Potential Output, the Output Gap, and the Labor Wedge

Transcription

1 Potential Output, the Output Gap, and the Labor Wedge Luca Sala Ulf Söderström Antonella Trigari June 21 Preliminary and incomplete Abstract We estimate potential output and the output gap in a quantitative business cycle model with nominal price and wage rigidities. We establish three closely related results. First, we show that estimates of potential output and the output gap are highly sensitive to the interpretation of the labor market shocks in the estimated model: the estimated output gap is strongly procyclical under one interpretation of the shocks but essentially acyclical under an alternative interpretation. Second, we demonstrate that the output gap is closely related to the labor wedge (the wedge between households marginal rate of substitution and firms product of labor), and we show how a decomposition of the labor wedge can help interpret the output gap estimates. Third, we show that the model does not provide a satisfactory explanation of the joint dynamics of hours and wages: fluctuations in hours are essentially exogenous, and the endogenous markup of the real wage over the marginal rate of substitution (that is due to wage rigidities) is acyclical. Keywords: Business cycles, Efficiency, Monetary Policy, Labor markets, Model misspecification. JEL Classification: E32, E52, E24, C11. Sala and Trigari: Department of Economics and IGIER, Università Bocconi, Milan, Italy, luca.sala@unibocconi.it, antonella.trigari@unibocconi.it; Söderström: Research Division, Monetary Policy Department, Sveriges Riksbank, Stockholm, Sweden, and CEPR, ulf.soderstrom@riksbank.se. We have benefited from discussions with and comments from Ferre De Graeve, Simona Delle Chiaie, Michael Evers, Jordi Galí, Tommaso Monacelli, Lars Svensson, Anders Vredin, and Karl Walentin. We also thank seminar participants at Sveriges Riksbank, Norges Bank, the Norwegian School of Management, Oxford University, University College London, ESSIM 29, the 8th Macroeconomic Policy Research Workshop on DSGE Models at Magyar Nemzeti Bank, the Royal Economic Society 21 Annual Conference, and the Konstanz Seminar on Monetary Theory and Policy 21. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank.

2 1 Introduction To what extent are business cycle fluctuations efficient? And how should economic policy respond to such fluctuations? These are classic questions in macroeconomic, and different schools of thought have provided very different answers. Traditional Keynesian theory seemed to imply that all business cycle fluctuations are inefficient and therefore should be counteracted by economic policy. 1 In contrast, real business cycle theory (for example, Kydland and Prescott (1982); Long and Plosser (1983); King, Plosser, and Rebelo (1988)) suggested that a large part of fluctuations may well be driven by the efficient responses of firms and households to exogenous shifts in technology and preferences, reducing the scope for economic policymakers to dampen business cycle fluctuations, Modern monetary business cycle models (starting with Yun (1996); Goodfriend and King (1997); and Rotemberg and Woodford (1997)) extend the real business cycle framework to include imperfect competition and nominal rigidities. These models typically imply that economic policy should counteract fluctuations due to nominal rigidities, but accommodate fluctuations due to shifts in real factors (for example, technology and preferences). In other words, optimal policy should act to make the real economy mimic the RBC model at the core of the monetary model (Goodfriend and King (1997)). Recent developments of the monetary business cycle model have demonstrated that medium-sized versions of the model that incorporate more frictions and shocks are able to well match the behavior of aggregate macroeconomic variables (Smets and Wouters (27)). These models are therefore potentially useful to estimate the degree to which business cycle fluctuations are efficient and advice policymakers about the appropriate response to such fluctuations. Much work has therefore been aimed at estimating potential output within this class of models. Potential output is then defined as the level of output that would appear in an allocation without nominal rigidities and where the degree of imperfect competition is constant. 2 This is also the level of output towards which optimal monetary policy should try to steer the economy in order to maximize household welfare. 3 Recently, Sala, Söderström, and Trigari (28) and Justiniano and Primiceri (28) have shown that potential output implied by a typical estimated business cycle model is similar to a smooth measure of trend output, and that the output gap (the percent deviation of output from potential) is closely related to traditional measures of the business cycle. 4 In this paper, we use an empirical structural business cycle model to estimate potential GDP and the output gap in the U.S. economy. We focus on the extent to which these 1 [Add references.] 2 The potential level of output is thus at a constant difference from the efficient level of output, which is the allocation with perfect competition, the distance being determined by average wage and price markups. The actual level of output is affected also by exogenous shocks to price and wage markups that affect the degree of imperfect competition. Kiley (21) gives an overview of different definitions and uses of potential output. 3 See, for instance, Neiss and Nelson (23), (25), Edge, Kiley, and Laforte (28), Sala, Söderström, and Trigari (28), Justiniano and Primiceri (28), Basu and Fernald (29), or Coenen, Smets, and Vetlov (29). 4 Other authors have obtained estimates of the output gap that are more volatile and less closely related to the business cycle (see, for instance, Walsh s (25) discussion of Levin, Onatski, Williams, and Williams (25)). 1

3 estimates are sensitive to alternative structural interpretations of the model, in particular the interpretation of structural shocks. The model implies that shocks to the disutility of supplying labor are observationally equivalent to shocks to the markup of wages over the households marginal rate of substitution between consumption and leisure. We show that our estimates are highly sensitive to the structural interpretation of these shocks: the estimated output gap is strongly procyclical under one interpretation of the labor market shocks but essentially acyclical under the alternative interpretation. This is true even when we allow for measurement errors in the real wage and inflation, following Justiniano and Primiceri (28). Even though the measurement errors remove much of the high-frequency volatility in the shocks, the low- and medium-frequency volatility is still important. As these different output gap estimates have very different implications for monetary policy and welfare, our results suggest that models in this class are not entirely reliable to answer normative questions. We then move on to interpret our estimates. Over business cycle frequencies, the inefficient fluctuations in output should be closely related to inefficiencies in the allocation of labor. We therefore study how our estimated output gaps compare with a direct measure of labor market inefficiency: the wedge between households marginal rate of substitution and firms marginal product of labor. This labor wedge is exactly proportional to the output gap in a simple version of our model without capital. We show that the wedge and the gap are approximately proportional also in our larger empirical model with capital accumulation. To better interpret our estimates of the output gap, we therefore study how different shocks and frictions directly or indirectly determine the labor wedge. The existing literature has studied the labor wedge in simple versions of the standard business cycle model (see, for instance, Hall (1997), Chari, Kehoe, and McGrattan (27), Galí, Gertler, and López-Salido (27), and Shimer (29)). Their measure of the labor wedge is essentially determined by the labor input, hours worked. We define a fundamental wedge in our model and show that this wedge is very similar to the wedge in the simple model. As a consequence, it is also closely related to hours. In principle, this fundamental wedge can be determined by endogenous price and wage markups, exogenous markup shocks, or exogenous shocks to household preferences. Galí, Gertler, and López-Salido (27) show that the wedge is mainly determined by the markup of wages over the marginal rate of substitution, and provide some indirect evidence that the wedge is endogenous. We are able to provide more direct evidence, and we show that the fundamental wedge (and therefore the wage markup) is largely exogenous in our model, driven by labor market shocks. The endogenous component of the fundamental wedge, given by endogenous movements in the wage markup, is essentially acyclical. Under one interpretation of the labor market shocks, the total wage markup is countercyclical, but this is entirely due to exogenous wage markup shocks. The importance of the exogenous labor market shocks also implies that their interpretation matters for normative issues. We conclude that the model does not provide a satisfactory explanation of the joint dynamics of hours and wages: fluctuations in hours are essentially exogenous, and the endogenous markup of the real wage over the marginal rate of substitution (that is due to wage rigidities) is acyclical. This failure of the neoclassical model to reconcile the behavior of hours and the real wage is an old issue in macroeconomics, going back at least to Hall (198), Altonji (1982), and 2

4 Mankiw, Rotemberg, and Summers (1985). Our results show that the issue is still not resolved in modern business cycle models, casting doubts on their usefulness to study business cycle dynamics. A separate problem originates in the identification of labor disutility shocks and wage markup shocks. Galí, Smets, and Wouters (21) approach this identification problem by reinterpreting the model, showing that the rate of unemployment is proportional to the wage markup. Their approach solves the identification problem, but the larger problem remains: in their estimated model most variations in unemployment are due to the direct impact of exogenous markup shocks rather than the endogenous propagation of other shocks. In future work we will approach these two problems using a model with search and matching frictions in the labor market and nominal price and wage rigidities, based on Gertler, Sala, and Trigari (28). A version of that model with an intensive and an extensive margin of labor supply solves the identification problem, although through mechanisms very different from Galí, Smets, and Wouters (21). Furthermore, in that model the intensive margin is efficient but search frictions may create inefficiencies in the extensive margin, due to inefficient job creation. At the same time, the extensive margin is not subject to the Barro (1977) critique: once firms and workers have met, their decision to stay together or separate is efficient as long as the wage stays within the bargaining set. This model therefore has the potential of giving a different account of the dynamics of the labor input, the labor wedge, and the output gap. The paper is organized as follows. We begin by presenting our model framework in Section 2. In Section 3 we show how the output gap and the labor wedge are related in the theoretical model. We then move on to estimate the model in Section 4. Section 5 provides our main results. We examine the robustness of our results by analyzing some alternative specifications of the empirical model in Section 6. Finally, we offer some concluding remarks in Section 7. 2 The model economy Our model is a monetary Dynamic Stochastic General Equilibrium (DSGE) framework, and is similar to many models used in the literature. This particular specification builds closely on Smets and Wouters (27) and Justiniano, Primiceri, and Tambalotti (21). The model consists of five sectors: households, a competitive final goods sector, a monopolistically competitive intermediate goods sector, employment agencies, and a government sector. The model features habit formation, investment adjustment costs, variable capital utilization, monopolistic competition in goods and labor markets, and nominal price and wage rigidities. The model also includes growth in the form of a non-stationary technology shock, as in Altig, Christiano, Eichenbaum, and Lindé (25). 2.1 Households The model is populated by a continuum of households, indexed by j [, 1]. Each household consumes final goods, supplies a specific type of labor to intermediate goods firms via employment agencies, saves in one-period nominal government bonds, and accumulates physical 3

5 capital through investment. It transforms physical capital to effective capital by choosing the capital utilization rate, and then rents the effective capital to intermediate goods firms. Household j chooses consumption C t (j), labor supply L t (j), bond holdings B t (j), the rate of capital utilization ν t, investment I t, and physical capital K t to maximize the intertemporal utility function { E t β s ε b t+s s= [ log (C t+s (j) hc t+s 1 (j)) ε l t+s L t+s (j) 1+ω ] }, (1) 1 + ω where β is a discount factor, h measures the degree of habits in consumption, ω is the inverse Frisch elasticity of labor supply, ε b t is an intertemporal preference shock, and ε l t is a shock to the disutility of supplying labor. The intertemporal preference shock has mean unity and is assumed to follow the autoregressive process log ε b t = ρ b log ε b t 1 + ζ b t, ζ b t i.i.d. N(, σ 2 b ), (2) while we explore different processes for the labor disutility shock in the estimated model in Section 4 below. The capital utilization rate ν t transforms physical capital Kt into efficient capital K t according to K t = ν t Kt 1, (3) and the efficient capital is rented to intermediate goods firms at the nominal rental rate R k t. The cost of capital utilization per unit of physical capital is given by A(ν t ), and we assume that ν t = 1 in steady state, A(1) =, and A (1)/A (1) = η ν, as in Christiano, Eichenbaum, and Evans (25) and others. Physical capital accumulates according to K t = (1 δ) K t 1 + ε i t [ ( )] It 1 S I t, (4) I t 1 where δ is the depreciation rate of capital, ε i t is an investment-specific technology shock with mean unity, and S( ) is an adjustment cost function which satisfies S (γ z ) = S (γ z ) = and S (γ z ) = η k >, where γ z is the steady-state growth rate. The investment-specific technology shock follows the process log ε i t = ρ i log ε i t 1 + ζ i t, ζ i t i.i.d. N(, σ 2 i ). (5) Let P t be the nominal price level, R t the one-period nominal (gross) interest rate, A t (j) the net returns from a portfolio of state-contingent securities, W t the nominal wage, Π t nominal lump-sum profits from ownership of firms, and T t nominal lump-sum transfers. Household j s budget constraint is then given by P t C t +P t I t +B t = T t +R t 1 B t 1 +A t (j)+π t +W t (j)l t (j)+r k t ν t Kt 1 P t A (ν t ) K t 1. (6) 4

6 Assuming that households have access to a complete set of state-contingent securities, consumption and asset holdings are the same for all households. The first-order conditions for consumption, bond holdings, investment, physical capital, and efficient capital are then given by Λ t = ε b t C t hc t 1 βhe t Λ t = βr t E t { 1 = Q t ε i t P t Λ t+1 P t+1 ( It [ 1 S I t 1 { ε b t+1 }, (7) C t+1 hc t }, (8) ) I ( )] { t S It Λ t+1 + βe t Q t+1 ε i t+1 I t 1 I t 1 Λ t ( It+1 I t ) 2 ( ) } S It+1, (9) { [ ]} Λ t+1 R k Q t = βe t+1 t ν t+1 A (ν t+1 ) + (1 δ)q t+1, (1) Λ t P t+1 R k t = P t A (ν t ), where Λ t is the marginal utility of consumption and Q t is Tobin s Q, that is, the marginal value of capital relative to consumption. 2.2 Final goods producing firms A perfectly competitive sector combines a continuum of intermediate goods Y t (i) indexed by i [, 1] into a final consumption good Y t according to the production function [ 1 Y t = ] ε p t Y t (i) 1/εp t di, (12) where ε p t is a time-varying measure of substitutability across differentiated intermediate goods. This substitutability implies a time-varying (gross) markup of price over marginal cost equal to ε p t that is assumed to follow the process log ε p t = ( 1 ρ p ) log ε p + ρ p log ε p t 1 + ζp t, ζp t i.i.d. N(, σ2 p), (13) where ε p is the steady-state price markup. Profit maximization by final goods producing firms yields the set of demand equations [ ] Pt (i) ε p t /(εp t 1) Y t (i) = Y t, (14) P t where P t (i) is the price of intermediate good i and P t is an aggregate price index given by [ 1 P t = ] ε p P t (i) 1/(εp t 1) t 1 di. (15) I t (11) 5

7 2.3 Intermediate goods producing firms Each firm in the intermediate goods sector produces a differentiated intermediate good i using capital and labor inputs according to the production function { } Y t (i) = max K t (i) α [Z t L t (i)] 1 α Z t F,, (16) where α is the capital share, Z t is a labor-augmenting productivity factor, whose growth rate ε z t = Z t /Z t 1 follows a stationary exogenous process with steady-state value ε z which corresponds to the economy s steady-state (gross) growth rate γ z, and F is a fixed cost that ensures that profits are zero. The rate of technology growth is assumed to follow log ε z t = (1 ρ z ) log ε z + ρ z log ε z t 1 + ζ z t, ζ z t i.i.d. N(, σ 2 z). (17) Thus, technology is non-stationary in levels but stationary in growth rates, following Altig, Christiano, Eichenbaum, and Lindé (25). We assume that capital is perfectly mobile across firms and that there is a competitive rental market for capital. and Cost minimization implies that nominal marginal cost MC t is determined by MC t (i) = W t (1 α)z 1 α t (L t (i)/k t (i)) α (18) MC t (i) = r k t αzt 1 α α 1, (19) (K t (i)/l t (i)) so nominal marginal cost is common across firms and given by MC t = [ α α (1 α) 1 α] 1 (Wt /Z t ) 1 α ( r k t ) α. (2) Prices of intermediate goods are set in a staggered fashion, following Calvo (1983). Thus, only a fraction 1 θ p of firms are able to reoptimize their price in any given period. The remaining fraction are assumed to index their price to a combination of past inflation and steady-state inflation according to the rule P t (i) = P t 1 (i)π γ p t 1 π1 γ p, (21) where π t = P t /P t 1 is the gross rate of inflation with steady-state value π and γ p [, 1]. If the indexation parameter γ p is equal to zero, firms index fully to steady-state inflation, as in Yun (1996), while if γ p = 1, firms index fully to lagged inflation, as in Christiano, Eichenbaum, and Evans (25). Firms that are able to set their price optimally instead choose their price P t (i) to maximize the present value of future profits over the expected life-time of the price contract: E t { s= (βθ p ) s Λ t+s Λ t { Π t,t+s P t (i)y t+s (i) W t+s L t+s (i) R k t+sk t+s (i)} }, (22) 6

8 where 1 for s =, Π t,t+s = s k=1 πγ p t+k 1 π1 γ p for s 1., (23) subject to the demand from final goods producing firms in equation (14), As all firms changing their price at time t face the same problem, they all set the same optimal price P t. The first-order condition associated with their maximization problem is E t { s= [ (βθ p ) s Λt+s ( Y t,t+s Πt,t+s Pt ε p t+s Λ MC ) ]} t+s =, (24) t where Y t,t+s is demand facing the firm at time t, given the price P t. In the limiting case of full price flexibility (θ p = ) the optimal price is P t = ε p t MC t, (25) that is, a markup on nominal marginal cost. With staggered price setting, the price index P t evolves according to P t = ( ) [(1 θ p ) (P t ) 1/(εpt 1) + θ p π γ 1/(ε p p t 1 π1 γ pp 1)] ε p t 1 t t 1. (26) 2.4 The labor market As in Erceg, Henderson, and Levin (2), each household is a monopolistic supplier of specialized labor L t (j), which is combined by perfectly competitive employment agencies into labor services L t according to [ 1 εw t L t = L t (j) 1/εw t dj], (27) where ε w t is a time-varying measure of substitutability across labor varieties that translates into a time-varying (gross) markup of wages over the marginal rate of substitution between consumption and leisure. As for the labor disutility shock, we will explore different stochastic processes for the wage markup shock below. Profit maximization by employment agencies yields the set of demand equations [ ] Wt (j) ε w t /(ε w t 1) L t (j) = L t, (28) W t for each j, where W t (j) is the wage received from employment agencies by the household supplying labor variety j, and W t is the aggregate wage index given by [ 1 W t = ] ε w t 1 W t (j) 1/(εw t 1) dj. (29) In any given period, a fraction 1 θ w of households are able to set their wage optimally. 7

9 Similar to the price indexation scheme, the remaining fraction indexes their wage to the steady-state growth rate γ z and a combination of past inflation and steady-state inflation according to 5 W t (j) = W t 1 (j)γ z π γ w t 1 π 1 γ w. (3) The optimizing households choose the wage to maximize { [ E t (βθ w ) s W t (j) Λ t+s L t+s (j) ε b P t+sε l L t+s (j) 1+ω ] } t+s, (31) t+s 1 + ω s= subject to the labor demand equation (28). All optimizing households then set the same optimal wage W t to satisfy the first-order condition { E t (βθ w ) s Λ t+s L t,t+s [Π w Wt t,t+s ε w P t+sε b t+sε l t+s t+s s= L ω t,t+s Λ t+s ] } =, (32) where L t,t+s is labor demand facing the household at time t given the wage W t, and Π w 1 for s =, t,t+s = (33) s k=1 γ zπ γ w t+k 1 π 1 γ w for s 1. The limiting case of full wage flexibility (θ w = ) implies that W t P t = ε w t ε b tε l L ω t t, Λ t (34) so the real wage is set as a markup over the marginal rate of substitution. With staggered wages, the aggregate wage index W t evolves according to W t = 2.5 Government [ (1 θ w ) (W t ) 1/(εw t 1) + θ w ( γz π γ w t 1 π 1 γ ww t 1 ) 1/(ε w t 1) ] ε w t 1. (35) The government sets public spending G t according to G t = [ 1 1 ] ε g Y t, t (36) where ε g t is a spending shock with mean unity that follows the process log ε g t = ρ g log ε g t 1 + ζg t, ζg t i.i.d. N(, σ2 g). (37) 5 Justiniano, Primiceri, and Tambalotti (21) and Justiniano and Primiceri (28) assume that wages are partly indexed to past nominal productivity growth π t 1ε z t 1 and partly to the nominal steady-state growth rate πγ z. Our specification instead follows Smets and Wouters (27). 8

10 The nominal interest rate R t is set using the monetary policy rule R t R = ( Rt 1 R ) ρs [( ) rπ ( ) πt Yt /Y ry ] 1 ρs t 1 ε r t, (38) π t where π t is a time-varying target for inflation, which follows γ z log π t = (1 ρ ) log π + ρ log π t 1 + ζ t, ζ t i.i.d. N(, σ 2 ). (39) and ε r t is a monetary policy shock which is i.i.d. (in logarithms) with mean unity and variance σ 2 r. Thus the monetary policy rule is affected by two different shock processes: one persistent and one i.i.d. Although we will call the persistent shock an inflation target shock, it could in principle represent any persistent deviation from the monetary policy rule, such as errors in the perception of the long-run growth rate γ z. We specify the monetary policy rule in terms of output growth rather than the output gap, defined as the deviation of output from potential. Thus, we implicitly assume that the central bank either is unable to observe potential output or is unwilling to let monetary policy depend on its estimate of potential output. As a consequence, our estimates of the benchmark model are independent of the evolution of potential output Market clearing Finally, to close the model, the resource constraint implies that output is equal to the sum of consumption, investment, government spending, and the capital utilization costs: Y t = C t + I t + G t + A (ν t ) K t 1. (4) 2.7 Model summary The complete model consists of 17 endogenous variables determined by 17 equations: the capital utilization equation (3), the capital accumulation equation (4), the households firstorder conditions (7) (11), the production function (16), the marginal cost equations (18) (19), the optimal price and wage setting equations (24) and (32), the aggregate price and wage indices (26) and (35), the rules for government spending and monetary policy in (36) and (38), and the resource constraint (4). In addition there are nine exogenous shocks: to household preferences ε b t, the disutility of labor ε l t, labor-augmenting technology ε z t, investment-specific technology ε i t, government spending ε g t, the price and wage markups εp t and εw t, the inflation target π t, and monetary policy ε r t. Output, the capital stock, investment, consumption, government spending, and the real wage all share the common stochastic trend introduced by the non-stationary technology shock ε z t. Therefore, the model is rewritten on stationary form by normalizing these variables 6 Justiniano and Primiceri (28) specify the monetary policy rule in terms the four-quarter growth rate of output and the four-quarter inflation rate. Smets and Wouters (27) and Justiniano, Primiceri, and Tambalotti (21) use a rule with the deviation of quarterly inflation from a constant inflation target, the level and first difference of the output gap, and an AR(1) monetary policy shock. 9

11 by the non-stationary technology shock, and then log-linearized around its steady state. The stationary model, the steady state, and the log-linearized model are shown in Appendix A. 7 3 The output gap and the labor wedge in the theoretical model One advantage of a fully specified structural model is that it can answer both positive and normative questions about the economy. For instance, the model can be used to evaluate whether fluctuations in economic activity are due to the efficient response of households and firms to disturbances to preferences and technology, or the result of different distortions in the economy, such as imperfect competition or price or wage rigidities. The model can then also be used to study the optimal design of economic policy in the face of such fluctuations. 3.1 Efficient and potential output Our model economy implies an efficient allocation where competition is perfect and there are no price and wage rigidities. That is, prices and wages are flexible and markups are zero. In this allocation all variables are at their efficient levels and fluctuate over time as agents efficiently respond to structural disturbances to technology and preferences. This hypothetical economy therefore is affected by five of our nine shocks: those to technology, investment, consumer preferences, the disutility of labor, and government spending. We will label these efficient shocks. The remaining four shocks to the wage and price markups, monetary policy, and the inflation target are instead labelled inefficient. Due to imperfect competition (and thus the presence of average wage and price markups), the steady-state level of output in the efficient allocation is higher than in the model with sticky prices and wages. For purposes of economic policy, and in particular monetary policy, a more relevant concept is the potential level of output, defined as the allocation with flexible prices and wages but imperfect competition and constant markups, so there are no shocks to price and wage markups. 8 In the log-linearized model, this allocation has the same average (steady-state) level of output as in the model with sticky wages and prices, so any deviation between the actual and potential levels of output is zero in steady state. The focus of our analysis is on this measure of potential output, and we define the output gap as the percent deviation of actual output from potential. Following Woodford (23) and Adolfson, Laséen, Lindé, and Svensson (28), we distinguish between two different measures of potential output. The first is derived from the allocation where prices and wages have been flexible forever, and thus uses the state variables from this allocation. 9 We call this the unconditional potential output. The second measure 7 In addition, we supplement the log-linearized model with a block of equations that determines the allocation with flexible prices and wages. While this block is not used when estimating the model, it is useful when constructing different measures of potential output in Section 5. 8 This definition follows Woodford (23). A closely related concept is the natural level of output, which is the allocation with flexible prices and wages, but including exogenous shocks to price and wage markups. See also Justiniano and Primiceri (28). 9 In the model with flexible prices and wages, the state variables are the physical stock of capital, lagged 1

12 instead uses the state variables in the allocation with sticky prices and wages. This measure, which we call conditional potential output, is taken from an allocation where prices and wages have been sticky in the past, and then unexpectedly become flexible and are expected to remain flexible in the future. While it is straightforward to derive the behavior of the unconditional potential output by setting price and wage rigidities and inefficient shocks to zero, the conditional potential output is more involved. Appendix C describes how we calculate the conditional potential output from the solution of the model. The existing literature focuses on the unconditional measure. 1 Neiss and Nelson (23) motivate this choice by the fact that the conditional potential output depends not only on the efficient shocks, but also on past inefficient shocks (for instance, to monetary policy), through their effect on the current state. Therefore, if monetary policy is set as a function of the conditional output gap, a mistake in monetary policy today is not fully offset in the future, as it has affected both actual and potential output, and may have had a small effect on the output gap. Woodford (23) instead argues that the conditional potential output is more closely related to the efficient level, which should depend on the current state of the economy. In our model where monopolists profits are zero due to the fixed cost, there is indeed a constant distance between the efficient and the conditional potential levels of output. Another argument in favor of the conditional measure is that monetary policy that is determined by the unconditional output gap depends not only on the current state of the economy and the current shocks, but also on the entire path of historical shocks, as these affect the (unobserved) state variables in the allocation with flexible prices and wages. We will not take a strong position regarding which measure is a better guide for monetary policy. Instead we will estimate both measures and show that there is very strong comovement between the conditional and the unconditional measures. Thus, as guides for monetary policy, the two measures will give very similar qualitative advice, even if quantities differ. We will estimate the path of potential output (conditional and unconditional) and the related output gaps (the deviation of actual GDP from potential GDP) in our model. Importantly, the output gap defined in this way is not necessarily a measure of the business cycle, for instance as defined by the NBER. Instead, the output gap should be seen as a measure of inefficiency in the economy, that is, the output fluctuations that are due to nominal rigidities and inefficient shocks. 3.2 The output gap and the labor wedge Our model embodies a standard production function where capital and labor inputs are converted into output. Fluctuations in output over the business cycle are then largely due to fluctuations in the labor input and in the stochastic process for technology growth, as the capital stock adjusts slowly over time. Shocks to technology mainly generate fluctuations in the efficient level of output. Inefficient fluctuations in output should therefore be closely related to inefficient fluctuations in the labor input. consumption, and lagged investment. 1 Examples include Neiss and Nelson (23), (25), Edge, Kiley, and Laforte (28), Sala, Söderström, and Trigari (28), or Justiniano and Primiceri (28). Coenen, Smets, and Vetlov (29) instead focus on the conditional measure. 11

13 In the standard competitive model with only efficient fluctuations, the labor input is determined by the equality between households marginal rate of substitution between consumption and leisure and firms marginal product of labor. Inefficient fluctuations in the labor input are then generated by deviations from this efficiency condition. Such deviations (labeled the labor wedge by Chari, Kehoe, and McGrattan (27)) have been studied extensively in the literature (recent examples include Hall (1997), Chari, Kehoe, and McGrattan (27), Galí, Gertler, and López-Salido (27), and Shimer (29)). Letting x t denote the log deviation of any variable X t from its steady-state level, the log-linearized version of our model implies that the marginal rate of substitution and the marginal product of labor are given by 11 mrs t = ω l t λ t + ε b t + ε l t, (41) mpl t = α k t α l t, (42) and the labor wedge is then determined as wedge t mrs t mpl t = (α + ω) l t λ t α k t + ε b t + ε l t. (43) The wedge defined as in equation (43) is a measure of inefficiency in the labor market. We can write the labor wedge as the sum of a fundamental component and two unobservable shocks to household preferences: wedge t = wedge t + ε b t + ε l t, (44) where the fundamental component is given by wedge t = (α + ω) l t λ t α k t. (45) This fundamental wedge is closely related to the wedge studied by Hall (1997), Galí, Gertler, and López-Salido (27), Shimer (29), and others. These authors typically use a simple model without habits and fixed costs in production, so mrs t = ω l t + ĉ t and mpl t = ŷ t l t. The labor wedge is then given by wedge t = (1 + ω) l t + ĉ t ŷ t, (46) and can be measured from observable data, given an estimate of the labor supply elasticity ω. 12 The wedge in the simple model is proportional to hours, adjusted for the consumption/output ratio. We will show below that the fundamental wedge is roughly proportional 11 As output, investment, the capital stock, consumption, the marginal utility of consumption, and the real wage are non-stationary in our model, so are the MRS and the MPL. The log-linearized model is defined in terms of the ratios of these variables to the non-stationary technology shock that generates the common stochastic trend. 12 Hall (1997) and Galí, Gertler, and López-Salido (27) also include a low-frequency preference shifter that is similar to our labor disutility shock. However, our shock is not restricted to low frequencies. 12

14 to hours also in our estimated model. 13 Therefore, by understanding movements in the fundamental wedge we are able to interpret movements in the labor input. To relate the labor wedge to the output gap, we use the fact that the labor wedge will be zero in the allocation of the log-linearized model with flexible prices and wages and constant markups. Then we can write the wedge in equation (43) in terms of the deviations of hours lt, the marginal utility of consumption λ t, and the effective capital stock k t, from their levels with flexible prices and wages (indexed by f ) as ) ( λt λ f t wedge t = (α + ω) ( lt l f t ) α ( kt k f t ). (47) The production function (16) implies that the log-linearized output gap is given by ŷ t ŷ f t = Y + F Y [ α ( kt k ) f t + (1 α) ( lt l )] f t, (48) and combining these two expressions we can write the output gap in terms of the labor wedge as ŷ t ŷ f t = Y + F Y [ 1 α α + ω wedge t + ( λt λ f t ) α(1 + ω) + ( kt 1 α k ) ] f t. (49) In a simple version of the model without capital, a government sector, and habits, the capital gap is zero and the marginal utility gap is inversely related to the output gap. Then the output gap is exactly proportional to the wedge, see Erceg, Henderson, and Levin (2) and Galí, Gertler, and López-Salido (23). In our more elaborate model, the output gap also depends on the gaps for the marginal utility consumption and the effective capital stock. We will show, however, that the output gap is close to proportional to the labor wedge also in our estimated model. This strong relationship between the gap and the wedge implies that by understanding the labor wedge we may gain insights into the determinants of the output gap. It also allows us to understand if the output gap is mainly due to exogenous disturbances or the endogenous effects of price and wage rigidities. 3.3 Interpreting the labor wedge To understand and interpret fluctuations in the labor wedge, it is useful to further decompose the fundamental wedge. First, imperfect competition in labor and product markets implies that the real wage is set as a markup over the marginal rate of substitution, and the price is set as a markup over nominal marginal cost, given by the nominal wage adjusted for the marginal product of labor. By adding and subtracting the real wage from the expression for the labor wedge, we can write the wedge in terms of the wage and price markups (denoted 13 Strictly speaking, the fundamental wedge is not directly observable in our model, due to the presence of the marginal utility of consumption and the capital stock. However, it behaves very similarly to the observable simple wedge. 13

15 µ w t and µ p t ):14 wedge t = mrs t mpl t = ( mrs t ŵ t ) + (ŵ t mpl ) t = ( µ w t + µ p t ). (5) Furthermore, each markup can be decomposed into an endogenous component (due to price and wage rigidities) and an exogenous component (equal to the markup shock, or the desired markup in the absence of price and wage rigidities) as µ w t = µ w t + ε w t, µ p t = µ p t + εp t, (51) (52) where µ w t and µ p t are the endogenous markups. The literature on the labor wedge focuses on how to explain the fundamental wedge, that is, the apparent inefficient fluctuations in the labor input. In principle, these fluctuations can be due to movements in exogenous preference shocks, endogenous markups, or exogenous markup shocks (see, for example, Shimer (29) and Galí, Gertler, and López-Salido (27)). Shimer (29) criticizes accounts of the labor wedge that rely on exogenous shocks to preferences and markups. Galí, Gertler, and López-Salido (27) provide some evidence that the observed wedge is endogenous. First, they show evidence of a Granger causal link from different macroeconomic variables to the wedge. Second, they show that the wedge responds negatively to a contractionary monetary policy shock in an identified VAR. Our estimated model gives a complete description of the data and therefore all variables in the model, including the labor wedge. It is therefore well suited to interpret movements in the labor wedge and give a more precise answer to what explains fluctuations in the wedge. Using the markup decomposition, we can decompose the fundamental wedge in two different dimensions as Efficient Inefficient {}}{{ ( }} ){ wedge t = ( µ w t + µ p t ) (εw t + ε p t ) ε b t + ε l t ) = ( µ w t + µ p t }{{} ) (ε w t + ε p t (ε ) b t + ε l t }{{} Endogenous Exogenous (53) In one dimension, the fundamental wedge is decomposed into an efficient component (due to preference shocks) and an inefficient component (due to the total markups). In another dimension, it is decomposed into an endogenous component (due to price and wage rigidities) and an exogenous component (due to disturbances). The inefficient component is, of course, the total labor wedge in equation (43), and is in part endogenous and in part exogenous. The efficient component is entirely exogenous and due to intertemporal preference and labor disutility shocks. The endogenous component is entirely inefficient and due to wage and price 14 See also Galí, Gertler, and López-Salido (27). In our estimated model, the price markup is typically small (as the real wage moves closely with the marginal product). Thus, the total wedge is largely determined by the wage markup. 14

16 rigidities, while the exogenous component includes both efficient and inefficient shocks. The decomposition into the endogenous and exogenous components is independent of whether the shocks are efficient or inefficient: the exogenous component depends only on the sum of the shocks. But the interpretation of shocks crucially affects the decomposition into the efficient and inefficient components. The same is true for the estimates of the output gap and the inefficient labor wedge. 3.4 Interpreting the structural shocks In some cases, the structural interpretation of the estimated shocks is straightforward: shocks to technology are clearly efficient, while shocks to monetary policy do not affect the efficient allocation, and these shocks are unambiguously identified in the estimated model. In other cases, this interpretation is less clearcut. It is well known that shocks to the disutility from supplying labor and shocks to the wage markup are observationally equivalent in the loglinearized version of this class of models. These two shocks enter only in the equation relating the real wage to the marginal rate of substitution. In the log-linearized model this equation is given by ŵ t = γ b [ŵ t 1 π t + γ w π t 1 ε z t ] + γ o [ω l t λ ] t + ε b t + ε l t + ε w t +γ f [ŵt+1 + π t+1 γ w π t + ε z t+1], (54) where γ b, γ o, and γ f are convolutions of the structural parameters (see Appendix A). The real wage is driven by movements in the marginal rate of substitution, given by ω l t λ t + ε b t + ε l t, adjusted for shocks to the wage markup, ε w t. Thus, in the log-linearized model the two shocks are observationally equivalent, and if both shocks are present they can only be separately identified if they are assumed to follow different stochastic processes. For our estimation, the exact interpretation of these two shocks is not important. Each shock could be interpreted as either a labor disutility shock or a wage markup shock. For normative questions, however, the interpretation of the two shocks is important; the labor disutility shock is efficient and therefore does not directly drive the inefficient wedge or the output gap, while the wage markup shock is inefficient (as markups are zero in the efficient allocation) and does directly affect the wedge and the output gap. From a monetary policy perspective, the labor disutility shock should be accommodated by monetary policy, while the wage markup shock should be counteracted. We will explore below exactly how the interpretation of these two shocks affects our estimates of the output gap and the labor wedge Similar issues of interpretation also apply to the price markup shock, which can be interpreted as an efficient relative-price shock to a flexible-price sector in a two-sector model. See Smets and Wouters (2xx) [reference to be added]. 15

17 4 Estimation 4.1 Data and estimation technique We estimate the log-linearized version of the model using quarterly U.S. data from 196Q1 to 29Q2 for seven variables: (1) output growth: the quarterly growth rate of per capita real GDP; (2) investment growth: the quarterly growth rate of per capita real private investment plus real personal consumption expenditures of durable goods; (3) consumption growth: the quarterly growth rate of per capita real personal consumption expenditures of services and nondurable goods; (4) real wage growth: the quarterly growth rate of real compensation per hour; (5) employment: hours of all persons divided by population; (6) inflation: the quarterly growth rate of the GDP deflator; and (7) the nominal interest rate: the quarterly average of the federal funds rate. Many of our results will be driven by the behavior of hours over the business cycle. We use data on hours from Francis and Ramey (29) that refer to the total economy (rather than then non-farm business sector) and are adjusted for low-frequency movements due to changes in demographics. These data therefore display less low-frequency behavior than unadjusted data. Data definitions and sources are available in Appendix B. We estimate the model using Bayesian likelihood-based methods (see An and Schorfheide (27) for an overview). Letting θ denote the vector of structural parameters to be estimated and Y the data sample, we use the Kalman filter to calculate the likelihood L(θ, Y), and then combine the likelihood function with a prior distribution of the parameters to be estimated, p(θ), to obtain the posterior distribution, L(θ, Y)p(θ). We use numerical routines to maximize the value of the posterior, and then generate draws from the posterior distribution using the Random-Walk Metropolis-Hastings algorithm. We use growth rates for the non-stationary variables in our data set (output, consumption, investment, and the real wage, which are non-stationary also in the theoretical model) and we write the measurement equation of the Kalman filter to match the seven observable series with their model counterparts. Thus, the state-space form of the model is characterized by the state equation X t = A(θ)X t 1 + B(θ)ε t, ε t i.i.d. N(, Σ ε ), (55) where X t is a vector of endogenous variables, ε t is a vector of innovations, and θ is a vector of parameters; and the measurement equation Y t = C(θ) + DX t + η t, η t i.i.d. N(, Σ η ), (56) where Y t is a vector of observable variables, that is, Y t = 1 [ log Y t, log I t, log C t, log W t, log L t, log π t, log R t ], (57) and η t is a vector of measurement errors. The model contains 18 structural parameters, not including the parameters that characterize the exogenous shocks and measurement errors. We calibrate four parameters using standard values: the discount factor β is set to.99, the capital depreciation rate δ to.25, 16

18 the capital share α in the Cobb-Douglas production function is set to.33, and the average ratio of government spending to output G/Y to.2. We estimate the remaining 14 structural parameters: the steady-state growth rate, γ z ; the elasticity of the utilization rate to the rental rate of capital, η ν ; 16 the elasticity of the investment adjustment cost function, η k ; the habit parameter h and the labor supply elasticity ω; the steady-state wage and price ε w and ε p ; the wage and price rigidity parameters θ w and θ p ; the wage and price indexing parameters γ w and γ p ; and the monetary policy parameters r π, r y, and ρ s. In addition, we estimate the autoregressive parameters of the exogenous disturbances, as well as the standard deviations of the innovations and measurement errors. 4.2 Labor market shocks in the estimated model We will estimate two versions of the model that differ in their assumptions about the labor market shocks. In the first model, there is only one shock to the labor market, denoted ε 1,t, which follows the auto-regressive process log ε 1,t = (1 ρ 1 ) log ε 1 + ρ 1 log ε 1,t 1 + ζ 1,t, ζ 1,t i.i.d. N(, σ 2 1). (58) In the second model, there is also a second shock, denoted ε 2,t, which is assumed to be i.i.d. normal: log ε 2,t = log ε 2 + ζ 2,t, ζ 2,t i.i.d. N(, σ 2 2). (59) The exact interpretation of these two shocks is not important for the estimated model: each shock could be interpreted as either a labor disutility shock or a wage markup shock. As discussed above, however, for our estimates of the output gap and the labor wedge the interpretation of the two shocks comes to the forefront. 4.3 Priors Before estimation we assign prior distributions to the parameters to be estimated. These are summarized in Table 1. Most of the priors are standard in the literature; see, for example, Smets and Wouters (27) and Justiniano, Primiceri, and Tambalotti (21). The prior distribution for the steady-state growth rate, γ z is Normal with mean 1.4 and standard deviation.1. The prior mean is close to the average gross quarterly growth rate of GDP over the sample period. The utilization rate elasticity ψ ν and the habit parameter h are both assigned Beta priors with mean.5 and standard deviation.1, while the capital adjustment cost elasticity η k is assigned a Normal prior with mean 4 and standard deviation 1.5. The labor supply elasticity ω (the inverse of the Frisch elasticity) is given a Gamma prior with mean 2 and standard deviation.75. The two Calvo parameters for wage and price adjustment, θ w and θ p, are assigned Beta priors with means 3/4 and 2/3, respectively, and standard deviation.1, while the indexation parameters γ w and γ p are given Uniform priors over the unit interval. The two steady-state 16 Following Smets and Wouters (27), we define ψ ν such that η ν = (1 ψ ν ) /ψ ν and estimate ψ ν. 17

19 wage and price markups are both given Normal priors centered around 1.15, with a standard deviation of.5. The coefficient r π on inflation in the monetary policy rule is given a Normal prior with mean 1.7 and standard deviation.3, while the coefficient r y on output growth is given a Gamma prior with mean.125 and standard deviation.1. The coefficient on the lagged interest rate, ρ s, is assigned a Beta prior with mean.75 and a standard deviation of.1. All these are broadly consistent with empirically estimated monetary policy rules. All persistence parameters for the shocks are given Beta priors with mean.5 and standard deviation.15. Finally, for the standard deviations of the shock innovations, we assign Gamma priors. We use Gamma priors instead of Inverse Gamma priors as in Smets and Wouters (27), Justiniano, Primiceri, and Tambalotti (21) and others, in order to allow for very small (or zero) standard deviations. The Inverse Gamma distribution, by construction, puts no probability mass on zero, but we want to allow for the possibility that some shocks have zero variance. We do not normalize any of the shocks before estimation, and we assign Gamma priors with mean and standard deviation of 5. percent for all innovation standard deviations except for the monetary policy and inflation target shocks that have mean and standard deviation equal to.15 percent. These priors imply reasonable volatility of all innovations: the unconditional prior standard deviation of the AR(1) shocks is 5.77 percent, except for the monetary policy shocks that have standard deviations of.17 percent, i.e., 17 basis points in quarterly terms. However, in one respect these priors contrast with the common approach in the literature. A standard procedure is to normalize some of the shocks (in particular, the price and wage markup shocks) to have a unit impact on some observable variable (for instance, the rate of inflation and the real wage). Then a prior distribution is assigned to the normalized shock. For instance, in our model the AR(1) labor market shock would be normalized by defining ε 1,t = (1/γ o ) ε 1,t, where γ o is a convolution of structural parameters. The normalized shock then has a unitary impact on the real wage, see equation (54). One advantage of the normalization is that it is easier to assign a prior distribution based on the volatility of the real wage. In addition, the normalization allows for correlation between the shock and the parameters in γ o, which can be useful. However, this procedure can lead to unreasonable prior distributions for the structural shock, in this case ε 1,t, if this shock has a very small direct impact on the real wage. The direct impact of the labor market shocks on the real wage in the log-linearized equation (54) is governed by the coefficient γ o, which is a function of four parameters: the discount factor β, the labor supply elasticity ω, the Calvo wage parameter θ w, and the steady-state wage markup ε w. Our calibration of β and our prior means for ω, θ w, and ε w imply a value for γ o of.26. Smets and Wouters (27) and Justiniano, Primiceri, and Tambalotti (21) both use a prior mean of.1 on the normalized labor market shock. In our model, such a prior on the normalized shock would imply a standard deviation of.1/.26 = 38.5 percent on the structural innovation, and a standard deviation of 44 percent of the structural shock ε 1,t The prior assumptions used by Justiniano and Primiceri (28) would imply even larger volatility of the 18