Nominal Wage Rigidities in a New Keynesian Model with Frictional Unemployment

PA P ER SERI ES D I SCU SSI O N IZA DP No. 2528 Nominal Wage Rigidities in a New Keynesian Model with Frictional Unemployment Vincent Bodart Gregory De Walque Olivier Pierrard Henri R. Sneessens Raf Wout ers December 2006 Forschungsinst it ut zur Zukunf t der Arbeit Inst it ut e f or t he St udy of Labor Nominal Wage Rigidities in a New Keynesian Model with Frictional Unemployment Vincent Bodart Université catholique de Louvain Gregory De Walque National Bank of Belgium Olivier Pierrard Central Bank of Luxembourg and Université catholique de Louvain Henri R. Sneessens Université catholique de Louvain, Université catholique de Lille and IZA Bonn Raf Wouters National Bank of Belgium Discussion Paper No. 2528 December 2006 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: iza@iza.org Any opinions expressed here are those of the author(s) and not those of the institute. Research disseminated by IZA may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit company supported by Deutsche Post World Net. The center is associated with the University of Bonn and offers a stimulating research environment through its research networks, research support, and visitors and doctoral programs. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 2528 December 2006 ABSTRACT Nominal Wage Rigidities in a New Keynesian Model with Frictional Unemployment In this paper, we propose a search and matching model with nominal stickiness à la Calvo in the wage bargaining. We analyze the properties of the model, first, in the context of a typical real business cycle model driven by stochastic productivity shocks and second, in a fully specified monetary DSGE model with various real and nominal rigidities and multiple shocks. The model generates realistic statistics for the important labor market variables. JEL Classification: Keywords: E31, E32, E52, J64 DSGE, search and matching, nominal wage rigidity, monetary policy Corresponding author: Olivier Pierrard Banque centrale du Luxembourg 2 boulevard Royal L-2983 Luxembourg E-mail: olivier.pierrard@bcl.lu 1 Introduction Real wage and labor market dynamics are crucial for understanding the inflation process. Standard new-Keynesian models contain only a highly abstract description of the labor market which does not allow for involuntary unemployment and real wage rigidity. Two keys issues that are central when monetary policy is faced with complicated trade-off decisions. Search and matching models, on the other hand, provide a more realistic framework that can be used to analyze unemployment and wage bargaining situations. For these models to match the stylized properties of the data, some degree of wage rigidity is necessary. In this paper, we propose a search and matching model with nominal stickiness à la Calvo in the wage bargaining. We analyze the properties of the model, first, in the context of a typical real business cycle model driven by stochastic productivity shocks and second, in a fully specified monetary DSGE model with various real and nominal rigidities and multiple shocks. The model generates realistic statistics for the important labor market variables. Standard new-Keynesian DSGE models approach the labor market as a duplicate of the goods market: households supply differentiated services in a monopolistic competitive market which provide them with monopoly power over the wage. The resulting wage is determined as a mark-up over the marginal rate of substitution between consumption and leisure, where the mark-up may vary due to nominal stickiness. At the given wage, firm decide on their optimal demand for labor and workers will deliver the requested labor service. For realistic parameters of labor supply and nominal wage stickiness, these model reproduce the observed volatility in hours worked and the relative smooth behavior of real wages over the business cycle (see for instance Shimer (2002) or Shimer (2004) for empirical evidence on the cyclical behavior of labor market variables). However these models are ignorant among other things on the concept and the role of unemployment and other labor market flows, on the specific nature of continuing labor contracts and the resulting wage bargaining, and on labor adjustment along the intensive and the extensive margins. Therefore, these standard new-Keynesian models can hardly be considered as realistic characterizations of the labor market and any normative analysis based on the welfare implications of these models might result in misleading conclusions. Search models à la Pissarides-Mortensen overcome some of the weaknesses of the standard newKeynesian labor market models by starting from the specific nature of the labor market. Matching workers and firms is costly and this results in a surplus for existing jobs and a bargaining situation over the wage and possibly broader working conditions. Merz (1995) and Andolfatto (1996) integrated this search and matching setup in a general equilibrium model and illustrated its relative success to explain cyclical behavior in wages and employment fluctuations. More recently, Hall (2005) and Shimer (2004) showed that these models fail to generate the observed volatility in unemployment and job vacancies. The reason is that under standard parameterization, new vacancies induce a strong reaction in the real wage that erode the profitability of new job creation. Wage rigidity, especially for new jobs (see Bodart, Pierrard, and Sneessens (2005)), can overcome this reaction and boost the sensitivity of labor market variables. Following up on this idea, Gertler and Trigari (2006) introduce wage staggering à la Calvo in the bargaining solution, and show how the spill over effects of the slowly adjusting aggregate wage mitigate the change in the new contract wage. For realistic contract durations, this mechanism produces the observed relative smooth wage response while doing fine on the volatility of vacancies and unemployment as well. Gertler and Trigari execute their exercise in a basic real business cycle model that is exclusively driven by productivity shocks and where no explicit distinction is necessary between nominal and real wage setting. Another series of papers - Walsh (2005), Trigari (2004), Moyen and Sahuc (2005) - have studied the role of labor-search frictions for inflation dynamics and the monetary policy transmission mechanism. These models combine the labor matching function in a wholesale production sector with sticky nominal prices in the final retail sector. By altering the wage formation process, compared to the standard new-Keynesian framework, these models also change the cyclical behavior of the marginal cost and inflation. In particular, these models are able to show how institutional factor, such as the bargaining power and the replacement benefit for unemployed workers, can affect inflation. Trigari (2004) also point out that the marginal cost can behave differently depending on whether the required labor adjustment takes place along the intensive margin, i.e. via changes in hours worked, or along the extensive, i.e. employment margin. 2 The integration of wage rigidities and nominal price stickiness in the labor-search models has been analyzed by Krause and Lubik (2005), Christoffel and Linzert (2004), Blanchard and Gali (2005) and Christoffel, Kuster, and Linzert (2006). Krause and Lubik claim that the real wage rigidity is important for matching the labor market volatilities but that wage rigidity is not crucial for the inflation dynamics. This follows from the argument by Goodfriend and King (2001) that the period-by-period wage looses its allocative role in the marginal cost in the context of long term labor relations, which are implicitly assumed in the matching labor market setup. Christoffel et al. integrate various forms of wage rigidities and nominal price stickiness in a fully specified DSGE model and estimate this model to German data. Their results show that important labor market shocks are necessary to fit the wage and employment data but these shocks have a limited role on the overall dynamics of output and inflation. Blanchard and Gali analyze the implication of real wage rigidity for monetary policy. The inefficient reaction of wages and employment to productivity shocks complicates the stabilizing task for monetary policy because it creates a conflict between inflation targeting and employment stabilization. In this paper we extend the work of Gertler and Trigari (2006), by incorporating the wage staggering a la Calvo in a model with nominal price and wage setting together with a series of other frictions that are often considered as necessary to capture the cyclical dynamics in consumption, investment and production (Christiano, Eichenbaum, and Evans (2005), Smets and Wouters (2003)). In addition, firms have the possibility to adjust the labor input along the intensive and extensive margin. In order to get sufficient persistence in the labor market dynamics, we also evaluate alternative cost schedules for the vacancy and hiring decisions: a standard recurrent vacancy cost à la Mortensen-Pissarides, a variable recurrent hiring cost à la Gertler-Trigari and a Fujita-Ramey type of sunk cost for vacancy posting. First, we evaluate the cyclical properties of our model by concentrating exclusively on the laborsearch friction in combination with nominal wage staggering. The volatility, the persistence and the cyclical nature of the labor market variables in our model are directly compared to the results in Gertler and Trigari. We show that the results depend on the writing of the vacancy costs and that the model with sunk cost does well (and at least as well as Gertler and Trigari) at reproducing the main variable cyclical properties. Then we consider the extended version of 3 our model that contains the complete set of nominal and real frictions typically used in the new generation of monetary DSGE models, and we evaluate whether this model is still able to perform well on the labor market statistics both for real and nominal shocks. The extensive model, and especially the model with a sunk vacancy cost, proves also successful in reproducing the typical labor market volatilities and correlations. However, some drawbacks remain as the too strong reaction of hours and the inability of the model to generate a lagged response of inflation to a monetary shock. These simulation exercises point out the important parameters and frictions that are at work in the model. We consider these exercises as a necessary preliminary step before taking the model to the data in a more elaborate estimation procedure. 2 The Model We consider three broad categories of agents: households, firms and the government, together with three types of markets: goods, labor and capital. We distinguish two types of goods producers, final goods and intermediate goods. Perfect competition is assumed on the final goods market and monopolistic competition on the intermediate goods market. On the capital market, the supply is determined by the stock of capital previously accumulated by the household. The return on capital adjusts to make the quantity demanded by the representative final firm equal to the predetermined capital stock. We introduce labor market frictions à la MortensenPissarides. We assume a single representative household. Consumer-workers may be employed or unemployed1 . 2.1 Labor Market Flows Let Nt represent the total number of jobs. Normalizing the total labor force to one yields the following accounting identity: Nt + Ut = 1 1 (1) This representative household formulation amounts to assuming that workers are perfectly insured against the unemployment risk. This simplification is common in the literature (see for instance Merz (1995) or Andolfatto (1996)) and reflects the current state of the art. Taking into account workers heterogeneity due to imperfect insurance markets would make the model totally intractable. 4 where Ut denote the number of unemployed job-seekers. Let the number of job matches be denoted by Ht . We assume that the number of matches is a function of the number of job vacancies Vt and effective job seekers Ut , that is, we use the following matching function: Ht = H (Vt , Ut ) = h̄ Vtφ Ut1−φ , (2) assumed to be linear homogeneous. The probability of finding a job can be written as follows: pt = Ht . Ut (3) Similarly, the probability of filling a vacancy is given by: qt = Ht . Vt (4) We assume an exogenous job destruction rate s, implying the following employment dynamics: Nt+1 = (1 − s) Nt + qt Vt , (5) = (1 − s) Nt + pt Ut . (6) 2.2 Households There is a continuum of households indexed by τ . Each household maximizes an intertemporal utility function represented by: ∞ X t=0 βt U µ Ctτ , C̄t , Mtτ Pt ¶ , where β is the subjective discount factor. Instantaneous utility U is a function of current consumption Ctτ and real cash balances Mtτ /Pt . External consumption habit effects are introduced by C̄t . We assume the following separable utility function: ¶ µ τ ¶1−νm µ ¡ τ ¢ Mt χνm Mtτ τ . = log Ct − e C̄t−1 + U Ct , C̄t , Pt 1 − νm Pt (7) External consumption habits are represented by an effect of past aggregate consumption. Each household (worker) is looking for a full-time job and can be employed or unemployed. Following Christiano, Eichenbaum, and Evans (2005), we assume that there exist state-contingent securities that insure the households against variations in household specific labor income. With 5 perfect insurance markets and with separability between consumption and leisure, employed and unemployed worker will have the same marginal utility of wealth and choose the same optimal consumption level and money demand. Individual behaviors can then be analyzed in terms of the representative household’s optimization program. We normalize total population to 1 and define Nt as the fraction of workers hired at time t − 1 or before and productive at time t. The representative household’s optimization program can then be written as follows: ( µ ¶1−νm ) ∞ νm X χ M t β t log (Ct − e Ct−1 ) + max , 1 − νm Pt t=0 subject to : Mt−1 Bt−1 (1 + Rt−1 ) Mt Bt + + Ct + It + Tt = + + Wt + bt (1 − Nt ) , Pt Pt Pt Pt (8) ∀t≥0 . Wt stand for aggregate income received by employed workers; bt is an unemployment benefit2 ; Tt stand for total lump-sum taxes. The inflation rate πt = Pt /Pt−1 − 1 determines inflation taxes and Rt is the nominal interest rate between t − 1 and t: (1 + Rt ) = (1 + rt )(1 + πt+1 ) . (9) We allow for variations in labor working time (or hours) ht and for variations in the capacity utilization rate zt , at a cost Ψ(zt ). We also take into account capital installation costs, measured by a function of investment changes Φ(∆It /It−1 ). This leads to the following aggregate income and capital accumulation equations: o n o n wt + Cth Nt + (rtk + δ) zt − Ψ(zt ) Kt−1 + Πt , ´ c0 ³ 1+c1 = ht −1 , 1 + c1 n o = 1 − Φ(∆It /It−1 ) It − δ Kt−1 . Wt = (10) Cth (11) ∆Kt (12) The employed worker’s labor income is made of two parts, a base wage wt plus an overtime work compensation Cth . Normal working time is normalized to 1 and overtime compensation is proportional to the difference between actual and normal working time3 . Hours are decided by 2 It could alternatively be interpreted as the income generated by the domestic activities of an unemployed worker. 3 An alternative modelization would be to introduce hours in the (des)utility function. This formulation would allow hours to vary with the marginal utility. 6 the firms and, at the steady state, overtime compensation is equal to zero. Total capital income is equal to the return on utilized capital net of capacity utilization costs Ψ(zt ). The normal utilization rate is normalized to 1. Πt are the profits redistributed by the intermediate goods producers. We assume the following cost functions: Ψ(zt ) = ¶ µ ∆It = Φ It−1 i d0 h 1+d1 −1 , zt 1 + d1 ¶ µ ϕ ∆It 2 . 2 It−1 (13) (14) The consumer’s optimal decisions are then given by: (15) Bt : UCt = β (1 + rt ) UCt+1 , zt : rtk + δ = d0 ztc1 , (16) · µ ¶¸ ( µ ¶2 ) ∆It+1 It+1 ∆It ∆It It , (17) − βt+1 pkt+1 ϕ 1 = pkt 1 − Φ − pkt ϕ It−1 It−1 It−1 It It n o k pkt = βt+1 zt+1 (rt+1 (18) + δ) − Ψ(zt+1 ) + (1 − δ) pkt+1 , µ ¶−1/νm ³ ´−1/νm Rt Mt =χ UCt (19) . Pt 1 + Rt It : Kt : Mt : (20) pkt is the shadow price of capital at time t; βt+1 in (18) and (17) is a discount factor defined by: βt+1 = β 2.3 2.3.1 UCt+1 , UCt where UCt = 1 . Ct − eCt−1 (21) Goods Producers Final Goods We assume a CES production technology: ¾1/λx ½Z 1 λx , [Xt (i)] di Yt = (22) 0 with λx positive but smaller than unity to ensure decreasing marginal productivity. The profit maximization program gives a first order optimality condition which can be recast as a demand for intermediate goods: · ¸ 1 Pt (i) − 1−λx Xt (i) = Yt , Pt ∀ i ∈ [0, 1] . (23) 7 The price index is given by: Pt = ½Z 1 λx − 1−λ [Pt (i)] 2.3.2 x x 0 x ¾− 1−λ λ di . (24) Intermediate Goods The production function of an intermediate goods producer i is Cobb-Douglas with constant returns to scale: iα h i1−α h , Xt (i) = ǫt K̃t (i) hθt Nt (i) (25) where K̃t (i) = zt Kt−1 (i); ǫt is an aggregate exogenous productivity shock and θ ≤ 1 (productivity concave in hours). Hours of work Intermediate good firms rent the desired quantity of workers from labor service firms, at a price dt per worker determined by the market. Hours of work may vary over time. Overtime work is paid Cth (see equation (11)). The no-arbitrage condition between the firm’s internal and external margins implies: c0 hct 1 = θ dt + Cth , ht (26) implying: ht = ( 1 1− θ 1+c1 θ c0 µ c0 dt − 1 + c1 1 ¶) 1+c 1 , (27) with c1 ≥ 0. To obtain h = 1 at stationary equilibrium, we set c0 = θ d. For c1 → ∞, hours of work are constant. Marginal Cost At given selling price Pt (i) (and corresponding output level Xt (i)), the intermediate goods producer’s optimization program is a standard cost minimization program, implying the same optimal capital-labor ratio for all intermediate goods producers: #−1 " K̃t (i) (rtk + δ)/α ¢ , ∀i . = ¡ Nt (i) dt + Cth /(1 − α) 8 (28) Because we assumed constant returns to scale and price taking behavior on the input markets, the (real) marginal cost Λxt is independent of the price and production levels and given by: Λxt 1 = ǫt µ dt + Cth (1 − α) hθt ¶1−α µ rtk + δ α ¶α . (29) When c1 → ∞, ht ≡ 1, which implies the following standard marginal cost equation: 1 ǫt Λxt = µ dt 1−α ¶1−α µ rtk + δ α ¶α . Optimal prices All intermediate goods producers who are allowed to reset optimally their selling price at time t face exactly the same optimization problem. Let us denote Pt∗ the optimal price reset at time t. In a Calvo contract framework, the optimal price decision is determined by the following optimization program4 : max ∗ Pt ∞ X ξpj βt+j j=0 · (1 + π̄)j Pt∗ − Λxt+j Pt+j ¸µ (1 + π̄)j Pt∗ Pt+j ¶−1/(1−λx ) Yt+j , (30) where ξp is the probability that the price cannot be reset from one period to the next (perfect price flexibility is thus obtained for ξp = 0). The discount factor βt+j is compatible with the pricing kernel used by consumers-shareholders: βt+j = β j UCt+j . UCt (31) The first-order optimality condition can be written as (after rearrangements): ∞ X j=0 ξpj βt+j Yt+j · (1 + π̄)j Pt∗ Pt+j ¸−1/(1−λx ) ½ 1 x (1 + π̄)j Pt∗ − Λ Pt+j λx t+j ¾ =0. (32) A transitory increase in the aggregate demand (following a monetary shock e.g. ) will thus lower the current average markup rate, both because some intermediate goods prices are not reset (a fraction ξp of them) and because reset prices do not fully adjust to transitory cost changes. 4 The computation of the optimal price Pt∗ is based on the information available at time t. A more careful notation should thus include the conditional expectation operator Et . Our simplified notation is easier to read. One has to bear in mind though that all future variables are actually conditional expectations. For instance Zt+j stands for Et (Zt+j ), where Z may be any variable or combination of variables. It is worth noticing that our notation is in line with the conventions used in the Dynare software (cf. http://www.cepremap.cnrs.fr/dynare/). 9 2.3.3 Aggregate Price and Quantity Indices Aggregate demand for labor and capital services All intermediate goods producers use the same production technology (capital-labor ratio). With constant returns to scale, the demand for labor and capital is linear in output. The aggregate demand for labor and capital is thus proportional to aggregate output, even though different firms may have different production levels. Aggregate intermediate goods production is determined by: Xt = Z 1 Xt (i) di , (33) 0 where Xt (i) is given by equation (23). Substituting in (33) yields Z 1 1/(1−λx ) Pt (i)−1/(1−λx ) di Xt = Pt Yt 0 = µ P̄t Pt ¶−1/(1−λx ) Yt , (34) where the index P̄t is defined by: −1/(1−λx ) P̄t = Z 1 Pt (i)−1/(1−λx ) di . (35) 0 The value of P̄t can be computed by using the property that in any period t − j (with j ≥ 0) j a fraction (1 − ξw ) of all prices is reset and remains unchanged till time t with probability ξw . This yields5 : −1/(1−λx ) P̄t = (1 − ξp ) ∞ X j=0 h = (1 − ξp ) Pt∗ h i−1/(1−λx ) ∗ ξpj (1 + π̄)j Pt−j i−1/(1−λx ) h i−1/(1−λx ) + ξp (1 + π̄) P̄t−1 . (36) Aggregate Price Index The aggregate price index is given by equation (24). With Calvo contracts, a fraction (1 − ξp ) of previous period prices is reset optimally, while a fraction ξp is simply indexed to trend inflation 5 The second expression can be obtained directly by using the fact that reset prices (a fraction (1 − ξw ) of all prices) are chosen at random. The price index computed over unchanged prices (a fraction ξw of all prices) is thus equal, up to the indexation factor, to the previous period price index. 10 π̄. Because the individual prices that can or cannot be revised are chosen randomly, the value of the price index aggregating all prices that are not reset optimally is equal to the aggregate price index of the previous period, scaled up for trend inflation. The new aggregate price level is thus determined by: Z 1 λx λx − 1−λ x Pt [Pt (i)]− 1−λx di = 0 λx λx = (1 − ξp ) [Pt∗ ]− 1−λx + ξp [(1 + π̄) Pt−1 ]− 1−λx . (37) Log-linearizing around steady-state values yields : p̃t = (1 − ξp ) p̃∗t + ξp p̃t−1 . (38) Final Goods Production The previous results (see equation (34)) can be used to write the “final goods production function” as follows6 : µ ¶1/(1−λx ) P̄t Xt Yt = Pt µ ¶1/(1−λx ) h i1−α P̄t = ǫt K̃tα hθt Nt . Pt 2.4 (39) Labor Services Labor services are offered to intermediate goods firms by “labor packers”. Labor packers are perfectly competitive intermediaries who rent labor services from households and sell these services to intermediate goods producers at a rate dt . The wage paid by existing labor service firms is not bargained again in every period. We assume instead a Calvo framework, wherein only a fraction (1 − ξw ) of all existing wage contracts is renegotiated every period. All other nominal wages are simply adjusted for trend inflation π̄. New jobs are paid either the average or the freely negotiated wage. The respective proportions are κ and (1 − κ). Full nominal wage flexibility obtains for ξw = κ = 0. With κ = 0 and 6 Notice though that the log-linearized form of equation (36) is similar to that of the true price index (see equation (38)). The distinction between P̄t and Pt and between Xt and Yt thus vanishes in a log-linearized model. 11 ξw = 1, the nominal wage of all new jobs would be freely negotiated while the nominal wage of all existing jobs would simply be indexed on trend inflation. Although they all have the same productivity, different workers may thus be paid different wages, depending on the time they entered the labor market and on the time their wage was (re-)negotiated. Let wt∗ = Wt∗ Pt represent the real value of the nominal wage negotiated at time t; wt stands for the average real wage observed at time t. Let Nt (xt−j ) represent the number of workers employed at time t at a wage fixed at time t − j and since then simply indexed on trend ∗ , w inflation, where xt−j ǫ {wt−j t−j } represents the real value of the wage at time t − j. Total employment is equal to: ∞ n o X ∗ ) + Nt (wt−j ) . Nt (wt−j Nt = j=0 By definition of wt , we have: ∞ o X (1 + π̄)j Pt−j n ∗ ∗ wt Nt = wt−j Nt (wt−j ) + wt−j Nt (wt−j ) . Pt j=0 2.4.1 Value of a job for a labor service firm We assume that when a job is destroyed, it definitively disappears and its asset value is therefore equal to zero. The asset value AFt (wt∗ ) of a job with wage renegotiated at time t is then given by: h i ¡ ∗ ¢ AFt (wt∗ ) = (dt − wt∗ ) + βt+1 (1 − s) (1 − ξw ) AFt+1 wt+1 + ξw AFt+1 (wt∗ ) . (40) where the discount factor βt+1 is compatible with the pricing kernel used by consumers-shareholders (see (31)). It will prove convenient to recast this value in marginal utility terms by multiplying both sides of the above expression by UCt . Let us define AFt+j = UCt+j AFt+j . We thus obtain: h i ¡ ∗ ¢ + ξw AFt+1 (wt∗ ) . AFt (wt∗ ) = UCt (dt − wt∗ ) + β (1 − s) (1 − ξw ) AFt+1 wt+1 (41) The second term inside the square brackets is the value of a job whose wage was negotiated one period earlier. This value is determined by: ( ) j ¡ ∗ ¢ (1 + π̄) P t AFt+j (wt∗ ) = UCt+j dt+j − wt∗ + β (1 − s) (1 − ξw ) AFt+j+1 wt+j+1 Pt+j + β (1 − s) ξw AFt+j+1 (wt∗ ) , 12 (42) for j ≥ 1. Substituting repetitively in equation (41) yields: AFt (wt∗ ) = ∞ h X β (1 − s) ξw j=0 − +  ∞ h X  ij UCt+j dt+j β (1 − s) ξw j=0 ∞ X j=0 h β (1 − s) ξw ij ij  (1 + π̄) Pt  ∗ UCt+j w  t Pt+j j (43) ¡ ∗ ¢ β (1 − s) (1 − ξw ) AFt+j+1 wt+j+1 . Let St1 represent the value of the summation term between the curly brackets: St1 = ∞ h X β (1 − s) ξw j=0 ij UCt+j (1 + π̄)j Pt Pt+j h i (1 + π̄) P t 1 = UCt + β (1 − s) ξw St+1 . Pt+1 (44) One can similarly define Std = ∞ h X β (1 − s) ξw j=0 ij UCt+j dt+j i h d . = UCt dt + β (1 − s) ξw St+1 (45) Using these definitions into (43) and rearranging, one obtains: AFt (wt∗ ) = = h h Std − St1 wt∗ i + i ∞ h X β (1 − s) ξw j=0 ij ¡ ∗ ¢ β (1 − s) (1 − ξw ) AFt+j+1 wt+j+1 i h ∗ d 1 ∗ + β (1 − s) AFt+1 (wt+1 ). Std − St1 wt∗ − β (1 − s) ξw St+1 − St+1 wt+1 (46) The value AFt (wt ) of a new job starting with a wage equal to the average wage wt can be obtained in the same way. Starting from: h i ∗ AFt (wt ) = (dt − wt ) + βt+1 (1 − s) (1 − ξw ) AFt+1 (wt+1 ) + ξw AFt+1 (wt ) , (47) and proceeding as before yields in marginal utility terms: i h i h d 1 − St+1 wt+1 + β (1 − s) AFt+1 (wt+1 ) . AFt (wt ) = Std − St1 wt − β (1 − s) ξw St+1 2.4.2 (48) The free entry condition Let AN t represent the asset value of a new job, which can be written as follows: F ∗ F AN t = (1 − κ) At (wt ) + κ At (wt ) . (49) 13 The asset value of a vacant job AVt is then given by: V AVt = −ct + βt+1 qt AN t+1 + βt+1 (1 − qt ) At+1 , (50) where ct is the recurrent cost of opening a vacancy. We alternatively consider three cases for ct , yielding three variants of the model : (i) a constant recurrent cost, (ii) a variable recurrent cost, and (iii) a sunk cost. Constant recurrent cost In this variant, we assume that ct = a1 , as in the standard Mortensen and Pissarides (1999) framework. The free entry condition implies that AVt = 0 and equation (50) can be recast in: a1 = qt βt+1 AN t+1 . (51) Total vacancy costs are given by: vt = ct Vt . (52) It is worth noting that the average cost per hiring is ct /qt , that is a1 /qt in this setup. The variant of the model derived under this assumption is denoted MP-model hereafter. Variable recurrent cost In this second case we follow Gertler and Trigari (2006) and drop the assumption of a fixed recurrent cost. Instead, we suppose that the total cost of adjusting the workforce is a2 x2t Nt , where xt is the hiring rate. Because the hiring rate xt = Ht Nt , this total hiring cost can be recast as a2 xt Ht . The cost per hiring is thus proportional to the hiring rate and equal to a2 xt . This expression can be substituted for ct /qt in equation (50), where ct is the variable recurrent cost. With the free entry condition AVt = 0, equation (50) then becomes: a2 Ht = βt+1 AN t+1 . Nt (53) Total vacancy costs are still given by equation (52). The model variant adopting this variable recurrent cost is labeled GT-model in the sequel. Sunk cost 14 In the third variant of equation (50), we drop the recurrent cost (ct = 0). Instead, as Fujita and Ramey (2005), we assume that a sunk cost SC has to be paid only once when the new vacancy is created. The sunk cost may differ across firms and let the continuous function F (SC) give the total mass of firms that have a sunk cost no greater than SC. Then equation (50) can be written as: V AVt = βt+1 qt AN t+1 + βt+1 (1 − qt ) At+1 . (54) The standard free entry condition is replaced by: nt = Z AV t dF (SC) , (55) 0 and the law of motion of vacancies is: Vt = (1 − qt ) Vt−1 + nt . (56) Finally, total vacancy costs are given by: vt = Z AV t SC dF (SC) . (57) 0 In the rest of the paper we will refer to this variant of the model as being the FR-model. 2.4.3 Value of a job for the worker The household optimization program discussed in section 2.2 can be recast in terms of a value function WtH . In this alternative setup, the household’s optimization program can be written as a Bellmann equation: WtH χνm = max log (Ct − e Ct−1 ) + 1 − νm n µ Mt Pt ¶1−νm £ H ¤o + β Et Wt+1 , (58) implying the optimality conditions detailed in section 2.2. The value of WtH is a function of all state variables: WtH µ ¶ Mt ∗ ∗ =W Nt (wt ), Nt (wt−1 ), ... , Nt (wt ), Nt (wt−1 ), ... , . Pt H (59) H Let AH t (xt−j ) = ∂Wt /∂Nt (xt−j ) denote the marginal utility value at time t of a job whose ∗ or wt−j ) and has never wage was fixed at time t − j (with j ≥ 0) at a value xt−j (either wt−j 15 since been renegotiated. From equation (58) and the envelope theorem, one obtains: µ ¶ (1 + π̄)j Pt−j H h At (xt−j ) = UCt (xt−j − bt−j ) + Ct Pt i h ∗ + β (1 − s) (1 − ξw ) − β (1 − κ) pt AH t+1 (wt+1 ) (60) H − β κ pt AH t+1 (wt+1 ) + β (1 − s) ξw At+1 (xt−j ) . We assume that, as wages, the unemployment benefit is indexed on long-run inflation7 . Because of its impact on the outcome of the wage negotiation, we are most interested in the marginal value of a job whose wage is currently renegotiated. By combining the above expressions, this marginal value can be shown to be equal to: ∗ AH t (wt ) = ∞ h X β (1 − s) ξw j=0 ∞ X + − j=0 ∞ X j=0 ij UCt+j h β (1 − s) ξw ij h h β (1 − s) ξw ij h µ (1 + π̄)j Pt h (wt∗ − bt ) + Ct+j Pt+j ¶ i ∗ β (1 − s) (1 − ξw ) − β (1 − κ) pt+j AH t+1+j (wt+1+j ) (61) i β κ pt+j AH t+1+j (wt+1+j ) . The value of the first part of the first summation term appearing on the left-hand side of (61) has already been defined (see definition of St1 in (44)). We furthermore define: Stc = ∞ h X β (1 − s) ξw j=0 ij h UCt+j Ct+j i h c . = UCt Cth + β (1 − s) ξw St+1 ∗ Introducing St1 , Stc and next subtracting β (1 − s) ξw AH t+1 (wt+1 ) from both sides of (61) yields after rearrangements: ª © 1 ∗ c ∗ 1 ∗ AH t (wt ) = St (wt − bt ) − β (1 − s) ξw St+1 (wt+1 − bt+1 ) + St i h ∗ H + β 1 − s − (1 − κ) pt AH t+1 (wt+1 ) − β κ pt At+1 (wt+1 ) . 7 It means the real value of an unemployment benefit decreases in time of high inflation. 16 (62) The marginal utility value of a new job paid at the average wage wt can be obtained in a similar fashion. Starting from: AH t (wt ) = ∞ h X β (1 − s) ξw j=0 + − ∞ h X j=0 ∞ X j=0 h ij UCt+j µ (1 + π̄)j Pt h (wt − bt ) + Ct+j Pt+j ¶ β (1 − s) ξw ij h i ∗ β (1 − s) (1 − ξw ) − β (1 − κ) pt+j AH t+1+j (wt+1+j ) β (1 − s) ξw ij h i β κ pt+j AH t+1+j (wt+1+j ) , (63) one obtains: ª © 1 c 1 AH t (wt ) = St (wt − bt ) − β (1 − s) ξw St+1 (wt+1 − bt+1 ) + St i h ∗ + β (1 − s)(1 − ξw ) − (1 − κ) pt AH t+1 (wt+1 ) h i + β (1 − s) ξw − κ pt AH t+1 (wt+1 ) . 2.4.4 (64) Wage Determination Let parameter ψ measure the individual worker’s bargaining power. The bargained wage comes from the maximization problem: £ H ∗ ¤ψ £ F ∗ ¤1−ψ max At (wt ) At (wt ) , ∗ (65) wt F ∗ F ∗ ∗ H ∗ where AH t (wt ) = At (wt )/UCt and At (wt ) = At (wt )/UCt denote the asset value (measured in units of the final goods) of a job, calculated from the worker’s and the firm ’s point of view respectively. The first-order optimality condition implies the following sharing rule: ∗ F ∗ (1 − ψ) AH t (wt ) = ψ At (wt ). (66) The economy wide average wage wt satisfies: h h i i 1 + π̄ Nt wt = (1 − s) Nt−1 ξw wt−1 + (1 − ξw ) wt∗ + Ht−1 κ wt + (1 − κ) wt∗ , 1 + πt (67) where Ht is the number of new jobs (hirings) created at time t and Nt = (1 − s) Nt−1 + Ht−1 . In the particular case where κ = 1 (all new jobs have wage equal to the average wage), one obtains: wt = (1 − γ) 1 + π̄ wt−1 + γ wt∗ . 1 + πt 17 2.5 Monetary Policy and Government Consumption The interest rate is determined by a reaction function that describes monetary policy decisions: 1 + Rt = ft (1 + Rt−1 )0.9 " 1 + π̄ β µ 1 + πt 1 + π̄ ¶1.5 #0.1 , (68) where ft is an exogenous monetary policy shock. In this simplified Taylor rule, monetary authorities respond to deviation of inflation from its objective π̄. We also keep the simplest possible representation for government consumption: we assume no non-monetary public debt. Government expenditures are thus tax and/or monetary financed. Public consumption is exogenously determined. The government flow budget constraint is: ∆Mt + Tt , Pt Mt πt Mt−1 = ∆ + + Tt , Pt 1 + πt Pt−1 gt Yt = (69) where gt is an exogenous government consumption shock. The government chooses Tt so as to satisfy its budget constraint. 2.6 Exogenous shocks To close the model, we need to precise equations governing the monetary, government consumption and productivity shocks: ft = (1 − ρf ) f¯ + ρf ft−1 − vtf , (70) gt = (1 − ρg ) ḡ + ρg gt−1 + vtg , (71) ǫt = (1 − ρe ) ǭ + ρe ǫt−1 + vte . (72) 3 3.1 Results Calibration Production function As Gertler and Trigari (2006), we choose a monthly calibration. As usual, we have an annual 18 capital depreciation rate of 10% (δ = 10%/12) and an elasticity of production with respect to capital α = 1/3. Labor market Parameters related to the labor market are identical to Gertler and Trigari (2006). Since there is no strong evidence on the bargaining power, we assign equal power to both workers and firms (ψ = 0.5). And as usual, the worker bargaining power is equal to the match elasticity to unemployment (ψ = 1 − φ). The separation rate s = 0.035 is standard and supported by strong empirical evidences. The unemployment benefit is supposed constant bt = b̄ and we choose this replacement ratio b̄/w to be 0.4. We impose two restrictions: both the job finding rate and vacancy filling rate must be 0.45 at the steady state. These restrictions yield the values h̄ = 0.45 and a1 = 1.63 (MP-model). Parameters for the GT- and FR- models are derived to keep the same steady state. More precisely, in the GT-model, a2 H/N = a1 /q. In the FR-model, we define F (SC) = SC/γ and we choose γ to reproduce the same level of vacancies8 . Preferences and interest rate We use the results of Smets and Wouters (2003) to calibrate the utility function9 . We set the habit formation parameter e = 0.85, the money demand parameters νm = 5 and χ = 1.98. Setting β = 0.997 implies an annual real interest rate of 3.7% ∼ = (1/0.997)12 − 1. We assume an annual inflation of 2%, which gives π̄ = 2%/12. Utilization rates and investment cost We suppose a quadratic overtime income (c1 = 1) and we choose c0 = θd to normalize normal working time to 1. The productivity of hours is concave and θ = 0.5. Similarly, we suppose a quadratic capital utilization cost (d1 = 1) and we choose d0 = r + δ to normalize capital utilization rate to 1. Finally, as in Smets and Wouters (2003), the investment cost is ϕ = 12. Nominal rigidities Most of the these parameter values are borrowed from Smets and Wouters (2003). The elasticity of substitution between intermediate goods is 10, the average duration of a price contract is slightly more than two years whereas the average duration of a wage contract is less than one year. More precisely, we have λx = 0.9, ξp = 0.962 and ξw = 0.888. Moreover, we assume that 8 N β A It is easy to show that γ = V (1−β(1−q)) . 9 The quarterly parameters are transformed in monthly ones. 19 the probability to bargain a new wage is the same that the probability to bargain an old wage, that is κ = ξw . Shocks We use conventional values for all these parameters. The monetary policy shock is centered around 1 and has no persistence , the government consumption shock is centered around 0.15 (government expenditures represent 15% of GDP) and has an autoregressive parameter ρg = 0.901/3 , and the productivity shock is centered around one and has an autoregressive parameter ρe = 0.951/3 . Finally, we define the innovation vector [vtf , vtg , vte ] ∼ N (0, Σ), where Σ11 = ′ (0.005/12)2 , Σ22 = Σ33 = 0.0052 and Σij = 0. 3.2 Simulations We examine the behavior of the model taking the technology shock as the exogenous driving force. Then we look successively at the model responses to an interest rate shock and to the three shocks (productivity, monetary, government) all together. 3.2.1 Productivity shock Base model with only real wage rigidities Gertler and Trigari (2006) modify the standard Mortensen-Pissarides framework, by allowing for staggered multi-period wage contracts and by dropping the assumption of a fixed vacancy cost. They show that their model can reasonably well reproduce the cyclical behavior of the labor market observed in the data. The first point we want to examine is how our model behaves with respect to their. For this, we remove all the frictions (except the frictions in the labor market) in our model to get something similar to Gertler and Trigari (2006): e = 0 (no consumption habit), c1 → ∞ (no overtime work), d1 → ∞ (no variable capital utilization rate), ϕ = 0 (no capital adjustment cost) and ξp = 0 (flexible prices). For the rest, (labor market block) we adopt a similar calibration (see previous sub-section). Table 1 shows the relative standard deviation, the contemporaneous correlation with output and the serial autocorrelation for the key - labor market - variables. The statistics reported are for US data (taken from Gertler and Trigari (2006)), the original Gertler and Trigari (2006) model - GT (2006) hereafter -, and our 20 model with the three different types of free entry condition described above: the MP-, GT-, and FR-models. Firstly, we see that we have small differences between the GT (2006) original results and our model with a similar type of vacancy cost (GT-model). They are explained by remaining differences between the two approaches, as for instance the specific way to introduce the variable vacancy costs. Secondly, it is well known that wages bargained every period lead to too highly volatile and procyclical wages (see for instance Shimer (2004)). Here, because of the staggered wage setting, all models are able to reproduce a realistic wage dynamics. However, the standard matching MP-model (with a constant vacancy opening cost) fails to reproduce the volatility and the autocorrelation for the main labor market variables (employment, vacancies and tensions). The poor performance of this variant of the model can be explained by the rapid adjustment of vacancies following a shock. By introducing an opening cost proportional to the hiring rate (GT-model) or a sunk cost (FR-model), we allow vacancies to adjust sluggishly and we increase the standard deviation and the persistence. Overall, the GT- and especially the FR-model (with wage rigidities and specific entry costs) do well in capturing the basic features of the data. In particular, the FR-model variant performs as well the original GT (2006) model. It produces less realistic volatilities but performs better in reproducing data correlations and persistences. Complete model with nominal rigidities We now conduct the same simulations but with the complete model including the other frictions and nominal rigidities. But rather than comparing our model (the three versions) to the original GT (2006), we compare it to a more standard DSGE model with monopolistic competition on both the goods and the labor market, denoted MC-model hereafter.10 The calibration is identical for the MC-model and the three versions of our matching model. The results are displayed in 10 See for instance Smets and Wouters (2003) or Christiano, Eichenbaum, and Evans (2005)) 21 Table 211 . Monopolistic competition models are successful at explaining a number of phenomena, but suffer from some shortcomings related to their simplified representation of the working of the labor market. More precisely: (i) there is no explicit discussion of equilibrium unemployment fluctuations, (ii) there is no distinction between employment and hours of work change, and (iii) they do not generate enough persistence in employment. The matching model generates unemployment, makes an explicit distinction between employment and hours (extensive vs. intensive margins) and, since it incorporates a sluggish labor reallocation process, generates a higher employment persistence. However, the matching model with standard constant vacancy costs (MP-model) is unable to amplify the productivity shock and still generates a too low volatility for main variables (at the exception of total hours). As explained before, alternative vacancy costs (GT- and FR-models) allow to strongly improve the results. Further insights into the differences between models are given in Figures 1 and 2. The employment reaction depends on the way we write the vacancy costs. In the standard MP model, the amplification is quite weak. When we remove the usual free entry condition (GT- and FRmodels), we increase the amplification as well as the persistence. In a monopolistic competition model, a positive productivity shock first decreases total hours. The matching models also give an initial decrease (due to the initial fall in hours, see Figure 312 ) but the decrease is less pronounced and the subsequent positive effect is more persistent. Globally, the FR-model does quite well in capturing the basic features of the data. However, comparing last columns of Table 1 and Table 2, we see that the FR model statistics are slightly better in the base model. Adding nominal rigidities and other frictions creates new interactions with the rest of the model and may deteriorate the statistics. 11 In the matching models, we are able to make the distinction between employment Nt and hours ht . Total hours are then defined as the product of employment and hours: ht Nt . We are also able to make the distinction between the base wage wt and the overtime compensation Cth . The hourly compensation is defined as the base wage plus the overtime pay, divided by hours: (wt + Cth )/ht . 12 Impulse responses with the FR model. But similar graphs would be generated with the MP or GT models. 22 3.2.2 Monetary shock The main motivation for introducing the additional nominal and real frictions is that it is interesting to consider a broader DSGE setup to investigate the implications for monetary policy. The standard framework for analyzing monetary policy is indeed the Smets and Wouters (2003) or the Christiano, Eichenbaum, and Evans (2005) type of models that provides a realistic picture for the aggregate demand reaction to monetary policy shocks. Christiano, Eichenbaum, and Evans (2005) and Trigari (2004) represent the dynamic responses of the US economy to an expansionary monetary policy shock. Their main findings are that: (i) output responds with a high persistence, (iii) individual hours do not react strongly, (iii) the response of total hours is even more persistent than output but with a lower amplitude, (iv) the reaction of inflation is lagged and highly persistent, (v) wages react very weakly, and (vi) the real interest rate declines sharply but quickly returns to its initial level. In Figure 4, we plot the IRF’s to a monetary shock for our different models13 . We see that the three models respond similarly to a monetary shock. The lower interest rate reduces savings and increase demand, which requires higher employment. Since employment cannot immediately respond in the MP- and FR-models, labor adjustment is first realized through hours. It is a promising result that the search models can do as well as standard models with monopolistic competition on the labor market, for the monetary transmission toward wages and prices (and at the same time give a more realistic description of labor market flows). However, both the MP- and FR-models still fail to reproduce the weak reaction of hours and the delayed reaction of inflation, as it is also the case for the MC-model. A complete summary of statistics is reported in Table 3. As already mentioned, all the models behave quite similarly. 3.2.3 All three shocks together In this section, we conduct the same simulation exercises as previously, but with the three shocks (productivity, monetary and government shocks) all together. Table 4 displays the results 13 We only consider the MC, MP and FR models. The GT model results are very close to the FR model. 23 and we see they are quite similar to those displayed in Table 2, stressing the importance of the productivity shock in cyclical fluctuations. Focusing now on the FR model, it is worth noting that the statistics are even slightly better in the setup with the three shocks than in the setup with only the productivity shock (volatilities deteriorate but correlation with output and persistence improve). This suggests FR is a promising model and that a complete estimation of this model could be of great interest. 4 Conclusion Our model performs well on reproducing the stylized facts of real wage and labor market variables both in its simple and extended version, although the last one needs further fine-tuning of the parameters. A complete estimation of the model will offer further insight on what frictions and what type of shocks are crucial for maximizing the explanatory power of the model. Additional data on labor market flow variables are needed to identify the exact nature of the vacancy costs. The use of hours worked and employment data is necessary to pin down the relative cost of labor adjustment along the intensive or extensive margin. Detailed data on wages are needed to evaluate whether the adjustment costs along the intensive margin are cyclical, for instance via the marginal utility of households, or not. Another remaining issue is the importance of cyclical employment adjustments via endogenous separation and on-the-job search behavior. All these issues might have an impact on the wage-price dynamics. In these labor-search models with ongoing employee-employer relations, the marginal cost appears as a complicated function of current and future contract wages as well as of overtime premiums and employment adjustment costs. As a result, the interaction between inflation and wages becomes more complicated and dependent on the labor market tightness. In our model, the price decision, the marginal cost and the labor cost formation are determined in three separate sectors: the retail firms, the wholesale production firms and the labor service firms. It is not yet clear whether and how the integration of these three sectors might affect further the connection between wage and price decisions. The work by Kuster (2006), where the search friction is integrated into the price setting sector, offers useful additional insight for the interaction between price and wage setting. More theoretical 24 and empirical work is needed to clarify all these issues. A major limitation for further empirical work is the availability and the quality of data on labor market flows and detailed wage costs. Especially within the euro area, the lack of statistical data impede further empirical research. 25 References Andolfatto, D. (1996): “Business Cycles and Labor-Market Search,” American Economic Review, 86(1), 112–132. Blanchard, O., and J. Gali (2005): “Real Wage Rigidities and the New Keynesian Model,” CEPR DP 5375. Bodart, V., O. Pierrard, and H. Sneessens (2005): “Calvo Wages in a Search Unemployment Model,” DNB Working Paper. Christiano, L., M. Eichenbaum, and C. Evans (2005): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113(1-45), 346– 353. Christoffel, K., K. Kuster, and T. Linzert (2006): “The Impact of Labor Markets on the Transmission of Monetary Policy in an Estimated DSGE Model,” ECB Working Paper 635. Christoffel, K., and T. Linzert (2004): “The Role of Real Wage Rigidity and Labor Market Frictions for Unemployment and Inflation Dynamics,” ECB Working Paper 556. Fujita, S., and G. Ramey (2005): “The Dynamic Beveridge Curve,” FRB Philadelphia Working Paper 05-22. Gertler, M., and A. Trigari (2006): “Unemployment Fluctuations with Staggered Nash Wage Bargaining,” Proceedings FRB San Francisco. Goodfriend, M., and R. King (2001): “The Case for Price Stability,” NBER DP 8423. Hall, R. (2005): “Job Loss, Job Finding, and Unemployment in the U.S. Economy over the Past Fifty Years,” Paper prepared for the NBER Macro Annual Conference, April 2005. Krause, M., and T. Lubik (2005): “On-the-Job Search and the Cyclical Dynamics of the Labor Market,” mimeo. Kuster, K. (2006): “Real Prices and Wage Rigidities in a Model with Matching Frictions,” mimeo. 26 Merz, M. (1995): “Search in the Labor Market and the Real Business Cycle,” Journal of Monetary Economics, 36, 269–300. Mortensen, D., and C. Pissarides (1999): “Job reallocation, Employment Fluctuations and Unemployment,” in Handbook of Macroeconomics, ed. by J. Taylor, and M. Woodford, vol. 1, chap. 18, pp. 1171–1228. Amsterdam: North-Holland. Moyen, S., and J. Sahuc (2005): “Incorporating Labour Market Frictions into an OptimisingBased Monetary Policy Model,” Economic Modelling, 22, 159–186. Shimer, R. (2002): “The Cyclical Behavior of Equilibrium Unemployment and Vacancies, and Wages: Evidence and Theory,” mimeo. (2004): “The Consequences of Rigid Wages in Search Models,” Journal of the European Economic Association, 2, 469–479. Smets, F., and R. Wouters (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1(5), 1123–1175. Trigari, A. (2004): “Equilibrium Unemployment, Job Flows, and Inflation Dynamics,” ECB Working Paper 304. Walsh, C. (2005): “Labor Market Search, Sticky Prices, and Interest Rate Policies,” Review of Economic Dynamics, 8, 829–849. 27 4.1 Appendix 1: Summary of the main equations If we define mt ≡ Mt /Pt , P̄t ≡ P̄t /Pt and Pt∗ ≡ Pt∗ /Pt , then our final model is: Households i d0 h 1+d1 −1 , zt 1 + d1 ¶ µ ϕ ∆It 2 , 2 It−1 n o = 1 − Φ(∆It /It−1 ) It − δ Kt−1 , Ψ(zt ) = µ ¶ ∆It Φ = It−1 (73) ∆Kt (75) Yt = It + Ct + gt Yt + vt + ψ(zt )Kt−1 , UCt = β (1 + rt ) UCt+1 , (74) (76) (77) rtk + δ = d0 ztc1 , ¶¸ ( µ ¶ ) · µ pkt+1 ∆It+1 It+1 2 ∆It k ∆It It k − pt ϕ − ϕ 1 = pt 1 − Φ It−1 It−1 It−1 1 + rt+1 It It o n k pkt = zt+1 (rt+1 + δ) − Ψ(zt+1 ) + (1 − δ) pkt+1 /(1 + rt ), ¶−1/νm µ Rt (UCt )−1/νm , mt = χ 1 + Rt 1 , UCt = Ct − eCt−1 (78) 1 + Rt = (1 + rt )(1 + πt+1 ) . (83) 28 (79) (80) (81) (82) Goods Producers ¢ c0 ¡ 1+c1 h −1 , 1 + c1 dt + Cth , = θ ht " #−1 (rtk + δ)/α ¡ ¢ = , dt + Cth /(1 − α) µ ¶α ¶1−α µ k dt + Cth 1 rt + δ , = ǫt (1 − α) hθt α Cth = c0 hct 1 zt Kt−1 Nt Λxt Et1 = λx Pt∗ Et2 , µ ¶ −1 1−λx 1 + π̄ 1 Et+1 , 1 + πt+1 µ ¶ −λx 1−λx 1 + π̄ 2 2 Et = Yt UCt + ξp β Et+1 , 1 + πt+1 · ¸−1/(1−λx ) h i−1/(1−λx ) 1 + π̄ −1/(1−λx ) ∗ + ξp P̄t = (1 − ξp ) Pt P̄t−1 , 1 + πt · ¸ h i−λx /(1−λx ) 1 + π̄ −λx /(1−λx ) ∗ 1 = (1 − ξp ) Pt + ξp , 1 + πt Et1 = Yt UCt Λxt + ξp β (84) (85) (86) (87) (88) (89) (90) (91) (92) 1 Yt = P̄t1−λx ǫt (zt Kt−1 )α Nt1−α . (93) 29 Labor Services h i 1 + π̄ St1 = UCt + β(1 − s)ξw S1 , 1 + πt+1 t+1 h i d Std = dt UCt + β(1 − s)ξw St+1 , i h i h ∗ ∗ 1 d ), wt+1 − St+1 + β(1 − s)AFt+1 (wt+1 AFt (wt∗ ) = Std − St1 wt∗ − β(1 − s)ξw St+1 i h i h d 1 − St+1 wt+1 + β(1 − s)AFt+1 (wt+1 ), AFt (wt ) = Std − St1 wt − β(1 − s)ξw St+1 = (1 − κ)AFt (wt∗ ) + κAFt (wt ), i h c , Stc = UCt Cth + β (1 − s) ξw St+1 © 1 ∗ ª ∗ 1 ∗ AH St (wt − bt ) − β(1 − s)ξw St+1 (wt+1 − bt+1 ) + Stc t (wt ) = i h ∗ H +β 1 − s − (1 − κ)pt AH t+1 (wt+1 ) − βκpt At+1 (wt+1 ), © 1 ª 1 AH St (wt − bt ) − β(1 − s)ξw St+1 (wt+1 − bt+1 ) + Stc t (wt ) = i h ∗ +β (1 − s)(1 − ξw ) − (1 − κ)pt AH t+1 (wt+1 ) h i +β (1 − s)ξw − κpt AH t+1 (wt+1 ), AN t UCt AVt M-P version: G-T version: F-R version: = −ct + qt AVt+1 AN t+1 + (1 − qt ) , 1 + rt+1 1 + rt+1 ct = a1 ; AVt = 0 ; vt = ct Vt , (94) (95) (96) (97) (98) (99) (100) (101) (102) (103) (104) (105) (106) Ht ; AVt = 0 ; vt = ct Vt , Nt Z AVt dF (SC) ; ct = 0 ; Vt = (1 − qt−1 )Vt−1 + ct = a2 qt (107) (108) 0 vt = Z AV t (109) SC dF (SC). 0 30 Wage determination and labor flows ∗ ψAFt (wt∗ ) = (1 − ψ)AH t (wt ), h 1 + π̄ i h i Nt wt = (1 − s)Nt−1 ξw wt−1 + (1 − ξw )wt∗ + Ht−1 κwt + (1 − κ)wt∗ , 1 + πt (110) (111) Ht = h̄Vtφ (1 − Nt )1−φ , (112) Nt = (1 − s)Nt−1 + Ht−1 , (113) qt = pt = Ht , Vt Ht . 1 − Nt (114) (115) Policies and exogenous shocks 0.9 1 + Rt = ft (1 + Rt−1 ) gt Yt = mt − mt−1 + " 1 + π̄ β µ 1 + πt 1 + π̄ πt mt−1 + Tt , 1 + πt ¶1.5 #0.1 , (116) (117) ft = (1 − ρf ) f¯ + ρf ft−1 − vtf , (118) gt = (1 − ρg )Ḡ + ρg gt−1 + vtg , (119) ǫt = (1 − ρe )ǭ + ρe ǫt−1 + vte . (120) 31 three types of free entry conditions US Data GT (2006) MP GT FR relative standard deviation output 1.00 1.00 1.00 1.00 1.00 wage 0.52 0.56 0.56 0.61 0.59 labor share 0.51 0.57 0.51 0.49 0.63 employment 0.60 0.35 0.24 0.44 0.32 unemployment 5.15 4.46 3.12 5.63 4.11 vacancies 6.30 5.83 4.83 7.48 5.25 tensions 11.28 9.88 7.61 12.80 9.20 productivity 0.61 0.71 0.80 0.65 0.76 contemporaneous correlation with output output 1.00 1.00 1.00 1.00 1.00 wage 0.56 0.60 0.70 0.70 0.67 labor share -0.20 -0.56 -0.78 -0.39 -0.54 employment 0.78 0.77 0.87 0.88 0.83 - 0.86 -0.77 -0.87 -0.90 -0.87 vacancies 0.91 0.91 0.64 0.84 0.91 tensions 0.90 0.94 0.76 0.89 0.91 productivity 0.71 0.97 0.99 0.95 0.97 output 0.87 0.84 0.79 0.84 0.84 wage 0.91 0.95 0.93 0.94 0.95 labor share 0.73 0.65 0.61 0.58 0.67 employment 0.94 0.90 0.74 0.88 0.91 unemployment 0.91 0.90 0.74 0.88 0.91 unemployment serial correlation vacancies 0.91 0.83 0.45 0.80 0.88 tensions 0.91 0.88 0.60 0.85 0.90 productivity 0.79 0.76 0.79 0.75 0.76 US Data: taken from Gertler and Trigari (2006), GT (06): simulation results directly taken from Gertler and Trigari (2006), MP: own simulations with free entry condition à la Mortensen and Pissarides (1999), GT: own simulations with free entry condition à la Gertler and Trigari (2006), FR: own simulations with free entry condition à la Fujita and Ramey (2005). Table 1: Productivity shock: summary of the base model statistics 32 three types of free entry conditions US Data MC MP GT FR 1.00 1.00 1.00 1.00 0.26 0.29 0.25 relative standard deviation output 1.00 base wage hourly compensation 0.52 0.41 0.28 0.31 0.27 labor share 0.51 0.70 0.45 0.52 0.54 0.22 0.42 0.37 employment total hours 0.60 0.74 0.79 0.79 unemployment 5.15 0.90 2.89 5.35 4.77 vacancies 6.30 3.64 6.58 5.83 tensions 11.28 6.43 11.80 10.49 productivity 0.61 0.57 0.50 0.51 1.00 1.00 1.00 1.00 0.85 0.80 0.72 contemporaneous correlation with output output 1.00 1.00 base wage hourly compensation 0.56 0.90 0.99 0.85 0.81 labor share -0.20 0.01 -0.26 -0.09 -0.20 0.99 0.98 0.95 0.82 0.88 0.86 employment total hours 0.78 0.72 unemployment -0.86 -0.99 -0.98 -0.95 vacancies 0.91 0.89 0.98 0.99 tensions 0.90 0.94 0.99 0.98 productivity 0.71 0.49 0.69 0.61 0.63 0.87 0.94 0.94 0.94 0.94 0.93 0.96 0.97 hourly compensation 0.91 0.96 0.94 0.96 0.96 labor share 0.73 0.71 0.66 0.73 0.76 0.93 0.95 0.95 0.86 0.89 0.89 0.93 0.95 0.95 serial correlation output base wage employment total hours 0.94 unemployment 0.91 0.75 vacancies 0.91 0.87 0.93 0.94 tensions 0.91 0.91 0.94 0.95 productivity 0.79 0.77 0.73 0.76 0.73 Same definitions as in table 1, MC: monopolistic competition model à la Smets and Wouters (2003). Table 2: Productivity shock: summary of the full model statistics 33 Employment 0.45 MP GT FR 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 Figure 1: Impulse response functions to an aggregate productivity shock Total Hours 0.8 0.6 0.4 0.2 0 −0.2 −0.4 MP GT FR MC −0.6 −0.8 0 10 20 30 40 50 60 70 80 90 100 Figure 2: Impulse response functions to an aggregate productivity shock FR model 0.4 0.3 0.2 0.1 0 −0.1 employment overtime base wage −0.2 −0.3 −0.4 0 50 100 150 200 Figure 3: Impulse response functions to an aggregate productivity shock 34 GDP Employment 0.6 0.2 0.4 0.15 0.2 0.1 0 0.05 −0.2 0 20 40 Total Hours 0 60 0 20 40 Nominal Hourly Compensation 60 0 20 40 Yearly Real Interest Rate 60 0.2 0.4 0.15 0.2 0.1 0.05 0 0 20 40 Yearly Inflation Rate 0 60 0.15 0.2 0.1 0 0.05 −0.2 0 −0.4 −0.05 0 20 40 −0.6 60 MP FR MC 0 20 Figure 4: Impulse response functions to a monetary shock 35 40 60 three types of free entry conditions US Data MC MP GT FR 1.00 1.00 1.00 1.00 0.15 0.33 0.23 relative standard deviation output 1.00 base wage hourly compensation 0.52 0.24 0.18 0.34 0.23 labor share 0.51 0.37 0.25 0.42 0.28 0.40 0.66 0.52 employment total hours 0.60 1.07 1.11 1.08 unemployment 5.15 1.16 5.18 8.45 6.63 vacancies 6.30 7.63 11.20 8.49 tensions 11.28 12.33 19.21 14.85 productivity 0.61 0.08 0.12 0.09 1.00 1.00 1.00 0.98 0.66 0.22 0.17 contemporaneous correlation with output output 1.00 1.00 base wage hourly compensation 0.56 0.81 0.99 0.70 0.35 labor share -0.20 0.97 0.99 0.82 0.57 0.97 0.98 0.91 1.00 1.00 1.00 employment total hours 0.78 1.00 unemployment -0.86 -0.97 -0.98 -0.91 vacancies 0.91 0.74 0.95 0.99 tensions 0.90 0.87 0.98 0.98 productivity 0.71 -0.94 -0.88 -0.91 -0.87 0.87 0.88 0.85 0.85 0.86 0.87 0.94 0.94 hourly compensation 0.91 0.93 0.87 0.94 0.95 labor share 0.71 0.91 0.86 0.93 0.95 0.79 0.88 0.91 0.85 0.86 0.87 0.79 0.88 0.91 serial correlation output base wage employment total hours 0.94 unemployment 0.91 0.88 vacancies 0.91 0.56 0.81 0.88 tensions 0.91 0.68 0.85 0.90 productivity 0.79 0.83 0.89 0.92 0.88 Same definitions as in table 1, MC: monopolistic competition model à la Smets and Wouters (2003). Table 3: Monetary shock: summary of the full model statistics 36 three types of free entry conditions US Data MC MP GT FR 1.00 1.00 1.00 1.00 0.21 0.26 0.22 relative standard deviation output 1.00 base wage hourly compensation 0.52 0.31 0.23 0.28 0.23 labor share 0.51 0.51 0.36 0.45 0.45 0.24 0.43 0.37 employment total hours 0.60 0.85 0.88 0.88 unemployment 5.15 1.02 3.14 5.59 4.72 vacancies 6.30 4.34 7.09 5.85 tensions 11.28 7.26 12.48 10.44 productivity 0.61 0.45 0.41 0.43 1.00 1.00 1.00 0.84 0.73 0.56 0.51 contemporaneous correlation with output output 1.00 1.00 base wage hourly compensation 0.56 0.76 0.88 0.76 0.63 labor share -0.20 0.25 -0.12 0.04 -0.12 0.96 0.94 0.88 0.89 0.92 0.91 employment total hours 0.78 0.88 unemployment -0.86 -0.96 -0.94 -0.88 vacancies 0.91 0.84 0.96 0.96 tensions 0.90 0.92 0.97 0.93 productivity 0.71 0.20 0.50 0.47 0.50 0.87 0.85 0.86 0.87 0.88 0.93 0.96 0.96 hourly compensation 0.91 0.96 0.93 0.96 0.96 labor share 0.73 0.72 0.66 0.76 0.76 0.85 0.92 0.94 0.79 0.83 0.83 0.85 0.92 0.94 serial correlation output base wage employment total hours 0.94 unemployment 0.91 0.76 vacancies 0.91 0.67 0.88 0.92 tensions 0.91 0.77 0.90 0.93 productivity 0.79 0.76 0.74 0.76 0.73 Same definitions as in table 1, MC: monopolistic competition model à la Smets and Wouters (2003). Table 4: All three shocks together: summary of the full model statistics 37