Cherry-picking in Labor Markets with Imperfect Information

Transcription

1 Cherry-picing in Labor Marets with Imperfect Information Shuaizhang Feng Bingyong Zheng April 13, 2009 Abstract We study a competitive labor maret with imperfect information. In our basic model, the labor maret consists of heterogeneous worers and ex ante identical firms who have imperfect information about worers productivities. Firms compete by posting wages each round to win the right to mae job offers, and tae turns to cherry-pic more productive worers. We also consider extensions of the model where firms differ in either productivity or information technology. The model predicts many important empirical regularities, including non-degenerated firm size distribution, persistent wage dispersions, positive sorting of firms and worers, and employer size-wage premium. The main insight of this paper is that identical worers can get different wages in a competitive maret when they are pooled with coworers of different average productivities. Keywords: Imperfect information, cherry-picing, wage dispersion, size-wage premium. JEL classification codes: D83; J31. We would lie to than Han Farber, Hanming Fang, Peter Gottschal, Lars Lefgren, Alan Krueger, Andrew Marder, David McAdams, Bruce Meyer, Jim Smith, Wing Suen, Cheng Wang, and Tao Zhu for comments. We also than participants at the Princeton IRS Lunch seminar. All errors are our own. Princeton University and IZA. shuaizha@princeton.edu. Shanghai University of Finance and Economics. bingyongzheng@gmail.com.

2 1 Introduction Larger firms pay more on average to worers of similar characteristics than smaller firms. A large number empirical studies have found the wage differentials to be statistically significant and practically large even after controlling for worers observable characteristics and unobservable heterogeneity. To put the magnitude of the size-wage premium into perspective, we quote Brown and Medoff (1989), if a typical worer went from an establishment with employment one standard deviation below average to an establishment with employment one standard deviation above average, the employee would enjoy a wage increase of 8-12 percent, about as large as the union-nonunion differential in these data. Why should worers of apparently same ability get different wages at similar jobs? Various theories have been advanced to explain the size-wage premium puzzle, with the search friction model (e.g. Burdett and Mortensen 1998) being the most commonly invoed explanation. Suppose ex ante identical worers do not now the wages offered by firms, but now the distribution of wage offers. At random time intervals, a worer receives information about new jobs. As jobs are identical apart from the wages associated with them, employed worers move from lower to higher paying jobs as the opportunity arises. Since worers are more liely to leave a low paying job but less liely to leave a high paying job, firms can offer higher wages to retain worers and also to attract new worers. In equilibrium, firms that offer higher wages earn lower profit per worer but hires more worers, while firms that offer lower wages earn higher profit per worer but hires fewer worers; thus yielding the same total profit for all firms. Hence, persistent wage differentials can exist as a result of search frictions. While search friction may explain part of what we observe in the labor maret, it can not be the whole story. The search friction model concentrates on search frictions, arising because worers do not now firms wage offers, which is not always plausible. Further, the search friction model predicts a positive relationship between firm size and wages offered, which is only one form of wage dispersions. But it does not explain other form of wage dispersions. In some labor marets, for example, the academic job maret, worers of similar ability can get different pay in different firms, but there is not necessary a positive relationship between firm sizes and wages. Finally, the search friction model predicts that unemployment decreases with the increase of minimum wages in a competitive maret. In this paper we focus on information imperfection instead of search friction. That is, the employer may not now the productivities of potential employees. We demonstrate this type of imperfection can result in wage dispersion as well. In many job marets, the employer is not sure of the productive capabilities of an individual worer at the time of hiring. To learn more about the worer s productivities, the employer can mae an assessment based on observable characteristics, such as the individual s sex, race, age, education, etc. But this assessment is rarely perfect. A productive worer may also try to inform the employer her true type through some costly signaling (e.g., Spence 1973). However, in a world where there are more types than the number of signals available for this means, signaling has only a limited use 2

3 and is not always effective at separating worers of various productive capabilities. 1 As a result, employers only have imperfect information of potential worers productivities when maing hiring decisions. We show that imperfect information about worer productivity may lead to persistent wage dispersions, including the employer size-wage premium in a competitive job maret. Our basic model considers a one period labor maret with infinite number of firms that are ex ante identical, and a continuum of worers who are either of high productivity type or of low productivity type. Because firms only have limited information about worers productivities, they mae job offers conditional on the (unbiased) expectations, or beliefs, of worer s type. Firms tae turns to cherrypic worers that they thin are of high productive capabilities. They compete for the right to hire each round by posting high wages, with the firm with the highest wage getting the right to hire in any round. In equilibrium, firms hire worers of different expected (average) productivities at different wages. Firms that hire in earlier rounds pay higher wages and draw disproportionately more high type worers. As the wage a firm pays equals the average productivity of its labor force, worers of the same productivity will be paid differently when they are pooled with coworers of different average productivities. Consequently, all equilibria of the model are characterized by persistent wage dispersion. When we restrict attention to equilibria in which firms hire all worers that they thin are of high type, there is a positive relationship between firm sizes and wages in equilibrium. That is, firms who post higher wages and thus hire earlier will also hire more worers. 2 This is so as they select from a better applicant pool in terms of average productivity. Our model predicts that productive worers can remain unemployed as firms can not tell them apart from other less productive ones. The equilibrium unemployment is increasing in minimum wage. This prediction is consistent with traditional wisdom and what we observe in real world. One drawbac of our basic model is perhaps the multiplicity of equilibria. As firms are ex ante identical, the order of hiring is not unique, as any firm can tae any position in the sequence of hiring. However, when firms are different in their respective productivities, there is a unique equilibrium with wage dispersion, as we demonstrate in Section 3. In this equilibrium, firms with higher productivities hire earlier, hire more worers that are of higher average productivities, offer higher wages, and mae more profits. Thus, there is a size-wage premium for worers conditional on type. Also, even though high productivity worers and firms are not perfectly matched, there exists positive sorting in the sense that the proportion of high productivity worers in the worforce is increasing in firm productivity. In another extension we consider a labor maret where firms have different in- 1 A necessary condition for signaling to be effective is the negative correlation relationship between productivities and signaling cost. Only under this condition, productive worers may be able to successfully separate themselves from unproductive worers through costly signaling. See Spence (1973) for detailed discussion. 2 In fact, this is true for all equilibria in which hiring firms hire the same portion of worers they thin as high type. 3

4 formation technology the ability to judge worer s true types. We show that when an equilbrium exists, firms with superior information technology always earn higher profits. However, the order of hiring depends on relative scarcity of high productivity worers. When the proportion of high type worer is sufficiently low, marginal benefits of cherry picing is high, and firms with better information technology will hire earlier. This results again a unique equilibrium with size-wage premium. When the proportion of high type worer is sufficiently high, marginal benefits of cherry picing is low, equilibrium may or may not exists in that case. We therefore have demonstrated that worers of same sills can get different pay, not because of search friction, but because they are pooled with coworers of different average productivities. In addition, we predict that wage differentials exists in any labor maret in which firms may have imperfect information about potential employees abilities, even if no clear size-wage relationship can be identified. Given firms always have some way to screen worers, is information imperfection really that important? We believe so, especially in some labor maret for silled worers. The academic job maret for new Ph.D.s is the best example, typically with job candidates nowing more about hiring departments while the recruiting institutions have difficulties judging a candidate s abilities. In our model, worer now the wages posted by firms but firms do not now the types of worers, an thus, a firm pays the average productivity of its labor force. This is the source of wage dispersion here. In the search friction model, firms have no difficulty judging worers productivity. However, worers do not now the wages offered by firms, and this friction leads to wage dispersion. In this sense, both the search friction model and our model are looing at some type of imperfect nowledge (or imperfect information) by maret participants in explaining persistent wage dispersions. As to the nature of imperfection, however, our model and the search friction model part company. This paper falls into a large literature on labor maret information that dates bac at least to Stigler (1962). Previously, Spence (1973) shows how high productivity job applicants could send costly signals to prospective employers in order to be separated from low productivity worers. However, signaling only plays a limited role in the labor maret because the available number of signals is far less than number of worer types, due to both individual heterogeneity and uncertainty in the production process. 3 More recently, Grossman (2004) demonstrates that the combination of imperfect information with national differences in the dispersion of worer talents can be an independent source of comparative advantage, and lead to trade in two otherwise identical countries. Grossman (2004) motivates imperfect information with team wor in production and labels this incomplete labor contract, as no contract can be written conditional on each individual worer s marginal product. This paper is also closely related to the statistical discrimination literature (for example, Coate and Loury 1993, Moro and Norman 2004), which focus on informational friction to explain group based inequality in labor maret. In their models, worers 3 Also, a necessary condition for signaling to be effective is the negative correlation relationship between productivities and signaling cost. 4

5 sills are unobservable, but employers may observe a noisy signals of sill. Thus, employer s inability to judge worer s type also plays a vital role in their models. However, there is main difference between our model and the statistical discrimination modes. There, the wage inequality is rationalized as a coordination failure, and thus, the inequality can be solved by policies aimed at reversing discrimination. Whereas in our model, the wage inequality among worers of similar sill is a result of pooling with coworers of different average productivities. Hence, remedy comes only when firms have better information technology in judging worers sills. The rest of the paper is structured as follows. Section 2 presents our basic model. Section 3 extends the model by introducing productivity heterogeneity. In Section 4, we consider an extension in which firms have different information technology. Section 5 discusses the relationship of this paper to some others in the literature. Section 6 concludes. 2 The basic model 2.1 Setup We consider a competitive labor maret with no barriers to entry and exit. Firms are ex ante identical, and have a constant return to scale production technology. They produce the same output that are sold in a competitive maret at price P which we normalize to 1. There is a continuum of worers of measure one, that are of either high or low type. High type and low type worers account for proportion α and 1 α of the population, respectively. A high type worer produces one unit of output if employed, while a low type worer produces nothing. The proportion of the two types and productivities of each type are common nowledge to all maret participants. However, an individual worer s type is unnown to all parties, including herself 4. Worers maximize their wages subject to a reservation wage r. There is no application cost and worers are free to apply to all hiring firms. When a worer applies to a firm, the firm maes a private assessment of the job applicant s type. With probability β H, a high type worer is taen as high type by the firm, thus labeled as h. With probability 1 β H, she is mistaenly recognized as low type, or labeled as l. Similarly, a low type worer is labeled l with probability β L and h with probability 1 β L. We assume all firms have the same ability to differentiate high type worers from low type ones, i.e., they have the same β H and β L. The private assessments of different firms are independent, 5 thus a worer recognized as low type by one firm may be taen as high type by another. 4 An alternative assumption is that worers now their own types, but are prohibited from sending signal to firms. Our model thus departs from the signaling literature but resembles statistical discrimination models such as Coate and Loury (1993). 5 Our main results do not change as long as these assessments are not perfectly correlated. Assuming independence simplifies our analyses, though. 5

6 For simplicity, we let β H = β L = β. 6 When β = 1/2, firms have no ability to differentiate high type worers from low type worers. When β = 1, firms can perfectly distinguish high type from low type worers. When 1 > β > 1/2, firms have some but less than perfect ability to judge worer types, and this is the case we focus on in this paper. Because all individuals have the same reservation wage, firms offering w < r will not be able to hire anyone. Therefore, if the reservation wage is too high, no firm would hire in the maret. To eliminate this uninteresting case, throughout the paper we assume r < α, which implies that a firm could mae a profit by hiring all worers or a random subset at their reservation wage r. The job maret wors in a sequential manner, consisting of many rounds of auctions among hiring firms. Each round, firms that have not hired yet post wages they are willing to offer to at least some worers in the remaining pool of job applicants, and the one with the highest wage wins the right to hire. In case several firms tie at the highest wage, one firm is randomly chosen to hire in that round. The firm who wins then offers to hire selected worers from the remaining pool of applicants at the posted wage. We assume firms always mae offers to a positive measure of worers when it wins the right to hire. Upon receiving an offer from the firm, worers decide whether to accept the offer or not. After that, the game moves to the next round in which the same process is repeated for firms who have not won any round and for worers who remain unemployed. The game ends when no firm finds it profitable to post a wage that will be accepted. That is, firms are unwilling to hire any unemployed worer at a wage greater than their reservation wage. Production then starts, and firms who have hired a positive measure of worers realize their profits. Sequential wage posting leads to all standard perfect competition results in a frictionless setting. It therefore offers a basic framewor upon which maret frictions such as imperfect information can be built and studied. Also, many actual labor marets seem to display this sequential nature and clear upside-down. Tae the U.S. academic job maret as an example, it consists three segments: the preemptive maret, which is conducted prior to the Allied Social Science Association (ASSA) meetings and only participated by top candidates and departments; the primary maret, which holds interviews at the ASSA meetings and campus visits shortly after; and the secondary maret, which starts after the primary maret has begun to clear. Even for the primary maret where the majority of applicants and hiring departments participate, although applications are processed at roughly the same time, in reality, top programs usually get priority in hiring their ideal candidates. For example, less prestigious programs frequently give good job candidates a grace period of at least two wees to respond to their offers, so that the job candidates can contact their preferred departments. Often candidates are willing to give up less prestigious program s offer to wait for better offers when they have positive feedbacs from them (See Cawley 2006.). In this model, an equilibrium is characterized by the number of active firms K, unemployment rate u, and wage distribution F(w), such that the following conditions 6 None of our substantive results will change without this assumption. 6

7 are satisfied: 1. Worers maximize the wage income they receive, provided the wage offered is greater than their reservation wage r, 2. Firms mae wage offers to maximize profit, 3. Firms update beliefs about worers types using Bayes rule. 2.2 Main results To start, note that because firms are ex ante identical, they must mae the same level of total profit in any equilibrium. Otherwise, firms that mae lower level profit will mimic the behavior of those earning higher profit. A firm actually maes two choices in the recruiting process: to post a wage, and decide whom to hire at the posted wage if it wins the right to hire. From the firm s perspective, it faces two groups of worers, those labeled as h and those labeled as l. After winning the right to hire, the firm can hire in three different ways, to hire randomly irrespective of the signals received from worers, to hire only those it receives a signal h, or to hire only those with a signal l. Competition between firms, however, forces a firm to hire only those it thins are of high type. Claim 1. In any equilibrium, a firm that wins the right to hire at any round maes offers only to those it labels as h. Proof. See Appendix A. The intuition for this result is straightforward. As firms compete for the right to hire in a round, they have pushed the winning bid so high, such that hiring those l label worers would decrease profit given the posted wage. As a result, we observe firms tae turns cherry picing more productive worers in equilibrium. Claim 2. In any equilibrium, firms mae zero profit. Proof. See Appendix A. The above two claims suggest that firms expected profit from each worer hired is zero. Therefore, a firm does not have to hire every worer it thins as high type. This implies that there are multiple equilibria in this model. In an equilibrium, it is optimal for firm to hire any δ (0, 1] proportion of those from whom it labels as h at the posted wage. We summarize our results so far in the following proposition. Proposition 1. There exist multiple equilibria. In any equilibrium, firms pay worers their average productivities. Also, both average productivities and wages decline in, the order of hiring. 7

8 Proof. The results follow directly from Claim 1 and Claim 2. Let firm mae offers to δ proportion of those it receives good signals h, with 0 < δ 1. It suffices to show that H > H +1, as w = H and w +1 = H +1. Let the applicant pool firm hires from be R 1, which consists ñ measure of high worers and m measure of low type worers. The average productivity of worers at firm is H = ñ β ñ β + m (1 β). The applicant pool firm +1 will choose from consists of ñ +1 = (1 δ β)ñ measure of high type worers and m +1 = [1 δ (1 β)] m measure of low type worers. Hence, H +1 = (1 δ β)ñ β (1 δ β)ñ β + [1 δ (1 β)] m (1 β). Clearly, H > H +1 given that 1 > β > 1/2. It remains to show that there exists at least two active firms in the maret. This is guaranteed by the assumption that r < α. Note that the average productivity of worers with good signals for firm 2 is H 2 = (1 δ 1 β)αβ (1 δ 1 β)αβ + [1 δ 1 (1 β)](1 α)(1 β) > α, which is greater than the reservation r, so firm 2 will be active. We focus our attention to the case when firms hire the same portion of worers they thin are of high types, i.e., δ = δ for all. It turns out that for any δ (0, 1], the corresponding equilibria have the same properties. For simplicity, we assume that δ = 1, that is, firms hire every worer they thin as hight type. This is formally stated in the following assumption 7. Assumption 1. Firms treat observationally identical worers in the same way. That is, if a firm maes an offer to any worer, then it must also mae the same offer to other worers who have the same expected productivity from the firm s perspective. Proposition 2. Under Assumption 1, the equilibrium is characterized by employer size-wage premium. Firms that offer higher wages (both unconditionally and conditional on type) also hire more worers. Proof. Note this is just the special case with δ = 1 as considered in Proposition 1. In general, the average productivity of worers at firm is H = αβ(1 β) 1 αβ(1 β) 1 + (1 α)β 1 (1 β). The measure of worers firm hires equals N = α(1 β) 1 β + (1 α)β 1 (1 β). Clearly, both H and N decrease as increases. Hence, there is a positive relationship between firm size and wage. 7 In an environment when firms now worer types perfectly, Shimer (2005) similarly assumes that firms wage offers may be conditional on a worer s type but not on her individual identity. 8

9 Under Assumption 1, the equilibrium is unique up to renaming of the firms. Again, note that maing the alternative assumption, δ = δ < 1 for all, also results in equilibrium with size-wage effect, but with H and N formulated differently. After firms 1, 2,, 1, have hired, the measure of worers remains to be hired equals R = α(1 β) + (1 α)β, and the average productivity of the remaining worers will be α +1 = α(1 β) α(1 β) + (1 α)β = α[(1 β)/β] α[(1 β)/β] + (1 α). Note that both R and α +1 are strictly decreasing in. As more and more firms have hired, the proportion of high type worers in the remaining pool decreases, so does the measure of unemployed worers. At certain point, it becomes unprofitable for another firm to hire, as the expected benefit from hiring a worer the firm recognizes as high type drops below the reservation wage r firm needs to pay. This implies that the equilibrium number of firms K is determined by the condition H K r > H K+1. (1) Claim 3. In the equilibrium characterized in Proposition 2, there exists a finite number of active firms. Also, number of active firms K is decreasing in worers reservation wage r, while equilibrium unemployment rate is increasing in r. Proof. The first part of the claim follows immediately from the fact that H strictly decreases in, and that K is determined by the condition H K r > H K+1. To show the second part, we note that the measure of worers remain unemployed after round equals R = α(1 β) +(1 α)β. The measure of unemployed worers equals R K. As K decreases in r, R K increases in r. Hence, we conclude that the measure of unemployed worers, also the unemployment rate, strictly increases in the reservation wage r. In the equilibrium, there are α(1 β) K measure of unemployed high type worers and (1 α)β K measure of unemployed low type worers, both increasing with the reservation wage r. In addition, the proportion of high type worers among unemployed worers, α(1 β) K /[α(1 β) K + (1 α)β K ] decreases in K and thus, increases in the reservation wage r. As one interpretation of r could be the minimum wage, the prediction therefore implies that an increase in minimum wage results higher proportion of high productivity worers in the unemployed pool. 3 Heterogeneity in Firm s Productivity Because firms are identical in our basic model, no predictions can be made with respect to the order of hiring by each firm. In this section we consider an extension of the model where firms are ex ante different in terms of their productivities. For convenience, we index firms by the ran order of their productivities, thus firm i has productivity ψ i which rans ith among all firms. Worer type and firm productivity enter the production function multiplicatively. A high type worer hired by firm i produces ψ i units of output, while a low type worer still produces nothing. 9

10 Claim 4. In any equilibrium, active firms profits strictly increase in their production technology ψ i. For two firms, firm i with ψ i and firm j with ψ j, both of which hire a positive measure of worers in an equilibrium, ψ i > ψ j implies Π i > Π j. Proof. Suppose there is an equilibrium in which firm j hires in the -th round at w and employs n measure of high type worers and m measure of low type worers, and generate a profit of Π j = n ψ j ( n + m )w. Firm i could offer w + ǫ and hire in the -th round, mimic firm j s behavior and hire the same measure of high type and low type worers, as the two firms have the same β. Thus it must be that Π i Π i = n ψ i ( n + m )w > Π j. Claim 5. In any equilibrium, the wage offers by active firms strictly decrease. That is, if w is the wage offer for the -th round, then w > w +1 for any. Proof. We prove this result by contradiction. Suppose there were one equilibrium in which the equilibrium wage offers for two consecutive rounds are such that w w +1. Let the applicant pool for the -th round consists of ñ measure of high type, m measure of low type worers. Let firm j be the one that hires in the ( + 1)-round, facing ñ +1 measure of high type and m +1 measure of low type worers. Clearly ñ +1 + m +1 < ñ + m and ñ +1 / m +1 < ñ / m because the firm that hires in round proportionally select better worers from the pool. Thus firm j has an incentive to deviate by offering w + ǫ and get the right to hire in round, where it faces strictly better applicants pool (in terms of both total measure of worers and proportion of high productivity worers) and lower wages. Since firms have to pay at least the reservation wage r > 0, Claim 5 implies that there exists finite number of active firms, which we denote as K. In addition, Claim 4 implies that those active firms will be firms 1, 2,...,K. The next assumption is technical which rules out the possibility that some firms dominate the maret because their productivities are far more superior than others. Assumption 2. The production technology difference is not too large, specifically, { } ψi α(1 β)β + (1 α)β 2 max Ω(α,β) i<j α(1 β)β + (1 α)(1 β) 2. ψ j Claim 6. Under Assumption 2, in equilibrium, no active firms hire any worer they label as l when it is their turn to hire. Proof. Consider firm that hires in the -th round, facing an applicant pool with α proportion of high type and 1 α proportion of low type worers. It is necessary that α w βψ K+1 α β+(1 α, where firm K + 1 is an inactive firm, as otherwise it could offer a )(1 β) wage w + ε and realizes a positive profit. In this case, however, it will not be profitable for any firm to hire worers they label as l signal. This because the expected productivities of those l worers are strictly less than the wage. α (1 β)ψ α (1 β) + (1 α )β < α βψ K+1 α β + (1 α )(1 β) w. 10

11 ( ) The first inequality holds because ψ ψ ψ K+1 max i i<j ψ j < Ω(α,β) < Ω(α,β). Note that for any > 1 we have α < α as the average productivities of the applicant pool deteriorate over time. Claim 7. In any equilibrium, high productivity firms always hire before low productivity firms. Proof. Note first that Claim 4 ensures that all active firms except the last one firm K maes strictly positive profits. This combined with Claim 6 means that those firms will hire all h worers when it is their turn in order to maximize profit. We prove this claim by contradiction again. Suppose this is not the case, then there must exist at least one equilibrium in which one firm s productivity is greater than that of the firm that hires immediately before it. Without loss of generality, let firm j hires in the -th round and firm i hires in the + 1-th round, with ψ j < ψ i. The applicant pool facing firm j at the -th round consists of ñ measure of high type, m measure of low type worers. The expected profit for firm j is Π j = ñ βψ j [ñ β + m (1 β)]w. Suppose firm j wait and post w +1 + ǫ in the + 1-th round, its profit would be Π +1 j Meanwhile,firm i s profit is: = ñ (1 β)βψ j [ñ (1 β)β + m β(1 β)](w +1 + ǫ). Π +1 i = ñ (1 β)βψ i [ñ (1 β)β + m β(1 β)]w +1, and it could post w + ǫ and hire in the -th round, and get a profit of Π i = ñ βψ i [ñ β + m (1 β)](w + ǫ). Equilibrium requires that both firms be maximizing their profits, thus it has to be true that Π i Π +1 i and Π j Π +1 j. Substituting the above profit equations and combining the two inequalities leads to ψ j ψ i, which clearly violates our assumption that ψ j < ψ i. We now characterize the determination of K and the behavior of firm K. Because all firms hiring before K cherry pic the worers that they recognize as high type, the applicant pool left for the K-th round consists of (1 β) K 1 α high type worers and β K 1 (1 α) low type worers. This indicates that the number of active firms is determined by the condition α(1 β) K 1 βψ K α(1 β) K 1 β + (1 α)β K 1 (1 β) r > α(1 β) K βψ K+1 α(1 β) K β + (1 α)β K (1 β). (2) In any equilibrium, the wage offered at each round is such that the firm hires in the next round is indifferent between hiring in the current round and the next. For 11

12 example, in competing the right to hire in the K-th round, firm K + 1, who has the highest productivity among those not hiring, is willing to offer w K K+1 = α(1 β) K 1 βψ K+1 α(1 β) K 1 β + (1 α)β K 1 (1 β). Consequently, any active firm who wins the right to hire in this round has to offer at least this much. However, if wk+1 K happens to be less than worers reservation wage, then the firm has to offer at least r. Clearly, the firm does not have any incentives to offer more than the maximum of r and wk+1 K either. Hence, { } α(1 β) K 1 βψ K+1 w K = max r,. (3) α(1 β) K 1 β + (1 α)β K 1 (1 β) Firm K maes a strictly positive profit unless r = α(1 β) K 1 βψ K α(1 β) K 1 β + (1 α)β K 1 (1 β). (4) In that case, K could hire any positive proportion of worers it labels as h, otherwise it will hire them all. Similarly, profit maximization ensures that w in general is determined by the condition Π +1 = Π+1 +1 such that the firm hires in the + 1 round has no incentive to deviate to one round earlier. The following proposition proves formally that this is an equilibrium. Proposition 3. Under Assumption 2, there exists a unique equilibrium with the following properties: (a) There are K active firms determined by the condition (2). (b) Firms hire according to the ran order of their productivities. That is, firm with ψ hires in the -th round. (c) All active firms hire all worers they thin are of high type but none they thin are of low type. The last hiring firm K, though, hire any proportion of worers it considers as high type should condition (4) holds. (d) Wages decrease in each round. Further, they are recursively determined starting from w K in (3). In general w is determined such that firm + 1 is just indifferent between hiring in the +1-th round at wage w +1 and hiring in the -th round at wage w. (e) All active firms mae positive profits except firm K, which maes zero profit if Condition (4) happens to hold. Total profits decrease strictly with. (f) The equilibrium exhibits a positive relationship between firm size (the measure of worers) and wages. Proof. The uniqueness of such an equilibrium, if it exists, is guaranteed by the previous discussions and claims. Thus we only need to establish existence. 12

13 Given other firms strategies, we show that there is no incentive for any firm to deviate. First, firm has no incentive to hire earlier than the -th round. For any h 0, we have: Π h Π h 1 =(Π h Π h h ) (Π h 1 Π h 1 h ) = β(1 β) h 1 α(ψ ψ h ) β(1 β) h 2 α(ψ ψ h ) > 0 Next, firm has no incentives to hire later than the -th round. For any h 0, wages w +h and w +h+1 are such that Π +h Π +h+1 Π +h +h+1 ) (Π+h+1 Π +h+1 +h+1 ) = β(1 β) +h 1 α(ψ ψ +h+1 ) β(1 β) +h α(ψ ψ +h+1 ) > 0 =(Π +h Given our previous discussions, firm hires β(1 β) 1 α high type worers and (1 β)β 1 (1 α) low type worers. The only exception is that for firm K, if condition (4) holds, firm K maes zero profit, thus it could hire δβ(1 β) K 1 α high type worers and δ(1 β)β K 1 (1 α) low type worers in which 0 < δ 1. (f) simply follows from (c) and (d). 4 Heterogeneity in Firm s Information Technology In this section we extend our basic model in a different direction by assuming firms have different information technologies. Firms are now indexed by the ran order of their information technology, i.e, for two firms i and j, i < j implies that β i > β j. Also, there is imperfect information as for all i, β i (1/2, 1). Claim 8. In any equilibrium, no firm hires all remaining worers when it wins the right to hire. Proof. We show this by contradiction. Suppose firm j hires all worers at round at wage w, then there must be some firms end up being inactive. Also, it must be that w = α, the average productivities of the applicant pool, such that firm j only breas even. Otherwise an inactive firm could bid w + ǫ, hire all worers at round, and mae strictly positive profit. In this case, an inactive firm could deviate by offering w + ǫ at round. It could then hire all worers it labels as h and mae a strictly positive profit, as the average productivities of those worers will be greater than α. Proposition 4. In any equilibrium, profits of active firms strictly increase in their information technology. That is, for two firms i and j that both hire a positive measure of worers, β i > β j implies Π i > Π j. 13

14 Proof. Without loss of generality, suppose there is an equilibrium in which firm j hires in the -th round from an applicant pool with ñ measure of high type worers and m measure of low type worers, at a wage w. Suppose firm j hires δ > 0 proportion of worers it labels as h, and hires δ proportion of those it labels as l. Based on claim 8, it must be that profits from l worers are zero even when those worers are hired. Thus, firm j s total profit is As β i > β j, it can be shown that Π j = δ ñ β j δ [ñ β j + m (1 β j )]w 0. ñ β i [ñ β i + m (1 β i )]w > ñ β j [ñ β j + m (1 β j )]w 0. This is because ñ ñ w + m w 1 β j β j > ñ w > ñ w m w. Therefore, in equilibrium firm i could at least offer w at the -th round and hire all worers it labels as h. Hence Π i Π i = ñ β j [ñ β j + m (1 β j )]w > Π j, and firm i earns a higher profit than firm j in equilibrium. Claim 9. In any equilibrium, number of active firms K is finite. In addition, all active firms except the one with worst information technology among them (which is firm K) hire all worers they label as h. No active firm hires any worer it labels as l. Proof. Following proposition 4, all active firms except the one with lowest β must earn strictly positive profit. Thus to maximize profit, it is necessary for them to hire all h worers given the wage offers. This ensures that worer average productivity strictly deteriorates each round (except for maybe when firm K hires). As reservation wage r is positive, number of active firms K must be finite. To see why firms don t hire l worers, note that in any round it is necessary that wage w is greater than the average productivity of all worers α, otherwise an inactive firm can enter by offering w + ǫ and earn a positive profit by hiring all worers. As the average productivity of worers the firm recognizes as low type is strictly less than α, it is not profitable for the firm to hire any one of them at w. Claim 10. Wage offers strictly decrease in any equilibrium. That is, for any consecutive rounds and + 1, w > w +1. Proof. We prove this by contradiction. Suppose w w +1. Let the applicant pool for the -th round consists of ñ measure of high type and m measure of low type worers. Let firms i and j hire in the -th and ( +1)-th rounds, respectively. Firm j s profit from hiring in the ( + 1)-th round equals Π +1 j = ñ (1 β i )β j [ñ (1 β i )β j + m β i (1 β j )]w +1. As w w +1, it can shown that firm j would deviate to hire in the -th round at w + ǫ and earn a higher level of profit. Π j = ñ β j [ñ β j + m (1 β j ](w + ǫ) > Π +1 j. 14

15 This is because Π j Π+1 j β j ñ β i [ñ β i + m (1 β i ) 1 β j β j ]w > Π i > 0. Hence, there exists no equilibrium such that w w +1. As in the previous section, equilibrium wage offers are determined in a bacward direction. Given Proposition 4, we now that if any equilibrium exists, then it must be firms 1, 2,...,K that are active in the maret, but we are still not sure about the order of hiring. Let the firm who hires last be firm j 1, 2,...,K, then { α[ K wj K =1, j = max r, (1 β } )]β K+1 α[ K =1, j (1 β )]β K+1 + (1 α)[ K =1, j β, ](1 β K+1 ) As firm j has to pay at least the maximum of reservation r and the highest possible offer that firm K + 1 (which has the best information technology among all inactive firms) can mae to brea even. Similarly, the equilibrium wage offered for the (K 1)-th round will be such that the firm that hires in the K-th round is indifferent between hiring in the K-th round and the (K 1)-th round. In general, we now that for a firm to win the right to hire in the -th round, it has to offer a wage no less than the maximum wage firms who hire after the -th round are willing to offer for the -th round. In order to determine the order of hiring in each round, we have the following results: Claim 11. At any round, let the measure of high type and low type worers be ñ and m, respectively. Suppose in equilibrium firm i and j hire at round and + 1, respectively. If ñ < (>) m, then it must be that β i > (<)β j. Proof. See Appendix B. In general, firms remaining in the maret mae a decision regarding whether to bid the highest wage and win the right to hire at the current round, or wait until the next round. Waiting lowers the wage the firm has to pay, but also results an applicant pool with lower average productivity. As the previous claim suggest, the trade-off to some extent depends on the composition of high type and low type worers. In what follows, we separately discusses the two cases when there are at least as many low type worers as high type ones in the maret initially, α 1/2, and when there are more high type worers, α > 1/ Low type worers less than or equal to high type worers When α < 1/2, there are relatively few high type worers in the pool. Thus securing a better applicant pool becomes the main concern for firms, and those with informational advantage will bid aggressively for the right to hire earlier. As a result, firms with larger βs hire earlier than firms with smaller ones. Thus the order of hiring can be uniquely determined. The following proposition summarizes the results. 15

16 Proposition 5. When α < 1/2, there exists a unique equilibrium in which firms hire in the order of information technology. Firms 1, 2,...,K are active in the maret, where K is determined by the condition Proof. See Appendix B. α K 1 (1 β i)β K α K 1 (1 β i)β K + (1 α) K 1 β i(1 β K ) r > (5) α K (1 β i)β K+1 α K (1 β i)β K+1 + (1 α) K β i(1 β K+1 ) Definition 1. The distribution of information technology is not too dispersed if for any two consecutively raned firms i,i+1 with β i and β i+1 respectively, (1 β i )/(1 β i+1 ) β i. Claim 12. When α < 1/2, if the distribution of information technology is not too dispersed, then there exists a positive relationship between firm size and wage. Proof. See Appendix B. When α = 1/2, hiring in the order of information technology (1, 2,...K) is still an equilibrium. However, in certain cases there also exists another equilibrium in which firm 2 hires first, i.e, firms hire in the order (2, 1,...K). Claim 13. When α = 1/2, there exists no equilibrium in which the hiring order is other than (1, 2,...K) or (2, 1,...K). Proof. See Appendix B. Claim 14. When α = 1/2, in the case of K > 2, the number of active firms K is determined by the condition in (5). Hiring in the order of (1, 2,...K) is always an equilibrium. In addition, under certain conditions, there is another equilibrium in which firms hire in the order (2, 1,...K). Proof. See Appendix B. 4.2 More high type worers When α > 1/2, there are relatively abundant high type worers initially in the applicant pool. Thus, firm 1 may find it profitable to wait until later rounds to hire, as suggested by claim 11. While waiting to later rounds costs the firm in terms of the average quality of applicants, the benefit of paying a lower wage may more than offset the cost. Proposition 6. When 1/2 < α β 2, there exists a unique equilibrium in which firms hire in the order of (2, 1,...K), where the number of active firms K is determined by the condition in (5). 16

17 Proof. See Appendix B. In this case, firm 1 is willing to let firm 2 hire first. Because firm 2 s ability in selecting the good candidates is not as good as firm 1 s, firm 1 finds the wage w 2 a bargain price given the quality of the remaining pool. when α > β 2, we consider cases with different K. To start, when K = 1, obviously only firm 1 is active in the maret. This happens under condition that { α(1 β r > 1 )β 2 r, αβ 1 αβ 1 +(1 α)(1 β 1 ) αβ 1 αβ 1 +(1 α)(1 β 1 ). Firm 1 pay a wage of w α(1 β 1 )β 2 +(1 α)β 1 (1 β 2 ) 1 = max When K = 2, both firm 1 and 2 are active in the maret. Based on claim 11, the only possible hiring order is that firm 2 hires before firm 1, then α(1 β 2 )β 1 α(1 β 2 )β 1 + (1 α)β 2 (1 β 1 ) r > α(1 β 1 )(1 β 2 )β 3 α(1 β 1 )(1 β 2 )β 3 + (1 α)β 1 β 2 (1 β 3 ) firm 1 maes a profit of { w 2 = max r, α(1 β 2 )β 3 α(1 β 2 )β 3 + (1 α)β 2 (1 β 3 ) }, Π 2 1 = α(1 β 2 )β 1 [α(1 β 2 )β 1 + (1 α)β 2 (1 β 1 )]w 2, Note that w 1 is determined by the condition Π 1 1 = Π 2 1, where Π 1 1 = αβ 1 [αβ 1 + (1 α)(1 β 1 )]w 1. Thus w 1 = αβ 1β 2 + [α(1 β 2 )β 1 + (1 α)β 2 (1 β 1 )]w 2, αβ 1 + (1 α)(1 β 1 ) Π 1 2 = αβ 2 [αβ 2 + (1 α)(1 β 2 )]w 1, Π 2 2 = α(1 β 1 )β 2 [α(1 β 1 )β 2 + (1 α)β 1 (1 β 2 )]w 2, We can chec that Π 1 2 > Π 2 2, thus as long as the condition Π holds, three exists an equilibrium in which firm 2 hires before firm 1. Otherwise no equilibrium exists. When K 3, no equilibrium exist. This is summarized by the following claim. Proposition 7. When α > β 2, no equilibrium exists if the number of active firms K 3, in which K is determined by condition (5). Proof. See Appendix B. In this case, no order of hiring is stable; no equilibrium exists. However, the significance of this result should not be overemphasized. Note that α(1 β 2 ) > (1 α)β 2 if and only if β 2 < α, which requires the proportion of high type worers in the maret to be high while at the same time the information technology β of firms other than firm 1 not very accurate. Hence, this case is at most a rarity rather than a norm. }. 17

18 5 Related Literature Our paper is closely related to the vast literature of labor maret search and matching. Although these models sometimes generate similar implications as we do, the types of frictions under consideration are quite different. Random search models typically assume that worers do not fully now potential jobs, thus have to either wait for a time period or incur a direct cost to sample from the pool of job offers. In the onthe-job search model of Burdett and Mortensen (1998), worers search randomly and gradually move from low paying jobs to high paying jobs. Identical firms offer different wages in the equilibrium, as those who offer higher wages attract more worers at the expense of enjoying lower per worer profit. Thus the model also generates size-wage premium in the equilibrium, a well-established but puzzling empirical regularity in labor maret 8. However, their underlying intuition is quite different than ours. Because worers are homogeneous in their model, large firms pay more to eep a larger worforce. In our model, large firms are willing to pay more because they want to select better worers from a heterogeneous population of job candidates. The directed search literature restricts the number of jobs a worer can apply at a time, thus creates a coordination problem among fellow job seeers. In Shimer (2005), heterogeneous worers and firms interact in a static environment. The model generates wage dispersion as well as positive sorting of worers and firms. However, it does not predict size-wage premium as a firm is just characterized by one job vacancy. On the other hand, Shimer (2005) captures the coexistence of unemployment and job vacancies, which is beyond the scope of this paper. Shimer (2005) also consider more general production functions. Similarly, Acemoglu and Shimer (2000) study how labor maret coordination friction also induce ex ante firms to adopt different technologies. Albrecht et al. (2006) allow worers to mae multiple (but still fixed number of) applications. Many macroeconomic models of labor maret rely on an aggregate matching function without specifying the exact source of maret frictions (see Pissarides 2000). The micro-foundations of a well-performed matching function includes random search (urn-ball model), coordination friction, the stoc-flow analysis of Coles and Smith (1998), and model of Lagos (2000) which considers optimal decisions across spatially distinct locations. Our paper also falls into the large literature on labor maret information that dates bac at least to Stigler (1962), who studies information about jobs for worers and initiated the whole search literature. Spence (1973) shows how high productivity job applicants could send costly signals to prospective employers in order to be 8 Moore (1911) first documented this phenomenon for worers at Italian textile mills. For studies in U.S., see Lester (1967), Personic and Barsy (1982), Mellow (1982), Brown and Medoff (1989), Brown et al. (1990), Idson and Feaster (1990), Trose (1999). For studies in U.K. and other European countries, see Main and Reilly (1993), Green et al. (1996), Abowd et al. (1999), Albæ et al. (1998), Winter-Ebmer and Zweimüller (1999). Similar findings are also reported in developing countries, see e.g. Velenchi (1997), Schaffner (1998), and Söderbom et al. (2005). The seminal paper of Brown and Medoff (1989) examine a variety of potential explanations but still find sizable wage differentials unexplained. 18

19 separated from low productivity worers. Similarly, the statistical discrimination literature (for example, Coate and Loury 1993, Moro and Norman 2004) analyzes how personal characteristics such as race and gender can be used to form conditional expectations in labor marets with imperfection information, and lead to self-fulfilling vicious cycles. More recently, Grossman (2004) demonstrates that the combination of imperfect information with national differences in the dispersion of worer talents can be an independent source of comparative advantage, and lead to trade in two otherwise identical countries. Grossman (2004) motivates imperfect information with team wor in production and labels this incomplete labor contract, as no contract can be written conditional on each individual worer s marginal product. Recently, there is also a growing literature exploring the role of private information in the framewor of search. This includes Guerrieri (2008), Faig and Jerez (2005), and Guerrieri et al. (2008). These models combine search (coordination) friction with imperfect information, thus are different from our model. Their predictions are also different from ours. 6 Conclusions Information plays a very important role in many job maret. With perfect information, the law of one price must hold, with worers of identical productive capabilities being paid equal wages. However, real world labor contract are seldom perfect as employer may have imperfect nowledge of worers productivities. In this case, competing force is not enough to ensure worers are paid according to their talents. Worers of different abilities can be paid the same wage, while worers of the same ability can be paid differently. In this paper we exploit this intuition and analyze a model of cherry picing by firms that have imperfect nowledge of worers productivities. The cherry-picing process by firms most resembles the junior faculty recruiting process at the North American job maret for economists. We show that the combination of competition and imperfect information will generate wage dispersion. In addition, there is a positive relationship between firm size and wage. Therefore, our model has offered an alternative way of understanding the puzzling size-wage premium phenomenon in the labor maret. Although we consider firms in a given industry in our model, the analysis can be extended to understand inter-industry wage structures (See e.g. Krueger and Summers 1988). Treating one industry as a firm, if some industries have higher productivity, then they will draw from the worer pool first, but only imperfectly. Thus, controlling for worer heterogeneity, there will be a industry wage premium not explained under competitive setting. Note that this ind of industry wage differentials are hard to be rationalized under alternative model such as Burdett and Mortensen (1998) and Shimer (2005). 19