Economics Letters 92 (2006) 163 – 169

Wage and effort dispersion Joel Shapiro * Universitat Pompeu Fabra, Departament D’Economia i Empresa, Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain Received 20 April 2005; received in revised form 9 January 2006; accepted 25 January 2006 Available online 27 April 2006

Abstract We construct a simple model where ex-ante homogeneous firms offer jobs with different wages and effort levels to ex-ante homogeneous workers. A minimum wage is shown to be welfare reducing because of its effect on the distribution of wages and effort. D 2006 Elsevier B.V. All rights reserved. Keywords: Wage posting; Productivity dispersion; Minimum wage JEL classification: D83; J24; J41

1. Introduction In this paper, we use a matching model to illustrate simultaneous wage and effort dispersion among ex-ante identical firms and workers. The model is very tractable and allows for easy analysis of policy issues. By generating differences in productivity among identical jobs, the model can address how factors affecting wages will affect productivity. We perform a simple analysis demonstrating that an increase in the minimum wage will change the effort and wage distributions and reduce overall productivity. Wage dispersion models typically have accounted for differences in productivity by assuming technological heterogeneity among firms (for example, see Burdett and Mortensen, 1998). The main

* Tel.: +34 93 542 2718; fax: +34 93 542 1746. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.01.034


J. Shapiro / Economics Letters 92 (2006) 163–169

exception is Acemoglu and Shimer (2000), who get heterogeneity in productivity from allowing firms to choose how much capital to purchase. While they address the efficiency of firm investment and the holdup problem, we look specifically at worker productivity holding firm technology fixed. The model consists of two parts. First we assume that a moral hazard problem exists and look at optimal contracting conditional on a firm and a worker matching. Then we analyze the matching problem.

2. The moral hazard problem The moral hazard problem that we analyze is a limited liability problem with risk neutral agents. Workers and firms are ex-ante homogeneous. Firms post contracts that specify wages conditional on outcomes. Since prospective employees compare contracts in terms of utility, equilibrium requires that a firm cannot increase their profits by changing the contract to (i) offer a different utility level or (ii) offer the same utility level but different wages. In this section, we address (ii), by formulating a moral hazard ¯. problem where the principal maximizes given a fixed utility level U We formulate the problem as simply as possible, in order to obtain closed form results in the rest of the paper.1 A worker exerts a continuous effort e, which yields one of two levels of output. With probability e, output is high and equal to 1, and with probability 1  e, output is low and equal to 0. The 2 cost of effort is equal to ce2 . The worker’s utility is therefore: ewH þ ð1  eÞwL 

ce2 2

L The first order approach holds here and the worker chooses effort e ¼ wH w c . The principal chooses ¯ . In addition, there is a wages w H and w L to maximize its profits while guaranteeing a utility level of U minimum wage equal to w min that the firm must respect. The firm’s maximization problem is:  wH  wL wH  wL  max ð1  wH Þ þ 1  ð0  wL Þ wH ;wL c c


w  w   wH  wL  c  wH  wL  2 H L ¼ U¯ wH þ 1  wL  2 c c c

wL zwmin It is easy q toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi show that the minimum wage constraint binds, which implies that w L = w min and  the high wage varies with the wH ¼ wmin þ 2c U¯  wmin . Hence, the low wage remains fixedqand ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 U¯ w amount of utility that is offered to the worker. Effort is equal to ð c Þ and also increases with the utility level offered. A contract specifying a wage for high output and low output therefore implicitly specifies an effort level and a utility level for the worker. As the utility level increases, so do the high qffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ min ¯  2U¯ þ wmin . Profits are concave in U wage and effort exerted. Lastly, firm profits are equal to U w c min


This moral hazard problem is used quite often for its tractability; for example, see Ghatak et al. (2001).

J. Shapiro / Economics Letters 92 (2006) 163–169


1 and have a maximum at wmin þ 8c ; for low utility levels providing more utility increases effort more than the cost of providing incentives, while for high utility levels the reverse is true.

3. Labor market frictions We assume that an exogenous application/search process2 matches firms with workers. The application process that we specify is similar to the model of nonsequential search in Burdett and Judd (1983) and was established in Shapiro (2004).3 Consider the labor market interactions of a mass M of firms and a mass 1 of workers. Each firm has one position available. Workers apply to n firms drawn at random from the available pool. If all workers only apply to one firm, then firms will set wages at the workers’ reservation utility since workers will not have the option to refuse an offer. This is the dDiamond-paradoxT. However, if workers apply to more than one firm, some will have a positive probability of being able to refuse a low offer, putting pressure on firms to raise their wages in order to increase the probability of getting a worker. This creates a wage distribution in equilibrium. In our model, the probability of a worker’s application yielding a job offer is between 0 and 1, implying that more than one application will allow some workers to compare wages. To obtain wage dispersion with closed form solutions, we limit the number of applications to two. Since workers are homogeneous in terms of their productivity, firms choose randomly among them. If a worker gets more than one offer, she decides which offer to accept. This is enough to allow us to define 1e 2 successful application probabilities. The probability that an application leads to a job offer is k ¼ 2 M and 2 M the probability that a firm receives no applications is e M . This is a well-known result (albeit in a slightly 4 different context); a proof is provided in Shapiro (2004). ¯ (or equivalently, wage levels) and match with workers Firms in the labor market post utility levelsqU ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ  2U¯ þ wmin . Expected profits when posting a as described above. A match yields profits 2ðU¯ w c ¯ depend on the probability of matching with workers and equals: utility level U min



0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ffi    2 U¯  wmin  2U¯ þ wmin A 1  k þ kG U¯ @ c


The first term equals the probability that the firm receives an application at all. Given that it does receive at least one application, the second term represents the probability that the randomly chosen ¯ , meaning that the worker has no offer larger than U ¯ (where G(U ¯) worker will accept the firm’s offer of U represents the distribution function for utilities). Firms are homogeneous ex-ante, so expected profits ¯ . The distribution of utilities is pinned down in the following lemma. must be equal for all levels of U


Many search models assume an exogenous search process, e.g., Varian (1980) and Stahl (1989). Unlike our model, they obtain wage dispersion from the ex-ante heterogeneity of agents. 3 The model of matching with multiple applications by Albrecht et al. (2004) is very similar. The model by Gautier and Moraga-Gonzalez (2004) generalizes Shapiro (2004) by endogenizing the application process and looking at different possible selection criteria for firms. 4 The first to use this urn-ball process was Butters (1977).


J. Shapiro / Economics Letters 92 (2006) 163–169

Proposition 1. Utilities will be distributed along a continuous distribution [wmin þ 81c , U top ] and expected payoffs for the firm along this distribution are equal. Proof. 1 . Remember that ex-post profits from a match (i) q The minimum utility offered will be wmin þ 8c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2ðU¯ wmin Þ 1  2U¯ þ wmin are concave and have a maximum at wmin þ 8c . Consider a firm that c 1 1 offers a utility below wmin þ 8c. If the firm deviates to wmin þ 8c , it will get at least as many workers in expectation if not more, and will have larger profits conditional on matching. So no 1 . Now suppose that the minimum utility offered was firms will offer a utility below wmin þ 8c 1 1 , a firm would have the same probability of matching above wmin þ 8c . By deviating to wmin þ 8c (only workers with no other offers would accept) and would have higher profits conditional on matching. 1 , then there is a profitable deviation upward by e, (ii) If all firms only offered utility level wmin þ 8c where the profits conditional on matching are smaller, but the probability of the worker having another offer which is weakly preferred jumps discretely to 0. (iii) Suppose there is a gap in the distribution of utilities offered. Then a firm offering a utility at the top of the gap can lower utility by q, have higher profits conditional on matching and have the same probability of attracting workers. (iv) Expected payoffs along the utility distribution must be equal. If not, a firm will deviate to the utilities, which offer higher payoffs. 5

Using the above lemma, we can pin down profits and the utility distribution. Making use of the fact 2 1 1 ) = 0, profits must equal (1  e M )(1  k)( 4c  wmin ). A necessary condition for that G(wmin þ 8c existence of the equilibrium is that profits are positive. We formalize this in the following assumption. 1 Assumption 1. wmin b 4c .

The equal profits condition defines the cumulative distribution function: 1 0 1   1kB C 4c  wmin ffi G U¯ ¼  1A @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2ðU¯ wmin Þ  2U¯ þ wmin c

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðUtop wmin Þ

The upper bound of the distribution, U top, is implicitly defined by  2Utop þ c 1  wmin ). We have now defined the utility distribution; wage and effort distributions wmin = (1  k)( 4c immediately follow. Higher wages accompany higher effort and higher utility. Since an individual with a higher wages and effort has higher utility, the balance between wages and effort is not fully internalized in a compensating differential. One last point remains to complete the model. We implicitly assumed that effort e a [0, 1], but we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¯ min Þ , it is easy to show that must demonstrate that this is true in equilibrium. Using the fact that e ¼ 2ðU w c 1 the lowest effort of the equilibrium pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieffort distribution is ebottom ¼ 2c , which is greater than zero. The highest effort is etop ¼ 1þ kð14cwmin Þ , which we derive in the following lemma. 2c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ kð14cwmin Þ Lemma 1. The top effort is defined by etop ¼ . 2c

J. Shapiro / Economics Letters 92 (2006) 163–169


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðUtop wmin Þ

1 Proof. We defined U top implicitlyqby  2U top + w min = (1  k)( 4c  w min). Transforming the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 2ðU w Þ 1 gives e top  ce top  w min = (1  k)( 4c  wmin ). Using the expression by substituting etop ¼ pc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1F k ð 14cw Þ min . Note that 1  4cw min is positive from Assumption 1. quadratic formula gives us two roots: 2c 1 , we have proved the lemma. 5 Since we know ebottom ¼ 2c top


A sufficient condition for e top to be less than 1 is that the cost parameter c is greater than or equal to 1.5 We assume this for the rest of the paper. Assumption 2. c z 1.

4. Labor market conditions and productivity By incorporating a frictional labor market, we are able to depict simultaneous wage and effort dispersion for ex-ante homogeneous firms and workers. This is not only a theoretical innovation, but it lays the groundwork for understanding the interaction between labor market conditions and productivity. In this section, we address this in a simple way, by examining the effect of a minimum wage on the effort distribution and on overall productivity. ¯ + p. Summing up over all the matches A match between a worker and a firm generates a surplus of U that occur (substituting for p) yields total surplus: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1   Z Utop       2 U¯  wmin 2 @ U¯ þ wmin AM 1  e M 1  k þ kG U¯ g U¯ d U¯ 1 c wmin þ 8c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðU¯ wmin Þ We can simplify this expression by using a change of variable, substituting effort e ¼ . To c ¯ (e)), not g(U ¯ (e)) (also, we will write G(e) rather save on notation, define g(e) as the derivative of G(U ¯ (e))). The expression becomes: than G(U  Z etop  ce2 e 2kð1  k þ kGðeÞÞgðeÞde ð2Þ 2 ebottom where e top and e bottom are as defined in the previous section. This expression has an easy interpretation— surplus in the economy is defined by production (1 multiplied by the probability e) minus effort costs. Now we can prove that surplus decreases with the minimum wage. Proposition 2. Given that Assumptions 1 and 2 hold, an increase in the minimum wage changes the wage and effort distributions in a way that decreases total surplus. We prove this in Appendix A. A minimum wage decreases surplus by changing the distribution  1 of effort. An increase in w min 2 4c eþce increases the distribution function G(e) (where GðeÞ ¼ 1k 2 k ece wmin . In fact, we can almost claim V b w min W , G(e|w min V ) first order stochastically dominates first order stochastic dominance, i.e. for w min W ). The reason that this claim is balmostQ true is that the ranges are not the same, e top shrinks with G(e|w min


This comes from setting w min to its lowest possible level, zero, and using the fact that ka(0, 1).


J. Shapiro / Economics Letters 92 (2006) 163–169

w min (though e bottom remains fixed). This implies that lower efforts are weighted more for a larger w min. The reason lower efforts receive more weight is because the increase in the minimum wage reduces the firms rents conditional on matching, and it does so more at the bottom of the distribution than at the top dp (we can show that dwmin dU¯ is positive). The change in the minimum wage also affects the wage distribution, although the direction of the change depends on the parameters. Labor market tightness also changes the wage and effort distributions. Defining the worker–firm ratio as q ¼ M1 , we find that both G(e|q) and G(w|q) increase with q (and the upper bound of each distribution decreases with q).6 This result clearly derives from the fact that as relative amount of workers increases, their probability of having an outside offer decreases, allowing firms to impose lower wages. In response, the effort levels imposed must also decrease. Acknowledgements I thank the editor and an anonymous referee for helpful comments. I gratefully acknowledge financial support from DGES and FEDER under BEC2003-00412 and Barcelona Economics (CREA). Appendix A. Proof of Proposition 2 Proof. We simplify Eq. (2) by breaking it up into two integrals and integrating by parts, yielding: !  1  Z etop ce2top    e þ ce2 2 ð1  ceÞ 4c 2 2kð1  kÞ þ k  2ð1  kÞ2 etop  de 2 e  ce  wmin ebottom 2

 ð1  kÞ


 e þ ce2 ð1  ceÞ e  ce2  wmin ebottom etop

1 4c

2 de


The derivative shows that the minimum wage only affects surplus through the integrand, not e top. Using the definition that G(e top) = 1, we find that the derivative is equal to: 1  1  Z etop 2 2 4c  e þ ce 4c  wmin ð1  ceÞ  2ð1  kÞ de ð4Þ ðe  ce2  wmin Þ3 ebottom All four terms of the integrand of Eq. (4) are non-negative for all values of e, so the integral must be positive. 5 References Acemoglu, D., Shimer, R., 2000. Wage and technology dispersion. Review of Economic Studies 67, 585 – 607. Albrecht, J., Tan, S., Gautier, P., Vroman, S., 2004. Matching with multiple applications revisited. Economic Letters 84, 311 – 314. Burdett, K., Judd, K.L., 1983. Equilibrium price dispersion. Econometrica 51, 955 – 969.


dk A proof is straightforward given that dq

b0. This point was proven in Shapiro (2004).

J. Shapiro / Economics Letters 92 (2006) 163–169


Burdett, K., Mortensen, D.T., 1998. Wage differentials, employer size, and unemployment. International Economic Review 39, 257 – 273. Butters, G.R., 1977. Equilibrium distributions of sales and advertising prices. Review of Economic Studies 44, 465 – 491. Gautier, P., Moraga-Gonzalez, J.L., 2004. Strategic wage setting and coordination frictions with multiple applications, mimeo. Ghatak, M., Morelli, M., Sjostrom, T., 2001. Occupational choice and dynamic incentives. Review of Economic Studies 68, 781 – 810. Shapiro, J., 2004. Income taxation in a frictional labor market. Journal of Public Economics 88, 465 – 479. Stahl, D.O., 1989. Oligopolistic pricing with sequential consumer search. American Economic Review 79, 700 – 712. Varian, H.R., 1980. A model of sales. American Economic Review 70, 651 – 659.

Wage and effort dispersion

choose how much capital to purchase. While they address the ... the paper.1 A worker exerts a continuous effort e, which yields one of two levels of output. With .... it will get at least as many workers in expectation if not more, and will have larger profits conditional on matching. So no firms will offer a utility below wmin ю 1. 8c.

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