Cherry-picking in Labor Markets with Imperfect Information ∗ Shuaizhang Feng† Bingyong Zheng‡ October 21, 2011

Abstract We consider labor markets where firms have imperfect information about worker productivity, and compete for better workers by offering higher wages. The model predicts both wage differentials and positive sorting. In equilibrium, firms that offer higher wages always hire more productive workers. Meanwhile, due to firms’ inability to perfectly differentiate high productivity workers from low productivity ones, some high productivity workers are employed in low wage firms while some low productivity workers are employed in high wage firms. We show that introducing employer learning does not change the results. Keywords: Imperfect information, wage differentials, sorting. JEL classification codes: D83; J31.

∗

We thank John Abowd, Melvyn Coles, Hank Farber, Hanming Fang, Peter Gottschalk, Oleg It-

skhoki, Larry Kahn, Alan Krueger, Lars Lefgren, Andrew Marder, David McAdams, Bruce Meyer, Steeve Mongrain, Arthur Robson, Robert Shimer, Wing Suen, Cheng Wang, Simon Woodcock, Tao Zhu, and seminar participants at CUHK, Essex, Princeton, Simon Fraser, Tsinghua, and the 2009 Society of Labor Economists conference for comments. Remaining errors are our own. † Shanghai University of Finance and Economics. Email: [email protected]. ‡ Shanghai University of Finance and Economics. Email: [email protected].

1

Introduction Empirical studies have found large and persistent wage differentials that appar-

ently can not be explained by differences in worker or job characteristics. These studies show that large firms, exporting firms, and certain “high-wage” industries pay more to observationally identical workers, even after controlling for job and worker heterogeneities. Meanwhile, they have also identified positive sorting; that is, large firms, exporting firms, and high-wage industries are usually the more productive ones, and tend to hire better workers (Brown and Medoff (1989), de Melo (2008)). These empirical regularities are robust, yet puzzling to economists. In competitive labor markets, the law of one price should hold. Wages should only reflect worker productivity and not be correlated with employer characteristics, such as their workforce size, industry affiliations, or exporting status. So the natural question that arises is: why are apparently identical workers paid differently? Many have attempted to give an answer. One of the most prominent explanation is the on-the-job search model of Burdett and Mortensen (1998), which attributes persistent wage differences among observationally identical workers to search friction workers experienced in the process of looking for new jobs. Whereas search friction clearly affects job searches of less skilled workers whose ability levels are more homogeneous, it is probably less important in markets for skilled workers whose ability levels are more heterogeneous, for example, academic job market. This paper attempts to formulate a theory of wage dispersion that may better describe the skilled workers for whom the main difficulty in job search is not search friction, but rather inability of prospective employers to observe their productive abilities. When firms are unable to observe workers’ productivities and pay them accordingly, a firm may wish, then, to reward workers based on performance. But there are at least two constraints on the use of performance-based pay. First, most workers work in settings where an objective output measure does not exist, thereby precluding

2

compensations based on objective performance measures.1 Moreoever, even the more limited contracts that tie a worker’s pay to subjective performance may be restricted when workers can not readily observe firm’s claim of performance.2 Thus, firms may have no choice but to pay similar compensation to workers who appear identical but may differ in abilities. We show that both wage differences and positive sorting can be explained by strategic competition for productive workers in an environment where firms have imperfect information about workers’ abilities. In the environment we consider, firms know the distribution of worker ability, and observe some but not all relevant aspects of a worker’s ability. Given the information, firms play a “wage-posting” game to attract better workers. As workers who receive multiple offers always accept the one with the highest wage, the firm that offers the highest wage will get all workers it makes an offer, who are, on average, more productive than those not receiving its offer. Hence, the average quality of workers in the pool of unemployed keeps declining as more firms have hired. Furthermore, to attract workers a firm must pay a wage commensurate with the average productivity of its workforce that goes down in the order of hiring. There is thus dispersed wage distribution in equilibrium. And more importantly, due to firms’ inability to perfectly distinguish between more productive workers and less productive ones, workers of the same ability can be paid different wages, depending on the expected productivity of their co-workers. We begin by analyzing a simple one-period model with homogeneous firms, which provides the main insight of this paper. The bare-bones model shows that firm’s inability to tell a high productivity worker from a low productivity worker has important consequence on workers’ job market outcome; two otherwise identical workers 1

MacLeod and Parent (1999) have found that only about one to five percent of U.S. workers receive

performance pay in the form of commissions or piece rates. Far more common, especially in positions that require team work, are long-term relational contracts that reward or punish workers on the basis of subjective performance measures. 2 See Prendergast (1999) and Levin (2003) for more discussion on issues that may arise where workers are paid based on some subjective performance measures.

3

can be paid differently simply because they are in firms whose labor forces have different average productivities. The empirical implication of this is that wage differentials exist even after econometricians have controlled for worker’s unmeasured ability with the fixed effects approach. Meanwhile, the model predicts that high ability workers are more likely to be at high wage firms than low ability workers. It therefore helps explain an important empirical puzzle. In a classic and influential article, Gibbons and Katz (1992) showed that, while unmeasured ability alone does not account for inter-industry wage differences in the data, the pre-displacement wage of exogenously displaced workers nevertheless has a large impact on post-displacement wage. The finding is hard to rationalize by existing theories on wage differentials, but is consistent with our imperfect information model. Next, we investigate two extensions of the basic model. In the first extension we explore the influence of employer learning on wage differentials. The dynamic model shows that as long as learning is not perfect, wage differences among identical workers would persist. Moreover, a worker’s current wage depends on his past wage, and a worker who is fortunate to be employed by a high wage firm will continue to receive higher wages in the future than another worker of the same ability but who has been less fortunate early in career. In the second extension we analyze a model in which some firms are more productive than others. In addition to wage differentials, the model with heterogeneous firms brings in new prediction on sorting, i.e., it is the more productive firms that are paying higher wages and hiring better workers. While search friction can explain the positive association between firm size and the wage paid, it fails to explain other forms of wage differentials and related stylized facts, such as the findings in Gibbons and Katz (1992). Moreover, the stark prediction that large firms always pay higher wages is inconsistent with empirical evidence showing that size-wage relationship is not uniform across industries.3 Finally, it does 3

Using NLSY data, Feng and Zheng (2010) have found that in certain industries, it is actually the

smaller firms that are paying wage premiums, and those smaller firms also hire better workers in terms of education and AFQT (Armed Forces Qualification Test) score.

4

not give an explanation for positive sorting. By contrast, the current model can explain all these empirical findings. The rest of the paper is structured as follows. The next section briefly reviews the empirical literature on wage differentials. Section 3 sets up the basic model. Section 4 presents a dynamic model. In Section 5 we consider a model in which firms differ in productivities. Section 6 discusses the relationship of the current work to existing literature. Section 7 concludes. The appendix collects omitted proofs.

2

Empirical regularities There exist three separate but interrelated literatures on wage differentials, in-

cluding inter-industry wage differentials, employer size-wage premium, and exporter wage premium. Two major stylized facts have been documented. First, “genuine” wage differences exist even with control on both observed and unobserved worker heterogeneities. Second, there is positive sorting in the sense that firms/industries paying higher wages tend to be more productive ones and also hire workers that are, on average, more productive.

2.1 Inter-industry wage differentials Early studies on inter-industry wage differences include Slichter (1950) and Cullen (1956). Using U.S. data, Dickens and Katz (1987b), and Krueger and Summers (1988) find large and persistent inter-industry wage differentials for seemingly identical workers on similar jobs, even after controlling for worker unobserved heterogeneities with fixed effects approach. Gibbons and Katz (1992) point out that endogenous jobchange decisions can create important self-selection biases even in first-differenced estimates, and therefore suggest using data on exogenously displaced workers. With CPS data on workers exogenously displaced by plant closings, they find that firstdifferenced differentials are significant even for these workers and thus conclude that

5

unmeasured ability alone can not explain inter-industry wage differentials. Meanwhile, the evidence shows that a worker’s pre-displacement industry has a large impact on his or her post-displacement earnings, suggesting that ability that helps a worker find employment in a high-wage industry once is likely to do so again. Furthermore, these studies have documented positive sorting. Dickens and Katz (1987a) examine industry characteristics and show that workers in high-wage industries also have higher average education levels, and inter-industry wage differentials decrease after worker heterogeneity has been controlled for. The evidence indicates that high-wage industries on average hire better workers, both in the observed and unobserved dimensions.

2.2 Employer size-wage premium Moore (1911) is the first to document wage gains associated with working in large firms or establishments using data from Italian textile mills. Subsequent studies report similar findings for U.S. (see Idson and Feaster (1990)), the European countries (see Abowd et al. (1999)), and also for developing countries (see Velenchik (1997)). The size-wage premium cannot be explained by job or worker characteristics, as substantial wage premiums exist even with control on observable worker characteristics (Gibson and Stillman (2009)) and unobserved ones using the fixed effects approach or matched employer-employee data (Troske (1999)). Similarly, there is evidence of positive assortative matching. After controlling for worker heterogeneities, these studies show that wage premiums associated with large firms or establishments decline (see e.g. Brown and Medoff (1989)). Using NLSY79 data, Feng and Zheng (2010) have found that sorting seems to appear concurrently with wage premiums. For industries displaying positive size-wage premium, large firms also hire workers with more education and higher AFQT scores than small firms. For industries that do not display positive size-wage premium, such as health services, large firms on average hire workers with less education and lower AFQT

6

scores.

2.3 Exporter wage premium Bernard and Jensen (1995) first document the existence of exporter wage premium. Using U.S. manufacturing data, they show that exporting plants pay 14 percent higher wages than non-exporting plants, which can not be explained by observable differences, such as plant heterogeneity, region and industry factors (See also Bernard and Jensen (1997, 2004)). Moreover, Schank et al. (2007) show that, though wage premium becomes smaller after controlling for worker unobserved effects, it nevertheless remains significant. These studies also show that on average, exporting firms are larger, more productive, and more capital-intensive. Frias et al. (2009) report that across plants and within industries, approximately one-third of the higher level of wages in exporting plants is explained by sorting. Schank et al. (2007) also find that wage premium decreases considerably after controlling for worker heterogeneity.

3

The basic model

3.1 Environment We consider a competitive labor market with a large number of ex ante identical firms who have a constant return to scale production technology. Firms produce a single consumption good with labor as the only factor of production. They are riskneutral, infinitely lived and can contractually commit to a wage policy. Each period, a new generation of workers of measure one enters the market. Workers live for one period and has a preference represented by utility function u(·), which depends on income only. We also assume workers are risk averse, with u′ > 0 and u′′ < 0, and maximize their expected-utilities. When a worker first enters the labor market, his ability is not known with certainty to prospective employers and himself. 7

But it is common knowledge that a worker can be of either high or low type, and that each generation consists of α proportion of high type and 1 − α proportion of low type workers. For simplicity we assume a high type worker can produce one unit of output when employed and a low type worker can produce none. Here we use the word “output” to represent an individual worker’s marginal contribution to a firm. In general, this “output” may not be observable to the employer. This is true of many workers, in particular, those work in teams, where output measures are not the outcome of the inputs of an individual worker, but rather reflect the joint contributions of many workers. There is no cost of applying for jobs, so workers can apply to all potential employers. After receiving a worker’s application, each firm makes an assessment of the worker’s ability. We assume the assessment yields a private signal that imperfectly reflects a worker’s type. Conditional on a worker being of high type, a firm receives an “h” signal with probability βH and “l” with probability 1 − βH . Conditional on a worker being of low type, a firm receives “l” with probability βL and “h” with probability 1−βL . For simplicity, we assume that firms have the same ability in differentiating high type workers from low type ones and that βH = βL = β ∈ (1/2, 1). Signals received by different firms are independent and therefore a worker deemed as low type by one firm could be taken as high type by another.4 Workers have the same reservation utility u0 from staying unemployed, and any wage offer has to give them utility of at least u0 . Before proceeding we first discuss the type of contract firms can commit. Since workers are risk averse and face uncertainty about their own type, the optimal contract involves firms paying each worker a fixed wage. To see this point, suppose that an objective measure on worker’s performance exists and all firms offer workers a compensation contract based on performance. In this case, one firm can make a profit by offering workers a more attractive offer with expected wage cost less than that of the performance pay scheme, and the other firms would be compelled to follow suit in 4

Our main result does not change as long as signals received by different firms are not perfectly

correlated. Assuming independence simplifies our analyses, though.

8

due course. Henceforth we take the competition between firms as offering wages to attract workers. Every period, firms play a sequential “wage-posting” game, which consists of many rounds of competition among hiring firms. Each round, firms that have not hired yet post wages they are willing to offer to some of the unemployed workers. Those who post the highest wage win the right to make offers to a selected group of workers in the pool of unemployed at the posted wage. We exclude the case in which a winning firm makes offer to no workers, so firms posting the highest wage in a round always make offers to a positive measure of workers. Upon receiving an offer, workers decide whether to accept the offer. Since signals on a worker’s type are independent across firms, if several firms happen to post the same highest wage, some workers would receive more than one offer. In that case, they randomly pick a firm’s offer to accept. After the first round, the game moves to the next round with the same process repeated for firms who have not hired and for workers who remain unemployed. The same wage-posting is repeated in each round, until no firm finds it profitable to post a wage greater than worker’s reservation wage r = u−1 (u0 ). At that point, the job market closes. Production then starts and active firms – those who have hired a positive measure of workers – realize their profits. Workers exit at the end of the period. An equilibrium in this competitive labor market is a set of offer choices, including the wage posted and selection of workers who receive the wage offer, such that, when workers choose offers to maximize expected utility, (i) no offer choices in the set makes negative expected profits; and (ii) no offer choices outside the set that, if made, will make a non-negative profit. Hence, the equilibrium concept is still of the Nash type; every round a firm that haven’t hired yet makes its offer choice to maximize own profit while taking the choices of the other firms as given. Note that the assumption of sequential wage-posting is not essential for our main results. A model of simultaneous wage-posting in which firms make wage offers to 9

selected workers at the same time while workers accept the offer that gives them the highest wage would still produce qualitatively similar result as the sequential wage-posting game. The key mechanism underlying this “cherry-picking” model, as it will be clear soon, is that one firm’s selection of workers always results in a worse pool of unemployed in terms of productivity, which in turn lowers the other firms’ willingness to pay for workers not being selected. Indeed, as long as workers takes the best offer, the firm offering the highest wage will get every worker it wants, but those offering lower wages will get only a fraction of workers whom they wish to hire, which also means that those accepting their offers are less productive. So the same cherry-picking process would take place in a simultaneous-move game. The problem with a simultaneous wage-posting game is that no pure strategy equilibrium exists, and a mixed strategy equilibrium, if it exists, would be hard to solve.5 We therefore adopt the sequential wage-posting framework to simplify our analysis.

3.2 Main results The first result is on dispersed wage distribution. We show that firms always offer different wages in equilibrium. Lemma 1. In no equilibrium do two or more firms tie at the highest wage in any round. The intuition for the result is similar to that of Burdett and Mortensen (1998). In Burdett and Mortensen, because of search friction, firms get positive profit from each worker and care about the size of their workforce. Moreover, the measure of workers at a firm is discontinuous in wage if all firms are offering the same wage. 5

The appendix provides a simple two-firm example of simultaneous wage-posting model. The ex-

ample is similar to the sales model of Varian (1980), which, to explain sales behavior between pricecompeting firms, analyzes a mixed strategy equilibrium in which firms randomize over prices. The firm happens to offer the lowest price wins all the informed consumers while all firms take equal shares of uninformed consumers.

10

Thus, should all firms offer the same wage w, any one can deviate by offering slightly more than w, which would significantly enlarge its workforce and increase total profit. This rules out a single market wage in equilibrium in their model. In the imperfect information model developed here, workers are heterogeneous and firms compete for better workers instead of the size of their workforce. And the quality of workers hired by a firm is discontinuous in wage whenever two firms offer the same wage w, because workers receiving more than one offer would randomly pick an offer and such workers are also the more productive ones. As a result, whenever two firms offer the same wage, one would have an incentive to offer w + ǫ to get better workers, i.e., those receiving offers from both firms. The same reasoning also rules out more than two firms offering the same wage. Intuitively, since the market is competitive, firms all get zero expected profit in equilibrium, and thus the winning wage in any round can be no less than the expected productivity of those workers a firm observes an “h” signal. But the expected productivity of a worker with a signal “h” is strictly greater than the expected productivity of one with an “l” signal, so it is unprofitable for any firm to hire a worker with an “l” signal. Lemma A1 in the appendix proves this result. As a winning firm expects zero profit from each new hire, it can hire any portion of those it observes an “h” signal. Let δk ≤ 1 denote the fixed proportion of workers firm k hires among the unemployed from whom it observes an “h” signal. In general, the constant δk can be less than one, and multiple equilibria exist in the game. Let ∆ ≡ (δ1 , δ2 , . . .) be a vector such that δk ∈ (0, 1] for all k. We present our main result in the following proposition. Proposition 1. In any equilibrium, firms pay workers their average productivity. Both average productivity and wage decline in the hiring round k. Proof. Lemma 1 and Lemma A1 show that in any round k, only one firm wins the right to hire, who makes offer to workers with a signal “h” and earns zero per worker expected profit. Thus, the first part of the result is straightforward, firm k pays each 11

of its new hires the average productivity Hk , and expects zero profit from each worker. Next we show that average productivity of workers hired by firm k (also denoted by Hk (∆)) is greater than that of firm k + 1 (denoted by Hk+1 (∆)). Let the applicant pool facing firm k consists n ˜ k measure of high type workers and m ˜ k measure of low type workers. We have Hk (∆) =

n ˜kβ . n ˜kβ + m ˜ k (1 − β)

The pool of workers firm k + 1 will choose from consists of n ˜ k+1 = (1 − δk β)˜ nk measure of high type workers and m ˜ k+1 = [1 − δk (1 − β)]m ˜ k measure of low type workers. Hence, Hk+1 (∆) =

(1 − δk β)˜ nk β . (1 − δk β)˜ nk β + [1 − δk (1 − β)]m ˜ k (1 − β)

Since 1 > β > 1/2 and 0 < δk ≤ 1, we have Hk (∆) > Hk+1 (∆). On overage, workers at firm k are more productive than those at firm k + 1. Hence wage differences and positive sorting are distinct features of any equilibrium. First, workers of the same type are paid differently in different firms; those who are “lucky” to get favorable assessment from firms hiring in earlier rounds are paid more. Second, firms that offer higher wages and hire in earlier rounds also draw disproportionately more high type workers than firms offering lower wages, although they all earn zero profit. Furthermore, wage differences and positive sorting - - in the sense that high-wage firms hire better workers on average - - always occur at the same time. In equilibrium the number of firms K that hire a positive measure of the new generation of workers is determined by the condition HK ≥ r > HK+1 , where r = u−1 (u0 ). That is, when the expected productivity of unemployed workers gets too low, firms who have not hired will have no incentives to make an offer to those unemployed workers that they observes an “h” signal. As a result, some productive workers will remain unemployed in equilibrium.

12

4

Dynamic model The basic model characterizes equilibrium allocation in a labor market where firms

have imperfect information about productivity of individual workers. Though a pretty simple model, it nevertheless generates a number of predictions consistent with the evidence. On the other hand, because of the static nature of the model, one may wonder what happens after firms hire workers and gradually learn about their productivity. In particular, will wage then be equal to marginal product of labor as information is accumulated and competitive pressure mounts? To address this concern we develop a dynamic model that includes employer learning, and analyze its effect on equilibrium wage dispersion. Compared with the basic model, workers now live for two periods and has the same reservation utility of u0 each period. Note that while the sequential framework significantly simplifies our analysis and produces similar results as the mixed strategy equilibrium in a simultaneous wageposting model, it nevertheless introduces new complications. That is, the sequential framework now allows firms that post lower wages and hire in latter rounds to acquire more information on the productivity of workers employed by high-wage firms. This is innocuous in the basic model when workers stays in the market only for one period, but it brings in an uninteresting feature that a low-wage firm may poach workers away from a high-wage firm in a dynamic model. To avoid the problem, we assume that firms have short memory and do not recall the first-period assessments they have on workers who have applied but are not currently working with them, which does not seem implausible in a large autonomous economy.6 At the end of the first period, firms observe a worker’s performance. Learning is 6

Alternatively, we can assume that firms do not observe signals on workers’ type before the hiring

game. Instead, each round the firm announcing the highest wage wins the right to interview all unemployed workers to make a private assessment of worker’s ability, the result of which yields a noisy signal on the worker’s type. In case several firms tie at the highest wage, one firm is randomly chosen as the winner. As a result, a firm hiring in latter round does not has more information on workers hired in earlier rounds.

13

symmetric in the sense that, at any point in time firms all have the same knowledge of a worker’ performance. Knowledge of a worker’s performance in the first period conveys information about his ability and helps predict his future performance. We assume that, conditional on a worker being of high type, firms have a good performance evaluation with probability ρ ∈ (0.5, 1], and conditional on a worker being of low type, firms have a good evaluation with probability 1 − ρ. Firms are infinitely lived and can credibly commit to a wage policy. The wage contract now specifies the first-period wage w0 , and the second period wage w(g) if the worker has a good first-period performance and w(b) if the worker has a bad performance. Thus, firms now compete for a generation of new workers by offering a bundle {w0 , w(g), w(b)}, and those that offer the best contract, in this case, providing a worker with the highest expected utility, win the right to hire in a round. A firm’s expected profit from a new worker is the expected total output minus total wage cost. First, we find the optimal contract that firms can commit in an equilibrium. A firm’s wage strategy includes choosing salaries to be offered to different group of workers in both periods. Suppose that firm k hires a group of workers who are observationally equivalent to firm k with average productivity Hk . To simplify notation we also assume that firms do not discount the second period profit. This gives the firm an expected total revenue of 2Hk from one worker, provided the worker stays with the firm also in the second period. Let πk (g) = ρHk + (1 − Hk )(1 − ρ) be the ex ante expected probability of such a worker receiving a good performance evaluation at the end of period one, and πk (b) = Hk (1 − ρ) + (1 − Hk )ρ be the probability of a bad evaluation. Since workers are free to quit, firm k needs to offer each worker a secondperiod wage at least equal to his alternative offer. Workers face borrowing constraints but can save first-period income for consumption in the second, and we let s ≥ 0 be a worker’s first-period saving. Conditional on workers taking its wage offer, firm k’s

14

problem becomes: max

wk0 ,wk (g),wk (b)

Πk = 2Hk − [wk0 + πk (g)wk (g) + πk (b)wk (b)]

s.t. U (k) ≡ u(wk0 − s) + [πk (g)u(wk (g) + s) + πk (b)u(wk (b) + s)] ≥ 2u0 ; ρHk ; Hk ρ + (1 − Hk )(1 − ρ) (1 − ρ)Hk wk (b) ≥ . Hk (1 − ρ) + (1 − Hk )ρ

wk (g) ≥

This is equivalent to maximizing worker’s expected utility, max

wk0 ,wk (g),wk (b),s

u(wk0 − s) + [πk (g)u(wk (g) + s) + πk (b)u(wk (b) + s)]

s.t. U (k) ≥ 2u0 ; U (k) ≥ U (j)

∀j 6= k;

ρHk ; Hk ρ + (1 − Hk )(1 − ρ) (1 − ρ)Hk wk (b) ≥ . Hk (1 − ρ) + (1 − Hk )ρ

wk (g) ≥

Setting up Lagrangian and solving the optimization problem yields the solution: s = 0; wk (g) =

ρHk , Hk ρ + (1 − Hk )(1 − ρ)

wk (b) = wk0 =

2Hk − Hk ρ . 1 + Hk (1 − ρ) + (1 − Hk )ρ

(1)

Thus, the optimal wage policy stipulates that a firm pays those with a bad performance the same wage in both periods, as showed in Freeman (1977) and Harris and Holmstr¨om (1982). The intuition for the downward rigid wage is simple. Whereas firms may want to base each worker’s current compensation on his past performance, doing so imposes downward risk on workers, which is costly to firms through higher wages. From this perspective pay for performance is constrained by worker’s ability to handle risk. On the other hand, since workers can not be prevented from quitting to accept a better offer, the wage contract does not insure a worker against risk associated with a positive realization of his ability. 15

Taking this result as given, we now turn our attention to competition between firms in the first period. Since firms can commit to the same downward rigid wage policy, there is no loss of generality in supposing that they compete by posting the highest first-period wage wk0 , which is also the lowest wage they promise to pay in the second period. And workers take the offer of firms that post the highest wk0 , knowing that they will be paid their expected productivity if their performance is good and wk0 otherwise. With this simplification, competition for talents becomes identical as the one period model. Therefore, in what follows we will proceed in an informal way, presenting the main results without giving any proofs. It is still true that in equilibrium, firms make zero profit, and every round, the firm that posts the highest first-period wage hires only workers it observes an “h” signal. This indicates that the average wage cost to firm k that hires in the k-th round is equal to 2Hk , assuming that no one leaves in the second period. This is generally true since by definition, wk (b) = wk0 ≥ r for all k, so workers whose performance is bad in the first period will stay in the labor market and with their current employer. Further, workers whose performance is good can not get better offers either, as what they get from current employer are already the best they can get in the labor market. 0 ≥ r but The number of hiring firms in equilibrium is determined by the condition wK 0 wK+1 < r. We summarize the main result in the following proposition

Proposition 2. When learning is not perfect, persistent wage differentials exist in equilibrium. Moreover, 1. every firm earns zero total profit in the two periods; 2. workers with a bad performance evaluation in the first period receive the same wage wk0 in both periods; 3. workers with a good performance evaluation in the first period receive a second period wage wage wk (g) that is equal to their expected productivity; 4. average productivity Hk and wages decline in the hiring round k. 16

This result clearly demonstrates that introducing employer learning does not eliminate the key feature of our basic model, and the existence of wage differentials is not a peculiar feature of the one period model. Our focus is on the case of ρ < 1; the performance measure observed by firms may not reflect a worker’s true performance and reveal his type. Whereas the economic literature has been mainly focused on workers for whom output measures are easily observed, in real world, as pointed out in Prendergast (1999, pp.57), “most people don’t work in jobs like these.” Instead, most worker are evaluated by their supervisors who provide some subjective evaluations on their performance. Furthermore, considerable evidence in the personnel literature (e.g., Murphy and Cleveland (1991)) shows that supervisors tend to distort subjective performance ratings by not sufficiently differentiating good performance from bad performance. In particular, two forms of wage compression are identified in the literature: “centrality bias” and “leniency bias.” Centrality bias refers to the practice where supervisors have a tendency to judge their workers as above average, resulting in performance evaluations that are more compressed and less variable than actual performance. Leniency bias refers to the practice that supervisors overstate the performance of poor performers (See also Prendergast (1999)). Hence, imperfect learning probably applies to the majority of workers. In this case, wage differences among apparently identical workers persist; a worker at firm k receives higher pay in both periods than another of the same type but at firm k + 1 for all k ≤ K − 1. In fact, wage differences would persist even in the very special case when firms can perfectly learn about worker’s true ability. Note that in this case, while high type workers receive the same second-period wage, low productivity workers still receive different pay in the second period. To see this, simply replace ρ = 1 in (1) to have: wk (g) = 1,

wk (b) = wk0 =

Hk 2 − Hk

for all k.

The dynamic model therefore corroborates the findings of the basic model. In addition, the dynamic model generates some new prediction. 17

Corollary 1. Conditional on worker type, workers who receive higher wages in the first period also receive higher wages in the second period. As there is no interaction between a worker’s productivity and starting wage with control on type, this result implies that a worker’s current wage also depends on something other than his productivity, in particular, his starting wage. Hence, if one is fortunate to have a good start early in career, he is also more likely to fare better in future than others who are equally capable but not as fortunate. While we know of no direct evidence in support of this prediction, there is evidence that workers’ current wages depend on their employment history. For example, using individual data from CPS and PSID Beaudry and DiNardo (1991) find that history of labor market conditions experienced by workers has an important effect on their current wages (see also McDonald and Worswick (1999)).

5

Heterogeneous firms In real world, firms may not be ex ante identical. Empirical studies have shown

that more productive firms/industries are paying higher wages and hiring a larger proportion of more productive workers. In view of this, we consider another extension in which firms differ in productivities. For notational purpose, we index firms by the rank order of their productivity parameter, or production technology. Firm i has production technology ψi which ranks ith among all firms. Like previous studies (e.g., Shimer (2005)), we assume that worker and firm productivities are complements. In particular, the two enter into the production function multiplicatively; a high type worker hired by firm i produces ψi units of output, while a low type worker still produces none. Except that firms now differ in productivities, the setup of the model is otherwise the same as the basic model. Because firms are now heterogeneous, some will get positive profit in equilibrium. Consider two firms i and j, if ψi > ψj , firm i’s expected profit, denoted as Πi , is no less than firm j’s; and if in addition, both i and j are active (hire a positive measure of 18

workers in equilibrium), then Πi > Πj . The intuition is simple. The more productive firm can earn higher profit by simply mimicking the behavior of the less productive one. On a similar note, if firm j is active, then firm i with ψi > ψj must also be active, since an inactive firm gets zero profit. Therefore, if we denote the total number of active firms by K, then active firms must be the most productive ones - - firms 1, 2, ..., K. By the same argument as in the homogeneous firm case, in equilibrium, there will only be one winning firm in each round. Moreover, a winning firm would always select better workers from the applicant pool, i.e., it would always prefer hiring “h” workers over “l” ones. This also guarantees that wages strictly declines in hiring order, as the following lemma shows. Lemma 2. In equilibrium, wage offers by active firms strictly decrease in k. Denoting wk as the winning wage in the k-th round, then wk > wk+1 for all k ≤ K. When differences in production technology are too large, less productive firms can no longer compete with more productive ones and the labor market becomes essentially monopsonistic. To rule out this uninteresting case, we make the following assumption: Assumption 1. The difference in productivity is not too large, for all j   α(1 − β)β + (1 − α)β 2 ψ1 ≤ Ω(α, β) ≡ . max ψj α(1 − β)β + (1 − α)(1 − β)2 We can now further pin down firm’s hiring behavior and the order of hiring by the following lemmas. Lemma 3. Under Assumption 1, no firm hires any worker it receives an “l” signal. Lemma 4. In equilibrium, high productivity firms hire before low productivity firms. By Lemma 3 no firms would hire “l” workers. In addition, all active firms, perhaps except the least productive one among them, must make positive profit from each new hire. As a result, active firms would hire all “h” workers when it is their turn to hire. 19

We now consider how wages are determined in equilibrium. Note that because every firm hires all workers they receives an “h” signal, the pool of unemployed in the k-th round consists of αk = (1 − β)k−1 α/[(1 − β)k−1 α + β k−1 (1 − α)] proportion of high type workers and 1 − αk proportion of low type. In competing for the right to hire in the k-th round, firm k + 1, the most productive one among firms who have not hired, is willing to offer k wk+1 =

α(1 − β)k−1 βψk+1 . α(1 − β)k−1 β + (1 − α)β k−1 (1 − β)

Thus, any active firm who wins the right to hire in this round has to offer at least this k much, wk = wk+1 for all k.

In general, equilibrium wages are determined in this fashion: winning wage wk in the k-th round is such that the firm hiring in the k + 1 round would be indifferent between hiring in the k-th and k + 1-th round. The only possible exception is in the K is less than workers’ reservation wage, last round K, in which it is possible that wK+1

then the firm has to offer at least r. Thus,   α(1 − β)K−1 βψK+1 . wK = max r, α(1 − β)K−1 β + (1 − α)β K−1 (1 − β)

(2)

After firm K hires, it is no longer profitable for firm K + 1 to offer a wage at least r to the remaining workers that it receives an “h” signal, and the labor market closes. Therefore, number of active firms is determined by the following condition.

α(1 − β)K−1 βψK α(1 − β)K βψK+1 ≥ r > . α(1 − β)K−1 β + (1 − α)β K−1 (1 − β) α(1 − β)K β + (1 − α)β K (1 − β)

(3)

Proposition 3. Under Assumption 1, there exists a unique equilibrium such that: 1. There are K active firms, with K determined by the condition (3); 2. Firm k (≤ K) with ψk hires in the k-th round; 3. In any round k ≤ K, firm k hires only workers with a signal “h” but none with a signal “l”; 20

4. Wages decrease in the round of hiring k, i.e., wk < wk−1 for all k ≤ K − 1; 5. Active firms make positive profits, and their expected profits decrease in the round of hiring k. Note that while in general active firms expect positive profit and hire every worker with a signal “h”, in the special case of r=

α(1 − β)K−1 βψK , α(1 − β)K−1 β + (1 − α)β K−1 (1 − β)

(4)

firm K makes zero profit and may hire any portion of workers it observes an “h” signal. When firms differ in productivity, Proposition 3 demonstrates that there is a unique equilibrium that exhibits both wage dispersion and positive sorting. More productive firms pay higher wage, hire earlier and thus, hire workers that are more productive on average. Meanwhile, workers of the same ability will be paid differently at different firms. Of course, the unique equilibrium result depends crucially on the assumption that no two firms have the same productivity ψ. If some firms have the same ψ, then we are back to the case of homogeneous firms and there will be multiple equilibria again, with the hiring order indeterminate between the identical firms. On the other hand, the main feature of the current model - - wage dispersion and positive sorting - remains. Thus, the predictions of a more general cherry-picking model will be similar as the heterogeneous model and consistent with the empirical findings.

6

Discussion While on-the-job search models show that large firms pay more to obtain a larger

workforce, the current work demonstrates that firms may pay more to attract better workers. The empirical implications of the two are also different. The Burdett and Mortensen (1998) model has no prediction on the quality of workers at firms of 21

different sizes. But our model predicts that firms paying higher wages also hire better workers, which is consistent with the empirical evidence on positive assortative matching (e.g., Brown and Medoff (1989), de Melo (2008)). Moreover, our prediction that high ability workers are more likely to be employed in high wage firms/industries supports the conjecture of Gibbons and Katz (1992) that “the traits that help a worker find employment in a high-wage industry once are likely to do so again.” Note that directed search models such as Shimer (2005) can also generate wage differentials and positive sorting.7 On the other hand, because of the restriction that each firm has one vacancy only, the directed search model can not explain the sizewage premium puzzle. Further, it can not explain wage differentials and positive sorting in markets where coordination friction does not seem to be an issue, for example, the academic job market, where job candidates can apply to any hiring institutions. But wage differences and positive sorting clearly exist in such markets. The current work is also related to Grossman (2004), who demonstrates that the combination of imperfect contracting with national differences in the distribution of worker talents can be an independent source of comparative advantage, and lead to trade in two otherwise identical countries. Grossman has also observed that realworld labor contract is rarely efficient because of two factors. First, prospective employers usually do not observe a worker’s ability at the time of hiring and thus can not offer contracts contingent on ability. Moreover, contracts that link workers’ pay to performances are restricted because an objective and verifiable performance measure may not exist. Like Grossman (2004), we believe imperfect information is an important imperfection in labor markets, in particular, markets for skilled workers whose ability levels 7

Shimer (2005) considers a model with complementarity and coordination friction to explain the

coexistence of unemployed workers and unfilled job vacancies. In particular, firm productivity and worker productivity are complementary, which produces sorting. Additionally, since workers can apply to one firm only, with further restriction that they can not coordinate their applications, workers randomly apply to firms, which generates wage dispersion.

22

are heterogeneous and hard to judge by potential employers (see also Lazear (1984)). For example, in the academic job market for junior economists, both search and coordination frictions do not seem to be an issue, as job candidates know the potential offers from different academic institutions, and can apply to any employers at a nominal cost. On the other hand, imperfect information about applicants’ quality seems to be important. The assumption of imperfect information is certainly not new. In fact, our basic intuition goes back at least to Alfred Marshall who “ recognized that workers were frequently not paid on the basis of tasks performed. One of the reasons for this is the inability to observe the tasks perfectly – either the inputs or outputs.” (Stiglitz (2000)). The current work falls in a large literature, initiated by Weiss (1980), on wage and employment determination in frictionless markets where worker productivity is not observable to employers. One interesting strand of this literature allows firms to get partial information about worker productivity either by testing workers with some error (see Guasch and Weiss (1981)) or by learning over time (see Greenwald (1986)). In addition to wage differentials and positive sorting, imperfect information about worker productivity also provides a basis for alternative explanation of other labor market puzzle. In a classical and influential paper, Kasper (1967) has reported that unemployed workers are willing to reduce their asking wage, and asking wage declines over the course of unemployment at a rate of about 0.3 per cent per month. With a sequential search model, Gronau (1971) has tried to explain the empirical finding by declining return to search as workers approach their retirement age. Yet, as Mortensen (1986, pp. 859) points out, given the relatively short duration of unemployment spells, declining return to search is unlikely to cause the relative large rate of decrease in reservation wage that Kasper and others have reported for relatively young workers. But an imperfect information model where both employers and workers learn about worker type from the duration of a worker’s employment history is perfectly consistent with this piece of evidence (see also Lockwood (1991)). 23

7

Conclusion Information plays an important role in many labor markets. With perfect informa-

tion, the law of one price must hold, with workers of identical productive capabilities being paid equal wages. However, in real world, employers often only have imperfect knowledge about workers’ productivity, and employment contracts linking a worker’s compensation to performance may not be possible due to the unavailability or unverifiability of performance measures. As a result, competition does not ensure that workers are paid according to their talents. Workers of different abilities can be paid the same wage, while workers of the same ability can be paid different wages. In this paper we formalize this intuition by proposing a “cherry-picking” model in an environment with imperfect information. In this environment, firms/industries compete by offering higher wages to attract better workers. This mechanism leads to both positive sorting and wage differentials. Our model is highly stylized and meant to be so. We assume that firms’ production technology is constant returns to scale and the product market is characterized by constant marginal revenue. One may extend our analyses to model monopolistic competition in product market jointly with the labor market. However, we believe the “cherry-picking” mechanism would still produce wage differentials and sorting in a more elaborate model. As well, there is no turn-over of workers between firms. But there are extensions that would generate turn-over. For example, one alternative assumption on learning could be that incumbent firm as well as outside firms observe a worker’s type with certain probability every period, as in Lazear (1984). Such a model would generate turn-over when a worker’s ability is observed by outside firms but not by the incumbent firm. However, as the main purpose of this paper is to provide a simple model to show how information imperfection can lead to persistent wage differentials in competitive labor markets, such an extension is best left for future research.

24

Appendix A Proof of Lemma 1. We prove this result using contradiction. Suppose in round k, there are n firms posting the same highest wage w ≥ r, and also let αk be the proportion of high type workers in the pool of unemployed before this round starts. First, we show that it is not optimal for a winning firm to make offer only to workers with an “l” signal, i.e, if firm i makes offer to an “l” signal worker, then it must also make offer to all “h” signal workers. Note that because the private signals are independent, workers can be divided into 2n−1 groups, each with the same evaluation from the n − 1 firms other than firm i. For example, the best group only consists of workers that receive “h” evaluations from all the n − 1 firms. Among the 2n−1 groups, let’s consider a generic group j, with average productivity αkj . Because β > 1/2, within group j, the average productivity of workers from whom firm i receives an “l” signal equals

αkj (1−β) αkj (1−β)+(1−αkj )β

< αkj , whereas the average produc-

tivity of those from whom firm i receives an “h” is

αkj β αkj β+(1−αkj )(1−β)

> αkj . Because all

workers in the same group must receive the same offers from the other n − 1 firms, and workers having multiple offers randomly choose one to accept, regardless of how the other firms make offers, it is always more profitable for firm i to offer to a worker with an “h” signal than to offer to one with an “l” signal.8 Next, since firms are ex ante identical, every firm that posts w in the round must expect the same total profit. Let Πki ≥ 0 be the profit of firm i. We consider two cases. First, suppose w is low enough such that it is profitable for firms to make offers to workers with “l” signal as well, indicating Πki > 0. But in this case, it is profitable for firm i to offer w + ǫ to every unemployed worker, which would bring in a total profit of nΠki for firm i. Second, consider the case when w is so high that it is unprofitable to make an 8

In group j, the average productivity of workers with “h” signal and accepting firm i’s offers is

always strictly greater than αkj , while the productivity of those with an “l” signal and accepting i’s offer is strictly less than αkj . The same argument holds for all groups, and even if any of the other firms randomly make offers to workers conditional on the private signals it observes.

25

offer to workers with an“l” signal and only workers with an “h” signal receive offer(s). In this case, average productivity of workers who receive multiple offers is strictly greater than the average productivity of workers who accept firm i’s offers. To see this, note that the average productivity of workers firm i observes a signal “h” equals Hkh =

αk β , αk β + (1 − αk )(1 − β)

but the average productivity of those who has m offers (with an offer from firm i as well as the other m − 1 firms) is equal to Hkmh =

αk β m > Hkh . αk β m + (1 − αk )(1 − β)m

Because only 1/m of those workers would accept firm i’s offer, the average productivity of those not accepting firm i’s offer is higher than the average productivity of those taking its offer. By offering w + ǫ to all workers with a signal “h,” firm i can attract the more productive workers at a slightly higher per worker wage cost to increase its total profit. This concludes the proof of this result.

Lemma A1. When firms are ex ante identical, firms make zero profit in equilibrium. In each round, the firm that posts the highest wage hires only workers it observes an “h” signal. Proof. To prove the first part, we use contradiction. Suppose that in a round k, the firm hiring in that round, denoted as firm k, posts the highest wage wk . For firm k to make a positive profit, it has to be true that wk < Hkh , where Hkh is the average productivity of workers with a signal “h”. If this were the case, profit maximization ensures that firm k hires every worker with a signal “h”. In addition, for the other firms not to undercut firm k, they must obtain the same expected profit as well in equilibrium. But this clearly can not be true. To see this, note that if in every round, the hiring firm earns a positive profit from workers with a signal “h”, it would, like firm k, hire all workers with an “h” signal. As a consequence, after k rounds, the proportion of high type workers in the remaining 26

pool of unemployed becomes αk+1 =

α(1−β)k . α(1−β)k +(1−α)β k

The best a firm can do is to select

the “h” signal workers from the remaining worker pool, with the average productivity equal to h Hk+1 =

α(1 − β)k β . α(1 − β)k β + (1 − α)β k (1 − β)

h Since Hk+1 goes to zero in the limit as k goes to ∞, for any r > 0, there exists a k such h that for all k ≥ k, Hk+1 < r. Firms hiring after the k-th round would suffer a loss

should they offer at least r to attract any worker. Hence, we conclude that firms make zero expected profit in equilibrium, and in any round, the winning wage is equal to Hkh . Since the expected productivity of a worker with a signal “l” is equals to Hkl =

αk (1−β) αk (1−β)+(1−αk )β

< αk < Hkh for all k, no firm

would hire any worker with a signal “l” at its posted wage wk = Hkh . This concludes the proof of this result. Proof of Lemma 2. We establish the first result by contradiction. Suppose there were one equilibrium in which the equilibrium wage offers for two consecutive rounds are such that wk ≤ wk+1 . Let the applicant pool for the k-th round consists of n ˜k measure of high type, m ˜ k measure of low type workers. Let firm j be the one that hires in the (k + 1)-round, facing n ˜ k+1 measure of high type and m ˜ k+1 measure of low type workers. Clearly n ˜ k+1 + m ˜ k+1 < n ˜k + m ˜ k and n ˜ k+1 /m ˜ k+1 < n ˜ k /m ˜ k because the firm that hires in round k proportionally select better workers from the pool. Thus firm j has a clear incentive to deviate by offering wk + ǫ and get the right to hire in round k, where it faces strictly better applicants pool (in terms of both total measure of workers and proportion of high productivity workers) and lower wages. Proof of Lemma 3. Consider firm i that hires in the k-th round, facing a pool of unemployed consisting of αk proportion of high type and 1 − αk proportion of low type. It is necessary that wk ≥

αk βψK+1 , αk β+(1−αk )(1−β)

where firm K +1 is an inactive firm; otherwise

firm K + 1 could offer wk + ε to those with a signal “h” to obtain a positive profit. In this case, however, it will not be profitable for any firm to hire workers with an

27

“l” signal, since the expected output of those workers is strictly less than the wage, αk βψK+1 αk (1 − β)ψi < ≤ wk . αk (1 − β) + (1 − αk )β αk β + (1 − αk )(1 − β)   ψi The first inequality holds because ψK+1 ≤ maxj ψψ1j ≤ Ω(α, β) < Ω(αk , β). Note

that for all k > 1, αk < α; the average productivity of the pool of unemployed deteriorates over time. Proof of Lemma 4. We prove the lemma using contradiction. Suppose there exists one equilibrium in which firm j with ψj hires in the k-th round and firm i with ψi > ψj

hires in the (k + 1)-th round. Also, the pool of unemployed in the k-th round consists n ˜ k measure of high type, m ˜ k measure of low type workers. Given the wage wk+1 , high enough to ensure the firm hiring in the (k + 2)-th round is indifferent between hiring in the (k + 1)-th and (k + 2)-th round, and in the case of k + 1 = K, to ensure the most productive inactive firm is indifferent between hiring in this round and stay inactive, firm i expects to get a profit of Πk+1 =n ˜ k (1 − β)βψi − [˜ nk (1 − β)β + m ˜ k β(1 − β)]wk+1 i Moreover, for firm i not to hire in the k-th round, firm j has to post a wage wk such that Πk+1 ≥ Πki = n ˜ k βψi − [˜ nk β + m ˜ k (1 − β)]wk . i

(A.1)

If this were the case, however, firm j would strictly prefer to hire in the (k + 1)th round at wage wk+1 , rather than in the k-th round at wk , contradicting this is an equilibrium. This is so as Πk+1 − Πkj = −˜ nk β 2 ψj + [˜ nk β + m ˜ k (1 − β)]wk − [˜ nk (1 − β)β + m ˜ k β(1 − β)]wk+1 , j which is strictly greater than zero if the inequality (A.1) holds. Thus, we conclude that firm i with ψi hires after firm j with ψj < ψi can not be part of any equilibrium. Proof of Proposition 3. The uniqueness of the equilibrium, if it exists, is guaranteed by the lemmas in the text. We still need to establish its existence. 28

Given other firms’ strategies, we show that there is no incentive for any firm k to deviate. First, firm k has no incentive to hire earlier than the k-th round. For any h ≥ 0, we have: k−h k−h−1 Πkk−h − Πkk−h−1 = (Πkk−h − Πk−h ) − (Πkk−h−1 − Πk−h )

= β(1 − β)k−h−1 α(ψk − ψk−h ) − β(1 − β)k−h−2 α(ψk − ψk−h ) > 0. Next, firm k has no incentives to hire later than the k-th round. For any h ≥ 0, wages wk+h and wk+h+1 are such that k+h k+h+1 Πkk+h − Πkk+h+1 = (Πkk+h − Πk+h+1 ) − (Πkk+h+1 − Πk+h+1 )

= β(1 − β)k+h−1 α(ψk − ψk+h+1 ) − β(1 − β)k+h α(ψk − ψk+h+1 ) > 0. Given our previous discussions, each round, firm k hires β(1 − β)k−1 α measure of high type workers and (1 − β)β k−1 (1 − α) measure of low type workers, and obtain a positive profit. The only exception is perhaps with firm K and in the case when condition (3) holds, when firm K expects zero profit from each new hire and thus, can hire δβ(1 − β)K−1 α high type workers and δ(1 − β)β K−1 (1 − α) low type workers with 0 < δ ≤ 1.

Appendix B Here we consider a simultaneous wage-posting model. Free entry implies that firms make zero profit in equilibrium. There is a continuum of workers of two types, α proportion of high type and 1−α proportion of low type. Firms make simultaneous job offers to workers they intend to hire at offered wages. Upon receiving offers, workers decide which offer to take. Workers always take the highest wage offer, provided that it is greater or equal to the reservation wage r. Production starts after the firmworker matching process is completed. We now show that no pure strategy equilibrium exists. First, Lemma 1 shows that no two firms offer the same wage in equilibrium. Second, since no two firms 29

offer the same wage and workers accept the highest offer, the equilibrium outcome must resemble that of the sequential wage posting model. That is, firms who post higher wages have the priority to select good workers. Let the firm who offer the highest wage (w1 ) and the second highest wage (w2 ) be denoted as firm 1 and firm 2, respectively. Clearly, firm 1 will send offer to those workers it labels as “h” (or a random subset), and hire workers with average productivity H1 =

αβ . αβ+(1−α)(1−β)

The

zero profit condition guarantees that w1 = H1 . Suppose firm 1 hires δ (0 < δ ≤ 1) proportion of workers it labels as “h”, then average productivity of workers hired by firm 2 will be H2 =

α(1−δβ)β . α(1−δβ)β+(1−α)[1−δ(1−β)](1−β)

Note that H2 < H1 and H2 = w2 .

However, in this case firm 1 could deviate by offering only w2 + ǫ and still be able to hire the same workers, thus make a positive profit. Now we give a simple example of two active firms in equilibrium. For simplicity we assume α, β are such that α(1 − β)β = r. α(1 − β)β + (1 − α)β(1 − β) This simple model is very close to the random sales model of Varian (1980). Therefore, some analyses are similar to Varian. There exists mixed strategy NE, with firms randomizing on the wages offered to workers they think are of high type. For firms ¯ where i = 1, 2, its mixed strategy is a CDF Fi (x) = prob(wi < x) in the support [w, w], ¯ has a positive probability to be set. Note identical firms implies that any wi ∈ [w, w] Fi (x) = F (x) for i = 1, 2. Proposition A1. f (w) = 0 for w < r or w > H1 . Proof. As workers have a reservation wage of r, no one will accept an offer less than r. This implies w = r. Meanwhile, as the expected productivity of a worker can not be greater than H1 , offering a wage wi > H1 is a strictly dominated strategy. Thus, w¯ ≤ H1 . On the other hand, w¯ can not be less than H1 either, as if this were the case, a firm could simply offer a wage wi ∈ (w, ¯ H1 ) and makes a positive profit. Hence, wi ∈ [r, H1 ]. 30

In the mixed strategy equilibrium, any wage wi ∈ [r, H1 ] brings the same ex ante expected profit, zero for firm i given the mixed strategies of the other firms F−i (·). That is, for all w ∈ [r, H1 ] which may be offered, P r(wi < w)[αβ − (αβ + (1 − α)(1 − β))w]+ P r(wi > w)[α(1 − β)β − (α(1 − β)β + (1 − α)β(1 − β))(w + ε)]. There is no point mass in the equilibrium wage strategies. Proposition A2. There are no point masses in the equilibrium wage strategies Proof. The proof is a revised version of Varian (1980, Proposition 3). Note that whenever some wage w were offered with positive probability, there would be a positive probability of a tie at w. In this case, a firm could profitably deviate by offering a different wage with positive probability. At first, we note H1 will never be offered with positive probability as it is the highest wage to be offered. If both firms offer H1 with a positive probability, then there would be a positive probability of a tie at H1 , which brings negative profits. Now suppose w < H1 is offered with positive probability. The number of points of positive mass in any probability distribution must be countable, so there is an arbitrarily small ε such that w + ε is offered with probability 0. In this case, if one firm i offer w + ε with the same positive probability with which w is offered in the equilibrium strategy, the increased profit for firm i would be (j 6= i) P r(wj < w + ε)[αβ − (αβ + (1 − α)(1 − β))(w + ε)]+ − P r(wj < w)[αβ − (αβ + (1 − α)(1 − β))w] + P r(wj > w + ε)[α(1 − β)β − (α(1 − β)β + (1 − α)β(1 − β))(w + ε)] − P r(wj > w)[α(1 − β)β − (α(1 − β)β + (1 − α)β(1 − β))w] + P r(wj = w, wi = w + ε)[αβ − (αβ + (1 − α)(1 − β))(w + ε)] − P r(wi = w, wi = w)[α − α(1 − β)2 − (α − α(1 − β)2 + (1 − α)(1 − β 2 ))w]/2

31

As ε approaches zero, the sum of the first four terms goes to zero, whereas the sum of the last two terms remains positive. This implies firm i can increase profit by offering w + ε for small ε, contradicting that offering w with positive probability is an equilibrium strategy. To see that the sum of the last two terms remains positive for small ε, we look at the limit when ε approaches zero. In this case, the sum equals P r(wj = w, wi = w)[αβ 2 − (αβ 2 + (1 − α)(1 − β)2 )w]. As αβ − (αβ + (1 − α)(1 − β))w ≥ 0 and β > 1 − β, the sum is strictly positive in the limit. Since there are no point masses in the equilibrium density, the cumulative distribution function will be continuous on (r, H1 ). Given F (w), the expected profit of firm i equals Z

H1

{πh F (w) + πl (1 − F (w))}dF (w), r

where πh = αβ −(αβ +(1−α)(1−β))w and πl = α(1−β)β −(α(1−β)β +(1−α)β(1−β))w. All wages that are offered with positive density must yield the same expected profit zero for the firm. Otherwise, the firm could profitably increase the frequency with which the more profitable wage were charged. This condition indicates πh F (w) + πl (1 − F (w) = 0. Simplification gives F (w) =

α(1 − β)β + (1 − α)β(1 − β))w − α(1 − β)β . αβ 2 − (αβ 2 + (1 − α)(1 − β)2 )w

Note that F (r) = 0, and F (H1 ) = 1. In equilibrium, both firms play mixed strategy, offering w following the distribution F (w). Since F (·) is a continuous distribution, the case of a tie can be ignored without loss of generality. In any equilibrium outcome, the two firms will offer different wages. As workers who receives offers from both firms choose the higher one to 32

accept, the firm offering higher wages gets to hire every one of its selected workers, while the other firm hires the rest. Hence, the outcome resembles that of the sequential wage posting model, with firms offering higher wages appearing to hire earlier than firms offering lower wages.

References Abowd, J. M., F. Kramarz, and D. N. Margolis (1999). High wage workers and high wage firms. Econometrica 67, 251–333. Beaudry, P. and J. DiNardo (1991). The effect of implicit contracts on the movement of wages over the business cycle: evidence from micro data. Journal of Political Economy 99, 665–6988. Bernard, A. B. and B. J. Jensen (1995). Exporters, jobs, and wages in US manufacturing: 1976-87. Brookings Papers on Economic Activity: Microeconomics 1995, 67–112. Bernard, A. B. and B. J. Jensen (1997). Exporters, skill upgrading and the wage gap. Journal of International Economics 42, 3–31. Bernard, A. B. and B. J. Jensen (2004). Why some firms export. Review of Economics and Statistics 86, 561–569. Brown, C. and J. Medoff (1989). The employer size-wage effect. Journal of Political Economy 97, 1027–1059. Burdett, K. and D. T. Mortensen (1998). Wage differentials, employer size, and unemployment. International Economic Review 39, 257–273. Cullen, D. E. (1956). The inter-industry wage structure, 1899-1950. American Economic Review 46, 353–369.

33

de Melo, R. L. (2008). Sorting in the labor market: Theory and measurement. working paper. Dickens, W. T. and L. F. Katz (1987a). Inter-industry wage differences and industry characteristics. In K. Lang and J. S. Leonard (Eds.), Unemployment and the Structure of Labor Markets, pp. 48–89. Basil Blackwell. Dickens, W. T. and L. F. Katz (1987b). Inter-industry wage differences and theories of wage determination. NBER working paper No. 2271. Feng, S. and B. Zheng (2010). Imperfect information, on-the-job training, and the employer size-wage puzzle: Theory and evidence. IZA working paper No. 4998. Freeman, S. (1977). Wage trends as performance displays productive potential: a model and application to academic early retirement. The Bell Journal of Economics 8, 419–443. Frias, J. A., D. S. Kaplan, and E. A. Verhoogen (2009). Exports and wage premia: Evidence from Mexican employer-employee data. Columbia University Working Paper. Gibbons, R. and L. F. Katz (1992). Does unmeasured ability explain inter-industry wage differentials? Review of Economic Studies 59(3), 515–535. Gibson, J. and S. Stillman (2009). Why do big firms pay higher wages? evidence from an international database. Review of Economics and Statistics 91(1), 213–218. Greenwald, B. C. (1986). Adverse selection in the labor market. Review of Economic Studies 53, 325–347. Gronau, R. (1971). Information and frictional unemployment. American Economic Review 61, 290–301. Grossman, G. M. (2004). The distribution of talent and the pattern and consequences of international trade. Journal of Political Economy 112(1), 209–239. 34

Guasch, L. J. and A. Weiss (1981). Self-selection in the labor market. American Economic Review 71, 275–284. Harris, M. and B. Holmstr¨om (1982). A theory of wage dynamics. Review of Economic Studies 49, 315–333. Idson, T. L. and D. J. Feaster (1990). A selectivity model of employer-size wage differentials. Journal of Labor Economics 8(1), 99–122. Kasper, H. (1967). The asking price of labor and the duration of unemployment. The Review of Economics and Statistics 49, 165–72. Krueger, A. B. and L. H. Summers (1988). Efficiency wages and the inter-industry wage structure. Econometrica 56, 259–293. Lazear, E. P. (1984). Raids and offer-matching. NBER working paper No. 1419. Levin, J. (2003). Relational Incentive Contracts. American Economic Review 93, 835– 57. Lockwood, B. (1991). Information externalities in the labour market and the duration of unemployment. Review of Economic Studies 58, 733–753. MacLeod, W. B. and D. Parent (1999). Jobs characteristics and the form of compensation. Research in Labor Economics 18, 177–242. McDonald, J. T. and C. Worswick (1999). Wages, implicit contracts, and business cycle: evidence from Canadian micro data. Journal of Political Economy 107, 884–892. Moore, H. L. (1911). Laws of Wages. Macmillan, New York, reprinted 1967. Mortensen, D. (1986). Job search and labor market analysis. In O. Ashenfelter and R. Laylard (Eds.), Handbook of Labor Economics, Volume 2, Chapter 15, pp. 849– 919. Elsevier Science B.V.

35

Murphy, K. R. and J. Cleveland (1991). Performance appraisal: an organizational perspective. Boston: Allyn and Bacon. Prendergast, C. (1999). The Provision of Incentives in Firms. Journal of Economic Literature 37, 7–63. Schank, T., C. Schnabel, and J. Wagner (2007). Do exporters really pay higher wages? first evidence from German linked employer-employee data. Journal of International Economics 72, 52–74. Shimer, R. (2005). The assignment of workers to jobs in an economy with coordination frictions. Journal of Political Economy 113(5), 996–1025. Slichter, S. (1950). Notes on the structure of wages. Review of Economics and Statistics 32, 80–91. Stiglitz, J. E. (2000). The contributions of the economics of information to twentieth century economics. Quarterly Journal of Economics 115(4), 1441–1478. Troske, K. R. (1999). Evidence on the employer size-wage premium from workerestablishment matched data. The Review of Economics and Statistics 81(1), 15–26. Varian, H. R. (1980). A model of sales. American Economic Review 70(4), 651–659. Velenchik, A. D. (1997). Government intervention, efficiency wages, and the employer size wage effect in Zimbabwe. Journal of Development Economics 53(2), 305–338. Weiss, A. (1980). Job queues and layoffs in labour markets with flexible wages. Journal of Political Economy 88, 526–538.

36