction between the ‘direct’ effects of a policy intervention, which are the usual reactions when the regime is held fixed, and the ‘expectation-formation’ effects that are introduced by changes in agents’ beliefs about the policy regime. In their language, the simulation shown in Figure 7.1 is assumed to be a ‘modest’ intervention, where there are no expectation-formation effects. (6) If all prices were completely flexible, then temporary monetary policy shocks would lead to an immediate change in all nominal prices, without any effect on real expenditure. The long-run effects of the shock would be observed immediately. (7) The exchange rate appreciation follows from the fact that higher domestic interest rates, relative to interest rates on equivalent foreign-currency assets, make sterling assets more attractive to international investors. Uncovered interest parity implies that the exchange rate moves to a level where investors expect a future depreciation just large enough to make them indifferent between holding domestic and foreign-currency assets. When used for forecasting, alternative paths for the nominal exchange rate are typically used, as discussed in the box ‘The exchange rate in forecasting and policy analysis’ on page 48 of the November 1999 Inflation Report. 129 The Bank of England Quarterly Model Figure 7.1: Effects of an interest rate shock Shock: nominal interest rate Percentage points 1.2 Private sector output Per cent 1.0 0.8 0.5 0.4 0.0 0.0 -0.5 -0.4 -0.8 0 1 2 Years 3 CPI annual inflation 4 5 -1.0 0 1 2 Years 3 Unemployment rate Percentage points 0.3 4 5 Percentage points 0.2 0.1 0.0 0.0 -0.1 -0.3 -0.2 -0.3 -0.6 0 1 2 Years 3 4 Real consumption wage 0 5 1 2 Years 3 Real interest rate Per cent 0.9 4 5 Percentage points 1.5 1.2 0.6 0.9 0.3 0.6 0.0 0.3 -0.3 0.0 -0.6 0 1 2 Years 3 4 Real exchange rate 5 -0.3 0 Per cent 1.0 1 2 Years 3 Imported consumption annual inflation 4 5 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -1.0 0 130 1 2 Years 3 4 5 -0.9 0 1 2 Years 3 4 5 Model properties All of these effects act to reduce aggregate demand. In the face of unchanged potential supply, firms immediately reduce factor utilisation and start to reduce employment and investment. Though prices are sticky, they are not fixed and inflation falls below the starting rate as firms react to the fall in demand. Real consumption wages rise in the very first period, but fall back quickly as a result of lower demand for labour. Fiscal policy reacts by increasing the lump sum tax rate to preserve tax revenue as nominal GDP falls in the short run. The shock lasts for four periods. In the fifth period, monetary policy reverts to its standard reaction function. Faced with a negative output gap and inflation below target, nominal interest rates gradually fall below their long-run equilibrium to bring inflation back to target. The economy is close to equilibrium after five years. The responses shown here accord qualitatively with the broad conclusions from the empirical literature on monetary policy shocks: following an unanticipated monetary policy tightening, interest rates rise, output contracts, profits and real wages fall, and inflation (eventually) falls. Output, consumption, investment and inflation are ‘hump-shaped’ and highly persistent, with the peak effect on inflation lagging the peak effect on real variables. (8) The behaviour of the model for this shock is broadly similar to that of the previous MTMM model (9) and qualitatively close to other studies of the UK economy. (10) The simulation shown in Figure 7.1 is based on the assumption that the unexpected change in interest rates does not affect agents’ long-run inflation expectations. But the response of the economy to a change in interest rates depends on the credibility of the inflation target. In particular, as inflation expectations become more firmly anchored around the inflation target – the target becomes more credible – a change in the short-term interest rate is likely to have less impact. To illustrate the sensitivity of these simulations to assumptions about expectations, Figure 7.2 shows how the effect on inflation and private sector output differs if agents wrongly perceive that the unexpected increase in interest rates may have been triggered by a reduction in the targeted rate of inflation. The blue line is the same as in Figure 7.1 and the black line is based on the assumption that agents revise down their expectation of the targeted rate of inflation and expect a prolonged period of tighter policy in order to achieve that perceived lower target. (11) (8) See, for example, Leeper, Sims and Zha (1996), Leeper and Zha (2003) and Christiano, Eichenbaum and Evans (1997, 1999). (9) See Bank of England (2004). (10) See, for example, Batini, Harrison and Millard (2003), Bean, Larsen and Nikolov (2002) and Dale and Haldane (1995). In many of these studies, the responses are very persistent – more than we judge to be plausible. Bean (1998) estimates an IS curve and Phillips curve instead of a VAR; in that model output drops immediately and there is no effect on inflation for one year. But the signs and ordering of output and inflation responses in these studies are otherwise the same. (11) This simulation is sensitive to the precise assumptions made about the change in expected inflation. Here we assume that the unexpected increase in interest rates causes agents to revise down their expected level of inflation in the long run by around 0.2 percentage points by the end of the first year. 131 The Bank of England Quarterly Model Figure 7.2: How expectations can affect shock responses CPI annual inflation Percentage points 0.3 Private sector output Per cent 0.1 0.0 credible policy -0.1 0.0 -0.2 -0.3 -0.3 changed inflation expectations -0.4 -0.5 -0.6 0 1 2 Years 3 4 5 0 1 2 Years 3 4 5 The response of inflation when agents believe the target has been reduced is much sharper, reflecting the effects of lower expected inflation on price setting. After the first four quarters, the absence of further policy surprises (relative to an unchanged target inflation rate) leads agents to correct their expectations gradually towards the true target. (12) In BEQM, the effect of expectations about the long-run inflation rate mainly comes through nominal variables. The effect on real variables is somewhat smaller but consistent with that on inflation: the initial fall in private sector output is sharper, though there is little difference between the two lines by the end of the third year. The larger response of inflation when agents perceive that the target has been reduced both illustrates how different assumptions about expectations can affect key model properties and underlines the importance of monetary policy credibility in determining the sensitivity of the economy to changes in interest rates. (12) We use the approach of Erceg and Levin (2003) to model how agents realise their mistake and gradually correct their expectations over time. 132 Model properties 7.2.2 A productivity shock Figure 7.3 shows the effects of a permanent 1% increase in the level of labour productivity. (13) This simulation is of particular interest because it highlights the importance of stock-flow dynamics and the wealth dimension of the model, and illustrates how demand and policy react to supply shifts. In the long run, with the new steady-state shown by the dashed line in Figure 7.3, the increase in labour productivity is accompanied by a rise in wages paid by firms. Desired capital and potential supply also increase (although proportionately less than the size of the productivity shock, due to the second round effects on supply discussed below). Households increase their consumption, given higher wages. Aggregate import levels rise with the increase in consumption and capital investment. The economy must export more to restore external balance, and a real exchange rate depreciation is required. Steady-state external balance is brought about by an increase in the trade surplus (or reduction in deficit), offset by a fall in net foreign asset levels. (14) We could also tell this story equivalently in terms of stocks. The increase in productivity raises the long-run value of equities (15) and corporate debt; and government debt rises broadly in line with output. Some of the increase in domestic financial assets is sold to overseas residents, which results in a decrease in net foreign assets. An increase in net exports is needed to service the extra debt and so the real exchange rate depreciates. This depreciation has two ‘second-round’ effects on the real economy, through labour supply and capital, which reduce the overall impact on long-run supply. First, we make a distinction in BEQM between real consumer wages and real producer wages. The long-run real consumption wage rises unambiguously, but by less than the real producer wage, because the exchange rate depreciation causes an increase in total consumption prices (including imported goods) relative to domestic producer prices. At the same time, the reservation wage rises fully in line with the increase in labour productivity. (16) The combined effect is sufficient to cause a small long-run fall in labour supply and employment, which mitigates the output increase from the productivity shock. Second, the long-run price of imported goods rises relative to domestically produced goods. This reduces slightly the ratios of imported to domestically produced consumption and capital goods, but imports of capital goods still rise in absolute terms. (17) This relative price effect raises the overall cost of capital goods, which also offsets part of the increase in capital and output from the productivity shock. (18) (13) It is implemented as a 1 percentage point increase in the growth rate of labour productivity for one period. (14) Figure 7.1 shows that there is little change in the steady-state ratio of net foreign assets to GDP. The initial equilibrium has a negative level of net foreign assets. It is the combination of a fall in the level of net foreign assets (thus becoming more negative) and a rise in GDP that leaves the steady-state ratio of the two virtually unchanged. (15) The long-run value of the firm is equal to the value of the capital stock plus the expected stream of discounted supernormal profits. The value of equity increases by slightly less than the value of the capital stock because the exchange rate depreciation means that value-added prices fall relative to consumer prices. This reduces real average profitability in terms of consumer prices. (16) We assume that unemployment benefits move in line with the real product wage, and hence labour productivity. This is a necessary assumption to prevent the long-run natural rate of unemployment trending above 100% or below zero in the long run. (17) Imported and domestically produced capital goods are assumed to be complements. (18) The amount of the increase in the desired capital stock depends on the share of imported capital in the capital aggregator, the elasticity of substitution between imported and domestic capital goods, and the elasticity of export demand. 133 The Bank of England Quarterly Model The exchange rate movement is associated with some permanent shifts in relative prices, but inflation and nominal interest rates return to the initial equilibrium implied by the inflation target and world nominal interest rates, leaving the real interest rate unchanged. There is little fiscal policy reaction: government transfers spending and debt are all assumed to rise in line with output, and tax revenues rise broadly in line with output too. (19) In the long run, therefore, there is no change in the inflation rate and nominal interest rates, with an overall increase in output flows. The economy exports more and holds more foreign debt, with a depreciation in the real exchange rate. The fact that output does not rise by exactly 1% is due to the second-round impacts of the real exchange rate on desired labour supply and capital. Turning to the short run, firms are immediately able to produce more output, for given inputs of capital and labour. But sluggish adjustment means that aggregate demand does not immediately increase and so factor utilisation falls sharply. This encourages firms both to reduce prices to stimulate demand and to reduce their demand for labour in the short run. The real exchange rate immediately depreciates close to its new long-run level and this is gradually passed through into higher import price inflation. Consumer price inflation falls because the effect of temporarily reduced domestic prices outweighs the effect of higher import prices. Unemployment rises in the short run, as firms begin to reduce employment in response to the fall in factor utilisation. The fall in employment (and rise in unemployment) is relatively short lived, however, as higher real wages mean that demand rises steadily, bringing utilisation back to normal levels. Monetary policy responds to low inflation and output below potential by cutting interest rates to stimulate demand. This prevents inflation from falling further, and the inflation rate gradually rises back towards the target. There is a large body of empirical literature that aims to identify the responses to a productivity shock. The responses here are consistent with the findings from that literature: investment, consumption and output all increase. (20) However, there is some disagreement over labour market responses, especially whether hours worked rise or fall. (21) The responses here are also consistent with the initial responses from the MTMM model: output rises gradually to a higher level, and inflation falls quickly before rising back to its starting point. Work applying the methods of Bayoumi and Eichengreen (1992) and Monticelli and Tristani (1999) to UK data also corroborates this finding in the data. (19) Real lump-sum taxes do not move by exactly the same amount as real domestic output, because the government spending and debt targets are expressed in terms of ratios to the value of private sector output, and the GDP deflator changes a little because of the permanent effects on the real exchange rate. (20) See, for example, Christiano, Eichenbaum and Evans (2001). (21) In a widely cited paper, Galí (1999) reports that hours worked fall after a positive productivity shock. Several studies have affirmed this proposition – see, for example, Basu, Fernald and Kimball (1998). But Christiano, Eichenbaum and Vigfusson (2003) argue that hours worked rise, consistent with the neoclassical model. 134 Model properties Figure 7.3: Effects of a productivity shock Shock: labour productivity (λ) Per cent Private sector output 1.2 Per cent 1.0 Steady state 1.0 0.5 0.8 0.6 0.0 0.4 -0.5 0.2 0.0 0 2 4 Years 6 CPI annual inflation 8 10 -1.0 0 2 4 Years 6 Unemployment rate Percentage points 0.3 8 10 Percentage points 0.2 0.1 0 0.0 -0.1 -0.3 -0.2 -0.3 -0.6 0 2 4 Years 6 8 Real consumption wage 0 10 2 4 Years 6 Nominal interest rate Per cent 8 10 Percentage points 1.2 0.9 0.8 0.6 0.3 0.4 0.0 0.0 -0.3 -0.4 -0.8 -0.6 0 2 4 Years 6 8 Real exchange rate 0 10 2 4 Years 6 Imported consumption annual inflation Per cent 1.0 8 10 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -0.9 -1.0 0 2 4 Years 6 8 10 0 2 4 Years 6 8 10 135 The Bank of England Quarterly Model Figure 7.3: (continued) Effects of a productivity shock Factor utilisation Non-durable consumption Per cent 0.6 Per cent 0.8 Steady state 0.4 0.4 0.2 0.0 0.0 -0.2 -0.4 -0.4 -0.8 -0.6 -1.2 -0.8 0 2 4 Years 6 8 Exports 0 10 Per cent 1.5 2 4 Years 6 8 Imports 10 Per cent 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 0 2 4 Years 6 8 Real equity price -1.5 0 10 Per cent 0.6 2 4 Years 6 8 Capital investment 10 Per cent 2 0.4 1 0.2 0.0 0 -0.2 -1 -0.4 -0.6 0 2 4 Years 6 Net foreign assets (% of GDP) 8 10 -2 0 Percentage points 2 4 Years 6 8 Real product wage 10 Per cent 1.0 4 0.8 2 0.6 0 0.4 -2 0.2 -4 0 136 2 4 Years 6 8 10 0.0 0 2 4 Years 6 8 10 Model properties 7.2.3 A government spending shock Figure 7.4 illustrates the effect of a 5% increase in the (exogenous) target level of government spending on private sector goods and services, defined as a ratio of nominal government spending to nominal private sector output. (22) The fiscal instrument – here, for simplicity, modelled as the lump-sum tax on households – then ensures that this new spending target is sustainable. This simulation is particularly interesting because of the shock’s direct effect on demand. In the long run, the model shows conventional crowding-out effects, with reduced consumption because taxes must rise to pay for higher government spending. The higher lump-sum taxes cause households to reduce both non-durable consumption and dwellings investment. In many textbook open economy models, output would be unchanged in the long run. (23) However, potential supply in BEQM can be affected (albeit only slightly in practice) by movements in the real exchange rate. The long-run change in the real exchange rate depends on the net impact of the shift from private to public consumption on the long-run demand for imports. The import content of total government spending (including factor payments) is relatively low, but the intermediate import propensity of government procurement is higher than that of domestically produced private non-durable consumption. So the long-run demand for imports rises, but there is also a fall in direct imports of consumption goods that partially offsets this effect. Overall, the net effect is to increase slightly the long-run demand for imports. As a result, the real exchange rate depreciates to raise exports and maintain current account balance. This depreciation has the same ‘second round’ effects as seen in the productivity shock, though the effects here are quantitatively very small. First, increases in the price of imported consumption goods lower the real consumption wage and, hence, labour supply and employment. Second, increases in the price of imported capital goods raise the overall cost of capital goods and lower the long-run level of the capital stock. Output consequently falls slightly in the long run. Despite a fall in the capital stock, the valuation of the corporate sector rises because of an increase in the price of capital goods, which raises the overall (relative) value of the capital stock. (24) There is a slight rebalancing of portfolios – net foreign asset levels fall – but the effect is quantitatively small. In the short run, government procurement increases immediately, but the crowding out of private consumption takes longer to come through. This reflects the non-Ricardian nature of the model (see Section 3.2.1) and also the existence of habits. The net effect is an immediate rise in aggregate demand and factor utilisation. Higher factor utilisation stimulates the demand for labour (though more gradually) and encourages firms to increase margins. Domestic inflation rises as a result of the increase in factor costs and demand pressures. Imported inflation also rises somewhat due to the depreciation of the real exchange rate. The peak effect on CPI inflation occurs at around six quarters, reflecting nominal rigidities in domestic price setting and gradual pass through of the exchange rate to import prices. Monetary policy responds to the short-run increase in aggregate demand by raising the nominal interest rate to bring inflation back to target. After some time, the crowding-out effect on consumption also (22) Government spending on private sector goods and services (‘procurement’) is only part of total government expenditure. The final increase in total government consumption, as measured by the National Accounts, is much less than 5%, because procurement does not include wages and salaries paid to government employees or the government’s gross operating surplus. For this simulation, the government spending rule was modified to bring spending immediately up to the new target level. (23) The only net effect would be a reduction in the share of consumption in total expenditure. (24) There is also a technical effect that comes from our assumption of different mark-ups across expenditure categories as a way of incorporating relative price trends with goods produced by a single production function (as explained in Section 6.3). 137 The Bank of England Quarterly Model reduces aggregate demand, which fully adjusts within four years. Factor utilisation returns to normal but unemployment adjusts more gradually because of rigidities in the labour market. While there is increasing consensus on the responses of the economy to monetary policy and technology shocks, there is less consensus on the effects of a fiscal policy shock. A major difficulty is in identifying unanticipated shocks to fiscal policy, rather than systematic movements in spending through the business cycle. Moreover, the responses are sensitive to whether an increase in government spending is expected to be matched by higher future taxation or by lower future spending: changes in government spending in practice could be associated with very different beliefs about what will happen in the future and when. So such simulations are not always a good guide to what happens in practice. Nevertheless, the responses here are consistent with results from event studies of government procurement: increases in government spending are associated with short-lived increases in private sector output. (25) In some studies, evidence is found for short-run increases in consumption as well as investment. (26) In BEQM, private consumption falls immediately. (27) However, robust findings are difficult to come by for the United Kingdom. (28) The trade balance immediately worsens, in line with Roubini (1988), but in general there is little agreement on the effect on the current account. (29) A textbook result is that output is crowded out by higher public expenditure; this is seen here and in the MTMM model. But ‘crowding-in’ is a possibility in an open economy. (30) (25) See Burnside, Eichenbaum and Fisher (BEF) (2002), following the methodology of Ramey and Shapiro (1998). The results in BEF indicate that real wages fall following an increase in government procurement. This is not the case here, as there is a rise initially in real consumption wages. (26) See Blanchard and Perotti (2002). (27) This accords with the empirical findings reported in Cavallo (2002). (28) For example, Perotti (2002) finds that the short-run effects of government spending on inflation and output in five OECD economies (including the United Kingdom) has changed substantially over time. (29) In contrast to Roubini (1988), Roubini and Kim (2003) report that expansionary fiscal shocks tend to improve the current account. See Erceg, Guerrieri and Gust (2004) for an assessment and discussion. (30) See Barry and Devereux (2003). 138 Model properties Figure 7.4: Effects of a government spending shock Shock: government procurement target (µ gy ) Per cent 6.0 Private sector output Per cent 1.0 5.0 0.5 4.0 0.0 3.0 Steady state 2.0 -0.5 1.0 0.0 0 2 4 Years 6 CPI annual inflation 8 10 -1.0 0 2 4 Years 6 Unemployment rate Percentage points 0.3 8 10 Percentage points 0.2 0.1 0 0.0 -0.1 -0.3 -0.2 -0.6 0 2 4 Years 6 8 Real consumption wage 10 -0.3 0 Per cent 0.9 2 4 Years 6 Nominal interest rate 8 10 Percentage points 1.2 0.6 0.8 0.3 0.4 0.0 0.0 -0.3 -0.4 -0.8 -0.6 0 2 4 Years 6 8 Real exchange rate 0 10 2 4 Years 6 Imported consumption annual inflation Per cent 1.0 8 10 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -1.0 0 2 4 Years 6 8 10 -0.9 0 2 4 Years 6 8 10 139 The Bank of England Quarterly Model Figure 7.4: (continued) Effects of a government spending shock Factor utilisation Non-durable consumption Per cent Per cent 0.8 0.6 0.4 0.4 0.2 0.0 0.0 -0.2 -0.4 -0.4 -0.8 -0.6 Steady state -1.2 -0.8 0 2 4 Years 6 8 Exports 0 10 Per cent 1.5 2 4 Years 6 8 Imports 10 Per cent 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 0 2 4 Years 6 8 Real equity price 10 -1.5 0 2 4 Years 6 8 Government procurement 10 0.6 Per cent 6 0.4 5 0.2 4 0.0 3 Per cent 2 -0.2 1 -0.4 -0.6 0 2 4 Years 6 Total managed expenditure (% of GDP) 8 10 Percentage points 0 0 2 4 Years 6 Total taxation (% of GDP) 8 10 Percentage points 1.0 0.8 0.7 0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0.0 0.2 -0.2 0.1 0.0 0 140 2 4 Years 6 8 10 -0.4 0 2 4 Years 6 8 10 Model properties 7.2.4 A terms of trade shock Figure 7.5 shows the impact of a permanent, unanticipated 1% decrease in the level of the world price of imported goods (leaving the world price of exported goods unchanged). The model is constructed to allow the basket of goods and services imported by the domestic economy to differ from those exported, so a shift in the relative world price of these different goods and services can have an effect on the domestic economy. Here we look at the effects of the price of imported goods falling relative to the price of exported goods. This simulation is interesting because it highlights how relative price movements lead to reallocations of expenditure. If the fall in world import prices were matched by a fall in world export prices, the nominal exchange rate would immediately depreciate to offset completely the impact of falling world prices on domestic currency import and export prices, leaving relative prices and activity unchanged. Here, however, we consider an alternative shock that changes the relative world prices of import and export goods. An exchange rate depreciation cannot therefore offset the fall in the world price of imported goods without changing the world price of exported goods. The shock represents an improvement in the terms of trade. Domestic output is now more valuable relative to world production, which increases domestic wealth and hence domestic demand in the long run. In particular, non-durable consumption rises with the increase in wealth. The higher level of long-run consumption is sustained by a higher net foreign asset position, financed by a fall in the long-run value of the trade balance brought about by a rise in the real exchange rate. The long-run fall in the relative price of imports increases the demand for these goods. Within total imports, imported consumption responds most, reflecting the higher price elasticity of this component. In the long run, the capital stock and labour force participation rise due to the second round effects described in Section 7.2.2, and total output increases. In the short run, falling world import prices are gradually passed through into falling domestic currency import prices and inflation falls. But the rise in wealth leads to an increase in demand in the medium term. This puts pressure on supply capacity and increases factor utilisation, leading to an increase in margins and labour demand, which reduces unemployment. The upward pressure on domestic prices outweighs the effect from lower import prices over the medium term, leading to an increase in inflation. Monetary policy responds initially by cutting interest rates to offset lower inflation, before tightening in response to the increase in inflation over the medium term. Empirical studies differ on the contribution of terms of trade shocks to the business cycle. The responses from VAR analysis are highly dependent on the exchange rate regime. (31) Evidence cited in De Gregorio and Wolf (1994) supports the notion that the real exchange rate appreciates following a terms of trade shock. (31) See Broda (2001). 141 The Bank of England Quarterly Model Figure 7.5: Effects of a terms of trade shock Shock: relative import price (wmargin) Level 0.000 Private sector output Per cent 1.0 -0.002 0.5 -0.004 Steady state -0.006 0.0 -0.008 -0.5 -0.010 -0.012 0 2 Years 4 6 8 CPI annual inflation 10 -1.0 0 Percentage points 0.3 2 4 Years 6 8 10 Percentage points 0.2 Unemployment rate 0.1 0 0.0 -0.1 -0.3 -0.2 -0.3 -0.6 0 2 4 Years 6 8 Real consumption wage 0 10 2 4 Years 6 10 Percentage points Nominal interest rate Per cent 0.9 8 1.2 0.6 0.8 0.3 0.4 0.0 0.0 -0.3 -0.4 -0.8 -0.6 0 2 4 Years 6 8 Real exchange rate 0 10 2 4 Years 6 Imported consumption annual inflation Per cent 1.0 8 10 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -1.0 0 142 2 4 Years 6 8 10 -0.9 0 2 4 Years 6 8 10 Model properties Figure 7.5: (continued) Effects of a terms of trade shock Factor utilisation Non-durable consumption Per cent 0.6 Per cent 0.8 Steady state 0.4 0.4 0.2 0.0 0.0 -0.2 -0.4 -0.4 -0.8 -0.6 -1.2 -0.8 0 2 4 Years 6 8 Exports 0 10 Per cent 1.5 2 4 Years 6 8 10 Imports Per cent 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 0 2 4 Years 6 Net foreign assets (% of GDP) 8 -1.5 0 10 2 4 Years 6 8 Imported consumption 10 4 Per cent 4 2 3 0 2 -2 1 Percentage points 0 -4 0 2 4 Years 6 8 10 0 2 4 Years 6 8 10 143 The Bank of England Quarterly Model 7.2.5 A world demand shock Figure 7.6 shows the effects of a permanent, unanticipated 1% increase in the level of world demand for domestic goods. This simulation is particularly useful for highlighting the role of the real exchange rate and imported capital. In a standard neoclassical model of a small open economy model, long-run supply is anchored by the world real interest rate. An increase in export demand for UK goods and services would not be met by a rise in output but would be completely offset by an increase in the real exchange rate. Neither is there a short-run effect in many simple ‘textbook’ models, because the real exchange rate appreciation simply offsets the impact of higher world demand. However, as described in Section 7.2.2, BEQM allows for the possibility that real exchange rate movements can affect long-run supply. In the case of a world demand shock, the real exchange rate appreciates, which acts to reduce the cost of capital goods and raise the desired capital stock. Similarly, real consumption wages rise and labour supply increases. Both effects lead to an increase in output. The long-run increase in real consumption wages and employment allows greater expenditure by households on non-durable consumption and investment in dwellings. Consumption, investment and government spending move proportionally by slightly more than output. (32) Holdings of net foreign assets rise and the value of imports is lower in the long run (reflecting a relatively low price elasticity of import volumes), so a higher real exchange rate and a fall in export volumes is needed to balance the current account. The shock effectively improves the economy’s relative earnings capability and net foreign asset position. In the short run, the foreign currency price of domestic exports is slow to adjust to the increase in world demand. This fuels a short-run increase in export demand and, given slow adjustment of other demand components, output immediately overshoots its long-run level. Factor utilisation rises as supply capacity is relatively slow to adjust. Firms increase their demand for labour, reducing unemployment, and also increase prices. The exchange rate immediately appreciates close to its new long-run level, though the pass through to lower import prices is gradual. The net effect is for CPI inflation to rise a little above target, to be brought down gradually over the longer term by a rise in interest rates. (32) The model has a target for the ratio of nominal government spending to nominal private sector output. The movement of relative prices in this shock means that there are slightly different movements in the volumes of government spending and total output. 144 Model properties Figure 7.6: Effects of a world demand shock Shock: world demand (cf ) Per cent Private sector output Per cent 1.0 1.2 1.0 0.5 0.8 0.0 0.6 Steady state 0.4 -0.5 0.2 0.0 0 2 4 Years 6 CPI annual inflation 8 -1.0 0 10 Percentage points 0.3 2 4 Years 6 Unemployment rate 8 10 Percentage points 0.2 0.1 0 0.0 -0.1 -0.3 -0.2 -0.6 0 2 4 Years 6 8 Real consumption wage 10 -0.3 0 2 4 Years 6 Nominal interest rate Per cent 0.9 8 10 Percentage points 1.2 0.6 0.8 0.3 0.4 0.0 0.0 -0.3 -0.4 -0.8 -0.6 0 2 4 Years 6 8 Real exchange rate 10 0 2 4 Years 6 Imported consumption annual inflation Per cent 1.0 8 10 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -0.9 -1.0 0 2 4 Years 6 8 10 0 2 4 Years 6 8 10 145 The Bank of England Quarterly Model Figure 7.6: (continued) Effects of a world demand shock Factor utilisation Non-durable consumption Per cent 0.6 Per cent 0.8 Steady state 0.4 0.4 0.2 0.0 0.0 -0.2 -0.4 -0.4 -0.8 -0.6 -1.2 -0.8 0 2 4 Years 6 8 Exports 0 10 Per cent 1.5 2 4 Years 6 8 Imports 10 Per cent 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 0 2 4 Years 6 Net foreign assets (% of GDP) 8 0 10 2 4 Years 6 8 Capital investment 10 4 Per cent 2 2 1 0 0 -2 -1 Percentage points -2 -4 0 146 2 4 Years 6 8 10 0 2 4 Years 6 8 10 Model properties 7.2.6 A labour market participation rate shock Figure 7.7 shows the impact of a permanent decrease in labour market participation, such that the participation rate is lower at all real wage rates. (33) The simulation is interesting for the shock’s direct effects on supply and the equilibration of demand. In a standard small open economy model, in which labour supply is exogenous and there is no imported capital, a given drop in employment would be matched in the long run by a proportional fall in capital and output. In BEQM, however, labour supply is endogenous and some final capital goods are imported. Therefore, the interaction between the real exchange rate and long-run supply discussed in Section 7.2.2 means that the decrease in labour supply, employment, capital stock and output is slightly less than 1%. That is because the real consumption wage rises and the cost of capital goods falls. Long-run aggregate labour income falls because the rise in the real consumption wage is less than the fall in employment. Consumption and housing investment fall, along with investment by firms and government spending. (34) As total consumption and capital expenditures are reduced, imports fall too. But lower exports are needed to preserve external balance, so the real exchange rate appreciates. As with the other shocks described in this chapter, these long-run effects on expenditure are reflected in changes in asset positions. In addition to lower labour income, margins (and therefore profits and dividends) are squeezed as firms attempt to hold on to market share, so equity values fall. Net foreign assets are also reduced (even though there is a marginal rise in the ratio to output): the aggregate economy, in effect, borrows from overseas in an attempt to support aggregate consumption. In the short run, the exchange rate appreciates immediately, to close to its new steady-state level. Participation and employment fall sharply over the first two years, along with consumption and investment. The supply of workers at every wage rate falls, so wages rise initially as firms try to prevent employment falling by as much as participation. The fall in labour participation is faster than that of employment, so the unemployment rate falls too in the short term. As employment and demand fall, firms initially reduce factor utilisation. Import price inflation falls initially, because of the higher exchange rate. But domestic inflation rises as firms have to bid wages up, and the real product wage rises substantially within the first two years. This effect dominates the effect of cheaper imported goods and CPI inflation rises. Interest rates rise to bring inflation back to the target. The results shown here accord qualitatively with the effects of a labour supply shock identified in Peersman and Straub (2004) for the euro area: hours and output fall, prices and real wages rise, and interest rates rise temporarily. Despite the observed variation of unemployment with output over the business cycle, there are few models that examine the joint behaviour of employment, unemployment and labour market participation: Veracierto (2002) is one exception, using a matching framework, but the results did not match observed dynamics in the US labour market. (33) Technically, the shock is implemented as a reduction by 0.01 in the intercept term in the (log) labour supply schedule (equation (A.14) in Appendix A). (34) In this shock, the government spending target is expressed as a ratio to output, so spending falls in line with output. 147 The Bank of England Quarterly Model Figure 7.7: Effects of a participation shock Shock: participation intercept (κ ) l Level 0.000 Private sector ouput Per cent 1.0 -0.002 0.5 -0.004 -0.006 0.0 -0.008 -0.5 -0.010 Steady state -0.012 0 2 4 Years 6 8 CPI annual inflation 10 -1.0 0 Percentage points 0.3 2 4 Years 6 Unemployment rate 8 10 Percentage points 0.2 0.1 0 0.0 -0.1 -0.3 -0.2 -0.3 -0.6 0 2 4 Years 6 8 Real consumption wage 10 0 2 4 Years 6 Nominal interest rate Per cent 0.9 8 10 Percentage points 1.2 0.6 0.8 0.3 0.4 0.0 0.0 -0.3 -0.4 -0.8 -0.6 0 2 4 Years 6 8 Real exchange rate 0 10 2 4 Years 6 Imported consumption annual inflation Per cent 1.0 8 10 Percentage points 0.6 0.3 0.5 0.0 0.0 -0.3 -0.5 -0.6 -0.9 -1.0 0 148 2 4 Years 6 8 10 0 2 4 Years 6 8 10 Model properties Figure 7.7: (continued) Effects of a participation shock Factor utilisation Non-durable consumption Percentage points 0.6 Per cent 0.8 0.4 0.4 0.2 0.0 0.0 -0.2 -0.4 Steady state -0.4 -0.8 -0.6 -0.8 0 2 4 Years 6 8 Exports -1.2 0 10 Per cent 1.5 2 4 Years 6 8 Imports 10 Per cent 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 0 2 4 Years 6 8 Real equity price 0 10 2 4 Years 6 8 Participation Per cent 10 Per cent 0.0 0.6 0.4 -0.3 0.2 -0.6 0.0 -0.2 -0.9 -0.4 -0.6 0 2 4 Years 6 8 10 Capital investment -1.2 0 2 4 Years 6 8 Private sector employment Per cent 2 10 Per cent 0.0 1 -0.3 0 -0.6 -1 -0.9 -1.2 -2 0 2 4 Years 6 8 10 0 2 4 Years 6 8 10 149 The Bank of England Quarterly Model 7.3 Summary This chapter illustrates the properties of the new model by describing a series of model simulations. The results must be used carefully, because they are stylised experiments that look at the marginal effects of a single change, starting from an equilibrium position. And the effects of changes can be affected significantly by different assumptions about how policy reacts and about agents’ expectations. The reactions to a temporary shock to interest rates correspond to the broad conclusions of the empirical literature on monetary shocks: unanticipated policy tightening reduces output and then inflation. The behaviour is broadly similar to that of the previous MTMM model and close to other studies. As this is a temporary shock, there are no long-run effects on real variables or inflation. The effect of an unexpected change in interest rates on inflation is larger if agents believe that it is associated with a change in the targeted rate of inflation. We then illustrate the interaction of long-run properties and short-run adjustment through shocks to productivity, government spending, the terms of trade, world demand and labour market participation. As well as the conventional effects that would be expected from simple models, the results show how movements in the real exchange rate can affect potential supply through changes in labour force participation and the desired capital stock: other things being equal, a real exchange rate depreciation leads to a slight reduction in potential supply. Short-term movements in the face of shocks illustrate the effects of immediate changes in asset prices coupled with sluggish adjustment of employment, capital and prices. The simulations also show how the policy reaction functions that we use outside of the forecast act to ensure fiscal solvency and to bring inflation back to target over the medium term. 150 Chapter 8 Final remarks This book describes a new macroeconomic model that has been created for use in preparing the Monetary Policy Committee’s quarterly economic projections. Building the new model has been a substantial investment for the Bank, and in this book we attempt to explain the model in some detail. The new macroeconomic model is by no means the only input into the forecasting and policy processes: the Committee continues to draw on a range of other information, including the Bank’s suite of models. Indeed, one of the intended benefits from the way in which the new model was built is that it should be a better complement to the suite of models – it should be easier to compare assumptions and to incorporate off-model information and analysis. The design of the new macroeconomic model reflects the demands of forecasting and policy analysis at the Bank of England – a central bank where interest rate decisions are decided by the votes of a committee to meet a specified inflation target. A key question in the design of the new model is where to position the model in terms of a trade-off between theoretical rigour and accounting for observed correlations in the data. Two further issues are how to parameterise a large system such as this, and how to use the model for forecasting (such as the need to incorporate judgement and to ‘fix’ the paths of some endogenous variables). All of these posed challenges to the model design. There are certainly alternative solutions to these problems, and it is likely that advances in computing technology will make some of them (such as systems estimation) easier to deal with in the future. One clear difference from the Bank’s previous macroeconomic model is the shift to a more micro-founded theoretical structure, as embodied in the core model. We do not assume that the model is automatically made believable by the use of micro-foundations; instead it meets the need for a clear and consistent structure as the basis for differentiating between several possible explanations for the observed behaviour of the economy. The full forecasting model does not rely exclusively on the micro-founded core model. We can think of the forecast as a process that weights together three types of information: a structural story that comes from the core model; additional variables and robust correlations in the data that are not captured in the core theory but that we might want to project forward; and direct adjustment made on the basis of judgement and ‘off-model’ information. This structure implies a progressive modelling agenda: if ad hoc features are consistently important in producing a forecast, then an obvious aim is to incorporate these features into the structural core model if possible. This book documents the new macroeconomic model used by the MPC in the preparation of its economic projections. The new model does not represent a change in the Committee’s view of how the economy works or of the role of monetary policy. The model is not fixed, and will evolve over time: continuing maintenance and development will be required to deal with new issues and puzzles. The Bank has committed resources to this and also to the development of complementary models. 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Most variables have an equation that describes their definition or determining behaviour – in many cases this is clear from looking at the left-hand side of the equation. However, there are some exceptions. First, as described in Chapter 3, the core model is a general equilibrium model in which endogenous variables are determined by the equilibration of demand and supply in the markets for goods, labour and assets. Therefore, not all endogenous variables have a natural association with a single equation. Some variable are determined by a market clearing condition rather than a decision rule: for example, real marginal costs (rmc) are determined by the condition that all production by domestic firms is allocated across the markets to which they sell (equation (A.42)), even though the variable rmc does not appear in this equation. Similarly, we reference private sector employment, e, with the first-order condition for private sector employment (equation (A.31)). Second, there are cases in which a variable may be associated with more than one equation. The inflation rate of the numeraire price, ṗ, is not a decision variable of any agent in the core model. In the presence of nominal rigidities, inflation is jointly determined by the interaction of all agents in the model. Nevertheless, here we reference ṗ with the monetary reaction function (equation (A.110)), which acts to ensure that inflation is at the target rate in the long term. Finally, some equations are associated with more than one variable. The monetary reaction function could also be seen as determining the short-term nominal interest rate, rg, as well as inflation. Both domestically produced and imported consumption (ch and cm) are referenced by two equations ((A.8) and (A.9)), because both equations are important for each variable. And the production-clearing condition (equation (A.42)) is provided as a reference for both real marginal cost, rmc, and private sector value added, y. A number of these choices are to some extent arbitrary, but our guiding principle has been to identify the most relevant core model equation (or equations) for each variable. Table A.1: Endogenous variables a avh ben bf bg bgtar bk c ch chv cir cm Net financial wealth of the household sector (A.158) Private sector average hours worked (A.35) Unemployment benefit (A.70) Stock of foreign bonds denominated in terms of foreign consumption goods (A.4) Stock of government bonds (A.79) Government debt target (A.81) Stock of corporate bonds (A.47) Volume of consumption goods (A.1) Domestically produced consumption goods (A.8) and (A.9) Value-added component of domestically produced consumption goods (A.142) Volume of actual and imputed rents (A.148) Volume of directly imported consumption goods (A.8) and (A.9) 165 The Bank of England Quarterly Model cmod d d4cpi dels duser dv dw e ecost ecostg eg eh en f g gc gl gosgexp gtar gv hab habd hw id ig igtar ik ikh ikhv ikm io iov k kh km l mi mon mpc µeg nfa ṗ ṗch ṗchv ṗcm ṗmin 166 Sum of domestically produced and imported consumption (A.159) Stock of dwellings (A.10) Four-quarter growth rate of the CPI (A.138) Stockbuilding (including alignment adjustment) (A.46) User cost of dwellings (A.11) Dividend payments to households (A.25) Stock of dwellings brought into the current period (A.7) Private sector employment index (A.31) Rate of employers’ total social contributions, private sector (A.101) Rate of employers’ total social contributions, general government (A.102) General government employment (A.84) Private sector hours worked (A.71) Private sector employment (A.65) Production function output (A.26) Volume of government procurement of private sector goods and services (A.82) Volume of government procurement of private sector goods and services (consumption goods) (A.88) Government demand for resources (opportunity cost of government labour) (A.112) General government gross operating surplus (A.89) Government procurement target (A.83) Value-added component of total general government procurement (A.145) Habit level for non-durable consumption goods (A.18) Habit level for dwellings (A.19) Human wealth (A.5) Volume of investment in dwellings (A.12) Volume of government procurement of investment goods (A.86) Government procurement of investment goods target (A.87) Volume of total business investment (A.160) Volume of domestically produced investment (A.32) Value-added component of domestically produced investment (A.143) Volume of directly imported investment (A.33) Volume of other investment (A.149) Value-added component of other investment (A.144) Capital stock (A.27) Volume of domestically produced capital goods (A.28) Volume of directly imported capital goods (A.29) Labour supply (participation) (A.14) Volume of intermediate imports of goods and services (A.113) Stock of money holdings (A.13) Marginal propensity to consume (A.2) Share of government employment in total labour supply (A.68) Stock of foreign bonds denominated in terms of consumption goods (A.150) Quarterly inflation rate of consumption goods (excluding actual and imputed rents) (A.110) Quarterly rate of inflation of domestically produced consumption goods (A.155) Quarterly rate of inflation of the value-added component of domestically produced consumption goods (A.153) Quarterly rate of inflation of directly imported consumption goods (A.156) Quarterly inflation rate of intermediate imports (A.154) The core model pbpa pc pch pchv pcm pcmnew pdv pg pgv pio piov pkh pkhv pkm pkmnew pmin pminew psv px pxv py pym q rg rk rmc s star surp τ lumpc tax taxd taxee taxef taxeu taxf taxind taxk taxlumpc taxlumpk taxw trans transc transec transf Relative price of the basic price adjustment (A.141) Numeraire price (consumption) (A.130) and (A.131) Relative price of domestically produced consumption goods (A.132) Relative price of the value-added component of domestically produced consumption goods (A.36) Relative price of directly imported consumption goods (A.116) Newly set imported consumption goods price (A.119) Relative price of dwellings investment (A.40) Relative price of government procurement of private sector goods and services (A.135) Relative price of the value-added component of government procurement of private sector goods and services (A.38) Relative price of other investment (A.134) Relative price of the value-added component of other investment (A.137) Relative price of domestically produced capital goods (A.133) Relative price of the value-added component of domestically produced capital goods (A.37) Relative price of directly imported capital goods (A.117) Newly set imported capital goods price (A.120) Relative price of intermediate imports (A.115) Newly set intermediate import goods price (A.118) Relative price of stockbuilding (A.41) Relative price of exports (A.136) Relative price of the value-added component of exports (A.39) Relative price of private sector value added at basic prices (A.140) Relative price of private sector value added at market prices (A.139) Real exchange rate using consumer prices (A.15) Short-term nominal interest rate (A.110) Nominal corporate bond yield (A.17) Real marginal cost (in terms of consumption goods) (A.42) Stock of inventories (A.44) Target stock of inventories (A.43) Measure of firms’ surplus used in wage bargaining (A.69) Effective lump sum tax rate on households (A.80) Total taxation receipts (A.90) Revenue from tax on dwellings (A.93) Employees’ National Insurance Contributions (A.92) Employers’ National Insurance Contributions (A.94) Indirect taxes minus subsidies paid to EU (A.100) Tax revenue from overseas residents (A.98) Revenue from indirect taxation (A.99) Revenue from corporation tax (A.95) Revenue from lump sum taxes on households (A.96) Revenue from lump sum taxes on firms (A.97) Revenue from labour income taxes (A.91) Total general government transfers (A.103) General government transfer payments to households excluding unemployment benefit (A.104) Employers’ other social contributions, general government (A.109) General government transfers to overseas sector (A.107) 167 The Bank of England Quarterly Model transfp transk transkc transkf transkp transksubs transu trw u v w ẇ wa wealth wg wgtar wl wnew x xm xmca xv y yd ym ystar z Net overseas transfers to households (A.20) General government transfers to firms (A.106) Supernormal profit transfers from firms to households (A.48) Net transfers from firms to overseas (A.50) Employers’ other social contributions, private sector (A.49) General government subsidies on products (A.108) Total unemployment benefits (A.105) Transfer wealth (A.6) Unemployment rate (A.66) Value of equities (A.16) Private sector real wage (A.64) Quarterly growth rate of nominal private sector wages (A.157) Alternative wage used in wage bargain (A.61) Total household wealth (A.3) Government wage (A.67) Target level of government’s wage bill (A.85) Expected return from labour market participation (A.62) Newly set private sector real wage (A.63) Volume of exports (A.114) Net expenditure on overseas goods and services (A.151) Current account balance, plus net capital transfers from overseas (A.152) Value-added component of export volumes (A.146) Private sector value added (A.30) and (A.42) Volume of final demand (A.45) Private sector value added at market prices (A.147) Potential output (A.111) Capital utilisation (A.34) Table A.2: Exogenous variables avhstar cf cpiwedge λ̇ ss λ̇ ṅ ṅss ṗ f ṗ f ss ṗss pxf rf rfprem rgprem τc τd τ ee 168 Long-run average weekly hours worked in the private sector World trade Wedge between the non-durable consumption deflator and CPI inflation rates Labour-augmenting productivity growth Steady-state labour-augmenting productivity growth Population growth Steady-state population growth Overseas rate of consumer price inflation Overseas inflation target Domestic inflation target Relative price of world exports M6 short-term nominal interest rate Premium on overseas interest payments to households Premium on government interest payments to households Effective net indirect tax rate (ratio of basic price adjustment to value added) Effective tax rate on dwellings Effective rate of employees’ National Insurance Contributions The core model τ ef τ eu τf τ knd τ lumpk τw tfp trc trec trf trfp trk trkf trkp trksubs wmargin ẏ ẏss Effective rate of employers’ National Insurance Contributions Effective tax rate on EU net indirect taxes Effective tax rate on revenue from overseas residents Effective corporation tax rate Effective lump sum tax rate on firms Effective income tax rate Total factor productivity term in production function General government transfer rate to households (excluding unemployment benefit) Rate of employers’ other social contributions, general government General government transfer rate to overseas Rate of net transfers from overseas to households General government transfer rate to firms Transfer rate from firms to overseas Transfer rate from firms to households Subsidy rate from general government to firms Margin of UK import prices over sterling-denominated world export prices Growth rate of productive potential (λ̇ and ṅ) Steady-state output growth Table A.3: Parameters α β β hw χd χ dels χ kh χ km χl χ pch χ pd χ pg χ pkh χ px χz δd δ kh δ km kh km pchdot pcm pcmdot pddot pgdot pkhdot pkm Share parameter for capital in production function Households’ discount factor Human wealth overdiscounting parameter Weight on adjustment costs for dwellings investment Weight on adjustment costs for stockbuilding Weight on adjustment costs for domestically produced capital investment Weight on adjustment costs for directly imported capital investment Weight on adjustment costs for employment Weight on price adjustment costs (domestically produced consumption goods) Weight on price adjustment costs (investment in dwellings) Weight on price adjustment costs (government procurement) Weight on price adjustment costs (domestically produced capital goods) Weight on price adjustment costs (exports) Weight of adjustment costs for capital utilisation Depreciation rate on dwellings Depreciation rate on domestically produced capital goods Depreciation rate on directly imported capital goods Weight on lagged investment in target investment rate for domestically produced capital goods Weight on lagged investment in target investment rate for directly imported capital goods Weight on lagged inflation in target price increase for domestically produced consumption goods Weight on world prices for directly imported consumption goods Weight on lagged inflation in target price increase for directly imported consumption goods Weight on lagged inflation in target price increase for investment in dwellings Weight on lagged inflation in target price increase for government procurement Weight on lagged inflation in target price increase for domestcially produced capital goods Weight on world prices for directly imported capital goods 169 The Bank of England Quarterly Model pkmdot pmidot pxdot wdot ηc ηcm ηd ηg ηk ηkm ηl ηmi η px ηw ηx γ γk γ mi γ pcm γ pkm γu γw κc κ chv κ gv κ ikhv κ iov κl κ pcm κ pkm κ pmi n κx κ xv µbenw µbgy µbkv µgy µigy µs µwg µwgy φ φc φk φm φz ψc ψ cir 170 Weight on lagged inflation in target price increase for directly imported capital goods Weight on lagged inflation in target price increase for imported intermediate goods Weight on lagged world inflation in target price increase for exports Weight on lagged private sector wage inflation in target nominal private sector wage increase Price elasticity of demand for domestically produced consumption goods Price elasticity of demand for directly imported consumption goods Price elasticity of demand for investment in dwellings Price elasticity of demand for government procurement Price elasticity of demand for domestically produced capital goods Price elasticity of demand for directly imported capital goods Slope parameter in reservation wage distribution Price elasticity of demand for imported intermediate goods Real exchange rate elasticity of overseas demand for domestic exports Elasticity of demand for labour from different unions Price elasticity of demand for exports Household over-discounting parameter (probability of survival) Firms’ over-discounting parameter Probability of resetting imported intermediate prices Probability of resetting directly imported consumption prices Probability of resetting directly imported capital good prices Transition probability for unemployment Probability of rebargaining wage Scale parameter on consumption basket Share of value added in domestically produced consumption goods Share of value added in government procurement Share of value added in domestically produced investment goods Share of value added in other investment goods Scale parameter in labour supply curve Scale parameter for relative price of directly imported consumption Scale parameter for relative price of directly imported capital Scale parameter for relative price of imported intermediates Scale parameter in export demand equation Share of value added in exports Replacement ratio Steady-state government debt to output ratio Corporate sector debt-equity ratio Steady-state government procurement to private sector output ratio Steady-state government investment to private sector output ratio Stockbuilding to final demand ratio Ratio of government wages to private sector wages Steady-state government wage bill to output ratio Share parameter for capital in production Share parameter for non-durable consumption in consumption aggregator Share parameter for domestically produced capital in capital aggregator Share parameter for directly imported consumption goods in consumption aggregator Elasticity of utilisation costs Share parameter for non-durables in consumption aggregator Ratio of actual and imputed rentals to the stock of dwellings The core model ψe ψ gosg ψ hab ψ habd ψ io ψk ψm ψ mon ψ pi o ψs ψ snp ψu ρ ch ρg ρ i kh ρx σc σd σk σm σy θ bg θ bk θ bp θ dbg θg θ pdot θ rg θ wg θy Union preferences of employment over wages Ratio of government gross operating surplus to private sector output Weight of non-durable consumption habits in utility function Weight of dwellings habits in utility function Share of other investment in private sector output Share parameter for home capital in capital aggregator Share parameter for directly imported consumption goods in consumption aggregator Inverse of weight on real money balances in utility function Parameter determining the relative price of other investment goods Elasticity of target inventory stocks level Share of supernormal profits transferred to consumers from firms Union bargaining power Weight of intermediate imports in domestically produced consumption goods Weight of intermediate imports in government procurement Weight of intermediate imports in domestically produced capital goods Weight of intermediate imports in exports Intertemporal substitution of consumption Elasticity of substitution between consumption and dwellings in utility bundle Elasticity of substitution between imported and domestically produced capital in the capital aggregator Elasticity of substitution between imported and domestically produced consumption in consumption basket Elasticity of substitution between capital and labour in private sector production Coefficient on government debt gap in fiscal reaction function Partial adjustment coefficient in the corporate debt rule Partial adjustment coefficient for the relative price of the basic price adjustment Coefficient on government debt change in fiscal reaction function Partial adjustment coefficient in the government procurement reaction function Coefficient on inflation gap in monetary reaction function Interest rate smoothing coefficient in monetary reaction function Partial adjustment coefficient in the government wage bill reaction function Feedback from output gap in monetary reaction function Table A.4: Working variables ξ cm ξ cmmc ξd ξ dsur p ξ gai n ξ hwndot ξ kh ξ km ξ kmmc ξl ξ mi n ξ mi nmc Discounted flow of expected imported consumption demand (A.125) Discounted flow of expected imported consumption costs (A.124) Relative price contribution to dwellings demand (A.21) Derivative of firms’ surplus used in wage bargaining (A.78) Expected value of excess returns on asset holdings (A.23) Population adjustment for human wealth equation (A.24) Costs of adjusting domestically produced capital (A.53) Costs of adjusting imported capital (A.54) Discounted flow of expected imported capital costs (A.127) Costs of adjusting labour (A.52) Discounted flow of expected imported intermediate demand (A.122) Discounted flow of expected imported intermediate costs (A.121) 171 The Bank of England Quarterly Model ξ mon ξ pch ξ pcmdot ξ pd ξ pg ξ pkh ξ pkm ξ pkmdot ξ pmidot ξ px ξ sur p ξw ξ wa ξ wdot ξ wlag ξ wnew ξ yd ξ ydot Relative price component of the real money balances-consumption ratio (A.22) Costs of adjusting domestically produced consumption goods prices (A.55) Weight on lagged imported consumption price inflation in average price of imported consumption goods (A.126) Costs of adjusting the price of dwellings (A.59) Costs of adjusting the price of government procurement (A.58) Costs of adjusting the price of domestically produced capital goods (A.56) Discounted flow of expected imported capital demand (A.128) Weight on lagged imported capital goods price inflation in average price of imported capital goods (A.129) Weight on lagged intermediate import price inflation in average price of imported intermediates (A.123) Costs of adjusting export prices (A.57) Discounted flow of expected firms’ surpluses used in wage bargaining (A.77) Value to unions of discounted flow of expected returns from newly set wage (A.72) Value to unions of discounted flow of expected returns from outside wage (A.73) Indexation factor for unadjusted wage rates (A.76) Effect of unadjusted wages on average wage (A.75) Effect of newly set wages on average wage (A.74) Expected sales in stocks equation (A.51) Expected growth scalar adjustment (A.60) A.2 Core model equations This section presents the equations of the core model. Here we often refer to equations by names that are conventional in the literature. However, these labels are somewhat arbitrary in a simultaneous system: consumption is not determined solely by wealth, even though we refer to equation (A.1) as the ‘consumption function’, since the core paths for consumption and wealth are jointly determined by the core model. Nevertheless, we find it convenient to group certain equations under headings such as ‘households’ and ‘firms’, to collect together the equations arising from the decision problems of these agents. A number of technical points are worth noting: • the equations are written in detrended model units. As explained in Box 12 on page 88, this means that terms involving leads and lags will generally be scaled according to growth and/or inflation. We do not comment on these growth terms unless they are particularly important for the interpretation of the equation; • similarly, Chapter 5 describes how we choose (without loss of generality) the price of the non-durable consumption bundle (PC) as the numeraire. This implies that the relative price of non-durable consumption, pc, is 1 in every period – see equation (A.130) below. Often this is imposed implicitly in the equations (ie we eliminate terms in pc), but it is included explicitly when particularly important for intuition. For example, the consumption function (A.1) is a statement about consumption expenditure, so we include pc in this case; • we use the steady-state values of some variables, which we denote by the superscript ss; and • we make use of a number of working variables in the equation listing. These variables – bearing the prefix ξ – are generally used to make other expressions more compact. 172 The core model A2.1 Households The consumption function (A.1) arises from the maximisation problem for individuals (see Box 2 on page 30 for the household maximisation problem). It expresses desired total consumption of non-durable goods (c) as a simple linear function of wealth, given preferences for other expenditures that yield utility, such as dwellings and money. ct = mpct wealth t pct (A.1) Equation (A.2) is the marginal propensity to consume out of wealth (mpc), which is derived from the first order conditions of households. For compactness it is expressed as a forward difference equation, but can also be expressed as a function of the discounted sum of present and future interest rates (rg), consumption prices ( pc), the user cost of dwellings (duser), as well as the structural preference parameters for households’ discount factor (β), expected survival rate (γ ), the elasticity of intertemporal substitution (σ c ), and the weight on real money balances in utility (ψ mon ). mpct−1 ξ mon dusert d = 1+ ξ t + t mon pct ψ −σ c −1 +mpct+1 γ γ 1− 1 + rgt c ξ mon ψ hab 1−σ σc β t+1 mon (1 + ẏt+1 ) ξt σc 1−σ c pct 1 + rgt pct+1 1 + ṗt+1 (σ c −1) (A.2) Wealth (A.3) is derived in the conventional way by iterating over individuals’ period-by-period budget constraints, and then aggregating over the whole population. An individual’s wealth is defined as all the resources that are available for expenditure at the start of a period. Financial assets include money (mon), the domestic currency value of foreign bonds (b f /q), corporate bonds (bk), government bonds (bg), and shares (valued at price v), plus the interest returns and dividends (dv) arising from holding these instruments over from the previous period. Non-financial assets include human wealth (hw), transfer wealth (trw), and the value of dwellings (dw). One non-standard term appearing in the consumption function is the expected excess returns from asset holdings – this term (ξ gai n ) is given by equation (A.23) and described in more detail below. wealth t = 1 + rgt−1 pgt−1 bgt−1 1 + rkt−1 bkt−1 1 + r f t−1 pct b f t−1 + + f 1 + ṗt 1 + ẏt 1 + ṗt 1 + ẏt 1 + ṗt qt 1 + ẏt pct−1 mon t−1 gain + + v t + dv t + hwt + trwt + dwt + ξ t (A.3) (1 + ẏt ) (1 + ṗt ) The household budget constraint (A.4) is an aggregation over all individuals’ constraints. Households receive post-tax income from participating in the labour market (wl · l), and net transfers (which may be positive or negative) from the government (transc), from firms (transkc, transkp) and from foreigners (transkf, transec). They carry forward financial assets from the previous period inclusive of interest payments and dividends, as well as real money balances. The terms rfprem and rgprem represent net returns from foreign and government bond investment over and above the interest return. (1) With these resources households make expenditures on consumption, net dwellings (d) investment, new money stocks, and new investments in financial assets. Finally, they make tax payments to the government, consisting of a lump sum tax (taxlumpc) and a tax on the dwellings stock (levied at rate τ d ). The stock of dwellings is part of the definition of wealth (A.3), in terms of resources fungible for consumption, but (1) These terms therefore represent unanticipated ‘excess returns’ relative to the no-arbitrage conditions in the previous period. 173 The Bank of England Quarterly Model is not wealth in terms of the household budget constraint, where dwellings investment is an expenditure choice. b ft pct + pgt bgt + v t + bkt qt = 1 + r f t−1 b f t−1 pct pct−1 mon t−1 − pct mon t + f (1 + ẏt ) (1 + ṗt ) 1 + ṗt 1 + ẏt qt 1 + rgt−1 pgt−1 bgt−1 1 + rkt−1 bkt−1 + + 1 + ṗt 1 + ẏt 1 + ṗt 1 + ẏt +v t + dv t + wlt · lt − pct ct + transct + transkct +transkpt + transfpt + rfpremt + rgprem t −taxlumpct + transect (1 − δ dt )dt−1 − pdv t dt − 1 + ẏt − τ dt pdv t dt (A.4) Human wealth (A.5) is aggregated over individual consumers, defined as the present value of post-tax labour income, factoring in expected lifetimes. The aggregation includes adjustments for the way that wage profiles vary over age groups (β hw ) and the population distribution (ξ hwndot ), which is described in equation (A.24) below. wlt lt + β hw γ 1 + λ̇t+1 hwt = ξ hwndot t 1 + ṗt+1 hwt+1 hwndot 1 + rgt ξ t+1 (A.5) Transfer wealth (A.6) is derived, in a similar way to human wealth, as the present value to individuals of future transfer payments from firms and the government. Also included are additional bond returns (which represent transfers from/to the government and overseas). trwt = transct + transkct + transkpt + transfpt + rgprem t + rfpremt 1 + ṗt+1 trwt+1 −taxlumpct + transect + γ (1 + λ̇t+1 ) 1 + rgt (A.6) The value of dwellings wealth (A.7) is the current value of the dwellings stock held over from the previous period, net of depreciation. The aggregate expression is derived under the assumption that the dwellings of those who die are distributed directly to newborn individuals. dwt = (1 − δ dt ) pdv t dt−1 1 + ẏt (A.7) Non-durable consumption (A.8) is defined as a constant elasticity of substitution (CES) aggregate of domestically produced consumption (ch) and imported consumption (cm). The parameters φ m and ψ m are important in determining the share of overall consumption expenditure on home and imported consumption and the parameter σ c measures the elasticity of substitution between the two. ct = κ c 1 − ψm 1 − φ m ch t 1− 1m σ + ψ m φ m cm t 1− 1m σ σm σ m −1 (A.8) The home-imported consumption ratio (A.9) is derived from the utility-maximising allocation, given a budget constraint and the assumption of CES preferences in equation (A.8). Hence this expression depends on their relative prices and the parameters governing the preferences for each. ch t = cm t 174 1 − φm φm σ m −1 pcm t 1 − ψ m pch t ψ m σm (A.9) The core model The dwellings stock (A.10) is derived from the first-order optimality conditions for consumption of non-durables and dwellings services, given CES preferences over both. The variable ξ d , defined in equation (A.21), determines how the stock of dwellings responds to movements in relative prices and the user cost of dwellings. dt = ξ dt ct (A.10) The user cost of dwellings (A.11) is a conventional function of the capital gain from price changes adjusting for depreciation and taxation. This expression helps to compact some of the other equations. dusert = pdv t 1 − 1 − δ dt+1 γ pdv t+1 1 + ṗt+1 pdv t 1 + rgt + pdv t τ dt (A.11) The demand for housing investment (A.12) is defined as the net addition to the dwellings stock, adjusted for depreciation. (1 − δ dt )dt−1 idt = dt − (A.12) 1 + ẏt The demand for money (A.13) relates the desired money stock to the flow of non-durable consumption. The variable ξ mon , defined in equation (A.22), determines how the ratio of real money balances to consumption is affected by the weights on money and consumption in utility, the elasticity of intertemporal (consumption) substitution, the user cost of dwellings, and interest rates. Money balances are not insured, so the optimal level will depend on the survival probability (γ ). As with dwellings, to derive the aggregate equation, we assume that money balances of those who die at the end of the previous period are transferred to newborns. ξ mon mon t = t mon ψ −σ c γ 1− 1 + rgt −σ c ct (A.13) The labour supply curve (A.14) is a simple function of a household’s expected return from participation in the labour market (wl), relative to the price of the non-durable consumption bundle ( pc). The parameter ηl represents the elasticity of participation with respect to this real wage measure. Underlying this function is an assumption about the distribution of reservation wages among households. log lt = κ l + ηl log wlt pct (A.14) Each household’s first-order conditions for asset holdings deliver a number of no-arbitrage conditions. The first-order conditions for government and foreign bonds deliver the conventional real uncovered interest parity condition (A.15) which says that the expected change in the real exchange rate (q) is equal to the real interest rate differential between foreign and government bonds. (2) f qt = qt+1 pct 1 + rgt 1 + ṗt+1 pct+1 1 + ṗt+1 1 + r f t (A.15) (2) Here rg is the nominal interest rate on government bonds so that (1 + rgt ) pct / 1 + ṗt+1 pct+1 represents the (gross) expected real return. The nominal interest rate on foreign bonds is r f and foreign inflation is ṗ f . 175 The Bank of England Quarterly Model The value of equity is priced by the conventional dividend discount model and equation (A.16) is derived from the combination of the first-order conditions for equity holdings and government bonds. This equation can be rewritten to express the value of equity as being the present value of future dividend payments (dv). Equation (A.16) includes an adjustment for individual households’ over-discounting, to reflect the assumption that equity holdings are uninsured, as explained in Box 4 on page 42. 1 + ṗt+1 vt = γ (A.16) (v t+1 + dv t+1 ) (1 + ẏt+1 ) 1 + rgt Equation (A.17) combines the first-order conditions for government and corporate debt. The nominal interest rate on corporate debt (rk) is equal to the yield on riskless government debt, adjusted for a premium reflecting the fact that corporate debt claims are assumed to be uninsurable against death (again, see Box 4). 1 + rgt 1 + rkt = (A.17) γ As described in Chapter 3, the core model incorporates external habit formation. The habit variable (A.18) is equal to be the previous period’s consumption level. (1 + ẏtss )ct−1 (A.18) habt = 1 + ẏt The dwellings habit variable (A.19) is set equal to the non-durable habit variable. This simplifying assumption means that the demand for housing is not a function of the lagged consumption-dwellings ratio. habdt = habt (A.19) Transfers from overseas to households (A.20) are defined by an effective rate applied to private sector output. transfpt = trfpt pyt yt (A.20) A number of working variables are used in the equations describing the behaviour of households. Equation (A.21) describes the relative price component of the desired dwellings relationship (A.10). The relative price component (ξ d ) depends on the relative levels of non-durable and dwellings habits and on the user cost of housing relative to the price of non-durable consumption. The expression also depends on parameters that determine the relative importance of non-durables and dwellings in utility (ψ c , φ c ), as well as the elasticity of substitution between dwellings and non-durable consumption in the utility function (σ d ). ξ dt = 1 − ψc ψc σd ψ hab (1 − φ c )habt σ d −1 ψ habd φ c habdt σd pct dusert (A.21) Equation (A.22) describes the relative price component of the real money balances-consumption ratio used in equation (A.13). The key determinants are the relative habit levels for non-durable and dwellings consumption and the relative price component of the desired dwellings relationship (ξ d ). This equation is derived by substituting the equation determining the optimal ratio of dwellings and non-durable consumption (A.10) into an expression combining the first-order conditions for consumption and real money balances. ξ mon = ψc t 176 φc ψ hab habt 1− 1d σ + 1 − ψc (1 − φ c ) ξ dt ψ habd habdt 1− 1d σ σ c −σ d σ c σ d −1 ( ) ψc φc ψ hab habt 1− 1d σ (A.22) The core model Equation (A.23) gives the expected value of excess returns on asset holdings. It can be rewritten as the expected discounted flow of excess returns. When the no-arbitrage conditions (A.15), (A.16) and (A.17) hold, it can be seen that the expressions in square parentheses are equal to zero and so there are no excess returns. However, this term is required for the consumption function (A.1) to remain valid when these conditions are overwritten, perhaps when imposing conditioning assumptions for the forecast (as described in Chapter 5 and in Box 11 on page 80). (3) ξt gai n ⎡ ⎤ q + r f + ṗ b ft 1 + rkt ) ) (1 (1 t t t+1 = ⎣ − 1⎦ + γ − 1 bkt f qt 1 + rgt qt+1 (1 + rgt ) 1 + ṗt+1 + γ (A.23) 1 + ṗt+1 v t+1 + dv t+1 (1 + ṗt+1 ) (1 + ẏt+1 ) gain − 1 vt + γ ξ t+1 (1 + ẏt+1 ) 1 + rgt vt 1 + rgt Equation (A.24) describes the adjustment to the human wealth equation (A.5) for changes in the composition of human wealth in response to population growth. This reflects an underlying assumption about age-dependence of the distribution of reservation wages (reflected in the parameter β hw ). However, this parameter is used as a technical device to allow greater control over the desired asset holdings in the steady state, and is not intended to match life-cycle effects in the data. ξ thwndot = hw 1 + ṅ ss β hw γ hwndot t − β γ 1 + ṅ t − γ + ξ 1 + ṅ ss 1 + ṅ t 1 + ṅ t t−1 t −γ (A.24) A2.2 Firms Firms act to maximise the expected present value of dividend flows (see Box 7 on page 54 for the firm’s maximisation problem). Equation (A.25) shows that dividends are a function of cash flow (sales less expenses) and taxation on firms’ income. Sales revenue depends on sales volumes in each market – home consumption goods (chv), dwellings investment (id), home and other investment (ikhv, iov), government procurement (gv) and exports (xv) – and the prices that the firm receives for these ( pchv, pdv, pkhv, piov, pgv, pxv). The firm aims to sell at a time-varying mark-up over its real marginal costs, as described in the discussion of the pricing equations (A.36)-(A.40)) below. Expenses include labour costs (the gross wage bill including employers’ social contributions, (1 + ecost) · w · e), total investment ( pkh · ikh + pkm · ikm + piov t iov t ), debt servicing, transfers overseas and to households, and lump-sum taxes. dv t = pchv t chv t + pdv t idt + pkhv t ikhv t + piov t iov t + psv t delst + pgv t gv t + pxv t xv t (1 + rkt−1 ) bkt−1 − (1 + ecostt ) wt et − pkh t ikh t − pkm t ikm t − piot iot − psv t delst − (1 + ẏt ) (1 + ṗt ) +bkt − taxkt − taxlumpkt − transkf t + transkt − transkct (A.25) (3) If the no-arbitrage conditions (A.15), (A.16) and (A.17) are violated, consumers will want to borrow unlimited amounts of the lowest return asset to lend unlimited amounts at the highest rate of return. This implies that the optimal portfolio choice will be driven to a corner solution. Technically, the adjustment factor ξ gai n is only valid under the assumption that consumers hold all domestically supplied assets, even if this is suboptimal. 177 The Bank of England Quarterly Model Firms operate a CES production function (A.26) which combines inputs of labour (e) and a CES bundle of home and imported capital (k). The capital stock can be utilised at a variable rate (z) that is chosen by the firm. In the standard CES function, as the elasticity of factor substitution σ y tends to zero, the shares of capital and labour in the total product each approach one half. To match income shares and marginal products, we use an extended functional form that includes an extra parameter, φ, to control the shares. f t = tfpt (1 − α) {(1 − φ) et avh t }1− σ y + α 1 φz t kt−1 1 + ẏt ) σy σ y −1 1− σ1y (A.26) Equation (A.27) describes the CES function defining the capital index as a function of home and imported capital. In the same way that the production function (A.26) includes an additional parameter to control factor shares as the substitution elasticity approaches zero, the capital index includes the additional parameter ψ k . kt = ψ k φ kh t k 1− 1k σ + 1−ψ 1−φ k k km t σk σ k −1 1− 1k σ (A.27) The domestically produced capital stock cumulates according to a perpetual inventory condition (A.28). The effective depreciation rate depends on the utilisation rate (z) so that the capital stock depreciates faster (slower) if it is used more (less) intensively than the ‘normal’ rate of utilisation (z ss ). At the normal utilisation rate, the depreciation rate is given by δ kh . χz 1+φ z − z tss z zt 1+φ kh t = ikh t + 1 − δ kh t − 1+φ z kh t−1 1 + ẏt (A.28) The imported capital stock (A.29) cumulates in the same way as domestically produced capital. km t = ikm t + 1 − δ km t − χz 1+φ z − z tss z zt 1+φ 1+φ z km t−1 1 + ẏt (A.29) Private sector output (A.30) is given by production ( f ) net of quadratic costs of adjusting capital (ξ kh , ξ km ) and labour (ξ l ). The form of these cost variables is described in the discussion of equations (A.53), (A.54) and (A.52) below. yt = f t − χ kh 1 + ẏtss kh t−1 kh ξt 2 1 + ẏt 2 − χ km (1 + ẏtss )km t−1 km ξt 2 1 + ẏt 2 − χl yt ξ lt 2 2 (A.30) Equation (A.31) is derived from the first-order condition for private sector employment, e. The cost of labour is equated to the marginal product, adjusted for the the firm’s mark-up in product markets (here captured by real marginal cost, rmc) and labour adjustment costs (ξ l ). (1 + ecostt ) wt avh t = rmct (1 − φ) · tfpt +rmct+1 178 1−α 1 σ y −1 ft et avh t 1 σy − rmct χ l l (1 + ṅ t ) yt ξ avh t t (1 + ṅ ss t ) et−1 χ l 1 + ṗt+1 ξ lt+1 yt+1 (1 + ẏt+1 ) (1 + ṅ t+1 ) et+1 2 avh t 1 + rgt (1 + ṅ ss t ) (et ) (A.31) The core model The expression for desired domestically produced capital (A.32) is a first-order condition that relates the marginal product of home capital to the cost of capital. The expression contains the conventional Jorgensonian user-cost elements of the change in home capital goods prices and the depreciation rate. Analogously to the labour demand condition (A.31), adjustments are included for the firm’s mark-up in product markets and for capital adjustment costs. pkh t + rmct χ kh ξ kh 1 + ξ kh t t = ⎡ γ k (1 + ṗt+1 ) ⎢ ⎢ ⎢ ⎣ 1 + rgt 1 + ẏtss kh t−1 (1 + ẏt ) kh t pkh t+1 1 − δ kh t+1 − χz 1+φ z z 1+φ ss − z t+1 z t+1 kh +rmct+1 χ kh ξ kh t+1 1 + ξ t+1 +rmct+1 αψ k (φ k ) 1− 1k σ (φ·tfpt+1 z t+1 ) σ y 1 −1 ss 1 + ẏt+1 f t+1 (1+ ẏt+1 ) kt 1 σy ⎤ 1+φ z kt kh t 1 σk ⎥ ⎥ ⎥ ⎦ (A.32) The expression for desired imported capital (A.33) has the same formulation as the previous equation. = 1 + ẏtss km t−1 pkm t + rmct χ km ξ km 1 + ξ km t t (1 + ẏt ) km t ⎡ z 1+φ z χz ss 1+φ pkm t+1 1 − δ km − z t+1 z t+1 − 1+φ z t+1 ⎢ km ss γ k (1 + ṗt+1 ) ⎢ +r mct+1 χ km ξ km ⎢ t+1 (1 + ξ t+1 )(1 + ẏt+1 ) 1 ⎢ 1 1− (1 + rgt ⎣ f t+1 (1+ ẏt+1 ) σ y α(1−ψ k )(1−φ k ) σ k kt +rmct+1 1 −1 kt km t y σ {φ·tfpt+1 zt+1 } ⎤ 1 σk ⎥ ⎥ ⎥ (A.33) ⎥ ⎦ The first order condition for capital utilisation (A.34) equates the marginal product of utilisation (adjusting for the product market mark-up) to the implied depreciation cost. r mct α tfpt · φ 1− σ1y 1 + ẏt ft z t kt−1 1 σy = χ z z tφ z pkh t kh t−1 + pkm t km t−1 kt−1 (A.34) Average hours (avh) are assumed to be determined exogenously (by the series avhstar) as described by equation (A.35). avh t = avhstart (A.35) The equation for (value-added) domestically produced consumption relative prices (A.36) is derived from a first-order condition of the firm’s optimisation problem. This is a mark-up condition expressing the price as a mark-up on real marginal cost (rmc). The mark-up is a function of the demand elasticity for home consumption goods (ηc ) and terms reflecting price adjustment costs (ξ pch ). A log-linearised version of this equation could be rearranged to look like a New Keynesian Phillips Curve expression, as shown in Box 5 on page 46, though the specification of the adjustment costs in this case implies that lagged inflation appears in the equation. The price adjustment costs (ξ pch ) are described in the discussion of equation (A.55) below. The equation is dynamic because of price adjustment costs – in the absence of price adjustment costs (χ pch = 0) and in the steady state (which implies ξ pch = 0), the condition reduces to a constant mark-up ηc / (ηc − 1) over real marginal cost. ⎡ ⎤−1 pch c pch pch − 1 + χ ξ η 1 + ξ t t ⎦ pchv t = ηc rmct ⎣ (A.36) pch ṗt+1 pchv t+1 chv t+1 pch pch −γ k 1+ ξ + 1 χ ξ + ẏ ) (1 t+1 t+1 t+1 1+rgt pchv t chv t 179 The Bank of England Quarterly Model The pricing equations for the value-added components of home capital goods prices (A.37), government procurement prices (A.38) and export prices (A.39) have the same form as (A.36), reflecting the incorporation of the same form of price adjustment costs in the optimisation problem. ⎡ pkhv t = ηk rmct ⎣ ηk − 1 + χ pkh ξ t pkh pgv t = η rmct pkh ṗt+1 t+1 ikhv t+1 −γ k 1+ (1 + ẏt+1 ) χ pkh ξ t+1 + 1 ξ t+1 pkhv 1+rgt pkhv t ikhv t pkh ṗt+1 −γ k 1+ 1+rgt pxv t = η r mct x pkh η g − 1 + χ pg ξ t 1 + ξ t pg pg t+1 gv t+1 (1 + ẏt+1 ) χ pg 1 + ξ t+1 ξ t+1 pgvpgv t gv t pg g 1 + ξt pg η x − 1 + χ px ξ t (1 + ξ t ) px px k 1+ ṗt+1 px −γ 1+rgt χ (1 + ẏt+1 ) ξ t+1 1 + ξ t+1 px ⎤−1 ⎦ (A.37) −1 (A.38) −1 px pxv t+1 xt+1 pxv t xt (A.39) The expression for the relative dwellings price (A.40) contains additional terms capturing the assumption that firms also face costs of adjusting their supply, which are introduced to capture the assumption that the short-run supply of housing investment is relatively inelastic. ⎡ ⎤−1 pd ẏt ) idt (1+ ẏt ) pd pd ηd − 1 − ηd χ d idi dt (1+ ξ − 1 + χ 1 + ξ ss ss t t idt−1 (1+ ẏt ) t−1 (1+ ẏt ) ⎢ ⎥ pd pd (1+ ẏt+1 ) pdv t+1 i dt+1 ⎢ ⎥ k 1+ ṗt+1 pd d ξ t+1 + 1 ξ t+1 −γ 1+rgt χ pdv t = η r mct ⎢ ⎥ (A.40) pdv t idt ⎣ ⎦ 2 i dt+1 (1+ ẏt+1 ) pdv t+1 k 1+ ṗt+1 d d idt+1 (1+ ẏt+1 ) +γ 1+rgt η χ −1 idt ·(1+ ẏtss ) i dt (1+ ẏtss ) pdv t The price of inventories (A.41) is not determined by optimisation by firms, but is set equal to the average price of the other goods sold by firms. psv t = ( pchv t chv t + pdv t idt + pkhv t ikhv t + piov t iov t + pgv t gv t + pxv t xv t ) /ydt (A.41) Equation (A.42) is the production clearing condition which says that all output from production must be allocated among the markets in which firms sell. When deciding optimal prices and factor demand (as described above), an individual firm is constrained by this condition. The real marginal cost (rmc) variable that appears in the first-order conditions of the model represents the Lagrange multiplier on this constraint. yt = chv t + idt + ikhv t + iov t + delst + gv t + xv t (A.42) Target inventory stocks (star) are based on firms’ choice, balancing between the costs of forgone sales and of storage. As explained in Chapter 3, there is no role for inventories as insurance against unanticipated demand movements, so equation (A.43) is instead imposed, based on the model of Kahn (1987). start + ydt − ξ t−1 ξ t−1 / (1 + ẏt ) µs ydt yd 180 ydot 1 ψs = 1− γk 1 − rmct / pyt (1 + ṗt+1 ) rmct+1 / {pyt+1 (1 + rgt )} (A.43) The core model The level of actual inventories (s) is generally not at the target level in each period and equation (A.44) describes the determination of inventory stocks given quadratic costs of adjustment. The parameter χ dels measures the size of adjustment costs (so that setting χ dels = 0 implies that inventories are always at target). st 1 + χ dels + χ dels 1 + ṗt+1 1 + ẏtss 1 + rgt 1 + ṗt+1 (1 + ẏt+1 ) χ dels st+1 1 + rgt 1 + ẏtss +χ dels st−1 (A.44) 1 + ẏt = start + The level of private sector demand excluding stockbuilding is given by equation (A.45). ydt = chv t + idt + ikhv t + iov t + gv t + xv t The rate of change of stocks – stockbuilding (dels) – is defined by the identity (A.46). st−1 delst = st − 1 + ẏt (A.45) (A.46) Corporate debt issuance (A.47) is imposed to avoid ‘corner solutions’ in which the firm is financed entirely by either debt or equity. We assume that the stock of corporate bonds depends on the value of equity and the level of bonds in the previous period. bkt = 1 − θ bk µbkv v t + θ bk bkt−1 (A.47) Supernormal profit transfers from firms to consumers (A.48) are given by a proportion (ψ snp ) of the value of private sector output. transkct = ψ snp pyt yt (A.48) Employers’ (other) private social contributions to households (A.49) are given by the application of a transfer weight (trkp) to the pre-tax private sector wage bill. transkpt = trkpt · wt et (A.49) Transfers from firms to overseas are defined by equation (A.50). transkf t = trkf t · pyt yt (A.50) We now turn to the working variables that appear in the firms’ equations. The desired stock level – star described in equation (A.43) – depends on the lagged expectation of demand (excluding stockbuilding) one period ahead (A.51), the model-consistent solution for the level prevailing next period, absent any shocks. yd ξ t = ydt+1 (A.51) The labour adjustment cost (A.52) is defined as the net change in employment. ξ lt = (1 + ṅ t ) et −1 (1 + ṅ ss t ) et−1 (A.52) 181 The Bank of England Quarterly Model The domestically produced capital adjustment cost (A.53) is a function of the net and lagged changes in the home capital stock. The parameter kh determines the extent to which lagged changes in the capital stock affect adjustment costs. When kh = 0 capital adjustment costs depend only on the change in the capital stock, but when 0 < kh < 1 the lagged change in the capital stock also matter for adjustment costs. This specification means that the first order condition for home capital can depend on lagged investment and is similar to the specification of external habit formation in consumption. (1 + ẏt ) kh t /kh t−1 ξ kh −1 (A.53) t = kh kh 1− {kh t−1 (1 + ẏt−1 ) /kh t−2 } (1 + ẏtss ) The imported capital adjustment cost (A.54) is defined analogously to the cost for home capital – as a function of the net and lagged changes in the imported capital stock. (1 + ẏt ) km t /km t−1 −1 (A.54) ξ km t = km km 1− (1 + ẏtss ) (km t−1 (1 + ẏt−1 ) /km t−2 ) The cost of adjusting domestically produced consumption prices (A.55) has the same form as the capital adjustment costs of equations (A.53) and (A.54). The cost depends on the deviation between nominal home consumption price inflation ((1 + ṗt ) pchv t / pchv t−1 ) and a weighted average of steady-state inflation and lagged home consumption price inflation. The parameter pchdot controls the weighting. When pchdot = 0, the pricing equation (A.36) can be log-linearised to deliver a New Keynesian Phillips curve representation in a very similar way to the example in Box 5 on page 46. When pchdot takes a value between 0 and 1, the log-linearised relationship also contains a term in lagged home consumption price inflation. (1 + ṗt ) pchv t / pchv t−1 pch ξt = −1 (A.55) pchdot ss 1− pchdot (1 + ṗt ) ((1 + ṗt−1 ) pchv t−1 / pchv t−2 ) The cost of adjusting domestically produced capital goods prices (A.56) has the same form as the equation for ξ pch . ξt pkh = (1 + ṗtss )1− (1 + ṗt ) pkhv t / pkhv t−1 pkhdot ((1 + ṗt−1 ) pkhv t−1 / pkhv t−2 ) pkhdot −1 (A.56) The cost of adjusting export prices (A.57) has a similar form, though two differences are important. First, the adjustment cost is defined in terms of the foreign currency price of export goods, reflecting the assumption that exporters set their prices in foreign currency. Second, when px takes a value between 0 and 1, the lagged price inflation that matters is world export price inflation (in foreign currency). This reflects the assumption that exporters respond to pricing developments in world markets. px ξt f = 1 + ṗt 1+ f ss ṗt 1− pxv t qt / ( pxv t−1 qt−1 ) pxdot 1+ f ṗt−1 pxf t−1 /pxf t−2 pxdot −1 (A.57) The costs of adjusting prices of government procurement (A.58) and dwellings prices (A.59) have the same form as the equation for ξ pch . (1 + ṗt ) pgv t / pgv t−1 pg ξt = −1 (A.58) pgdot pgdot (1 + ṗtss )1− ((1 + ṗt−1 ) pgv t−1 / pgv t−2 ) (1 + ṗt ) pdv t / pdv t−1 pd ξt = −1 (A.59) pddot ss 1− pddot (1 + ṗt ) ((1 + ṗt−1 ) pdv t−1 / pdv t−2 ) 182 The core model The desired stock level – star described in equation (A.43) – depends on the lagged expectation of supply growth one-period ahead (A.60), which is the model-consistent solution for the level prevailing next period, absent any shocks. ydot ξt = 1 + ẏt+1 (A.60) A2.3 Wage bargaining Wage bargaining takes place to maximise a Nash maximand of firms’ and unions’ surpluses – see Box 8 on page 58 for a description of the optimisation problem. The outside wage (A.61) is a weighted average of the post-tax wages from (private and public sector) employment and unemployment benefit. The weights are related to the unemployment rate (u), the rate of employment in the public sector (µeg ) and a parameter (γ u ) that summarises the likelihood of unemployed workers re-entering the pool of available labour. wat = 1 − γ u u t − γ u µt eg ee wt + γ u u t ben t + γ u µt 1 − τw t − τt eg ee wgt 1 − τw t − τt (A.61) Equation (A.62) shows that the expected wage from participation is a weighted average of the post-tax wages from (private and public sector) employment and unemployment benefit. The weights are given by the rates of employment (in private and public sectors) and unemployment respectively. When the parameter γ u = 1 in equation (A.61), then wa = wl. wlt = 1 − u t − µt eg ee wt + u t ben t + µt 1 − τw t − τt eg ee wgt 1 − τw t − τt (A.62) The first-order condition for the wage bargain (A.63) determines the equilibrium real wage. The expression is derived from the equilibrium condition of the Nash bargain and equates the marginal benefit to unions of an increase in the wage rate (left-hand side of the expression) to the marginal cost to firms of that increase (right-hand side). In a static framework, the marginal benefit would be an increasing function of the difference between the bargained wage and the outside alternative. Given that nominal wage contracts are negotiated infrequently, the marginal benefit depends on the expected wat evolution of the difference between ξ w and this also explains the extensive use of working t and ξ t variables. The marginal cost to firms is a function of the expected evolution of labour costs relative to revenues. dsur p w e wa (1 − ηw ψ e ) ξ w t + η ψ ξt u ξt ψu = − 1 − ψ (A.63) sur p wa ξw ξt t − ξt The average wage is a weighted average of newly set wages and unadjusted wages. Equation (A.64) transforms this relationship using working variables described below – see equations (A.74) and (A.75). Newly set wages (on which ξ wnew depends) arise out of the wage bargain; unadjusted wages (on which ξ wlag depends) are assumed to be indexed by a weighted average of steady-state and past nominal wage inflation. The weight on newly set wages, γ w , is the Calvo adjustment probability (the probability that wages are reset in any given period). When γ w = 1 nominal wages are perfectly flexible and newly set and average wages coincide. wlag 1 = γ w ξ wnew + (1 − γ w ) ξ t t (A.64) 183 The Bank of England Quarterly Model The theoretical structure of the core model assumes that total employment is an index over heterogeneous labour types (each represented by a union – see Box 8 on page 58). Equation (A.65) shows how this index is related to private sector employment. The Calvo adjustment probability plays a role here as well, when; γ w = 1 (flexible nominal wages) the employment index and the private sector employment rate will coincide. en t = γ w wnewt wt −ηw et et + (1 − γ ) et−1 w wt (1 + ṗt ) 1 + λ̇t ηw ξ wdot wt−1 t en t−1 (A.65) The unemployment rate (A.66) measures the proportion of the participating labour force (l) not employed in the private sector (en) or public sector (eg). ut = lt − en t − egt lt (A.66) The public sector wage (A.67) is related to the private sector wage according to the factor µwg . wgt = µwg wt The share of public sector employment in participation (A.68) is defined as: egt eg µt = lt (A.67) (A.68) The firm’s surplus (A.69), over which the wage bargain is made, is defined as sales revenue less wage costs. The definition of the firm’s surplus is somewhat open, but equation (A.69) is a standard assumption that also prevents transfers influencing the wage bargain, which would be the case, say, if dividends were used as the measure of surplus. sur pt = pchv t chv t + pdv t idt + pkhv t ikhv t + piov t iov t + psv t delst +gv t pgv t + pxv t xv t − (1 + ecostt )wt et (A.69) We assume that unemployment benefits (A.70) are an exogenous fraction of the private sector real wage. So µbenw represents the replacement ratio. This implies that unemployment benefit grows in line with wages (and hence labour productivity). (This is a requirement in the long run, so that the natural rate of unemployment does not exhibit a trend, but we need not assume this in the short run.) ben t = µbenw wt (A.70) The labour input (A.71) into production is measured in hours with a simple equation converting the employment index into hours. (A.71) eh t = et avh t Turning to working variables, as noted in the discussion of equation (A.63), the bargained real wage depends on the expected benefits to unions from negotiating the newly set wage. The expected marginal benefit to unions of the newly set wage is given by equation (A.72), which shows that this benefit depends on the expected flow of future post-tax wages and employment levels. When nominal wages are completely flexible (γ w = 1) the second (dynamic) term disappears. 184 The core model ξw = t wnewt pct ee 1 − τw t − τt −ηw wnewt wt ψe et (A.72) 1−ψ e ηw ξ wdot t+1 wnewt 1 + ṗt+1 +γ (1 − γ w ) 1 + rgt 1 + λ̇t+1 (1 + ṅ t+1 )ψ ξ w t+1 e 1 + λ̇t+1 (1 + ṗt+1 ) wt+1 Weighed against the benefits of the newly set wage rate is the expected marginal value of the outside wage (A.73) that union members could expect to receive if not employed at the newly set wage. This value depends on the outside wage and the employment level. As in equation (A.72), the expression for ξ wa simplifies when wages are fully flexible (γ w = 1). = ξ wa t wat pct wnewt wt −ηw ψe et 1 + ṗt+1 +γ (1 − γ ) 1 + rgt w (A.73) −ηw ψ e ξ wdot t+1 wnewt 1 + λ̇t+1 (1 + ṗt+1 ) wt+1 1 + λ̇t+1 (1 + ṅ t+1 )ψ ξ wa t+1 e The newly set wage enters the determination of the private sector real wage equation (A.63) via the term ξ wnew which is given by equation (A.74). This depends on the elasticity of substitution, ηw , between labour types in the employment index. ξ wnew t = wnewt wt 1−ηw (A.74) The role of unadjusted wages in the equation determining private sector real wages (A.63) is given by ξ wlag (A.75), which is a function of nominal wage growth relative to an indexation factor (ξ wdot ) which specifies how unadjusted wage contracts evolve over time. wlag ξt ξ wdot wt−1 t = wt (1 + ṗt ) 1 + λ̇t 1−ηw (A.75) The indexation factor (A.76) is a function of steady-state nominal wage inflation and lagged nominal wage inflation. This assumption is similar to the adjustment cost formulation for prices and capital in equations (A.53) to (A.59). The parameter wdot controls the extent to which non-renegotiated wage contracts are adjusted to account for lagged average wage inflation. This effect is absent when wdot = 0, but when wdot lies between 0 and 1, wage contracts that are not renegotiated are (partially) adjusted in line with past private sector wage inflation. ξ wdot = t 1+ ss λ̇t 1 + ṗtss 1− wdot wt−1 (1 + ṗt−1 ) 1 + λ̇t−1 wt−2 wdot (A.76) The discounted flow of surpluses to the firm is defined in equation (A.77). As in equations (A.72) and (A.73) above, this collapses to a static equation when nominal wages are flexible (γ w = 1). ξt sur p = sur pt + γ (1 − γ w ) 1 + ṗt+1 sur p (1 + ẏt+1 ) ξ t+1 1 + rgt (A.77) 185 The Bank of England Quarterly Model The discounted flow of costs associated with the newly set wage (technically the derivative of the surplus with respect to the newly set wage) is given by equation (A.78). dsur p ξt = − (1 + ecostt ) wnewt +γ (1 − γ w ) w wnewt −η et wt (1 + ẏt+1 ) ξ wdot t+1 wnewt 1 + ṗt+1 dsur p ξ t+1 1 + rgt (1 + ṗt+1 ) 1 + λ̇t+1 wnewt+1 (A.78) A2.4 Government The government budget constraint is given by equation (A.79). Previous stocks of government bonds (bg) and money (mon) are rolled over and (net) additional issues of bonds and money are required to finance that part of government spending that is not covered by tax revenue. Total spending consists of procurement ( pg · g), public sector wage spending ((1 + ecost) wg · eg), the public sector gross operating surplus (gosgexp) (4) and transfer payments (trans). Each of these components is discussed below. pgt bgt + pct mon t = pct−1 mon t−1 1 + rgt−1 pgt−1 bgt−1 + + pgt gt 1 + ṗt 1 + ẏt (1 + ṗt ) (1 + ẏt ) + (1 + ecostgt ) wgt egt + gosgex pt + transt − taxt (A.79) We assume that the government stabilises its debt at a target level in the long run by following a simple fiscal reaction function (A.80). We assume that the fiscal instrument is a lump sum tax on consumers (levied at rate τ lumpc ) that is adjusted in response to deviations of government debt from target (bgtar) and the change in the debt stock as a proportion of the value of private sector output. (5) The reaction function therefore has elements of both differential and integral control and ensures that the government meets its intertemporal budget constraint as long as the feedback coefficients θ bg and θ dbg are parameterised to ensure that the evolution of the debt stock converges on target. τt lumpc = τ t−1 + θ bg lumpc bgt − bgtart + θ dbg pyt yt bgt−1 bgt − pyt yt pyt−1 yt−1 (A.80) The target government debt level (A.81) appearing in the fiscal reaction function (A.80) is set as an exogenous ratio to nominal private sector output. pgt bgtart = µbgy pyt yt (A.81) The government procurement reaction function (A.82) assumes some inertia in reaching a target level of procurement, according to a partial adjustment equation. The parameter θ g is negative to ensure that government procurement growth rises when procurement is below target, and falls when procurement is above target. g 1 + ẏtss 1 + ṗtss pgt gt gt−1 θ = (A.82) pgt−1 gt−1 (1 + ẏt ) (1 + ṗt ) gtart−1 (4) The definition of total tax revenue (A.90) also includes gosgexp, so that it nets out of the government budget constraint. (5) The choice of fiscal instrument is arbitrary. Many other components of the government budget constraint, either singly or in combination, could also be chosen. 186 The core model The target government procurement (A.83) in the procurement reaction function (A.82) is set as an exogenous ratio to nominal private sector output. gtart = µgy pyt yt pgt (A.83) The government wage spending reaction function (A.84) is similar to the procurement reaction function and assumes some inertia in reaching target wage spending, according to a partial adjustment equation. The parameter θ wg is negative to ensure that spending growth rises when spending is below target, and falls when above target. 1 + ẏtss 1 + ṗtss (1 + ecostgt ) wgt egt = (1 + ecostgt−1 ) wgt−1 egt−1 (1 + ẏt ) (1 + ṗt ) (1 + ecostgt−1 ) wgt−1 egt−1 wgtart−1 θ wg (A.84) The target government wage spending (A.85) in the wage spending reaction function (A.84) is set as an exogenous ratio to nominal private sector output. wgtart = µwgy pyt · yt (A.85) We assume that government investment (A.86) simply moves to target immediately. igt = igtart (A.86) Target government investment (A.87) is set as an exogenous ratio to private sector output. igtart = µigy yt (A.87) Government procurement on consumption goods (A.88) is defined as the residual between total government procurement, g, and government investment, ig. gct = gt − igt (A.88) The government gross operating surplus (A.89) is modelled as a simple proportion of the value of private sector output. gosgex pt = ψ gosg pyt · yt (A.89) Total tax revenue (A.90), is the sum of revenues from labour income tax (taxw), employees’ National Insurance Contributions (taxee), taxes on dwellings (taxd), employers’ National Insurance Contributions (taxef ), corporation tax (taxk), lump-sum taxes on consumers and firms (taxlumpc and taxlumpk), tax revenue from overseas (taxf ), and indirect taxation (taxind). Total tax revenue also includes the government gross operating surplus (gosgexp), in accordance with National Accounts measurement of government revenues. taxt = taxwt + taxeet + taxdt + taxef t + taxkt + taxlumpct +taxlumpkt + taxf t + taxindt + gosgex pt (A.90) Income tax revenue (A.91) is given by the effective income tax rate, τ w , applied to private and public sector wage bills. (A.91) taxwt = τ w t (wt et + wgt egt ) 187 The Bank of England Quarterly Model Employees’ National Insurance Contributions (A.92) are given by the relevant effective rate, τ ee , applied to private and public sector wage bills. taxeet = τ ee t (wt et + wgt egt ) (A.92) Revenue from tax on dwellings (A.93) is defined as an exogenous effective tax rate, τ d , applied to the value of the stock of dwellings. taxdt = τ dt pdv t dt (A.93) Employers’ National Insurance Contributions (A.94) are given by the relevant effective rate, τ e f , applied to private and public sector wage bills. taxef t = τ t (wt et + wgt egt ) ef (A.94) Corporation tax revenue is defined by an exogenous effective rate applied to the flow of private sector output (A.95). taxkt = τ knd pyt yt (A.95) t Lump sum tax revenue from consumers (A.96) is given by the application of the effective tax rate τ lumpc to the value of private sector output. This is the instrument that we assume is used in the fiscal reaction function (equation (A.80)). taxlumpct = τ t lumpc pyt · yt (A.96) Lump sum tax revenue from firms (A.97) is given by the application of the effective tax rate τ lumpk to the tax base (the value of private sector output). taxlumpkt = τ t lumpk pyt · yt (A.97) Tax revenue from overseas (A.98) is defined by an effective rate applied to the value of private sector output. f taxf t = τ t pyt yt (A.98) Indirect tax revenue (A.99) includes the basic price adjustment, which captures taxes on products. It is given by the basic price adjustment ‘deflator’ multiplied by the difference between market price and basic price measures of private sector output (see equations (A.42) and (A.147)). Other terms are net subsidies to firms and he indirect tax payments to the EU. taxindt = pbpat (ym t − yt ) + transksubst − taxeu t (A.99) Net indirect taxes paid to the European Union (A.100) are defined by an exogenous rate applied to the value of private sector output. taxeu t = τ eu (A.100) t pyt yt The total wage tax rate for private sector firms (A.101) and the total wage tax rate for government (A.102) combine employers’ National Insurance Contributions and the pensions transfer rates paid to households. ef ecostt = τ t + trkpt (A.101) ecostgt = τ t + trect ef 188 (A.102) The core model Total transfer payments from government (A.103) consists of transfers to households (transc, rgpr em), to the unemployed (ie benefits, transu), to firms (transk, transksubs) and overseas (transf ). transt = transct + transu t + transkt + transf t + transksubst + rgprem t (A.103) Transfers to consumers (excluding unemployment benefits) (A.104) are defined by a rate applied to the value of private sector output. transct = trct pyt yt (A.104) Total unemployment benefits (A.105) are defined by the rate from (A.70) applied to the number of unemployed. transu t = ben t u t lt (A.105) Government transfers to firms (A.106), transfers overseas (A.107), and transfer subsidies on products (A.108) are defined by a rate applied to the value of private sector output. transkt = trkt pyt yt (A.106) transf t = trf t pyt yt (A.107) transksubst = trksubst pyt yt (A.108) Social contributions paid to government employees (A.109) are defined by a rate applied to the value of private sector output. They do not appear in the definition of total transfers (A.103) because they are incorporated in the government wage bill, through the definition of ecostg in equation (A.102). transect = trect wgt egt (A.109) A2.5 Monetary authority The nominal side is anchored by a monetary reaction function (A.110). In the default setting, this is a Taylor-type rule where the short-term nominal rate is used as an instrument to ensure that inflation returns to its target, ṗtss . rgt = 1 − θ rg rgtss + θ pdot d4cpi t − ṗtss + θ y log yt + glt ystart + θ rg rgt−1 (A.110) The reaction function responds to the deviations of a four-quarter inflation measure (d4cpi, defined in equation (A.138) below) from the inflation target ṗss and to a measure of output relative to potential. Potential output (A.111) is derived from the private sector production function using the steady-state levels of total (i.e. private and public sector) employment and capital utilisation. ss ss ystart = tfpt (1 − α) (1 − φ) 1 − u ss t lt avh t 1− σ1y +α φ z tss kt−1 1 + ẏt 1− σ1y σy σ y −1 (A.111) 189 The Bank of England Quarterly Model The level of output entering the policy rule is given by private sector output (y) plus the government demand for resources expressed in terms of private sector output. Equation (A.112) shows that this is calculated by evaluating the private sector production function using actual total employment (e + eg) and private capital when utilised at the steady-state rate (z ss ); and then subtracting the value of private sector output that would be produced using only private sector employment and capital, when capital utilisation is at its steady-state rate. glt = tfpt (1 − α) {(1 − φ) (et + egt ) avhstart }1− σ y + α 1 −tfpt (1 − α) {(1 − φ) et avhstart } 1− σ1y φz tss kt−1 1 + ẏt φz tss kt−1 +α 1 + ẏt 1− σ1y 1− σ1y σy σ y −1 σy σ y −1 (A.112) A2.6 External Imported intermediates (A.113) combine expenditure components where the ρ parameters denote the imported intermediate shares of individual expenditure components. As described in Chapter 3, final output is produced by combining private sector output (produced by domestic firms), imported intermediate goods and tax payments to the government, using a Leontief technology. (6) The remaining imports into the economy are directly imported consumption (determined by the consumption index (A.8)) and directly imported investment (see equation (A.33)). mi t = ρ ch ch t + ρ ikh ikh t + ρ g gt + ρ x xt (A.113) The demand for exports (A.114) is a conventional downward-sloping curve in the price of domestic exports relative to (exogenous) world export prices, pxf. The elasticity of demand is given by the parameter η px . World demand is given by the exogenous variable c f . Though we do not model the decisions of agents in the rest of the world from first principles, the demand function (A.114) is consistent with the assumption that consumers in the rest of the world allocate their consumption among domestically produced goods and goods produced in the rest of the world according to CES preferences similar to the ones specified for domestic consumers in equation (A.8). xt = κ x pxt qt pxf t −η px c ft (A.114) The relative price of imported intermediate goods (A.115) is the average of newly set and unadjusted prices. The parameter γ mi is the Calvo price adjustment probability: imported intermediates goods prices are adjusted with probability γ mi each period, and under perfectly flexible imported intermediate prices (γ mi = 1), the average price and newly set price coincide. Equations (A.116) and (A.117) show that the relative prices of imported consumption goods and of imported capital goods are given analogously. ⎡ ⎤ 1 1−ηmi 1−ηmi pmidot mi ξ pmin t−1 t ⎦ pmin t = ⎣γ mi pminewt1−η + 1 − γ mi (A.115) 1 + ṗt (6) Technically, the ρ parameters are the inverse of the imported intermediate coefficients in the Leontief technologies. 190 The core model ⎡ cm ⎡ 1−ηkm pcm t = ⎣γ pcm pcmnewt1−η + 1 − γ pcm pkm t = ⎣γ pkm pkmnewt + 1 − γ pkm pcmdot ξt 1−ηcm ξt 1−ηkm pcm t−1 1 + ṗt pkmdot pkm t−1 1 + ṗt ⎤ 1−η1cm ⎦ ⎤ (A.116) 1 1−ηkm ⎦ (A.117) The newly set imported intermediate price (A.118) is a function of the expected discounted costs of production (captured by ξ minmc ) relative to the expected discounted demand for the good (captured by ξ min ). (7) The newly set imported consumption price (A.119) and the newly set imported capital price (A.120) are given analogously. (8) pminewt = pmin t ξ minmc t ξ min t (A.118) pcmnewt = pcm t ξ cmmc t ξ cm t (A.119) pkmnewt = pkm t ξ kmmc t (A.120) ξt pkm Turning to working variables, the expected flow of imported intermediate costs (A.121) shows that each period the costs are given by the world price of exports in domestic currency (pxf /q) adjusted for an exogenous ‘margins effect’ (wmargin) and an exogenous scaling factor (κ pmin ). The margins effect is included because movements in measured world trade prices can have different effects on the world prices of UK imports and of UK exports. The scaling factor reflects the fact that the price of imported intermediates may not be well captured by the world export price (since the imported intermediate bundle is likely to consist of different goods to the bundle of goods traded on world export markets). The expected flow is equal to the discounted sum of future costs, where the discount factor depends on the Calvo adjustment probability γ mi . nmc ξ mi = mi t κ pmin (1 + wmargin t ) t + 1−γ f mi 1 + ṗt 1 + r ft pxf t pct pmin t qt pmin t+1 (1 + ṗt+1 ) ηmi +1 pmidot pmin t ξ t+1 ξ minmc t+1 (A.121) The expected flow of imported intermediate demand (A.122) is equal to the expected discounted flow of imported intermediates. As before, the discount factor depends on the Calvo price adjustment probability γ mi . ξ min = mi t + 1 − γ mi t f 1 + ṗt 1 + r ft pmin t+1 (1 + ṗt+1 ) pmin t ξ t+1 pmidot ηmi n ξ mi t+1 (A.122) (7) These working variables are given by equations (A.121) and (A.122) below. (8) The relevant working variables are described by the pairs of equations (A.124)-(A.125) and (A.127)-(A.128) respectively. 191 The Bank of England Quarterly Model We assume that for the duration of the pricing contract, unadjusted imported intermediates prices are increased in line with an imported intermediates price indexation factor (A.123) that depends on steady-state inflation and lagged imported intermediate price inflation. The parameter pmidot determines the extent to which the indexation factor depends on lagged imported intermediate price inflation. pmidot pmi dot (1 + ṗt−1 ) pmin t−1 pmidot ss 1− ξt = 1 + ṗt (A.123) pmin t−2 The expected flow of imported consumption costs (A.124) has the same form as (A.121). However, in this case, the cost may also depend on the price of domestic consumption goods. This is designed to capture (in a non-structural way) the possibility that competition effects cause importers to interact strategically with domestic producers, such that the import price is affected by the prices of domestically produced goods. The extent of these effects is controlled by the parameter pcm . ξ cmmc = cm t t pcm + 1−γ · κ pcm (1 + wmargin t ) pcm t+1 (1 + ṗt+1 ) f pcm pxf t pct + 1− pcm t qt 1 + ṗt 1 + r ft ηcm +1 pcmdot pcm t ξ t+1 · pcm pch t pcm t ξ cmmc t+1 (A.124) The expected flow of imported consumption demand (A.125) is equal to the expected discounted flow of imported consumption. As before, the discount factor depends on the Calvo price adjustment probability γ pcm . pcm ξ cm t = cm t + 1 − γ pcm t+1 (1 + ṗt+1 ) f 1 + ṗt 1 + r ft pcm t ξ t+1 pcmdot ηcm ξ cm t+1 (A.125) Analogous to the treatment of imported intermediates prices, we assume that unadjusted imported consumption prices are increased in line with an imported consumption price indexation factor (A.126). pcmdot pcmdot (1 + ṗt−1 ) pcm t−1 pcmdot ss 1− ξt = 1 + ṗt (A.126) pcm t−2 The expected flow of imported capital costs (A.127) is analogous to equation (A.124). ξ kmmc = ikm t t pkm · κ pkm (1 + wmargin t ) + 1 − γ pkm pxf t pct + (1 − pkm t qt pkm t+1 (1 + ṗt+1 ) f 1 + ṗt 1 + r ft ηkm +1 pkm t ξ t+1 pkmdot pkm )· pkh t pkm t ξ kmmc t+1 (A.127) The expected flow of imported capital demand (A.128) is analogous to equation (A.125). pkm ξt 192 = ikm t + 1 − γ f pkm 1 + ṗt 1 + r ft pkm t+1 (1 + ṗt+1 ) pkmdot pkm t ξ t+1 ηkm ξ t+1 pkm (A.128) The core model Analogous to the treatment of imported intermediate and imported consumption prices, we assume that for the duration of the pricing contract, unadjusted imported capital prices are increased in line with an imported capital price indexation factor (A.129). pkmdot ξt = 1+ 1− ṗtss pkmdot (1 + ṗt−1 ) pkm t−1 pkm t−2 pkmdot (A.129) A2.7 Prices and inflation As explained in Chapter 5, the core model is written in stationary units, which requires a numeraire price relative to which all other prices are measured. Without loss of generality, we choose the non-durable consumption price (excluding actual and imputed rents) to be the numeraire. By definition this means that the relative price of non-durable consumption (A.130) is equal to 1 in every period. pct = 1 (A.130) The relative price of non-durable consumption is also related to the (relative) prices of home and imported consumption by equation (A.131), which is simply the definition of total expenditure on consumption (measured in detrended model units). pct ct = pcm t cm t + pch t ch t (A.131) The (relative) price of home consumption (A.132) is the weighted average of the prices of domestically produced home consumption, imported intermediates and indirect taxes. (9) These relative prices are given by equations (A.36), (A.115) and (A.141) respectively. pch t = κ chv pchv t + ρ ch pmin t + 1 − κ chv − ρ ch pbpat (A.132) Equation (A.133) for the relative price of home capital is analogous to the expression for the relative market price of home consumption. pkh t = κ ikhv pkhv t + ρ i kh pmin t + 1 − κ ikhv − ρ i kh pbpat (A.133) The relative price of other investment (A.137), has no imported intermediate component, so the price is a weighted average of the relative price of domestically produced other investment and the component for the basic price adjustment. piot = κ iov piov t + 1 − κ iov pbpat (A.134) The relative price of government procurement (A.135) and the market price of exports (A.136) are analogous to the relative market price of home consumption. pgt = κ gv pgv t + ρ g pmin t + 1 − κ gv − ρ g pbpat (A.135) pxt = κ xv pxv t + ρ x pmin t + 1 − κ xv − ρ x pbpat (A.136) The (relative) basic price of other investment (A.137) is not modelled as part of the firm’s optimal pricing decision but is instead linked by a simple equation to the basic price of home consumption. piov t = ψ pio pchv t (A.137) (9) The weights correspond to the (inverse) coefficients of the Leontief technology (discussed above) that combines domestic value added, imported intermediates and the basic price adjustment. 193 The Bank of England Quarterly Model The four-quarter change in the CPI (A.138) is approximated by the four-quarter change in the price of non-durable consumption with an (exogenous) adjustment for the historical wedge between these series. This measure abstracts from seasonal effects. 3 d4cpi t = 0.25 i=0 ( ṗt−i − cpiwedget−i ) (A.138) The market price of private sector output (A.139) is given by the value of marketed output, measured at market prices and adjusted for imported intermediates, divided by the volume of private sector output at market prices, ym. pym t ym t = pch t ch t + pdv t idt + pkh t ikh t + piot iot + psv t delst + pgt gt + pxt xt − pmin t mi t (A.139) The basic price of private sector output (A.140) is defined by subtracting the value of the basic price adjustment from the market value of private production. This gives the basic price value of private sector output which can then be divided by private sector output to calculate the price. pyt yt = pym t ym t − pbpat (ym t − yt ) (A.140) The relative price of the basic price adjustment (A.141) depends on its own lag and the exogenous tax rate τ c , applied to the ratio of the basic price value of private sector output and the basic price adjustment (ym − y). The parameter θ bp controls the influence of the lag. When θ bp = 0, the price is such that the tax revenue from the basic price adjustment (representing taxes on products) is a fraction τ c of the (basic price) value of private sector output. pyt yt pbpat = θ bp pbpat−1 + 1 − θ bp τ ct (A.141) ym t − yt A2.8 Quantities The (basic price) value-added component of domestically produced consumption (A.142) is derived from the assumption that domestic value added, imported intermediates and basic price adjustment are combined to produce final output using a Leontief technology, so there is a simple linear relationship between the value-added quantity of home consumption and final home consumption. The parameter controlling this relationship, κ chv , appears in the underlying Leontief technology. chv t = κ chv ch t (A.142) The (basic price) value-added components of home investment (A.143), other investment (A.144), government procurement (A.145) and exports (A.146) are treated in the same way as home consumption. ikhv t = κ ikhv ikh t (A.143) iov t = κ i ov iot 194 (A.144) The core model gv t = κ gv gt (A.145) xv t = κ xv xt (A.146) The level of private sector output measured at market prices (A.147) adds the basic price adjustment to basic price private sector output. ym t = yt + 1 − κ chv − ρ ch ch t + 1 − κ i khv − ρ ikh ikh t + 1 − κ iov iot + 1 − κ gv − ρ g gt + 1 − κ xv − ρ x xt (A.147) Imputed and actual rents (A.148) are not modelled as part of the household optimisation problem and are defined as a simple function of the dwellings stock. cirt = ψ cir dt (A.148) Other investment (A.33) is not modelled as part of the firm’s optimisation problem and is defined as a simple function of private sector output. iot = ψ i o yt (A.149) A2.9 Accounting and reporting We list here a number of variables that are not required for the solution of the core model but that are also calculated. Net foreign assets (A.150) are calculated as the domestic currency value of foreign bonds. pct b f t (A.150) nfat = qt The value of the trade balance (A.151) is the value of exports less the value of total imports . xm t = pxt xt − pcm t cm t − ikm t pkm t − pmin t mi t (A.151) The value of the current account (A.152) is the trade balance plus net taxes and transfers overseas and interest payments on the stock of net foreign assets. xmcat = xm t + taxf t − transf t − transkf t + transfpt − taxeu t r f t−1 nfat−1 +rfpremt + (1 + ṗt ) (1 + ẏt ) (A.152) Inflation rates (A.153) to (A.156) for the value-added price of home consumption, imported intermediates prices, the market price of home consumption and imported consumption prices are defined as the change in the relative price multiplied by the inflation rate of the numeraire. 1 + ṗtchv = (1 + ṗt ) pchv t pchv t−1 (A.153) 195 The Bank of England Quarterly Model (1 + ṗt ) pmin t pmin t−1 (A.154) 1 + ṗtch = (1 + ṗt ) pch t pch t−1 (A.155) 1 + ṗtcm = (1 + ṗt ) pcm t pcm t−1 (A.156) 1 + ṗtmin = Nominal private sector wage inflation (A.157) is calculated as the change in the real wage multiplied by the numeraire inflation rate and the change in labour productivity. 1 + ẇt = (1 + ṗt ) 1 + λ̇t wt wt−1 (A.157) Total asset holdings (A.158) includes the values of equities corporate bonds, government bonds, real money balances and net foreign assets. at = v t + bkt + pgt bgt + pct mon t + nfat (A.158) A simple measure of total consumption (A.159) is given by the sum of home and imported consumption. cmodt = cm t + ch t (A.159) A simple measure of total investment (A.160) is given by the sum of home and imported investment. ikt = ikh t + ikm t 196 (A.160) Appendix B The non-core equations B.1 Mnemonics In the tables below, we describe the mnemonics in terms of detrended model units. Other versions of these variables would be indicated by suffixes as follows: cp current prices, kp (chained) volume, de f implicit deflator, and ex p expenditure volume in detrended model units. Some variables may only be included in the model in current price terms, for instance National Accounts concepts that are simple transformation, but we omit the cp suffix to help clarity. Similarly, some labour market variables can be denoted in heads, hds, or hours worked, hrs. Details of these simple transformations and data sources are listed in Appendix C. We give references to the relevant non-core model equations for the endogenous variables. In general, the mapping from variables to equations is clearer for non-core equations than for the core model. For instance, there is an estimated non-core equation for private sector employment, e, rather than the first-order condition in the core model. Some variables have more than one referenced equation, where there are non-core equations for two forms of that variable. For instance, there are equations for divident payments, dv, in detrended model units (B.116) and in current prices (B.161). Table B.1: Endogenous variables a aa avh avhrs avhrsagg avhrsg ben bf bg bgtar bk bpa c ch chv cir cm cna comp compg cpi cpidot cpixr cpixrdot cpr cprdot Net financial wealth of the household sector (B.120) Alignment adjustment (B.145) Private sector average hours worked (B.67) Average weekly hours worked in the private sector (B.154) Average weekly hours worked in the whole economy (B.156) Average weekly hours worked in the general government sector (B.155) Unemployment benefit (B.131) Stock of foreign bonds denominated in terms of foreign consumption goods (B.114) Stock of government bonds (B.70) Government debt target (B.72) Stock of corporate bonds (B.63) Basic price adjustment (B.110) Volume of consumption goods (B.1) Domestically produced consumption goods (B.11) Value-added component of domestically produced consumption goods (B.102) Volume of actual and imputed rents (B.112) and (B.113) Volume of directly imported consumption goods (B.9) National Accounts measure of consumption (B.167) and (B.168) Total compensation of private sector workers (including self-employed) (B.157) Total compensation of general government employees (B.158) Consumer Prices Index, (1996=100) (B.52) Quarterly CPI inflation rate (B.42) Consumer Prices Index (excluding rents), (1996=100) (B.50) Quarterly inflation rate of the CPI (excluding rents) (B.40) Rents component of CPI, (1996=100) (B.51) Quarterly inflation rate of the rents component of the CPI (B.41) 197 The Bank of England Quarterly Model d d4cpi dels dv e eagg ecost ecostg eer eg eh en fu g gc gdp gdpbp ggnb gl gons gosg gosp gv id ig ik ikh ikhv ikm io iov ipdf ipdg ipdp k kh km gap λ̇ l ly mi mips mipsdot mon µeg ṅ gap 198 Stock of dwellings (B.121) Four-quarter growth rate of the CPI (B.18) Stockbuilding (including alignment adjustment) (B.5) Dividend payments to households (B.116) and (B.161) Private sector employment index (B.66) Aggregate employment (B.127) Rate of employers’ total social contributions, private sector (B.88) Rate of employers’ total social contributions, general government (B.89) Nominal sterling effective exchange rate (1990=100) (B.143) General government employment (B.73) Private sector hours worked (B.69) Private sector employment (B.129) Private sector factor utilisation (B.14) Volume of government procurement of private sector goods and services (B.74) Volume of government procurement of private sector goods and services (consumption goods) (B.76) GDP at market prices (B.146) and (B.150) GDP at basic prices (B.147) and (B.151) General government net borrowing (B.175) Opportunity cost of government labour (B.101) Total ONS-measured general government final consumption and investment expenditure (B.148) and (B.149) General government gross operating surplus (B.77) Household sector gross operating surplus (B.163) Value-added component of total general government procurement (B.105) Volume of investment in dwellings (B.2) Volume of government procurement of investment goods (B.75) Volume of total business investment (B.3) Volume of domestically produced investment (B.12) Value-added component of domestically produced investment (B.103) Volume of directly imported investment (B.10) Volume of other investment (B.4) and (B.111) Value added component of other investment (B.104) Net interest payments from overseas (B.160) Interest payments on general government debt (B.159) Total net interest payments to household sector (B.162) Capital stock (B.124) Volume of domestically produced capital goods (B.122) Volume of directly imported capital goods (B.123) Productivity growth adjustment term (B.137) Labour supply (participation) (B.68) Real post-tax labour income (B.125) Volume of intermediate imports of goods and services (B.8) MIPS component of the RPI (January 1987=100) (B.55) Quarterly growth rate of the MIPS component of the RPI (B.47) Stock of money holdings (B.115) Share of government employment in total labour supply (B.132) Population growth adjustment term (B.138) The non-core equations nfa nlf nlgg nlp ṗ ṗ ch ṗ chv ṗ cm ṗ mi n pbpa pc pch pchv pcm pcna pdv pg pgc pgdp pgdpbp pgv phse pio piov pkg pkh pkhv pkm pmin psv px pxv py pym q rg rhpi rk rpcc rpccdot rph rphdot rpi rpidot rpidotsa Stock of foreign bonds denominated in terms of consumption goods (B.133) Overseas sector net lending (B.176) General government net lending (B.174) Household sector net lending (B.172) Quarterly inflation rate of consumption goods (excluding actual and imputed rents) (B.13) Quarterly rate of inflation of domestically produced consumption goods (B.142) Quarterly rate of inflation of the value-added component of domestically produced consumption goods (B.140) Quarterly rate of inflation of directly imported consumption goods (B.141) Quarterly inflation rate of intermediate imports (B.22) Relative price of the basic price adjustment (B.37) Numeraire price (consumption) (B.23) and (B.144) Relative price of domestically produced consumption goods (B.19) Relative price of the value-added component of domestically produced consumption goods (B.25) Relative price of directly imported consumption goods (B.20) Relative price of the National Accounts measure of consumption (B.169) Relative price of dwellings investment (B.29) Relative price of government procurement of private sector goods and services (B.30) Relative price of government procurement (consumption goods) (B.36) Relative price of GDP at market prices (B.152) Relative price of GDP at basic prices (B.153) Relative price of the value-added component of government procurement of private sector goods and services (B.26) Relative price of housing (adjusted for trend productivity) (B.17) Relative price of other investment (B.34) Relative price of the value-added component of other investment (B.28) Relative price of government procurement (investment goods) (B.31) Relative price of domestically produced capital goods (B.33) Relative price of the value-added component of domestically produced capital goods (B.27) Relative price of directly imported capital goods (B.32) Relative price of intermediate imports (B.21) Relative price of stockbuilding (B.35) Relative price of exports (B.16) Relative price of the value-added component of exports (B.24) Relative price of private sector value added at basic prices (B.39) Relative price of private sector value added at market prices (B.38) Real exchange rate using consumer prices (B.60) Short-term nominal interest rate (B.99) Total available household resources (B.170) Nominal corporate bond yield (B.62) Council tax component of the RPI (B.54) Quarterly inflation rate of the council tax component of the RPI (B.45) Housing depreciation component of the RPI, (January 1995=100) (B.53) Quarterly inflation rate of the housing depreciation component of the RPI (B.44) Retail Prices Index (January 1987=100) (B.59) Quarterly inflation rate of the RPI (B.48) Quarterly inflation rate of the RPI, seasonally adjusted (B.49) 199 The Bank of England Quarterly Model rpix rpixdot rpxc rpxch rpxchdot s savr τ lumpc tax taxd taxee taxef taxeu taxf taxind taxk taxlumpc taxlumpk taxp taxw tme trans transben transc transct transec transf transfp transk transkc transkf transkp transksubs transu u v w ẇ wagg wg wl x xm xmca xv y 200 RPI excluding mortgage interest payments (January 1987=100) (B.58) Quarterly inflation rate of RPIX (B.46) Retail price index. excluding mortgage interest payments and council tax (January 1987=100) (B.57) RPI excluding MIPS, council tax and housing depreciation (January 1987=100) (B.56) Quarterly inflation rate of RPI, excluding MIPS, council tax and housing depreciation (B.43) Stock of inventories (B.6) Household sector saving ratio (B.171) Effective lump sum tax rate on households (B.71) Total taxation receipts (B.78) Revenue from tax on dwellings (B.81) Employees’ National Insurance Contributions (B.80) Employers’ National Insurance Contributions (B.82) Indirect taxes minus subsidies paid to EU (B.90) Tax revenue from overseas residents (B.86) Revenue from indirect taxation (B.87) Revenue from corporation tax (B.83) Revenue from lump sum taxes on households (B.84) Revenue from lump sum taxes on firms (B.85) Total tax payments of household sector (B.166) Tax revenue from labour income taxes (B.79) Total managed general government expenditure (B.173) Total general government transfers (B.91) Total general government transfer payments to households (B.164) General government transfer payments to households excluding unemployment benefit (B.92) Total transfer payments to households (B.165) Employers’ other social contributions, general government (B.98) General government transfers to overseas sector (B.96) Net overseas transfers to households (B.93) General government transfers to firms (B.95) Supernormal profit transfers from firms to households (B.119) Net transfers from firms to overseas (B.118) Employers’ other social contributions, private sector (B.117) General government subsidies on products (B.97) Total unemployment benefits (B.94) Unemployment rate (B.126) Value of equities (B.61) Private sector real wage (B.64) Quarterly growth rate of nominal private sector wages (B.139) Aggregate real wage (B.128) Government wage (B.65) Expected return from labour market participation (B.130) Volume of exports (B.7) Net expenditure on overseas goods and services (B.134) Current account balance, plus net capital transfers from overseas (B.135) Value-added component of export volumes (B.106) Private sector value added (B.107) The non-core equations ycap yd ẏ gap ym ystar Private sector capacity output (B.15) Volume of final demand (B.108) Output growth adjustment term (B.136) Private sector value added at market prices (B.109) Potential output (B.100) Table B.2: Exogenous variables avhstar cf cpiwedge dumq1 dumq2 dumq3 dumq4 λ λ̇ ss λ̇ µcc µcpr µmi p µr ph ṅ ṅss nhds ṗ f ṗss pgons pxf pxfdef rf rfprem rgprem rpixwedge sd τc τd τ ee τ ef τ eu τf τ knd τ lumpk τw tfp Long-run average weekly hours worked in the private sector World trade Wedge between the non-durable consumption deflator and CPI inflation rates Seasonal dummy Seasonal dummy Seasonal dummy Seasonal dummy Labour-augmenting technical progress Labour-augmenting productivity growth Steady-state labour-augmenting productivity growth Weight on council tax in the RPI Weight on rents in the CPI Weight on MIPS in the RPI Weight on housing depreciation in RPI Adjustment factor for effect of value of housing on consumption Population growth Steady-state population growth Population aged 16+, thousands Overseas rate of consumer price inflation Domestic inflation target Relative price of total ONS-measured final general government consumption and investment expenditure Relative price of world exports M6 export prices, using sterling ERI weights, index (1998=100) M6 short-term nominal interest rate Premium on overseas interest payments to households Premium on government interest payments to households Wedge between the RPI (excluding housing components) and CPI inflation rates Statistical discrepancy Effective net indirect tax rate (ratio of basic price adjustment to value added) Effective tax rate on dwellings Effective rate of employees’ National Insurance Contributions Effective rate of employers’ National Insurance Contributions Effective tax rate on EU net indirect taxes Effective tax rate on revenue from overseas residents Effective corporation tax rate Effective lump sum tax rate on firms Effective income tax rate Total factor productivity term in production function 201 The Bank of England Quarterly Model trc trec trf trfp trk trkf trkp trksubs ẏ ẏss yf General government transfer rate to households (excluding unemployment benefit) Rate of employers’ other social contributions, general government General government transfer rate to overseas Rate of net transfers from overseas to households General government transfer rate to firms Transfer rate from firms to overseas Transfer rate from firms to households Subsidy rate from general government to firms Growth rate of productive potential (λ̇ and ṅ) Steady-state output growth Volume of world imports, using UK trade weights, (2000=100) Table B.3: Parameters α δd δ kh δ km κ chv κ gv κ ikhv κ iov κ xv µbenw µbgy µbkv µgy µwg φ φk ψ cir ψ gosg ψio ψk ψ snp ρ ch ρg ρ ikh ρx σk σy θ bg θ dbg θ pdot θ rg θy 202 Share parameter for capital in production function Depreciation rate on dwellings Depreciation rate on domestically produced capital goods Depreciation rate on directly imported capital goods Share of value added in domestically produced consumption goods Share of value added in government procurement Share of value added in domestically produced investment goods Share of value added in other investment goods Share of value added in exports Replacement ratio Steady-state government debt to output ratio Corporate sector debt-equity ratio Steady-state government procurement to output ratio Ratio of government wages to private sector wages Share parameter for capital in production Share parameter for domestically produced capital in capital aggregator Ratio of actual and imputed rentals to the stock of dwellings Ratio of government gross operating surplus to private sector output Share of other investment in private sector output Share of home capital in capital aggregator Share of supernormal profits going to consumers directly from firms Weight on intermediate imports in domestically produced consumption goods Weight on intermediate imports in government procurement Weight on intermediate imports in domestically produced capital goods Weight on intermediate imports in exports Elasticity of substitution between imported and domestically produced capital in capital aggregator Elasticity of substitution between capital and labour in private sector production Coefficient on government debt gap in fiscal reaction function Coefficient on government debt change in fiscal reaction function Coefficient on inflation gap in monetary reaction function Interest rate smoothing coefficient in monetary reaction function Feedback from output gap in monetary reaction function The non-core equations B.2 Non-core equations This section sets out the non-core equations under general headings, highlighting and collecting together equations that determine similar variables (such as demand components, prices and inflation and labour market variables), rather than following the order of the core model equations in Appendix A. Econometric results for the estimated equations are reported and discussed in Section 6.4.2. A number of technical points are worth noting first: • most of the equations are written in detrended model units (as are all of the core model equations listed in Appendix A). Box 12 on page 88 explains that this means that terms involving lags will generally be scaled according to some combination of productivity growth, population growth and inflation. We do not comment on these growth terms unless they are particularly important for the interpretation of the equation; • we therefore need to add ‘growth adjustment’ terms so that our specifications imply partial gap adjustment in actual (ie not model) units. These growth adjustment terms are ẏ gap , ṅ gap and λ̇ and are defined in equations (B.136), (B.137) and (B.138). These terms are zero over our historical sample, and hence not included in the estimation results reported in Chapter 6. But they will be non-zero in the event of productivity or population shocks; • we also add a number of recursive ‘post-transformation’ equations, which convert solutions in model units into units that are more directly comparable to National Accounts measures. Many of these are simple reversals of transformations listed in Appendix C. Others, including those that require some explanation, are listed in Section B2.8; • the price of non-durable consumption bundle (PC) is the numeraire. This implies that the relative price of non-durable consumption, pc, is set equal to 1 in every period – see equation (B.23) below. As in the core model listing, we often implicitly impose this condition in the equations (ie we eliminate terms in pc) but it appears explicitly when its inclusion may help to interpret the equation; • We use the symbol to indicate a change operator – so that xt = xt − xt−1 , and xt−1 = xt−1 − xt−2 . Solutions from the core model are denoted by the core superscript and non-core equation residuals are denoted by the res superscript (see equation (B.4) for a simple example) and steady-state values are denoted by the ss superscript; and • accounting relationships are often identical to core model equations, with the exception of an additional residual term. We describe such equations as identical to the core version, despite the presence of the residual. The residuals in the non-core versions of the equations also act as a cross check on the data transformations process used to generate the data set. B2.1 Demand components The aggregate (non-durable) consumption equation (B.1) contains an error correction to non-durable consumption. An extra argument in assets is included to ensure convergence in net foreign assets. Proxy variables for effects missing from the core include changes in the value of the housing stock, as a proxy for housing collateral effects; changes in household income, as a proxy for the existence of rule-of-thumb individuals; changes in the employment rate, as a proxy for confidence and uncertainty effects; and changes in nominal interest rates, as a proxy for credit and cash-flow effects. The variable t is an exogenous variable that allows us to change the size of the effect on consumption of changes in the value of housing. The box on pages 12-13 of the November 2004 Inflation Report explained how, in the recent past, the association between house price inflation and consumption has 203 The Bank of England Quarterly Model been less strong than in earlier years. As noted in previous Inflation Reports, that led the MPC to expect that a slowing of house price inflation would not necessarily imply a substantial weakening of household spending. gap log ct + ẏt gap = 0.197 + log eaggt + ṅ t · 0.194 t − 1.125 rgt + 0.221 gap log lyt + ẏt core log ( phset dt−1 ) + ẏt−1 − 0.125 log ct−1 − log ct−1 gap core +0.006 log at−1 − log at−1 + ctres (B.1) The equation for dwellings investment (B.2) includes an error correction term for dwellings investment. Extra variables include consumption growth and changes in nominal interest rates (to proxy the role of cash-flow effects on short-term housing demand). gap log idt + ẏt = −6.859 rgt−1 + 1.667 gap log ct−2 + ẏt−2 core −0.153 log idt−1 − log idt−1 + idtres (B.2) The equation for total capital investment (B.3) includes error correction to total capital investment. Extra variables include lagged changes in total investment; a steady-state capital gap term; and an accelerator term in the change in output growth. Lagged investment is included to capture long decision lags that are not fully captured by adjustment costs in the core model. The steady-state capital gap term implies that investment is boosted when there is an increase in the long-run desired capital stock. gap log ikt + ẏt = 0.346 · gap log yt−1 + ẏt−1 + 0.067 −0.107 log ikt−1 − core log ikt−1 gap log ikt−1 + ẏt−1 − 0.255 log kt−1 − ss log kt−1 (B.3) + iktres Other investment (B.4) adds a residual to the core solution for other investment. iot = iotcore + iores t (B.4) Stockbuilding (B.5) is a perpetual inventory condition, replicating the core model equation (A.46). st−1 + delstres (B.5) delst = st − 1 + ẏt Inventory stocks (B.6) adds a residual to the core solution. st = stcore + stres (B.6) The equation for exports (B.7) specifies an error correction term in exports and also includes changes in world demand. gap log xt + ẏt = 0.730 gap log c f t + ẏt core − 0.172 log xt−1 − log xt−1 + xtres (B.7) The non-core equation for imported intermediates (B.8) applies the imported intermediate shares parameters (ρ) to the individual demand components, as in the core equation (A.113). mi t = ρ ch ch t + ρ ikh ikh t + ρ g gt + ρ x xt + mi tres 204 (B.8) The non-core equations The equation for imported consumption (B.9) includes an error correction term in the ratio of imported to total (non-durable) consumption. Extra variables include changes in total consumption and changes in the relative price of imported and home-produced consumption goods. These effects capture short-run income and substitution effects on the demand for imported consumption. gap log cm t + ẏt = 1.134 gap log ct + ẏt + 0.221 gap log ct−1 + ẏt−1 − 0.187 · core −0.080 log (cm t−1 /ct−1 ) − log cm core t−1 /ct−1 + cm res t log pcm t pch t (B.9) The equation for imported capital investment (B.10) contains an error correction term in imported capital investment. Extra variables are lagged changes in imported capital investment, and changes in the relative price of imported and domestically produced capital goods. The lags in imported capital investment capture the sluggishness in the investment data; the relative price term captures substitution effects between domestic and imported capital goods. gap log ikm t + ẏt = 0.242 gap log ikm t−1 + ẏt−1 − 0.526 log res −0.056 log ikm t−1 − log ikm core t−1 + ikm t pkm t pkh t (B.10) Domestically produced consumption (B.11) is defined as a residual of total (non-durable) consumption less imported consumption. ch t = ct − cm t (B.11) Similarly, domestically produced capital investment (B.12) is defined as a residual of total capital investment less imported capital investment. ikh t = ikt − ikm t (B.12) B2.2 Prices and inflation The equation for (non-housing) consumption deflator inflation (B.13) specifies error correction to the inflation rate. Lagged inflation changes reflect sluggish inflation adjustment; there is an activity effect from the deviation from the steady state in aggregate employment; a term in private sector factor utilisation captures cyclical variations in margins; and there is an additional effect from imported intermediate price inflation. ṗt = 0.138 f u t−1 eaggt−1 min − 0.421 ṗt−1 − 0.350 ṗt−2 + 0.047 ṗt−1 + 0.072 log ss ss 100 et−1 + egt−1 core + ṗtres −0.332 ṗt−1 − ṗt−1 (B.13) Factor utilisation (B.14) is defined as the percentage difference between the level of private sector output and a measure of capacity output (ycap). yt − ycapt (B.14) f u t = 100 ycapt 205 The Bank of England Quarterly Model The measure of capacity output (B.15) is calculated by evaluating the private sector production function using actual levels of employment (e) and capital (k) at long-run levels of average hours worked and capital utilisation (avhstar and z ss respectively). ycapt = tfpt (1 − α) {(1 − φ) et avhstart }1− σ y + α φ 1 z tss kt−1 1− σ1y σy σ y −1 1 + ẏt (B.15) The equation for (relative) export prices (B.16) specifies sluggish error correction. log pxt = 0.373 log pxt−1 + 0.119 log pxt−3 core −0.046 log pxt−1 − log pxt−1 + pxtres (B.16) The equation for the (relative) house price index (B.17) features error correction to the housing investment deflator (with an estimated adjustment for the differences in the levels of these series). Additional variables include lagged house price and interest rate changes, the latter proxying for credit gap effects. The presence of adjustments for deviations in productivity growth from trend (λ̇ terms) reflects our assumption that the house price index, phsedef, grows in line with real wages in the long run. This means that phse is constructed by dividing phsedef by the level of labour productivity as well as the numeraire price level (see Appendix C). log phset + λ̇t gap = 0.611 +0.071 log phset−1 + λ̇t−1 + 0.192 gap log phset−2 + λ̇t−2 log phset−3 + λ̇t−3 − 1.452 rgt−2 gap −0.019 (log phset−1 − log pdv t−1 + 2.029) + phsetres gap (B.17) The four-quarter change in the CPI (B.18) is identical to its core model counterpart (A.138). 3 d4cpi t = 0.25 i=0 ( ṗt−i − cpiwedget−i ) (B.18) As in the core model (A.131), the relative price of non-durable consumption is related to the (relative) price of home consumption (B.19) by the definition of total expenditure on consumption. pct ct − pcm t cm t + pch res (B.19) pch t = t ch t The (relative) price of imported consumption (B.20) adds a residual to the core solution. + pcm res pcm t = pcm core t t (B.20) The (relative) price of imported intermediates (B.21) adds a residual to the core solution. + pminres pmin t = pmin core t t (B.21) The imported intermediates inflation rate (B.22) is the change in the relative imported intermediates price multiplied by the inflation rate of the numeraire, as in the core model (A.154). 1 + ṗtmin = 206 (1 + ṗt ) pmin t pmin t−1 (B.22) The non-core equations As in the core model, the non-durable consumption price is the numeraire, so by definition the relative price of non-durable consumption (B.23) is equal to 1. pct = 1 (B.23) The (relative) market price of exports (B.24) is given by the weighted average of the prices of domestically produced consumption, imported intermediates and indirect taxes. The core model equation (A.136) is identical. pxt = κ xv pxv t + ρ x pmin t + 1 − ρ x − κ xv pbpat (B.24) Equations for the market price of home consumption (B.25), the market price of government procurement (B.26), the market price of home capital (B.27) and the market price of other investment (B.28) are constructed analogously. The core model equivalents are equations (A.132),(A.135), (A.133) and (A.137). pch t = κ chv pchv t + ρ ch pmin t + 1 − κ chv − ρ ch pbpat (B.25) pgt = κ gv pgv t + ρ g pmin t + 1 − κ gv − ρ g pbpat (B.26) pkh t = κ i khv pkhv t + ρ ikh pmin t + 1 − κ ikhv − ρ ikh pbpat piot = κ iov piov t + 1 − κ iov pbpat (B.27) (B.28) The (relative) price of dwellings (B.29) adds a residual to the core solution. The logarithmic equation form implies that residuals can be treated as percentage differences between the core and non-core solutions. log pdv t = log pdv tcore + pdv tres (B.29) The equations for the (relative) prices of government procurement (B.30), government procurement of investment goods (B.31), imported capital (B.32), home capital (B.33), other investment (B.34) and inventory stocks (B.35) add residuals to the core solutions. This means that variations in the relative prices of the components of government procurement are modelled by residual adjustment. pgt = pgtcore + pgtres (B.30) pkgt = pgtcore + pkgtres (B.31) pkm t = pkm core + pkm res t t (B.32) pkh t = pkh core + pkh res t t (B.33) piot = piotcore + piores t (B.34) psv t = psv tcore + psv tres (B.35) 207 The Bank of England Quarterly Model The (relative) price of government procurement of consumption goods (B.36) is derived from the difference between the values of total and investment procurement. ( pgt · gt − pkgt · igt ) pgct = + pgctres (B.36) gct The (relative) price of the basic price adjustment (B.37) moves to ensure that revenue from the basic price adjustment (product taxation) is an exogenous fraction τ c of the value of private sector output. This corresponds to a long-run version of the core model equation (A.141). pyt yt + pbpatres (B.37) pbpat = τ ct ym t − yt The market price of private sector output (B.38) is given by the value of marketed output measured at market prices, adjusted for imported intermediates and divided by the volume of private sector output at market prices, ym. This replicates the core model (A.139) apart from the addition of a term for the statistical discrepancy between the average and expenditure measures of GDP. pym t ym t = pch t ch t + pdv t idt + pkh t ikh t + piot iot + psv t delst + pgt gt + pxt xt − pmin t mi t + sdex pt (B.38) The basic price of private sector output (B.39) is defined by subtracting the value of the basic price adjustment from the market value of private production, then dividing by private sector output. This is identical to the core model equation (A.140). pym t ym t − pbpat (ym t − yt ) pyt = + pytres (B.39) yt The inflation rate of Consumer Prices Index (CPI) excluding rents (B.40) is the inflation rate of non-durable consumption prices adjusted for the average wedge between that and CPI inflation, plus a residual, which mainly captures the seasonal component of cpi xr. cpi xrdott = ṗt − cpiwedget + cpi xrdottres (B.40) The inflation rate of the rents component of the CPI (B.41) is given by the inflation rate of non-durable consumption prices adjusted for the average wedge between that and CPI inflation, plus a residual. The residual again captures the seasonal components of the index, but also any deviation of the inflation rate of rents from the other components in the CPI basket. cprdott = ṗt − cpiwedget + cpr dottres (B.41) CPI inflation (B.42) is given by the weighted average of the rents and non-rents components. The residual mainly captures differences in the rounding of sub-indices used to construct the overall index. cpidott = µcpr cprdott + 1 − µcpr cpi xrdott + cpidottres (B.42) Inflation of the Retail Prices Index (RPI) excluding various housing factors (council tax, housing depreciation and mortgage interest payments) (B.43) adjusts the CPI inflation rate for the wedge between the two measures. This wedge is mainly driven by the ‘formula effect’ arising from the use of arithmetic averaging of price quotes to construct the RPI, compared with geometric averaging used to construct the CPI. r pxchdott = cpidott + r pi xwedget + r pxchdottres (B.43) 208 The non-core equations The housing depreciation component of the RPI (B.44) is an exponentially smoothed house price index. This implies that housing depreciation inflation is a weighted average of current and past house price inflation and past housing depreciation inflation. The residual captures the fact that the model’s house price index, phsedef, is not the house price index that underlies r ph. r phdott phset phset−1 (1 + ṗt ) 1 + λ̇t − 1 + 0.25 (1 + ṗt−1 ) 1 + λ̇t−1 − 1 phset−1 phset−2 +0.25 · r phdott−1 + r phdottres (B.44) = 0.5 The council tax component of RPI inflation (B.45) is the rate of growth of nominal council tax revenue (taxd) adjusted for the growth in the housing stock, multiplied by a seasonal dummy, dumq2. (1) r pccdott = dumq2t 1 + λ̇t (1 + ṅ t ) dt taxdt (1 + ṗt ) − taxdt−1 dt−1 + r pccdottres (B.45) RPIX inflation (which measures RPI inflation excluding mortgage interest payments) (B.46) is the weighted averages of the sub-indices r pccdot, r phdot and r pxchdot. The residual captures the effect of the different rounding conventions used in the sub-indices. r pi xdott = µr ph µcc r pccdot + r phdott t 1 − µmi p 1 − µmi p µr ph + µcc r pxchdott + r pi xdottres + 1− 1 − µmi p (B.46) Inflation of mortgage interest payments (B.47) is proxied using the rate of inflation of interest payments on the nominal value of the housing stock, evaluated using the house price index phse. mi psdott = rgt −1 + rgt−1 + phset (1 + ṗt ) 1 + λ̇t −1 phset−1 dt 1 + λ̇t (1 + ṅ t ) − 1 + mi psdottres dt−1 (B.47) RPI inflation (B.48) is the weighted average of the sub-indices mi psdot and r pi xdot. Again, the residual captures rounding effects. r pidott = µmi p mi psdott + 1 − µmi p r pi xdott + r pidottres (B.48) Seasonally adjusted RPI inflation (B.49) adjusts RPI inflation for seasonal effects using seasonal dummy variables. r pidotsat = r pidott + 0.0017dumq1t − 0.005dumq2t +0.0035dumq3t − 0.0002dumq4t + r pidotsatres (B.49) (1) This dummy takes the value 1 in second quarter of each year and zero otherwise. It is used here because changes in council tax rates are ordinarily implemented in the second quarter of each year. 209 The Bank of England Quarterly Model The levels of the non-rent component (B.50) and rent component (B.51) of the CPI are given by cumulating the inflation rates of these series. cpi xrt = cpi xrt−1 (1 + cpi xrdott ) (B.50) cprt = cprt−1 (1 + cprdott ) (B.51) The level of the CPI (B.52) is a chain-weighted index of cpi xr and cpr, where the residual arises from an approximation error owing to chaining the indices quarterly rather than monthly. (1 − µcpr ) cpi xrt 4 cpi t = i=1 dumqi t cpi t−i × 4 i=1 dumqi t cpi xrt−i + µcpr cprt 4 i=1 dumqi t cprt−i × 1 + cpi tres (B.52) The levels of the housing depreciation component (B.53), council tax component (B.54), mortgage interest payments component (B.55) of RPI and RPI excluding these components (B.56) are given by cumulating the inflation rates of these variables. r ph t = r ph t−1 (1 + r phdott ) (B.53) r pcct = r pcct−1 (1 + r pccdott ) (B.54) mi pst = mi pst−1 (1 + mi psdott ) (B.55) r pxch t = r pxch t−1 (1 + r pxchdott ) (B.56) The level of the RPIX excluding council tax (B.57) is a chain-weighted index of r ph and r pxch, where the residual arises from an approximation error owing to chaining the indices quarterly rather than monthly. r pxct = (dumq1t r pxct−4 + dumq2t r pxct−1 + dumq3t r pxct−2 + dumq4t r pxct−3 ) ⎡ r ph ×⎣ + 1− r ph t µ 1−µmi p −µcc dumq1t r ph t−4 +dumq2t r ph t−1 +dumq3t r ph t−2 +dumq4t r ph t−3 r pxch t µr ph 1−µmi p −µcc dumq1t r pxch t−4 +dumq2t r pxch t−1 +dumq3t r pxch t−2 +dumq4t r pxch t−3 × 1 + r pxctres ⎤ ⎦ (B.57) The level of the RPIX excluding mortgage interest payments (B.58) is a chain-weighted index of r pcc and r pxc, where the residual arises from an approximation error owing to chaining the indices quarterly rather than monthly. r pi xt = (dumq1t r pi xt−4 + dumq2t r pi xt−1 + dumq3t r pi xt−2 + dumq4t r pi xt−3 ) × + 1 r pcct µcc 1−µmi p dumq1t r pcct−4 +dumq2t r pcct−1 +dumq3t r pcct−2 +dumq4t r pcct−3 r pxct µcc − 1−µ mi p dumq1t r pxct−4 +dumq2t r pxct−1 +dumq3t r pxct−2 +dumq4t r pxct−3 × 1 + r pi xtres 210 (B.58) The non-core equations The level of the RPI (B.59) is a chain-weighted index of r pi x and mi ps, where the residual arises from an approximation error owing to chaining the indices quarterly rather than monthly. We can use a similar method to construct other measures, such as the RPIY. r pi t = (dumq1t r pi t−4 + dumq2t r pi t−1 + dumq3t r pi t−2 + dumq4t r pi t−3 ) × mi pst µmi p dumq1t mi pst−4 +dumq2t mi pst−1 +dumq3t mi pst−2 +dumq4t mi pst−3 xt + 1 − µmi p dumq1t r pi xt−4 +dumq2t r pi xt−1r pi+dumq3 t r pi x t−2 +dumq4t r pi x t−3 × 1 + r pi tres (B.59) B2.3 Asset prices The real exchange rate (B.60), the value of equities (B.61) and the interest rate on corporate debt (B.62) all take the core model value plus a residual. qt = qtcore + qtres (B.60) v t = v tcore + v tres (B.61) rkt = r ktcore + rktres (B.62) Corporate debt issuance (B.63) is a given ratio (µbk ) of the equity value plus a residual. This equation corresponds to a long-run version of the core model equation (A.47). bkt = µbkv v t + bktres (B.63) B2.4 Labour market The equation for private sector real wages (B.64) includes an error correction term in core private sector real wages. Extra variables include lagged real wage growth and changes in RPI inflation, which are likely to influence wage setting; terms in RPI and the change in the consumption deflator, reflecting the wedge between the two measures (in effect the equation is estimated on nominal wages deflated by RPI); and the gap between the unemployment rate and steady-state unemployment, to capture cyclical influences on wage setting. log wt + ṗt − r pidotsat + λ̇t gap = 0.508 log wt−1 + λ̇t−1 + ṗt−1 − r pidotsat−1 gap −0.488 r pidotsat − 0.599 r pidotsat−1 −0.519 r pidotsat−2 − 0.120 u t − u ss t core −0.420 log wt−1 − log wt−1 + wtres (B.64) The equation for public sector real wages (B.65) includes an error correction term in private sector real wages, adjusted for the wedge between public and private sector wages. An additional term in lagged public sector real wage growth captures sluggishness in real wage adjustment. log wgt + λ̇t gap = 0.348 log wt−1 + λ̇t−1 gap −0.166 log(wgt−1 ) − log µwg wt−1 + wgtres (B.65) 211 The Bank of England Quarterly Model The equation for private sector employment (B.66) specifies an error correction to private sector employment. Extra variables include lagged changes in private sector employment and changes in private sector output. These terms proxy sluggish employment adjustment and labour demand effects respectively. A lagged term in factor utilisation is also included, as increases in factor utilisation tend to raise subsequent employment growth. gap log et + ṅ t gap gap log yt + ẏt log et−1 + ṅ t−1 + 0.094 f u t−1 core − 0.031 log et−1 − log et−1 + etres +0.053 100 = 0.649 (B.66) The equation for average hours (B.67) includes an error correction term for average hours. Additional dynamic terms include lagged growth in average hours and private sector output – reflecting sluggish adjustment and labour demand effects respectively. = 0.560 log avh t−1 + 0.037 log avh t gap log yt + ẏt res −0.048 log avh t−1 − log avh core t−1 + avh t + 0.055 gap log yt−1 + ẏt−1 (B.67) The equation for participation (B.68) includes an error correction term in participation. Extra variables include lagged participation growth (reflecting sluggish labour market adjustment) and changes in average real wages (as a proxy for the return to entry into the labour market). gap log lt + ṅ t gap gap log lt−2 + ṅ t−2 log lt−1 + ṅ t−1 + 0.249 wt−2 et−2 + wgt−2 egt−2 gap +0.044 log + λ̇t−2 eaggt−2 core + ltres −0.099 log lt−1 − log lt−1 = 0.499 (B.68) Private sector employment in hours (B.69) is a logarithmic version of the core equivalent (A.71). log eh t = log et + log avh t + eh res t (B.69) B2.5 Government The government budget constraint (B.70), fiscal reaction function (B.71) and government debt target (B.72) are the same as the core equivalents (A.79), (A.80), (A.81). pgt bgt τt lumpc = 1 + rgt−1 pgt−1 bgt−1 pct−1 mon t−1 + pgt gt + (1 + ecostgt ) wgt egt + 1 + ṗt 1 + ẏt (1 + ṗt ) (1 + ẏt ) +gosgex pt + transt − taxt − pct mon t + bgtres (B.70) = τ t−1 + θ bg lumpc bgt − bgtart + θ dbg pyt yt bgtart = µbgy 212 bgt−1 bgt − pyt yt pyt−1 yt−1 pyt yt + bgtartres pgt + taulumpctres (B.71) (B.72) The non-core equations The non-core equations for government wage spending (B.73), government procurement (B.74) and government investment procurement (B.75) run off the core model solutions (A.84), (A.82) and (A.86). (B.73) (1 + ecostgt ) wgt egt = (1 + ecostgt ) wgtcore egtcore + egtres pgt gt = pgtcore gtcore + gtres (B.74) igt = igtcore + igtres (B.75) Government consumption procurement (B.76) subtracts investment procurement from total procurement. gct = gt − igt + gctres (B.76) The government’s gross operating surplus (B.77) adds a residual to the core model solution (A.89). gosgex pt = gosgex ptcore + gosgex ptres (B.77) Total tax revenue (B.78) is the same as the core equivalent (A.90). taxt = taxwt + taxeet + taxdt + taxef t + taxkt + taxlumpct + taxlumpkt + taxf t +taxindt + gosgex pt + taxtres (B.78) Income tax revenue (B.79), employees’ National Insurance Contributions (B.80), dwellings tax revenue (B.81), employers’ National Insurance Contributions (B.82), corporation tax revenue (B.83), lump sum tax revenue from consumers (B.84), lump sum tax revenue from firms (B.85), tax revenue from overseas (B.86), indirect tax revenue (B.87) the total wage tax rate for private sector firms (B.88), total wage tax rate for government (B.89), net indirect taxes paid to the European Union (B.90), total transfer payments from government (B.91), transfers to consumers (B.92), transfers from overseas (B.93), total unemployment benefits (B.94), transfers to firms (B.95), transfers to overseas (B.96), transfers subsidies to firms on products (B.97) and social contributions paid to government employees (B.98) are all defined in the same way as the core model. The corresponding core model equations are (A.20) and (A.91) to (A.108). res taxwt = τ w t (wt et + wgt egt ) + taxwt (B.79) res taxeet = τ ee t (wt et + wgt egt ) + taxeet (B.80) taxdt = τ dt pdv t dt + taxdtres (B.81) taxef t = τ t (wt et + wgt egt ) + taxef res t (B.82) pyt yt + taxktres taxkt = τ knd t (B.83) ef taxlumpct = τ t lumpc yt pyt + taxlumpctres (B.84) 213 The Bank of England Quarterly Model taxlumpkt = τ t lumpk yt pyt + taxlumpktres (B.85) taxf t = τ t yt pyt + taxf res t f (B.86) taxindt = pbpat (ym t − yt ) + transksubst − taxeu t + taxindtres (B.87) ecostt = τ t + tr kpt + ecosttres (B.88) ecostgt = τ t + trect (B.89) res taxeu t = τ eu t pyt yt + taxeu t (B.90) ef ef transt = transct + transu t + transkt + transf t + transksubst + rgprem t + transtres (B.91) transct = trct yt pyt + transctres (B.92) transfpt = trfpt yt pyt + transfpres t (B.93) transu t = ben t u t lt + transu res t (B.94) transkt = trkt yt pyt + transktres (B.95) transf t = trf t yt pyt + transf res t (B.96) transksubst = tr ksubst yt pyt + transksubstres (B.97) transect = trect wgt egt + transectres (B.98) B2.6 Monetary authority The monetary reaction function (B.99) takes the same form and coefficients as the core model equation (A.110). The only difference is that the non-core reaction function depends on the core model solution for the nominal interest rate (rather than the steady-state nominal interest rate). rgt = 1 − θ rg rgtcore + θ pdot d4cpi t − ṗtss + θ y log yt + glt ystart + θ rg rgt−1 + rgtres (B.99) The policy reaction function depends on potential output (B.100), which is measured in a similar way to the core model equation (A.111). ss ystart = tfpt (1 − α) (1 − φ) 1 − u ss t lt avhstart 214 1− σ1y +α φ z tss kt−1 1 + ẏt 1− σ1y σy σ y −1 (B.100) The non-core equations As in the core model (A.112), the level of demand entering the policy rule is given by private sector output (y) plus the government demand for resources (B.101) expressed in terms of private sector output. glt = tfpt (1 − α) {(1 − φ) eaggt avhstart } 1− σ1y z ss kt−1 +α φ t 1 + ẏt −tfpt (1 − α) {(1 − φ) et avhstart }1− σ y + α φ 1 z tss kt−1 1− σ1y 1− σ1y 1 + ẏt σy σ y −1 σy σ y −1 (B.101) B2.7 Accounting and reporting The (basic price) value-added components of home consumption (B.102), home capital investment (B.103), other investment (B.104), government procurement (B.105) and exports (B.106) are identical to the core model equations (A.142) to (A.146). chv t = κ chv ch t (B.102) ikhv t = κ ikhv ikh t (B.103) iov t = κ iov iot (B.104) gv t = κ gv gt (B.105) xv t = κ xv xt (B.106) The level of demand for private sector output at basic prices (B.107) is defined as the sum of market price domestic demand components net of imports and the basic price adjustment. The statistical discrepancy (sd) is also included in this definition of demand. yt = ct + idt + ikt + iot + delst + gt + xt − cm t − ikm t − mi t +sdt − bpat + ytres (B.107) The level of demand for private sector output excluding stockbuilding (B.108) is identical to the core model counterpart (A.45). ydt = chv t + idt + ikhv t + iov t + gv t + xv t (B.108) The level of demand for private sector output at market prices (B.109) is given by the demand for private sector output at basic prices, plus the basic price adjustment. ym t = yt + bpat + ym res t (B.109) 215 The Bank of England Quarterly Model The basic price adjustment (B.110) is defined as the sum of the contributions from each expenditure category. It is given by the difference between the market price volume and the sum of domestically produced value added (proportion κ) and imported intermediate components (proportion ρ). bpat = 1 − κ chv − ρ ch ch t + 1 − κ i khv − ρ ikh ikh t + 1 − κ iov iot + 1 − κ gv − ρ g gt + 1 − κ xv − ρ x xt + bpatres (B.110) Expenditure on other investment (B.111) is given by the volume of other investment multiplied by the relative price. ioex pt = piot iot + ioex ptres (B.111) The volume of imputed and other rents (B.112) is equal to the core solution plus a residual. cirt = cirtcore + cirtres (B.112) Expenditure on imputed and other rents (B.113) is a proportion of the value of the dwellings stock. cirex pt = ψ cir pdv t dt + cirex ptres (B.113) The household budget constraint (B.114) replicates the core model equation (A.4). b f t pct + pgt bgt + v t + bkt qt = pct−1 mon t−1 1 + r f t−1 b f t−1 pct − pct · mon t + f (1 + ẏt ) (1 + ṗt ) 1 + ṗt 1 + ẏt qt 1 + rgt−1 pgt−1 bgt−1 1 + rkt−1 bkt−1 + + v t + dv t + 1 + ṗt 1 + ẏt 1 + ṗt 1 + ẏt +wlt lt − pct ct + transct + transkct + transkpt + transfpt +rfpremt + rgprem t − taxlumpct + transect − pdv t dt − 1 − δ dt dt−1 1 + ẏt − τ dt pdv t dt + b f tres (B.114) The level of real money balances (B.115) relative to consumption is equal to the ratio in the core model (adjusted for a residual). mon core t mon t = ct + monres (B.115) t ctcore The definition of dividends (B.116) is identical to the core model expression (A.25). dv t = pchv t chv t + pkhv t ikhv t + pdv t idt + piov t iov t + psv t delst + gv t · pgv t + pxv t xv t − (1 + ecostt ) wt et − pkh t ikh t − psv t delst − transkct − pkm t ikm t − piot iot rkt−1 bkt−1 bkt−1 − + bkt − (1 + ẏt ) (1 + ṗt ) (1 + ẏt ) (1 + ṗt ) (B.116) −taxkt − taxlumpkt − transkf t + transkt Employers’ private social contributions (B.117) to consumers are given by the application of a transfer weight (trkp) to the pre-tax private sector wage bill, as in the core model equation (A.49). transkpt = trkpt wt et + transkptres 216 (B.117) The non-core equations Transfers from firms to overseas (B.118) is defined identically to the core model equation (A.50). transkf t = trkf t pyt yt + transkf res t (B.118) Transfers from firms to consumers (B.119) are given by a proportion of the value of private sector output as in the core model (A.48). transkct = ψ snp pyt yt (B.119) Total asset holdings (B.120) include the values of equities corporate bonds, government bonds, real money balances and net foreign assets, as in the core model (A.158). at = v t + bkt + pgt bgt + pct mon t + nfat (B.120) The cumulation equation for the dwellings stock (B.121) is identical to the core model equation (A.12). 1 − δ dt dt−1 + idt + dtres 1 + ẏt dt = (B.121) The cumulation equations for the home capital stock (B.122) and the imported capital stock (B.123) are similar to the core model, but the effective depreciation rates are not adjusted for capital utilisation. kh t = km t = 1 − δ kh kh t−1 t + ikh t + kh res t 1 + ẏt (B.122) 1 − δ km km t−1 t + ikm t + km res t 1 + ẏt (B.123) The CES function defining the capital index (B.124) as a function of home and imported capital replicates the core model equation (A.27). kt = ψ k φ k kh t 1− 1 σk + 1 − ψk 1 − φ k km t 1− 1 σk σk σ k −1 (B.124) Total labour income (B.125) is the effective wage rate from participating in the labour market multiplied by the participation rate, plus transfers to consumers. lyt = wlt lt + transct (B.125) The unemployment rate (B.126) is defined as in the core model (A.66). ut = lt − en t − egt lt (B.126) Aggregate employment (B.127) is the sum of private and public sector employment. eaggt = en t + egt (B.127) The aggregate wage rate (B.128) is a weighted average of public and private sector wage rates. waggt = egt wgt + (eaggt − egt ) wt eaggt (B.128) 217 The Bank of England Quarterly Model Private sector employment in heads (B.129) is set equal to the employment index (e). en t = et (B.129) The expected wage from participation (B.130) is identical to the core model equation (A.62). wlt = 1 − u t − µt eg ee wt + u t ben t + µt 1 − τw t − τt eg ee wgt 1 − τw t − τt (B.130) Unemployment benefits (B.131) are given by an exogenous ratio of the private sector pre-tax wage, as in the core model (A.70). ben t = µbenw wt (B.131) The share of public sector employment in participation (B.132) is defined analogously to the core model equation (A.68). egt eg µt = (B.132) lt Net foreign assets (B.133), the trade balance (B.134) and the current account are identical to the core model equations (A.150), (A.151) and (A.152). nfat = b f t pct + nfares t qt (B.133) xm t = pxt xt − pcm t cm t − pkm t ikm t − pmin t mi t xmcat = (B.134) pxt xt − pcm t cm t − pkm t ikm t − pmin t mi t + taxf t − transf t − transkf t r f t−1 nfat−1 + xmcatres (B.135) +transfpt − taxeu t + rfpremt + (1 + ṗt ) (1 + ẏt ) As noted in the introductory remarks to this section, the non-core model equations make use of technical gap terms ẏ gap , λ̇ and ṅ gap to ensure that the partial adjustment mechanisms operate in actual units rather than model units. The expressions for these growth adjustments are given by equations (B.136), (B.137) and (B.138). These growth adjustments are simply equal to the deviation between the observed growth rate and the steady-state growth rate, denoted by the superscript ss. ẏt = log (1 + ẏt ) − log 1 + ẏtss (B.136) λ̇t = log 1 + λ̇t − log 1 + λ̇t (B.137) = log (1 + ṅ t ) − log 1 + ṅ ss t (B.138) gap gap gap ṅ t ss Nominal private sector wage inflation (B.139) is calculated as the change in the real wage multiplied by the numeraire inflation rate and the change in labour productivity, as in the core model (A.157). 1 + ẇt = 218 (1 + ṗt ) 1 + λ̇t wt wt−1 (B.139) The non-core equations As in the core model, nominal inflation rates for prices can be calculated as the change in the relative price multiplied by the inflation rate of the numeraire. Inflation rates for value-added home consumption prices, imported consumption prices and domestically produced consumption prices are shown in equations (B.140), (B.141) and (B.142). 1 + ṗtchv = (1 + ṗt ) pchv t pchv t−1 (B.140) 1 + ṗtcm = (1 + ṗt ) pcm t pcm t−1 (B.141) 1 + ṗtch = (1 + ṗt ) pch t pch t−1 (B.142) B2.8 Post-transformation equations The majority of post-transformation equations reverse transformations that are documented in Appendix C. That is, these equations construct measures that can be compared to (eg) National Accounts data, from the output of the non-core equations in detrended model units. Details of the mapping are outlined in Chapter 6. The equations below show how we would derive variables from the detrended model unit series for exports, xt , and export relative prices, pxt . These show the relationships for series in current prices (cp); chained volume constant prices (kp); the implicit price deflator (de f ); and the inflation rate of export prices ( ṗtx ). xcpt = xt · λt · nhdst · pxt · pcde f t xkpt = xt · λt · nhdst pxde f t = pxt · pcde f t ṗtx = (1 + ṗt ) pxt −1 pxt−1 In this appendix, we list only those post-transformation relationships that do not follow the standard pattern or that require some explanation of how they are derived. Nominal exchange rates and prices The nominal effective exchange rate (B.143) is calculated recursively from the real exchange rate and relative rates of domestic and overseas consumer price inflation. f qt pcde f t−1 1 + ṗt eert = eert−1 qt−1 pcde f t (B.143) The consumption deflator (excluding actual and imputed rents) (B.144) cumulates the rate of consumer price inflation given an initial starting point for the deflator. pcde f t = pcde f t−1 (1 + ṗt ) (B.144) 219 The Bank of England Quarterly Model National Accounts The quarterly alignment adjustment (B.145) in the National Accounts is calculated as the difference between the ONS measure of stockbuilding including the quarterly alignment adjustment, dels, and the underlying accumulation of stocks. (2) st−1 λt nhdst (B.145) aakpt = delst − st − 1 + ẏt As discussed in Chapter 6, the values of GDP at market prices (B.146) and basic prices (B.147) can be constructed in a straightforward way, by adding the ONS measure of the government’s value added and expenditure on actual and imputed rents to the value of private sector output. gd pex pt = pym t ym t + cirex pt + (1 + ecostgt )wgt egt + gosgex pt + gdpex ptres (B.146) gdpbpex pt = pyt yt + cirex pt + (1 + ecostgt )wgt egt + gosgex pt + gd pex ptres (B.147) The value of the ONS measure of total government expenditure on goods and services (B.148) is given by the sum of government net procurement expenditure and value added. gonsex pt = pgt gt + (1 + ecostgt )wgt egt + gosgex pt (B.148) The ONS measure of real government expenditure on goods and services (B.149) divides nominal government spending by the ONS government deflator. gonsex pt (B.149) gonst = pgonst The volume of GDP at market (B.150) and basic prices (B.151) are defined using the implicit ONS measure of the government’s real value added, gonst − gt . gdpt = ym t + cirt + gonst − gt + gdptres (B.150) gdpbpt = yt + cirt + gonst − gt + gdptres (B.151) The relative prices corresponding to the National Accounts GDP deflators at market prices (B.152) and at basic prices (B.153) can be calculated by dividing the respective GDP values by volumes. gdpex pt (B.152) pgdpt = gdpt pgdpbpt = gdpbpex pt gdpbpt (B.153) Labour market transformations Private sector average hours (B.154), expressed in units of weekly hours worked, are calculated recursively from the index of private sector average hours avh. avh t avhrst = avhrst−1 (B.154) avh t−1 (2) This could also be rewritten as aakpt = delstres λt nhdst given the equation for (B5). 220 The non-core equations Average hours of government employees (B.155) are assumed to move in line with steady-state private sector average hours, plus a residual to allow for any divergence. avhr sgt = avhrsgt−1 avh ss t + avhrsgtres avh ss t−1 Aggregate average hours (B.156) are given by identity. egt et + avhrsgt avhrsaggt = avhrst eaggt eaggt (B.155) (B.156) Private sector compensation of workers (B.157) (including the labour component of self employment income) and government compensation of workers (B.158) add the rate of employers’ social contributions to the private sector wage bill. These can then be used to define measures of the labour and profit share for the private sector and the whole economy. compcpt = (1 + ecostt ) · wt · et · λt · nhdst · pcde f t compgcpt = (1 + ecostgt ) · wgt · egt · λt · nhdst · pcde f t (B.157) (B.158) Household income and financial balances As noted in Chapter 6, a number of auxiliary equations are needed to calculate National Accounts financial balance measures and the household sector saving ratio. Interest payments on government debt (B.159) and interest payments on overseas assets (B.160) are given by multiplying the stock of assets by the relevant interest rate. The premium term is added to match the data. i pdgcpt = rgpremcpt + rgt−1 bgcpt−1 + i pdgcptres (B.159) i pd f cpt = r f premcpt + r f t−1 n f acpt−1 + i pd f cptres (B.160) National Accounts household sector dividend payments (B.161) in current prices include a residual term that captures the difference between the National Accounts definition and the model concept. dvcpt = dv t λt nhdst pcde f t + dvcptres (B.161) Total household sector net property income (B.162) includes all property income payments to households. The variable transkccp is used in a similar way to the interest rate premium terms on government and overseas assets, to ensure that the model matches the data. (3) i pdpcpt = i pdgcpt + i pd f cpt + dvcpt + rkt−1 bkcpt−1 − transkccpt + i pdpcptres (B.162) The household sector gross operating surplus (B.163) is assumed to move in line with actual and imputed rents. circpt gospcpt = gospcpt−1 + gospcptres (B.163) circpt−1 (3) This is reflected in the calibration of ψ snp . 221 The Bank of England Quarterly Model Total government transfers to households (B.164) are the sum of unemployment benefits and other transfer payments to households. transbencpt = transccpt + transucpt (B.164) Total transfer payments to households (B.165) are the sum of government transfer payments, employers’ pensions contributions and transfers from overseas. transctcpt = transbencpt + transkpcpt + transeccpt + trans f pcpt (B.165) Total tax payments by households (B.166) are given by the sum of income taxes, employees’ National Insurance Contributions, lump sum taxes and taxes on dwellings (council tax). tax pcpt = taxwcpt + taxeecpt + taxlumpccpt + taxdcpt (B.166) National Accounts measures of nominal (B.167) and real household consumption (B.168) are derived by adding back in actual and imputed rents to consumption. The ratio of the two is the National Accounts consumption deflator (B.169). cnacpt = ccpt + circpt (B.167) cnakpt = ckpt + cirkpt (B.168) pcnade f t = cnacpt cnakpt (B.169) The measure of household available resources (B.170) used to construct the ONS household saving ratio includes transfers, the operating surplus, interest payments, total wages and salaries less household taxes. Real household resources can be obtained by dividing by the National Accounts consumption deflator. A residual is used to account for the conceptual differences between ONS data and BEQM variables. wscpt + wsgcpt + gospcpt + transctcpt + i pdpcpt − tax pcpt r hpikpt = 1 + r hpikptres pcnade f t (B.170) The saving ratio (B.171) is saving divided by available household resources. pcnade f t r hpikpt − cnacpt savrt = pcnade f t r hpikpt (B.171) Household sector net lending (B.172) is given by current saving less household investment in dwellings, plus a residual to capture capital transfers and taxes. nlpt = pcnade f t r hpikpt − cnacpt − idcpt + nlptres (B.172) Total managed expenditure by general government (B.173) is the sum of government spending on private sector goods and services, government spending on factor payments, government transfers to different sectors and government interest payments. tmecpt = gcpt + transccpt + transucpt + trans f cpt + transkcpt + transksubscpt +i pdgcpt + compgcpt + gosgcpt + tmecptres 222 (B.173) The non-core equations General government net lending (B.174) is the difference between total tax revenues and total managed expenditure. General government net borrowing (B.175) is the reverse. nlggt = taxcpt − tmecpt + nlggtres (B.174) ggnbcpt = tmecpt − taxcpt + ggnbcptres (B.175) Overseas net lending (B.176) is the reverse of the current account of the balance of payments plus net capital transfers. nl f t = −xmcacpt + nl f tres (B.176) 223 Appendix C Data transformations and sources This appendix lists the data transformations and sources for BEQM variables. As set out in Chapter 6, BEQM data are transformed into ‘model units’ and then transformed out again into levels. Most of the sources are either another BEQM variable or an original source – unless otherwise indicated, this is the code from the Office for National Statistics database (four letter identifiers shown in capitals). An asterisk indicates that some of the backdata for the series have been constructed, and the source used for this is indicated in the third column. To ease the exposition, a few additional variables are included here that do not feature in the model equations reported above – for example, aggregate wages and salaries (wsagg) appears in the expression for post-tax labour income (ly) and as the denominator for some effective tax rates. Tables 6.1 and 6.2 illustrate the notation we use for data, actual units and detrended model units. A BEQM quantity variable can have up to four associated concepts. To avoid confusion with ONS database identifiers and to help distinguish between the related concepts, we use lower case italic notation for most BEQM variables as follows: • volume measure in detrended model units; • current price (nominal) measure, with suffix cp; • chained volume measure (CVM), with suffix kp; and • (implicit) price deflator, with suffix def. For some prices, we also calculated the associated inflation rate, usually quarterly, which we denote either by a dot above the variable or (for longer names) by the suffix dot. The BEQM variables that are not in lower case italics are generally parameters, shown as Greek letters. Most monetary variables here are in terms of £ million. Chained volume measures are in the prices of the reference year, currently 2001. Table C.1: Data sources and transformations for BEQM NAME DESCRIPTION CALCULATION aacp Alignment adjustment, current prices DMUN aakp Alignment adjustment, CVM DMUM acp Net financial wealth of household sector, current prices NZEA (*ALDZ) avh Index of private sector average hours (1995=1) avhrs/avhrs[1995 value] avhrs Average weekly hours worked in the private sector BoE constructed data, derived from LFS microdata avhrsagg Average weekly hours worked in the whole economy 1000·eagghrs/eagghds avhrsg Average weekly hours worked in the general government sector 1000·eghrs/eghds ben Unemployment benefits, detrended model units transucp/(λ·uhds·pcdef ) 225 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION bf Stock of foreign bonds denominated in terms of foreign consumption goods, detrended model units nfa·q bg Stock of government bonds, detrended model units bgcp/(λ·nhds·pgdef ) bgcp Market value of general government gross debt, current prices MDQE bk Stock of corporate bonds, detrended model units bkcp/(λ·nhds·pcdef ) bkcp Stock of net corporate sector debt, current prices NLBE + NLBI + NKZA – NKJZ (*RHHS + AMXE + REWK) bpa Basic price adjustment, detrended model units bpakp/(λ·nhds) bpacp Basic price adjustment, current prices NTAP bpakp Basic price adjustment, CVM NTAO c Volume of consumption goods, detrended model units ckp/(λ·nhds) ccp Consumption expenditure (excluding actual and imputed rents), current prices cmcp + chcp cf World trade, detrended model units 5532·yf /(λ·nhds) ch Domestically produced consumption goods, detrended model units chkp/(λ·nhds) chcp Expenditure on domestically produced consumption goods, current prices cnacp – circp – cmcp chkp Expenditure on domestically produced consumption goods, CVM cnakp – cirkp – cmkp chv Value-added component of domestically produced consumption goods, detrended model units ch·κ chv cir Volume of actual and imputed rents, detrended model units cirkp/(λ·nhds) circp Expenditure on actual and imputed rents, current prices cirkp·(GBFJ + ZAVP)/(GBFK + ZAVQ) cirexp Expenditure on actual and imputed rents, detrended model units circp/(λ·nhds·pcdef ) cirkp Expenditure on actual and imputed rents, CVM QTPS[2001 value]·GDQL/400 ckp Consumption expenditure (excluding actual and imputed rents), CVM cmkp + chkp 226 Data transformations and sources NAME DESCRIPTION CALCULATION cm Volume of directly imported consumption goods, detrended model units cmkp/(λ·nhds) cmcp Expenditure on directly imported consumption goods, current prices 0·798·BQAR + 0·498·ENGD + ENGE + 0·369·IKBC cmkp Expenditure on directly imported consumption goods, CVM 0·798·BPIA + 0·498·ENGT + ENGU + 0·369·IKBF cna Volume of consumption goods (National Accounts measure), detrended model units cnakp/(λ·nhds) cnacp Consumption expenditure (National Accounts measure), current prices ABJQ + HAYE cnaexp Consumption expenditure (National Accounts measure), detrended model units cnacp/(λ·nhds·pcdef ) cnakp Consumption expenditure (National Accounts measure), CVM ABJR + HAYO compcp Total compensation of private sector workers (including self-employed), current prices wscp·(1 + ecost) compgcp Total compensation of general government employees, current prices BoE constructed data, based on seasonally adjusted QWRY + QWPS cpi Consumer Prices Index, (1996=100) CHVJ (*BoE constructed data) cpidot Quarterly CPI inflation rate cpi/cpit−1 – 1 cpiwedge Wedge between quarterly inflation rates of the consumption expenditure deflator (excluding actual and imputed rents) and CPI (excluding rents) BoE constructed data cpixr Consumer Prices Index (excluding rents), (1996=100) BoE constructed data cpixrdot Quarterly inflation rate of the CPI (excluding rents) cpixr/cpixrt−1 – 1 cpr Rents component of CPI, (1996=100) CHWC (*BoE constructed data) cprdot Quarterly inflation rate of the rents component of the CPI cpr/cprt−1 – 1 d Stock of dwellings, detrended model units dkp/(λ·nhds) dcp Net stock of dwellings, current prices dels Stockbuilding (including alignment adjustment), detrended model units pddef ·dkp delskp/(λ·nhds) delscp Stockbuilding (including alignment adjustment), current prices CAEX 227 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION delskp Stockbuilding (including alignment adjustment), CVM CAFU dkp Net capital stock of dwellings, CVM Interpolation of GUCX annual data dvcp Dividend payments to households, current prices (NHOL/(NHOK + NHOL))·ROYP + ROYN e Private sector employment index ehds/nhds eagg Aggregate employment en + eg eagghds Aggregate employment in heads, thousands MGRZ eagghrs Aggregate total weekly hours worked, millions YBUS ecost Rate of employers’ total social contributions, private sector ecostg Rate of employers total social contributions, general government NMXR/(NMXS – NMXR) eer Nominal sterling effective exchange rate (1990=100), quarterly average of daily data AGBG eg General government employment eghds/nhds eghds General government employment, thousands eagghds – ehds eghrs Total weekly hours worked by general government employees, millions eagghrs – ehrs eh Private sector hours worked, per head of population avh·e ehds Private sector employment (including self-employed), thousands BoE constructed data, derived from LFS microdata ehrs Total weekly private sector hours worked, millions avhrs·ehds/1000 en Private sector employment ehds/nhds g Volume of government procurement of private sector goods and services, detrended model units gkp/(λ·nhds) gc Volume of general government procurement of private sector goods and services (consumption goods), detrended model units gckp/(λ·nhds) gccp General government procurement of private sector goods and services (consumption goods), current prices BoE constructed data, based on NMRP, seasonally adjusted NMXV and seasonally adjusted QWRY + QWPS 228 τ e f + trkp Data transformations and sources NAME DESCRIPTION CALCULATION gckp General government procurement of private sector goods and services (consumption goods), CVM gccp/pgcdef gconscp ONS-measured general government final consumption expenditure, current prices NMRP gconskp ONS-measured general government final consumption expenditure, CVM NMRY gcp General government procurement of private sector goods and services, current prices gccp + igcp gdp Volume of GDP at market prices, detrended model units gdpkp/(λ·nhds) gdpbp Volume of GDP at basic prices, detrended model units gdpbpkp/(λ·nhds) gdpbpcp GDP at basic prices, current prices gdpcp – bpacp gdpbpexp Value of GDP at basic prices, detrended model units gdpbpcp/(λ·nhds·pcdef ) gdpbpkp GDP at basic prices, CVM ABMM gdpcp GDP at market prices, current prices YBHA gdpexp Value of GDP at market prices, detrended model units gdpcp/(λ·nhds·pcdef ) gdpkp GDP at market prices, CVM ABMI ggnbcp General government net borrowing, current prices tmecp – taxcp gkp Total general government procurement of private sector goods and services, CVM gckp + igkp gons Volume of total ONS-measured general government final consumption and investment expenditure, detrended model units gonskp/(λ·nhds) gonscp Total ONS-measured general government final consumption and investment expenditure, current prices gconscp + igcp gonskp Total ONS-measured general government final consumption and investment expenditure, CVM gconskp + igkp gosgcp General government gross operating surplus, current prices gconscp – compgcp – gccp gosgexp General government gross operating surplus, detrended model units gosgcp/(λ·nhds·pcdef ) 229 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION gospcp Household sector gross operating surplus, current prices CAEN gv Value-added component of total general government procurement, detrended model units g·κ gv id Volume of investment in dwellings, detrended model units idkp/(λ·nhds) idcp Private sector investment in dwellings, current prices GGAG idkp Private sector investment in dwellings, CVM DFEA ig Volume of government procurement of investment goods, detrended model units igkp/(λ·nhds) igcp General government procurement of private sector goods and services (capital goods), current prices RPZG (*seasonally adjusted AAYE) igkp General government procurement of private sector goods and services (capital goods), CVM DLWF (*DFED) ik Volume of total business investment, detrended model units ikkp/(λ·nhds) ikcp Business investment, current prices NPEK ikexp Total business investment expenditure, detrended model units ikcp/(λ·nhds·pcdef ) ikh Volume of domestically produced investment, detrended model units ikhkp/(λ·nhds) ikhcp Business investment (domestically produced), current prices ikcp – ikmcp ikhkp Business investment (domestically produced), CVM ikkp – ikmkp ikhv Value-added component of domestically produced investment, detrended model units ikh·κ i khv ikkp Business investment, CVM NPEL ikm Volume of directly imported investment, detrended model units ikmkp/(λ·nhds) ikmcp Business investment (directly imported), current prices ENGG ikmkp Business investment (directly imported), CVM ENGW io Volume of other investment, detrended model units iokp/(λ·nhds) 230 Data transformations and sources NAME DESCRIPTION CALCULATION iocp Other investment expenditure, current prices TLNI + TLOP + NPJQ (* DFBH) ioexp Other investment expenditure, detrended model units iocp/(λ·nhds·pcdef ) iokp Other investment expenditure, CVM DLWH + DLWI + NPJR (*DFDW) iov Value-added component of other investment, detrended model units io·κ iov ipdfcp Net interest payments from overseas, current prices HBOJ ipdgcp Interest payments on general government debt, current prices ROXY ipdpcp Total net interest payments to household sector ROYL – ROYT kh Domestically produced business sector capital, detrended model units khkp/(λ·nhds) khcp Domestically produced business sector capital, current prices khkp·pkhdef khkp Domestically produced business sector capital, CVM BoE constructed data km Imported business sector capital, detrended model units kmkp/(λ·nhds) kmcp Imported business sector capital, current prices kmkp·pkmdef kmkp Imported business sector capital, CVM BoE constructed data l Participation rate lhds/nhds Labour-augmenting technical progress (productivity) Estimated from production function λ λ̇ Growth rate of labour-augmenting technical progress λ/λt−1 – 1 lhds Participation (labour supply), thousands MGSF ly Real post-tax labour income, detrended model units lykp/(nhds·λ) lykp Real post-tax labour income, CVM (wsaggcp – taxwcp – taxeecp + transbencp)/pcdef mcp Total imports of goods and services, current prices IKBI mi Volume of intermediate imports of goods and services, detrended model units minkp/(λ·nhds) 231 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION mincp Intermediate imports of goods and services, current prices mcp – cmcp – ikmcp minexp Expenditure on intermediate imports, detrended model units mincp/(λ·nhds·pcdef ) minkp Intermediate imports of goods and services, CVM mkp – cmkp – ikmkp mips MIPS component of the RPI (January 1987=100) DOBQ (*HKFN) mipsdot Quarterly growth rate of the MIPS component of the RPI mips /mipst−1 – 1 mkp Total imports of goods and services, CVM IKBL mon Stock of money holdings, detrended model units moncp/(λ·nhds·pcdef ) moncp Stock of notes and coins in circulation (break-adjusted measure), current prices BoE constructed data Weight of council tax in the RPI CZXF Weight of rents in the CPI CJVC Weight of MIPS in the RPI CZXE Weight on housing depreciation in RPI DOGX ṅ Quarterly rate of population growth nhds/nhdst−1 – 1 nfa Stock of foreign bonds denominated in terms of consumption goods, detrended model units nfacp/(λ·nhds·pcdef ) nfacp Net stock of external assets, current prices HBQC nhds Population aged 16+, thousands MGSL nlf Overseas sector net lending, current prices RQCH (*–AAVA – AAVB – AAVD – AAVF – AAVG – AAVH) nlgg General government net lending, current prices RPZD nlp Household sector net lending, current prices RPZT (*AAVH) ṗ Quarterly inflation rate of consumption goods (excluding actual and imputed rents) pcdef /pcdef t−1 – 1 ṗch Quarterly rate of inflation of domestically produced consumption goods pchdef /pchdef t−1 – 1 µcc µcpr µ mi p µ r ph 232 Data transformations and sources NAME DESCRIPTION CALCULATION chv Quarterly rate of inflation of the value-added component of domestically produced consumption goods (pchv/pchvt−1 )·(1+ ṗ) – 1 ṗcm Quarterly rate of inflation of directly imported consumption goods pcmdef /pcmdef t−1 – 1 ṗ f Quarterly inflation rate of foreign consumption goods pcfdef /pcfdef t−1 – 1 ṗ min Quarterly inflation rate of intermediate imports pmindef /pmindef t−1 – 1 pbpa Relative price of the basic price adjustment bpacp/(bpakp·pcdef ) pc Numeraire price pcdef /pcdef pcdef Non-housing consumption implied deflator (2001=1) (excludes actual and imputed rent) ccp/ckp pcfdef M6 consumer prices, using sterling ERI weights, index (1998=100) BoE constructed data pch Relative price of domestically produced consumption goods pchdef /pcdef pchdef Expenditure on domestically produced consumption goods, implied deflator (2001=1) chcp/chkp pchv Relative price of the value-added component of domestically produced consumption goods pcm Relative price of directly imported consumption goods pcmdef /pcdef pcmdef Expenditure on directly imported consumption goods, implied deflator (2001=1) cmcp/cmkp pcnadef Consumption expenditure (National Accounts measure), implied deflator (2001=1) cnacp/cnakp pddef Private sector investment in dwellings, implied deflator (2001=1) idcp/idkp pdv Relative price of dwellings investment pddef /pcdef pg Relative price of government procurement of private sector goods and services pgdef /pcdef pgc Relative price of government procurement (consumption goods) pgcdef /pcdef ṗ (1/κ chv )·(pch – ρ ch ·pmin – (1 – ρ ch – κ chv )·pbpa) 233 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION pgcdef General government procurement of private sector goods and services (consumption goods), implied deflator (2001=1) BoE constructed data pgdp Relative price of GDP at market prices pgdpdef /pcdef pgdpbp Relative price of GDP at basic prices (gdpbpcp/gdpbpkp)/pcdef pgdef Total general government procurement of private sector goods and services, implied deflator (2001=1) gcp/gkp pgdpdef GDP at market prices, implied deflator (2001=1) gdpcp/gdpkp pgons Relative price of total ONS-measured general government consumption and investment expenditure pgonsdef /pcdef pgonsdef Total final general government consumption and investment expenditure, implied deflator (2001=1) gonscp/gonskp pgv Relative price of the value-added component of government procurement of private sector goods and services phse Relative price of housing (adjusted for trend productivity) phsedef /(pcdef ·λ) phsedef Average of the Nationwide and Halifax house price indices (1990=1) BoE constructed data pio Relative price of other investment piodef /pcdef piodef Other investment, implied deflator (2001=1) iocp/iokp piov Relative price of the value-added component of other investment pkg Relative price of government procurement (investment goods) pkgdef /pcdef pkgdef General government procurement of private sector goods and services (investment goods), implied deflator (2001=1) igcp/igkp pkh Relative price of domestically produced capital goods pkhdef /pcdef pkhdef Business investment (domestically produced), implied deflator (2001=1) ikhcp/ikhkp 234 (1/κ gv )·(pg – ρ g ·pmin – (1 – ρ g –κ gv )·pbpa) (1/κ iov )·(pio – (1 – κ i ov )·pbpa) Data transformations and sources NAME DESCRIPTION CALCULATION pkhv Relative price of the value-added component of domestically produced capital goods pkm Relative price of directly imported capital goods pkmdef /pcdef pkmdef Business investment (directly imported), implied deflator (2001=1) ikmcp/ikmkp pmdef Total imports of goods and services, implied deflator (2001=1) mcp/mkp pmin Relative price of intermediate imports pmindef /pcdef pmindef Intermediate imports of goods and services, implied deflator (2001=1) mincp/minkp psdef Stockbuilding (including alignment adjustment), implied deflator (2001=1) delscp/delskp psv Relative price of stockbuilding psdef /pcdef px Relative price of exports pxdef /pcdef pxdef Exports of goods and services, implied deflator (2001=1) xcp/xkp pxf Relative price of world exports 0·958·(pxfdef /pcfdef ) pxfdef M6 export prices, using sterling ERI weights, index (1998=100) BoE constructed data pxv Relative price of the value-added component of exports py Relative price of private sector value added at basic prices pydef /pcdef pydef Private sector value added at basic prices, implied deflator (2001=1) ycp/ykp pym Relative price of private sector value added at market prices ymcp/(ymkp·pcdef ) q Real exchange rate using consumer prices (1995=1) (eer·pcdef /pcfdef )/ ((eer·pcdef /pcfdef ) [1995 value]) rf M6 short-term nominal interest rate, using UK trade weights, quarterly rate BoE constructed data rfprem Premium on overseas interest payments to households, detrended model units rfpremcp/(λ·nhds·pcdef ) rfpremcp Premium on overseas interest payments to households, current prices ipdfcp – nfacpt−1 ·rf t−1 (1/κ ikhv )·(pkh – ρ ikh ·pmin – (1 – ρ ikh – κ ikhv )·pbpa) (1/κ xv )·(px – ρ x ·pmin – (1 – ρ x – κ xv )·pbpa) 235 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION rg Short-term nominal interest rate, quarterly rate ((1 + (rga/100)) 4 – 1) rga Short-term nominal interest rate, annual rate, per cent AMIH rgprem Premium on government interest payments to households, detrended model units rgpremcp/(λ·nhds·pcdef ) rgpremcp Premium on government interest payments to households, current prices (ipdgcp – rgt−1 ·bgcpt−1 ) rhpikp Total available household resources, CVM (RPHQ + RPQJ)·cnakp/cnacp rpcc Council tax component of the RPI DOBR (*HFKO) rpccdot Quarterly inflation rate of the council tax component of the RPI rpcc/rpcct−1 – 1 rph Housing depreciation component of the RPI, (January 1995=100) CHOO rphdot Quarterly inflation rate of the housing depreciation component of the RPI rph/rpht−1 – 1 rpi Retail Prices Index (January 1987=100) CHAW (*CBAB) rpidot Quarterly inflation rate of the RPI rpi/rpit−1 – 1 rpidotsa Quarterly inflation rate of the RPI, seasonally adjusted BoE constructed data rpix Retail Prices Index excluding mortgage interest payments (January 1987=100) CHMK (*RYYW) rpixdot Quarterly inflation rate of RPIX rpix/rpixt−1 – 1 rpixwedge Wedge between quarterly growth rate of the RPI, excluding council tax and housing depreciation and the CPI, excluding rents BoE constructed data rpxc Retail Prices Index excluding mortgage interest payments and council tax (January 1987=100) DQAD rpxcdot Quarterly inflation rate of the RPI, excluding mortgage interest payments and council tax rpxc/rpxct−1 – 1 rpxch Retail Prices Index, excluding MIPS, council tax and housing depreciation (January 1987=100) BoE constructed data 236 1 Data transformations and sources NAME DESCRIPTION CALCULATION rpxchdot Quarterly inflation rate of RPI, excluding MIPS, council tax and housing depreciation rpxch/rpxcht−1 – 1 s Stock of inventories, detrended model units skp/(λ·nhds) savr Household sector saving ratio, per cent 100·(1 – cnakp/rhpikp) sd Statistical discrepancy, detrended model units sdkp/(λ·nhds) sdcp Statistical discrepancy, current prices GIXM sdexp Value of statistical discrepancy, detrended model units sdcp/(λ·nhds·pcdef ) sdkp Statistical discrepancy, CVM GIXS skp Stock of inventories, CVM BoE constructed data Effective net indirect tax rate (ratio of basic price adjustment to value added) bpacp/ycp Effective tax rate on dwellings taxdcp/dcp Effective rate of employees’ National Insurance Contributions taxeecp/wsaggcp Effective rate of employers’ National Insurance Contributions taxefcp/wsaggcp Effective tax rate on EU net indirect taxes taxeucp/ycp Effective tax rate on revenue from overseas residents taxfcp/ycp Effective corporation tax rate taxkcp/ycp Effective lump sum tax rate on households taxlumpccp/ycp Effective lump sum tax rate on firms taxlumpkcp/ycp Effective income tax rate taxwcp/wsaggcp tax Total taxation receipts, detrended model units taxcp/(λ·nhds·pcdef ) taxcp Total taxation receipts, current prices GZXX taxd Tax revenue from tax on dwellings, detrended model units taxdcp/(λ·nhds·pcdef ) taxdcp Tax revenue from tax on dwellings (council tax), current prices RNTO taxee Employees’ National Insurance Contributions, detrended model units taxeecp/(λ·nhds·pcdef ) taxeecp Employees’ National Insurance Contributions, current prices AIIV – CUCT τ c τd τ ee τ ef τ eu τf τ knd τ lumpc τ lumpk τw 237 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION taxef Employers’ National Insurance Contributions, detrended model units taxefcp/(λ·nhds·pcdef ) taxefcp Employers’ National Insurance Contributions, current prices CUCT taxeu Indirect taxes minus subsidies paid to EU, detrended model units taxeucp/(λ·nhds·pcdef ) taxeucp Indirect taxes minus subsidies paid to EU, current prices CGDR – FKNG (*FJWB – FJWJ) taxf Tax revenue from overseas residents, detrended model units taxfcp/(λ·nhds·pcdef ) taxfcp Tax revenue from overseas residents, current prices FHDM (*FJKI + FKKL) taxind Tax revenue from indirect taxation, detrended model units taxindcp/(λ·nhds·pcdef ) taxindcp Tax revenue from indirect taxation, current prices bpacp + transksubscp – taxeucp taxk Tax revenue from corporation tax, detrended model units taxkcp/(λ·nhds·pcdef ) taxkcp Tax revenue from corporation tax, current prices Seasonally adjusted ACCJ + ACCD + EYNK (*AIAY + AIFS + ADRV) taxlumpc Tax revenue from lump sum taxes on households, detrended model units taxlumpccp/(λ·nhds·pcdef ) taxlumpccp Tax revenue from lump sum taxes on households, current prices RNGQ + RPHT – RNTO (*RNGQ + CFGE) taxlumpk Tax revenue from lump sum taxes on firms, detrended model units taxlumpkcp/(λ·nhds·pcdef ) taxlumpkcp Tax revenue from lump sum taxes on firms, current prices taxcp – taxwcp – taxindcp – taxlumpccp – taxkcp – taxfcp – taxdcp – taxeecp – taxefcp – gosgcp taxpcp Total tax payments of household sector, current prices taxwcp + taxeecp + taxlumpccp + taxdcp taxw Tax revenue from labour income taxes, detrended model units taxwcp/(λ·nhds·pcdef ) taxwcp Tax revenue from labour income taxes, current prices RPHS (*AIIU) tmecp Total managed general government expenditure, current prices GZWA + NMXO – EQJW trans Total general government transfers, detrended model units transc + transu + transk + transf + transksubs + rgprem transbencp Total general government transfer payments to households, current prices GZVX 238 Data transformations and sources NAME DESCRIPTION CALCULATION transc General government transfer payments to households excluding unemployment benefit, detrended model units transccp/(λ·nhds·pcdef ) transccp General government transfer payments to households excluding unemployment benefit, current prices transbencp – transucp transctcp Total transfer payments to households, current prices transbencp + transkpcp + transfpcp + transeccp transec Employers’ private social contributions, general government, detrended model units transeccp/(λ·nhds·pcdef ) transeccp Employers’ private social contributions, general government, current prices wsgcp·trec transf General government transfers to overseas sector, detrended model units transfcp/(λ·nhds·pcdef ) transfcp General government transfers to overseas sector, current prices FLUD – FNTL (*BoE constructed) transfp Net overseas transfers to households, detrended model units transfpcp/(λ·nhds·pcdef ) transfpcp Net overseas transfers to households, current prices FNTP + FKIL – FKIQ transk General government transfers to firms, detrended model units transkcp/(λ·nhds·pcdef ) transkc Supernormal profit transfers from firms to households, detrended model units transkccp/(λ·nhds·pcdef ) transkccp Supernormal profit transfers from firms to households, current prices transkcp General government transfers to firms, current prices -(ROYM + ROYQ – ROYU – ROYV + (NHOK/(NHOK + NHOL))·ROYP – ipdfcp – ipdgcp – rkt−1 ·bkcpt−1 ) tmecp – gcp – transccp – transucp – transfcp – transksubscp – ipdgcp – compgcp – gosgcp transkf Net transfers from firms to overseas, detrended model units transkfcp/(λ·nhds·pcdef ) transkfcp Net transfers from firms to overseas, current prices –FNTC – CGDR + FKNG – FNTQ – FNTR – FNTS + FKIL – FKIQ transkp Employers’ other social contributions, private sector, detrended model units transkpcp/(λ·nhds·pcdef ) 239 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION transkpcp Employers’ other social contributions, private sector, current prices ROYK – transeccp – taxefcp transksubs General government subsidies on products, detrended model units transksubscp/(λ·nhds·pcdef ) transksubscp General government subsidies on products, current prices NTAG – NTAP – FKNG (*NTAG – NTAP – seasonally adjusted FJWJ) transu Total unemployment benefits, detrended model units transucp/(λ·nhds·pcdef ) transucp Total unemployment benefits, current prices trc General government transfer rate to households (excluding unemployment benefit) trec Rate of employers’ other social contributions, general government trf General government transfer rate to foreigners transfcp/ycp trfp Rate of net transfers from foreigners to households transfpcp/ycp trk General government transfer rate to firms transkcp/ycp trkf Transfer rate from firms to overseas transkfcp/ycp trkp Transfer rate from firms to households transkpcp/wscp trksubs Subsidy rate from general government to firms transksubscp/ycp u Unemployment rate uhds/lhds uhds Unemployment, thousands MGSC ulc Private sector unit labour costs ur Unemployment rate, per cent wdef ·(1 + ecost)/prodhds 100·u v Value of equities, detrended model units vcp/(λ·nhds·pcdef ) vcp Nominal value of equities, current prices acp – bgcp – bkcp – nfacp – moncp w Private sector real wage, detrended model units wdef /(λ·pcdef ) Quarterly growth rate of nominal private sector wages wdef /wdef t−1 – 1 Aggregate real wage, detrended model units waggdef /(λ·pcdef ) ẇ wagg 240 µbenw ·u·lhds·wdef transccp/ycp ecostg – τ e f Data transformations and sources NAME DESCRIPTION CALCULATION waggdef Nominal aggregate wage per worker (including self-employed) wsaggcp/eagghds wdef Nominal private sector wage per worker (including self-employed) wscp/ehds wg Government wage, detrended model units wgdef /(λ·pcdef ) wgdef Nominal government wage per worker wsgcp/eghds wsaggcp Aggregate wages and salaries (including self-employment income), current prices wscp + wsgcp wscp Private sector wages and salaries (including self-employment income), current prices (ROYJ – wsgcp)·ehds/(ehds – MGRQ) (*MGRQ projected back with DYZN before 1992 Q2) wsgcp Total government wages and salaries, current prices compgcp/(1 + ecostg) x Volume of exports, detrended model units xkp/(λ·nhds) xcp Exports of goods and services, current prices IKBH xkp Exports of goods and services, CVM IKBK xm Net expenditure on overseas goods and services, detrended model units xmcp/(λ·nhds·pcdef ) xmca Current account balance, plus net capital transfers from overseas, detrended model units xmcacp/(λ·nhds·pcdef ) xmcacp Current account balance, plus net capital transfers from overseas, current prices –nlf xmcp Trade balance, current prices xcp – mcp xmkp Trade balance, CVM xkp – mkp xv Value-added component of export volumes, detrended model units x·κ xv y Private sector value added, detrended model units ykp/(λ·nhds) ycp Private sector value added at basic prices, current prices gdpbpcp – circp – gonscp + gcp yd Volume of final demand, detrended model units chv + id + ikhv + iov + gv + xv ydkp Value of final demand, CVM yd·λ·nhds yf Volume of world imports, using UK trade weights, (2000=100) BoE constructed data 241 The Bank of England Quarterly Model NAME DESCRIPTION CALCULATION ykp Private sector value added at basic prices, CVM gdpbpkp – cirkp – gonskp + gkp ym Private sector value added at market prices, detrended model units ymkp/(λ·nhds) ymcp Private sector value added at market prices, current prices ycp + bpacp ymkp Private sector value added at market prices, CVM ykp + bpakp 242 Appendix D Parameter and exogenous values Table D.1 reports the values of core model parameters and exogenous variables. As noted in Chapter 6, some of the factors proxied by these parameters may have changed over the past, so some parameters are assumed to change over the past too. The parameter values in Table D.1 are those for 2003 Q4 – the end of our estimation period – which were used to generate the simulation results reported in Chapter 7. The values for exogenous variables are those expected to prevail in the long run, rather than actual values for 2003 Q4. For example, cpiwedge = 0 which is consistent with the balanced growth steady state in which all prices grow at the same rate. Table D.1: Parameter values Households β 0.998 hw β 0.994 δd 0.004 γ 0.975 c κ 3.307 0.873 φc m φ 0.5 0.852 ψc ψ hab 0.7 ψ habd 0.7 0.26 ψm ψ mon 15.539 0.2 σc σd 0.5 m 1.77 σ Labour market avhstar 0.96 wdot 0.9 ηl 0.1 5.437 ηw u γ 1 0.5 γw l κ -0.33 0.141 µbenw e ψ 1 0.39 ψu Government & monetary authority 2.644 µbgy 0.151 µgy µigy 0.025 0.752 µwg µwgy 0.168 0.016 ψ gosg ṗss 0.005 0.16 τc 0.005 τd τ ee 0.053 0.067 τ ef τ eu 0.004 0.004 τf τ knd 0.052 0.035 τ lumpk τw 0.202 0.5 θ bg θ bp 0.5 1.5 θ dbg θg -0.1 1.5 θ pdot 0.65 θ rg θ wg -0.1 0.125 θy trc 0.166 trec 0.129 trf 0.014 trfp 0.001 trk 0.039 trkf 0 trkp 0.075 trksubs 0.005 243 The Bank of England Quarterly Model Table D.1: (continued) Parameter values Firms α χd χ dels χ kh χ km χl χ pch χ pd χ pg χ pkh χ px χz δ kh δ km kh km pchdot pddot pgdot pkhdot pxdot ηc ηd ηg ηk ηx γk µbkv µs φ φk φz ψk ψs ψ snp σk σy θ bk 244 0.31 10 10 165 35 0 400 0 218.139 375.809 54.879 0.017 0.006 0.015 0.7 0.7 0.5 0 0.5 0.5 0.5 11 7.777 6.453 10.395 14.72 0.999 0.287 0.822 0.436 0.5 0.1 0.973 0.1 -0.14 0.4 0.317 0.9 External cf 1.154 pcm 1 pcmdot 0.9 pkm 1 pkmdot 0.9 pmidot 0.9 cm 5.1 η 5.1 ηkm mi η 5.1 1.5 η px mi γ 0.15 0.15 γ pcm pkm γ 0.15 1.272 κ pcm pkm κ 1.066 1.241 κ pmin x κ 0.17 0.005 ṗ f ṗ f ss 0.005 pxf 0.894 0.139 ρ ch ρg 0.31 0.5 ρ ikh ρx 0.327 rf 0.013 Exogenous and technical cpiwedge 0 0.691 κ chv κ gv 0.578 ikhv κ 0.397 κ iov 0.618 xv 0.651 κ λ̇ 0.006 ss λ̇ 0.006 ṅ 0.001 0.001 ṅ ss rfprem 0.005 rgprem 0.001 tfp 1.001 cir 0.021 ψ 0.018 ψio pio ψ 1.434 wmargin 0 ẏ 0.007 ss 0.007 ẏ