The Vanishing Procyclicality of Labor Productivity

We document three changes in postwar US macroeconomic dynamics: (i) the procyclicality of labor productivity vanished, (ii) the relative volatility of employment rose, and (iii) the relative (and absolute) volatility of the real wage rose. We propose an explanation for all three changes that is based on a common source: the decline in labor market turnover, which reduced hiring frictions. We develop a simple model with hiring frictions, variable effort, and endogenous wage rigidities to illustrate the mechanisms underlying our explanation. We show that the decline in turnover may also have contributed to the observed decline in output volatility.


Introduction
The nature of business cycle uctuations changes over time. There is a host of evidence for changes in the dynamics of postwar US macroeconomic time series (Blanchard and Watson (1986), McConell and Pérez-Quirós (2000), Watson (2002), Hall (2007), Galí and Gambetti (2009)). The present paper documents and discusses three aspects of these changes.
The correlation of labor productivity with output or labor input has declined, by some measures dramatically so. 1 The volatility of labor input measures has increased (relative to that of output). 2 The volatility of real wage measures has increased, both in relative and absolute terms. 3 All three of the above observations point towards a change in labor market dynamics.
While each may be of independent interest and have potentially useful implications for our understanding of macro uctuations, our goal in the present paper is to explore their possible connection. In particular, we seek to investigate the hypothesis that all three changes may be driven by the decline in labor market turnover in the US over this period, which reduced hiring frictions and allowed rms to adjust their labor force more easily in response to various kinds of shocks. In order to illustrate the mechanism behind this explanation, we develop a stylized model of uctuations with labor market frictions and investigate how its predictions vary with the level of labor market turnover.
2 To the best of our knowledge, Galí and Gambetti (2009) were the rst to uncover that nding, but did not provide the kind of detailed statistical analysis found below. Independently, Hall (2007) o¤ered some evidence on the size of the decline in employment in the most recent recessions that is consistent with our nding.
3 As far as we know, this nding was not known previously, although it is reported in independent work by Gourio (2007) and Champagne and Kurmann (2013). interesting question aside. 4 We model the decrease in labor market turnover as a decline in the employment outow probability or separation rate, which as in much of the literature is exogenous in our model. In response to the decline in turnover labor market frictions decrease endogenously. This e¤ect arises because adjustment costs in employment in our model are convex, a relatively uncontroversial assumption. We show that the observed decline in turnover is su¢cient to quantitatively generate the reduction in frictions needed to explain the changes in labor market dynamics.
The main intuition for our qualitative results is easy to describe. The idea goes back to a literature, starting with Oi (1962) and Solow (1964), which attributes the procyclicality of productivity to variations in e¤ort, resulting in seemingly increasing returns to labor. 5 Suppose that rms have two margins for adjusting their e¤ective labor input: (observed) employment and (unobserved) e¤ort, which we respectively denote (in logs) by n t and e t . 6 Labor input (employment and e¤ort) are transformed into output according to a standard production function, where a t is log total factor productivity and is a parameter measuring diminishing returns to labor.
Measured labor productivity, or output per person, is given by Labor market frictions make it costly to adjust employment n t . Since these adjustment costs are convex, frictions are higher when the average level of hiring is higher. E¤ort e t provides an alternative margin of adjustment of labor input and is not subject to those frictions (or to a lesser degree). Thus, the larger the frictions, the less employment uctuates and the more volatile uctuations in e¤ort. As a result, a decline in turnover reduces the average amount of hiring, reduces frictions, decreases the volatility of e¤ort and therefore increases the relative volatility of employment with respect to output.
The increased volatility of n t also makes labor productivity less procyclical, and, in the presence of shocks other than shifts in technology, may even make productivity countercyclical, consistent with the evidence reported below.
In addition, as emphasized by Hall (2005), the presence of labor market frictions generates a non-degenerate bargaining set for the wage, i.e. a wedge between rms and workers reservation wages. Any wage within that bargaining set is consistent with labor market equilibrium. That feature makes room for wage rigidities. We model wages as rigid within the bargaining set, adjusting only when approaching its bounds.
In our model, the reduction in labor market frictions resulting from the decline in labor market turnover narrows the bargaining set and therefore endogenously makes wages more sensitive to shocks, increasing the volatility of uctuations in wages. If the rigidity is extended to the wages of newly hired workers, then the increased exibility of wages may dampen the volatility of output and employment in response to shocks. 7 That feature may help explain the observed decline in the volatility of those two variables in the recent US experience. 8 Other authors have also argued that the Great Moderation may have been driven at least in part by increased wage exibility (Gourio (2007), Champagne and Kurmann (2013), Nucci and Riggi (2011)). However, this paper is the rst to show that such an increase in wage exibility can arise endogenously from the decline in labor market turnover. 9 The remainder of the paper is organized as follows. Section 2 documents the changes in the patterns of uctuations in labor productivity, employment and wages. Section 3 develops the basic model. Section 4 describes the outcome of simulations of a calibrated version of the model, and discusses its consistency with the evidence. Section 5 concludes.

Changes in Labor Market Dynamics
We document three stylized facts regarding postwar changes in US economic uctuations. The three changes that motivate our investigation pertain to the cyclical behavior of labor productivity, labor input and wages. None of the facts we report are new. However, and to the best of our knowledge, this paper is the rst to provide a unied explanation for the three changes.
We use quarterly time series over the period 1948:1-2007:4 drawn from di¤erent sources (see below for a detailed description). To illustrate the changes in the di¤erent statistics considered, we split the sample period into two subperiods, pre-84 (1948:1-1983:4) and post-84 (1984:1-2007:4). The choice of the break date is fairly arbitrary, motivated by existing evidence on the timing of the Great Moderation (McConell and 7 This is clearly true for technology shocks. As argued in Blanchard and Galí (2007b), increased wage exibility may also dampen the sensitivity of GDP and ination to oil price shocks. 8 A more exible labor market does of course not make the economy immune to very large shocks like the recent nancial crisis. Under our hypothesis, if the labor market were as rigid as it was in the early 80s, the current recession might have been even substantially more severe. 9 Uren (2008) develops a model, in which a reduction in search frictions decreases output volatility. However, the reduction in frictions in that paper is exogenous and the mechanism, by which output volatility decreases (increased assortative matching), is completely di¤erent. various changes in labor market dynamics.
Our evidence makes use of alternative measures of output and labor input. In all cases labor productivity is constructed as the ratio between the corresponding output and labor input measures. Most of the evidence uses output and hours in the private sector from the BLS Labor Productivity and Cost (LPC) program. We also use GDP as an economy-wide measure of output, with the corresponding labor input measures being total hours or employment. The time series for economy-wide hours is an unpublished series constructed by the BLS and used in Francis and Ramey (2009). The employment series is the usual one from the Current Employment Statistics (CES) establishment survey. In all cases we normalize the output and labor input measures by the size of the civilian noninstitutional population (16 years and older).
We apply three alternative transformations on the logarithms of all variables in order to render the original time series stationary. Our preferred transformation uses the bandpass (BP) lter to remove uctuations with periodicities below 6 and above 32 quarters, as in Stock and Watson (1999). We also apply the fourth-di¤erence (4D) operator, which is the transformation favored by Stock and Watson (2002) in their analysis of changes in output volatility, as well as the more common HP lter with smoothing parameter 1600. 10 Figure 1 shows the uctuations at business cycle frequencies in labor productivity in the US over the postwar period. It is clear from the graph that in the earlier part of the sample, productivity was signicantly below trend in each recession. However, in the later years this is no longer the case. When we calculate the correlation of productivity with output or employment, as in Figure 2, it is clear that there is a sharp drop in the cyclicality of productivity. The correlation of productivity with output, which used to be strongly positive, fell to a level close to zero, while the correlation of productivity with employment, which was zero or slightly positive in the earlier period of the sample, became negative.

The Vanishing Procyclicality of Labor Productivity
These ndings are formalized in Table 1, which reports the contemporaneous correlation between labor productivity and output and employment, for alternative transformations and time periods. In each case, we report the estimated correlation for the pre and post-84 subsamples, as well as the di¤erence between those estimates. The standard errors, reported in brackets, are computed using the delta method. 11 We now turn to a 10 We also showed the results are robust to using shorter sample periods, centered around the break date 1984. These estimates, which are suppressed for brevity, are available in the online appendix to this paper. 11 We use least squares (GMM) to estimate the second moments (variances and and covariances) of each pair of variables, as well as the (asymptotic) variance-covariance matrix of this estimator. Then, short discussion of the results in this Table.

Correlation with Output
Independently of the detrending procedure, the correlation of output per hour with output in the pre-84 period is high and signicantly positive, with a point estimate around 0:60. In other words, in the early part of the sample labor productivity was clearly procyclical.
In the post-84 period, however, that pattern changed considerably. The estimates of the productivity-output correlation dropped to a value close to (and not signicantly di¤erent from) zero. The di¤erence with the corresponding pre-84 estimates is highly signicant. Thus, on the basis of those estimates labor productivity has become an acyclical variable (with respect to output) over the past two decades.
When we use an employment-based measure of labor productivity, output per worker, the estimated correlations also drop substantially but remain signicantly greater than zero in the post-84 period. This should not be surprising given that hours per worker are highly procyclical in both subperiods and that their volatility relative to employmentbased labor productivity has increased considerably. 12

Correlation with Labor Input
The right-hand side panels in Table 1 display several estimates of the correlation between labor productivity and labor input. The estimates for the pre-84 period are low, and in the case of output per hour, insignicantly di¤erent from zero. Thus, labor productivity was largely acyclical with respect to labor input in that subperiod. This near-zero correlation is consistent with the evidence reported in the early RBC literature, using data up to the mid 80s. 13 As was the case when using output as the cyclical indicator, the estimated correlations between labor productivity and employment decline dramatically in the post-84 period. In fact these correlations become signicantly negative for output per hour, with a point estimate ranging from 0:40 to 0:54, depending on the lter. The change with we calculate the standard errors for the standard deviations, the relative standard deviations and the correlation coe¢cient using the delta method. 12 Letting n and h denote employment and total hours respectively, a straightforward algebraic manipulation yields the identity: Thus, even in the case of acyclical hours-based labor productivity, i.e. (y h; y) ' 0, we would expect (y n; y) to remain positive if hours per worker are procyclical, i.e. (h n; y) > 0. 13 Christiano and Eichenbaum (1992) used data up to 1983:4 (which coincides with the cut-o¤ date for our rst subperiod), but starting in 1955:4. Their estimates of the correlation between labor productivity and hours were 0:20 when using household data and 0:16 using establishment data.
respect to the pre-84 period is highly signicant. In other words, labor productivity in the past two decades has become strongly countercyclical with respect to labor input.

The Rising Relative Volatility of Labor Input
The left-hand panel of Table 2 displays the standard deviation of several measures of labor input in the pre and post-84 periods, as well as the ratio between the two.
The variables considered include employment in the private sector, hours in the private sector (employment times hours per worker) and economy-wide hours. The decline in the volatility of hours, like that of other major macro variables, is seen to be large and highly signicant, with the standard deviation falling between 35% and 49% and always signicantly so.
A more interesting piece of evidence is the change in the relative volatility of labor input, measured as the ratio of the standard deviation of labor input to the standard deviation of output. These estimates are presented in the right-hand panel of Table   2. Without exception, all labor input measures have experienced an increase in their relative volatility in the post versus pre-84 period. In other words, the decline in the variability of labor input has been less pronounced than that of output. The increase in the relative volatility of hours worked ranges from 30% to 48% in the private sector and from 7% to 30% in the total economy. The corresponding increase for employment is slightly smaller, ranging from 23% to 43% in the private sector, but is still statistically signicant.
The previous evidence points to a rise in the elasticity of labor input with respect to output. Put di¤erently, rms appear to have relied increasingly on labor input adjustments in order to meet their changes in output.

The Rising Volatility of Wages
Next we turn our attention to the volatility of (real) wages, both in absolute and relative terms. We consider four di¤erent wage measures. The rst three are constructed as real compensation per hour. The rst di¤erence is in the measure of compensation, which is measured either from the national income and product accounts (NIPA) or as earnings from the CES establishment survey. The second di¤erence is in the measure of hours, which refers to the private sector or to the total economy. 14 The fourth measure is usual hourly earnings (or usual weekly earnings divided by usual weekly hours) from the Current Population Statistics (CPS). For all measures, compensation or earnings are deated using the compensation deator from the LPC, but the results are robust to deating with the consumer price index (CPI-U) as we show below.

Average Wages
The left-hand panel of Table 3 displays the standard deviation for each wage measure for di¤erent detrending procedures. Our statistics uncover a surprising nding: despite the general decline in macro volatility associated with the Great Moderation, the volatility of several wage measures has increased in absolute terms. The estimated increase the standard deviation is fairly large, between 10 and 42% and mostly signicant for the bandpass ltered NIPA-based wage measures. Using fourth di¤erences, the increase is much smaller and no longer signicant and by some periods there seems to have been a (small) decrease in wage volatility.
Using earnings per hour from the CES, however, there seems to be a large and highly signicant reduction in wage volatility. 15 One di¤erence between the two measures is that the NIPA compensation measure includes non-wage payments and, in particular, employee stock options. Mehran and Tracy (2001) have argued that since these options are recorded when they are realized rather than when they are handed out to employees, the NIPA measure gives a misleading picture of the evolution and volatility of compensation in the 90s. However, using the CPS measure of usual hourly earnings, presented in Table 4, which includes non-wage compensation but not stock options, we again observe a fairly large increase in the volatility of wages. 16 Given the short time series available for these data, it is remarkable that the increase in volatility is (borderline) signicant for the bandpass and HP ltered series. Using fourth di¤erences, wage volatility seems roughly constant.
Our nding that wage volatility increased or at least did not decrease around the time of the Great Moderation, although with a caveat, is consistent with the results in Champagne and Kurmann (2013), who also use the CPS to show that the increase in wage volatility is not driven by compositional changes in the labor force. To the best of our knowledge, this result was not previously known. 17 An immediate implication of the previous nding, and the one that we want to emphasize here, is the possibly very large increase in the relative volatility of wages 15 This nding is not driven by the fact that the CES earnings measure is only available after 1965. If we restrict the sample for the NIPA based wage measures to the 1965-2004, the volatility statistics for these measures look very similar those in the table. 16 We use data from the CPS outgoing rotation groups. Since these data are available only from 1979 onwards, we compare the volatility over the 1980-1984 period (allowing for fourth di¤erences) with that of the 1985-2005 period. For comparison, the rst panel of Table 4 presents the volatility of our baseline measure for compensation per hour for this period. The second panel presents comparable statistics from the CPS series. Because the CPS wage series is based on a fairly small cross-section of workers, there is substantial measurement error in these series. Therefore, the standard deviations of the fourth di¤erenced data are biased upward, see Haefke, Sonntag, and van Rens (2013) for details. There is no reason however, why the ratio of the standard deviations before and after 1984 would be biased. In addition, the bandpass ltered data, which do not include the high frequencies induced by the measurement error, are not subject to this bias. 17 Stock and Watson (2002) uncover breaks in the volatility of a long list of macro variables, but they do not provide evidence for any wage measure.
with respect to to output or labor input, as shown in the right-hand panels of Tables 3 and 4. The relative volatility of wages with respect to output more than doubled for the NIPA-based measures and for the CPS wage. We interpret this evidence as being consistent with a decline in the signicance of real wage rigidities. 18

Wages of Newly Hired Workers
In a frictional labor market, the average wage is not allocative, since the frictions drive a wedge between the reservation wages of rms and workers. Therefore, in order to assess the implications of increased wage exibility for other labor market variables, we also consider the volatility of the wage of newly hired workers as suggested by Haefke, Sonntag, and van Rens (2013) and Pissarides (2009). Table 4 presents volatility statistics for the wage of new hires, constructed from the CPS as in Haefke, Sonntag, and van Rens (2013). 19 The rst thing to notice is that the wage of newly hired workers is much more volatile than the average wage in the entire labor force. This is consistent with the results in Haefke, Sonntag, and van Rens (2013), who argue that, in the post-84 period, wages of newly hired workers are perfectly exible, in the sense that they respond one-to-one to changes in labor productivity. Here, we focus on the change in the volatility of wages over time.
The absolute volatility of the wage of newly hired workers, unlike the average wage, decreased substantially and signicantly between the pre and post-84 periods. As a result, the increase in the relative volatility with respect to output is much smaller for new hires, ranging between 3% and 69%, depending on whether we use the mean or median wage and on the lter used. Although the increase in the relative volatility of the wage of newly hired workers is much less pronounced, there is some evidence that wages uctuated more between recessions and booms also for this group of workers.
This nding is consistent with the evidence presented in Haefke, Sonntag, and van Rens (2013, section 3.4) and points towards a decrease in wage rigidity that may be important for employment uctuations.

Correlation of Wages with Productivity and the Labor Wedge
In macroeconomic models with a perfectly competitive labor market and a standard production function, wages are proportional to labor productivity. 20 Our evidence makes 18 Blanchard and Galí (2007b) argue that a reduction in the rigidity of real wages is needed in order to account for the simultaneous decline in ination and output volatility, in the face of oil price shocks of a similar magnitude.
19 But unlike in that paper, we do not correct uctuations in the CPS wage series for changes in the composition of the labor force by demographic characteristics, education level and experience for comparability with the other wage measures. Doing so however, makes very little di¤erence for the conclusions presented here. 20 With competitive labor markets, the wage equals the marginal product of labor, and with a Cobb-Douglas production function, the marginal and average product of labor are proportional to eachother. clear that the extent to which this is true in the data depends very much on the period one looks at. From Table 3, we see that the relative standard deviation of wages with respect to labor productivity was about 0:3 0:7 in the the pre-84 period. In the post-84 period, this relative standard deviation more than doubled to about 1:2 1:3. This is consistent with the evidence in Haefke, Sonntag, and van Rens (2013, section 3.4), who show that the elasticity of wages with respect to productivity in 1984 increases from 0:2 to 0:4 for all workers, and from 0:3 to 0:8 for newly hired workers.
The increased responsiveness of wages to movements in productivity has implications for the volatility of the labor wedge. However, the direction of the e¤ect is ambiguous and depends on the relative importance of technology and non-technology shocks. If business cycles are driven by technology shocks, then more exible wages dampen the volatility of employment uctuations and make the labor wedge less countercyclical. If business cycles are primarily due to non-technology shocks, then more exible wages amplify the volatility of employment and the labor wedge. In the data, the labor wedge became more volatile over time, see Shimer (2010, Figure 1.2), suggesting that the latter e¤ects dominates the former.

Evidence for Europe
Although in this paper we focus on the US, it is worth exploring whether the basic patterns hold for other countries as well. In particular, given the importance of labor market frictions for our story, one might expect similar or even stronger changes in labor market dynamics in European countries, where labor market frictions are generally believed to be more important. For many countries, data are not available for our sample period. However, Ohanian and Ra¤o (2012) collected data on output, employment and hours worked from the OECD Economic Outlook database and national statistics o¢ces, for some countries starting in 1960. Table 5 reports the cyclicality of labor productivity and the relative volatility of labor input for the four major European economies using these data. 21 For comparison, we also report the statistics for the US over the same period. The changes in labor market dynamics look very similar for these ve countries. Unfortunately, lack of data does not allow us to investigate whether the European countries, like in the US, experienced a decline in labor market turnover that may have driven these changes. 22 This well-known argument, which does not rely on business cycles being driven by productivity shocks, was used recently by Rogerson and Shimer (2010, section 1.3.2). 21 To the best of our knowledge, quarterly data for employee compensation are not available for these countries over this period. Therefore, we are not able to explore whether the volatility of wages rose in these countries. 22 International time series data for worker ows for these countries were calculated by Elsby, Hobijn, andŞahin (2013), but for most countries (including Germany and Italy) these data start in 1983. For France and the UK the data start in 1975 and 1969 respectively and do not show a secular decline in the employment exit probability over this period.

Conclusions
Summarizing, we showed that labor productivity in the US became less procyclical or acyclical with respect to output, and countercyclical with respect to employment. In addition, the relative volatility of both employment and wages increased. For completeness, we also report that the relative volatility of labor productivity increased, and the correlation between employment and output decreased slightly. 23 These changes in business cycle dynamics, which seem to have happened in European countries as well, coincided with the reduction in volatility in output and most other macroeconomic aggregates, the so called Great Moderation.
From the vantage point of the early 80s the period when the seminal contributions to RBC theory were written the procyclicality of labor productivity was a well established empirical fact. This observation lent support to business cycle theories that assigned a central role to technology shocks as a source of uctuations. The relative volatility of labor input in these models was lower than in the data, which posed one of the main challenges for these models, see King and Rebelo (1999) or Hall (1997).
From todays perspective, things look distinctly worse for real business cycle theory.
The relative volatility of labor input increased even further and productivity is barely procyclical anymore. A possible reason is that over time non-technology shocks became more important drivers of business cycles, as Barnichon (2010) argues. However, this does not explain why similar changes in dynamics are observed also when conditioning on particular shocks, as in Galí and Gambetti (2009).
In the remainder of this paper, we explore whether the observed changes in business cycle dynamics may be explained by a structural change in the labor market. We show that the vanishing procyclicality of labor productivity and the increasing relative volatility of employment, can be explained by the decline in labor market turnover, which resulted in a reduction in hiring costs. If wages are set competitively, then a reduction in hiring costs, which makes employment faster to adjust, should make wages relatively less volatile. Although the relative standard deviation of wages decreased in some datasets, we argued above that the majority of the evidence seems to point towards a large increase in the relative volatility of wages. Therefore, we explore how the model can be extended with endogenous wage rigidities in order to generate this result.

A Model of Fluctuations with Labor Market Frictions and Endogenous E¤ort
Having documented in some detail the changing patterns of labor productivity, labor input, and wages, we turn to possible explanations. More specically, and as anticipated in the introduction, we explore the hypothesis that all three observed changes documented above may have, at least partly, been caused by the same secular change: a decline in labor market turnover.
To formalize this explanation, we develop a model of uctuations with labor market frictions, modelled as adjustment costs in employment (hiring costs). The crucial element in this model is an endogenous e¤ort choice, which provides an intensive margin for labor adjustment that is not subject to the adjustment costs. Since the purpose of the model is to illustrate the main mechanisms at work, we keep the model as simple as possible in dimensions that are likely to be orthogonal to the factors emphasized by our analysis. Thus, we abstract from endogenous capital accumulation, trade in goods and assets with the rest of the world, and imperfections in the goods and nancial markets. We also ignore any kind of monetary frictions, even though we recognize that these, in conjunction with changes in the conduct of monetary policy in the Volcker-Greenspan years, may have played an important role in accounting for the decline in macro volatility. 24

Households
Households are innitely-lived and consist of a continuum of identical members represented by the unit interval. The household is the relevant decision unit for choices about consumption and labor supply. Each household members utility function is additively separable in consumption and leisure, and the household assigns equal consumption C t to all members in order to share consumption risk within the household. Thus, the households objective function is given by, 25 where 2 (0; 1) is the discount factor, 2 [0; 1] is the inverse of the intertemporal elasticity of substitution, 0 can be interpreted as a xed cost of working and Z t is a preference shock. The second term in the period utility function is disutility from e¤ective labor supply L t , which depends on the fraction N t of household members that 24 See, e.g. Clarida, Galí, and Gertler (2000) for a discussion of the possible role of monetary policy in the Great Moderation. 25 We assume utility is linear in e¤ective labor for simplicity. The implication that the Frisch elasticity of labor supply is innity is of course counterfactual, but our results are very similar if we assume a Frisch elasticity of 0:25, as advocated by Chetty, Guren, Manoli, and Weber (2012). are employed, as well as on the amount of e¤ort E it exerted by each employed household member i. Formally, where the second equality imposes the equilibrium condition that all working household members exert the same level of e¤ort, E it = E t for all i. The parameter 0 measures the importance of e¤ort for the disutility of working, and the elasticity parameter 0 determines the degree of increasing marginal disutility from exerting e¤ort. For simplicity we assume a constant workweek, thus restricting the intensive margin of labor input adjustment to changes in e¤ort.
The household maximizes its objective function above subject to the sequence of budget constraints, where t represents rms prots, which are paid out to households in the form of lump-sum dividends, and W it are wages accruing to employed household member i.
The household takes into account the e¤ect of its decisions on the level of e¤ort exerted by its members.

Firms
Firms produce a homogenous consumption good using a production technology that uses labor and e¤ort as inputs, where Y t is output, E it is e¤ort exerted by worker i, 2 (0; 1) is a parameter that measures diminishing returns to total labor input in production, 2 [0; 1] measures additional diminishing returns to e¤ort, and A t is a technology shock common to all rms. Since all rms are identical, we normalize the number of rms to the unit interval, so that Y t and N t denote output and employment of each rm as well as aggregate output and employment in the economy. The second equality imposes the equilibrium condition that all workers in a rm exert the same level of e¤ort, E it = E t for all i.
Firms choose how many workers to hire H t in order to maximize the expected discounted value of prots, where the function g (:), with g 0 > 0 and g 00 > 0, represents the costs (in terms of output) of hiring new workers, subject to a law of motion for employment implied by the labor market frictions, where is the gross separation rate (employment exit probability) and Q 0;t is the stochastic discount factor for future prots. The stochastic discount factor is dened recursively as Q 0;t Q 0;1 Q 1;2 :::Q t 1;t , where measures the marginal rate of substitution between two subsequent periods. Like the household, the rm takes into account the e¤ect of its decisions on the level of e¤ort exerted by its workers.

E¤ort Choice and Job Creation
The household and the rm jointly decide the wage and the level of e¤ort that the worker will put into the job. In equilibrium, the e¤ort level of all workers is set e¢ciently, maximizing the total surplus generated by each match. 26 This e¢cient e¤ort level, in each period and for each worker, equates the cost of exerting more e¤ort, higher disutility to the household, to the benet, higher production and therefore prots for the rm.
Consider a worker i, who is a member of household h and is employed in rm j. The marginal disutility to the household from that worker exerting more e¤ort, expressed in terms of consumption, is obtained from equation (2) for total e¤ective labor supply and equals: The marginal product of that additional e¤ort to the rm is found from production function (4): In equilibrium, the marginal disutility from e¤ort must equal its marginal product for all workers i. Also, because all rms and all households are identical, it must be that C ht = C t and N jt = N t in equilibrium. Therefore, it follows that all workers exert the same level of e¤ort in equilibrium, E it = E t for all i. Imposing this property, we obtain the following equilibrium condition for e¤ort, (1 ) 26 Suppose not. Then, household and rm could agree on a di¤erent e¤ort level that increases total match surplus, and a modied surplus sharing rule (wage) that would make both parties better o¤. or, using production function (4) to simplify: When considering whether to hire a worker, rms take into account the impact of the resulting increase in employment on the e¤ort level exerted by their workers. Thus, the marginal product of a new hire is given by, 27 where F = 1+ (1 ) measures the additional (negative) e¤ect from a new hire on output that comes from the endogenous response of the e¤ort level in the rm.
Maximizing the expected net present value of prots (5), where output is given by production function (4) and the stochastic discount factor by (7), subject to the law of motion for employment implied by the matching technology (6) and the equilibrium condition for e¤ort (11), gives rise to the following rst order condition, where S F t is the marginal value to the rm of having an additional worker in period t, which is given by, where the second equality follows from iterating forward (and dening Q t;t = 1). This is a job creation equation, which states that the marginal costs of hiring a new worker g 0 (H t ), must equal the expected net present value of marginal prots (additional output minus the wage) of the lled job, S F t .

The Bargaining Set
Employment relationships generate a strictly positive surplus. This property of our model comes from the assumption that wages and e¤ort levels are determined after 27 With a slight abuse of notation, Ejt denotes the e¤ort level exerted by all workers (from di¤erent households) in a particular rm j. Firm j considers employing Njt workers, given that all other rms employ the equilibrium number of workers Nt. Because there are innitely many rms, rm js decision to employ Njt 6 6 = Nt workers does not a¤ect the fraction of household hs members that are employed, so that by the assumption of perfect risk-sharing within the household, the consumption of workers in rm j, C ht = Ct, is not a¤ected. Therefore, the relation between e¤ort and employment that the rm faces if all other rms (and all households) play equilibrium strategies, is given by equation (10), keeping Ct xed. See appendix A for details on the derivation of equation (12). employment adjustment costs are sunk: if rm and worker cannot agree to continue their relationship, then the rm has to pay the hiring costs again in order to nd another worker to match with. We make this timing assumption in order to generate wage setting under bilateral monopoly, as in a search and matching model, which we believe to be a realistic feature of the labor market. 28 Firms and households bargain over the wage as a way to share the match surplus. These negotations are limited only by the outside option of each party. The lower bound of the bargaining set is given by the reservation wage of the household, the wage o¤er at which the household is indi¤erent between accepting the o¤er and looking for another job. Similarly, the upper bound of the bargaining set is the reservation wage of the rm, the wage o¤er that makes the rm indi¤erent between accepting the o¤er and hiring a di¤erent worker. Within these bounds, any wage is consistent with equilibrium, see Hall (2005). Clearly, the bounds of the bargaining set are endogenous variables, which we now derive before introducing an equilibrium selection rule for the wage within the bargaining set.
The part of the match surplus that accrues to the rm S F t , as a function of the wage, is given by equation (14). In order to derive a similar expression for the households part of the surplus S H t , we must rst calculate the marginal disutility to the household of having one additional employed member, taking into account the endogenous response of e¤ort. This marginal disutility of employment, expressed in terms of consumption, is given by, 29 where the second equality follows from substituting equation (11), and where H =

1+
(1 )(1+ ) 1+ captures the e¤ect on utility of one more employed member in the household through the endogenous response of e¤ort. Using this expression, we can take a derivative of the households objective function (1) with respect to N t and divide by the marginal utility of consumption, to obtain the following expression for S H t .
The value to the household of having one more employed worker, equals the wage minus 28 Specically, the within-period timing we assume is the following. First, aggregate shocks realize and a randomly chosen fraction of employed workers is separated from their jobs. Second, rms that want to hire pay employment adjustment costs g (Ht) and are randomly matched with Ht non-employed workers. Third, rm and worker bilaterally and with full commitment decide on the e¤ort the worker will put into the job and the wage she will be paid for doing it. If a rm and a worker cannot agree, the worker is placed back into the unemployment pool and the rm pays g 0 (Ht) in order to get another random draw from that pool. Since all unemployed workers are identical, this never happens in equilibrium. When a rm and worker do reach an agreement, the worker is hired and added to the pool of employed workers. Finally, production, consumption and utility are realized. 29 The derivation of this expression is similar to that of equation (12), see appendix A for details. the disutility expressed in terms of consumption, plus the expected value of still having that worker next period, which is discounted by the probability that the worker is still employed next period.
The upper bound of the bargaining set W U B t is the highest wage such that S F t 0, whereas the lower bound W LB t is the lowest wage such that S H t 0. Using equations (14) and (17), we get S F t = W U B t W t and S H t = W t W LB t . Substituting back into equations (13), (14) and (17), we can explicitly write the equilibrium of the model in terms of the wage and the bounds of the bargaining set.
Everything else equal, the more rigid is the wage in response to technology or preference shocks that shift the bounds of the bargaining set, the more volatile is hiring H t and therefore employment N t in response to those shocks, see equation (18). We now turn to various possibilities for how wages are determined within the bargaining set.

Wage Determination
One possible criterion for wage determination that we can interpret as exible wages in a model with a frictional labor market, is period-by-period Nash bargaining. Nash bargaining assumes that the wage is set such that the total surplus from the match is split in equal proportions between household and rm. 30 It is straightforward to see that in our framework, this assumption implies that the wage is the average of the lower and upper bounds of the bargaining set, where W t denotes the Nash bargained wage.

Shimer (2005) and Hall (2005), among others, have argued that period-by-period
Nash bargaining generates too volatile a wage in equilibrium, relative to what is observed in the data. As discussed below, in our model period-by-period Nash bargaining leads to uctuations in the (log) wage of the same amplitude as labor productivity, and perfectly correlated with the latter. This is at odds with the data, where wages are about half as volatile as labor productivity in the pre-84 period, with the correlation between the two variables much smaller than one. Both the relative volatility of wages and their 30 The symmetry assumption is not crucial, but simplies the solution of the model substantially. The online appendix presents shows that our results are virtually unchanged for bargaining power well below and above 0:5. correlation with labor productivity increases signicantly in the post-84 period. This motivates the introduction of a wage setting mechanism that departs from period-byperiod Nash bargaining. 31 We use the following wage determination process, where R t measures the degree of wage rigidity, which is endogenous.
Only wages within the bargaining set are renegotiation-proof and therefore consistent with equilibrium, see Barro (1977). The maximum degree of wage stickiness that respects this condition, is for the wage to remain xed inside the bargaining set, but to be adjusted when it hits either bound. This is the wage determination mechanism in Thomas and Worrall (1988), MacLeod and Malcomson (1993) and Hall (2003). We use a convex version of this wage rule, in order to be able to solve the model using perturbation methods, and assume the following reduced-form equation for wage rigidity, where 2 N + 0 . This wage rule captures the idea that the wage is more likely to adjust when it is closer to the bounds of the bargaining set. The parameter captures the degree of non-linearity in this relation. For = 0, R t = 0 and W t = W t , i.e. wages are exible. For 2 N + , the degree of wage rigidity is endogenous, with wages being perfectly exible at the upper or lower bound of the bargaining set and most rigid at the Nash-bargained wage W t . As becomes larger, wages are rigid in a larger part of the bargaining set. The limiting case for ! 1 and R = 1 captures the case where the wage is xed within the bargaining set but adjusts when it has to in order to avoid ine¢cient match destruction as in Hall (2003). We consider a exible wage regime with = 0 and regime with endogenous wage rigidity, 2 N + and R = 1.
The crucial insight for our purposes is that with this type of wage rule, the degree of wage rigidity depends endogenously on the size of the frictions. If frictions decrease, the width of the bargaining set decreases as well, so that there is less room for wage rigidity. Notice also that this type of wage rigidity can never lead to ine¢cient match destruction.

Equilibrium
We conclude the description of the model by listing the conditions that characterize the equilibrium. Vacancy posting decisions by rms are summarized by the job creation equation (18).
The equilibrium level of e¤ort is determined by e¢ciency condition (11), and wage negotations are described by equations (22) and (23) for the equilibrium selection rule, and stochastic di¤erence equations for the upper and lower bounds of the bargaining set (19) and (20).
Employment evolves according to its law of motion (6).
Finally, goods market clearing requires that consumption equals output minus hiring costs.
Output is dened as in production function (4), and the stochastic discount factor as the marginal rate of intertemporal substitution (7).
and the parameters F = 1+ (1 ) and H = 1+ (1 )(1+ ) 1+ are functions of the structural parameters. In total, we have 8 equations in the endogenous variables H t , , W LB t , N t and C t , or 10 equations including the denitions for Y t and Q t;t+1 .
Without an endogenous e¤ort choice ( = 0 so that e¤ort is not useful in production, F H = 0, and E t = 0 for all t in equilibrium), and with exible wages (R = 0 so that R t = 0 for all t), the model reduces to a standard RBC model with labor market frictions. However, unlike in the standard model, uctuations in our model are driven by technology shocks as well as non-technology shocks or preference shocks. The two driving forces of uctuations, log total factor productivity a t log A t and log preferences over consumption z t log Z t follow stationary AR(1) processes, where " a t and " z t are independent white noise processes with variances given by 2 a and 2 z respectively.

Implications of the Decline in Labor Market Turnover
We now proceed to use this model to analyze the possible role of the decline in labor market turnover in generating the observed changes in the cyclical patterns of output, labor input, productivity, and wages. We start by looking at a version of the model with zero separation rate. In this case, small employment adjustments are costless so that the labor market e¤ectively becomes frictionless. This model provides a useful benchmark that we can solve for in closed form. Then, we rely on numerical methods to simulate the model for di¤erent values of the separation rate.

Zero Labor Market Turnover
Consider the limiting case of an economy with a separation rate of zero. In this case, hiring costs are zero in steady state. As a result, the width of the bargaining set collapses to zero, and the job creation equation (24) and the wage block of the model, equations (26), (27), (28) and (29), imply for all t. Employment becomes a choice variable, so that its law of motion (30) is dropped from the system and employment is instead determined by the static condition (36).
Substituting into the equilibrium condition for e¤ort (25), we obtain implying an e¤ort level that is invariant to uctuations in the models driving forces.
Since e¤ort has stronger diminishing returns in production and stronger increasing marginal disutility than employment, this intensive margin of adjustment is never used if the extensive margin is not subject to frictions.
Without hiring costs, the aggregate resource constraint (31) reduces to C t = Y t .
Combining the resource constraint and equations (37) and (38) with the production function (32), we can derive closed-form expressions for equilibrium employment, output, wages and labor productivity. Using lower-case letters to denote the natural logarithms of the original variables, ignoring constant terms and normalizing the variance of the shocks, 32 we get: A useful benchmark is the model with logarithmic utility over consumption ( = 1). In this case, employment uctuates in proportion to the preference shifter z t but does not respond to technology shocks. 33 From the previous equations, it is straightforward to calculate the models implications for the second moments of interest. In particular we have In the absence of labor market frictions, labor productivity is unambiguously countercyclical in response to preference shocks. The intuition for this result is that output responds to preference shocks only through employment, and this response is less than proportional because of diminishing returns in labor input ( 0). Since productivity is unambiguously procyclical in response to technology shocks, the unconditional correlations depend on the relative variances of the shocks and the model parameters. For a wide range of parameter values, e.g. with logarithmic utility over consumption ( = 1), productivity is procyclical with respect to output but countercyclical with respect to employment.
The relative volatility of employment and wages with respect to output are given by 32 If the original shocks areãt andzt, then we dene at at and zt zt, where 1= [1 (1 ) (1 )]. 33 This result is an implication of the logarithmic or balanced growth preferences over consumption in combination with the absence of capital or any other intertemporal smoothing technology, and is similar to the neutrality result in Shimer (2010). the following expressions: The size of the relative volatility measures above depends again on the relative importance of the shocks, as well as on the size of , the parameter determining the degree of diminishing returns to labor.

Innite Labor Market Turnover
We can contrast the predictions of the frictionless model above, with the opposite extreme case of an innitely large separation rate, resulting in employment being xed at zero, see equation (30). In this case, no workers will be hired, so that by the aggregate resource constraint (31) C t = Y t , as in the case of zero turnover. Combining the production function (32) with the equilibrium condition for e¤ort (25), and taking logarithms, ignoring constant terms and normalizing the variance of the shocks, 34 we get: Since employment is xed, e¤ort is now procyclical in response to both types of shocks, as all of the adjustment of labor input occurs on the intensive margin. With an innitely large separation rate, labor productivity is perfectly (positively) correlated with output.
The correlation between productivity and employment, as well as the relative volatility of employment with respect to output equal zero. Finally, since the bargaining set is now innitely wide, wages may be arbitrarily rigid, depending on the model parameters, so that the relative volatility of wages is also arbitrarily close to zero.

Preview of the Results
Comparing the predictions of the model with zero separations to the model with an innitely large separation rate, it is clear that for a su¢cienly large decline in labor market turnover: 1. Labor productivity becomes less procyclical with respect to output.
2. Labor productivity goes from acyclical to countercyclical with respect to employment, depending on parameter values (a su¢cient condition is logarithmic utility 34 In this case, the normalization factor is 1= [1 + (1 ) (1 ) ]. over consumption).
3. The relative volatility of employment increases.

The relative volatility of wages increases.
All four of these predictions are consistent with the data, as we documented in section 2.
Four elements of our model are crucial for these results: convex employment adjustment costs, multiple shocks, endogenous e¤ort and endogenous wage rigidity.
We are not arguing, of course, that labor market turnover fell from innity to zero.
Rather, the argument so far is meant to illustrate that if the decline in labor market turnover was large enough, it can qualitatively explain the patterns we observe in the data. To answer the question whether we can also quantitatively match those patterns for reasonable parameter values, we now turn to a numerical analysis of the full model.

Calibration
We simulate data at quarterly frequency and calibrate accordingly. The calibration is summarized in Table 6. Many of the models parameters can be easily calibrated to values that are standard in the literature. In this vein, we set the discount factor equal to 0:99, assume logarithmic utility over consumption ( = 1), and assume = 1=3 for the curvature of the production function to match the capital share in GDP. In the model there is no di¤erence between unemployment and non-participation. Therefore, we set the marginal utility from leisure to match the employment-population ratio.
Since the amount of labor market frictions a¤ects this ratio as well, we calibrate to an employment-population ratio of 0:7 in the frictionless model.
The calibration of the labor market frictions is crucial for the simulation exercise.
Estimates of the convexity of employment adjustment costs vary, with the exponent 1+ of the cost function g (H) = 1+ H 1+ ranging from 1.6 to 3.4. The lower end of this range corresponds to a specication, in which we interpret the adjustment costs as search frictions, vacancy posting costs are linear and the matching function has an elasticity with respect to unemployment of 0.6, as in Mortensen and Nagypal (2007). The upper end of the range is the point estimate of the convexity of employment adjustment costs in Merz and Yashiv (2007). In our benchmark specication, we use the midpoint of this range and assume an exponent of 1 + = 2:5, but we explore the implications for our results if adjustment costs are less or more convex than that. 35 We calibrate such that 35 Despite the estimates in Merz and Yashiv (2007), in later work Yashiv (2012) and Faccini, Millard, and Yashiv (2012) favor a less convex specication closer to our benchmark specication. Two important points should be noted here. First, a crucial insight from Merz and Yashiv (2007) and Yashiv (2012) is that it is important to allow for an interaction in adjustment costs in capital and labor in order to get correct estimates for these adjustment costs. We ignore this interaction because our model does not have capital. This should not cause any problems because we are not estimating the model. Second, by assuming adjustment costs are convex, we are ruling out a large class of adjustment cost functions. hiring costs are 3% of output in calibration for the pre-84 period, consistent with the estimates in Silva and Toledo (2009), see also Hagedorn andManovskii (2008, p.1699).
The employment outow rate declined by about 50%, from 4% per month in the early 1980s to 2% per month in the mid-1990s (Davis, Faberman, Haltiwanger, Jarmin, and Miranda (2010), Fujita (2011), Cairó andCajner (2013)). 36 Using these estimates, we calibrate the gross separation rate in our model to 35% per quarter for the pre-84 subsample and to 20% per quarter for the post-84 period. 37 In equilibrium, the decline in the separation rate implies a decline in job creation, because the amount of replacement hiring that is necessary to maintain a certain level of employment decreases. This e¤ect is dampened, however, by the lower cost of hiring, which raises equilibrium employment by about 14%.
For the models driving forces, we assume high persistence in both shocks, setting a = 0:97 to match the rst-order autocorrelation in Solow residuals, and z = 0:97 to make sure that none of the results are driven by di¤erences in persistence. Given those values, we calibrate 2 a and 2 z so that the frictionless version of the calibrated model matches the relative volatility of employment and predicts a standard deviation of log output of 1%. The rst target is justied by the observation that in this very simple model, preference shocks are a stand-in for all sources of misspecication that result in the unemployment volatility puzzle. The second target is arbitrarily chosen to emphasize that we consider this model mostly illustrative and not able to generate realistic predictions for the overall level of volatility in the economy.
For the parameters related to e¤ort, we have very little guidance from previous literature. We normalize = 0 and such that e¤ort is expressed in utility units and equals 1 in the frictionless steady state. We treat the curvature of the production function in e¤ort as a free parameter. Since we are mostly interested to illustrate This assumption is justied by various studies that nd that, while non-convexities are important at the plant level, convex adjustment costs provide a good approximation for the aggregate dynamics for capital (Cooper and Haltiwanger (2006), Khan and Thomas (2008)) and employment (Cooper and Willis (2004)). 36 The estimates in Fujita (2011) di¤er from those in Davis, Faberman, Haltiwanger, Jarmin, and Miranda (2010) and Cairó and Cajner (2013) because Fujita calculates worker ows from matching the labor force status of workers in the monthly CPS les, whereas the other two studies use data on unemployment duration following Shimer (2012). The size of the proportional decline in the separation rate is very similar in both approaches, but the level of the separation rate is di¤erent. Starting with Shimer (2005), it is common in the literature to calibrate models to the level of the separation rate as calculated from the unemployment duration data, resulting in a post-war sample average of about 3% per month. 37 The quarterly separation probability is the probability that a worker who is employed at the beginning of the quarter is no longer employed at the end of the quarter. Using a monthly job nding probability of fm = 0:45, see Shimer (2012), and a monthly separation probability of sm = 0:04, we get a quarterly separation probability of s = s m (1 f m ) 2 + (1 s m ) s m (1 f m ) + (1 s m ) 2 s m + s 2 m f m = 0:07 and a quarterly job nding probability of f = fm (1 sm) 2 + (1 fm) fm (1 sm) + (1 fm) 2 fm + f 2 m sm = 0:80. The gross separation rate is the average number of times that a worker who is employed at the beginning of the quarter loses her job over the quarter. Since workers that are separated in a given quarter may nd another job within that quarter, the quarterly gross separation rate is given by = s= (1 f ) = 0:35. the qualitative changes in the business cycle moments that the model can generate, we set this parameter fairly abitrarily to = 0:3, so that the model roughly replicates the second moments in the data. The testable prediction here is not whether the model can quantitatively match some or most of the second moments, but whether it can qualitatively generate all observed changes, changing only the separation rate.

Simulation Results
We now simulate the calibrated model in order to calculate the second moments of interest. We start with the model with exible wages and show that a decline in labor market turnover of the same size as observed in the US, roughly matches the change in the cyclicality of labor productivity and the relative volatility of labor input in the data.
Then we consider endogenous wage rigidity and show that the model can also generate an increase in the relative volatility of wages and a reduction in output volatility, although not of the same magnitude as observed in the data.
The model with exible wages is close to log-linear and a rst-order approximation captures well the dynamics generated by the model. However, the model with endogenous wage rigidity is non-linear, because the wage depends on the past wage W t 1 and the bounds of the bargaining set W U B t and W LB t , multiplied by the degree of rigidity R t , which itself depends again on W t and W U B t and W LB t , see (26) and (27). If we were to log-linearize this wage rule, it would reduce to a partial adjustment rule with a constant degree of wage rigidity. Therefore, we use a second-order approximation of the policy functions. As an accuracy check, Figure 3 shows that a second-order approximation captures well the non-linear wage rule for = 1, but for = 2 or larger, a higher-order approximation is needed. We simulate the second-order approximation of the model 201; 000 periods, discarding the rst 1; 000 observations to eliminate the e¤ect of the initial conditions. The results of this exercise are reported in Table 7.

Flexible Wages
Labor productivity is strongly procyclical in terms of its correlation with output in the model and its procyclicality falls substantially as we reduce labor market turnover. The correlation of productivity with employment also falls, from around zero in the labor market with high turnover to a negative value in the calibration with low turnover. Both observations are qualitatively as well as quantitatively consistent with the evidence. The reason for the decline in the procyclicality of productivity, is the increase in the relative volatility of employment, a result that is consistent with the data as well.
Three elements in the model are crucial for these results. First, the convexity of the employment adjustment costs implies that hiring costs fall from 3% to around 1% of output with the decline in labor market turnover. Second, the e¤ort choice provides an intensive margin of adjustment for labor input. As frictions fall, it becomes optimal to adjust labor more through employment and less through e¤ort. Thus, the volatility of employment increases more than that of output. Note that the model also predicts that the volatility of e¤ort falls, but this predictions is not testable without observable measures of e¤ort. 38 The third element in the model that is important for the results is that uctuations in the model are driven by two types of shocks: technology shocks and preference shocks or labor supply shocks. In a one-shock model, the correlations between all variables would be close to either 1 or 1. 39 In addition, if uctuations were driven only by technology shocks then productivity could never be countercyclical, since employment would only uctuate because of changes in labor demand, and the direct e¤ect of technology on productivity would always prevail over the indirect e¤ect of employment. It is important to stress, however, that our results are not driven by changes in the relative importance of both shocks, which we keep constant, but by the reduction in frictions, which changes the response of the economy conditional on each shock.
The model also predicts a small decrease in the relative volatility of wages and a small increase in the volatility of output. As argued in section 2.3, the rst prediction is arguably not consistent with the data. The second one clearly is in contradiction with the well-documented reduction in output volatility, the so called Great Moderation. The decrease in the relative volatility of wages is driven by the fact that the wage is approximately proportional to the marginal product of labor. 40 Since the marginal product of labor is proportional to output, but inversely proportional to employment, an increase in the relative volatility of employment must necessarily also decrease the relative volatility of wages. The increase in the volatility of output simply stems from the fact that reducing adjustment costs amplies uctuations in employment and therefore in output as well. In the next subsection we show that endogenous wage rigidities can reverse the predictions of the model for wages and possibly also bring the prediction for output volatility closer to the evidence.

Endogenous Wage Rigidity
The third panel in Table 7 presents the simulated second moments for the model with endogenous wage rigidity ( = 1 and R = 0:95). Comparing these moments to those for 38 Some authors have used hours per worker as an observable proxy for e¤ort, see e.g. Basu, Fernald, and Kimball (2006). However, the crucial characteristic of the e¤ort margin in our model is that it is not subject to adjustment costs, or at least that adjusting this margin is less costly than adjusting along the extensive margin. It is not clear that adjusting work hours is costless. In fact, van Rens (2012) argues that adjustment costs in hours per worker may be larger rather than smaller than adjustment costs in employment in OECD countries. 39 This is exactly true in a static, linear model. Our model is close to (log)linear and the version without capital and with exible wages has only one state variable (employment), which has very fast transition dynamics. 40 In a frictional labor market, the wage is not equal to marginal product of labor, but as long as is not too large, they are still proportional since workers and rms share the match surplus in equal proportions. the exible wage model, we see that the previously described predictions of the model for the cyclicality of labor productivity and the relative volatility of employment remain largely unchanged. The reason is that the fact that wages adjust when they get close to the bounds of the bargaining set mitigates the allocative e¤ect of wage rigidity.
The prediction of the model for the relative volatility of wages, however, is reversed.
The reduction in labor market frictions now increases the volatility of wages, although not by as much as in the data. To understand the mechanism behind this result, it is useful to consider the extreme case of endogenous wage rigidity ( ! 1), in which wages are completely xed within the bargaining set, but adjust when they hit the bounds of the bargaining set. If frictions are high enough, so that the bargaining set is very wide, wages never adjust and their volatility is zero. On the other extreme, in a frictionless labor market, the bargaining set reduces to a point and wages behave as if they were exible. Of course this e¤ect is counteracted by the fact that the bounds of the bargaining set themselves are less volatile when frictions are lower. However, for our calibration of the parameters, the rst e¤ect dominates, as illustrated in Figure 4.
The small increase in output volatility in response to the reduction in labor market frictions is reversed to a (very small) decrease for the model with endogenous wage rigidity. The reason is that increased wage exibility dampens uctuations in output in response to technology shocks. 41 This result is consistent with a possible role of a decline in labor market frictions as a source of the Great Moderation. However, the e¤ect is again much smaller than in the data. Thus, whereas we nd this last result intuitively compelling, it is not clear whether it is quantitatively important.

Conclusions
In this paper, we documented three changes in labor market dynamics over the postwar period in the US: the strong procyclicality of labor productivity has vanished, the volatility of employment has increased with respect to output, and the volatility of wages has increased relative to output and possibly even in absolute terms. We presented a model to argue that the decline in labor market turnover, modelled as a reduction in the employment exit probability, can explain all three facts. In addition, we showed that in principle the decline in turnover may also have contributed to the reduction in output volatility, which happened around the same time.
The intuition for why a decline in labor market turnover increases the relative volatility of employment and reduces the procyclicality of labor productivity is straightforward and compelling. If employment adjustment costs are convex, then lower turnover implies lower hiring costs. If there is another input into production that can be used at least partly as a substitute for labor, then a reduction in hiring frictions will make that input less volatile, so that employment becomes more volatile with respect to output. In this paper, we refer to this other factor input as e¤ort, but a very similar argument can be made for capacity utilization of capital. Given that capital does not uctuate much at business cycle frequencies, the fact that the comovement of labor and output and therefore labor productivity has changed almost unavoidably leads to the conclusion that there must be another input into the production process.
Our argument that the decline in labor market turnover is also responsible for the increase in the relative volatility of wages is more contentious. Our simulations of the model with endogenous wage ridigity show that this e¤ect is qualitatively present, and can be made to dominate the direct e¤ect on wages for reasonable parameter values.
However, the increase in wage volatility in our simulations is much smaller than in the data. 42 It is possible that this e¤ect could be amplied by using a more non-linear wage rule, but we do not pursue this this possibility. While there is a compelling argument that wage rigidity is, at least to some degree, endogenous, it remains an open question quantitatively how important this mechanism is.
What have we learned about the possible causes of the Great Moderation? We showed that a decline in labor market turnover, through a decrease in hiring frictions and an endogenous decrease in wage rigidity, can lead to a decrease in output volatility. However, this e¤ect is small in our simulations, partly because the decrease in wage rigidity is relatively small (at least much smaller than in the data), but also because there is a direct e¤ect of the reduction in hiring frictions that counteracts the e¤ect of decreased wage rigidity: reduced frictions make labor more volatile, reducing the volatility of its marginal product. A further caveat against the conclusion that the Great Moderation is driven by the decline in labor market turnover is that this story only works if technology shocks are relatively important as a source of business cycle uctuations. The reason is that preference shocks act as labor supply shocks, so that wage rigidity dampens rather than amplies uctuations in employment in response to these shocks. Finally, reduced wage rigidity can only lead to reduced output volatility if the wage is allocative, i.e. if the reduction in wage rigidity applies to the wage of newly hired workers. We documented an increase in the relative volatility of wages of new hires, see section 2.3.2, but that increase is substantially smaller than for the average wage in the economy, casting further doubt on the importance of this mechanism for the Great Moderation. 42 At least than the data from the Labor Productivity and Cost program and the Current Population Survey. The data from the Current Employment Statistics show a decrease in the relative volatility of wages, see section 2.3, which our model has no trouble generating.

A Marginal Product and Disutility of E¤ort
This appendix derives the marginal product of employment to the rm, equation (12), and the marginal disutility from employment, expressed in consumption terms, to the household, equation (16), if e¤ort adjusts endogenously. From equations (4) and (2), it is straightforward di¤erentation to decompose the total e¤ect of employment on output and total e¤ective labor supply into a direct e¤ect and an e¤ect through the endogenous response of e¤ort.
Here, E jt denotes the e¤ort of all workers i that are employed in rm j and E ht the e¤ort of all workers that are members of household h.
To nd the response of e¤ort to changes in employment that rm and household face, we use the condition that the marginal disutility from e¤ort of a given worker i (expressed in consumption terms) from equation (8), in equilibrium must equal the marginal productivity of that worker to the rm from equation (9).
First, suppose rm j considers employing N jt workers, given that all other rms employ the equilibrium number of workers N t . Because there are innitely many rms, rm js decision to employ N jt 6 6 = N t workers does not a¤ect the fraction of household hs members that are employed, so that by the assumption of perfect risk-sharing within the household, the consumption of workers in rm j is not a¤ected, C ht = C t . Substituting this, as well as the condition that all workers in rm j exert the same amount of e¤ort, E it = E jt for all i 2 [0; N jt ], the e¤ort condition becomes, so that the elasticity of e¤ort in a given rm j with respect to employment in that rm, is given by Substituting this elasticity into equation (48) above, gives expression (12) in the text.
Next, suppose household h considers having N ht employed workers, given that all other households have N t employed workers. Because there are innitely many households, households hs decision to have a fraction of N ht 6 6 = N t of its members employed, does not a¤ect the level of employment in any rm N jt = N t . Furthermore, although the e¤ort level of worker i may change because of household hs decision, e¤ort of all other workers in rm j, who are members of di¤erent households, is una¤ected, E it = E ht and E i 0 t = E t for i 0 6 = i. Thus, the e¤ort condition becomes, and the elasticity of e¤ort exerted by members of household h with respect to employment in that household, using equation (3), is given by, Substituting this elasticity into equation (49) above, gives expression (16) in the text.       1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Labor productivity (bandpass filter) Output per hour in the US private sector. Shaded areas are NBER recessions.  1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Correl prod with output (blue) and hours (red), cntrd 6-yr rolling window, bp Correlations are calculated in a centered 6-year rolling window of quarterly bandpassltered data.