Unemployment in an Estimated New Keynesian Model

Abstract

Previous articleNext article FreeUnemployment in an Estimated New Keynesian ModelJordi Galí, Frank Smets, and Rafael WoutersJordi GalíCREI, Universitat Pompeu Fabra, Barcelona GSE, and NBER. Search for more articles by this author , Frank SmetsEuropean Central Bank, CEPR, and University of Groningen. Search for more articles by this author , and Rafael WoutersNational Bank of Belgium Search for more articles by this author CREI, Universitat Pompeu Fabra, Barcelona GSE, and NBER.European Central Bank, CEPR, and University of Groningen.National Bank of BelgiumPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreI. IntroductionOver the past decade an increasing number of central banks and other policy institutions have developed and estimated medium-scale New Keynesian DSGE models.1 The combination of a good empirical fit with a sound, microfounded structure makes these models particularly suitable for forecasting and policy analysis. However, as highlighted by Galí and Gertler (2007) and others, one of the shortcomings of these models is the lack of a reference to unemployment. This is unfortunate because unemployment is an important indicator of aggregate resource utilization and a central focus of the policy debate. Recently, a number of papers have started to address this shortcoming by embedding in the basic New Keynesian model various theories of unemployment based on the presence of labor market frictions (e.g., Blanchard and Galí 2010; Christoffel et al. 2009; Gertler, Sala, and Trigari 2008; Christiano, Trabandt, and Walentin 2010, 2011; and de Walque et al. 2009).The present paper takes a different approach. Following Galí (2011b, 2011c), it reformulates the Smets and Wouters (2003, 2007; henceforth, SW) model to allow for involuntary unemployment, while preserving the convenience of the representative household paradigm. Unemployment in the model results from market power in labor markets, reflected in positive wage markups. Variations in unemployment over time are associated with changes in wage markups, either exogenous or resulting from nominal wage rigidities.2The proposed reformulation allows us to overcome an identification problem pointed out by Chari, Kehoe and McGrattan (2009; henceforth, CKM) and interpreted by these authors as an illustration of the immaturity of New Keynesian models for policy analysis. Their observation is motivated by the SW finding that wage markup shocks account for almost 50% of the variations in real GDP at horizons of more than 10 years. However, without an explicit measure of unemployment (or, alternatively, labor supply), these wage markup shocks cannot be distinguished from preference shocks that shift the marginal disutility of labor. The policy implications of these two sources of fluctuations are, however, very different. Variations in wage markup shocks are inefficient and a welfare-maximizing government should be interested in stabilizing output fluctuations resulting from those shocks (at least partly). In contrast, output and employment fluctuations driven by preference shocks shifting the labor supply schedule should in principle be accommodated. Put differently, the relative importance of those two shocks will influence the extent to which fluctuations in output during a given historical episode should or should not be interpreted as reflecting movements in the welfare-relevant output gap (i.e., the distance between the actual and efficient levels of output). By including unemployment as an observable variable, this identification problem can be overcome, and "correct" measures of the output gap can be constructed, as we show in Section IV.When we estimate the reformulated SW model using unemployment as an observable variable, we find a much diminished role for wage markup shocks as a source of output and employment fluctuations, even though those shocks preserve a large role as drivers of inflation. Our estimates lead us to classify the multiple shocks in the model in three categories (which we label "demand," "supply," and "labor market" shocks), on the basis of their implied joint comovement among output, employment, the labor force, unemployment, inflation, and the real wage, as captured by their associated impulse response functions (IRFs). In addition, we show how the implied measure of the welfare-relevant output gap is to a large extent the mirror image of the unemployment rate, and resembles conventional measures of the cyclical component of log GDP, based on statistical detrending methods (though the correlation is far from perfect).Our estimates of the reformulated SW model allow us to address a number of additional questions of interest that could not be dealt with using the model's original formulation. Thus, in Section V we assess quantitatively the relative importance of different shocks as sources of unemployment fluctuations and their role during specific historical episodes, including the recent recession. Also, our approach allows us to uncover a measure of the natural rate of unemployment (i.e., the flexible wage counterfactual) and to study its comovement with actual unemployment. That comovement is shown to be particularly strong at low frequencies, as expected, but the gap between the two caused by wage rigidities is estimated to be large and persistent. We also revisit the evidence on the joint behavior of inflation and unemployment under the lens of our estimated model. This allows us to give a structural interpretation to empirical Phillips curves, both for wage and price inflation. In Section VI we discuss the robustness of our findings to the use of alternative sample period and data. Section VII concludes.In addition to reformulating the wage equation in terms of unemployment, our model shows a number of small differences with that in SW (2007). First, and regarding the data on which the estimation is based, we use employment rather than hours worked, and redefine the wage as the wage per worker rather than the wage per hour. We do so since the model focuses on variations in labor at the extensive margin, in a way consistent with the conventional definition of unemployment. Given that most of the variation in hours worked over the business cycle is due to changes in employment rather than hours per employee, this change does not have major consequences in itself. We also combine two alternative wage measures in the estimation, compensation and earnings, and model their discrepancy explicitly. Second, we generalize the utility function in a way that allows us to parameterize the strength of the wealth effect on labor supply, as shown in Jaimovich and Rebelo (2009). This generalization yields a better fit of the joint behavior of employment and the labor force, as we discuss in detail. Third, for simplicity, we revert to a Dixit-Stiglitz aggregator rather than the Kimball aggregator used in SW (2007).The rest of the paper is structured as follows. Section II describes the modified Smets-Wouters model. Next, Section III presents the data and estimation. Section IV contains the discussion of the CKM critique. Section V analyzes different aspects of unemployment fluctuations, which the reformulation of the SW model makes possible. Section VI presents some robustness exercises and, finally, Section VII concludes.II. Introducing Unemployment in the Smets-Wouters ModelA. Staggered Wage Setting and Wage Inflation DynamicsThis section introduces a variant of the wage-setting block of the SW model, which is in turn an extension of that in Erceg, Henderson, and Levin (2000; henceforth, EHL). The variant presented here, based on Galí (2011b, 2011c), assumes that labor is indivisible, with all variations in hired labor input taking place at the extensive margin. That feature gives rise to a notion of unemployment consistent with its empirical counterpart.The model assumes a (large) representative household with a continuum of members represented by the unit square and indexed by a pair (i, j) ∈ 0, 1] × 0, 1]. The first dimension, indexed by i ∈ 0, 1], represents the type of labor service in which a given household member is specialized. The second dimension, indexed by j ∈ 0, 1], determines his disutility from work. The latter is given by if he is employed, zero otherwise, where is an exogenous preference shifter (referred to in the following as a "labor supply shock"), is an endogenous preference shifter, taken as given by each individual household and defined in the following, and is a parameter determining the shape of the distribution of work disutilities across individuals.Individual utility is assumed to be given by: where , with h ∈ 0, 1], and with denoting (lagged) aggregate consumption (taken as given by each household), and where 1t(i, j) is an indicator function taking a value equal to one if individual (i, j) is employed in period t, and zero otherwise. Thus, as in SW and related monetary dynamic stochastic general equilibrium (DSGE) models, we allow for (external) habits in consumption, indexed by h.As in Merz (1995), full risk sharing of consumption among household members is assumed, implying Ct(i, j) = Ct for all (i, j) ∈ 0, 1] × 0, 1] and t. Thus, we can derive the household utility as the integral over its members' utilities; that is: where Nt(i) ∈ 0, 1] denotes the employment rate in period t among workers specialized in type i labor and 3We define the endogenous preference shifter Θt, as follows: where Zt evolves over time according to the difference equation Thus, Zt can be interpreted as a "smooth" trend for (quasi-differenced) aggregate consumption. Our preference specification implies a "consumption externality" on individual labor supply: during aggregate consumption booms (i.e., when is above its trend value Zt), individual (as well as household-level) marginal disutility from work goes down (at any given level of employment).The previous specification generalizes the preferences assumed in SW by allowing for an exogenous labor supply shock, χt, and by introducing the endogenous shifter Θt (just described). The main role of the latter is to reconcile the existence of a long-run balanced growth path with an arbitrarily small short-term wealth effect. The latter's importance is determined by the size of parameter υ ∈ 0, 1]. As discussed later in detail, that feature is needed in order to match the joint behavior of the labor force, consumption, and the wage over the business cycle. That modification is related to, but not identical to, the one proposed by Jaimovich and Rebelo (2009) as a key ingredient in order to account for the economy's response to news about future productivity increases.4Note that under the previous preferences, the household-relevant marginal rate of substitution between consumption and employment for type i workers in period t is given by: where the last equality is satisfied in a symmetric equilibrium with Using lower-case letters to denote the natural logarithms of the original variables, we can derive the average (log) marginal rate of substitution by integrating over all labor types: where is (log) aggregate employment and .We assume nominal wages are set by "unions," each of which represents the workers specialized in a given type of labor, and acting in an uncoordinated way. As in EHL, and following the formalism of Calvo (1983), we assume that the nominal wage for a labor service of a given type can only be reset with probability 1 – θw each period. That probability is independent of the time elapsed since the wage for that labor type was last reset, in addition to being independent across labor types. Thus, and by the law of large numbers, a fraction of workers θw do not reoptimize their wage in any given period, making that parameter a natural index of nominal wage rigidities. Furthermore, all those who reoptimize their wage choose an identical wage, denoted by Wt*, since they face an identical problem. Following SW, we allow for partial wage indexation between reoptimization periods, by making the nominal wage adjust mechanically in proportion to past price inflation. Formally, and letting Wt+k|t denote the nominal wage in period t + k for workers who last reoptimized their wage in period t, we assume for k = 1, 2, 3, … and , and where denotes the (gross) rate of price inflation, Πp is its corresponding steady-state value, Πχ is the steady-state (gross) growth rate of productivity, and γw ∈ 0, 1] measures the degree of wage indexation to past inflation.When reoptimizing their wage in period t, workers (or the union representing them) choose a wage Wt * in order to maximize their respective households' utility (as opposed to their individual utility), subject to the usual sequence of household flow budget constraints, as well as a sequence of isoelastic demand schedules of the form , where Nt+k|t denotes period t + k employment among workers whose wage was last reoptimized in period t, and where ∈w,t is the period t wage elasticity of the relevant labor demand schedule.5 We assume that elasticity varies exogenously over time, thus leading to changes in workers' market power.The first-order condition associated with the wage-setting problem can be written as: where, in a symmetric equilibrium, is the relevant marginal rate of substitution between consumption and employment in period t + k, and is the natural (or desired) wage markup in period t; that is, the one that would obtain under flexible wages.Under the previous assumptions, we can write the aggregate wage index as follows: Log-linearizing (1) and (2) around a perfect foresight steady state and combining the resulting expressions allows us to derive (after some algebra) the following equation for wage inflation : Where , is the (log) natural wage markup, and is the (log) average wage markup; that is, the log deviation between the average real wage and the average marginal rate of substitution. As equation (3) makes clear, variations in wage inflation above and beyond those resulting from indexation to past price inflation are driven by deviations of average wage markup from its natural level, because those deviations generate pressure on workers currently setting wages to adjust those wages in one direction or another.One might argue that the previous model provides, if interpreted literally, an unrealistic description of wage setting in the United States. We view it instead as a simple modeling device, consistent with the labor market block of the medium-scale DSGE models currently used for policy analysis (as exemplified by the SW model), and embedding three features of actual labor markets: (1) nominal wage rigidities, (2) staggered wage-setting, and (3) the presence of average wage levels above their perfectly competitive counterparts, resulting from different sources of market power by workers that prevent their underbidding by the unemployed.B. Introducing UnemploymentConsider an individual specialized in type i labor and with disutility of work χtΘtjφ. Using household welfare as a criterion, and taking as given cur rent labor market conditions (as summarized by the prevailing wage for his labor type), that individual will find it optimal to participate in the labor market in period t if and only if Evaluating the previous condition at the symmetric equilibrium, and letting the marginal supplier of type i labor be denoted by Lt(i), we have: Taking logs and integrating over i we obtain where can be interpreted as the (log) aggregate participation or labor force.Following Galí (2011b, 2011c), we define the unemployment rate ut as: Note that under our assumptions, the unemployed thus defined include all the individuals who would like to be working (given current labor market conditions, and while internalizing the benefits that this will bring to their households) but are not currently employed. It is in that sense that one can view unemployment as involuntary.6Combining (4) with (5) and (6), the following simple linear relation between the average wage markup and the unemployment rate can be derived which is also graphically illustrated in figure 1.Fig. 1. The wage markup and the unemployment rateView Large ImageDownload PowerPointFinally, combining (3) and (7) we obtain an equation relating wage inflation to price inflation, the unemployment rate, and the wage markup. Note that in contrast with the representation of the wage equation found in SW and related papers, the error term in (8) captures exclusively shocks to the wage markup, and not preference shocks (even though the latter have been allowed for in our model). That feature, made possible by reformulating the wage equation in terms of the (observable) unemployment rate, allows us to overcome the identification problem raised by CKM in their critique of New Keynesian models. We turn to this issue later, when we discuss our empirical findings.Finally, note that we can define the natural rate of unemployment, , as the unemployment rate that would prevail in the absence of nominal wage rigidities. Under our assumptions, that natural rate will vary exogenously in proportion to the natural wage markup, and can be determined using the simple relation: The remaining equations describing the log-linearized equilibrium conditions of the model are presented in the appendix. Those equations are identical to a particular case of the specification in SW (2007), corresponding to logarithmic consumption utility. In addition to the wage markup and labor supply shocks just discussed, the model includes six additional shocks: a neutral, factor-augmenting productivity shock; a price markup shock; a risk premium shock; an exogenous spending shock; an investment-specific technology shock; and a monetary policy shock.III. Data and EstimationA. DataWe estimate our model on US data for the sample period 1966Q1– 2007Q4 using Bayesian full-system estimation techniques as in SW (2007). We end our estimation period in 2007Q4 to prevent our estimates from being distorted by the nonlinearities induced by the zero lower bound on the federal funds rate and binding downward nominal wage rigidities during the most recent recession.7 In Section V we nevertheless use the estimated model to interpret the behavior of unemployment in the recent recession; that is, beyond the estimated period. Section VII on robustness discusses the of our model over an sample period in of the data used by SW (2007) are also used GDP, consumption, GDP inflation, and the federal funds rate, with the first three in per terms and log As the SW model is reformulated in terms of employment (given our interest in we use per employment rather than hours The main results are not if we use hours as discussed in Section In addition, we with two wage The first one is compensation per from the of and The second one is from the we the unemployment rate as an additional observable In the following we the model estimated with and without the latter as an observable of both wage are This is illustrated in figure 2, which their nominal growth First, average wage inflation based on compensation per is than that based on per Given average price inflation, the compensation more with a balanced growth path in which real wages at the rate as real output, consumption, and Second, the compensation is much more than the over the past two The deviation of wage inflation based on compensation is to for the the correlation between both wage inflation measures is low at wage inflation Large ImageDownload our estimation, we use both wage as measures of the wage This is by error to the corresponding equations and allowing for a trend in the In the section on we discuss the estimation results when we only use the compensation In the rest of the we focus on the model with both wage and 1 the estimated structural of the model with and without unemployment being used as an observable As discussed unemployment allows us to wage markup and labor supply In addition, it allows us to the model's of between the unemployment rate and the wage markup equation in order to and estimate the elasticity of substitution between different labor which in turn determines the steady-state wage markup. In the model without unemployment this parameter is not we it to be very to the of the estimate in the model with observable 1. for the with and without Unemployment as of is defined by the degree of effect of productivity on exogenous steady-state wage markup is not if the unemployment rate is not most of the estimated structural are very in the two on the that are important for the labor a number of findings are First, the estimated labor supply elasticity is one unemployment or not as an observable the of the elasticity from to as one includes unemployment. In the latter the steady-state wage markup is and estimated to be which is consistent with an average unemployment rate of about to some of the other that the wage Phillips the estimated degree of wage indexation is small and across the two models. The estimated Calvo probability of wages from to flexible wages with average of two the of unemployment as an observable to a wage Phillips the parameter the wealth on labor supply, changes from to this to a change from preferences to those in and Rebelo henceforth, by strong wealth on labor supply, to a specification to that in and In the latter wealth are to zero in the As discussed later, this that not only employment, but also the labor in response to most it is out that the monetary policy to the output gap as the deviation relative to the markup from to As discussed later, this is due to the lower of the output gap unemployment is used to wage markup to show the estimated impulse of output, inflation, the real wage, the interest rate, employment, the labor force, the unemployment rate, and the output gap to the structural focuses on the which include the technology shock, the risk premium shock, the exogenous spending shock, and the monetary policy We use the label to to those shocks because they all a positive comovement output, inflation, and the real It is particularly that employment and the labor in response to all those however, that the size of the labor response is much than that of employment, so that unemployment fluctuations are driven by changes in This is consistent with the second of data Galí as well as the empirical evidence on the of monetary policy shocks as shown in Trabandt, and Walentin to demand Large ImageDownload the dynamic to the labor supply and markup which we under the of "labor market" shocks generate a comovement of inflation and the real wage with wage markup a positive on price inflation and unemployment and a one on output, employment, and the output thus a for the other an labor supply on output, employment, and the output gap positive on but instead to a rise in the output gap and a in the unemployment rate, so that policy It is this different effect on unemployment and the output gap associated with the two labor market shocks that makes their identification so important from a policy as discussed to labor market Large ImageDownload the estimated model's implied impulse to a positive technology and a price markup We to those shocks as their feature being that they generate a real wage response and a response of inflation. It is in with much of the empirical evidence Galí in our estimated model a positive technology to a in employment and a rise in the unemployment This is in contrast with the of real business cycle or and models. and in a way to wage markup we that price markup shocks also a policy between stabilizing inflation and the output This is not the case for technology since they both these in the to supply Large ImageDownload to questions that can be with our estimated model, we to the importance of from conventional preferences in order to match aspects of the data. Note that under preferences = the labor supply equation (5) can be written as where is to the As by et al. the previous equation is at with their empirical estimates of the of monetary policy which show a response of – – with a response of the labor under the assumed preferences, a response of the labor is consistent with the model as as the wealth effect is implying a small of and a response of – – This is illustrated in figure which the impulse of employment, the labor force, and the unemployment rate to a monetary policy under (1) our estimated model and (2) an identical model with preferences to = Note that in the latter and in contrast with the the labor following an of monetary the response of the unemployment rate and as important a of the latter as policy shocks and the role of wealth Large ImageDownload Wage the CKM this section we address one of the CKM to an large of wage shocks and a large of the latter to output and employment fluctuations, implied by estimated DSGE models (e.g., SW As by that evidence cannot be of much use to since the SW model is not to between wage markup and labor supply are as a in the wage even discussed have very different policy discussed that problem of identification is overcome by our reformulation of the SW model using the unemployment rate as an observable In the estimated of the for the exogenous wage markup in 1 the latter's deviation from to unemployment is as an on equation (7) and the estimated labor