DISCUSSION PAPER SERIES DP16262 The Power of Open-Mouth Policies Vadym Lepetyuk, Lilia Maliar, Serguei Maliar and John B. Taylor MONETARY ECONOMICS AND FLUCTUATIONS ISSN 0265-8003 The Power of Open-Mouth Policies Vadym Lepetyuk, Lilia Maliar, Serguei Maliar and John B. Taylor Discussion Paper DP16262 Published 15 June 2021 Submitted 15 June 2021 Centre for Economic Policy Research 33 Great Sutton Street, London EC1V 0DX, UK Tel: +44 (0)20 7183 8801 www.cepr.org This Discussion Paper is issued under the auspices of the Centre’s research programmes: Monetary Economics and Fluctuations Any opinions expressed here are those of the author(s) and not those of the Centre for Economic Policy Research. Research disseminated by CEPR may include views on policy, but the Centre itself takes no institutional policy positions. The Centre for Economic Policy Research was established in 1983 as an educational charity, to promote independent analysis and public discussion of open economies and the relations among them. It is pluralist and non-partisan, bringing economic research to bear on the analysis of medium- and long-run policy questions. These Discussion Papers often represent preliminary or incomplete work, circulated to encourage discussion and comment. Citation and use of such a paper should take account of its provisional character. Copyright: Vadym Lepetyuk, Lilia Maliar, Serguei Maliar and John B. Taylor The Power of Open-Mouth Policies Abstract Central bank's announcements about future monetary policy make economic agents to react before the announced policy takes place. We evaluate the anticipation effects of such announcements in the context of a realistic dynamic economic model of central banking. In our experiments, we consider temporary and permanent anticipated changes in policy rules including changes in inflation target, natural rate of interest and Taylor-rule coefficients, as well as anticipated switches from inflation targeting to price-level targeting and average inflation targeting. We show that the studied nonrecurrent news shocks about future policies have sizable anticipation effects on the economy. Our methodological contribution is to develop a novel perturbation-based framework for constructing nonstationary solutions to economic models with nonrecurrent news shocks. JEL Classification: C61, C63, C68, E31, E52 Keywords: news shocks, turnpike theorem, time-dependent models, nonstationary models, Unbalanced growth, time-varying parameters, Regime switches, monetary policies, Price-Level Targeting, Average Inflation Targeting Vadym Lepetyuk - lepetyuk@gmail.com Bank of Canada Lilia Maliar - lmaliar@gc.cuny.edu The Graduate Center, CUNY, CEPR and CEPR Serguei Maliar - smaliar@scu.edu Santa Clara University John B. Taylor - johntayl@stanford.edu Stanford University Acknowledgements Lilia Maliar and Serguei Maliar acknowledge financial support from the NSF grants SES-1949413 and SES-1949430, respectively. Powered by TCPDF (www.tcpdf.org) The Power of Open-Mouth Policies Vadym Lepetyuk, Lilia Maliar, Serguei Maliar and John B. Taylor June 14, 2021 Abstract Central banks’ announcements about future monetary policy make economic agents to react before the announced policy takes place. We evaluate the anticipation e¤ects of such announcements in the context of a realistic dynamic economic model of central banking. In our experiments, we consider temporary and permanent anticipated changes in policy rules including changes in in‡ation target, natural rate of interest and Taylor-rule coe¢cients, as well as anticipated switches from in‡ation targeting to price-level targeting and average in‡ation targeting. We show that the studied nonrecurrent news shocks about future policies have sizable anticipation e¤ects on the economy. Our methodological contribution is to develop a novel perturbation-based framework for constructing nonstationary solutions to economic models with nonrecurrent news shocks. JEL classi…cation: C61, C63, C68, E31, E52 Key Words: news shocks; turnpike theorem; time-dependent models; nonstationary models; unbalanced growth; time-varying parameters; regime switches; monetary policies; price-level targeting; average in‡ation targeting Corresponding author: Serguei Maliar, maliars@stanford.edu. Lilia Maliar and Serguei Maliar acknowledge …nancial support from the NSF grants SES-1949413 and SES-1949430, respectively. The views expressed in this paper are solely those of the authors and may di¤er from o¢cial Bank of Canada views. No responsibility for them should be attributed to the Bank of Canada. 1 1 Introduction Central banks increasingly rely on communication to implement their monetary policy. Through their communication to the public, the monetary authorities indicate their future intentions as well as their views of the future states of the economy. For example, a central bank may promise to …x its interest rate for a certain number of periods before normalizing its policy (forward guidance) or it may announce a future change in the in‡ation target. Understanding the e¤ects of communication is essential from the policy evaluation perspective. In this paper, we assess the e¤ects of central bank communication by analyzing several monetary policies that are conjectured in the literature to be welfare improving. The studied policies include one-time and gradual anticipated changes in economic policies, as well as more complex scenarios in which various monetary policies happen with some probabilities. A distinctive feature of our model with communication is that the agents react to the news of a policy change even when the new policy will not take e¤ect until a later date. We …nd that such anticipation e¤ects can be very large. Our analysis is carried out in a realistically calibrated prototypical central banking model, a smaller replica of the Terms of Trade Model (ToTEM) used by the Bank of Canada for projection and policy analysis. Importantly, our “baby” ToTEM follows the full size ToTEM model as close as possible and generates very similar impulse response functions; see Dorich et al. (2013), and Lepetyuk, Maliar and Maliar (2020, henceforth, LMM) for a description of the ToTEM and “baby” ToTEM models, respectively. The policy experiments we consider include: (1) a gradual decline in the natural rate of interest; (2) a gradual change in the in‡ation target that happens in the future either with certainty or with some probability; (3) normalization of monetary policy regarding future nominal interest rates, when the economy is initially at a zero lower bound (ZLB) on nominal interest rates; (4) a switch to a more aggressive Taylor rule; (5) a switch to price-level targeting instead of in‡ation targeting; (6) a switch to average in‡ation targeting instead of in‡ation targeting. Our …ndings are as follows: (1) Being fully anticipated by agents, a gradual decline in the natural rate of interest of one percent over consecutive …ve years leads to a substantial (more than 1 percent) expansion of output over the whole period. (2) Postponing an increase in the in‡ation target by one year has a large (nearly 50%) increase in output over the transition to a new steady state. Even if the announced policy is implemented with some probability, there are still substantial anticipation e¤ects both before and after uncertainty is resolved. (3) When an economy is at ZLB on nominal interest rates, the central bank uses policy announcements (forward guidance) about its future return to the standard interest rate rule to direct the economy’s transition out of ZLB. The more it postpones such a return to the standard rule, the larger is output expansion over the transition; an initial jump in output is however invariant to the horizon of this forward guidance policy. Therefore, this experiment informs policy makers on optimal horizons of monetary policy normalizations after ZLB periods. (4) Nevertheless, a more aggressive (but realistic) behavior of the central bank toward targeting in‡ation and output is not translated into important anticipatory e¤ects on the side of economic agents. (5) Switching from in‡ation-level targeting to price-level targeting has smaller impacts with larger implementation lags. Price-level targeting was argued in the literature to be welfare improving. Therefore, a central bank that waits to implement the new policy in practice loses time, and the economy does not get earlier bene…ts from higher output. (6) Finally, a switch to average in‡ation targeting also has modest anticipation e¤ects; this is because average in‡ation targeting is in a middle ground between in‡ation targeting and price-level targeting. In sum, our analysis shows that the model’s implications about the importance of anticipation e¤ects depend on a speci…c experiment considered: there are substantial policy anticipation e¤ects present in our experiments (1)–(3), but such e¤ects are relatively small in experiments (4)–(6). On the methodological side, we argue that announcements about future economic conditions and policies can be modeled as non-recurrent news shocks. Such news shocks represent a challenge to economic dynamics because they result in time-dependent optimal decision rules that change from one period to another. Conventional perturbation methods cannot be used for analyzing the anticipatory e¤ects since they 2 are designed for constructing time-invariant (stationary) solutions. We modify the conventional perturbation framework to facilitate the construction of a sequence of time-dependent decision functions following the turnpike analysis; see Maliar, Maliar, Taylor and Tsener (2020, henceforth, MMTT) for a discussion and review of related literature. Our perturbation method is comparable in accuracy to a global projection method developed in MMTT (2020), but it is tractable in problems with much higher dimensionality, such as large-scale central-banking models. Our ubiquitous software is written using the popular Dynare platform combined with user-friendly MATLAB interface and can be easily adapted to other applications the reader might be interested in. There is a literature that studies economic models with recurrent news shocks; in particular, there is a perturbation-based method by Schmitt-Grohé and Uribe (2012) that can analyze the anticipatory e¤ects by augmenting the state space to include future shocks. In their analysis, news shocks follow a stationary Markov process and, hence, the resulting solution is stationary (time-invariant). In contrast, our future shocks are nonrecurrent, which leads to time-dependent decision rules. We view the two approaches as complementary: for example, our method can be used to study a pre-determined one-time accession or exit of a new member to the EU, while Schmitt-Grohé and Uribe’s (2012) method can be used to study countries that access and exit the EU with some stationary probability distribution. Quantitatively, we …nd that the two methods lead to very di¤erent solutions in some of the examples considered. The rest of the paper is organized as follows: Section 2 describes the large-scale central banking model and presents our perturbation methodology for analyzing anticipatory e¤ects. Section 3 analyzes our six policy experiments and presents comparison results with news shocks framework, and, …nally, Section 4 concludes. 2 Methodology In this section, we present a central banking model and outline the methodology of our numerical analysis. 2.1 The model Nowadays, the central banks, leading international organizations and government agencies, use largescale macroeconomic models for projection and policy analysis. A prominent example is the Terms of Trade Economic Model (ToTEM) of the Bank of Canada. That model includes several types of utilitymaximizing consumers, several pro…t-maximizing production sectors, monetary and …scal authorities, as well as a foreign sector. ToTEM is huge: it contains 356 equations and unknowns, including 215 state variables; see Dorich et al. (2013). In the paper, we consider a scaled-down version of ToTEM developed by LMM (2020). Like the fullscale model, the “baby” ToTEM (in short, bToTEM) is a small open-economy model. It features multiple new-Keynesian Phillips curves – one due to sticky prices in domestic production, one due to sticky wages and one due to sticky import prices. We incorporate the rule-of-thumb price settlers in line with Galí and Gertler (1999). We assume quadratic adjustment costs of investment and convex costs of capital utilization to generate more realistic model’s performance, in particular, with respect to monetary-policy transmission. The international trade consists of exporting domestic consumption goods and commodities and importing foreign goods for domestic production. Even though bToTEM is much smaller than ToTEM (it has only 47 equations and unknowns, including 21 state variables) it generates realistic impulse-responses of the Canadian economy to shocks, which are very similar to those produced by the full scale ToTEM model; see LMM (2020) for comparison results. Final-good production. Final consumption goods are produced in two stages. In the …rst stage, intermediate goods are produced competitively using labor, capital, commodities and imports. In the second stage, …nal goods are aggregated from di¤erentiated goods that are each produced by a monopolistically competitive …rm from the intermediate goods and from the …nal goods. The …nal goods can be consumed 3 by households. They can also be transformed using linear technologies into other types of goods, namely, investment goods and noncommodity exports goods. In the …rst production stage, a representative perfectly competitive …rm produces an intermediate good by solving the following pro…t maximization problem: max fZtg ;Ztn ;Lt ;Kt ;It ;COMtd ;Mt ;ut ;dt g s.t. Ztg = 1 l (At Lt ) E0 1 X R0;t Ptz Ztn Pti It Wt Lt Ptcom COMtd Ptm Mt t=0 + k (ut Kt 1 1) log At = 'a log At 1 Kt = (1 1 + com + (1 d t ) Kt 'a ) log A + 1 It It 1 i 2 + m (Mt ) a t; 1 1 ; (1) (2) + It ; (ut 1) dt = d0 + de Ztn = Ztg COMtd (3) ; (4) 2 1 It ; (5) where Ztg and Ztn are gross production and net (of adjustment costs) production of …nal goods; Lt , Kt , It COMtd , Mt , ut and dt are labor, capital, investment, commodity inputs, imports, capital utilization and depreciation rate, respectively; At is the level of labor-augmenting technology; at is a normally distributed variable, and 'a is an autocorrelation coe¢cient. The …rm discounts nominal payo¤s according to household’s stochastic discount factor Rt;t+j = j ( t+j = t ) (Pt =Pt+j ), where t is the household’s marginal utility of consumption, and Pt is the …nal good price. Investment goods and noncommodity exports are assumed to be produced from the …nal goods according to linear technology, Pti = i Pt and Ptnc = x Pt , where Pti and Ptnc are the price of investment goods and noncommodity exports goods, respectively. In the second stage of production, monopolistically competitive …rms produce a continuum of di¤erentiated good. Then, these di¤erentiated goods are aggregated into the …nal good by an aggregating …rm that solves the following cost minimization problem Z 1 min Pit Zit di fZit g 0 s.t. Zt = Z 1 " " 1 " 1 " Zit di 0 ; where Zt and Pit are given; Zit is a di¤erentiated good i, The cost minimization implies the following demand function for the di¤erentiated good i: Zit = Pit Pt " Zt ; with Pt Z 0 1 Pit1 " di 1 1 " : Each di¤erentiated good is produced from the intermediate goods and from the …nal goods using technology featuring perfect complementarity, Zitn Z mi Zit = min ; (6) ; it 1 sm sm where Zitn is an intermediate good and Zitmi is a …nal good input, and sm is a Leontief parameter. There are two types of the monopolistically competitive …rms producing di¤erentiated goods: ruleof-thumb …rms of measure ! and forward-looking …rms of measure 1 !. Both rule-of-thumb …rms and forward-looking …rms index their price to the in‡ation target t with probability as Pit = t Pi;t 1 . The 4 rule-of-thumb …rms partially index their price to lagged in‡ation and target in‡ation with probability 1 Pit = ( t 1) ( t )1 Pi;t 1: (7) Forward-looking …rms choose their prices Pt with probability 1 the price remains e¤ective max Et Pt 1 X j Rt;t+j j=0 j Y t+k Pt Zi;t+j s.t. Zi;t+j = Qj k=1 ! t+k Pt k=1 to maximize pro…ts generated when z sm ) Pt+j Zi;t+j (1 Pt+j , sm Pt+j Zi;t+j ! (8) " Zt+j : The production in the …rst stage Ztn and that in the second stages Zt are related via price dispersion t, Ztn where t R1 0 = Z 0 Pit Pt 1 Zitn di " = (1 sm ) Z 1 Zit di = (1 sm ) 0 Z 1 0 " Pit Pt Zt di = (1 sm ) t Zt ; (9) di. Production of commodities. Commodities are produced by a domestic …rm using …nal goods and land as inputs. They are sold domestically or exported to the rest of the world. The domestic …rm solves max Ztcom ;COMt fPtcom COMt s.t. COMt = (Ztcom )sz (At F )1 sz Pt Ztcom g com 2 Ztcom Ztcom 1 2 1 Ztcom ; (10) where Ztcom is the …nal good input, and F is a …xed production factor, which may be considered as land. Similar to production of …nal goods, the commodity producers incur quadratic adjustment costs when they adjust the level of …nal good input. The commodities are sold domestically or exported to the rest of the world, COMt = COMtd + Xtcom . They are sold at the world price adjusted by the nominal exchange rate, Ptcom = et Ptcomf , where et is the nominal exchange rate (i.e., domestic price of a unit of foreign currency), and Ptcomf is the world commodity price; in real terms, the latter price is given by pcom = st pcomf , t t comf comf f com com where pt Pt =Pt and pt Pt =Pt are domestic and foreign relative prices of commodities, f respectively, Pt is the foreign consumption price level, and st = et Ptf =Pt is the real exchange rate. Production of imports. The representative perfectly competitive …rm produces the …nal imported good Mt from a continuum of intermediate imported goods Mit and solves the following cost-minimization problem, Z 1 Pitm Mit di min fMit g 0 s.t. Mt = Z 0 1 "m 1 "m Mit "m "m 1 di ; where Mit is an intermediate imported good i. The demand for an intermediate imported good i is given by 1 Z 1 1 "m Pitm "m m 1 "m m (Pit ) di : Mt ; with Pt Mit = Ptm 0 5 Prices of the intermediate imported goods are sticky in a similar way as the prices of the di¤erentiated …nal goods. A measure ! m of the importers follows the rule-of-thumb pricing, and the others are forward looking. The optimizing forward-looking importers choose the price Ptm in order to maximize pro…ts generated when the price remains e¤ective ! j 1 X Y mf j m max Et ( m ) Rt;t+j et+j Pt+j Mi;t+j t+k Pt Mi;t+j m Pt j=0 k=1 Qj m k=1 t+k Pt m Pt+j Mi;t+j = ! "m Mt+j ; where Ptmf is the price of imports in the foreign currency. All importers face the same marginal cost given by the foreign price of imports. Households. Households maximize the lifetime utility by choosing holdings of domestic and foreigncurrency denominated bonds, labor and consumption, and they are subject to habits in consumption. Each household supplies a variety of di¤erentiated labor service to the labor market, which is monopolistically competitive. The representative household of type h solves the following utility-maximization problem: Bt et Btf s.t. Pt Ct + + Rt R f 1 + t f t Z +1 (1 ) 1 dh (Lht ) exp Ct 1 (1 + ) 0 Z 1 f Wht Lht dh + t ; = Bt 1 + et Bt 1 + log c t = 'c log max Ct ;Lht ;Bt ;Btf E0 1 X 1 t 1 t=0 Ct c t ; (11) 0 c t 1 c t; + (12) where Ct , Lht , Bt , Btf are consumption of …nal goods, labor service of type h, holdings of domestic and foreign-currency denominated bonds, respectively; Ct is the aggregate consumption, taken by the household as given; 2 (0; 1) is a subjective discount factor; and are the utility-function parameters; ct is a consumption demand shock, ct is a normally distributed variable, and 'c is an autocorrelation coe¢cient; Rt and Rtf are domestic and foreign nominal interest rate, respectively; ft is the risk premium on the foreign interest rate; Wht is the nominal wage of labor of type h; t is pro…ts paid by the …rms. The representative household supplies a variety of di¤erentiated labor service to the labor market, which is monopolistically competitive. Labor packer. A labor packer aggregates di¤erentiated labor services by solving Z 1 Wht Lht dh min fLht g 0 s.t. Lt = Z 0 1 "w 1 "w Lht "w "w 1 ; dh where Lt is aggregated labor demanded by …rms in the …rst stage of production. Cost minimization of the labor packer implies the following demand for individual labor: Lht = Wht Wt "w Lt ; with Wt Z 0 6 1 1 "w dh Wht 1 1 "w : (13) Labor unions. Labor unions set wages. There are two types of labor unions: rule-of-thumb unions of measure ! w and forward-looking unions of measure 1 ! w . Within each type, with probability w the labor unions index their wage to the in‡ation target t as follows Wit = Wi;t 1 . The rule-of-thumb unions that do not index their wage in the current period follow the rule w t 1 Wit = w ( t )1 w Wi;t 1: (14) A forward-looking unions that do not index its wage solves maxEt Wt 1 X ( 1 j w) Ct+j 1 j=0 Ct+j 1 Qj k=1 s.t. Lh;t+j = Pt+j Ct+j = exp j Y t+k Wt Wt+j t+k Wt (1 ) (1 + ) ! "w Lh;t+j dh + Z 1 +1 (Lht+j ) dh 0 c t+j (15) Lt+j ; (16) t+j ; (17) k=1 where t+j includes terms in budget constraints (11) other than Ct+j and Lh;t+j . Monetary authority. Rt = The central bank uses a Taylor rule to set the short-term nominal interest rate, r Rt 1 + (1 r) R+ ( t) t + Y log Yt log Yt + r t; (18) where r measures the degree of smoothing of the interest rate; R is the nominal neutral interest rate; measures a response to the in‡ation gap; t is the in‡ation target; Y measures a response to the output gap; Yt is potential output; rt is an interest rate shock following a process r t = 'r r t 1 + r t; where rt is a normally distributed variable, and 'r is an autocorrelation coe¢cient. Potential output changes with productivity according to log Yt = 'z log Yt 1 + (1 'z ) log At Y A : Foreign demand for noncommodity exports. By analogy with the demand for imports, the foreign demand function for noncommodity exports is assumed to be Xtnc = f Ptnc et Ptf ! where Ptnc is a domestic price of noncommodity exports; terms, we have st Xtnc = f pnc t Balance of payments. Ztf ; f (19) is the demand-function parameter. In real Ztf : (20) The balance of payments in nominal terms is given by et Btf Rtf 1+ f t et Btf 1 = Ptnc Xtnc + Ptcom Xtcom 7 Ptm Mt ; (21) where Btf is domestic holdings of foreign-currency denominated bonds, and Rtf is the nominal interest rate et Btf f t+1 Pt Y on the bonds. By normalizing the bonds holdings as bft becomes bft rtf 1+ f t bft 1 , the balance of payments in real terms st 1 Xtcom = (pnc X nc + pcom t st 1 Y t t pm t Mt ) ; (22) where rtf is the real interest rate on the foreign-currency denominated bonds. Rest-of-the-world economy. The rest of the world is speci…ed by three exogenous processes that, respectively, describe the evolution of foreign output Ztf , the foreign real interest rate rtf , and the foreign commodity price pcomf , t log Ztf = 'zf log Ztf log rtf = 'rf log rtf 'rf log r + = 1 log pcomf t 1 + 1 1+ 1 'comf log pcomf t 1 + 'zf log Z f + zf t ; rf t ; comf 'comf log p (23) (24) + comf ; t (25) comf rf are normally distributed random variables, and 'Zf , 'rf and 'comf are autocorwhere zf t , t and t relation coe¢cients. We impose an augmented uncovered interest rate parity condition 2 0 11 { 3 f f R 1 + t t 6 7 A et = Et 4(et 1 ){ @et+1 (26) 5: Rt Uncovered interest rate parity. Stationarity condition for the open-economy model. The risk premium of foreign assets f f bft ; t =& b f t is a decreasing function (27) where bf is the steady state level of the normalized bond holdings. This assumption ensures a decreasing rate of return to foreign assets. Summary of the model’s variables. For each period t, there are the following four types of variables in this model: 47 endogenous (or non-predetermined) variables, 8 9 w ; F w ; F m; F m; q ; ; s ; F1t ; F2t ; F1t < = 2t 1t 2t t t t m z Lt ; Kt ; It ; COMtd ; Mt ; ut ; dt ; Ztg ; Ztn ; Zt ; Ct ; Yt ; t ; rmct ; t ; m yt ; t ; t ; p t ; R t ; p t ; wt ; : w ; Y ; pcom ; pnc ; pmf ; py ; ; M P Kt ; Rtk ; pit ; ft ; bft ; Xtnc ; Xtcom ; COMt ; Ztcom ; w ; w ; t t t t t t t t w ; F w g, fF m ; F m g are supplementary variables in Phillips curves for prices, wages and where fF1t ; F2t g, fF1t 2t 1t 2t imports, respectively; qt is Tobin’s q; rmct and M P Kt are real marginal cost and marginal productivity of y capital, respectively; pnc t and pt are prices of noncommodity goods and output, respectively; 15 endogenous state variables n o w m m com f yt 1 Ct 1 ; Rt 1 ; st 1 ; t 1 ; t 1 ; wt 1 ; w t 1 ; t 1 ; pt 1 ; t 1 ; It 1 ; Zt 1 ; bt 1 ; Yt 1; Kt 1 ; 8 w , pm , m are wage in‡ation, wage dispersion, and price and in‡ation of imports; 19 where w t 1, t 1 t 1 t 1 endogenous forward variables + yt+1 w ;Fw ;Fm ;Fm ; F1t+1 ; F2t+1 ; F1t+1 2t+1 1t+1 2t+1 t+1 ; qt+1 ; ut+1 ; It+1 ; com w ; pcom ; pz ; M P K Zt+1 ; t+1 ; t+1 ; m ; t+1 ; st+1 t+1 t+1 t+1 t+1 and 6 exogenous state variables zt n At ; R c comf f ; rt ; Ztf t ; t ; pt o ; : In Appendix A, we describe our calibration procedure, which closely follows the calibration of the full scale ToTEM model. 2.2 Methodology of our numerical analysis LMM (2020) analyze conventional time-invariant (stationary) solutions to our model.1 Here, we construct novel time-dependent (nonstationary) solutions to the model. We consider announcements about economic policies that will be implemented at some future dates (non-recurrent news shock) and analyze a reaction of economic agents to such announcements. Assuming that we are at t = 0 and that a given policy will be implemented at T > 0, we obtain a sequence of optimal decision functions for periods t = 0; 1; :::; T that characterize anticipatory e¤ects (obviously, the optimal decision functions will depend on how far the economy is from the moment the policy is introduced). Below, we outline our perturbation-based framework for analyzing economies with non-recurrent news shocks. Let us consider an in…nite-horizon nonstationary equilibrium problem in which a solution is characterized by a set of equilibrium conditions for t = 0; 1; : : : ; Et G t y t + 1 ; yt ; yt+1 ; zt ; zt+1 zt+1 = Zt (zt ; = 0; t+1 ) ; (28) (29) where z0 ; y 1 is given; Et denotes the expectations operator conditional on information available at t; zt 2 Rdz is a vector of exogenous state variables at t; Zt is a time-dependent law of motion for zt ; yt 2 Rdy is a vector of endogenous variables; yt 1 2 Rdym is a vector of endogenous (random) state + variables at t; yt+1 2 Rdyp is a vector of endogenous forward variables at t; t+1 2 Rd is a vector of shocks; Gt is a continuously di¤erentiable vector function. Note that the latter function is time-dependent because the model is nonstationary (due to, for example, time-dependent parameters in policy rules, production function, utility function). A solution is given by a set of time-dependent equilibrium functions yt = Yt zt ; yt 1 that satisfy (28), (29) in the relevant area of the state space. Our perturbation analysis proceeds in the following two steps: Step I: solving a T -period stationary economy. Assume that in a very remote period T , the economy becomes stationary, i.e., Gt ( ) = G ( ) and Zt ( ) = Z ( ) for all t T . Therefore, the system (28), (29) becomes Et G yt + 1 ; yt ; yt+1 ; zt ; zt+1 zt+1 = Z (zt ; t+1 ) : Solving (30), (31) allows us to …nd the solution yT = YbT zT ; yT 1 = 0; (30) (31) 1 . LLM (2020) compare perturbation solutions with more accurate global projection solutions constructed using deep learning analysis. That paper …nds that high order perturbation solutions are su¢ciently accurate in the bToTEM model. Since our nonstationary analysis is more costly, we limit attention to perturbation solutions only. 9 Step II: constructing a function path. Using a T -period solution yT = YbT zT ; yT 1 as a terminal condition, iterate backward for T 1; :::; 1 on the corresponding equilibrium conditions to construct a sequence (path) of time-dependent value and decision functions fYT 1 ( ) ; :::; Y1 ( )g. For example, for period t, the system on which we iterate backward is Et G t yt + 1 ; yt ; Yt+1 zt+1 ; yt zt+1 = Zt (zt ; ; zt ; zt+1 = 0; t+1 ) ; Here, we solve for today’s endogenous variables yt , given tomorrow’s functions zt+1 = Zt (zt ; t+1 ) and + + + zt+1 ; yt is a subset of functions Yt+1 zt+1 ; yt . zt+1 ; yt , where Yt+1 = Yt+1 yt+1 In both steps, we use perturbation to …nd numerical approximations of the decision functions. For our central banking model, the variables zt , yt , yt , yt+ are de…ned in Section 2.1. In general, nonstationary models like ours have no deterministic steady state, so it is unclear around what point(s) decision rules must be approximated. To deal with this issue, we can augment the model’s equations with a time-varying growth rates xt that capture how much endogenous state variables grow from period t to t + 1 due to the time trend or the parameter change. We can …rst assume that growth rates f xt gTt=1 are given and then …nd those growth rates iteratively. (Note that growth rates should not be the same for all variables). One objective of this paper is to make the proposed perturbation framework ubiquitous and portable to other applications. To this purpose, we show how to construct time-dependent decision functions using Dynare. To explain how to solve a nonstationary model in Dynare, let’s …rst consider a standard secondorder perturbation solution to a stationary model around a deterministic steady state (v; 0), g (v; ) g (v; 0) + gx (v; 0) (v | {z 1 v) + gxx (v; 0) (v } |2 1st-order perturbation solution 1 v)2 + g 2 {z 2nd-order terms (v; 0) 2 ; } (32) where g (v; ) is a decision function to be approximated; v = (x; z) is a vector of endogenous and exogenous state variables; is a perturbation parameter that scales volatility of shocks; (v; 0) is a deterministic steady state; g (v; 0), gx (v; 0) and gxx (v; 0) are, respectively, steady state values, Jacobian and Hessian matrices of g; (v v) is a deviation from a steady state; (v v)2 (v v) (v v) is a tensor product of the deviations. Three observations are in order: First, a constant term of the policy function is given by g (v; 0) + 21 g (v; 0) 2 and hence, is a¤ected by variances of shocks. Second, the …rst-order perturbation solution does not depend on the degree of volatility , i.e., g (v; 0) = 0. Finally, the term g x (v; 0) is omitted as well because it is equal to zero; see Schmitt-Grohé and Uribe (2004). In Step I, a Taylor expansion of the policy functions in a stationary model is found around the deterministic steady state v of the model. In Step II, we consider two alternative options. The …rst option is to …nd solutions for vt+1 and vt around vt and vt 1 , respectively, such that vt = vt 1 vt ; in Dynare, it can be implemented by coding vt and vt+1 using the same variable names. The other option is to consider vt+1 and vt perturbed around vt and vt 1 , respectively, such that vt = vt 1 v;t 1 , where v;t 1 is a time-dependent growth rate; in Dynare, it can be implemented by coding vt and vt+1 with di¤erent variable names. In Appendix A, we illustrate our methodology of constructing time-dependent perturbation solutions by using a toy example of a neoclassical stochastic growth model with labor augmenting technological progress. A useful property of that model is balanced growth which allows us to construct an accurate reference solution. We use such a solution for accessing accuracy of our perturbation solutions which we obtain without relying on the property of balanced growth. In particular, we show how to iteratively construct time-dependent growth rates. In the main text, we apply the proposed methodology for the analysis of our large-scale open-economy model. 10 3 Analyzing nonstationary news shocks In this section, we show a series of policy experiments in which we consider anticipated changes in one or several model’s parameters. In all the …gures, the variables are shown in percentage deviations from the initial risky steady state, except for the interest rate and the in‡ation rate, which are both shown in percentage point deviations from the risky steady state and expressed in annualized terms.2 3.1 A decline in the real neutral interest rate Some future changes in economic environment could be envisioned by the public. A central bank’s announcements could play a signi…cant role by revealing information not just about its policy, but also about a central bank’s assessment of the economic outlook. In particular, Nakamura and Steinsson (2018) argue that Fed announcements contain information about the path of the natural rate. In this experiment, we model an anticipated gradual decline in the natural rate of interest. Namely, we assume that initial value of the real neutral interest rate is 3 percent and that it starts to go down to 2 percent gradually over 20 quarters. To model a decrease in the neutral rate, we exploit the fact that in steady state, this rate is equal to the inverse of the discount factor, and we translate the assumed decrease in the neutral interest rate into an increase in the discount factor. Empirical evidence indicates that long-term rates declined from the early 1960s through the mid-1970s, increased until the late 1980s, and declined again from that point on; see, e.g., Yi and Zhang (2019). Moreover, such a decline is not related to the Great Recession. The factors that are responsible for declining long-run rates include lower TFP growth, lower working-age population growth, long-run trends in marginal productivity of capital and risk premium. With this experiment, we investigate how a gradual reduction in the real interest rate captured by an increase in the discount factor a¤ects the economy. 0 Nominal Interest Rate Inflation 0 Output -0.5 -0.02 0.5 -0.04 0 -0.5 -1 -1 10 2 Exchange Rate 0 1 20 30 40 -0.06 10 Consumption 20 30 -0.5 40 Investment 0 4 1.5 -1.5 10 20 30 40 Commodity Export -1 -0.2 -2 -0.4 0 -3 1 10 20 30 40 Imports 1.5 0 2 0.5 1 10 20 30 40 0 40 -0.6 10 20 30 40 10 Capital 20 30 40 10 Labour 0.6 0 20 30 40 Real Interest Rate 0.4 -0.5 0.2 0 30 -4 3 1 20 Noncommodity Export 2 0.5 10 0 3 1 -2 0 10 20 30 -0.2 40 -1 10 20 30 40 10 20 30 40 Figure 1: A gradual decline in the real neutral interest rate As it is seen from Figure 1, a long-run gradual decrease in the real interest rate results in the corresponding gradual increase in consumption, investment, labor, capital, and imports. For example, investment 2 By risky steady state, we mean a state to which a stochastic economy converges in the absence of exogenous shocks. 11 increases by 5 percent at the peak. Because of signi…cantly higher investment and labor, output grows by more than 1 percent. The changes in in‡ation are so small that the nominal and real interest rates behave almost identically. The anticipation e¤ects are the largest in the commodity export and exchange rate which fall by 2 percent and 1.8 percent, respectively, when it became known that the real neutral interest rate will gradually decrease. The results of this experiment indicate that recent trends in the real rate of interest are bene…cial for growth. Note that we arrive at this result by assuming that the decline in the natural rate is caused by people becoming more impatient. However, if the decline is caused by other factors, our conclusion might not go through. 3.2 A change in the in‡ation target In this experiment, we consider a change in the in‡ation target that appears in the Taylor rule (18). In particular, we assume that the central bank announces in advance that it will increase the in‡ation target t and that everyone considers the announcement to be fully credible. Why is it a relevant policy experiment? During the Great Recession of 2007–2009, central bank’s nominal policy rates across a number of countries fell to a ZLB on nominal interest rates. There is ample literature arguing that the in‡ation target is a good policy instrument for dealing with ZLB episodes. For example, Summers (1991) and Fischer (1996) suggest to keep an in‡ation target as high as 2 or 3 percent if the economy hits ZLB. Krugman (1998) proposes to use a 4 percent in‡ation target in the Japanese economy to deal with persisting de‡ation. Furthermore, Blanchard, Dell’Arriccia and Mauro (2010), Williams (2009) and Ball (2013) argue that a higher in‡ation target would have prevented the interest rate from falling to the ZLB. In Canada, in‡ation-targeting framework was adopted in 1991, and since 1995, the in‡ation target was maintained at the level of 2 percent. The in‡ation target is reviewed and renewed every …ve years. In particular, the last review was in October of 2016, when the Bank of Canada decided to keep the target at the same level; this renewal covers the period from January 1st, 2017 to December 31st, 2021. There are two types of possible anticipation e¤ects here. First, we would have had a policy implementation lag leading to anticipation e¤ects if the Bank of Canada decided to change the target in October 2016. Second, in spite of the fact that the in‡ation target was not changed in 2016, anticipation e¤ects were still present as there were some chances that it would be changed given that Canada was close to the ZLB at that time and policymakers were seriously discussing this possibility. Figure 2 displays dynamics of the main model’s variables. We present the results for the method that …nds a perturbation solution obtained around a deterministic steady state (labeled as Method 2 in Appendix B; our sensitivity results for other methods predict similar patterns of behavior). We consider two cases: …rst, at t = 1, the central bank makes an announcement that starting from t = 1, it will gradually increase the in‡ation target t from 2 percent to 3 percent during a period of 8 quarters, and second, the same change takes place but starting from t = 5 (i.e., in one year); the in‡ation target remains at the new (higher) level after it is reached. When the in‡ation-target change begins at t = 1, in‡ation follows the same pattern as the target. What is the reason for such behavior of in‡ation? In our experiment, we assume full credibility of the in‡ationtarget policy. In‡ation repeats the pattern of the in‡ation target because agents determining the behavior of in‡ation are mainly non-optimizers who index their price by in‡ation target. As a result, the nominal interest rate gradually increases over the …rst …fteen periods by 1 percent, and it stays at the new level forever (see Figure 2; note that the real neutral rate is the same as before). Following the announcement, output, investment and commodity exports jump up, and over the transition, the economy experiences an investment- and export-driven growth with the peak increase of output of 0:2 percent. Output begins to descend toward its original level after one year. Consequently, there is only a temporary expansionary e¤ect on the economy due to a higher in‡ation target. When the in‡ation-target change is delayed for one year, the variables behave qualitatively similar. One visible di¤erence from the previous case is that most variables in the …gure experience larger increases at the peak (the exchange rate and noncommodity export are exceptions). Therefore, it pays for the central bank to announce this type of policy in advance as output increases more during the transition. The 12 Nominal Interest Rate Inflation 1 1 Output Exchange Rate 0.6 0.2 0.4 0.15 0.5 0.5 0.1 0.2 0.05 0 5 10 15 0 20 Consumption 0.15 10 15 20 5 Investment 1 0.2 10 15 20 Commodity Export 0.4 0.5 0.1 5 10 15 -0.5 20 Imports 0.4 0.2 0.3 0.15 5 10 15 Capital 0.2 0.1 0.1 0 5 10 15 20 Labour 0.4 10 15 20 Noncommodity Export -0.1 0.1 0.3 5 10 15 20 Real Interest Rate 0 0.2 0.1 0.2 0.3 0.2 0 20 5 0.3 0 0.05 0 0 0 5 -0.1 0.1 0.05 0.1 -0.2 0 0 5 10 15 0 20 5 10 15 20 5 effective immediately 10 15 20 -0.3 5 10 15 20 announced one year in advance Figure 2: A gradual increase in the in‡ation target Average Maximum Rt 4:66 4:30 t 5:25 5:17 Yt 4:09 4:03 Ct 4:47 4:38 It 5:11 5:03 Xtnc 4:10 4:08 Xtcom 3:17 3:13 Mt 5:10 4:98 Lt 4:42 4:37 Kt 5:91 5:84 Table 1: Residuals in the model’s equations on the simulated path, log10 units. Rt , t , Yt , Ct , It , Xtnc , Xtcom , Mt , Lt , Kt are the nominal interest rate, in‡ation, output, consumption, investment, noncommodity export, commodity export, imports, labor and capital, respectively. larger jumps in such variables as output, consumption, investment, capital are entirely due to anticipatory e¤ects. That is, agents expect the real interest rate to be lower in the near future, and they accumulate more capital in advance of the more favorable environment which has positive e¤ects on the economy’s output today. Table 1 contains the mean and maximum residuals in the model’s equations used for computing the corresponding variables in the table. As we can see, the maximum residuals range between 10 3:13 and 10 5:84 , i.e., between :07 percent and :0001 percent, which are very low.3 For the remaining experiments, the residuals in equations are of similar size so our solutions are very accurate (to save on space, these residuals are not reported). In the second experiment, we model a probabilistic setting in which agents rationally expect that the in‡ation target might change to two possible levels with some probabilities. Speci…cally, we assume that there is a 50-percent chance that starting from t = 5 the in‡ation target t gradually increases from 2 to 3 percent during 8 quarters; otherwise, the in‡ation target remains the same. Our computational method is easy to adapt to modeling more sophisticated anticipation scenarios like one considered in that experiment. When computing policies in period t = 4, we explicitly use the Dynare macro language to set the period 4 3 In LMM (2020), we reported larger residuals when the economy was hit by a large negative demand shock and the ELB was reached. 13 expectation functions to be equal to the weighted sums of expectations over the two possible realizations in period t = 5. Nominal Interest Rate 1 1 0.5 0.5 Inflation Output 5 10 15 20 5 Consumption 0.2 0.6 0.15 0.4 0.1 0.2 0.05 0 0 0 0 0.2 10 15 20 -0.2 5 Investment 0.4 10 15 20 0.2 0.2 0.05 0 0.1 -0.2 10 15 20 Imports 0.4 Noncommodity Export 0 10 15 20 -0.1 5 Capital 10 15 20 Labour 0.4 0.2 15 20 -0.1 0.1 0.05 10 Real Interest Rate 0 0.1 0.2 5 0.1 0.3 0.1 20 0.2 0 5 0.15 0.3 15 0.1 0.1 5 10 0.3 0.4 0 5 Commodity Export 0.6 0.15 Exchange Rate -0.2 0 0 0 5 10 15 20 -0.3 5 10 15 20 5 no change in the inflation target 10 15 20 5 10 15 20 a higher inflation target Figure 3: A gradual increase in the in‡ation target (50% probability) This experiment is plotted in Figure 3. There are two alternative transition paths di¤ering from period 5 onwards, one per each scenario, i.e., with and without an increase in t . Similar to the previous experiment, in‡ation mimics the behavior of the in‡ation target: it gradually rises to the new steady state level. Starting from the risky steady state at t = 1, all the variables experience mild increases, which are due to anticipatory e¤ects on the side of economic agents. Once it becomes known whether the target will go up or not, all the variables quickly return to the original steady state if the target does not increase, and they experience a more pronounced hump-shape behavior and return to a new steady state if the target increases. In Appendix B, we extend the latter experiment to vary the probability of switching to a higher in‡ation target at t = 5, namely, it is either 25 percent or 75 percent (instead of 50 percent). As those …gures show, in case of no in‡ation-target change, the transition back to the old steady state is signi…cantly faster for the 25-percent case than for the 75-percent case. 3.3 Monetary policy normalization During the Great Recession of 2007–2009, the nominal interest rate hit the ZLB. As a result, central banks could not rely on Taylor rules to conduct their monetary policy and resorted to forward guidance – an unconventional monetary policy consisting in announcing future interest-rate changes. As emphasized by the literature, central bank’s communication of the policy-rate’s future path is the main channel through which forward guidance policy a¤ects the economy. Eggertsson and Woodford (2003) demonstrate that a central bank’s promises to keep low interest rates for longer periods helps alleviate negative consequences of binding ZLB. As agents expect future interest rates to be lower than in the absence of forward guidance, they increase today’s investment and consumption, which stimulates today’s economy. Campbell et al. (2012) name this form of forward guidance Odyssean. Another form is Delphic forward guidance: a central 14 bank may have better information about the state of the shocks that hit the economy, and it communicates a forecasted path of policy rates.4 In the Odyssean case, future intentions are known, while in the Delphic case, forward guidance is implied – agents do not know its exact duration. In this paper, we assume that the central bank uses forward guidance to convey a policy change when lifting o¤ from an e¤ective lower bound (ELB) on the bank’s policy rates. In particular, we assume that initially the economy is at ELB and at t = 1 the central bank announces that it will keep the interest rate at that level for T periods and afterwards it will return to the standard Taylor rule (18). To model the central bank’s policy at the ELB periods, we assume that the nominal interest rate is given by Rt = Relb ; where Relb is the ELB. When the interest-rate policy is normalized after T periods, the Taylor rule’s coe¢cients return back to normal values, and the policy is described by the rule (18). Figure 4 presents the results for this experiment when the solutions are approximated around a deterministic steady state. The change in the interest rate rule announced at t = 1 is anticipated by agents. We compare three cases, depending on whether the interest-rate policy returns to normal (i) in one quarter (T = 1), (ii) in one year (T = 4), (iii) in two years (T = 8). In all the cases, the initial interest rate is below its risky steady state, however, it eventually returns to the steady state. Nominal Interest Rate 0 Inflation 0.1 Output 0.4 0.3 0.05 -0.5 Exchange Rate 1.5 1 0.2 0.5 0.1 0 -1 5 10 15 20 Consumption 0.3 0 0 5 10 15 20 5 Investment 0.6 10 15 20 Commodity Export 5 0.6 10 15 20 Noncommodity Export 1 0.2 0.5 0 0.1 0.4 0.4 0.2 0.2 -0.5 0 10 15 20 Imports 5 0.4 10 15 20 5 Capital 0.4 0.6 0 0 -1 5 0.4 0.2 0.2 0.1 0 15 20 5 Labour 0.6 0.3 10 10 15 20 Real Interest Rate 0 -0.5 0.2 0 0 5 10 15 20 -0.2 5 10 15 effective one quarter 20 -1 5 effective one year 10 15 20 5 10 15 20 effective two years Figure 4: Forward guidance When the policy is announced, the exchange rate, in‡ation, and the real variables jump up above the steady state. Local currency depreciation makes domestic exports more competitive, which leads to an increase in exports of both commodities and noncommodity goods. Domestic …rms bene…t from increased sales, which leads to immediate increases in output, labor, investment and capital. On the other hand, as households work more, they demand more of imported goods, so that imports go up as well. Evidently, an 4 Marinkov (2020) argues that during the ZLB period, agents may misjudge a central bank’s reaction function and bias their expectations. In this case, the central bank may want to use forward guidance as a guiding tool to correct agents’ beliefs. 15 output increase is the largest when the announced policy is kept for the longest horizon of eight periods. The peak increase in output is 70 percent higher than the one for the forward guidance horizon of four quarters. The di¤erences in output dynamics across the considered cases are only present over the transition but not in the initial period – in all three cases, an initial output jump is of equal size. The dependence of the initial reaction in output on the horizon of the forward guidance policy is known in the literature as a forward guidance puzzle. Even though there is no such dependence in the …gure, we do still see that the policy horizon matters for the total e¤ect on output: it reacts more if the policy change is postponed further away in the future. Our above experiment adds to the discussion on central bank’s communication strategy. After the Great Recession of 2007–2009, when the economic conditions improved, an important policy question was how and when to normalize the monetary policy, where normalizing means switching back to some Taylor rule; see Yellen (2015). In particular, the following questions arose after the end of the crisis: (1) Should the central bank normalize policy now or later? (2) Should the central bank do it gradually or all at once? (3) Should the regime shift be announced in advance? (4) Should the policy normalization be time or state dependent? All these questions are hard to address in the context of conventional stationary new Keynesian framework because by de…nition a monetary policy normalization is a nonstationary change. Nevertheless, the technique developed in this paper enables us to study these questions easily. In our above experiment, we compare the economy’s behavior under policies that di¤er in horizon of return to normal values, which corresponds to questions (1) and (3). We conclude that there are gains from announcing a future lift-o¤ in advance and we quantify these gains for di¤erent durations of forward guidance. Similarly, questions (2) and (4) can be answered using our techniques; we leave them for future research. An e¤ective commitment to keep the interest rate at the ELB implies that the rate should be kept at this low level longer than a Taylor rule would imply. In particular, during the COVID-19 pandemic, the interest rates reached ZLB across a number of developed economics. In the U.S., the Fed has already announced that it expects to keep its benchmark interest rate pinned near zero through 2023.5 Taylor (2021) argues that the Taylor rules considered by the Fed in the February 2021 Monetary Policy Report imply that the federal fund rate should be higher than the actual zero level and that the Fed “should now engage in a strategy or rule in which people and markets understand that it would raise the policy interest rate if economic growth increases and in‡ation rises as they are now forecast to do.” Our above analysis, however, plays up the importance of commitment to the announced policy on which hinges the desired monetary expansion. 3.4 Switching to a more aggressive Taylor rule In this experiment, we consider a one-time change in the sensitivity of the policy rate to in‡ation and the output gap in the Taylor rule (18), as measured by and Y , respectively.6 It di¤ers from previous experiments, in which the anticipated changes in the model’s parameters are gradual. Figure 5 plots the economy’s responses to two-time increases in either or Y or both, relative to benchmark parameterization. Note that this change in the coe¢cient values is quite large relative to what a central bank would typically consider. Switching to more aggressive Taylor rules is anticipated at t = 1 but occurs at t = 2, so that there are immediate anticipatory e¤ects in all the model’s variables. As we can see, both policies – a higher and a higher Y – are in‡ationary. However, a double increase in the sensitivity to in‡ation is more e¤ective in expanding the economy: output, consumption, investment, capital, labor are visibly higher both at peak and in the new steady state than in the old steady state; commodity production slightly drops, which is related a lower commodity exports. A double increase in the sensitivity to the output gap has more modest e¤ects however. When there is a stronger response to both in‡ation and the output gap, the quantitative expressions of the e¤ects are roughly in between the other two cases. That is, given that there is a trade o¤ between in‡ation and the output gap in 5 See https://apnews.com/9b9a335a1ce05d69fc97a1d6197371ab For example, Taylor (1999) argues that the Taylor rule with advocated in Taylor (1993) when = 0:5 and Y = 0:5. 6 16 = 0:5 and Y = 1 is more reasonable than the one 0.02 Nominal Interest Rate 10 0.01 10-3 Inflation Exchange Rate 0.06 8 0 Output 0.06 0.04 0.04 6 -0.01 0.02 4 0.02 -0.02 2 10 0.15 20 30 40 Consumption 0.15 0.05 0.1 0 0.05 -0.05 20 30 40 10 Investment 0.2 0.1 0.1 0 10 20 30 40 Imports 0.15 0 0 10 20 30 40 Commodity Export 20 30 0.02 -0.1 0.01 20 30 40 Labour 10 0.06 0.02 0.04 0 0.02 -0.02 0.06 0.1 40 0 10 Capital 0.08 30 Noncommodity Export 0 40 20 0.03 -0.2 10 10 20 30 40 Real Interest Rate 0.04 0.05 0.02 0 0 10 20 30 40 0 10 20 stronger response to inflation gap 30 40 -0.04 10 20 stronger response to output gap 30 40 10 20 30 40 stronger response to both Figure 5: A switch to more aggressive Taylor rules the policy rule, responding stronger to the output gap undoes the e¤ects of stronger responses to in‡ation. Overall, total e¤ects are not quantitatively important in our experiment: switching to a signi…cantly more aggressive Taylor rule has only minor e¤ects on the economy’s behavior when the economy is not hit by any shocks. 3.5 Switching from in‡ation targeting to price-level targeting Since the seminal paper of Svensson (1999), the literature argues that price-level targeting is a “free lunch” in a sense that it positively a¤ects a short-run trade o¤ between in‡ation and output variability (namely, it reduces in‡ation variability without an increase in output variability); see Hatcher and Minford (2016) and Ambler (2009) for surveys. Bernanke (2017) proposes to use a temporary price-level target when short-term interest rates are at (or near) ZLB. When ZLB prevents policymakers from providing adequate stimulus, in‡ation is below target. Price-level-targeting policymakers compensate for periods of low in‡ation below target by following a temporary surge in in‡ation The Bank of Canada has seriously considered the use of price-level targeting; see Kahn (2009) and Bank of Canada (2011). We …rst consider a scenario, in which central bank switches from the standard Taylor rule (18) targeting in‡ation to the one targeting a price-level gap, Rt = r Rt 1 + (1 r) R+ log Pt log Pt + Y log Yt log Yt + r t; (33) where Pt is the actual price level, and Pt is the target price level that grows at the rate of in‡ation target Pt = Pt 1 t . Therefore, price-level targeting does not suggest that policymakers pursue a constant price level but set a target for the price level that rises over time. An in‡ation-targeting central bank does not pay attention to temporary changes in in‡ation as long as in‡ation comes back to target after some time (“lets bygones to be bygones”). In contrast, price-leveltargeting central bank aims at reversing temporary deviations of in‡ation from target each time it misses 17 it (e.g., a central bank increases in‡ation when in‡ation falls below target). As a result, under in‡ation targeting, an in‡ation shock permanently shifts price path to a di¤erent level, while under price-leveltargeting, any movement in in‡ation above target is matched with an equal and opposite movement in in‡ation below target, so that the economy goes along a predetermined price path. Consequentially, with in‡ation targeting, agents will face a considerable amount of uncertainty about the future price level (the central bank treats past target misses as bygones and returns in‡ation to the target level gradually, without taking into account any impact on the price level), while with price-level targeting, agents will be much more con…dent on where the prices will be in the future, even with a positive average in‡ation. 0.08 Nominal Interest Rate 0.06 Inflation 0.04 Output Exchange Rate 0.15 0.15 0.03 0.1 0.1 0.02 0.05 0.05 0.04 0.02 0.01 0 5 10 15 20 Consumption 0.2 0 5 15 20 0 5 Investment 0.6 0.15 10 0.2 10 15 20 Commodity Export 0.15 0.4 0.1 5 0.06 10 15 20 Noncommodity Export 0.04 0.1 0.2 0.05 0 0 5 10 15 20 Imports 0.3 0 5 0.1 0 10 15 20 10 15 20 10 15 20 Labour 0.2 5 0.06 0.15 0.15 0.04 0.1 0.1 0.02 0.05 0.05 0 0 5 0 5 Capital 0.2 0.2 0.02 0.05 0 5 10 15 20 15 20 Real Interest Rate -0.02 5 effective immediately 10 10 15 20 5 10 15 20 announced one year in advance Figure 6: A switch to price-level targeting In Figure 6, we present the results for two policy experiments in which the policy change becomes e¤ective either immediately (at t = 1) or in one year after being announced (at t = 5). The new interestrate rule is associated with higher steady state levels for all the model’s variables in the …gure. Therefore, switching to price-level targeting has expansionary e¤ects on the economy. Moreover, for all of the variables (except of the nominal interest rate), the immediately implemented policy gives larger bene…ts than the policy announced one year in advance. That is, if the central bank postpones to implement the switch, the economy reaches the new steady state almost at the same time as the immediate policy, but over the transition the e¤ects are smaller. In the next experiment, we shock the economy, so that there appears a large output gap. In particular, we consider a permanent negative demand shock – a decrease in foreign demand, modeled as a negative innovation in the random-walk process for this shock. (A version of this experiment with a permanent decrease in productivity is presented in Appendix B.) In response, the central bank can either continue using a policy rule with in‡ation targeting or can switch to price-level targeting, which, as we saw, leads to higher steady state output. As a result, switching to price-level targeting can be viewed as an attempt to revive the economy. Figure 7 presents the results of this experiment when the switch is either implemented immediately or is delayed for one year (but it is still announced today). As we see, there is barely any di¤erence between 18 -0.2 Nominal Interest Rate Inflation 0.05 Output Exchange Rate 3.5 0 -0.4 0 -0.6 -0.8 -1 20 30 10 Consumption 20 30 10 Investment 10 1 0.4 0.5 8 0.2 0 6 0 -0.5 4 10 20 30 Imports 20 30 Commodity Export 10 -9.2 20 30 Noncommodity Export -9.3 -9.4 -9.5 -9.6 -1 -0.2 2.5 -1 -0.1 10 0.6 3 -0.5 -0.05 2 10 20 30 10 Capital 1 20 30 Labour 0 10 -0.2 20 30 Real Interest Rate -1 -0.5 -0.4 -1 -0.6 -1.5 -0.8 0.5 -1.5 -2 0 10 20 30 inflation targeting -2 10 20 30 -1 10 change to price level targeting 20 30 10 20 30 change to price level targeting announced one year in advance Figure 7: A negative foreign demand shock and a switch to price-level targeting the immediate and delayed changes in monetary policy for such variables as commodities and commodity export. For all other variables, the immediate policy change has larger impacts than the delayed policy change, which is in line with the previous experiment in Figure 6. Therefore, the anticipation e¤ects work in the direction of softening the e¤ects of the negative demand shock, with impulse responses lying between the cases of no-change and immediate change. In particular, the dynamics of the nominal interest rate is smoother in case of anticipated policy, which leads to smoother behavior of the remaining variables. The initial impact on the economy is signi…cant, e.g., output and labor fall by 1 and 1.5 percent in the three scenarios considered. With no switch in monetary policy, output recovers a bit but its new steady state is still below old steady state. With the policy switch, output is nearly the same as before the shock, and consumption is even higher. Note that each considered policy implies that the central bank tightens monetary policy, even in the economic downturn. 3.6 Switching from in‡ation targeting to average in‡ation targeting On August 27th 2020, the Fed’s Chair Jeromy Powell announced that Fed will switch from in‡ation targeting to average in‡ation targeting; see Powell (2020). However, as was stated by Richard Clarida, during his presentation at the Hoover Economic Policy Working Group on January 13, 2021, one month prior to that, there was evidence that Fed would introduce that framework, and as a result, there were substantial anticipatory price moves in the U.S. economy. In our experiment, we consider a switch from the in‡ation-targeting Taylor rule (18) to a rule that incorporates an average of the past in‡ation (including the actual in‡ation), 0 2 3 1 M X 1 @ 4 Rt = r Rt 1 + (1 log Yt 5 + rt : (34) t j t A + Y log Yt r) R + M +1 j=0 19 The policy of average in‡ation targeting shares many of the properties of price-level targeting. As was suggested by the previous literature, average in‡ation targeting is a middle ground between pricelevel targeting and in‡ation targeting; see Nessén and Vestin (2005). Under average in‡ation targeting, a central bank reacts to a deviation of today’s in‡ation averaged with previous in‡ation from target in‡ation. For example, if the in‡ation target is 2 percent, the averaging window is 3 years, and after consistently archiving 2 percent in‡ation in the past, in the most recent year, in‡ation deviates to 3 percent, the central bank will aim to achieve policy-induced in‡ation of 1 percent in the next year. As a result, in‡ation will oscillate around average in‡ation target and the average in‡ation target is achieved on average. The price level will stay close to its trend, even though the level will sometimes deviate from the …xed trend. 0.06 Nominal Interest Rate Inflation 0.014 10 10-3 Output 0.07 0.012 0.04 0.01 0.02 0.06 5 0.05 0.008 0 10 15 20 5 Consumption 0 0.04 0 0.006 5 Exchange Rate 0.08 10 15 20 5 Investment 0.3 10 15 20 Commodity Export 5 0.03 10 15 20 Noncommodity Export 0.02 -0.02 0.015 -0.04 0.01 -0.06 0.2 0.025 0.1 0.02 0.005 -0.08 0 5 10 15 20 0 5 10-3 Imports 0.01 10 15 20 Capital 0.015 5 15 20 Labour 0.025 0 10 0.04 10 15 20 Real Interest Rate 0.03 0.02 0 0.02 0.015 -5 5 0.01 -0.01 0.01 0 -10 -0.02 5 10 15 20 0.005 5 10 15 20 5 effective immediately 10 15 20 5 10 15 20 announced one year in advance Figure 8: A switch to average in‡ation targeting Figure 8 displays the results for two cases: one is when the switch happens immediately and the other when it is implemented with a lag of one year after it was announced. Amano et al. (2020) study optimal history dependence under average in‡ation targeting in the context of the standard new Keynesian model accounting for the ELB, and they …nd that optimal M ranges from 2 to 8. We assume M = 8 which is the largest number of lags found by Amano et al. (2020). It turns out that this policy change has very modest anticipation e¤ects on the economy in the absence of any shocks. In fact, when the policy becomes e¤ective immediately, there are larger responses in such variables as output, labor, imports, and noncommodity exports. That is, reacting to average in‡ation rather than in‡ation smooths out dynamics to a new steady state. Therefore, we would not expect the economy to experience any drastic changes in the course of transition to average in‡ation targeting. 4 Recurrent versus non-recurrent news shocks News shocks had been analyzed in several papers, including Barro and King (1984), Beaudry and Portier (2006, 2007), Jaimovich and Rebelo (2009). In particular, Schmitt-Grohé and Uribe (2012) introduce a tractable perturbation-based framework for solving models with recurrent news shocks. In their analysis, 20 news shocks follow a stationary Markov process and happen with a …xed periodicity and time horizon. As a result, models with recurrent news shocks have stationary time-invariant Markov solutions. An interesting question is how our nonstationary solutions constructed under the assumption of non-recurrent news shocks compare to stationary solutions constructed by Schmitt-Grohé and Uribe’s (2012) method under the assumption of recurrent shocks. To study this question, we consider a version of the experiment of Section 3.4 in which the central bank switches to a more aggressive Taylor rule; namely, we assume that the sensitivity to in‡ation in the Taylor rule (18) is doubled relative to its benchmark value: the change is announced at t = 1 and implemented at t = 2. The assumption of non-recurrent shocks provides a natural way of modeling this scenario. Namely, we construct a stationary solution for period t = 2 and we …nd a solution for period t = 1 that matches a given terminal condition (decision rule) constructed for period t = 2. Schmitt-Grohé and Uribe (2012) does not specify how their perturbation method can be used for analyzing non-recurrent anticipated shocks like the one we describe above. We tried out two ways of adapting periodic news shocks to our experiment: First, we consider a unit-root process for ;t , i.e., . In this speci…cation, the shock innovation, "t captures ;t = ;t 1 + "t 1 , in which initially ;0 = news that become known at period t and that have a direct impact at t + 1. In our experiment, "1 = at period t = 1, and at all other periods the shock innovation is zero. It implies that ;1 = and ;t = 2 for all t 2. Second, we consider the news shock to be temporary, i.e., ;t = + "t 1 . In that case, we get ;2 = 2 in period t = 2 and ;t = in all other periods. Figure 9 compares the second-order perturbation solutions constructed by our method with those produced by Schmitt-Grohé and Uribe’s (2012) method. The volatility of the news shock is assumed to be zero, so the initial risky steady state is the same for all three solutions. The following observations are in order: First, it appears that our solution with non-recurrent shocks is situated in between the two recurrent news-shock solutions. Second, both our solution and permanent recurrent news-shock solution converge to new (although di¤erent) risky steady states, while the temporary recurrent news-shock solution converges to the old steady state, given the temporary nature of the shock. Third, in the three cases, all the variables behave in qualitatively similar manner: a more aggressive central bank leads to an increase in in‡ation and a decrease in nominal and real interest rates, which raises output, investment, capital and imports. Fourth, the gap between our solution and permanent news shock solution depends on the initial condition: we observe in our sensitivity experiments (these experiments are not reported) that the gap is smaller if we start below steady state. This is because the anticipation e¤ects are mixed up with upward-sloping transition dynamics. Finally, the di¤erence between the two solutions with recurrent news shocks and our second-order perturbation solution comes from the di¤erences in slopes of the decision rules (like the term gxx (v; 0) in (32)). If we were to consider the …rst-order perturbation, the economy would remain at the deterministic steady state in the two recurrent news-shock solutions but not in our solution. Furthermore, it is important to emphasize that the permanent recurrent news-shock approach predicts dramatically larger e¤ects associated with the switch to a more aggressive Taylor rule than our approach (except for consumption). This is true both for the anticipation e¤ects and for di¤erences in steady states. For example, anticipation e¤ects in investment are …ve times larger at peak for the recurrent news shocks than for our perturbation solutions. We conclude that, the two approaches may lead to very di¤erent results: the recurrent news-shock approach signi…cantly overstates the importance of anticipated e¤ects. This is because we assume a unit-root process for ;t , which implies that once a news shock happens, its e¤ects will persist forever. With an autoregressive process for ;t , the e¤ect of news shocks critically depend on the stochastic process assumed. There is a simple intuition on why the solutions with non-recurrent news shock di¤er from those produced by assuming recurrent Markov news shocks. In the former case, the agent’s decision rule is constructed to be the best response to a given deterministic sequence of news shocks and in the latter case, it is constructed to be the best response to the given Markov stationary process. With recurrent shock, the response to news is determined not only by the news itself but also by the properties of the Markov stationary process that is assumed for constructing the solution, which is not the case in our analysis with 21 0.2 Nominal Interest Rate Inflation 0.1 Output Exchange Rate 0.4 0.2 0.3 0.1 0.15 0.05 0.2 0.1 0 0.1 0.05 -0.1 10 20 30 40 50 60 Consumption 0.15 0 0 0 10 20 30 40 50 60 10 Investment 1 20 30 40 50 60 Commodity Export 10 0.15 20 30 40 50 60 Noncommodity Export 1 0.1 0.1 0.5 0.5 0.05 0.05 0 0 0 0 10 20 30 40 50 60 Imports 0.4 10 20 30 40 50 60 10 Capital 0.4 20 30 40 50 60 Labour 0.4 10 0.3 0.3 0.05 0.2 0.2 0.2 0 0.1 0.1 0.1 -0.05 0 0 0 10 20 30 40 50 60 10 20 P-EFP 30 40 50 60 10 news shock, permanent 20 30 40 50 60 -0.1 30 10 20 30 news shock, temporary Figure 9: A switch to a more aggressive Taylor rule at t = 2 announced at t = 1. non-recurrent shocks. 22 40 50 60 Real Interest Rate 0.1 0.3 20 40 50 60 5 Conclusion The literature recognizes that private sector’s expectations are important for policy outcomes and central banks use the open mouth policies to anchor the expectations. However, little work has been done on evaluating the e¤ects of the open mouth policy within a DSGE framework. This paper …lls in this gap. We …nd that the anticipation e¤ects are the strongest for such time dependent economic policies as a gradual change in the natural rate of interest, policy-rate normalization in the aftermath of the ZLB crisis, and a gradual change in the in‡ation target level. The other time dependent policy changes like a switch to a more aggressive policy rate rule, a switch to price-level or average in‡ation targeting lead to more modest anticipation e¤ects. Our methodology is not limited to central-banking models. Many economic policies are announced ahead of being implemented. For instance, changes to taxes, tari¤s, minimum wage, pension reforms, Social Security are frequently signed into law well before they are put in practice. Other notable examples include an announcement about a new member state’s accession to the European Union (EU) or a member state’s exit from the EU (i.e., Brexit), an announcement of the outcome of presidential elections before the new elected president comes to power. Our perturbation-based framework for solving, calibrating, simulating and estimating of parameters provides a simple and tractable way of analyzing nonrecurrent transitions associated with such policy changes. Literally, our analysis makes it possible to construct a model-consistent path of real-world economies. References [1] Amano, R., S. Gnocchi, S. Leduc, J. Wagner, (2020). Average is good enough: averagein‡ation targeting and the ELB. Sta¤ Working Paper 2020-31. https://www.bankofcanada.ca/wpcontent/uploads/2020/07/swp2020-31.pdf [2] Ambler, S. (2009). Price-level targeting and stabilization policy: a survey. Journal of Economic Surveys 23(5), 974–997. [3] Ball, L. 2013. The case for four percent in‡ation. 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[28] Mitra, T. and Y. Nyarko, (1991). On the existence of optimal processes in nonstationary environments. Journal of Economics 53, 245–270. 24 [29] Nessén, M. and D. Vestin (2005). Average in‡ation targeting. Journal of Money, Credit and Banking 37(5) 837-863. [30] Nakamura, E. and J. Steinsson (2018). High-frequency identi…cation of monetary non-neutrality: The information e¤ect. Quarterly Journal of Economics 133(3), 1283–1330. [31] Powell, J., (2020). New economic challenges and the Fed’s monetary policy review. Economiv Policy Symposium "Navigating the Decade Ahead: Implications for Monetary Policy," https://www.federalreserve.gov/newsevents/speech/powell20200827a.htm At an economic policy symposium References [1] Schmitt-Grohé, S., and M. Uribe, (2003). Closing small open economy models. Journal of International Economics 61, 163-185. [2] Schmitt-Grohé S., and M. Uribe, (2004). Solving dynamic general equilibrium models using a secondorder approximation to the policy function. Journal of Economic Dynamics and Control 28 (4), 755-775. [3] Schmitt-Grohé S., and M. Uribe, (2012). What’s news in business cycles. Econometrica 80(6), 2733– 2764. [4] Summers, L., (1991). Panel discussion: price stability: how should long-term monetary policy be determined? Journal of Money, Credit and Banking 23 (3 Part 2): 625-631. [5] Svensson, L. (1999). Price-level targeting versus in‡ation targeting: a free lunch? Journal of Money, Credit and Banking, Vol. 31, No 3, 277-295. [6] Taylor, J. B. (1993). Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195–214. [7] Taylor, J., (1999). A historical analysis of monetary policy rules. In Monetary Policy Rules, (Editor), University of Chicago Press, pages 655-679. [8] Taylor, J.,(2021). The optimal reentry to a monetary policy strategy. Manuscript. [9] Williams, J., (2009). Heeding Daedalus: optimal in‡ation and the zero lower bound. Brookings Papers on Economic Activity 2, 1-37. [10] Yellen, J., (2015). Normalizing monetary policy: prospects and perspectives. Board of Governors of the Federal Reserve System. Speech. [11] Yi, K. and J. Zhang (2017). Understanding global trends in long-run real interest rates. Economic Perspectives 41(2). Appendix A In this section, we illustrate the implementation of perturbation-based method on a toy example – a neoclassical stochastic growth model with labor augmenting technological progress. We consider a version of the model that allows for balanced growth. To solve this model, we proceed as if growth was unbalanced, and then compare our solutions to those obtained by an accurate projection method that solves detrended (stationary) model. 25 The growth model with labor-augmenting technological progress. We consider the following neoclassical stochastic growth model with labor augmenting technological progress: # "1 X t u (ct ) (35) max 1 E0 fct ;kt+1 gt=0 t=0 s.t. ct + kt+1 = (1 ) kt + zt f (kt ; At ) ; ln zt+1 = ln zt + "t+1 , "t+1 N (0; 1) ; (36) (37) where (k0 ; z0 ) is given; Et is an operator of conditional expectation; ct 0 and kt 0 are consumption and capital, respectively; At = A0 tA is labor augmenting technological progress with the rate A 1; u and f are utility and production functions, respectively; and 2 (0; 1); 2 [0; 1]; 2 ( 1; 1); 2 (0; 1). Discussion. Why cannot we solve a nonstationary model with conventional solution methods? For (35)–(37), the Euler equations is given by u (ct ) = Et u0 (ct+1 ) (1 + zt+1 fk (kt+1 ; At+1 )) : The mainstream of economic literature considers stationary models. We can make our model stationary by setting at At = A for all t. The solution to such model is characterized by time-invariant (stationary) decision functions. Conventional solution methods iterate on the Euler equation until a …xed-point decision function for consumption ct = C (kt ; zt ) is found.7 However, if At grows over time, then the optimal decision function changes over time Ct ( ) 6= Ct+1 ( ), then there is no …xed-point solution C ( ) so that the conventional methods are not applicable. Under additional restrictions on preferences and technology, the model with labor augmenting progress has balanced growth and can be converted into stationary; see King et al. (1988). We will not focus on this special case but will approximate a sequence (path) of time-dependent functions fC0 ( ) ; C1 ( ) ; :::g. In doing this, we exploit a turnpike theorem; see Majumdar and Zilcha (1987) and Mitra and Nyarko (1991) for examples of turnpike theorems for nonstationary models. A turnpike theorem studies the convergence of …nite-horizon economies to in…nite horizon economies as the time horizon increases. For this model, the turnpike theorem is proven in MMTT (2020). Two important consequences of the turnpike theorem help us compute solutions in the nonstationary economy: First, the in…nite- and …nite-horizon solutions follow closely one another for a long time and diverge only when the economy approaches to a terminal condition. Second, two terminal conditions kT = k 0 and kT = k 00 that are close to the solution to nonstationary model make the …nite-horizon path closer to the in…nite-horizon path. The former allows us to approximate in…nite-horizon solutions by …nite-horizon solutions, while the latter tells us that it is important to select a good terminal condition, the one close to the in…nite-horizon equilibrium path. To implement the perturbation procedure described in the main text, in Step I, we construct a stationary (time-invariant) model of period T and construct the corresponding Markov decision rule for consumption CT ( ), and in Step II, we use CT to iterate backward on Euler equations in order to construct a sequence (path) of time-dependent value and decision functions fCT 1 ( ) ; CT 2 ( ) ; :::; C0 ( )g, respectively. As a …nal step, we check the turnpike theorem by verifying that the constructed …nite-horizon solution for initial periods converges periodwise to a limiting fC0 ( ) ; C1 ( ) ; :::; C ( )g as time horizon T increases, where is the …nal time period in which we want the solution to be accurate. We elaborate on Steps I and II in details below. First of all note that the model with growth has no natural steady state. To deal with this issue, we introduce time-varying growth rates of capital kt that capture how much this state variable grows from period t to t + 1 due to the time trend or the parameter change. 7 Solution to the growth model could equally well be expressed by a decision function for next period capital kt+1 = K (kt ; zt ). 26 Step I: Solving for terminal decision functions. In Step I, we aim to construct stationary Markov terminal condition in the form of a decision function for consumption cT = CT (kT ; zT ) which is as close as possible to unknown decision function of the in…nite horizon model. We assume balanced growth kT = cT = A , and feed the resulting two equations to a Dynare perturbation. Assuming cT = CT (kT ; zT ) and cT +1 = CT (kT +1 ; zT +1 ), we obtain the usual stationary solution to u0 (cT ) = ET u0 (cT +1 cT = (1 A ) (1 + zT +1 fk (kT +1 ) kT + zT f (kT ; AT ) kT +1 A ; AT A )) ; A: Unless A = 1, the model does not have a balanced growth and our approximation does not coincide with the in…nite horizon solution at T . But the turnpike theorem implies that the speci…c terminal condition assumed at T does not a¤ect signi…cantly the solution up to provided that T . There are ways of 8 constructing more accurate terminal conditions at additional costs. Step II: Finding a path of decision functions. In Step II, we start from the constructed terminal condition for T and proceed backward to compute the path of the decision functions for t = T 1; T 2; :::; 0 by iterating backward on u0 (ct ) = Et u0 (Ct+1 (kt+1 ; zt+1 )) (1 kt+1 = (1 + zt+1 fk (kt+1 ; At+1 )) ; ) kt + zt f (kt ; At ) ct : (38) (39) In particular, for period T 1, given cT = CT (kT ; zT ), Dynare produces the decision function for cT 1 = CT 1 (kT 1 ; zT 1 ), in period T 2, given cT 1 = CT 1 (kT 1 ; zT 1 ) we …nd cT 2 = CT 2 (kT 2 ; zT 2 ) and so on until the entire solution path is constructed. Perturbation solutions we construct are obtained around a deterministic growth path. We consider …ve alternative methods for constructing such a path. We either assume some exogenous growth rates or precompute the growth rates endogenously by shutting down uncertainty in the model. Also, our methods di¤er in a way the policy functions are speci…ed. In particular, for each deterministic growthpath speci…cation, we have two versions of the algorithm: one in which a next-period policy function takes into account the volatility of uncertainty , and the other in which it does not setting = 0. Why might we want to handle the volatility di¤erently? Perturbation policy functions of second and higher orders of approximation are not passing in general through a deterministic steady state of the model. Even in the balanced growth model, if true policy functions for period t + 1 are combined with the model’s equations written for period t, the deterministic steady state would not be a solution of the deterministic version of the combined system of equations. This feature can be overcome by recognizing explicitly that Ct+1 ( ) depends on and by setting to zero when computing the deterministic steady state. Methods 1 and 2. Methods 1 and 2 …nd local approximations of today’s consumption policy function Ct (kt ; zt ) in period t from equations (38) and (39) given the next-period function Ct+1 (kt+1 ; zt+1 ). The di¤erence between the two methods lies only in the point around which the local approximation is taken and it is related to our implementation in Dynare. In period t, Method 1 …nds local approximation around a point (kt ; 1) that solves the following system of two equations for ct and kt : u0 (ct ) = u0 (Ct+1 (kt ; 1)) [1 kt = (1 + fk (kt ; At+1 )] ; ) kt + f (kt ; At ) ct : (40) (41) 8 MMTT (2020) o¤er an alternative way of constructing a terminal condition. Namely, they assume that the solution is stationary in periods T; T + 1 and T + 2 provided that it is adjusted to growth. This gives 4 equations (Euler equation and constraint) for T and T + 1, which can be solved with respect to steady state kT , cT and growth rates kT and cT . 27 Here today’s and tomorrow’s capital are the same and equal to kt because we assume that the growth rate of capital is one. To understand Method 2, recall that the consumption decision function obtained by perturbation depends on the uncertainty parameter and is given by Ct+1 (:; :; ) in period t + 1; see (32) for a general representation.9 In Method 2, we perturb around a point that is computed taking Ct+1 (:; :; ) without the e¤ect of uncertainty, = 0; this approach is similar to …nding a deterministic steady state …rst (as = 0). In other words, the approximation is conducted around a point kty ; 1 that solves the following system of two equations for cyt and kty h u0 (cyt ) = u0 (Ct+1 kty ; 1; 0 ) 1 kty = (1 + fk kty ; At+1 ) kty + f kty ; At cyt : i ; (42) (43) Evidently, the …rst-order perturbation solutions obtained by Method 1 and 2 are identical, as such solutions do not depend on uncertainty.10 Methods 3 and 4. These two methods explicitly account for time-varying growth rates f kt gTt=1 (recall that for both Methods 1 and 2 we assume that growth rates are equal to unity). Similarly to the latter methods, our Methods 3 and 4 di¤er in points around which we …nd Taylor’s expansions and parallel to Methods 1 and 2, respectively. To construct a path of growth rates f kt gTt=1 , both Methods 3 and 4 solve a deterministic version of the model. Namely, we shut down uncertainty by assuming zt = 1 for all t, set c~T +1 and k~T +1 equal to the steady state of the stationary model in the terminal period, and solve the following system of equations:11 + f k~t+1 ; At+1 u0 (~ ct ) = u0 (~ ct+1 ) 1 k~t+1 = (1 ) k~t + f k~t ; At n oT , we compute the growth rates as Given the solution k~t+1 f t=1 T kt gt=1 ; c~t : kt = k~t+1 =k~t . Both Methods 3 and 4 take as given. In period t, Method 3 perturbs the solution around a point (kt ; 1) that solves for kt and ct the following system of two equations: u0 (ct ) = u0 (Ct+1 ( kt kt Note that a variable kt+1 is replaced by = (1 kt kt kt kt ; 1))(1 +f( ) kt + f (kt ; At ) kt kt ; At+1 ) ; (44) ct : (45) meaning that we take into account growth when computing the point of approximation. In turn, Method 4 …nds a perturbation solution around a point kty ; 1; 0 and …nds cyt and kty by solving u0 (cyt ) = u0 (Ct+1 y kt kt = (1 y kt kt ; 1; 0 )1 +f ) kty + f kty ; At y kt kt ; At+1 ; (46) cyt : (47) 9 Note that the dependence of Ct+1 (kt ; 1) on is implicit in Method 1, i.e., we mean Ct+1 (kt ; 1; ) there. Note, however, that higher-order approximations will di¤er between the two methods not only because the intercepts associated with uncertainty are distinct (equal to C ;t+1 (kt ; 1) 2 and C ;t+1 kty ; 1; 0 2 for Method 1 and Method 2, 10 respectively) but also because the points around we approximate di¤er. 11 To implement this step in Dynare, we just solve a system of equations backward in terms of variables 28 n o ~t . c~t ; k Method 5. Method 5 is close to Method 3, but the path for growth rates is computed iteratively. We T begin by exogenously …xing the path k;t t=1 and obtaining the policy functions for a stochastic version of the model; this is similar to Method 3. As a next step, we simulate the model with the realized values of shocks which are set to zero, we compute the growth rates of capital over this simulated path, and we obtain the policy functions for a stochastic version of the model. We can repeat this step as many times as necessary. We do not o¤er any counterpart of Method 5 (i.e., Method 6) that corrects for volatility as it is the case of the methods above because the stochastic growth path is computed in a stochastic version of the model, in which the growth path is obtained endogenously. Numerical results. In this section, we present the results of our numerical analysis. We assume the standard utility and production functions: u (c) = c1 1 1 ; f (k; A) = A1 k : For all the experiments, we …x the parameters f ; ; ; g at the following values: = 0:36; = 0:99; We vary the values of the remaining parameters f ; following values: = 5; " = 0:03; = 0:025; "; A ; T g; A = 1:01; = 0:95: in the benchmark case, we set them to the T = 200: We simulate the model’s solution for di¤erent values of the terminal date T . For all simulations, we use the same initial condition (k0 ; z0 ) and the same sequence of productivity shocks fzt gTt=1 . To see whether our perturbation-based method computes accurate solutions, we obtain an (almost) exact solution by exploiting the property of balanced growth. For this purpose, we …rst introduce laboraugmenting technical change into the model, then derive the …rst-order conditions, and …nally, detrend them. The resulting stationary model is solved by a very accurate standard projection method with Smolyak grid, third-order polynomial approximation, and 10-node Gauss-Hermite quadrature (the maximum residuals in the model’s equations are of order 10 9 in log10 units). We compare the simulated series generated by such a projection method with those of our perturbation method on a …xed sequence of shocks of length T . In Table 2, we report absolute unit-free mean and maximum di¤erences between our approximate and balanced growth (“exact”) solutions (in log10 units) on a simulated path [0; T ] with T 2 f50; 100; 150; 175; 200g. We consider both, …rst- and second-order approximations. As is evident from the table, the …rst-order perturbation solutions are signi…cantly less accurate than the second-order solutions; the di¤erence between the two can reach two orders of magnitude. However, in terms of running times (both solution and simulation), the two solutions are roughly comparable. It is not faster to obtain a …rst- than second-order solution because each perturbation step takes just few seconds and the largest share of time is spent on …nding di¤erent decision rules for each period. For second-order approximations, the most basic method, Method 1, yields very accurate solutions: the mean di¤erence from the exact solution is at most 1 percent across the considered simulation lengths, while the maximum di¤erence reaches 1:5 percent. The ranking of the methods in terms of accuracy varies with time horizon T . For example, for T = 200, Method 1 is the least accurate method, followed by Method 2, and then by Methods 4 and 3 (we look at the maximum errors). However, the ranking between Methods 2 and 4 reverses when the other T s in the table are considered. Methods 3 and 5 are about the same in terms of accuracy and they are the most accurate. Figure 10 plots our …rst- and second-order solutions for capital of the nonstationary model (produced by Method 5), as well as the exact solution of the balanced growth model (produced by the standard projection method); the left panel displays the growing solutions, while the right panel contains the detrended 29 Method 1 First-order solution Errors, in log10 units Horizon Mean [0; 50] -1.41 -1.24 [0; 100] [0; 150] -1.14 -1.07 [0; 175] [0; 200] -1.04 Running time, in seconds Solution 161.57 Simulation 0.0387 Second-order solution Errors, in log10 units Horizon Mean Max [0; 50] -2.28 -2.03 [0; 100] -2.12 -1.90 [0; 150] -2.05 -1.80 [0; 175] -2.00 -1.71 -2.04 -1.71 [0; 200] Running time, in seconds Solution 167.99 Simulation 0.0256 Method 2 Method 3 Max -1.13 -0.97 -0.77 -0.58 -0.58 Mean -1.50 -1.33 -1.25 -1.22 -1.19 157.60 0.0275 Mean -2.83 -2.77 -2.75 -2.71 -2.61 Max -2.53 -2.53 -2.53 -2.15 -1.79 294.28 0.0346 Mean -3.48 -3.30 -3.26 -3.14 -3.07 167.18 0.0249 Max -2.95 -2.93 -2.88 -2.23 -2.23 308.87 0.0326 Method 4 Max -1.22 -1.11 -1.08 -0.94 -0.94 Method 5 Mean -1.53 -1.35 -1.27 -1.23 -1.20 288.03 0.0293 Mean -2.79 -2.46 -2.33 -2.38 -2.43 Max -2.24 -2.05 -2.03 -2.03 -2.03 Max -1.23 -1.12 -1.08 -0.97 -0.97 317.38 0.0271 Mean -3.51 -3.30 -3.26 -3.14 -3.09 296.46 0.0312 Max -2.92 -2.91 -2.88 -2.23 -2.23 337.60 0.0348 Notes: Mean and Max are, respectively, the average and maximum of absolute di¤erence between the P-EFP and exact solutions (in log10 units) on a stochastic simulation of 200 observations. Table 2: Di¤erence of a simulated solution path from the balanced growth path in log10 units 30 Comparison of solutions 80 Comparison of detrended solutions Nonstationary, order 1 Nonstationary, order 2 Balanced growth 70 12 60 Detrended capital Growing capital 11 50 40 30 10 9 8 20 7 10 0 50 100 150 200 0 time 50 100 150 200 time Figure 10: Comparison of the nonstationary P-EFP solutions computed by Method 5 and the balanced growth solution solutions. One striking feature of our solutions is that its second-order approximation is virtually identical to the exact solution (blue and yellow lines coincide). In turn, the …rst-order solution is a visible upward shift of the other two solutions, and therefore, can imply substantial inaccuracy. An important question is: How does our perturbation solutions compare to the existing methods that can solve nonstationary models? In Table 3, we make a comparison of our perturbation method to three other methods, an extended path method of Fair and Taylor (1983) method, a naive method and a global EFP method of Maliar et al. (2020). Fair and Taylor’s (1983) method solves for a path of variables and not functions (as our method does). A naive method …nds a di¤erent solution for each period t under the assumption that the t-period level of technology prevails in each subsequent period. For each of the methods, we use T = 200 in the solution procedure, and we simulate the model for T 2 f50; 100; 150; 175; 200g. As is seen from the table, among the three alternative methods, the ranking of the methods is always the same: the naive method is the least accurate and the global EFP is the most accurate, with Fair and Taylor’s (1983) method being in between. The latter reaches a notorious accuracy of 0:0001 percent for T = 50; the residuals increase to 3:5% for T = 200. The main …nding in the table is that for T = 175 our second-order method is almost as accurate as third-degree solution obtained with the global EFP method, and for T = 200, the second-order solution overpasses the third-degree global EFP solution by a half order of magnitude. Moreover, our perturbation solution is not only more accurate for longer T but also much faster. This is because of perturbation used as a basis of the method. Appendix B In this section, we present sensitivity experiments. A gradual increase in in‡ation target implemented with probability. In Figures 11 and 12, we present the supplementary experiments for Section 3.2. Namely, we consider two experiments that are parallel to the one in Figure 3, where a gradual increase in the in‡ation target happens with probability 31 Table 3: Comparison of the P-EFP to the other methods Fair-Taylor (1983) Naive Global EFP P-EFP method method Type of approximation path path 3rd order 1st order 2nd order Maximum errors, in log10 units [0; 50] -1.29 [0; 100] -1.18 [0; 150] -1.14 -1.14 [0; 175] [0; 200] -1.14 -1.04 -0.92 -0.89 -0.89 -0.89 -6.35 -4.76 -3.22 -2.47 -1.51 -1.27 -1.11 -1.07 -0.94 -0.94 -2.24 -2.05 -2.03 -2.03 -2.03 Running time, in seconds Solution Simulation Total 28.9 2.6 31.5 199.4 0.0244 199.4 317.4 0.0271 317.4 337.6 0.0348 337.6 1.2(+4) 1.2(+4) Note: Maximum errors are the maximum of the absolute di¤erence between the given and exact solutions (in log10 units) on a stochastic simulation of T observations. of 50 percent. In Figures 11 and 12, such a gradual change occurs with probabilities 75 and 25 percent, respectively. As is seen from the …gures, a larger probability of implementing a higher in‡ation target leads to slightly larger expansionary e¤ects on output, consumption, investment, and commodity exports. Although the qualitative patterns are the same, the anticipation e¤ects (changes up to the …fth period when the actual change takes place) are visibly larger with 75 percent probability than with 25 percent probability. A negative supply shock and a switch to price-level targeting. In Figure 7, we focus on a switch to price-level targeting after a negative foreign demand shock. Here, we present a supplementary experiment for Section 3.5. namely, we consider a negative supply shock instead. 32 Interest Rate Inflation 1 1 Output 0.4 0.15 0.5 0.5 5 10 15 20 Consumption 0.2 5 10 15 0.2 0.05 0 20 -0.2 5 Investment 1 0.15 0.1 0 0 0 0.4 10 15 20 Commodity Export 0.3 0.5 0.1 Exchange Rate 0.6 0.2 5 0.2 10 15 20 Noncommodity Export 0.1 0.2 0 0.05 0 -0.5 5 10 15 20 0 5 Imports 0.4 0 0.1 10 15 20 -0.1 5 Capital 15 20 Labour 0.4 0.15 10 0.1 20 0.2 0.1 0.05 0.1 15 0.3 0.2 0.2 10 Commodities 0.4 0.3 0.3 5 0.1 0 0 0 5 10 15 20 0 5 10 15 20 5 no change in the inflation target 10 15 20 5 10 15 20 a higher inflation target Figure 11: A gradual increase in the in‡ation target (75% probability) Interest Rate Inflation 1 1 Output 0.2 Exchange Rate 0.6 0.15 0.4 0.1 0.5 0.5 0.2 0.05 0 0 0 0 5 10 15 20 5 Consumption 0.2 10 15 20 5 Investment 10 15 20 Commodity Export 5 0.6 0.4 0.15 0.4 0.3 0.2 0.1 0.2 0.2 0.1 0.05 0 0.1 0 0 0 5 10 15 20 Imports 0.4 5 10 15 20 15 20 Noncommodity Export -0.1 5 Capital 0.15 0.3 10 10 15 20 Labour 0.4 5 10 15 20 Commodities 0.3 0.3 0.3 0.1 0.2 0.2 0.2 0.1 0.05 0.1 0.1 0 0 0 5 10 15 20 0 5 10 15 20 5 no change in the inflation target 10 15 20 5 a higher inflation target Figure 12: A gradual increase in the in‡ation target (25% probability) 33 10 15 20 0.3 Nominal Interest Rate 0.2 Inflation Output 0.04 -0.2 0.03 -0.3 0.02 -0.4 0.01 -0.5 -0.2 -0.3 -0.4 0.1 0 0 10 0 20 30 Consumption -0.5 -0.6 10 20 30 -0.6 10 Investment 0 -0.4 Exchange Rate 20 30 10 Commodity Export 20 30 Noncommodity Export -0.1 -0.6 -0.2 -0.5 -0.15 -0.8 -0.4 -1 -1 -0.2 -0.6 -1.2 10 20 30 10 Imports 20 30 10 Capital 0 20 30 Labour 0.4 10 0.3 20 30 Real Interest Rate 0 -0.1 -0.2 0.2 0.2 -0.4 0 0.1 -0.2 -0.3 -0.4 -0.6 10 20 30 inflation targeting 0 -0.2 10 20 30 10 change to price level targeting 20 30 10 change to price level targeting announced one year in advance Figure 13: A negative supply shock and a switch to price-level targeting 34 20 30 Online appendix For the reader’s convenience, we provide a description of the calibration procedure, which is similar to LMM (2020). 5.1 Calibration The model contains 61 parameters to be calibrated. Whenever possible, we use the same values of parameters in the scaled-down model as those in the full-scale model, and we choose the remaining parameters to reproduce a selected set of observations from the Canadian time series data. In particular, our calibration procedure targets the ratios of six nominal variables to nominal GDP Pty Yt , namely, consumption Pt Ct , investment Pti It , noncommodity export Ptnc Xtnc , commodity export Ptcom Xtcom , import Ptm Mt , total commodities Ptcom COMt , and labor input Wt Lt . Furthermore, we calibrate the persistence of shocks so that the standard deviations of the selected bToTEM variables coincide with those of the corresponding ToTEM variables, namely, those of domestic nominal interest rate Rt , productivity At , foreign demand Ztf , foreign commodity price pcomf , and foreign interest rate rtf . The parameters choice is summarized in t Tables 4 and 5 below. Parameter Rates – real interest rate – discount factor – in‡ation target – nominal interest rate – ELB on the nominal interest rate Output production – CES elasticity of substitution – CES labor share parameter – CES capital share parameter – CES commodity share parameter – CES import share parameter – investment adjustment cost – …xed depreciation rate – variable depreciation rate – depreciation semielasticity – real investment price – real noncommodity export price – labor productivity Price setting parameters for consumption – probability of indexation – RT indexation to past in‡ation – RT share – elasticity of substitution of consumption goods – Leontie¤ technology parameter Price setting parameters for imports – probability of indexation – RT indexation to past in‡ation – RT share – elasticity of substitution of imports Price setting parameters for wages – probability of indexation 35 Symbol Value Source r 1.0076 0.9925 1.005 1.0126 1.0076 ToTEM ToTEM ToTEM ToTEM …xed A 0.5 0.249 0.575 0.0015 0.0287 20 0.0054 0.0261 4.0931 1.2698 1.143 100 ToTEM calibrated calibrated calibrated calibrated calibrated ToTEM ToTEM calibrated ToTEM ToTEM normalization ! " sm 0.75 0.0576 0.4819 11 0.6 ToTEM ToTEM ToTEM ToTEM ToTEM 0.8635 0.7358 0.3 4.4 ToTEM ToTEM ToTEM 0.5901 ToTEM R Relb l k com m i d0 d i x m m !m "m w w – RT indexation to past in‡ation 0.1087 ToTEM – RT share !w 0.6896 ToTEM – elasticity of substitution of labor service "w 1.5 ToTEM Household utility – consumption habit 0.9396 ToTEM – consumption elasticity of substitution 0.8775 ToTEM – wage elasticity of labor supply 0.0704 ToTEM Monetary policy – interest rate persistence parameter 0.83 ToTEM r – interest rate response to in‡ation gap 4.12 ToTEM – interest rate response to output gap 0.4 ToTEM y Other k – capital premium 0.0674 calibrated – exchange rate persistence parameter { 0.1585 ToTEM – foreign commodity price pcomf 1.6591 ToTEM – foreign import price pmf 1.294 ToTEM – risk premium response to debt & 0.0083 calibrated f – export scale factor 18.3113 calibrated – foreign demand elasticity 0.4 calibrated – elasticity in commodity production sz 0.8 calibrated – land F 0.1559 calibrated – share of other components of output 0.7651 calibrated z – share of other components of GDP 0.311 calibrated y – adjustment cost in commodity production 16 calibrated com – persistence of potential GDP 'z 0.75 calibrated Table 4: Calibrated parameters in endogenous model’s equations Parameter Shock persistence – persistence of interest rate shock – persistence of productivity shock – persistence of consumption demand shock – persistence of foreign output shock – persistence of foreign commodity price shock – persistence of foreign interest rate shock Shock volatility – standard deviation of interest rate shock – standard deviation of productivity shock – standard deviation of consumption demand shock – standard deviation of foreign output shock – standard deviation of foreign commodity price shock – standard deviation of foreign interest rate shock Table 5: Calibrated parameters in exogenous 36 Symbol Value Source 'r 'a 'c 'zf 'comf 'rf 0.25 0.9 0 0.9 0.87 0.88 ToTEM …xed …xed …xed calibrated calibrated 0.0006 0.0067 a 0.0001 c 0.0085 zf 0.0796 comf 0.0020 rf model’s equations r calibrated calibrated …xed calibrated calibrated calibrated