Keywords: Unemployment fluctuations, labor productivity, search and matching model, NewKeynesian model
Abstract:
JEL classifications: E32, E37, J63, J64
In a very influential paper, Shimer (2005) argues that the MortensenPissarides (MP) search model of unemployment lacks an amplification mechanism because it generates less than 10 percent of the observed business cycle fluctuations in unemployment given labor productivity shocks of plausible magnitude. In this paper, I argue that Shimer's (2005) estimate may be biased downward because of the endogeneity of labor productivity, and I estimate that a third, and possibly as much as 60 percent, of the Shimer puzzle is simply due to the misidentification of productivity shocks.
The Shimer puzzle has attracted a lot of interest in the literature, and a number of researchers have focused on ways to create more amplification so that small exogenous productivity movements generate large fluctuations in unemployment.^{1} However, there is substantial evidence that, perhaps due to labor hoarding and variable capacity utilization, some of the movements in productivity are in fact endogenous.^{2} For example, when the firm is demand constrained in the shortrun, firms can respond to changes in demand by adjusting their level of capacity utilization of inputs (capital or labor), and measured labor productivity fluctuates endogenously with aggregate demand and hence unemployment.^{3} By filtering out the trend component of output per hour to identify productivity shocks, Shimer (2005) may not identify the true productivity shocks but rather the endogenous response of productivity to unobserved disturbances. And because this endogenous response is small, this may explain why the cyclical component of measured labor productivity fluctuates less than unemployment.
To estimate the impact of exogenous changes in productivity on labor market variables, I impose longrun restrictions in a structural VAR model along the line of Gali (1999), and I find that a permanent productivity increase temporarily lowers labor market tightness (the vacancyunemployment ratio), while the MP model implies the opposite.^{4} Hence, before assessing the amplification properties of the MP model, I embed the search and matching model in a New Keynesian framework. In this setup, a permanent increase in productivity (i.e. a positive productivity shock) may temporarily raise unemployment and lower labor market tightness because aggregate demand does not adjust immediately to the new productivity level in the presence of nominal rigidities, and hence firms use less labor. The model also generates endogenous movements in productivity. Because hiring firms are demand constrained, an aggregate demand shock generates a transitory movement in productivity as firms vary their level of capacity utilization.
To estimate the proportion of Shimer's puzzle due to the endogeneity of productivity, I use a calibrated version of the model to control for endogenous productivity movements unrelated to productivity shocks, and I reproduce Shimer's (2005) exercise on data simulated from my model. With a standard calibration, simulated labor market tightness is 9 times more volatile than the cyclical component of labor productivity, while the ratio comes at about 26 in US data. I conclude that the MP model can account for about a third, rather than 10 percent, of labor market tightness fluctuations, and a sensitivity analysis suggests that this share could be as high as 60 percent.
The remainder of the paper is organized as follows: Section 2 discusses Shimer's (2005) puzzle; Section 3 presents and calibrates a NewKeynesian model with search unemployment and replicates Shimer's (2005) exercise on model generated data; and Section 4 offers some concluding remarks.
In this section, I reproduce Shimer's exercise (2005), and Table 1 presents summary statistics for unemployment, vacancies, labor market tightness and productivity.^{5} As originally argued by Shimer (2005), the volatility of productivity is only a fraction (here less than 4%) of the volatility of labor market tightness. Turning to the correlation matrix, unemployment and labor market tightness are weakly correlated with productivity with correlations of respectively 0.23 and 0.19.
In the context of a standard MP model where productivity movements are the central driving force of unemployment fluctuations, Shimer (2005) shows that the standard deviations of unemployment, vacancies and productivity are of the same order of magnitude, i.e. By estimating that productivity shocks are only 10% as volatile as unemployment fluctuations, Shimer (2005) concludes that the MP model can only account for less than 10% of unemployment fluctuations. Furthermore, Shimer (2005) notes that the MP model exhibits virtually no propagation as it implies a contemporaneous correlation between unemployment and productivity of 1 when the data show a contemporaneous and peak unemploymentproductivity correlation of respectively only 0.23 and 0.50 .
One way to reconcile the MP framework with the data is to modify the model so that it generates more amplification, i.e. that a given shock to productivity has a larger impact on unemployment. Mortensen and Nagypal (2006) provide a detailed review of the current effort in that direction, and I will only emphasize two influential examples. A first possibility, suggested by Hall (2005) and Shimer (2005), is to introduce real wage rigidity. In the standard MP model, the Nash bargaining real wage responds so much to movements in productivity that it effectively absorbs most of the changes in productivity. As a result, the surplus of the match responds only weakly to fluctuations in productivity. By introducing a degree of real wage rigidity, movements in productivity have a more substantial impact on the match surplus, on the incentives of firms to post vacancies and hence on equilibrium unemployment.
Another possibility, suggested by Hagedorn and Manovskii (2004), does not rely on real wage rigidity but uses a standard MP model with a different calibration than the one used in Shimer's. Hagedorn and Manovskii (2004) show that when the opportunity cost of employment is high, the job finding rate becomes very responsive to changes in productivity, and the MP model can quantitatively account for the magnitude of unemployment fluctuations. While this approach is different from the one proposed by Hall (2005) and Shimer (2005), the underlying philosophy is the same: one needs to modify the MP model (either its equations or its calibration) so that the surplus of the match becomes more responsive to changes in productivity.
The aforementioned literature generally considers productivity movements as exogenous. However, there is substantial evidence that, perhaps due to labor hoarding and variable capacity utilization, some of the movements in productivity are in fact endogenous.^{6}
To identify the impact of exogenous changes in productivity, I follow Galí (1999) and Blanchard and Quah (1989) and impose longrun restrictions in structural VAR models to identify technological disturbances. Technology shocks are the only shocks with a permanent impact on productivity, and I interpret transitory productivity movements as variations in capacity utilization. Specifically, I am interested in estimating the system
where is labor productivity defined as output per hours, the vacancyunemployment ratio, an invertible matrix polynomial and the vector of structural orthogonal innovations comprises technology shocks and nontechnology shocks .^{7}Figure 1 presents the impulse response functions. The Shimer puzzle is clearly apparent for both shocks: the standard deviation over the first two years after a technology shock is 16 times larger for labor market tightness than for output per hour, and after a nontechnology shock, the ratio is 21. However, as similarly emphasized in Gali (1999), Barnichon (2008) and Canova, LopezSalido and Michelacci (2008) a positive technology shock temporarily lowers labor market tightness, while the MP model implies the opposite.^{8} This implies that it is difficult to draw conclusions regarding the amplification properties of the baseline MP model since its transmission mechanism is likely to be incomplete.
To reassess the extent of Shimer's puzzle, it is important to extend the search and matching model so that it can (i) rationalize endogenous productivity movements, and (ii) account for the fact that permanent productivity increases temporarily lower labor market tightness. To do so, I follow Gali (1999) and Barnichon (2008), and I extend the MP model so that hiring firms are demand constrained in a NewKeynesian fashion.
In a neoclassical setting, firms post vacancies depending on the return of the match. However, this needs not be the case when firms have to satisfy a given level of demand for their products. In a NewKeynesian setting with monopolistically competitive firms and nominal rigidities, firms may have to hire more workers when demand is unexpectedly high even if productivity (and hence the match surplus) does not increase. Put differently, the number of posted vacancies could increase without any change in productivity. In practice, firms also respond to higher demand by increasing capacity utilization of inputs (capital or labor). As a result, measured labor productivity fluctuates with aggregate demand and hence unemployment.
A permanent increase in productivity (i.e. a technology shock) may temporarily raise unemployment because with nominal rigidities, aggregate demand does not adjust immediately to the new productivity level, and firms use less labor.
In the next subsections, I present and calibrate a NewKeynesian model with search unemployment, and I replicate Shimer's exercise on model generated data.
Following Barnichon (2008) and Krause and Lubik (2007), I extend the MP model by introducing nominal frictions so that hiring firms are demand constrained in a NewKeynesian fashion. In addition, I make a distinction between the extensive (number of workers) and the intensive (hours and effort) labor margins. In this framework, unemployment fluctuations are the product of two disturbances: technology shocks and monetary policy (or aggregate demand) shocks. A positive technology shock permanently raises productivity but may also temporarily raise unemployment and lower labor market tightness. A positive monetary policy shock decreases unemployment and increases measured productivity temporarily, because firms increase labor effort to satisfy demand in the short run. As a result, measured labor productivity is the product of two components: permanent and temporary disturbances.
The main ingredients of the model are monopolistic competition in the goods market, hiring frictions in the labor market and nominal price rigidities. There are three types of agents: households, firms and a monetary authority.
I consider an economy populated by a continuum of households of measure one and a continuum of firms of measure one. With equilibrium unemployment, exante homogenous workers become heterogeneous in the absence of perfect income insurance because each individual's wealth differs based on his employment history. To avoid distributional issues, I follow Merz (1995) and Andolfatto (1996) in assuming that households form an extended family that pools its income and chooses per capita consumption and assets holding to maximize its expected lifetime utility. There are unemployed workers who receive unemployment benefits in units of utility of consumption, and employed workers who receive the wage payment from firm for providing hours and effort per hour .^{9} Denoting the individual disutility from working, the representative family seeks to maximize
subject to the budget constraint with a positive constant, nominal money holdings, total transfers to the family and the composite consumption good index defined by where is the quantity of good consumed in period and is the price of variety is the elasticity of substitution among consumption goods. The aggregate price level is defined as . The disutility from supplying hours of work and effort per hour is the sum of the disutilities of the members who are employed. Following Bils and Cho (1994), the individual period disutility of labor takes the form: where and are positive constants. The last term reflects disutility from exerting effort with the marginal disutility of effort per hour rising with the number of hours. An infinite value for generates the standard case with inelastic effort.Each differentiated good is produced by a monopolistically competitive firm using labor as the only input. There is a continuum of large firms distributed on the unit interval. At date , each firm hires workers to produce a quantity where is an aggregate technology index, the effective labor input supplied by each worker and .^{10} I define effective labor input as a function of hours and effort per hour such that . Total effective labor input can be adjusted through three channels: the extensive margin , and the two intensive margins: hours and effort per hour . With variable effort, the model will be able to generate endogenous procyclical movements in productivity.
Being a monopolistic producer, the firm faces a downward sloping demand curve
and chooses its price
to maximize its value function given the aggregate price level and aggregate output . As is standard in NewKeynesian models, firms are subject to Calvotype price setting and can only reset their price at random dates. Each period a fraction of
randomly selected firms cannot reset its price.
In a search and matching model of the labor market, workers cannot be hired instantaneously and must be hired from the unemployment pool through a costly and timeconsuming job creation process. Firms post vacancies at a unitary cost, , and unemployed workers search for jobs. I assume that the matching function takes the usual CobbDouglas form so that the flow of successful matches within period is given by where is a positive constant, , denotes the number of unemployed and = the total number of vacancies posted by all firms. Accordingly, the probability of a vacancy being filled in the next period is where is the labor market tightness. Similarly, the probability for an unemployed to find a job is . Matches are destroyed at a constant rate , and the law of motion for a representative firm is given by .
When a firm and a worker meet, they must decide on the allocation of hours and effort to satisfy demand. I assume that both parties negotiate the hours/effort decision by choosing the optimal allocation and set hours and effort per hour to satisfy demand at the lowest utility cost for the worker. More precisely, they solve
subject to satisfying demand at date , and this implies that effort per hour is a function of total hours where is a positive constant. Thus, changes in hours can proxy for changes in effort, and I can write a reducedform relationship between output and hours with and . For , the production function displays short run increasing returns to hours, and endogenous labor productivity (i.e. output per hour) movements are procyclical.As is usual in the search literature, firms and workers bargain individually about the real wage and split the surplus in shares determined by an exogenous bargaining weight . Denoting the value of a matched worker to firm at date , and and the value for a worker of being respectively employed by firm and unemployed, the equilibrium wage satisfies arg and is a solution of the firstorder differential equation
Given the market wage and aggregate price level, firm will choose a sequence of price and vacancies to maximize the expected present discounted value of future profits subject to the demand constraint, the Calvo price setting rule, the hours/effort choice and the law of motion for employment. Formally, the firm maximizes its value
subject to the demand constraint the law of motion for employment and the bargained wageConsistent with the long run identifying assumption made in Section 2, the technology index series is nonstationary with a unit root originating in technological innovations. Hence, technology is comprised of a deterministic and a stochastic component: with and is a technology shock with a permanent impact on productivity.
Consistent with a growing economy and zero inflation in " steadystate", the money supply evolves according to with , and I interpret as an aggregate demand shock.
Averaging firms' employment, total employment evolves according to The labor force being normalized to one, the number of unemployed workers is Finally, as in Krause and Lubik (2007), vacancy posting costs are distributed to the aggregate households so that in equilibrium. To solve the model, I loglinearize the firstorder conditions around the (zeroinflation) long run equilibrium.^{13}
I now discuss the calibration of the parameters of the model, and Table 2 lists the parameter values. Whenever possible, I use values typically used in the literature. I set the quarterly discount factor to 0.99 and the returns to efficient labor to 0.64. I assume that the markup of prices over marginal costs is on average 10 percent, which amounts to setting equal to 11. I choose so that firms reset their price every 2 quarters, consistent with Bils and Klenow (2004). I set the growth rate of technology (and money supply) to a quarter so that the economy is growing by on average each year. I use a money growth autocorrelation parameter of following Krause and Lubik (2007). Turning to the labor market, I use a middle value for the matching function elasticity and set the bargaining weight following the Hosios (1990) condition. The scale parameter of the matching functions is chosen such that, as reported in den Haan, Ramey and Watson (2000), a firm fills a vacancy with a quarterly probability and, as used in Thomas (2008), a worker finds a job with probability . Following Shimer (2005), the separation rate is 10% so jobs last for about 2.5 years on average, and the income replacement ratio is set to 40%. I choose (i.e. an hours per worker elasticity of 0.5) and need to decide on to fix a value for . Bils and Cho (1994) build a model to account for the procyclicality of labor productivity. In doing so, they allow for variable effort and variable capital utilization. The present model does not consider capital explicitly but implicitly if one assumes a constant capitallabor ratio. A key hypothesis of Bils and Cho (1994) is that the capital utilization rate is proportional to hours. If a worker works longer hours and at a more intense pace, the utilization of the capital he operates will also tend to increase. As a result, changes in hours per worker proxy not only for variations in effort but also for unobserved changes in capital utilization. In that case, Schor's (1997) estimate for the elasticity of effort with respect to hours delivers a value for of . I set accordingly in order to match this estimate.^{14} Finally, and consistent with the aim of the paper to reassess Shimer's puzzle while controlling for the endogeneity of productivity, I set the standard deviations of technology and monetary policy shocks and equal to the standard deviations of technology and nontechnology shocks estimated with the structural VAR.^{15}
Figure 2 and 3 show the impulse response functions after technology shocks and monetary policy (or aggregate demand) shocks. A first observation is that this New Keynesian MP model fulfills the two necessary conditions to reassess Shimer's puzzle: it is successful at replicating the productivity responses to both shocks (or put differently, it can be used to control for the endogeneity of productivity), and it gets the sign of labor market tightness responses right. Nonetheless, the Shimer puzzle is apparent after both shocks: model labor market tightness moves a lot less than its empirical counterpart.
However, after a nontechnology shock, the standard deviation of model labor market tightness over the first two years after a technology shock is almost 9 times larger than for model output per hour. Since the empirical ratio is 21, the MP model explains in fact 40% of labor market tightness fluctuations following an aggregate demand shock. This backoftheenvelope calculation suggests that the misidentification of productivity shocks and the endogeneity of productivity may be responsible for some of the Shimer puzzle.
Using a calibrated version of the model, I simulate 50 years of data, and I repeat the exercise 5000 times. Following Shimer (2005), I detrend the model generated productivity series, and in Table 2, I report the summary statistics for the simulated labor market variables. Despite a baseline MortensenPissarides structure of the labor market and a standard calibration, simulated is 9 times more volatile than the cyclical component of labor productivity, while the ratio comes at about 26 in US data. I conclude that the MP model can account for about a third, rather than 10 percent, of labor market tightness fluctuations.
In other dimensions, the model performs remarkably well as the crosscorrelations have the right signs and are not far off the true values. In particular, unemployment is only weakly correlated with productivity ( 0.24) and matches quite closely its empirical counterpart ( 0.23). However, the autocorrelation of model vacancies is 0.42 instead of 0.90 for US data. This is due to the excessively rapid response of vacancies. This problem was already pointed out by Fujita and Ramey (2004) and incorporating sunk costs for vacancy creation as in Fujita and Ramey (2004) would presumably correct this shortcoming. Similarly, this excess sensitivity of vacancies can explain the slightly too high vacancyproductivity and labor market tightnessproductivity correlations (both 0.49, compared with empirical values of 0.25 and 0.19).
Since the main result of this paper comes out of a calibration exercise, I present in Table 4 the influence of key parameters on the ability of the extended MP model to generate fluctuations in labor market variables. First, I span the range of plausible values for the elasticity of the matching function from (Hall, 2005) to 0.72 (Shimer, 2005) and find that the MP model explains between roughly 25 and 50 percent of fluctuations in .^{16} The return to hours coefficient is also an important parameter, and Basu and Kimball (1998) estimates that it ranges between 1.3 and 1.6. Within this interval, the MP model accounts for between 30 and 60 percent of unemployment fluctuations. With a higher degree of price stickiness (one year), the MP model accounts for almost 50% of fluctuations in . Finally, varying the value of the income replacement ratio from 0.4 (Shimer, 2005) to the high value used in Hagedorn and Manovskii (2005) of improves the "amplification properties" of the New Keynesian MP model so much that it generates too much volatility in Similarly, lowering the worker's bargaining weight improves the performance of the MP model, and a low value as in Hagedorn and Manovskii (2005) allows the MP model to account for 40% of labor market tightness fluctuations.^{17}
In a very important paper, Shimer (2005) argues that the MortensenPissarides search model of unemployment lacks an amplification mechanism because it cannot generate the observed business cycle fluctuations in unemployment given labor productivity shocks of plausible magnitude.
In this paper, I show that because of the endogeneity of measured labor productivity, filtering out the trend component of output per hour as in Shimer (2005) may not correctly identify the shocks driving unemployment. In fact, using longrun restrictions in a structural VAR model to isolate exogenous productivity shocks, I find that a permanent increase in productivity lowers the vacancyunemployment ratio, while the MP model implies the opposite. I embed the MP model in a NewKeynesian framework to (i) account for this empirical evidence, and (ii) control for the endogeneity of productivity, and I estimate that the MP model can account for a third, and possibly as much as 60 percent, of fluctuations in labor market variables.
Interestingly, this finding is in line with the work by Pissarides (2007) who reconsiders the Shimer puzzle in the context of an MP model with endogenous job destruction. Pissarides (2007) reestimates the unemployment volatility puzzle downwards and claims that "with endogenous job destruction, the model fails to account for about half to two thirds of the volatility in unemployment" instead of the 90% originally estimated by Shimer (2005). If a third of the Shimer puzzle is due to the misidentification of productivity shocks and another 30 to 50 percent is due to the omission of endogenous job destruction, the low volatility of unemployment relative to that of productivity may be less of a problem than originally thought.
To analyze the behavior of the economy, I loglinearize the firstorder conditions around the (zeroinflation) long run equilibrium.
The optimal vacancy posting condition takes the form
(3) 
Loglinearizing the vacancy posting condition equation around the (zeroinflation) steady state, I get for any
with the value of a marginal worker given by With Calvotype price setting, a firm resetting its price at date will satisfy the standard Calvo price setting condition: where the optimal markup is and the firm's real marginal cost The firm will choose a price that is, in expected terms, a constant markup over its real marginal cost for the expected lifetime of the price.To derive the NewKeynesian Phillips curve, I loglinearize around the zero inflation equilibrium. However, because of firms' expost heterogeneity, the derivation is not as straightforward as with costly price adjustment. I follow Woodford's (2004) similar treatment of endogenous capital in a NewKeynesian model with Calvo price rigidity. In my case, employment is the state variable and plays the role of capital in Woodford's model. The pricesetting condition becomes
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Standard deviation  0.125  0.139  0.257  0.010 
Quarterly autocorrelation  0.87  0.90  0.89  0.69 
Correlation matrix:  1  0.95  0.97  0.23 
Correlation matrix:    1  0.98  0.25 
Correlation matrix:      1  0.19 
Correlation matrix:  1 
Notes: Seasonally adjusted unemployment u is constructed by the BLS from the Current Population Survey (CPS). The seasonally adjusted helpwanted advertising index v is constructed by the Conference Board. Labor market tightness is the vacancunemployment ratio. Average labor productivity p is seasonally adjusted real average output per person in the nonfarm business sector, constructed by the Bureau of Labor Statistics (BLS) from the National Income and Product Accounts and the Current Employment Statistics. All variables are reported in logs as deviations from an HP trend with smoothing parameter 1600.
Discount rate  
Matching function elasticity  
Bargaining weight  Hosios (1990)  
Probability vacancy is filled  q(=0.7  den Haan, Ramey and Watson (2000) 
Job finding probability  q(=0.6  Thomas (2008) 
Separation rate  Shimer (2005)  
Income replacement ratio  b=0.4  Shimer(2005) 
Returns to scale to efficient hours  
Disutility of hours  =2  
Shortrun increasing returns to hours  Schor (1997)  
Growth rate of 2% a year  a=0.5%  
Degree of price stickiness 
(2 quarters) 
Bils and Klenow (2004) 
Markup of 10%  
AR(1) process for money growth  =0.5  Krause and Lubik (2007) 
Standarddeviation of monetary policy shock  =0.0226  
Standarddeviation of technology shock  =0.006 
Standard deviation  0.042
(0.004) 
0.065
(0.004) 
0.089
(0.008) 
0.010
(0.001) 
Quarterly autocorrelation  0.85
(0.03) 
0.42
(0.06) 
0.74
(0.05) 
0.69
(0.05) 
Correlation matrix:  1  0.35
(0.07) 
0.72
(0.04) 
0.24
(0.10) 
Correlation matrix:    1  0.90
(0.01) 
0.49
(0.07) 
Correlation matrix:      1  0.49
(0.08) 
Correlation matrix:        1 
Notes: Standard errorsthe standard deviation across 5000 model simulationsare reported in parentheses.
Parameter value  (  Share of ( explained by MP model  

Matching function elasticity  0.24
0.72 
0.11
0.61 
47%
25% 

Return to hours  1.3
1.6 
0.15
0.07 
60%
27% 

Degree of price stickiness  0.75  0.12  47%  
Income replacement ratio  0.2
0.9 
0.61
0.29 
25%
116% 

Worker's bargaining weight  0.5
0.05 
0.85
0.10 
33%
40% 
Notes: MP( is the model generated standarddeviation of labor market tightness and US( is its empirical counterpart.