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Journal of Monetary Economics 51 (2004) 277–298

Unemployment and the business cycle Michelle Alexopoulos* Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ont. M5S 3G7, Canada Received 1 March 2001; received in revised form 30 August 2002; accepted 29 November 2002

Abstract This paper presents a dynamic general equilibrium model where labor effort is imperfectly observable and there is unemployment in equilibrium. In contrast to shirking models in the efficiency wage literature, detected shirkers are not dismissed. Instead, they face a monetary punishment because they forgo an increase in their compensation. Estimated versions of the model can generate the high variation in employment and low variation of real wages observed over the business cycle, and are consistent with existing qualitative evidence about the responses of the economy to fiscal policy shocks. r 2003 Published by Elsevier B.V. JEL classification: E24; E32; J41 Keywords: Business cycles; Unemployment; Efficiency wages

1. Introduction A classic challenge facing macroeconomists is to explain observed variations in employment and real wages over the business cycle. Early business cycle models, of the sort developed in Kydland and Prescott (1982), have been criticized on at least two dimensions. First, to account for the basic time series properties of aggregate U.S. employment and real wages they must assume a labor supply elasticity that is highly relative to microeconomic estimates.1 Second, all variation in employment

*Tel.: +1-416-978-4962. E-mail address: [email protected] (M. Alexopoulos). 1 According to studies such as MaCurdy (1981) and Pencavel (1986), the labor supply elasticity for males is near zero in the United States. 0304-3932/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/j.jmoneco.2002.11.002

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occurs at the intensive margin.2 These shortcomings have lead researchers to consider alternative models with labor market frictions. This paper presents a dynamic general equilibrium model in which labor effort is imperfectly observable and detected shirkers face a monetary punishment. The model, which I refer to as the shirking model, generates equilibrium unemployment and all employment fluctuations occur at the extensive margin. Consequently, it does not rely on the assumption that agents have a high labor supply elasticity to generate large fluctuations in employment along with small changes in real wages. To evaluate the model’s performance, I estimate the model and contrast its properties with the standard real business cycle models. In addition, I formally evaluate the model’s ability to account for various unconditional second moments of the data. The paper’s main results can be summarized as follows. First, the shirking model is better able to account for the observed post-war US fluctuations in employment and real wage than the standard divisible labor business cycle model. Second, the shirking model is consistent with existing evidence in Ramey and Shapiro (1998) and Edelberg et al. (1999) about the qualitative responses of the U.S. economy to a fiscal policy shock. Third, in response to a positive productivity shock the shirking model produces a large increase in employment alongside a small increase in real wages. Fourth, altering the insurance possibilities available to agents has a significant impact on consumption dynamics and the quantitative responses of employment and wages to shocks. With full income insurance the shirking model is observationally equivalent to the standard indivisible labor model developed by Hansen (1985) and Rogerson (1988). However, limiting the amount of income insurance available to individuals in the shirking model increases the variation of employment and decreases the variation of real wages, compared to the full income insurance case. As in shirking models associated with Shapiro and Stiglitz (1984) I assume that firms imperfectly observe workers’ effort levels. However, my model differs from others in the literature in two ways. First, instead of firing detected shirkers, I assume that firms punish them by withholding increases in their compensation. Second, I allow for different risk sharing arrangements between agents. In the partial insurance case, unemployed workers are ex-post worse of than employed workers. In the full income case, unemployed workers are ex-post better off than employed workers. This last result is not surprising given the observational equivalence between the full insurance version of the shirking model and the indivisible labor model. The finding that my shirking model performs well on empirical grounds contrasts with existing results in the efficiency wage literature.3 The reason for this difference is due to my assumption about how firms deal with detected shirkers. To obtain the desired amount of effort from workers in a shirking model, firms must offer them a wage that satisfies the following incentive compatibility constraint: the worker’s utility associated with providing this effort level is at least as great as the expected utility associated with shirking. The punishment associated with being a detected 2 Lilien and Hall (1986) report that the vast majority of changes in employment hours is accounted for by changes in the number of people employed. 3 See, e.g., Gomme (1999), Danthine and Donaldson (1995), Kimball (1994) and Strand (1992).

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shirker affects the properties of the incentive compatibility constraint. Therefore, it plays a crucial role in determining the properties of the model. Past shirking models have adopted Shapiro and Stiglitz’s (1984) assumption that firms punish detected shirkers by firing them. In this environment, the unemployment rate influences the wage rate firms must pay because it affects the severity of the punishment associated with shirking. As the unemployment rate falls so too does the expected duration of unemployment for a detected shirker. This means that the punishment associated with shirking falls. Other things equal, firms must respond by raising workers’ wages to prevent shirking. In the limit, as the unemployment rate goes to zero, there is no wage rate that supports positive effort. On net, these forces generate highly procyclical real wages. In my shirking model, if a worker is detected shirking, he receives only a fraction of the wages given to non-disciplined workers.4 This punishment does not fall disproportionately as the unemployment rate falls. Indeed there is a punishment associated with shirking even when there is no unemployment. These features help explain why my model does not embed the same forces that lead to highly procyclical wages in the standard Shapiro–Stiglitz model. Section 2 presents my model. Section 3 presents my empirical results. In this section I discuss the estimated models’ second moment properties and the way the model responds to technology and fiscal policy shocks. In addition, I compare my model with the standard divisible and indivisible labor real business cycle models. Finally, Section 4 contains concluding remarks.

2. The model The model economy is populated by three sets of agents: a government, a large number of identical families, each of which contains a continuum of workers/ consumers and a large number of perfectly competitive firms. Below I describe each agent’s problem. 2.1. The Government At time t; the government purchases Gt units of the economy’s final consumption good. These purchases are financed by identical lump-sum taxes, Taxt ; levied on each of the identical households/families in the economy. This implies that the government’s time t budget constraint is: Gt pTaxt :

4 There is empirical evidence to suggest that this punishment for shirkers is more plausible than Shapiro and Stiglitz’s. For example, evidence in papers such as Agell and Lundborg (1995), Hall (1993), Weiss (1990) and Malcomson (1998) indicate that: (1) the majority of firms report that they initially reprimand detected shirkers instead of firing them and (2) disciplined workers are not as likely to receive bonuses, raises or promotions.

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2.2. Families and individuals In order to maintain a representative agent framework in the economy with positive rates of unemployment, I make two assumptions. First, families own all the capital goods, Kt ; in the economy. They use the proceeds from their capital investments, rt Kt ; in period t to provide a stream of consumption for their members, ctf ; pay taxes, Taxt ; and purchase new investment goods, It : Second, individuals cannot transfer assets between periods.5 Families distribute ctf to each member before they know who will be hired by firms during the period. Each family also organizes a fully funded unemployment insurance scheme for its members where working family members transfer Ft to a fund for the unemployed.6 The nature of the unemployment insurance defines the risk sharing arrangement between agents in the model. I examine two different insurance schemes: partial income insurance and full income insurance. In the partial income insurance case, the amount of income transferred to an unemployed person is determined by the condition that unemployed family members are as well off as any family member caught shirking. In the full income insurance case, the income of an unemployed person is the same as the income of an employed member who is not detected shirking. This is the insurance scheme the family would choose to maximize its expected utility. Unemployed workers in the partial income insurance scheme will be involuntarily unemployed in the sense that an unemployed individual’s utility is lower than an employed person’s. Unemployed workers in the full income insurance case are voluntarily unemployed because their utility is higher that their employed counterparts since they have the same consumption, but do not incur the disutility associated with working. Individuals will always accept employment in the world with partial income insurance. However, they would not accept offers in the full income insurance case unless families can: (i) observe which members receive job offers and (ii) punish individuals who turn down job offers. As a result, I assume job offers can be observed and individuals rejecting offers become ineligible for any intra-family transfers. 2.2.1. Individuals Although individuals cannot directly engage in investment, they may increase their period’s consumption over the amount provided by their family, ctf ; by obtaining employment from firms. Firms hire workers by offering them a one period contract that specifies the fixed number of hours an employee must work, h; the required effort level, et ; a real hourly wage rate of swt to non-detected shirkers and wt to all 5 The main results of this paper are preserved if instead of using this family framework it is assumed that there are both entrepreneurs and workers in the economy, similar to Gomme (1999). However, to make the model consistent with a balanced growth path, workers must be endowed with a portion of the firm’s shares in the economy. See Alexopoulos (2001). 6 This intra-family transfer helps ensure that the differences between the employed and unemployed workers’ consumption are empirically plausible. The results of the analysis would not change if the government instead of the family ran the unemployment insurance.

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other workers. Firms agree to pay s-percent of the worker’s wage bill up front, but they retain the remaining ð1  sÞ percent until the end of the contract. Shirkers are detected by an exogenous monitoring technology with probability d: I assume that detected shirkers incur a monetary punishment because they forgo the payment of ð1  sÞwt h; while all other workers are given this amount at the end of the period. One interpretation of this setup is a compensation package that is comprised of a guaranteed base salary, swt h; and an extra payment of ð1  sÞwt h for non-disciplined workers through a bonus or promotion. For simplicity, I assume that firms will never punish individuals who do not shirk.7 The family classifies its members as either as employed or unemployed. Employed workers consume the amount the family gives them, ctf ; as well as their remaining wage income after the intra-family transfer, Ft : The exact amount of their consumption will depend on whether they are disciplined by firms during the period. Employed individuals who are not detected shirking have period t consumption ct ; while detected shirkers have a consumption level of cst : Unemployed members consume cut ; which is determined by the amount of family purchased consumption, ctf ; and their share of the unemployment fund. When all firms are identical, ctf ; ct ; cst ; and cut are determined according to the family’s budget constraint and following rules: ctf prt Kt  Taxt  It ;

ð1Þ

ct ¼ ctf þ wt h  Ft ;

ð2Þ

cst ¼ ctf þ swt h  Ft ;

ð3Þ

cut

¼

8 > < ctf þ

Nt Ft if the individual had no job offer; 1  Nt

> : c f if the family observes the individual rejecting a job offer: t

ð4Þ

Here Nt is the total number of family members employed in period t: The time t utility of a worker is lnðcit Þ þ y lnðT  Wð#et > 0Þðh#et þ xÞÞ; where yX0; e#t is the amount of effort supplied to the firm by the worker, WðÞ is an indicator function that takes on the value 1 if the worker is employed and providing effort and zero otherwise, and x is the fixed cost associated with providing any 7 Alternatively, one could assume that firms are monopolistic competitors that make positive profits and can get a reputation for cheating workers. It follows that if firms get a reputation for cheating their workers, non-shirking workers would not accept employment from these firms in the future and the firms would therefore loose their future profits. Simulations of a model similar to the one presented in this paper suggest that the results do not dramatically change when this more complex environment is used.

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positive level of effort. The variable cit takes on the following values: 8 u > < ct if unemployed; i ct ¼ cst if detected shirking; > : ct otherwise: The fixed cost associated with providing positive effort is unobservable and can be interpreted as the amount of time that an individual must initially spend organizing or planning his days in order to be productive. 2.2.2. The family’s problem Since all government expenditures are financed by lump-sum taxes levied on the families, Taxt is replaced by Gt in the families’ budget constraint. Letting Nts denote the actual number of shirkers hired in period t; and utilizing information about the firms’ detection technology, the family’s problem can be expressed as8 ( ( )) N s u s s X t ðNt  dNt Þlnðct Þ þ ð1  Nt Þlnðct Þ þ dNt lnðct Þ max E0 b þðNt  Nts Þy lnðT  h#et  xÞ þ ð1  Nt þ Nts Þy lnðTÞ fct f ;Ktþ1 gN t¼0 t¼0 subject to Eqs. (2)–(4) and ctf p½rt Kt  Gt  ½Ktþ1  ð1  dÞKt

; ( ð1  Nt Þswt h under partial income insurance; Ft ¼ ð1  Nt Þwt h under full income insurance:

ð5Þ

2.2.3. The worker’s problem After the family has made its investment decision and committed ctf to each of its members, individuals depart to the labor market. If an individual is offered a contract by a firm and chooses to accept employment, he faces the following problem: ( ) Wð#et Xet Þflnðct Þ þ y lnðT  h#et  xÞg max : e#t þWð0p#et oet Þfð1  dÞlnðct Þ þ d lnðcst Þ þ y lnðT  h#et  Wð#et > 0ÞxÞg Here WðÞ is an indicator function and et is the amount of effort specified in the contract. The worker maximizes his utility by choosing e#t ¼ et if he chooses to exert any effort, and e#t ¼ 0 otherwise. A worker will only choose not to shirk if the utility associated with providing effort et is at least as great as the expected utility associated with shirking, i.e., if his incentive compatibility constraint is satisfied. 8 Here it is assumed that families do not believe that their choices can affect the employment probability of their members. This assumption is made for simplicity. Alexopoulos (2001) provides assumptions that rationalize this assumption and leads to precisely the same allocations as in this model.

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2.3. The firms At time t; the final good, Yt ; is produced by a large number of perfectly competitive firms each of whom use the technology Yt ¼ At Kta ððNt  Nts Þh#et Þ1a : Here, 0oao1; At is the level of technology, e#t is the level of unobservable effort provided each non-shirking worker, and Nt ; Nts ; and Kt denote the number of workers hired, the number of shirkers hired, and the amount of the stock of capital rented by the firm in period t respectively. The level of technology follows the following process: ln At ¼ ð1  rA ÞlnA* þ rA ln At1 þ eA;t ; where jrA jo1 and eA;t is white noise. Firms hire workers using a one period contract where swt h is the compensation guaranteed to all workers and ð1  sÞwt h is the amount that is promised to workers not detected shirking. I assume that the parameter s is determined exogenously while firms choose the value of wt :9 Given the terms of the contract, and the fact that shirkers are detected with probability do1; it will not be profitable for firms to hire shirking workers. Therefore in equilibrium, profit maximizing firms will offer a wage that ensures workers will not shirk on the job, i.e., e#t ¼ et : This leads to the following expression of the representative firm’s period t problem:10 max

fwt ;Nt ;Kt ;et g

fðAt Kta ðhet Nt Þ1a Þ  wt hNt  rt Kt g

subject to the period by period individual rationality (IR) constraints and incentive compatibility (IC) constraints: uðct ; et ÞXuðcut ; 0Þ ðIRÞ; uðct ; et ÞXduðcst ; 0Þ þ ð1  dÞuðct ; 0Þ ðICÞ: Here uð; Þ is the representative worker’s utility for the period and d is the probability a shirking worker is detected. These constraints ensure that workers will voluntarily work for the firm and be indifferent between shirking and providing the effort required in the period’s contract. In equilibrium, the IC constraint holds with equality and defines the link between effort and wages in this environment since the IR constraint is not binding for the intra-family transfers considered.11 Therefore, 9 This case, where s is chosen exogenously, will deliver the same results as a model where there is a restriction on the minimum value of s; e.g., a legal restriction or an industry norm, and s is chosen endogenously by firms. If firms can choose s then they will set it to the lowest level possible. Therefore, the parameter s in the model can be considered this minimum value. In addition, it should be noted that it is possible to generate unemployment in the model even when s ¼ 0: 10 Although this model can produce equilibria without unemployment for certain parameter values, it is assumed that the firm is not constrained by the number of workers it can hire since the estimated parameter values for the U.S. economy reported in the next section will generate unemployment. 11 The IR constraint does not bind in the partial income insurance case since unemployment is involuntary. It does not bind in the full income insurance case since workers who refuse jobs would be worse off because they become ineligible for intra-family transfers.

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effort can be represented as a function of the real wage, the consumption provided by the family and the intra-family transfer using the distribution rule, fct ; cst g:  d=y ! T ct x 1 s ð6Þ et ¼ eðwt Þ ¼  : h h ct The first-order necessary conditions of the firm can be rewritten as follows under the assumption that firms take the amount of the family transfer parametrically when making wage decision,12 e0 ðwt Þwt ¼ 1; et Yt ð1  aÞ  hwt ¼ 0; Nt Yt a  rt ¼ 0: Kt The first equation is the classic Solow condition that implies the firm chooses the real wage to minimize the cost per unit effort. Substituting !  T d ct d=y1 cst  sct ct  cst 0 e ðwt Þ ¼ ¼ h and w t h y cst ð1  sÞh ðcst Þ2 into the Solow condition implies the following condition for ct =cst :13 ct wt h  Ft þ ctf ¼ ¼ const: ð7Þ cst swt h  Ft þ ctf Substituting Eq. (7) into Eq. (6) implies T x et ¼ e ¼ ð1  ðconstÞd=y Þ  : h h Finally, Eq. (5) and Eq. (7) imply 8 ðconst  1Þ f > > < 1  s þ sN ð1  constÞÞct in the partial income insurance case; t wt h ¼ > ðconst  1Þ > : c f in the full income insurance case: ð1  sÞconst þ Nt ð1  constÞÞ t

3. Empirical results To analyze the empirical properties of the shirking models, I first add growth to the model by assuming the presence of an exogenous labor augmenting technology, 12

A firm does not believe it could influence the value of Ft since an individual’s transfer level is based on the average compensation of his family’s workers. 13 The fact that the firms choose a wage to ensure that the ratio of consumption levels for a non-shirker and detected shirker is constant in equilibrium is not unique to the logarithmic form of the utility function. For example, a Cobb–Douglas utility function, Uðc; eÞ ¼ ðca ÞðT  eh  xÞ1a ; or any utility function of the form Uðc; eÞ ¼ lnðcÞ þ VðT  eh  xÞ has this property.

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Gt ; in the production function Yt ¼ At Kta ðet hNt Gt Þ1a ; where Gt ¼ gt and gX1: Balanced growth requires that government expenditures, Gt ; evolve according to Gt ¼ gt expðgt Þ;

ð8Þ

where gt ¼ Zg ð1  rg Þ þ rg gt1 þ egt ; and jrg jo1; Zg is a scalar, and egt is a serially uncorrelated process with mean zero and standard deviation sg : The remainder of this section is organized as follows. First, I discuss how I obtain the model’s parameter values. Second, I display the response of the model to technology and fiscal policy shocks. Finally, the models are tested using single hypothesis Wald tests to determine how the predicted second moments compare with those estimated from the U.S. data.14 3.1. The model’s parameters I did not estimate the model’s parameters T; h; and x: Instead, I fix T at a value of 1369 hours per quarter, and set x equal to 16 h per quarter.15 I set the fixed number of hours worked per person, h; to 1 since it does not affect the properties of the model. It is also necessary to make an additional assumption about the ratio ct =cst in order to identify the ratio d=y; and the parameter s; which appears in the IC constraint. I assume that the ratio of ct =cst is equal to 1.2853. This calibration is based on Gruber’s (1997) results that, absent government unemployment insurance, an unemployment spell results in a 22.2% drop in food consumption.16 The remaining parameters, fd=y; b; d; tg ; mg ; rg ; sg ; lnðAÞ; rA ; sA ; Ay ; lnðgy Þ; a; lnðg=yÞg and the second moments, fsc =sy ; si =sy ; sg =sy ; sw ; sn ; sy g; are estimated using the exactly identified GMM procedure.17 Table 1 reports the estimated parameter values for the partial income insurance case and the full income insurance case, respectively. Second moments implications for the shirking model are displayed in Tables 2 and 3. 14

A description of the Wald test is found in Christiano and Eichenbaum (1992). Only test single hypotheses are used given the problems associated with small sample properties of the generalized method of moments (GMM)-based Wald statistics. 15 T ¼ 1369 represents an endowment of 15 hours per day and x ¼ 16 represents a fixed cost of less than 10 min per day. The value of x was varied in other estimations, however, the results were not extremely sensitive to this value. 16 This calibration is based on the fact that cst ¼ cut in the partial income insurance case. The model’s sensitivity to this assumption was examined by varying the value of ct =cst since this value is never observed in equilibrium. In general, the findings suggest that small movements in ct =cst have little effect on the model’s second moments and responses to shocks. 17 The GMM procedure discussed in Appendix B and the data set described in Appendix A are used to calibrate the parameters and diagnose the performance of the model. The data used to estimate the second moments are detrended using a HP filter ðl ¼ 1600Þ: Here lnðg=yÞ is the ln of the ratio of government expenditures to output along the balanced growth * Furthermore, the parameters tg and mg emerge since the process for gts was path and ln A ¼ ð1  rA Þln A: altered to account for a trend seen in the data as in Burnside et al. (1993).

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Table 1 Parameter estimates for the models Parameter Estimates shirking model P.I.I. case b ln A rA sA Ay lnðgÞ d a lnðgyÞ d y y rg tg sg mg

0.9796 0.0013 0.9699 0.0074 1.9810 0.0024 0.0203 0.4574 1.6870

F.I.I. case

(0.0005) 0.9795 (0.0012) 0.0013 (0.0212) 0.9699 (0.0005) 0.0074 (0.0205) 1.9810 (0.0002) 0.0024 (0.0004) 0.0203 (0.0014) 0.4574 (0.0179) 1.6870

0.0475 (0.0004)

Estimates divisible labor

F.I.I. case

(0.0007) 0.9796 (0.0012) 0.0172 (0.0212) 0.9699 (0.0005) 0.0074 (0.0205) 1.9810 (0.0002) 0.0024 (0.0004) 0.0203 (0.0014) 0.4574 (0.0179) 1.6870

0.0468 (0.0005)

Estimates indivisible labor

(0.0005) 0.9795 (0.0122) 0.0172 (0.0212) 0.9699 (0.0005) 0.0074 (0.0205) 1.9810 (0.0002) 0.0024 (0.0004) 0.0203 (0.0014) 0.4574 (0.0179) 1.6870

n=a

P.I.I. case

(0.0007) 0.9795 (0.0122) 0.0172 (0.0212) 0.9699 (0.0005) 0.0074 (0.0205) 1.9810 (0.0002) 0.0024 (0.0004) 0.0203 (0.0014) 0.4574 (0.0179) 1.6870

n=a

(0.0008) (0.0122) (0.0212) (0.0005) (0.0205) (0.0002) (0.0004) (0.0014) (0.0179)

n=a

n=a n=a 1.7119 (0.0181) 2.6497 (0.0233) 2.7704 (0.0277) 0.9797 (0.0208) 0.9797 (0.0208) 0.9797 (0.0208) 0.9797 (0.0208) 0.9797 (0.0208) 0.0021 (0.0001) 0.0021 (0.0001) 0.0021 (0.0001) 0.0021 (0.0001) 0.0021 (0.0001) 0.0133 (0.0011) 0.0133 (0.0011) 0.0133 (0.0011) 0.0133 (0.0011) 0.0133 (0.0011) 0.4536 (0.0250) 0.4536 (0.0250) 0.4536 (0.0250) 0.4536 (0.0250) 0.4536 (0.0250)

Notes: Standard errors are in parentheses. P.I.I.: partial income insurance. F.I.I.: full income insurance.

To determine the relative success of the shirking model, I compare my results to those obtained using standard divisible and indivisible labor models.18 The parameters of these models are estimated using the GMM strategy discussed in Appendix B. The parameter estimates in Table 1 highlight that the main differences between the models are linked to the parameters governing the labor market, i.e., d=y and y:19 The other parameters are generally within the bounds commonly seen in the real business cycle literature.20 As a result, the differences in the second moments reported in Tables 2 and 3 reflect differences in the models’ structures rather than the parameter values per se. 18

The standard divisible labor model is the decentralized version of the model seen in Christiano and Eichenbaum (1992) with the time endowment normalized to 1 and the indivisible labor model follows Hansen (1985). 19 The variation in the estimates of the steady-state level of technology between the divisible labor model, and the shirking model can be attributed to the differences in the means of the employment series used for the estimation. However, since this constant does not play a large role in the normalized real wage and employment responses, and the other parameters are almost identical, the differences in the models’ second moments can be traced to the parameters, y and d=y: 20 The difference in a and b stems from the measure of real wages. Since the shirking model induces work effort using wages, the data for total wages were created by combining the reported wage data with the data on other labor income. This differs from the more traditional measure of labor compensation since it does not include the firm’s contributions to social security or government unemployment insurance.

Table 2 Second moments

U.S. data Shirking, partial income insurance p-value Shirking, full income insurance p-value Divisible labor p-value Indivisible labor, full income insurance p-value Indivisible labor, partial income insurance p-value

0.4984 0.3903 0.1986 0.4818 0.6039 0.5619 0.1214 0.4891 0.7765 0.4938 0.8870

si sy

(0.0323) (0.0850) (0.0042) (0.0329) (0.0118) (0.0075)

3.0575 3.3661 0.3341 3.1467 0.4424 3.0225 0.7960 3.1212 0.6634 3.0726 0.9185

sg sy

(0.1141) (0.2921) (0.0105) (0.0388) (0.0581) (0.0607)

1.2488 0.9302 0.0966 1.2036 0.8140 1.4515 0.3126 1.2132 0.8557 1.1807 0.7242

sn (0.2041) (0.1642) (0.1286) (0.1537) (0.1232) (0.1185)

0.0120 0.0179 0.1394 0.0099 0.1331 0.0051 0.0000 0.0097 0.1579 0.0105 0.3894

sw (0.0014) (0.0042) (0.0006) (0.0005) (0.0009) (0.0010)

0.0061 0.0065 0.5559 0.0069 0.0970 0.0081 0.0011 0.0070 0.1224 0.0069 0.1898

sy (0.0004) (0.0006) (0.0004) (0.0006) (0.0005) (0.0005)

0.0148 0.0186 0.1554 0.0144 0.8006 0.0119 0.0389 0.0143 0.7381 0.0147 0.9579

(0.0015) (0.0027) (0.0010) (0.0008) (0.0010) (0.0010)

Standard errors are in parentheses. Table 3 Second moments Moment

sn sy

U.S. data Shirking, partial income insurance p-value Shirking, full income insurance p-value Divisible labor p-value Indivisible labor, full income insurance p-value Indivisible labor, partial income insurance p-value

0.8107 0.9611 0.1003 0.6875 0.0154 0.4235 0.0000 0.6793 0.0532 0.7176 0.1868

(0.0469) (0.0947) (0.0108) (0.0426) (0.0484) (0.0518)

0.4110 0.3469 0.3456 0.4818 0.0709 0.6781 0.0000 0.4891 0.0554 0.4678 0.1714

rðw; nÞ (0.0397) (0.0646) (0.0042) (0.0047) (0.0118) (0.0145)

0.0052 0.0661 0.8167 0.4456 0.0012 0.6281 0.0004 0.4506 0.0188 0.3966 0.0484

rðw; yÞ (0.1335) (0.2178) (0.0592) (0.1355) (0.1574) (0.1703)

0.1811 0.2835 0.7342 0.7882 0.0000 0.9441 0.0000 0.7952 0.0001 0.7523 0.0005

rðn; yÞ (0.1425) (0.2732) (0.0284) (0.0277) (0.0741) (0.0878)

0.8470 0.9382 0.0163 0.9022 0.1105 0.8494 0.9654 0.8996 0.1886 0.9031 0.1553

rði; yÞ (0.0347) (0.0254) (0.0086) (0.0476) (0.0237) (0.0225)

0.8948 0.9872 0.0006 0.9859 0.0008 0.9845 0.0023 0.9861 0.0013 0.9863 0.0013

(0.0275) (0.0012) (0.0012) (0.0074) (0.0044) (0.0044)

287

Standard errors are in parentheses.

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Four main results emerge from Tables 2 and 3. First, both shirking models are better able to capture the high volatility of employment and low volatility of real wages than the standard divisible labor model. Interestingly, the shirking model actually overstates the volatility of employment. Second, the partial income insurance shirking model is better able to capture the correlation of wages and employment than the other models considered. Third, limiting the income insurance in the shirking model reduces the variation of wages and increases the variation of employment. Fourth, the full income insurance shirking model produces remarkably similar results to those obtained from the indivisible labor model. This occurs because the two models have identical reduced form representations (see Appendix C). In this sense they are observationally equivalent.21 The similarity between the standard indivisible labor model and the full income insurance shirking model leads to the question: What causes the improvement in the partial income insurance shirking model relative to the standard individual labor model? Is it the imperfectly observable effort or limitations on the amount of income insurance available to agents? To answer these questions I compare the results from my partial income insurance shirking model to the results derived from a model in which labor is indivisible but agents only have partial income insurance ( partial income insurance indivisible labor model). The structure of the model is as follows. There are four types of agents: households/families, insurance companies, firms and the Government. Individuals belong to families who make capital accumulation decisions. In addition to choosing ctf and Ktþ1 the family chooses the probability that their members will get a job during the period, Nt ; in this model. If a worker is employed he will receive a wage of wt h: However, an unemployed worker will receive only swt h since only s% of the wages are insurable in the model. The risk neutral insurance provider charges the premium ð1  Nt Þswt h for insuring s% of the wages, swt h: The family can affect its premium by altering its choice of Nt : This model is not observationally equivalent to the partial income insurance shirking model. The basic difference is that in the partial income insurance indivisible labor model a family takes into account that its choice of Nt affects its insurance premium. In contrast, families in the partial income insurance shirking model do not choose Nt and therefore do not affect the amount of the intra-family transfer, ð1  Nt Þswt h: Table 1 presents the estimated parameter values for the two models while Tables 2 and 3 report the corresponding second moment implications. The key result is the following: limiting the amount insurance in the indivisible labor model has a much smaller affect on the volatility of employment and wages than in the shirking model. Based on this, I infer that it is the imperfectly observed effort, rather than partial income insurance, that is primarily responsible for the low wage variation and high employment variation in the shirking model.

21

The minor differences between these models seen in the tables and the simulated impulse responses comes from the fact that the models are not estimated using their reduced form representation.

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3.2. Dynamic responses to fiscal policy and technology shocks Fig. 1 displays the dynamic responses of the models to a 1% unexpected exogenous increase in government purchases and Fig. 2 displays the dynamic responses of the models to a technology shock. Two main findings emerge from these figures. First, the assumption of imperfectly observable effort matters for the dynamic responses of macroeconomic variables to productivity and fiscal policy shocks. This result is seen by comparing the responses of the shirking model to those in standard divisible labor model. Compared to the standard divisible labor model, the shirking models produce: (i) much larger employment and output responses to both types of shocks, (ii) larger wage responses to a fiscal policy shock, and (iii) smaller wage responses to a technology shock. Second, although there are many similarities between the responses of the full income insurance shirking model and the partial income insurance shirking model, limiting the amount of insurance affects the magnitudes of the responses. For

Fig. 1. A comparison of the models’ impulse responses to a positive fiscal policy shock. —, partial insurance shirking model; o; full income insurance shirking model; - -, indivisible labor model; - -, divisible labor model.

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Fig. 2. A comparison of the models’ impulse responses to a positive technology shock. —, partial insurance shirking model; o; full income insurance shirking model; - -, indivisible labor model; - -, divisible labor model.

example, in the partial income insurance shirking model: (i) real wages respond much less to a technology shock, (ii) employment responses more to technology shock, and (iii) investment, real wages, employment, and output are more responsive to a fiscal policy shock than in the full income insurance shirking model. Figs. 3 and 4 compare the responses to a fiscal policy shock and technology shock in the indivisible labor model and shirking model, with full and partial income insurance. The findings illustrate that while the full insurance shirking model and the standard indivisible labor model produce identical responses to the shocks, the responses of the models differ when the amount of income insurance available to agents is limited. For example, the partial income insurance shirking model produces much large employment responses than the partial income insurance indivisible labor model. Therefore, although partial income insurance can help to increase the response of employment to shocks, the inclusion of imperfectly observed effort has important affects on the dynamic responses of the variables to productivity and fiscal policy shocks.

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Fig. 3. A comparison of the models’ impulse responses to a positive fiscal policy shock. —, partial income insurance shirking model; o; full income insurance shirking model; - -, full income insurance indivisible labor model; -}-, partial income insurance indivisible labor model.

3.2.1. Fiscal policy shocks In this subsection I discuss the intuition underlying the responses of the shirking model to a fiscal policy shock. When there is an exogenous increase in government purchases, it is financed by increasing the amount of the families’ tax burden. All else equal, the family’s budget constraint implies that the tax increase reduces the amount of consumption, c f ; and investment, It ; a family can purchase. This decrease in ctf causes the punishment associated with being detected shirking to rise because ct =cst increases. When ct =cst rises, workers strictly prefer not to shirk at the given wage and effort levels. Since the Solow condition ensures firms want to maintain the same effort level in equilibrium, firms decrease the wage offered to workers to the point where ct =cst is again equal to const: Employment then rises because the real wage has fallen, and output and the rental rate on capital rise because of the increase in employment.22 22

These responses are feasible so long as there is enough unemployment in the economy. If firms want to hire more workers then are available in the unemployment pool, although employment will increase, effort and wages will not be determined from the Solow condition. Specifically, to increase production further, firms must vary the level of effort required by workers during the period.

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Fig. 4. A comparison of the models’ impulse responses to a positive fiscal policy shock. —, partial income insurance shirking model; o; full income insurance shirking model; - -, full income insurance indivisible labor model; -}-, partial income insurance indivisible labor model.

When the shock induces a persistent increase in government purchases, the expected return on capital also rises. This induces the families to increase investment. In general, ctf falls in response to the shock since the increases in Taxt and It outweigh the affect of the increase in the return on capital. Therefore, real wages decrease, output, investment, and employment increase and consumption decreases due to the decline in the real wage and ctf : These result are qualitatively consistent with evidence presented in Edelberg et al. (1999) and Ramey and Shapiro (1998). 3.2.2. Technology shocks Similar intuition can be outlined for the model’s response to a positive technology shock. A positive technology shock has two effects. First, it increases the marginal product of labor. Second, it affects the level of family consumption, ctf ; through its affect on the return on capital, rt Kt ; and investment, It : The change in ctf affects ct =cst ; thereby altering the punishment associated with shirking. The response of wages and employment will depend on the change in ctf for the same reasons discussed in the previous subsection.

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Since the technology shock is persistent, a positive shock increases both the return on capital and the expected future return on capital. This latter affect induces the family to increase their investment in capital goods. In practice, ctf increases since the rise in rt Kt dominates the rise in It : As a result, ct =cst decreases causing workers to prefer shirking at the given wage rate. Since the Solow condition implies firms want to maintain the same effort level in equilibrium, they raise the wage offered to workers until ct =cst ¼ const: The amount of the wage movement depends on the change in ctf : For example, if the investment response to the shock is large, as it is in Fig. 2, the increase in ctf is small. Hence, the punishment associated with shirking only decreases slightly so firms will only require a small increase in the wage to induce the desired effort level from workers. In general, the increase in the marginal product of labor is larger than the increase in the marginal cost of labor, wt : Therefore, firms respond by increasing the size of their labor force. This in turn increases output and further increases the return on capital, which reinforces the upward pressure on wages. Consumption then rises due to the increases in employment, real wages and ctf :

4. Conclusions The paper formulates and estimates a general equilibrium model with imperfectly observed effort. In contrast to existing shirking models seen in the efficiency wage literature, firms do not dismiss workers for shirking. Instead, detected shirkers in the model incur a loss in wages. To evaluate this new model’s performance, I estimate the model and contrast its properties with the standard real business cycle models. In addition, I evaluate the model’s ability to account for various unconditional second moments of the data. The results suggest that my shirking model is better able to capture the second moments of the data than the standard divisible labor model. The shirking model can account for the high volatility of employment and the low volatility of real wages. The partial income insurance shirking model is able to capture the correlation of wages and employment over the business cycle. The model is also consistent with some existing evidence about the qualitative responses of the economy to fiscal policy shocks. The results also suggest that the presence of imperfectly observable effort and monetary punishments matter for the dynamics of macroeconomic variables. Although my shirking model has the same qualitative responses to technology and fiscal policy shocks as the standard models, the responses differ quantitatively. For example, in contrast to the magnitudes seen in the divisible labor model the shirking models predicts that following a positive technology shock: (i) real wages only mildly increase, (ii) employment increases almost as much as output, and (iii) investment increases. Although the results of the simple shirking model appear promising, more work is needed to determine if the results reported in this paper are robust to alternative specifications of the model. For example, it will be important to examine the

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model’s predictions when continuing matches between firms and workers are allowed for. In extensions of the model, richer labor contracts can be explored, as well as an environment where firms are also occasionally able to dismiss workers for shirking. These modifications will most likely decrease the variability of employment in the model. However, since the variation of employment in the simple model is higher than seen in the data, the affect of incorporating continued matches may in fact improve the model’s predictions, in addition to adding a more realistic structure. Other interesting extensions include cases where workers are heterogeneous, and cases where the form of the contract is optimally chosen, since these models may provide additional implications that are testable using microeconomic data.

Acknowledgements I would like to thank Martin Eichenbaum, Larry Christiano, Ian Domowitz, Dale Mortensen, Joe Altonji, Gadi Barlevy, Tim Conley, Marco Bassetto, Angelo Melino, Jon Cohen, an anonymous referee, Bob King, and seminar participants at Northwestern University, the University of Chicago, UCLA, the University of Pennsylvania, New York University, the University of Toronto, and Stanford Business School for helpful suggestions and comments. All errors and omissions are my own.

Appendix A. The data The data used for consumption, output, investment and government expenditures is obtained from Citibase Database and the Bureau of Economic Analysis. The official capital stock was obtained from the Bureau’s Survey of Current BusinessFixed Reproducible Tangible Wealth in the U.S. The capital stock, Kt ; is defined as the sum of the net stocks of consumer durables, producer structures, and equipment and private residential capital plus the Government non-residential capital. Private consumption, Ct ; is the sum of private-sector expenditures on non-durable goods and services plus the imputed service flow from the stock of consumer durable goods. Gross investment, It ; is measured as the sum of consumer expenditures on durable good, gross private non-residential (structures and equipment) and residential investment, as well as the change in the gross stock of government capital. Government expenditures, Gt ; was computed using federal, state and local expenditures on goods and services minus real government investment measure by the change in the gross stock of government capital. The measure of output at time t was then defined as real GDP. The wage series was created by combining data on wages and other labor income. The employment series was created using the total hours from establishments. All the data were converted to per-capita terms by dividing by the size of the labor force. Employment in the divisible and indivisible

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labor models is defined as Nt ¼

total hours total labor force 1369

to get employment per worker. Employment in the shirking models is defined as Nt ¼

total hours total labor force hours worked per person

to get the fraction of people employed. The number of hours worked per person used was chosen so that the implied unemployment rate from this series matched the average unemployment rate from the unemployment series.

Appendix B. GMM identifying restrictions The shirking model’s parameters, fd=y; b; d; tg ; mg ; rg ; sg ; lnðAÞ; rA ; sA ; Ay ; lnðgÞ; a; lnðgyÞg; are simultaneously estimated using the following exactly identifying restrictions:  wt Nt E ð1  aÞ  ¼ 0; Yt EðlnðAt Þ  ln A  rA lnðAt1 ÞÞ ¼ 0; EððlnðAt Þ  ln A  rA lnðAt1 ÞÞ lnðAt1 ÞÞ ¼ 0; EððlnðAt Þ  ln A  rA lnðAt1 ÞÞ2  s2A Þ ¼ 0; EðlnðYt Þ  Ay  t lnðgÞÞ ¼ 0; t E ðlnðYt Þ  Ay  t lnðgÞÞ ¼ 0; 150  ðKtþ1  It Þ  d ¼ 0; E 1 Kt   Ytþ1 þ ð1  dÞ ¼ 0; E muct  bmuctþ1 a Ktþ1   Gt E ln t  mg  t tg ¼ 0; g EðlnðGt Þ  lnðYt Þ  lnðg=yÞÞ ¼ 0; EðNt  N ss Þ ¼ 0;   Gt t ln t  mg  t tg ¼ 0; 150 g 0   1 Gt B ð1  rg LÞ ln gt  mg  t tg C B C   C ¼ 0; EB @ A Gt1

ln t1  mg  tg ðt  1Þ g

E

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 E

!   2 Gt 2 ð1  rg LÞ ln t  mg  t tg sg ¼ 0; g

EðlnðGt Þ  lnðYt Þ  lnðg=yÞÞ ¼ 0; EðNt  N ss Þ ¼ 0; where muct ; is the marginal utility of consumption, At ; is the Solow residual defined * and N ss is the steady-state employby At ¼ ðYt =Kta ðNt gt Þ1a Þ; ln A ¼ ð1  rA Þln A; ment rate implied by the model’s parameters. The process for gt was altered to account for a time trend seen in the data as in Burnside et al. (1993). The same identifying restrictions were used to estimate the divisible and indivisible labor models with the exception that: (i) the coefficient on the utility of leisure, y; was estimated instead of d=y and (ii) the marginal utility of consumption series and the steady-state value of employment adjusted to fit the models’ definitions. Finally, to test the models’ predictions for fsc =sy ; si =sy ; sg =sy ; sw ; sn ; sy g; HP filtered data was hp 2 2 2 used along with the equation Eððyhp t Þ ðsx =sy Þ  ðxt Þ Þ ¼ 0 for xAfc; i; gg and 2 2 Eððxhp t Þ  ðsx Þ Þ ¼ 0 for xAfn; y; wg:

Appendix C. The observational equivalence of models The results for the full income insurance shirking model and the indivisible labor model in the paper are virtually identical since they can be represented by the same reduced form. In the full income insurance shirking model, ct ¼ Ct so: Ct ¼ constðswt h  ð1  Nt Þwt h þ ctf Þ ¼ ðs  1Þconst wt h þ const Ct ðconst  1Þ : ) wt h ¼ Ct ð1  sÞconst Combining this with the firms’ Euler equation for Nt yields the equation: ð1  aÞ

Yt ¼ Ct C; Nt

where C ¼ ðconst  1Þ=ð1  sÞconst: The equilibrium allocations can be described by the following conditions: Yt ð1  aÞ ¼ Ct C; Nt    1 1 Ytþ1 Et b ð1  dÞ þ a ¼ 0; Ct Ctþ1 Ktþ1 Yt  A# t K a ðNt Þ1a ¼ 0; t

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Yt ¼ rt ; Kt Yt  Ct  Ktþ1 þ ð1  dÞKt  Gt ¼ 0;

a

where A# t ¼ At ðheÞ1a : In the standard indivisible labor model, individual’s choose unemployment insurance and a probability of working that perfectly insures them against unemployment. Moreover, the individual’s Euler equation for the probability of employment, Nt implies: yCt ¼ hwt : The remaining Euler equations imply that the equilibrium allocations in this model are described by the following equations:    1 1 Ytþ1 b ð1  dÞ þ a Et ¼ 0; Ct Ctþ1 Ktþ1 Yt  Ct  Ktþ1 þ ð1  dÞKt  Gt ¼ 0; Yt  At Kta ðNt Þ1a ¼ 0; ð1  aÞ

a

Yt ¼ yCt ; Nt

Yt ¼ rt : Kt

It follows that if the same data set is used to estimate the reduced forms for the full income insurance shirking model and the indivisible labor model, they should have the same empirical predictions about the variables’ second moments and impulse response functions.

References Agell, J., Lundborg, P., 1995. Theories of pay and unemployment: survey evidence from Swedish manufacturing firms. Scandinavian Journal of Economics 97, 295–307. Alexopoulos, M., 2001. Notes on Shirking Models. Manuscript, University of Toronto. Burnside, C., Eichenbaum, M., Rebelo, S., 1993. Labor hoarding and the business cycle. Journal of Political Economy 10, 245–273. Christiano, L., Eichenbaum, M., 1992. Current real-business-cycle theories and aggregate labor-market fluctuations. American Economic Review 82, 430–450. Danthine, J.P., Donaldson, J., 1995. Risk sharing in the business cycle. In: Cooley, T. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Edelberg, W., Eichenbaum, M., Fisher, J., 1999. Understanding the effects of a shock to government purchases. Review of Economic Dynamics 2, 166–206. Gomme, P., 1999. Shirking, unemployment and aggregate fluctuations. International Economic Review 40, 3–21.

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Gruber, J., 1997. The consumption smoothing benefits of unemployment insurance. American Economic Review 87, 192–205. Hall, J.D., 1993. The wage setters guide to wage rigidity. Master’s Thesis, University of Southampton. Hansen, G.D., 1985. Indivisible labor and the business cycle. Journal of Monetary Economics 16, 309–327. Kimball, M., 1994. Labor-market dynamics when unemployment is a worker discipline device. American Economic Review 82, 1045–1059. Kydland, F.E., Prescott, E.C., 1982. Time to build and aggregate fluctuations. Econometrica 50, 1345–1370. Lilien, D., Hall, R., 1986. Cyclical fluctuations in the labor market. In: Ashenfelter, O., Layard, R. (Eds.),, Handbook of Labor Economics, Vol. 2. North-Holland, Amsterdam, pp. 1001–1038. MaCurdy, T., 1981. An empirical model of labor supply in a life cycle setting. Journal of Political Economy 89, 1059–1085. Malcomson, J.M., 1998. Individual Employment Contracts. Discussion Paper 9804, University of Southampton. Pencavel, J., 1986. Labor supply of Men: a survey. In: Ashenfelter, O., Layard, R. (Eds.),, Handbook of Labor Economics, Vol. 2. North-Holland, Amsterdam, pp. 3–102. Ramey, V., Shapiro, M., 1998. Costly capital reallocation and the effects of government spending. Carnegie Rochester Conference on Public Policy 48, 145–194. Rogerson, R., 1988. Indivisible labor, lotteries and equilibrium. Journal of Monetary Economics 21, 1–16. Shapiro, C., Stiglitz, J.E., 1984. Equilibrium unemployment as a worker discipline device. American Economic Review 74, 433–444. Strand, J., 1992. Business cycles with worker moral hazard. European Economic Review 36, 1291–1303. Weiss, A., 1990. Efficiency wages: models of unemployment, layoffs, and wage dispersion. Princeton University Press, Princeton, NJ.