Why is Unemployment so Countercyclical?

We argue that wage inertia plays a pivotal role in allowing empirically plausible variants of the standard search and matching model to account for the large countercyclical response of unemployment to shocks.


Introduction
Wage rigidities play a critical role in many quantitative business cycle models. Estimated New Keynesian (NK) models consistently feature important nominal wage rigidities. 1 The newest generation of heterogeneous agent (HANK) models also assigns prominent roles to wage rigidities. 2 There is also a long tradition of wage rigidities in open-economy models of aggregate fluctuations. 3 Wage rigidities enable all of these models to generate employment fluctuations that are comparable to those observed in the data.
But do wage rigidities play an important role in understanding why unemployment is so countercyclical? The standard framework used to study unemployment features search and matching frictions of the sort emphasized by Diamond (1982), Mortensen (1982) and Pissarides (1985) (henceforth, DMP). Shimer (2005) showed that standard versions of these models cannot -at least with plausible parameter values -explain the countercyclical behavior of unemployment. This 'Shimer critique' has led to a very large literature whose goal is to account quantitatively for the dynamics of unemployment. There is intense disagreement within this literature about the role of wage inertia in the cyclical behavior of unemployment.
Authors like Hall (2005) and Rogerson and Shimer (2011) argue that wage inertia greatly increases the cyclical volatility of unemployment in search and matching models. Similarly, Gertler and Trigari (2009) and Christiano, Eichenbaum and Trabandt (2016, CET) stress the importance of wage rigidities in estimated New-Keynesian search and matching models.
In sharp contrast, Hagedorn and Manovskii (2008, HM) and Ljungqvist and Sargent (2017, LS) argue that the emphasis on wage rigidity is misplaced. Building on HM's analysis, LS stress the usefulness of what they call the 'fundamental surplus fraction' for understanding the response of unemployment to shocks. By the fundamental surplus fraction they mean the "upper bound on the fraction of a job's output that the invisible hand can allocate to vacancy creation". 4 LS argue that the fundamental surplus fraction must be small to account for the Shimer puzzle and that wage rigidities don't play any special role in this connection. We argue that wage rigidities do in fact play a pivotal role in allowing variants of the standard search and matching models to account for the large countercyclical response of unemployment to shocks. In fact wage rigidities are necessary in empirically plausible versions of those models for explaining the cyclical volatility of unemployment.
In section two we present a simple labor market search and matching model. In section three we proceed as in much of the literature and use comparative steady state analysis as a short-cut for analyzing model dynamics. We characterize the implications of the search and matching model for the volatility of unemployment by a standard statistic: the elasticity of steady state labor market tightness with respect to a shift in the steady state marginal revenue product of labor. We decompose that elasticity into a wage inertia component and a profit rate component. Our decomposition makes clear that wage inertia is a necessary condition for generating a large countercyclical response of unemployment to a shift in the marginal revenue product of labor. If the wage rate responds one-to-one to the marginal revenue product across steady states, then the unemployment rate remains exactly the same across those steady states. Our decomposition also makes clear that, other things equal, the less the wage rate responds to the marginal revenue product, the more responsive is the unemployment rate. We show that similar conclusions hold for changes in the discount rate of the sort considered in Hall (2017).
We apply our decomposition to analyze the role of wage inertia in di↵erent wage-setting models estimated by CET. Specifically, we consider versions of their DSGE model where wages are set according to Nash bargaining and alternating o↵er bargaining (AOB). Both models solve the Shimer puzzle because they have more wage inertia than does the model considered by Shimer. A sticky wage variant of the model, in which wages are constant across steady states, generates an enormous response of unemployment to a permanent shift in firms' marginal revenue product. As a by-product of our analysis, we provide a counter- 4 See LS, abstract. example to the claim in LS that search and matching models can only successfully address the Shimer critique by incorporating features that make the inverse of the fundamental surplus fraction large. In that section we also address the HM critique of wage inertia as a solution to the Shimer puzzle.
The comparative steady state approach can yield important insights about the role of wage rigidity in dynamics. However, it can also be very misleading. The approach assumes the underlying shock is close to a random walk and the economy does not have quantitatively important state variables. These two assumptions are satisfied in the simple search and matching model, e.g., Shimer (2005). But they are not satisfied in generalizations of that model which include transitory shocks and a rich assortment of state variables. Moreover, model features like adjustment costs and nominal rigidities give rise to additional sources of dynamics while leaving no trace in the steady state. A steady-state analysis could assign no role at all to wage rigidity even if in fact it plays an very important role.
In light of these considerations, section four focuses on the dynamic response of unemployment to shocks. We do so using variants of the Nash and AOB models estimated in CET. Our main results can be summarized as follows. First, wage inertia greatly magnifies the response of unemployment to shocks. That is, models which do well at matching the data like the estimated AOB model do poorly if one replaces the wage determination mechanism by one in which wages are less inertial. Models that do badly at matching the data, like the Nash model with plausible replacement ratios, do much better if we impose on them wage processes that are more inertial. Second, we show that steady-state based measures of the fundamental surplus are uninformative about the dynamic response of unemployment to shocks. Models which have identical steady states exhibit dynamic response functions of unemployment to shocks that are very di↵erent from each other. These di↵erences are driven by di↵erent degrees of wage inertia.
Section five contains concluding remarks.

A Simple Labor Market Model
In this section we consider a simple discrete time search and matching model developed in CET. There is a continuum of identical workers and firms. To map into the medium-sized DSGE model in section 4, we refer to these firms as wholesaler firms. Firms produce a good using labor as the sole input. A wholesaler firm that wishes to meet a worker in period t must post a vacancy at a cost s t , expressed in units of the consumption good. The vacancy is filled with probability Q t . Following Pissarides (2009), the firm must pay a fixed real cost,  t , before bargaining with the newly-found worker.
Let J t denote the value to the firm of a worker, expressed in units of the final consumption good: Here, # p t denotes the expected present discounted value, over the duration of the worker/firm match, of the marginal revenue product of the worker. The latter could be stochastic because technology shocks a↵ect the productivity of workers or because there are shocks to the price of the wholesale firm good. The variable w p t denotes the discounted value of the real wage. The latter is determined by worker-firm bargaining and is discussed below. In recursive form: Here ⇢ is the probability that a given firm/worker match continues from one period to the next. In equation (2), m t+1 is a discount factor which firms and workers view as an exogenous stochastic process with a well defined steady-state value. We allow m t to be stochastic in anticipation of the medium-sized DSGE model considered in section 4.
The law of motion for aggregate employment, l t , is given by: The term ⇢l t 1 denotes the number of workers that were attached to firms in period t 1 and remain attached at the start of period t. The variable x t denotes the hiring rate so that x t l t 1 represents the number of new firm/worker meetings at the start of period t. The number of workers searching for work at the start of period t is the sum of the number of unemployed workers in period t 1, 1 l t 1 , plus the number of workers that separate from firms at the end of t 1, (1 ⇢)l t 1 . Consequently, the probability, f t , that a searching worker meets a firm is given by: Free entry by wholesalers implies that, in equilibrium, the expected benefit of a vacancy equals the cost: Let V t denote the value to a worker of being matched with a firm. We write V t as the sum of the expected present discounted value of wages earned while the firm-worker match endures and the continuation value, A t , when the match terminates: Here and where D t denotes unemployment benefits. In addition,Ũ t denotes the continuation value of unemployment:Ũ vacancy filling rate, Q t , and the job finding rate for workers, f t , are related to t , as follows: where m > 0, 0 < < 1 and Here, v t l t 1 denotes the total number of vacancies posted by firms at the start of period t.
As soon as the l t matches are determined at the start of period t, each worker in l t engages in bilateral bargaining over the current wage rate, w t , with a wholesaler firm. Each worker-firm bargaining pair takes the outcome of all other period t bargains as given. In addition, they take as given the outcome of future wage agreements as long as the worker and firm remain matched. Because bargaining in period t applies only to the current wage rate, we refer to it as period-by-period bargaining. The bargaining problem of all worker-firm pairs is the same, regardless of how long they have been matched. 5 In the basic search and matching model, the match surplus J t + V t U t is split between a matched firm and worker according to Nash bargaining. The Nash-sharing rule is given by: where ⌘ is the share of total surplus going to the worker.
Following Hall and Milgrom (2008) and CET, we also consider a version of the model in which real wages are determined by alternating o↵er bargaining (AOB). We suppose that bargaining proceeds across M sub-periods within the period, where M is even. 6 The firm 5 This result follows from our assumptions that hiring costs, i.e. s t and  t , are sunk when bargaining occurs and the expected duration of a match is independent of how long a match has already been in place. 6 Our model di↵ers from Hall and Milgrom's in two ways. First, they assume alternating o↵ers are made in successive periods, t, t + 1, etc., and can potentially continue indefinitely. With this assumption, they must specify the time period of the model to be shorter than the quarterly or monthly rate over which many macroeconomic variables are measured. Our approach, which assumes that bargaining proceeds within a period, means that when we use standard time series estimation methods which use quarterly data, as in CET, we can avoid having to explicitly take into account temporal aggregation e↵ects. Second, we assume that a worker can go from one job to another without passing through unemployment. makes a wage o↵er at the start of the first sub-period. It also makes an o↵er at the start of a subsequent odd sub-period in the event that all previous o↵ers have been rejected. The cost to a firm of making an o↵er is t . Similarly, the worker makes a wage o↵er at the start of an even sub-period in case all previous o↵ers have been rejected. The worker makes the last o↵er, which is take-it-or-leave-it. In sub-periods j = 1, ..., M 1, the recipient of an o↵er has the option to accept or reject it. If the o↵er is rejected, the recipient may declare an end to the negotiations or she may plan to make a countero↵er at the start of the next sub-period. In the latter case, with probability bargaining breaks down.
As shown in CET, the solution to the AOB problem is given by: where (4) and (11) can be combined and written in the form of the AOB-sharing rule: with i = ↵ i+1 /↵ 1 , for i = 1, 2, 3. The Nash-sharing rule can be viewed as a special case of the AOB-sharing rule, in which 1 = (1 ⌘)/⌘ and 2 = 3 = 0.
We use (2) to solve for the real wage: w t = w p t ⇢E t m t+1 w p t+1 . In principle w p t is consistent with a wide variety of wage payments over the periods in which the worker and firm remain matched. We resolve this potential non-uniqueness in w t by assuming that each period's wage rate is the same time-invariant function of variables that are exogenous to the worker and firm.

Market Volatility
This section analyzes the role of wage inertia in generating labor market volatility using the type of comparative steady-state methods commonly used in the literature. We consider three models of wage determination: Nash bargaining, AOB, and sticky wages. In the latter case, w is simply a constant,w, like in Hall (2005).

A Wage-Inertia Decomposition
The free-entry condition and the bargaining equation play a central role in the steady-state equilibrium conditions of the model. Equation (8) implies that in steady state, the vacancy filling probability, Q, and market tightness, , are related by: where a time series variable without a time index denotes its steady-state value. The steadystate version of the free entry condition, (3) is: Here, is the steady-state value of the representative household's discount factor, m t . The right hand side of equation (13) corresponds to the steady-state expected profits from a filled vacancy.
Denote the elasticity of market tightness with respect to the marginal revenue product #, by ⌘ ,# : Shimer (2005) and much of the related literature use ⌘ ,# as a measure of the labor market volatility implied by a model.
Totally di↵erentiating (13) and rewriting implies: Expression (15) decomposes labor market volatility ⌘ ,# into a term that reflects the inverse profit rate and a term that is a function of dw/d#. Note that equation (15) uses only the free entry condition, so that it holds regardless of how wages are determined. Other things equal, a wage determination mechanism that implies greater wage inertia, i.e. a smaller value of dw/d#, implies a larger value of ⌘ ,# . The intuition is simple. When the wage rate is more inertial, firms receive a greater share of the rent associated with vacancies after a rise in the marginal revenue product (or technology) #. So the more inertial is the wage, the greater is the incentive of the firm to post vacancies in the wake of an increase in #. This increased incentive leads to a greater increase in market tightness and a larger drop in unemployment after an increase in #.
Expression (15) makes clear that some wage inertia is necessary for the model to generate a larger value of ⌘ ,# . If, across steady states, a change in # is fully reflected in the real wage rate (dw/d# = 1), then ⌘ ,# must be zero.

Fundamental Surplus-Based Decompositions
We now study the steady-state decomposition of ⌘ ,# developed in LS. This decomposition is based on what they call the fundamental surplus fraction, which we denote by F S. In contrast to our decomposition, an F S based decomposition is derived using both the free entry condition and the details of the wage-setting mechanism. That decomposition takes the following form, where ⌥ > 0. We show that this type of decomposition is in general not unique. In what follows we discuss two such decompositions for each of the Nash and AOB models. In the first decomposition F S is an analytic function of model structural parameters. In contrast, ⌥ generally involves steady-state variables. We refer to this type of decomposition as a structural decomposition. With the possible exception of the section on the financial accelerator model, LS focus on such decompositions. In the second decomposition F S depends on steady-state variables. We refer to this type of decomposition as a non-structural decomposition.

Nash Bargaining
In Appendix A.1.1 we derive the following structural decomposition for ⌘ ,# in the Nash bargaining model, where This decomposition e↵ectively coincides with the decomposition reported in section A.5 of the LS online technical appendix. 7 There is a slight di↵erence between our structural decomposition and the one in LS, reflecting timing di↵erences in job-to-job transitions. These timing di↵erences do not a↵ect the substance of our analysis. 8 A distinguishing feature of F S in equation (16) is that it does not involve endogenous variables like f . In contrast, ⌥ Nash involves f .
In Appendix A.1.2 we derive a non-structural decomposition for ⌘ ,# . It takes the form of (16) with F S Nash given in (18). The di↵erence is that ⌥ Nash and ⌧ Nash  in (17) are then given by: Notice that ⌧ Nash  and, hence, F S Nash , depend on the steady-state value of f . In the special 7 The equivalence between the two decompositions is easier to see using the fact that Q = m ⇣ ft m ⌘ 1 1 . 8 LS use the same timing assumptions adopted by Hall and Milgrom (see footnote 6). In their main text LS assume, as in Mortensen and Nagypal (2007), that  is paid after bargaining has occurred. In section A.5 of their online technical appendix, LS assume, as in Pissarides (2009), that  is paid before workers and firms bargain. We adopt the latter assumption. case,  = 0, the structural and non-structural decompositions coincide. 9 We now show that when  6 = 0 there is an important di↵erence between the structural and non-structural decompositions in terms of their usefulness for analyzing how changes in parameter values a↵ect ⌘ ,# . To understand the di↵erence, it is useful to distinguish between two types of model parameters. First, some parameters enter the decomposition explicitly.
Second, there may be other parameters that only enter via their impact f . In the case of our non-structural decomposition there is such a background parameter, m . In the presence of such a background parameter, the non-structural decomposition provides a simple and transparent way to perform a particular experiment: evaluate the impact on ⌘ ,# of a change in a model parameter holding f and the non-background parameters constant. This type of experiment is of interest to the extent that the analyst has flat priors about the value of m . 10 In the case of the structural decomposition stressed in LS, when  6 = 0 there are no background parameters that can implicitly be adjusted to keep f fixed. As a consequence the formula loses its transparency and simplicity. This loss is easy to see for the experiment described above: the analyst is now forced to solve a non-trivial problem to determine the required change in m .

Alternating O↵er Bargaining
We now consider the alternating o↵er bargaining case. In Appendix A.2.1 we derive the following structural decomposition for ⌘ ,# , where The coe cients 1 , 2 and 3 are defined after equation (11). Notice that F S AOB is not a function of endogenous variables like f .
In Appendix A.2.2 we derive a non-structural decomposition for ⌘ ,# in which: where Also, Here 1 , 2 and 3 are the same as above. Equations (20) and (24) continue to hold. Note that f now appears in F S AOB via ⌧ AOB  and ⌧ AOB (see (25)).
Recall that in discussing Nash bargaining, we discussed a particular experiment in which the analyst wants to study the e↵ect on ⌘ Nash ,# of a structural parameter, keeping f fixed.
We argued that it is easy to do so using the non-structural decomposition and is more com-plicated using the structural decomposition. The same result holds in the case of AOB bargaining. One interesting di↵erence is that in the Nash case the structural and non-structural decompositions are the same when  = 0. In the AOB case, the two decompositions are di↵erent, even when  = 0.

Sticky Wages
When wages are constant (dw/d# = 0) (15) implies: Here,w denotes the constant wage. LS interpret this expression as an F S based decomposition.

Using F S based Decompositions for Cross-model Comparisons
The These observations lead LS to conclude that researchers should look for models in which F S is small.
Critically, the version of the AOB model that LS consider di↵ers from ours along three important dimensions. First, the time needed to make an o↵er or a countero↵er is the same as the time needed to produce the good. Second, the bargaining process can in principle go on forever. Third, to solve their version of the AOB model, LS adopt a particular approximating assumption: the probability that a job is destroyed is the same as the probability, , that bargaining breaks down. This assumption has the important consequence that workers do not consider the value of their outside option when they decide whether or not to accept a wage o↵er from a firm. 12 Ljungqvist and Sargent (forthcoming) consider our version of the AOB model. However, in solving and analyzing the model they adopt the approximating assumption just discussed.
Recall that in our AOB model jobs can be destroyed at the beginning of a period, but not within a period. So the approximating assumption requires ! 0. As above, this assumption leads to the extreme implication that workers do not consider the value of their outside option when bargaining.
To see why setting to zero in our model has this extreme implication, it is useful to understand how a firm chooses its wage o↵er in bargaining round, j. Let w j,t denote a firm's wage o↵er in bargaining round j of period t, where j is an odd number between 1 and M 1.
The firm wants to set w j,t as low as possible, subject to not being rejected by the worker.
So, w j,t has the property that workers are indi↵erent between accepting and rejecting the o↵er: The term, w j,t +w p t , represents the present discounted value of the wages associated with accepting the firm's o↵er, w j,t . So, the left-hand side of (26) represents the value to the worker of accepting the firm's o↵er. Under our period-by-period bargaining assumption, the firm takesw p t as given. The right-hand side of the indi↵erence condition is the value to the worker of rejecting an o↵er and, with probability 1 , making a countero↵er in bargaining round, j + 1. The firm takes w j+1,t as given and understands that if its current o↵er is rejected, then w j+1,t will be accepted unless bargaining breaks down. The first term on the right-hand side of (26) is times the sum of two terms: (i) the pro-rata unemployment benefits received by the worker in the event that bargaining breaks down; and (ii) the continuation value of being unemployed (see (7)). The second term on the right-hand side is 1 times the sum of one sub-period's unemployment benefit plus the value to the worker of an accepted o↵er.
When = 0 the worker indi↵erence condition, (26) reduces to So, under Ljungqvist and Sargent (forthcoming)'s approximating assumption, workers in our AOB model do not consider the value of their outside option, U t , when considering a particular wage o↵er, w j,t .
In the special case of = 0, the structural and non-structural F S based decompositions coincide and are given by Here, ⌥ AOB = 1/ . So, our AOB model with the approximating assumption, = 0, fits the pattern of the models considered by LS in which ⌥ is bounded above by roughly 2. 13 While the arguments associated with the LS approximation assumption are elegant, we find the models above that embed this assumption unappealing. First, the implication that workers do not consider their outside option when considering a wage o↵er seems implausible on a priori grounds. Second, CET find that = 0 is empirically implausible. In particular, the 95 percent probability interval associated with CET's estimate of easily excludes a value of zero. 14 Third, CET report that the mode of the posterior distribution of implies 13 It is easily verified that ! 0 implies 1 ! 0, 2 ! M 2 2 , 3 ! 1 2 . Substitute out for J and Q in (3) using (12) and (8), respectively. Then, take the limit, ! 0. Finally, totally di↵erentiate (3) with respect to and # and rearrange, to obtain (27).
14 CET estimate their model using Bayesian methods. Notably, the lower bound of the posterior probability interval is substantially higher than the lower bound of the prior probability interval for that parameter. So, the data push the distribution of away from zero. See Table (4). that the total job destruction probability, conditional on no resolution to bargaining over a quarter, is meaningfully higher than zero, at roughly 10 percent. 15 Fourth, formula (27), which corresponds to the case = 0, provides a strikingly bad approximation to the value of ⌘ ,# implied by CET's estimated AOB model. The value of ⌘ ,# implied by the mode of the posterior distribution reported by CET is 24.1. 16 In sharp contrast, equation (27) implies a negative value, 28.14, for ⌘ ,# .

Quantitative Analysis
In this section we provide a quantitative analysis of the role played by the fundamental surplus and wage inertia on labor market volatility in di↵erent models.

F S based Decompositions
We begin by discussing the implications of CET's AOB model for ⌘ AOB ,# . Row one of Table 2 displays components of the structural F S based decomposition of ⌘ AOB ,# as well as the wageinertia decomposition. These are evaluated at the mode of the model parameters' posterior distribution and steady states (see Tables 1 and 4). Notice that ⌘ AOB ,# is roughly 24, so the model is successful in terms of generating a large elasticity of labor market tightness (and unemployment) to a change in the marginal revenue product of labor. The key factor underlying the model's success in generating a large value of ⌘ AOB ,# is the high value of ⌥ AOB , 7.08. The latter exceeds by a factor 3 the upper bound of ⌥ AOB emphasized by LS for the models that they consider. The inverse fundamental surplus fraction, 1/F S, plays a smaller role than ⌥ AOB in generating the large value of ⌘ AOB ,# . This finding provides a stark counterexample to the claim in LS that a small surplus fraction is crucial for a model to generate empirically plausible volatility in labor market variables.
Row four of Table 2 displays results for the estimated Nash model (for parameter and 15 Table (4) reports that the mode of is 0.0019. CET set M = 60, so P M j=1 (1 ) j 1 = 1 (1 ) M = 0.11. We thank Lars Ljungqvist for suggesting this way of interpreting the magnitude of . 16 The number is generated using equation (20).  Tables 3 and 4. Source: CET.
steady state values see Tables 1 and 4). The estimated Nash model also generates a value of ⌘ ,# in excess of 20. But, it does so with a low ⌥ Nash and a high 1/F S Nash . Taken together, the Nash and AOB model results imply that F S is not a reliable guide to understanding whether a model generates a high value of ⌘ ,# .
We now consider the role of the F S based decomposition in cross-model comparisons of ⌘ ,# . As CET point out, the estimated Nash model is able to generate a high elasticity because the estimated value of unemployment benefits D, reported in Table 1, is very high (88% of the steady-state wage). The last row of Table 2 presents results for what we call the restricted Nash model. This is a version of the estimated Nash model in which D is a more reasonable 37% of the steady-state wage, a replacement ratio that corresponds to the one in the estimated AOB model. 17 Notice that ⌘ ,# is only about 4 in the restricted Nash model, as opposed to 24 in the estimated AOB model. The lion's share of this six-fold increase in the elasticity is due to the higher value of ⌥ in the AOB model. So, the inverse surplus fraction is not a good guide for understanding why one model generates much higher labor market volatility than another.

Wage Inertia
We now turn to the role of wage inertia in generating high values of ⌘ ,# . Consider first the estimated AOB and Nash models. According to Table 2, wage inertia fully accounts for the fact that ⌘ AOB ,# is larger than ⌘ Nash ,# . The wage inertia e↵ect more than makes up for the fact that the inverse profit rate is lower in the AOB model than in the Nash model.
Although the wage inertia component may appear to be numerically small in the estimated AOB model, it is the percent change in that term that matters for ⌘ AOB ,# when wage inertia changes. This percent change can be large even if the level seems low.
Next, we compare the estimated and restricted Nash models. By construction, the inverse profit rate is the same in these two models. But, ⌘ ,# is much larger in the estimated Nash model. All of the di↵erence is due to the rise in wage inertia as we move from the restricted to the estimated Nash model.
We now consider our two sticky wage models. In the first one we fix the wage rate at its steady state value in the estimated AOB model. The non-bargaining parameters are the same as in the estimated AOB model. The results are reported in row two of Table 2. Notice that ⌘ ,# is now 3837 -orders of magnitude larger than in the estimated AOB model. As one might anticipate from LS, the inverse of the fundamental surplus increases dramatically compared to the estimated AOB model. But the interesting economic question is why? By construction the profit rate is the same in this model and in the estimated AOB model. So, all of the change in the fundamental surplus is due to the wage inertia component, which is equal to 1 by construction in the sticky wage model.
Results for the second sticky wage model appear in the third row of Table 2. That model has the same parameters and steady state wage as the estimated Nash model. Notice that the results of the two sticky wage models are similar. HM's finding depends on their assumption that wages are determined by Nash bargaining and their way of inducing wage inertia. To see this, recall our decomposition of ⌘ ,# , given by equation (15). That decomposition is derived using only the firms' free entry condition.
It holds regardless of how wages are determined. Equation (15) implies that, other things equal, wage inertia drives ⌘ ,# up. But other things aren't equal. HM induce wage rigidity in a way that drives the level of the wage down. According to (15), this level e↵ect drives ⌘ ,# down. The net e↵ect of these o↵setting forces is the negligible rise in ⌘ ,# that HM find.

Clearly HM's results do not imply that other ways of inducing wage inertia in other models
of wage determination will have small e↵ects on ⌘ ,# .

Alternative Driving Forces: Discount Rate Shocks
A variety of authors argue that variations in discount rates can be an important source of variations in unemployment. Hall (2017) models these variations as exogenous shocks to the stochastic discount factor in a search and matching model. A comparative steady-state analysis implies that wage inertia is a necessary condition for variations in the discount factor to induce movements in labor market tightness and unemployment.
For simplicity, we assume that the fixed hiring cost  is equal to zero. In steady state the value of a worker to a firm is equal to In addition, the free entry condition is given by: Substituting out J, totally di↵erentiating and re-arranging yields the elasticity of labor market tightness with respect to the discount rate, ⌘ , : To interpret this expression suppose that the steady-state value of # is una↵ected by the value of . This is the case in standard search and matching models. If dw d = ⇢J, then the right hand side of equation (28) is una↵ected by a change in . In equation (28), the term ⇢J is the discounted expected future value of a worker to the firm. If the discount rate change induces a change in w that is exactly equal to ⇢J, then the current value of a worker to the firm, J, is unchanged. Then the firm has no incentive to post more vacancies and the unemployment rate is not a↵ected by a change in . So, equation (29) shows that for ⌘ , to be positive, wages must be inertial in the sense that dw d < ⇢J. Of course the actual value of dw d is determined by the wage determination mechanism in the model, e.g. sticky wages, Nash bargaining or AOB.

Dynamic Analysis
In this section we consider the dynamic impact of wage inertia in generating labor market volatility in search and matching models. We make two key points. First, comparative steady-state analysis can be deeply misleading about the dynamic behavior of these models. Models which have identical fundamental surpluses can exhibit very di↵erent dynamic response functions. In these cases, the fundamental surplus is uninformative about the question of interest. Second, even conditioning on a low inverse fundamental surplus, wage inertia greatly magnifies labor market fluctuations in empirically plausible versions of search and matching models. In the first subsection we consider a simple dynamic example. In the second subsection, we analyze the impact of wage inertia in estimated DSGE models conditional on a given value of the fundamental surplus.

Wage Inertia: A Simple Dynamic Example
In this section we consider the role of wage inertia using a simple dynamic example. Suppose that the equilibrium wage rate is given by the following simple inertial wage rule: where > 0 and D  w t  # t . In addition, assume that 0 < (1 ) < . The latter assumption implies that a rise in # t generates a positive response in w t . The dynamics of w t depend on and . Note that has no impact on the steady-state value of w t . We can think of this wage rule as being a variant of the Hall (2005) wage norm, as long as D  w t  # t so that the firm and worker each have an incentive to produce. A larger value of means more inertia in w t : a given shock to # t is associated with a smaller change in w t . For simplicity, in this subsection, we abstract from the hiring cost, i.e.  = 0.
We seek to evaluate the generic formula for the steady-state elasticity of labor market tightness with respect to technology, i.e. equation (15), for the simple wage rule (30).
Note that in steady state dw d# = . Inserting the latter result together with the steady-state version of equation (30) into equation (15) yields the following expression for the steady-state elasticity: Note that the steady-state elasticity is independent of the fundamental surplus. Moreover, the elasticity is identical to one implied by the Nash bargaining model with ⌘ = D = 0.
To derive the actual dynamics of the simple model, we must take a stand on the law of motion for # t . To this end, we assume: where " t is uncorrelated over time and uncorrelated with # t u for u > 0.
In Appendix B we show that the equilibrium solution for the value of a worker to the firm, J t , is given by: Combining the free-entry condition (13) with the solution for J t we obtain: Totally di↵erentiating and rearranging yields the following expression for the dynamic elasticity of labor market tightness with respect to the marginal revenue product: Consider v close to unity, so that the marginal revenue product of a worker is close to a random walk.

Model
In this subsection we consider the model of CET who embed the labor market model of subsection 2 into a medium-sized DSGE NK model.

Households
The economy is populated by a large number of identical households. The representative household has a unit measure of workers which it supplies inelastically to the labor market.
We denote the fraction of employed workers in the representative household in period t by l t . An employed worker earns the nominal wage rate, W t and an unemployed worker receives D t goods in government-provided unemployment compensation. Each worker has the same concave preferences over consumption. Households provide perfect consumption insurance to their members, so that each worker receives the same level of consumption, C t . The preferences of the representative household are the equally-weighted average of the preferences of its workers: where the parameter 0  b < 1 controls the degree of habit formation in preferences. The representative household's budget constraint is: Here, T t denotes lump sum taxes net of profits, P t denotes the price of consumption goods, P I,t denotes the price of investment goods, B t+1 denotes one period risk-free bonds purchased in period t with gross return, R t and I t denotes the quantity of investment goods. The object R K,t denotes the rental rate of capital services, K t denotes the household's beginning of period t stock of capital, a(u K t ) denotes the cost, in units of investment goods, of the capital utilization rate, u K t and u K t K t denotes the household's period t supply of capital services. We discuss details about the capital utilization cost function in subsection 4.2.4. All prices, taxes and profits in equation (34) are in nominal terms. The representative household's stock of capital evolves as follows: where K denotes the depreciation rate and S(I t /I t 1 ) are convex investment adjustment costs. We discuss details about the latter in subsection 4.2.4.

Final Goods Producers
A final homogeneous good, Y t , is produced by competitive and identical firms using the following technology: where > 1. The representative firm chooses specialized inputs, Y j,t , to maximize profits: subject to the production function (35). Output, Y t can be used to produce either consumption goods or investment goods. The production of the latter uses a linear technology in which one unit of Y t is transformed into t units of I t .

Retailers and Wholesalers
The j th input good in (35) is produced by a retailer, with production function: Here k j,t denotes the total amount of capital services purchased by firm j and ' t represents a fixed cost of production which evolves according to an exogenous stochastic process that is consistent with balanced growth. We discuss details about the latter in subsection 4.2.4. The variable z t is a technology shock and h j,t is the quantity of an intermediate good purchased by the j th retailer. This good is purchased in competitive markets at the price P h t from a wholesaler discussed in subsection 2. To produce in period t, the retailer must borrow P h t h j,t at the gross nominal interest rate, R t . The retailer repays the loan at the end of period t after receiving sales revenues. The j th retailer sets its price, P j,t , subject to the demand curve for its good, and a Calvo sticky price friction. With probability 1 ⇠, the retailer can re-optimize his price and with probability ⇠, P j,t = P j,t 1 . 18 Wholesaler firms produce the intermediate good using labor which has a fixed marginal productivity of unity. The real price of the intermediate good is P h

Monetary Policy and Functional Forms
We adopt the following specification for monetary policy: Here, ⇡ denotes the monetary authority's target inflation rate. The monetary policy shock, " R,t , has unit variance and zero mean. Also, R is the steady-state value of R t . The variable, GDP t , denotes Gross Domestic Product (GDP), which equals C t + I t / t + G t and GDP ⇤ t denotes the value of GDP t along the non-stochastic steady-state growth path. We assume that the growth rate of neutral technological progress, lnµ z,t ⌘ ln(z t /z t 1 ), is i.i.d. and that the growth rate of investment-specific technological progress, lnµ ,t ⌘ ln( t / t 1 ), follows a stochastic first order autoregressive process.
The sources of growth in the model are neutral and investment-specific technological progress because t and z t grow over time. Let t = (↵/(1 ↵)) t z t denote the composite level of technology. To guarantee balanced growth in the non-stochastic steady state, we require that each element in [' t , s t ,  t , t , G t , D t ] grows at the same rate as t in steady state. To this end, we adopt the following specification: Here, ⌦ t is a diagonal matrix with the i th diagonal element, The cost of adjusting investment takes the form: Here, µ and µ denote the unconditional growth rates of t and t . Also, S 00 denotes the second derivative of S(·), evaluated in steady state. The cost associated with setting capacity utilization is given by: where a and b are positive scalars. For a given value of a , we select b so that the steady-state value of u K t is unity. Finally, we refer the reader to CET and CET's technical appendix for the market clearing conditions, the definition of equilibrium and the set of dynamic and steady-state equilibrium equations.

The Role of Wage Inertia in the Estimated Model
CET estimate the medium-sized DSGE NK model discussed above for various wage bargaining environments. Their estimation strategy is a Bayesian variant of the strategy in Christiano, Eichenbaum and Evans (2005) that minimizes the distance between the dynamic responses to three shocks in the model and the analog objects in the data. The shocks used by CET include a shock to monetary policy, a neutral technology shock, and an investmentspecific technology shock. The dynamic responses to those shocks are obtained using an identified VAR for post-war quarterly U.S. times series that include key labor market variables, see CET for further details.
Here we focus on the versions of the model in which wages are determined by Nash and AOB bargaining. Table 3 reports the values of parameters that CET set a priori. Table 4 reports the mean and 95 percent probability intervals for the priors and posteriors of the estimated parameters in the Nash and AOB bargaining models. Table 1 reports the implied steady states.
Note that the estimated values of the replacement ratio D/w in the Nash and AOB models are 0.88 and 0.37, respectively. CET argue that the estimated value of the replace-  This result depends critically on the fact that the model does a reasonably good job of  matching the inertial response of real wages to the shocks. To substantiate this claim we consider the following experiment. We impose on the model the assumption that real wages go up, in period 1, by 50% more than their peak response in the estimated Nash model and then stay at that level for three consecutive years. After the three years the economy returns to the wage rule implied by the restricted Nash-sharing rule. The dotted red lines in rows 1 and 2 of Figure 1  The key result here is that for both shocks the response of unemployment is much smaller, in absolute value, than in the estimated model. For the monetary policy shock, the response is counterfactually small, even taking VAR sampling uncertainty into account: without wage inertia, there is a Shimer-like puzzle. Note also that by construction the estimated Nash model and the estimated Nash model with the three-years elevated wage have the same steady state so the fundamental surplus is exactly the same in both models.
So, the fundamental surplus contains no information about the dynamic responses of the two economies being compared.
Rows 1 and 2 of Figure 2 reports the results for the restricted Nash model. From the solid blue lines we see that real wages respond by a counterfactually large amount to both shocks. Not surprisingly, the response of unemployment in this model is much smaller than in the estimated Nash model. Again, without wage inertia, there is a Shimer-like puzzle. The red dashed lines display the impulse response functions if we hold real wages fixed for three consecutive years and then let wages be determined by the Nash-sharing rule afterwards.
With inertial wages, unemployment now responds by much more. So, wage inertia allows a Nash model with a low replacement ratio to overcome the Shimer-like puzzle. Note again, that the two models being compared have identical steady states and fundamental surpluses.
Finally rows 1 and 2 of Figure 3 repeat the experiment presented in Figure 1 for the estimated AOB model. The results are consistent with those emerging from the estimated We conclude that one should be skeptical about the use of comparative steady-state analysis to understand the dynamics of empirically plausible models.

Conclusion
Wage inertia is widely recognized as playing an important role in business cycle fluctuations.
An important exception to this view is the search and matching literature where the role of wage inertia is the subject of an ongoing debate. In this paper we argued that wage inertia does in fact play a crucial role in allowing variants of standard search and matching models to account for the large countercyclical response of unemployment to shocks. We made this argument using comparative steady state and dynamic analyses of estimated DSGE models.
While the former mode of analysis is widely used and can generate useful insights, it can also be very misleading in the present context. Specifically, dynamic models with the same steady state and fundamental surplus can exhibit very di↵erent dynamic responses of unemployment to shocks. In the models that we investigate, large dynamic responses of unemployment to shocks always coincide with an inertial response of wages. The basic intuition is that if wages increase too much after a change in the marginal revenue product of labor, firms have little incentive to invest in new jobs.
Impose steady state, multiply out the last term on the right hand side, collect terms and re-arrange: Totally di↵erentiating: and Q = m so that: Defining ⌥ Nash and ⌧ Nash  yields: which are the expressions for ⌘ Nash ,# , ⌥ Nash and ⌧ Nash  in the main text. Note that the inverse fundamental surplus fraction does not contain endogenous variables.

A.1.2 Non-structural Fundamental Surplus Based Decomposition
As discussed in the main text, we derive a non-structural fundamental surplus based decomposition of ⌘ ,# for the Nash model in which the inverse fundamental surplus term involves the endogenous variable f . Note that the elasticity formula, equation (37), can be written as: Also, equation (36) can be solved for s: Substituting for s in the elasticity formula and simplifying yields: which are the expressions provided in the main text. Notice that ⌧  contains the endogenous variable f in this alternative expression for the inverse fundamental surplus.

A.2.1 Structural Decomposition
Recall the set of equilibrium labor market equations with the AOB-sharing rule: Impose steady state: Totally di↵erentiate: Solve (⇤) for c Q : Substitute into elasticity formula: Re-arrange to obtain the elasticity formula for the AOB model:

B Dynamic Elasticity Formula in Inertial Wage Rule
Here we derive the equilibrium solution for the value of a worker to a firm, J t , provided in subsection 4.1.
Note that the value of a worker to a firm is given by: Substituting out for w t using (30) gives: Next, we solve for J t using the method of undetermined coe cients. Guess that the solution takes the following form: where 0 and 1 are undetermined coe cients that are to be determined as a function of model parameters. Substituting (42) into (41) also making use of (31) gives: Setting the two square braskets to zero and solving for 0 and 1 gives: Next, combine the free-entry condition (13) with the solution for J t : Totally di↵erentiating: s Re-arranging and using s m = 0 + 1 # yields: which is the expression for the dynamic elasticity of labor market tightness with respect to the marginal revenue product (or technology) provided in subsection 4.1.