Comparison of Sticky Wages & Labor Supply Constraints: Demand vs. Dréeze Equilibrium, Study notes of Material Science and Technology

Download Comparison of Sticky Wages & Labor Supply Constraints: Demand vs. Dréeze Equilibrium and more Study notes Material Science and Technology in PDF only on Docsity! Sticky Wage Models and Labor Supply Constraints By Zhen Huo and José-V́ıctor Ŕıos-Rull∗ In sticky wages models (either à la Calvo or à la Rotemberg), la- bor is solely determined by the demand side. However, a change of circumstances may make labor demand higher than agents’ willing- ness to work. We find that workers are required to work against their will between 15 percent and 30 percent of the time (with 5 percent wage markup, less with higher markups and in Rotem- berg models). Estimating models with the minimum of the de- mand and supply of labor instead of the demand-determined quan- tity yields different and unappealing properties. Hence, special at- tention should be paid to possible violations of the labor supply constraint. JEL: E20, E32, E37, E52 Keywords: Sticky wages, New Keynesian model, Drèze equilibrium In New Keynesian models with sticky wages à la Calvo or à la Rotemberg, the quantity of labor is solely determined by the demand side, implicitly assuming that households are always willing to work at whatever wage rate is specified. This assumption can be justified if the amount of labor is not larger than what agents are willing to work, or what we refer to as the labor supply constraint is not violated. Unions with monopsony power set wages above agents’ marginal willingness to work, which provides a cushion that accommodates the effects of various shocks on the demand and supply of labor. If the shocks to the economy are not too large, the cushion is sufficient to guarantee that households happily accommodate the quantity of labor required. In this paper we document that the cushion may be too small in popular DSGE models: demand-determined labor often implies that some of the labor is provided against the will of the workers, a violation of the principle of voluntary exchange. What is the natural alternative to the violation of the labor supply constraint? Here we have taken the strict position of staying within the two types of models that we explore, rather than proposing a change of model. In the Calvo model trade occurs at non-market clearing prices. For these type of environments, Drèze (1975), following the notion of disequilibrium modeling of Barro and Grossman (1971) and Malinvaud (1977), posed that the amount traded is the minimum of the quantities supplied and demanded and that the agents are aware of the ∗ Huo: Yale University, 28 Hillhouse Ave, New Haven, CT 06510, zhen.huo@yale.edu. Ŕıos-Rull: University of Pennsylvania, 133 S 36th St, Philadelphia, PA 19104, vr0j@upenn.edu. We are also grateful for the comments of the editor and three referees. We are also thankful for discussions with V.V. Chari, Jean-Jacques Herings, Patrick Kehoe, Ellen McGrattan and Frank Schorfheide. Ŕıos-Rull thanks the National Science Foundation for Grant SES-1156228. 1 limitation in the availability of the trades.1 The Drèze equilibrium provides an outcome that satisfies individual rationality without changing any feature of the environment and hence we think that it is the natural equilibrium concept when wages are deemed to be fixed, as they are in sticky wage à la Calvo models. For the Rotemberg model, we think that the obvious alternative to ignoring the labor supply constraint is to let the unions internalize the constraint when they set prices, automatically inducing that the constraint is not violated. We start our analysis by first substituting the demand-determined quantity of labor that the log-linearization procedure delivers with the minimum of the quantity of labor demanded and the quantity that agents would like to work. We refer to this quantity as voluntary ex-post aggregate labor or just ex-post labor.2 The comparison between the two series tells us whether some agents are working against their will in demand-determined allocations. The ex-post labor is not part of an equilibrium (Drèze or otherwise), since agents made their decisions based on the demand-determined quantity of labor, but it does give us a preliminary account of the extent to which the demand-determined allocation is consistent with agents not working against their will. We carry out this comparison in two of the most standard models of the New Keynesian literature: Altig et al. (2011) and Smets and Wouters (2007). With Calvo pricing we find that the properties of the two measures of labor are quite different: In the Altig et al. (2011) model, the fraction of workers with labor sup- ply constraint being violated varies from 19 percent to almost zero as the wage markup moves from 5 percent to 25 percent. The variance of labor shrinks by 15 percent for a wage markup of 15 percent when we move from the demand- determined to the ex-post quantity of labor. In the Smets and Wouters (2007) model, wage markup shocks are important in accounting for wage and labor move- ments, but their magnitude and structural interpretation are controversial (as we discuss below). Therefore, we look at versions with and without these shocks. With wage markup shocks, the fraction of workers that have their labor supply constraint violated varies from 45 percent to 16 percent as the wage markup moves from 5 percent to 25 percent. In expansions, ex-post labor shrinks so much so that the volatility actually becomes larger than that in the demand-determined case. With a 15 percent wage markup, the variance increases by 50 percent. When wage markup shocks are excluded, the fraction of workers that have their labor supply constraint violated varies from 32 percent to 2 percent as the wage markup goes from 5 percent to 25 percent. The variance of labor is 15 percent smaller in the demand-determined model for a wage markup of 15 percent. 1This requires that all firms internalize that they are treated equally when facing a limited labor supply. We can see this as the result of assuming that firms send bids and that the available workers are equally distributed between all firms. That firms understand this is consistent with the model. An alternative that would add a lot of complexity without any substance is to pose a randomization mechanism. 2The calculation of ex-post labor is not a trivial endeavor: Calvo pricing implies that many different wages coexist at any point in time depending on the exact period when the wage was last set, and there is a different quantity of labor associated to each one of those wages. It can be argued along the lines of Barro (1977), that the form of wage rigidity in the Calvo model that the Drèze equilibrium maintains, is inconsistent with rational behavior,4 and that the Calvo assumption should be changed. A possible alternative to the strict Calvo pricing rigidity assumption could be to specify additional circumstances under which prices or wages could change (violation of the labor supply constraint is one them). We take this possibility as a change of the physical environment and hence we choose not to pursue it in this paper. That being said, note that in Rotemberg models wages are reset every period, which is not subject to the Barro (1977) critique. As a result, the Calvo setting and the Rotemberg setting could be potentially viewed as two extreme benchmarks for other wage setting protocols when incorporating wage rigidities. Our paper focuses on the willingness of agents to work, but a similar argument can also be made on the willingness of firms to produce goods at a fixed price. For example, Corsetti and Pesenti (2005) emphasize that firms should only produce if the ex-post price markup is larger than one. Bills (2004) and Alessandria, Kaboski and Midrigan (2010) consider firms’ inventory stockout problems, where firms’ sales have to be the minimum of the goods demanded and their existing inventory. Michaillat and Saez (2015) combine nominal rigidity with matching frictions in both goods and labor markets, where supply and demand jointly determine the outcome via affecting market tightness. Van der Laan (1980), Kurz (1982), Dehez and Drèze (1984), Drèze (1997), and Citanna et al. (2001) are all related to Drèze (1975)’s original work, and study the properties of supply-constrained economy and explore the connection between price distortion and coordination failure. Herings (1996, 2014) extend Drèze (1975)’s work to settings with more flexible primitives and to dynamic environments. Bénassy (1993) compares the original Drèze equilibrium with other closely related disequilibrium concepts, and explore their implications in a static monetary economy with fixed prices and a fixed wage. Our paper differs from the previous literature in two ways. First, the market structure in our paper is monopolistic competition instead of perfect competition. Therefore, in periods where the wages can be reset, they will be set by forward-looking unions rather than the market.5 Second, the previous literature focuses on equilibrium existence and multiplicity, while our paper explores the quantitative properties of Drèze equilibrium in a state-of-the-art DSGE model. Organization. — We discuss the implicit assumption made in New Keynesian models when there is trade at non-market-clearing prices in Section I in the con- text of a model with wage setting à la Calvo. We proceed to explore in Section II the extent to which agents work against their will—what we jocularly label as 4Trujillo (1985), on the other hand, argued that rationality of conjectures can be defended and suffices to yield existence of equilibrium. 5Bénassy (1993) also considers the case where private agents set the prices and wages in a static environment, and in our paper agents need to solve a more complicated dynamic pricing problem. slavery—in standard New Keynesian models (versions of Altig et al. (2011) and Smets and Wouters (2007)) and conclude that it happens too often to simply look the other way. Section III discusses what we think is the appropriate equilibrium concept for economies with wage settings à la Calvo, the Drèze equilibrium (Drèze, 1975), and compares its properties with those of the demand-determined alloca- tion used in New Keynesian models and with those of an approximation to the Drèze equilibrium in various economies that we can solve. We then proceed to estimate a version of the Altig et al. (2011) model using the approximated Drèze equilibrium and we show that we obtain quite different estimates than those ob- tained when using demand-determined allocations in Section IV. Section V poses wage settings a la Rotemberg and explores the extent to which the labor supply constraint is violated. Section VI concludes by arguing that the approximation to the Drèze equilibrium should be used in lieu of the demand-determined equi- librium when studying environments with sticky wages. I. The Labor Market in New Keynesian Models à la Calvo We pose a typical New Keynesian model with sticky wages, first introduced by Erceg, Henderson and Levin (2000). There is a continuum of differentiated labor varieties ni, i ∈ [0, 1], which firms combine into a final labor input n for production using a Dixit-Stiglitz aggregator with elasticity of substitution εw: n = [∫ n εw−1 εw i di ] εw εw−1 . The wage wi is set by unions that are specific to each labor variety i. Firms take all wages as given. Cost minimization, given wages and total labor n, yields demand schedules for each labor variety i, (1) ni = (wi w )−εw n, where w is an aggregate wage index w = [∫ w1−εw i di ] 1 1−εw that satisfies ∫ wi ni di = wn. A representative household consists of a continuum of workers, each one with different labor variety i that enjoys the same consumption level. The household’s utility is given by E0 { ∞∑ t=0 βt ( u(ct)− ∫ i v(ni,t)di )} . The union sets the wage to maximize agents’ utility. The opportunity to reset the wage occurs with probability 1 − θw (à la Calvo) every period. The union’s problem is max w∗i,t Et { ∞∑ k=0 (βθw)k [ u′(ct+k) w∗i,t pt+k ni,t+k − v(ni,t+k) ]} ,(2) subject to ni,t+k = ( w∗i,t wt+k )−εw nt+k. The first-order condition is (3) Et { ∞∑ k=0 (βθw)k [ ni,t+ku ′(ct+k) ( w∗i,t pt+k − εw εw − 1 v′(ni,t+k) u′(ct+k) )]} = 0. Although not stated explicitly, this problem assumes that firms can choose any quantity that they want of each labor variety, which requires that workers comply. Note that the worker is not choosing how much to work. If it did, it would choose `i to equate the real wage to the marginal rate of substitution (the standard intratemporal Euler condition): (4) wi,t pt = v′(`i,t) u′(ct) . We refer to the `i that solves equation (4) as the optimal labor supply under wage wi. In the absence of wage rigidity (θw = 0), the union sets the wage every period and condition (3) becomes (5) w∗i,t pt = εw εw − 1 v′(ni,t) u′(ct) , i.e. marginal revenue equals the marginal rate of substitution, or in standard parlance, the real wage is set to equal the marginal rate of substitution multiplied by the wage markup εw εw−1 . Standard values for the elasticity of substitution ensure that what we call the labor supply constraint, `i ≥ ni, that agents would like to work more than the quantity chosen by firms, is not violated and hence that the determination of the equilibrium quantity of labor via the quantity demanded is justified. Under wage stickiness, however, the wage set by equation (3) may imply an optimal supply of labor `i,t < ni,t, violating the labor supply constraint. In this case, the assumption that labor is demand-determined implies that workers are working against their will (i.e., slavery). What is the correct notion of equilibrium within the Calvo model in the context of a non-market-clearing price? Drèze (1975), following the disequilibrium models We discuss the details of how to construct the voluntary ex-post labor in Sec- tion II.A. This is not a trivial endeavor, because at any point in time, there are a large number of different wages, each one of them affecting a different group of workers who have different preferred labor choices. The quantitative analysis is in Section II.B. A. The Determination of the Voluntary Ex-post Aggregate Labor To determine the desired labor supply of workers we have to keep track, not only of the aggregate wage index of the economy, but also of the wages for all labor varieties i. Fortunately, this can be done by noting that all labor varieties that set the wage in a given period choose the same wage. We describe our procedure in three steps. Step 1: Construct the cross-sectional wage distribution. — The measure of workers that can reset their wages in the current period is µ0 = 1− θw, while the measure of workers with wage reset τ periods before is µτ = (1−θw)θτw, τ = 0, 1, 2, . . ., so µτ becomes negligible for τ large enough. The simulation of the log-linearized model with demand-determined labor yields the sequence of the aggregate wage index {wt}, which evolves according to (9) wt = [∫ w1−εw i,t di ] 1 1−εw = [ θw(wt−1)1−εw + (1− θw)(w∗t ) 1−εw] 1 1−εw , where w∗t is the newly set wage in period t. Since we already have the aggregate wage sequence {wk}tk=0, we can easily calculate the sequence of newly set wages {w∗k}tk=0 using Equation (9). The wages prevailing in period t are then {w∗t−τ}, with corresponding measure µτ , τ ≥ 0. Step 2: Construct cross-sectional labor Demand and labor Supply. — Given aggregate labor {nt}, the labor demand for workers with wage rate w∗t−τ is nτ,t = ( w∗t−τ wt )−εw nt. Agents that face wage rate w∗t−τ , have an optimal choice of labor given by the `τ,t that solves w∗t−τ pt = v′(`τ,t) u′(ct) . Aggregating both series over cohorts or wage groups, we obtain the aggregate demand for labor,7 nt = [∑∞ τ=0 µτ , n εw−1 εw τ,t ] εw εw−1 , and the aggregate supply of labor `t = [∑∞ τ=0 µτ , ` εw−1 εw τ,t ] εw εw−1 . Step 3: Construct aggregate labor. — Voluntary ex-post labor, ept (we use the superscript p to denote that it is an ex-post quantity), is the minimum of supply and demand at each wage,8 (10) ept = [ ∞∑ τ=0 µτ (min {nτ,t, `τ,t}) εw−1 εw ] εw εw−1 , We want to emphasize that ept is not an equilibrium object, both because when making decisions, neither firms nor unions or workers take this factor into con- sideration, and because the implied path of consumption, investment, and capital is that associated with the demand-determined allocation. However, it allows us to check whether the extent to which the labor supply constraint is violated. If nτ,t < `τ,t all the time, then nt = ept and it is correct to use demand-determined labor. If instead, nτ,t > `τ,t happens frequently and the difference between nτ,t and `τ,t is large, then nt will be substantially different from ept and the answers obtained by models that use demand-determined quantities of labor are question- able. B. Quantitative Analysis of the Altig et al. (2011) and Smets and Wouters (2007) Models Sticky wage models lack a straight identification of the steady-state wage markup, which affects the dynamics of labor and wages only through the slope of the wage Phillips curve that also depends on other deep parameters. The parameter εw that determines the steady-state markup is typically set exogenously. For exam- ple, Altig et al. (2011) sets the wage markup to be 5 percent, Smets and Wouters (2007) sets its value to 50 percent, and most DSGE models set this value between 5 percent to 25 percent.9 Gaĺı (2011) uses the relationship between the wage markup, the unemployment rate and the Frisch elasticity in (8), and explores 7Under log-linearization, an approximation error results in a negligible difference between aggregate labor and this expression. 8Quantitatively, the difference between the Dixit-Stiglitz aggregator and the linear average labor is negligible. 9Lewis (1986) surveys the literature on the wage premium for workers in a union, which corresponds to the wage markup in the model, and the value is between 10 percent to 20 percent. The steady-state wage markup is 5 percent in Christiano, Eichenbaum and Evans (2005), 15 percent in Chari, Kehoe and McGrattan (2002), and 20 percent in Levin et al. (2006). markups from 5 percent to 25 percent for an empirically relevant range of Frisch elasticity.10 In this paper we have chosen to estimate both of models, setting the wage markup to values in accordance with the recent literature, ranging from 5 percent to 25 percent. For values larger than 25 percent, the labor supply constraint turns out to be much less relevant. The Labor Supply Constraint in the Altig et al. (2011) Model . — Figure 1 displays sample paths of the demand-determined labor (nt) and of the volun- tary ex-post labor ept constructed as discussed in Section II.A for different wage markups. Note that demand-determined labor is always as large as the voluntary ex-post aggregate labor by construction. The difference between these two series is noticeable. Both series coincide in recessions, but the voluntary ex-post labor does not expand as much as the demand-determined labor in expansions. In fact, for a 5 percent wage markup, the voluntary ex-post labor actually declines when the demand-determined labor expands. This is because the expansion takes place by asking low-paid workers to supply a huge amount of labor which is no longer possible if the workers can choose not to meet the demand. Also, the smaller the wage markup, the larger the differences between these two series as the average distance between labor demand and labor supply shrinks. 0 100 200 300 400 500 -8 -6 -4 -2 0 2 4 6 5 percent wage markup 0 100 200 300 400 500 -6 -4 -2 0 2 4 6 10 percent wage markup 0 100 200 300 400 500 -6 -4 -2 0 2 4 6 25 percent wage markup Figure 1. Sample Paths in Altig et al. (2011) Table 1 summarizes the relevant statistics to compare both labor series for the Altig et al. (2011) economy for wage markups ranging form 5 percent to 25 percent. As discussed the violation of the labor supply constraint is more important the lower the wage markup. When the wage markup is 5 percent (the actual choice in Altig et al. (2011)), the voluntary ex-post labor is on average less than 1.41 percent lower than in the demand-determined. Almost 19 percent 10In an estimated version, Gaĺı, Smets and Wouters (2012) obtain the steady-state wage markup with an value 18 percent. Table 2—Smets and Wouters (2007) w/ and w/o Wage Markup Shock w/ Wage Markup Shock w/o Wage Markup Shock mean var corr w/ labor mean var corr w/ labor output violation output violation 5 percent wage markup Demand-determined — 1.16 0.81 44.57 — 1.02 0.80 31.63 Voluntary ex-post -6.73 11.76 0.01 — -1.29 1.04 0.27 — 10 percent wage markup Demand-determined — 1.14 0.81 37.73 — 0.98 0.79 12.95 Voluntary ex-post -3.53 3.24 0.16 — -0.50 0.66 0.53 — 15 percent wage markup Demand-determined — 1.13 0.81 29.98 — 0.96 0.79 5.79 Voluntary ex-post -2.14 1.73 0.32 — -0.23 0.70 0.68 — 25 percent wage markup Demand-determined — 1.12 0.81 16.38 — 0.94 0.78 1.57 Voluntary ex-post -0.84 1.07 0.59 — -0.06 0.85 0.76 — Note: All the variables except the mean are logged and HP filtered. The column labor violation corresponds to the average measure of workers whose labor supply constraint is violated. this shock can generate by itself large violations of the labor supply constraint.13 To address this concern, we also examine the labor supply constraint in the Smets and Wouters (2007) economy without the wage markup shocks. Figure 3 shows a sample path without markup shocks of the demand-determined labor and the voluntary ex-post labor series. We see immediately that the two series are much closer to each other, indicating that indeed the problem of violating the labor supply constraint may be much smaller without the labor markup shock. Table 2 compares the demand-determined labor with the ex-post labor with and without wage markup shocks. Things are quite different, yet even without the wage markup shocks the labor supply constraint is violated quite often; there is also a sizeable reduction of average hours worked and a weakening of the cor- relation between labor and output; and last but not least, there are important differences in the variance of the labor series. For middle markups (10 percent, 15 percent) the variance of the demand-determined series is between 37 percent and 13See Chari, Kehoe and McGrattan (2009) for a discussion for different possible interpretations of the wage markup shocks. 0 100 200 300 400 500 -8 -6 -4 -2 0 2 4 6 5 percent wage markup 0 100 200 300 400 500 -5 0 5 10 percent wage markup 0 100 200 300 400 500 -4 -3 -2 -1 0 1 2 3 4 5 25 percent wage markup Figure 3. Sample Paths in Smets and Wouters (2007) w/o Wage Markup Shocks 48 percent larger than the voluntary ex-post series. Curiously, for the 5 percent markup version of the economy the opposite is true and the variance of the vol- untary ex-post labor is slightly larger than that of the demand-determined series. The reason for this is, again, that for a small wage markup, voluntary ex-post labor sometimes shrinks to the point of moving in the opposite direction than the demand-determined, and hence what is an expansion under demand-determined labor may be a recession in terms of voluntary ex-post labor as the much lower correlation with output indicates. -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 demand-determined voluntary ex-post Altig et al. (2011) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.005 0.01 0.015 0.02 0.025 demand-determined voluntary ex-post Smets and Wouters (2007) Figure 4. Cross-sectional Dispersion of Labor Cross-sectional Dispersion of Labor across Sectors . — An implication of models with wage settings à la Calvo that is often ignored is that there is signifi- cant cross-sectional dispersion of hours across labor varieties i. In expansions, the increase in aggregate labor is mostly due to an increased labor demand for vari- eties that reset wages a while ago at a low level, and the varieties that reset wages more recently typically choose a high wage that can actually lead to a decline of the demand for their labor. The implied dispersion is especially large when the steady-state wage markup is relatively small. Figure 4 shows the cross-sectional distribution of hours for the Altig et al. (2011) and Smets and Wouters (2007) models for a 5 percent wage markup (without wage markup shocks). We see the large dispersion of the demand-determined labor and how the voluntary ex-post labor limits those varieties with high demand. Table 3—Standard Deviation of Cross-Sectional Labor Altig et al. (2011) Smets and Wouters (2007) without markup shocks Wage Markup 5 percent 15 percent 25 percent 5 percent 15 percent 25 percent Demand-determined 0.069 0.039 0.029 0.080 0.046 0.035 Voluntary ex-post 0.046 0.035 0.029 0.048 0.038 0.033 Table 3 also illustrates this point for a variety of wage markups by showing the standard deviations of the cross-sectional demand-determined sector-specific labor demands and of the cross-sectional voluntary ex-post sector-specific labor demands. The dispersion is much larger for all demand-determined labor varieties than for the voluntary ex-post labor varieties. III. Drèze Equilibrium So far, we have made the case that the use of demand-determined labor as the equilibrium condition is inappropriate because households want to work less quite often: the minimum of the amount of labor demanded and supplied (as standard theory considers the appropriate equilibrium condition) behaves very differently than the amount of labor demanded. However, the series that we have constructed (voluntary ex-post labor) is not an equilibrium object because it is constructed along a path defined by the demand-determined labor and its associated series: output, consumption, investment, prices, wages, and so on. Moreover, the forecasts of agents are those of the demand-determined allocation. Therefore, we need to compute the Drèze equilibrium explicitly. Unfortunately, log-linearization cannot be used to solve for the Drèze equilib- rium. Global methods are needed given that the equilibrium condition is based on the min operator. Recent developments in computational economics that allow us to deal effectively with corner solutions, e.g. Guerrieri and Iacoviello (2015) the behavior of all other parts of the economy. Workers cannot be made to work against their will, and the union takes into account that there is an upper bound on the amount of labor that will be provided in their sector. The union chooses a nominal wage that will be effective for Tw periods: max w∗t Et Tw−1∑ k=0 { βku′(ct+k) w∗t pt+k ei,t+k − e1+γ i,t+k 1 + γ } (12) subject to ei,t+k = min {( u′(ct+k) φ w∗t pt+k ) 1 γ ,Ψt+k(w ∗ t ) } ,(13) where Ψt+k(·) is the desired labor demand from the firm’s side and the labor supply function Φ(wi,t) is given by Φ(wi,t) = ( u′(ct+k) φ wi,t pt ) 1 γ . In the standard model, the constraint for the union is simply ei,t+k = Ψt+k (w∗t ). In summary, we have defined three objects: labor supply in variety i, `i,t = Φt(wi,t), labor demand in variety i, ni,t = Ψt(wi,t), and actual labor in variety i, ei,t = min{`i,t, ni,t}. To complete the model, we include a simple Taylor type monetary policy rule: logRt = log 1 β + φππt + φy log yt y∗ + ηt, where πt = log pt pt−1 and y∗ is the steady-state output level. The shock to the monetary policy rule follows an AR(1) process, ηt = ρmηt−1 + ζmt , ζmt ∼ N(0, σ2 m). The details of the numerical solution via global methods of this economy can be found in Appendix B. B. The Approximated Drèze Equilibrium Even in the simplified staggered wage model, computing the exact Drèze equi- librium is computationally intense. We therefore consider an approximation to the Drèze equilibrium which does not require the global solution. It has a much smaller computational burden and can be applied to medium-size DSGE models. As in our calculation of the voluntary ex-post aggregate labor, we also employ the log-linearized solution of the demand-determined allocation and then impose the ex post labor supply constraint. But unlike in the construction of the ex-post labor, the approximated Drèze equilibrium reconstructs all the main aggregate variables recursively, including capital, output, interest rate, and so on, guarantee- ing that the resources constraints are satisfied. In what follows, we compare this approximation with the exact Drèze equilibrium, and we find that the allocations are very similar. As a result, we argue that the approximated Drèze equilibrium can be used to address questions in medium-size DSGE models where computing the exact Drèze equilibrium is extremely hard. Specifically, the construction of the approximated Drèze equilibrium consists of the following four logical steps. Step 1: Log-linearize and solve the demand-determined equilibrium. — This is a standard step. The decision rules are required, not just a simulation. Step 2: Recursively construct a voluntary ex-post measure of labor. — This step is what we described in Section II.A. The key difference is that there we use the sequence of capital stocks yielded by the demand-determined equilibrium, which may not be feasible. Thus, at this stage we construct a measure of the voluntary labor one period at a time, denoted as eat . In this step we keep track of historical wages, wat , which also include the information about the cross-sectional wage distribution. Step 3: Recursively construct the main aggregate variables. — Here we use the labor in period t, eat , and the previous period series of capital kat to calculate output yat (which is also used to construct the output gap). We then use the same policy function as in the demand-determined equilibrium to determine the newly set wage and price level. This is an approximation, since in the true Drèze economy, agents will take into account the possibility that the labor supply constraint may be binding. The interest rate Rat is set by using the reconstructed output gap. This part is mechanical. Step 4: Determine consumption, investment, and next period capital. — This step is not mechanical. We have considered two possibilities: use the same consumption-to-output ratio or the same consumption of the demand-determined solution (investment is set residually to satisfy the resource constraint). We fi- nally chose the same consumption because choosing the consumption-to-output ratio sometimes leads to countercyclical consumption. More specifically, in the demand-determined economy, after a positive technology shock, the consumption- output ratio is below its steady-state level because agents understand that it is better to increase investment to take advantage of the temporary high produc- tivity. In the Drèze equilibrium, however, the response of labor is much more subdued with the same positive technology shock, which may lead to a much smaller expansion. If we used the low consumption-to-output ratio of the demand- determined allocation, there would be a recession rather than an expansion. We do not want to argue that our approximation strategy is conceptually ideal, and we are aware that the allocation obtained in this approximation is subject to the fact that agents are not fully rational. The usefulness of this approximated equilibrium is simply justified by its small distance to the true Drèze equilibrium as we will show next. C. A Comparison between the Drèze Equilibrium and Its Approximation We now specify the staggered wage model quantitatively and solve for the Drèze equilibrium and for its approximation. The model has a large number of state variables to keep track of the wage distribution (see Appendix B for more details). The model period is a quarter and the annual interest rate in the steady state is 4 percent. The implied Frisch elasticity is 0.75 ( 1 γ ), similar to estimates in Heathcote, Storesletten and Violante (2010). The labor share is 0.64, and the capital depreciation rate is 0.08 annually. The process for the TFP shock is similar to the one used in Ŕıos-Rull and Santaeulàlia-Llopis (2010). The monetary policy rule is the same as in Christiano, Eichenbaum and Rebelo (2011). The persistence of the monetary shock is 0.5, the same as in Gaĺı (2008). We set the standard deviation of the innovation to the monetary shock to be 0.004. As discussed earlier, the most important parameter is εw, which determines the wage markup. The one we use here implies a 10 percent wage markup. If we apply the logic of equation (8), our choice of εw and γ leads to a 6 percent average unemployment rate15. We choose the duration of the wage contract to be four model periods, or one year. 0 20 40 60 80 100 -6 -4 -2 0 2 4 6 Demand determined Dreze equilibrium Approximated Dreze Voluntary ex-post With TFP Shock 0 20 40 60 80 100 -6 -4 -2 0 2 4 6 Demand determined Dreze equilibrium Approximated Dreze Voluntary ex-post With Monetary Shock Figure 5. Sample Paths in the Staggered Wage Model 15Following Gaĺı, Smets and Wouters (2012), the unemployment rate in sector i (the economy-wide counterpart is immediate) is ui,t = log `i,t − log ei,t. prevents us from applying standard linear Kalman filter techniques in evaluating the model’s likelihood. The alternative nonlinear Kalman filter requires large computational power, which is only feasible for models with a relatively small number of state variables. We can, however, estimate the approximated Drèze equilibrium in Altig et al. (2011), the other central model in the New Keynesian literature. Altig et al. (2011) and its precedent Christiano, Eichenbaum and Evans (2005) estimate a medium-scale DSGE model by matching the impulse responses of various variables to different shocks. The impulse responses are recovered from the estimation of a certain structural vector autoregression (VAR) model. The identification strategy in Altig et al. (2011) is similar to that in Christiano, Eichenbaum and Evans (2005), where only nominal variables like the velocity of cash balances respond to contemporaneous monetary policy shocks but not the real variables such as hours, consumption, investment, and so on. It is also assumed that monetary policy is set conditional on the current values of real variables and only on the past values of nominal variables. In addition, innovations to technology (both neutral and capital embodied) are the only shocks that affect long-run labor productivity, and capital embodied technology shocks are the only shocks that affect the long-run relative price of investment goods. Crucial to this endeavour is the ability to identify the shocks, something that can be done with the three shocks in Altig et al. (2011). The parameters of the model are chosen in such a way that the model’s impulse responses to the structural shocks match their counterpart estimated from the data. In particular, three structural shocks are considered: a monetary shock, a neutral technology shock, and an embodied investment technology shock. The estimation method is generalized method of moments (GMM), which only requires the impulse response of the model.17 Because the likelihood of the model is not required, we can apply this estimation method to the approximated Drèze equilibrium using the same exogenously calibrated parameters than Altig et al. (2011). Table 6 shows the properties of the estimates of the approximated Drèze equi- librium and of the demand-determined allocation in the Altig et al. (2011) model with a 5 percent markup. Our interpretation of these very different sets of esti- mates is that the unwillingness of households in the Drèze equilibrium to work a lot under some circumstances requires that other pieces of the model have to do a lot more work to create the observed fluctuations: 1) The neutral technology shock is dramatically affected. To induce more movement in labor, the estimated shock is now both much more volatile and less persistent: the unconditional variance of the neutral technology shock is 0.039 in the Drèze equilibrium relative to 0.024 in the demand- determined allocation. A larger, but less persistent, shock makes households 17The weighting matrix of GMM is diagonal with the inverse of the standard deviations of the impulse responses estimated in the structural VAR. Table 6—Estimated Parameter Values Demand-Determined Approximated Drèze Std Dev of neutral tech shock, σµz 0.068 0.140 (0.046) (0.089) Autocor neutral tech shock, ρµz 0.902 0.697 ( 0.102) (0.240) Std Dev of monetary shock, σM 0.331 0.325 (0.084) (0.078) Autocor monetary policy shock, ρM -0.037 -0.040 (0.111) (0.130) Std Dev of embodied tech shock, σµΥ 0.303 0.286 (0.042) (0.046) Autocor embodied tech shock, ρµΥ 0.241 0.318 (0.224) (0.176) Wage rigidity, ξw 0.722 0.825 (0.123) (0.043) Price rigidity, γ 0.040 0.054 (0.029) (0.039) Variable capital utilization, σa 1.995 4.564 (2.222) (7.070) Investment adjustment cost, S′′ 3.281 4.752 (2.038) (2.378) Interest elasticity of money demand, ε 0.808 0.779 (0.208) (0.193) Habit formation, b 0.706 0.698 (0.045) (0.058) Effects of neutral tech shock on policy, ρxz 0.343 0.195 (0.266) (0.480) Effects of embodied tech shock on policy, ρxΥ 0.824 0.832 (0.154) (0.132) Scaling factor of neutral tech shock, cz 2.997 1.027 (2.310) (0.749) Scaling factor of neutral tech shock, cpz 1.327 0.665 (1.381) (0.650) Scaling factor of embodied tech shock, cpΥ 0.135 0.107 (0.244) (0.268) Scaling factor of embodied tech shock, cΥ 0.246 0.305 (0.244) (0.266) Note: The estimation is with 5 percent wage markup. The magnitude of shocks that generate the impulse response functions are set to their standard deviations. more willing to supply labor. 2) The rigidity of wages and prices is somewhat larger. The lower response of labor in the Drèze equilibrium also requires, perhaps a bit counterintuitively, larger rigidities in the model to generate more fluctuations. This is true both for wages, where the Drèze equilibrium is imposed, and for prices, where it is not. 3) Two other pieces of the model are now larger. The role of variable capital utilization is now value, as is investment adjustment cost parameter. Still, these two parameters are somewhat imprecisely estimated and we should not insist on them. Table 7—Labor Comparison with Different Estimation Strategies Estimated with Estimated with Demand-Determined Approximated Drèze mean var corr w/ labor mean var corr / labor output violation output violation Neutral Technology Shock Demand-Determined — 0.18 0.87 15.09 — 0.24 0.97 19.03 Approximated Drèze -1.57 1.16 0.96 — -2.59 1.41 0.95 — Investment Technology Shock Demand-Determined – 0.67 0.99 6.22 – 0.52 0.99 7.89 Approximated Drèze -0.42 0.34 0.98 — -0.55 0.32 0.99 — Monetary Shock Demand-Determined — 0.46 1.00 2.56 — 0.33 1.00 1.15 Approximated Drèze -0.07 0.33 0.99 — -0.01 0.30 1.00 — All Shocks Demand-Determined — 1.38 0.96 18.83 — 1.15 0.95 22.63 Approximated Drèze -2.28 2.06 0.98 — -3.41 1.99 0.96 — Note: Numbers are in percentages except for the correlation with output. Table 7 shows what the different solutions yield for each of set of estimates obtained. The left panel of the table shows the effects of the processes esti- mated via the demand-determined solution for labor when we look both at the demand-determined solution and at the approximated Drèze equilibrium. The right panel shows the effects of the processes estimated with the approximated Drèze equilibrium when we both look at the demand-determined solution and at the approximated Drèze equilibrium. The numbers in boldface are the properties of the economies when they are used to estimate the parameters. The Table shows some other important features of the differences between the demand-determined solution and the approximated Drèze equilibrium: 4) The estimates of the approximated Drèze equilibrium increase the role of the neutral technology shock. The variance of labor is larger using both V. Wage Adjustment Costs à la Rotemberg A popular alternative to nominal wage rigidity à la Calvo is the wage-adjustment cost mechanism proposed by Rotemberg (1982), where the nominal wage can be reset every period, but there is a quadratic adjustment cost when its value is changed. As shown in Born and Pfeifer (2016), by suitably choosing the adjust- ment quadratic cost parameter, the aggregate wage dynamics in the Rotemberg model and the Calvo model are identical up to a first order approximation. But being observationally equivalent at the aggregate level does not imply that the cross-sectional labor allocation and the extent to which the labor supply con- straint is violated is the same: in the Calvo model, the aggregate wage is less volatile than the wages for each labor variety, while in the Rotemberg model, a common wage prevails in the economy and is reset every period. In the Calvo model there are workers whose wage was set at a low level long time ago and hence they are likely to have their labor supply constraint violated making it natural to expect that the labor supply constraint will be binding more frequently than in the Rottemberg model. Meanwhile, as mentioned in the introduction, the Calvo pricing assumption may be too rigid and one may want to specify additional circumstances under which prices or wages could change (violation of the labor supply constraint being one of them). The Rotemberg model could be viewed as an example where wage rigidity is introduced without the strong assumption that wages cannot be changed no matter what. As a result, it is not subject to the Barro (1977) critique. To explore the extent to which violating the labor supply constraint is quantita- tively relevant in the Rotemberg model, we revisit the Smets and Wouters (2007) and Altig et al. (2011) models under a Rottemberg wage adjustment mechanism and we compare its performance relative to the restriction that the amount of labor cannot exceed what workers are willing to work. Consider economies like the ones described in the previous sections except in the fact that labor unions can change the wage every period subject to a quadratic wage adjustment cost. The variety i union’s problem differs from that in Equa- tion (1) and becomes max {wi,t+k} Et ∞∑ k=0 βk [ u′(ct+k) ( wi,t+k pt+k ni,t+k − ϑ 2 ( wi,t+k wi,t+k−1 − 1 )2 yt+k ) − v(ni,t+k) ] , subject to ni,t+k = ( wi,t+k wt+k )−εw nt+k, where parameter ϑ determines the size of the wage adjustment cost which is assumed to be proportional to nominal output ptyt. Here, the union implicitly assumes that workers are always willing to supply amount ni,t+k of labor. Due to the equivalence of the Rotemberg model and the Calvo model after linearization, we choose a Rotemberg cost parameter ϑ so that it will imply the same wage Phillips curve as that in the Calvo model. To see whether the standard characterization of Rottemberg pricing (the demand- determined solution) violates the requirement that agents do not work more than what they want we compare the demand-determined quantity of labor that solved the previous problem, n∗t , with the minimum of this demand-determined quantity and the actual amount of work that agents are willing to supply, this is, the `∗t that solves wt pt = v′(`t) u′(ct) . We denote this min{n∗t , `∗t } the ex-post labor. Note that in this case we do not need to construct the cross-sectional labor demand and labor supply as we did in Section II.A for the Calvo type economies since all labor types have the same wage in equilibrium. Table 8—Altig et al. (2011) with Rotemberg Adjustment Costs 5 percent wage markup 10 percent wage markup mean var corr w/ labor mean var corr w/ labor output violation output violation Demand-Determined — 1.38 0.96 5.37 — 1.35 0.96 0.06 Voluntary Ex-post -0.07 1.24 0.93 — 0.00 1.35 0.96 — Note: All the variables except the mean are logged and HP filtered. The column labor violation corresponds to the frequency of the labor supply constraint violation. The wage markup used is 5 percent. Tables 8 and 9 compare the properties of the quantities of labor allocations implied by ignoring workers willingness to work with those that arise when such constrained is taking into account. The labor supply constraint is still sometimes violated with Rottemberg adjustment costs, although as expected, less often than in economies with Calvo pricing. In the Altig et al. (2011) economy, the labor supply constraint binds 5 percent of the time, and labor volatility is 11 percent larger when ignoring the labor supply constraint than when imposing it with a 5 percent markup, while it is essentially identical with a 10 percent markup. In the Smets and Wouters (2007) without wage markup shocks, and with a 5 percent markup the labor supply constraint binds 19 percent of the time and has 23 percent larger labor volatility when we ignore the labor supply constraint. There are almost no differences with a larger markup. When the wage markup shocks are also included, the differences between the economy that ignores the labor supply constraint and that where we prevent that constraint to be violated are much more dramatic. For a 5 percent markup the constraint is violated 36 percent of the time, leading to a labor volatility 16 percent larger and a correlation Table 9—Smets and Wouters (2007) with Rotemberg Adjustment Costs w/o wage markup shock with wage markup shock mean var corr w/ labor mean var corr w/ labor output violation output violation 5 percent wage markup Demand-Determined — 1.01 0.80 18.48 — 1.16 0.82 35.79 Voluntary Ex-post -0.19 0.82 0.69 — -1.12 1.00 0.49 — 15 percent wage markup Demand-Determined — 0.95 0.79 0.19 — 1.13 0.81 15.55 Voluntary Ex-post 0.00 0.95 0.79 — -0.41 1.07 0.68 — 25 percent wage markup Demand-Determined — 0.93 0.78 0.00 — 1.12 0.81 5.50 Voluntary Ex-post 0.00 0.93 0.78 — -0.12 1.09 0.77 — Note: All the variables except the mean are logged and HP filtered. The column labor violation corresponds to the frequency of the labor supply constraint violation. with output also much larger in the unconstrained economy. With a 15 percent markup the constraint binds 16 percent of the time, volatility is still 5 percent larger and labor is clearly more correlated with output. Even with a the 25 percent markup the differences are noticeable: hours are 3 percent more volatile and the correlation is .04 larger than in the constrained economy. To summarize, with a Rotemberg wage setting mechanism the issue of agents working against their will is also present. In the Altig et al. (2011) economy this happens only when the markup is no higher than 5 percent. In the Smets and Wouters (2007) economy this happens also with markups no higher than 5 percent when we ignore wage markup shocks. But when we include them, the labor supply constraint becomes very relevant as ignoring it yields noticeably larger labor volatility and correlation between labor and output than when it is imposed. 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