Understanding Unemployment through the Lens of Search and Growth Theory:

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1 Understanding Unemployment through the Lens of Search and Growth Theory: Shirking and Unemployment Fluctuations 1 Norikau Tawara 2 August 2008 Preliminary Please do not cite without permission Abstract A variant of Diamond, Mortensen and Pissarides model with aggregate shocks to labor productivity, combined with a shirking model of e ciency wages, is examined. Incorporating the no-shirking condition (NSC) into the standard matching model of unemployment does not reduce uctuations of wages much, being consistent with the previous literature (Gomme 1999, e.g.), on one hand. On the other, it increases uctuations of unemployment and vacancies to the large extent, because the observed wage output ratio and the binding NSC implies the very large value of non-working (leisure). The key is that the binding NSC o ers tight disciplines in calibrating the parameter of the non-working (leisure) value. Key Words: Search and matching; Shirking models of e ciency wages; Fluctuations of vacancies, unemployment, and wages; The value of nonworking (leisure) JEL Classi cations: E24; E32; J41; J63; J64 1 All errors are my own. Other usual disclaimers apply. 2 Kanto-gakuen University, Department of Economics and Nihon University, Advanced Research Institute for Sciences and Humanities. address: nori.tawara@gmail.com. 1

2 1 Introduction A variant of Diamond, Mortensen and Pissarides model (Diamond 1982; Mortensen and Pissarides 1994; Pissarides 1985), combined with a shirking model of e ciency wages (Shapiro and Stiglit 1984), is examined. I am aware of two papers that do this. Rocheteau (2001) examines steady state of such model, in which wages are determined by maximiing the Nash product so that the no-shirking conditions are satis ed. Costain and Jansen (2006) examines a search matching model with aggregate shocks to productivity in which wage determination is similar to Rocheteau (2001). In both papers, the no shirking conditions, which say that employees surplus has to be suf- ciently large to motivate employees to exert e orts, are binding at some aggregate states and are slack at other aggregate states. In this paper, I examine a variant of search matching model with aggregate shocks to labor productivity in which the no-shirking conditions hold with equality at all aggregate states by assuming that rms unilaterally determine wages by taking into account the no-shirking conditions. One reason why not so many papers have examined this class of models in the past may be this. On the one hand, some previous work has demonstrated that introducing shirking models of e ciency wages (Shapiro and Stiglit 1984) into aggregate business cycle models does not help generating "rigid" wages (Gomme 1999, e.g.). On the other, some authors have argued that introducing mechanisms of making wages "rigid" into a search matching model of Diamond, Mortensen and Pissarides type may help to generate large uctuations of unemployment and vacancies as observed in data (Shimer 2004, 2005; Hall 2005). These two sub-literature seems to imply that introducing shirking models of e ciency wages into DMP models may not help in delivering large uctuations of vacancies and unemployment as observed in data. I argue in this paper that this reasoning may not be correct. Some recent papers (Hagedorn and Manovskii 2008; Pissarides 2007) demonstrate that introducing rigid wages may not be the key to addressing the issue raised by Robert Hall and Robert Shimer. Hagedorn and Manovskii argue that the appropriate value of non-working (leisure) should be much larger, and employees bargaining power should be much smaller than the values used by Hall and Shimer if the observed value of the productivity elasticity of wages 2

3 and the pro t-output ratio were used as target moments in calibrations. The key parameter is the value of non-working. Indeed, Hagedorn and Manovskii also illustrate that there is a value of workers bargaining power such that a model with such bargaining power and a leisure value of 0.95 (as a ratio of labor productivity) delivers large uctuations of both vacancy unemployment ratios and wages. Pissarides (2007) argues that the aggregate elasticity of wages with respect to labor productivity, as used by Hagedorn and Manovskii, is not relevant to search and matching models. He argues that what matters for search matching models is wages in new matches, not aggregate wages including wages in old matches. He argues that the relevant elasticity of wages in new matches with respect to labor productivity should be close to 1, way from 0.5, the value used by Hagedorn and Manovskii as one of target moments in their calibration. Therefore, the past business cycle literature, showing that the shirking mechanism of e ciency wages does not help in delivering rigid wages as observed in data, should not discourage researchers from investigating a variant of aggregate matching models in which a shirking model of e ciency wages is incorporated. I do this in the simplest possible setup in this paper. I nd in this paper that replacing the standard Nash bargaining wage determination by the no-shirking conditions in the standard matching model of unemployment does not reduce much uctuations of wages, being consistent with the previous literature (Gomme 1999, e.g.), on one hand. On the other, it increases uctuations of unemployment and vacancies to the large extent, partly because the observed wage output ratio and the binding no-shirking conditions imply the very large value of non-working, which lies between the value used by Shimer (0) and Hagedorn and Manovskii (0.55). Remarkably, my calibrated value (0.52) of non-working, using the no-shirking conditions as a discipline, is not way di erent from 0.43, the value independently estimated by Hall (2006). The result is important for the following reason. One commonly heard criticism against Hagedorn and Manovskii s calibrations is that their calibrated value of non-working (leisure) is "implausibly too high". Independently estimated values of non-working are helpful to pin down the empirically plausible range of the leisure value. Recent two studies deserve mentioning in this regard. In their in uential and often cited paper, Aguiar and Hurst (2005) nd that food consumption expenditures drop by about 17 percent, 3

4 while the time spent on food production increases by 53 percent. Hall (2006) uses empirical results from the consumption and labor supply literature at the micro level to obtain a value of non-working relative to labor productivity of about I nd that a reasonable value of wage-output ratio of 0.97 and the binding no-shirking conditions imply a value of non-working relative to productivity of about The key is that the binding NSCs o ers tight disciplines in calibrating the parameter of the non-working (leisure) time. Comparison of calibration procedures across three di erent papers (Shimer 2005; Hagedorn and Manovskii 2008; this paper) may help in understanding my point. Start with a standard DMP model with Nash bargaining wage determination. There are two nontrivial key parameters that must be calibrated: the sum of the leisure value and the unemployment bene t level, and employees bargaining power. Shimer implicitly assumes that the leisure value is ero and invokes the empirical replacement ratio of to assume that the unemployment bene t is 0.4. he invokes Hosios (Hosios 1990) condition to posit that employees bargaining power is equal to the elasticity of matches with respect to unemployment, which he estimates to be 0.72 by regressing the measured job nding probability on the measured vacancy unemployment ratio. Hagedorn and Manovskii calibrated these two parameters by explicitly matching the model s implications to data. They choose the value of the two parameters so that the model generates the empirically observed productivity elasticity of aggregate wages (0.499) and the pro t-output ratio (0.03). As argued above, there appears to be controversies about what the relevant empirical moment of productivity elasticity of wages is. It may be desirable not to use productivity elasticity of wages as a target moment in calibration. My matching model with e ciency wages does not contain a parameter of employees bargaining power, since wages are determined by the binding no-shirking condition. Therefore only the data of pro t-output ratio can identify the value of non-working. The paper is organied as follows. The next section builds a search matching model with e ciency wages with no aggregate shocks. Only steady state is considered. The model is calibrated and some numerical elasticities are computed. Section 3 adds aggregate shocks to labor productivity and the 3 Some susequent authors argue that this is too large, since only a part of the unemployed apply for the unemployment bene t. Hall and Milgrom (2008) argues that a reasonable number of the unemployment bene t is

5 model is calibrated and simulated. The statistics computed from the simulated data is compared with data. Section 4 concludes. 2 Model 2.1 Environment Time is discrete: t 2 f1; 2; 3; g. The beginning of period t is called date t 1, and its end date t. Only steady states are considered. Agents and Preferences. This economy is populated by a continuum of risk-neutral workers and rms, both of whom discount future payo s at the common discount factor 2 [0; 1). The preference of each worker is given by E 1X t [c t + (1 t=1 e t )] where c t is consumption, e t 2 f0; 1g the fraction of time devoted to work, which is also called the employee s e ort level, and b > 0 the value of leisure. The preference of each entrepreneur (or rm) is given by: E 1X t [ t I v k] t=1 where t is the pro t, and I v is the indicator function for the event that the rm is vacant. Parameter k > 0 is a ow cost of recruiting. Production Technology. A pair of a worker and rm during employment in tenure t 2 f1; 2; 3; g produces e t y, in which y 2 R ++ is match quality, which is assumed to be the same across employment periods. Each match separates with probability s 2 [0; 1] each period for exogenous reasons. Search Technology. Let v=u be the vacancy-unemployment ratio, where v is a measure of vacant rms and u unemployed job seekers. Each unemployed worker contacts a vacant rm with probability f() per period, where f 0 () > 0. Each vacant rm contacts a job-seeker with probability q() per period, where q 0 () < 0. Functions f() and q() are related in such 5

6 a way that f() = q(). I assume that lim!0 f() = 0, lim!1 f() = 1, lim!0 q() = 1, and lim!1 q() = Equilibrium Asset Equations. Now, I write down a bunch of asset equations. The value of unemployment U satis es the following equations: U = + b + f()w + [1 f()]u (1) where b is the amount of unemployment bene ts per period, W the value of employment for each employee. The value of employment for each employee, exerting high e ort, W must satisfy the following asset equation: W = w + (1 s)w + su (2) where w is the ow wage level. The value of each lled job J must satisfy the following equation: J = y w + (1 s)j (3) No-shirking Conditions. It is assumed for simplicity that monitoring employees e ort is perfect in the sense that the employee s e ort level is observed with no noise. However, it is assumed that e ort is unveri able. As a consequence, any agreements contingent on e ort levels are not enforceable. When the worker chooses high e ort (e=1), the employment continues unless the match receives exogenous separation shocks. When the worker chooses low e ort (e=0), the employment is terminated. The employee chooses high e ort (e=1) if w + (1 s)w + su w + + U, which is rewritten as: (1 s)(w U) (4) We call this inequality the no-shirking condition, or NSC in what follows. Wage Determination. When the worker and rm choose to match, the rm chooses the wage level, so that the no-shirking condition is satis ed. If the NSC is binding, the solution will be W U =, J = G, (1 s) (1 s) 4 One example satisfying these assumptions is q() = [1 exp( 1 )]=. Another example is q() = min[1; ], where 2 (0; 1). 6

7 where the match surplus G is de ned by G W +J (2) and (3), one can show that U. By using equations G = y (1 )U 1 (1 s) Free Entry of Vacancies. Each vacant rm incurs recruiting cost k > 0 per period in nding a worker. Entry into the labor market is free. This implies the following equation: (5) k q() = J = G (1 s) (6) Now, we can de ne a steady state equilibrium of a search model with workers shirking problems. De nition 1. A steady state equilibrium of a search matching model with workers shirking problems is de ned as a list fu; W; J; w; ; ug such that: (i) U, W, and J satisfy equations (1), (2) and (3); (ii) satis es equation (6); (iii) w satis es the binding NSC; and (iv) u satis es the steady state accounting: (1 u)s = uf(). The de nition given above is very standard except the part (iii), where wages must satisfy the NSC. A steady state equilibrium of a search model with workers shirking problems de ned above is characteried by a pair (; U) satisfying the following two equations: U = + f() + U (7) (1 s) k = G (8) q() (1 s) where the match surplus G is de ned above. Although my interest in this paper is aggregate uctuations of labor market aggregates under aggregate shocks, it is helpful to consider, rst, the steady state of the model. 7

8 2.3 Calibration in steady state My strategy for quantitative exercises below is to calibrate parameters in steady state and then quantify model generated moments using those parameter values. One remarkable thing is that one of important parameters,, can be calibrated with some disciplines: Given the observed value of, and the vacancy cost k, the binding no-shirking conditions o er the value of non-working time, 5. In steady state without aggregate shocks, we have: and These two imply that: k q() = G (1 s) G = y b + (1 s)g f() (1 s) k q() + (1 s) = y b f() 1 (1 s) Let f() = 1 and q() =. I choose the model period to be two weeks. I set = exp[log(1=1:05)=26] = 0:9981, so that it is consistent with the annual interest rate Shimer (2005) estimates that the monthly job nding rate from 1951 to 2003 on average is The corresponding biweekly job- nding probability is f = 0:2584. The average unemployment rate from 1951 to 2003 is This, along with the steady state accounting, implies that the bi-weekly job separation rate is s = uf=(1 u) = 0:0567 0:2584=(1 0:0567) = 0:0155. I normalie that y = 1. Hall (2005) found that the average value of the vacancy unemployment ratio during December 2000 to December 2002 was according to JOLTS. But this number may be too small since the economy was in downturns during that period. Hence I use = 0:6. This implies that q = f= = 0:2584=0:6 = 0:4307. The replacement ratio in the U.S. is roughly 0.4. Not all the unemployed quali ed for receiving the unemployment bene t are not receiving the bene t. So I choose b = 0:3, instead of 0.4, as has been common in the early literature. The values of two parameters k and remain to be chosen. The binding no-shirking condition gives as a function of k, which I denote as k(), with 5 Hall and Shimer assumes that = 0. Hagedorn and Manovskii calibrate + b = 0:96. 1 s (9) 8

9 some abuse of notation. Note that the ow wage w is given by: w b 1 [1 s f()] = (1 s) (10) Empirically, the value of w is very close to y, which is set to 0.97 for the following reason. Interpret the matching model above as the one with capital, following Pissarides (2000). Then, labor productivity y in my matching model can be interpreted as output minus capital income. The ratio of dividends to the GDP for the U.S. economy from 1946 to 2000 is around I assume that y w in my matching model roughly corresponds to the dividends payouts. This means that (y w)=(y + rk) = 0:02, where y = 1 by normaliation. Since capital income rk is about 0.3 of output y+rk = 1+rk, rk should be 3=7, and output 10=7. Hence, we have (1 w)=(1 + 3=7) = 0:02, implying that w ' 0:97. 6 I choose the value of so that the model generated w through equation (10) is The value of = 0:5234 does this job. Remarkably, this value is close to the value of non-working time calculated by Hall (2006). It is also slightly below the value calibrated by Hagedorn and Manovskii. Once the value of is obtained, one can compute the value of k satisfying equation (9). The value of k = 0:7424 does this job. This value is greater than the value of vacancy costs calibrated by Hagedorn and Manovskii (2008). The values of matching function parameters remain to be chosen. Shimer (2005) estimates = 0:72. Mortensen and Nagypal (2007) argue that should be smaller than Many authors use a value of 0.5. I choose = 0:5. I choose the value of so that the implied f = 0:5 is equal to = 0:2584=0:6 0:5 = 0: This number is close to the implied wage productivity ratio in standard DMP models. For instance, in Shimer s (2005) calibration, the implied w=y is

10 TABLE 1: Calibrated Parameters parameter interpretation value calibration strategy discount factor biweekly frequency y labor productivity 1 normaliation s match separation prob u = 0:0567 and f = 0:2584 matching function parameter 0.5 Petrongolo and Pissarides matching function parameter f = 0:2584 and = 0:6 b unemployment bene t 0.3 a replacement ratio of 0.4 the value of non-working (leisure) w = 0:97 k vacancy cost = 0:6 Now I do some numerical comparative statics in steady state before quantifying the model with aggregate shocks below. For y = 1, the model delivers = 0:6, and w = 0:97. For y = 1:01, the model delivers = 0:673, and w = 0:978. For y = 0:99, the model delivers = 0:53, and w = 0:9617. The elasticity of with respect to y is roughly when y changes from 1 to 1.01, and when y changes from 1 to The value of these implied elasticities are larger than 7.56, which is the regression coe cient of data of the deviations of log from its HP ltered trend on data of deviations of log productivity from its HP trend. The elasticity of wages with respect to productivity is when y changes from 1 to 1.01, and when y changes from 1 to This number is way greater than 0.449, which Hagedorn and Manovskii argue to be the empirical elasticity of wages with respect to productivity. However, Pissarides (2007) recently argues that the relevant empirical elasticity of wages with respect to productivity should be close to 1. TABLE 2: Comparative Statics in Steady State y w

11 TABLE 3: Data versus Implied Productivity Elasticities ;y w;y Model with Shirking 11.9 a 0.84 a Data 7.56 (U.S., ) a (BLS, ) Model (Shimer) Model (H and M) Note: xy denotes the elasticity of variable x with respect to variable y, where all variables are reported in logs as deviations from an HP trend with a smoothing parameter of either 1600 or The superscript a indicates a smoothing parameter of Data for xy is the regression coe cient of variable x on variable y. The value of ;y for Model (Shimer) is what is implied by his TABLE 3 ( ;y = (;y)() = 0:9990:035 = 1:75). (y) 0:02 My calibrated search model with e ciency wages delivers a su ciently large sie of productivity elasticity of the v-u ratio. It delivers a productivity elasticity of wages of 0.84, which is way larger than aggregate data (0.449), which Hagedorn and Manovskii (2008) estimate. If the goal of introducing shirking problems of e ciency wages into a search matching model were to deliver sticky wages whose elasticity with respect to productivity is 0.499, as in some previous papers such as Gomme(1999), then such attempt would be a failure, although it reduces the elasticity slightly compared with the benchmark calibration of Shimer(2005). However, Pissarides (2007) has recently argued that what is relevant for assessing a search matching model is behavior of wages in only new matches. He argues that a plausible empirical elasticity of wages in new matches with respect to productivity is close to one. In this respect, the productivity elasticity of wages generated by my calibrated model is not that di erent from what Pissarides (2007) considers empirically plausible. Comparison to Shimer and Hagedorn and Manovskii: The key to understanding my calibration is that the binding NSCs o ers tight disciplines in calibrating the parameter of the non-working (leisure) time. Comparison of calibration procedures across three di erent papers (Shimer 2005; Hagedorn and Manovskii 2008; this paper) may help in understanding my point. Start with a standard DMP model with Nash bargaining wage determination. There are two nontrivial key parameters that must be 11

12 calibrated: the sum of the leisure value and the unemployment bene t level, and employees bargaining power. Shimer implicitly assumes that the leisure value is ero and invokes the empirical replacement ratio of 0.4 to assume that the unemployment bene t is 0.4. He invokes Hosios (Hosios 1990) condition to posit that employees bargaining power is equal to the elasticity of matches with respect to unemployment, which he estimates to be 0.72 by regressing the measured job nding probability on the measured vacancy unemployment ratio. Hagedorn and Manovskii calibrated these two parameters by explicitly matching the model s implications to data. They choose the value of the two parameters so that the model generates the empirically observed productivity elasticity of aggregate wages (0.499) and the pro t-output ratio (0.03). As argued above, there appears to be controversies about what the relevant empirical moment of productivity elasticity of wages is. It may be desirable not to use productivity elasticity of wages as a target moment in calibration. My matching model with e ciency wages does not contain a parameter of employees bargaining power, since wages are determined by the binding no-shirking condition. Therefore only the data of pro t-output ratio can identify the value of non-working. 3 Aggregate Shocks to Productivity This section adds aggregate shocks to productivity to the model developed above. Let there be N possible aggregate states p 2 f1; 2; :::; Ng. Labor productivity in state p is denoted by y p, where y 1 < y 2 < ::: < y N. I assume that aggregate state p follows a N-state Markov chain with transition matrix p, whose ith row and jth column element ij indicates the probability that next period s state is j when the current state is i. 3.1 Equilibrium The value of being unemployed in aggregate state p, which is denoted by U p, satis es: U p = + b + f( p )E p (W p 0 U p 0) + E p U p 0 (11) where is the ow value of leisure, b is the unemployment bene t, and E p is the expectation operator conditional on that the current state is p. Note 12

13 that the value of employment for a worker and the vacancy unemployment ratio also depend on aggregate state p. p 0 in the equation indicates the next period s aggregate state. The value of employment for a worker in state p, which I denote as W p, satis es: W p = w p se p (W p 0 U p 0) + E p W p 0 (12) where it is assumed that the amount of the leisure time for each non-shirking employee is ero, w p is a ow wage in state p. The value of a lled job in state p, which is denoted by J p, satis es: J p = y p w p + (1 s)e p J p 0 (13) Free entry of vacant jobs in state p implies that k = q( p )E p J p 0 (14) where k > 0 is a ow cost of posting a vacant job. No-shirking conditions in state p is: (1 s)e p (W p 0 U p 0) (15) since the employee prefers not to shirk in state p if: w p + (1 s)e p W p 0 + se p U p 0 w p + + E p U p 0 Let the match surplus in state p be de ned by G p W p + J p U p. Assume that wages are unilaterally determined by the employer upon meeting 7. Then, the no-shirking condition is binding, as is the case of Shapiro and Stiglit. Then, one obtains that: G p = y p b + (1 s)e p G p 0 f( p ) (1 s) (16) This equation can be solved numerically for G p. Using G p, the free entry equation in state p is rewritten as: 7 This assumption is not empirically implausible in light of recent work of Hagedorn and Manovskii (2008). They calibrate the parameter of employees bargaining power to be close to

14 k q( p ) = E pg p 0 (1 s) (17) 3.2 Quantitative Exercises The stochastic process for labor productivity is described by the AR(1) process: log y t+1 = log y t + t+1 (18) where 2 (0; 1) and t+1 is a draw from N(0; 2 ). I follow Hagedorn and Manovskii (2008) in choosing = 0:9809 and = 0:0046. The mean of y is normalied to 1. I approximate the AR(1) process by a discrete 7-state Markov chain, following Tauchen (1986). The grid of log productivity is log y 2 f 0:0709; 0:0473; 0:0236; 0; 0:0236; 0:0473; 0:0709g. This means that y 2 f0:9316; 0:9538; 0:9767; 1; 1:0239; 1:0484; 1:0735g. The transition matrix is given by: 2 3 0:9886 0: :0028 0:9884 0: = :0038 0:9895 0: :0051 0:9898 0: :0067 0:9895 0: :0088 0:9884 0: :0114 0:9886 The standard deviation of the simulated data of log y, using the transition matrix above, is The standard deviation of log productivity, relative to its HP ltered trend, for the U.S. quarterly data is 0.02 when the smoothing parameter is 10 5 (Shimer 2005, ), and when the smoothing parameter is 1600 (Hagedorn and Manovskii 2008, 1951Q1-2004Q4). Note that the standard deviation in biweekly frequency should be larger than in quarterly frequency. The surplus vector G p, the vacancy unemployment ratio vector p, and the wage vector p for p 2 f1; 2; :::; 7g are computed as:

15 = (0:22; 0:3216; 0:4483; 0:6; 0:778; 0:9836; 1:2136) 0 G = (1:5761; 1:7957; 2:0248; 2:2596; 2:5; 2:7455; 2:9917) 0 w = (0:9158; 0:9332; 0:9514; 0:97; 0:9891; 1:0085; 1:028) 0 The simulations are undertaken as follows. Using the transition matrix above, I simulate the Markov chain with 7-state. In order to generate 212 "quarterly" data points, corresponding to quarterly data from 1951 through 2003, I need to generate 1272 "bi-weekly" data points, since my model works at biweekly frequency. To do so, for each simulation, I generate 2000 data points, and throw away the rst 728 data points to eliminate the dependence on the initial value chosen. Then, I aggregate the generated "biweekly-frequency" data to quarterly frequency, resulting in 212 "quarterly" generated data points. Then I detrend the log of the model-generated "quarterly" data using an HP lter with the smoothing parameter Then I compute the standard deviation of such model-generated data. I repeat this 10,000 times. The result is summaried by Table 1. TABLE 4: Data versus Model-generated Moments ()=(y) (u)=(y) (v)=(y) Proj(wjy) smoothing parameter Data (Shimer) 19.1(7.56) 9.5 (-3.9) 10.1 (3.67) 10 5 Data (H and M) 19.9 (7.83) 9.6 (-2.9) 10.7 (4.9) Model (this paper) Model (Shimer) Model (H and M) Note: (x) denotes the standard deviation of variable x 2 f; y; u; vg, where all variables are reported in logs as deviations from an HP trend with smoothing parameter mentioned in the Table. For data of ()=(y), I also report the regression coe cient of variable on variable y, which is Proj(wjy) is the regression coe cient of variable w on variable y. My calibrated search model with e ciency wages delivers a su ciently large uctuations of v-u ratios, the unemployment rate and vacancies, compared with data. TABLE 4 shows that the sie of uctuations of variable 8 This number of the smoothing parameter is standard in the RBC literature when detrending quarterly data. 15

16 , u and v, measured by the standard deviation, is greater than the empirical regression coe cient of each variable on productivity (the numbers in parentheses). They are also not way below the empirical standard deviations. Turning to the wage uctuations, my calibrated model with e ciency wages delivers a standard deviation of wages in relative to productivity of 0.83, which is way larger than data (0.449), which Hagedorn and Manovskii (2008) estimate. If the goal of introducing shirking problems of e ciency wages into a search matching model were to deliver sticky wages whose elasticity with respect to productivity is 0.499, as in some previous papers such as Gomme(1999), then such attempt would be a failure, since it reduces the wage uctuations only slightly compared with the benchmark calibration of Shimer(2005). However, as Pissarides (2007) has recently raised the issue, what is relevant for assessing a search matching model may be behavior of wages in only new matches. He argues that a plausible empirical elasticity of wages in new matches with respect to productivity is close to one. In this respect, the productivity elasticity of wages generated by my calibrated model is not that di erent from what Pissarides (2007) considers empirically plausible. 4 Discussion I have assumed that the NSC holds with equality, as in Shapiro and Stiglit(1984) and Gomme(1999). However, this is not the only possible way to determine wages in the presence of shirking problems. One alternative way is to assume that wages are determined by solving: w = arg max w 0 (W U) J 1, s.t. W U (1 s) In this case, the NSC may not always be binding. Rocheteau (2001) and Costain and Jansen (2006) examine the search matching model with this type of wage determination. In this section, the relationship between the likelihood that no-shirking conditions bind and the unemployment rate is examined. Addressing this question serves the purpose of better understanding 16

17 the relation between work incentive problems and labor market conditions. The equilibrium under this type of wage determination is given by: k q() U = + b + f() max[g; (1 s) ] = min[(1 )G; G (1 s) ] Examining the quantitative properties of this class of models may be important as well. 5 Conclusion The message of this paper is stated as follows. Introducing the shirking model of e ciency wages in the simplest possible setup into the standard search and matching model of Mortensen and Pissarides type helps greatly in delivering large uctuations of vacancies and unemployment, comparable to the empirical magnitude. This is not because introducing e ciency wages generate rigid wages, but because it provides a tight discipline in calibrating the value of non-working. Indeed, as being consistent with previous work such as Gomme (1999), my calibrated search matching model with e ciency wages deliver wage uctuations which are slightly smaller than the standard search and matching model. However, the model with the binding non-shirking conditions implies a value of non-working of 0.52, which is somewhat not very di erent from the Hall s (2006) independent estimate. The past business cycle literature, showing that the shirking mechanism of e ciency wages does not help in delivering rigid wages as observed in data, should not discourage researchers from investigating a variant of aggregate matching models in which a shirking model of e ciency wages is incorporated. This is what I did in the simplest possible setup in this paper. 17

18 6 References AGUIAR, Mark and HURST, Erik. "Consumption versus Expenditure." Journal of Political Economy 2005, 113 (5), pp COSTAIN, James and JANSEN, Marcel. "Employment Fluctuations with Downward Wage Rigidity." February 2006, mimeo, Bank of Spain and Univ. Carlos III de Madrid. DIAMOND, Peter. Wage Determination and E ciency in Search Equilibrium. Review of Economic Studies, 1982, 49, pp GOMME, Paul. "Shirking, Unemployment and Aggregate Fluctuations." International Economic Review, 1999, 40 (1), pp HAGEDORN, Marcus and MANOVSKII, Iourii. "The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited." American Economic Review, forthcoming. HALL, Robert E. "Employment Fluctuations with Equilibrium Wage Stickiness." American Economic Review, 2005, 95 (1), pp "Work-Consumption Preferences, Cyclical Driving Forces, and Unemployment Volatility." August 22, and MILGROM, Paul. "The Limited In uence of Unemployment on the Wage Bargain." American Economic Review, forthcoming. HOSIOS, A.J. "On the E ciency of Matching and Related Models of Search and Unemployment." Review of Economic Studies 1990, 57, pp MORTENSEN, Dale and NAGYPAL, Eva. "More on Unemployment and Vacancy Fluctuations." Review of Economic Dynamics 2007, 10, pp and PISSARIDES, Christopher. Job Creation and Job Destruction in the Theory of Unemployment. Review of Economic Studies, 1994, 61, pp and. "New Developments in Models of Search in the Labor Market.", November 16, 1998 PETRONGOLO, Barbara and PISSARIDES, Christopher. "Looking into the Black Box: A Survey of the Matching Function." 2001, Journal of Economic Literature 39 (2), pp PISSARIDES, Christopher. "The Unemployment Volatility Pule: Is Wage Stickiness the Answer?", November Equilibrium Unemployment Theory, 2000, second ed. The MIT Press.. "Short-Run Equilibrium Dynamics of Unemployment, Vacancies and Real Wages." American Economic Review, 1985, 75(4), pp

19 ROCHETEAU, Guillaume. "Equilibrium unemployment and wage formation with matching frictions and worker moral haard." Labour Economics, 2001, pp SHAPIRO, C. and STIGLITZ, J.E. "Equilibrium Unemployment as a Worker Discipline Device." American Economic Review, 1984, 74, pp SHIMER, Robert. "The Cyclical Behavior of Equilibrium Unemployment and Vacancies." American Economic Review, 2005, 95(1), pp "The Consequences of Rigid Wages in Search Models." Journal of the European Economic Association, 2004, 2(2-3): TAUCHEN, G. "Finite state Markov-chain approximation to univariate and vector autoregressions." Economics Letters 1986, 20, pp