Optimal coexistence of long-term and short-term contracts in labor Markets

Optimal Coexistence of Long-term and Short-term contracts in Labor Markets∗ Inés Macho-Stadler†David Pérez-Castrillo‡Nicolás Porteiro§ May 5, 2011 Abstract We consider a market where firms hire workers to run their projects and such projects differ in profitability. At any period, each firm needs two workers to successfully run its project: a junior agent, with no specific skills, and a senior worker, whose effort is not verifiable. Senior workers differ in ability and their competence is revealed after they have worked as juniors in the market. We study the length of the contractual relationships between firmsandworkersinanenvironmentwhere the matching between firms and workers is the result of market interaction. We show that, despite in a one-firm-one-worker set-up long-term contracts are the optimal choice for firms, market forces often induce firms to use short-term contracts. Unless the market only consists of firms with very profitable projects, firms operating highly profitable projects offer short-term contracts to ensure the service of high-ability workers and those with less lucrative projects also use short-term con- ∗We are grateful to the participants at the seminar at CREST (Paris, 2011) for their insightful comments. Financial support from Ministerio de Ciencia y Tecnología (ECO2008-04321 and ECO2009— 07616), Generalitat de Catalunya (2009SGR-169), Junta de Andalucía (SEJ-02936 and SEJ-04992), Barcelona Graduate School of Economics and ICREA Academia is gratefully acknowledged. The first two authors are fellows of MOVE. †Universitat Autonoma de Barcelona - Barcelona GSE; Dept. Economía e Hist. Económica; Edificio B; 08193 Bellaterra - Barcelona; Spain. Email: [email protected]. ‡Corresponding author. Universitat Autonoma de Barcelona - Barcelona GSE; Dept. Economía e Hist. Económica; Edificio B; 08193 Bellaterra - Barcelona; Spain. Email: david.p[email protected]. §Department of Economics (Universidad Pablo Olavide). Email: [email protected]. 1 tracts to save on the junior workers’ wage. Intermediate firms may (or may not) hire workers through long-term contracts. JEL numbers: D86, C78 2 1 Introduction Partners may establish relationships that last for several periods. In job contracts, for instance, some firms hire the same worker for several years with a long-term contract, while others may prefer to sign contracts period by period, sometimes with the same worker, sometimes with different workers over time. One question that arises is when is it better for the employer to write a long-term contract that covers the whole length of the relationship and when is a short-term contract signed for a certain period of time superior, knowing that once that period is over the firm has to offer another short-term contract. The contributions by Lambert (1983), Rogerson (1985), Malcomson and Spinnewyn (1988), and Chiappori et al. (1994), among others, address the previous question in settings characterized by moral hazard where one firm (the principal) enters into a longlasting relationship with a worker (the agent). In these settings, if the firm and the agent can commit to a long-term contract, the firm can design a long-term agreement that dominates the sequence of optimal short-term contracts. In fact, long-term contracts can always replicate the sequence of the optimal short-term contracts while the reverse is, in general, not possible. This result is robust to a number of different specifications and implies that, if commitment on the part of the participants is possible, we should expect firms to use mainly long-term contracts in practice.1However, this is not the case and thefactthatdifferent firms in similar markets follow different time-duration contracts suggests that there may be other explanations beyond lack of commitment. In this paper we argue that not all the characteristics of the optimal contract between an employer and a worker can be deduced from the analysis of this relationship in a one-worker, one-firm setting. As the recent contributions by Dam and Pérez-Castrillo (2006), Serfes (2008), Terviö (2008) and Alonso-Paulí and Pérez-Castrillo (forthcoming) have shown, when the analysis is enlarged to take into consideration market interaction, then the form of the optimal contracts may substantially differ from the agreements that one obtains for a given relationship studied in isolation. When heterogenous principals 1When commitment on the agent side is difficult, contracts may include, for example, non-compete clauses under which the agent agrees not to pursue a similar profession or trade in a firm in the same industry if he breaks the contract. Contracts may also include other clauses that will reduce mobility by increasing the cost of hiring the worker by another firm. 3 compete for heterogenous agents, the identity of the partners in each relationship (in addition to the contract signed) is endogenous, as it is the level of utility obtained by the agents. Hence, even if short-term contracts are not optimal when we consider an isolated relationship, they may arise as part of the market equilibrium where some firms may choose long-term contracts while other hire workers only on short-term agreements. We consider a market where heterogenous firms hire workers to run their projects. Firms differ in the profitability of their project. At any period, each firm needs two types of workers: a junior agent, with no specific skills, and a senior experienced, worker whose expertise is crucial for the good development of the project. Every worker starts as a junior agent in the first period he works in a firm. At this point in their lives, all workers are identical. After the period as an apprentice, the worker becomes senior for the second, and final, period of his job career. In our model, the first period has a training component: workers acquire the knowledge and experience needed to run a project when seniors. We assume that this period gives human capital specific to the industry. This allows any worker that was hired (and trained) by a firmasajuniortorunaprojectasaseniorin any firm of the market.2 The model is dynamic not only because the relationships (may) involve several periods, but also because information about workers’ characteristics changes over time: after the agent has worked for a firm as a junior, the firms and the worker himself learn his competence as senior, which was ex-ante unknown for all market participants. Therefore, while all junior workers are indistinguishable, this is not the case for senior workers as not only their responsibilities (the project they work on), but also their abilities may be quite different. Consequently, we model a very simple technology of training that combines two dimensions of learning. First, there is learning because the innate ability of the worker is revealed through his training as a junior and this information becomes common knowledge for the industry. Second, there is a learning-by-doing component since working as junior is a prerequisite to later running a project as a senior. In this respect, the paper is related, though different, to the literature on on-the-job talent discovery. Terviö (2009) also presents a situation where workers’ innate ability is unknown for the market (and 2In other words, we do not deal with firm-specific training in the model. If there is some firm-specific ability, workers who change jobs in the second period of their lives lose their firm-specific human capital and just keep their general training in the industry. This will tend to decrease the profitability of shortterm contracts. 4 the workers themselves) until they actually work for a firm. However, in Terviö’s paper theobjectiveisdifferentasheisconcernedbythepossiblitythatmarketimperfections hamper the process of discovering talent. In the model, a moral hazard problem is present as senior workers’ effort or decision is not contractible. On the contrary, and just for simplicity, we consider juniors’ effort to be contractible. All the participants are risk neutral and they all have the capacity to commit to a long-term contract.3However, workers are protected by limited liability: their salary when junior and their salary when senior cannot be lower than certain thresholds. We characterize equilibria in this market, which accounts for the type of contract offered by each firm and the characteristics of these contracts. Our equilibrium concept is close to the idea of “stability” used in the matching literature that has analyzed contracts in environments where the matching between firms and workers is endogenous.4To be an equilibrium, an outcome (that is, a matching and a set of contracts) must be immune to deviations. In our environment, at equilibrium, it must be the case that a firm cannot make more profit by changing its strategy, that is, by offering contracts to workers that make both the firm and the workers better-offthan before. We first show that firms signing equilibrium long-term contracts offer low salaries to junior workers together with the promise of high reward when senior. This allows the firms to alleviate the incentive problem they face with senior agents, improving the efficiency of the relationship and also their profits. Since they commit to do so, these firms will keep the agents when senior, irrespective of their ability. Firms that sign short-term contracts hire junior agents with no promise of continuation. They also sign short-term contracts with senior workers (who may or may not be the same they hired the previous period as juniors); the terms of the agreement may depend on the workers’ ability level. We have already argued that the optimal long-term contract always (at least weakly) 3If no participant can commit to a long-term contract, then all must be short-term contracts. If the participants in one of the sides of the market, say the firms, can commit while the others cannot, then there can still be room for long-term contracts, but they are typically less efficient than in the environment with full commitment. In terms of the commitment possibilities, we place ourselves in the best scenario for the prevalence of long-term contracts. 4Stability and competitive equilibrium are very close concepts. Any stable outcome is also a competitive equilibrium and vice-versa. For (early) matching models where the parties decide on money instead of contracts see, for instance, the original contribution by Shapley and Shubik (1972), and the excellent review of the literature by Roth and Sotomayor (1990). 5 dominates any sequence of short-term contracts when the identity of the parties matched in a job contract is predetermined. However, short-term contracts can be beneficial for firms when the firm-worker matching is endogenous. Short-term contracts allow the firms to screen workers before putting them in charge of leading a project. This noncommitment strategy gives firms the freedom to focus on the particular type of senior worker that fits their needs. Therefore, firmsfaceatrade-offbetween choosing the optimal contract for a given match (long-term are superior to short-term agreements) and selecting a contract that allows a better selection (hiring high-ability senior workers is more important for some firms than for others). The market equilibrium depends on the characteristics of the set of firms and the set of workers. We solve the model for markets where there is a large proportion of lowproductivity (normal) senior workers and a small proportion of high-productivity senior workers (stars) and where these highly-talented workers really make a difference in the firmstheyworkfor. When only firms with very profitable projects exist in the market, all of them sign long-term contracts at equilibrium. Each firm offers the same agreement that it would offer if no market would have existed. More interestingly, we show that, except in this case where the market only consists of firms with very profitable projects, there is always asetoffirms that sign short-term contracts with their junior workers and specialize in a particular type of seniors. Depending on the value they attach to their projects, some firms always look for high-ability while others hire low-ability senior workers. Firms with highly profitable projects give a great deal of relevance to hiring high-ability senior agents to run their projects and, hence, they are willing to offer high wages to attract them. As a result, the expected utility of junior workers when they accept short-term contracts becomes higher because, if they turn out to be of high ability, they will obtain a high reward when senior. The expectation of this potential reward leads workers to accept, when junior, a low wage. Firms with relatively poor projects take advantage of this reduction in the wage of junior workers. These firms put more weight on the savings on juniors’ wages than to the fact that they end up contracting with a low-ability senior worker. Therefore, at equilibrium, the firms with the most profitable projects use shortterm contracts to ensure the services of high-ability workers while the firms with the least profitable projects use short-term contracts to save in the cost of hiring junior workers. 6 In this sense, the matching between firms and senior agents is positive assortative, for the sets that choose short-term contracts.5 The trade-offbetween the advantages of long-term and short-term contracts is often solved in favor of the use of long-term contracts for firms with intermediate projects. The likelihood of the coexistence of the two types of contracts is higher as the distribution of firms is more biased toward good projects, the discount rate is lower, the difference between the reservation utility of junior workers and the minimum salary is lower, the difference in performance between high and low-ability senior workers is higher, and the cost of the workers’ effort is lower. Workers receive part of the increased surplus created by the optimal sorting of senior workers promoted by the short-term agreements. Indeed, although all junior workers are identical and they perform identical tasks, those who sign short-term contracts expect a higher utility than those signing long-term contracts. Long-term agreements allow the firms to avoid the competition for the best workers, who obtain high salaries under shortterm equilibrium contracts. In our analysis, we focus on markets with a small proportion of very talented senior workersthatmakeadifference for the firms they work for and whose level of ability is public for all the firms inside the industry. Moreover, the human capital acquired by seniors is industry-specific and not just firm-specific. This model can provide a schematic version of the university job market. The performance of researchers during the first years after the completion of their Ph.D., that we can associate to their “ability”, is public information since it can be measured, for instance, by their publication record. In this job-market some universities offer Ph.D. graduates a tenure track position that guarantees tenure if, after the probationary period, the candidate satisfies some predetermined performance criteria (in terms of publications and other measures). The tenure-track system corresponds in our model to short-term contracts. Other universities sign tenure contracts from the very beginning and take the commitment of keeping the researcher independent of the outcome of further evaluation (even if contract conditions may indeed depend on performance). This corresponds to a long-term contract. Arts and sports are also examples of markets where the ability of seniors is well-known, 5See Legros and Newman (2007) for conditions under which monotone matchings emerge in environments where utility is not fully transferable. 7 as it is subject to public scrutiny through their performance and where this human capital is mainly industry-specific. Singers or soccer players, for instance, may sign exclusive contracts (with a studio, a record company, or a club) for a long period in which they are prevented from recording an album for another company or playing with another club. Other companies, however, choose to offer shorter contracts, particularly to young singers or players. The stars of these markets, a few individuals, attain prominence and success and their value and earnings are significantly greater than the earnings of the standard worker in the markets. The same can be said about surgeons or creatives in advertising. Finally, the market for upper executives also shares some similar features: these high executives are well-known within their industry and their contracts may (or may not) include special clauses aimed at preventing them from moving to another firm. To the best of our knowledge, ours is the first paper to study how the choice of the contractual length may be determined by market interaction. There are other papers that have studied the implications of different contractual arrangements but in widely different set-ups. Rice and Sen (2008) show how a reduction in the length of a contract can help to alleviate the moral hazard problem when explicit incentives cannot be included in the terms of the contract. In their paper, the optimal choice for the principal depends on the balance between more incentives for effort (short-term contracts) and lower wages (long-term contracts). The contribution by Ghosh and Waldman (2010) compares two contractual arrangements: up-or-stay vs. up-or-out contracts in a setting with multiple firms competing for a worker. The paper does not address the issue of endogenous matching since it studies the firms’ Bertrand competition in wages to attract the single worker available in the market. They show that up-or-out prevails when firm-specific human capital is low and when highand low-level jobs are similar. Otherwise, standard (up-or-stay) practices are optimal. Similar to our paper, Ghatak et al. (2001) study an overlapping generations version of a principal-agent problem where contracts are determined in general equilibrium. In their model, all young workers are identical but have different investment possibilities when senior, depending on their performance. They do not allow for long term contracts because their the authors’ concern is to explain the seniors’ decision between becoming entrepreneurs or remaining workers. In our paper, short-term contracts act as a form of probationary period that allows 8 firms and workers to achieve a better matching. It is, therefore, not a way in which firms try to test if the worker is good enough for the job, but rather it allows senior workers to be matched with those firms where they are more productive. In this sense, the shortterm contract serves as a sorting device. A related, but different, argument can be found in Loh (1994) where it is argued that introducing an employment probation can serve as a sorting device as it will induce self-selection by workers. Firms offering probationary employment will tend to attract workers who are more confident about their capabilities. Finally, the coexistence of fixed payment schemes and incentive-based payment schemes related to the characteristic of the workers is also present in a static adverse selection framework where firms compete for agents. Matutes et al. (1994) study the choice of compensation schemes by two firms that compete in a labor market where agents are heterogenous and they have private information about their type. They show that, in equilibrium, if firms are not too different in the eyes of workers, one firm offers a wage rate and the other offers a piece rate. By proposing different compensation schemes, firms induce self-selection among workers, which thereby decreases the intensity of competition in the labor market. The remainder of the paper is organized as follows. Section 2 presents the model. Sections 3, 4 and 5 analyze the candidate long-term and short-term contracts for equilibrium. Section 6 characterizes the identity of the firms and workers that enter into the relationship, the equilibrium salaries, as well as the contracts that emerge as a result of the market interaction. Finally, Section 7 concludes. 2Model We model the economy as an overlapping generation model where at each period ,with =12 ,firms contract with workers to develop projects. Firms are infinite-lived players and the set of firms is constant for all periods. On the other hand, workers (agents) live for two periods. Both, firms and workers discount the future according to the discount factor ,where∈(01). Allparticipantsareassumedtoberiskneutral.Wealsoassumethataworker,atany age, enjoys limited liability over income. This constraint implies that his wage in any period and contingency cannot be lower than a certain threshold . 9 …wJ wJ wJ wJ … (wS, S)(wS, S)(wS, S)(wS, S) t – 1 t t + 1 t + 2 … wJw’Jw’Jw’J… (wS, S)(wS, S)(w’S, ’S)(w’S, ’S)… …wJ wJ wJ wJ … (wS, S)(wS, S)(wS, S)(wS, S) t – 1 t t + 1 t + 2 … wJw’Jw’Jw’J… (wS, S)(wS, S)(w’S, ’S)(w’S, ’S)… Figure 1: A firm considers changing the contract The situation is similar if a firm which is currently offering the contract  decides to switch to a series of ST contracts: it firstchangesthecontractitoffers to the junior agent to be able to fully implement the new strategy in the subsequent period. Also, we face the same situation if a firm is currently offering ST contracts and plans to switch to LT contracts: it needs to change the agreement with the junior agent today but still needs to hire a senior agent through an ST contract to be able to fully implement the change tomorrow. Finally, when a firm switches from ST contracts to another stream of ST contracts in a period, it can do it immediately, without waiting till the subsequent period. Indeed, it can keep hiring junior agents under the same conditions as before (that is, under the lowest salary that the agent is ready to accept). Whether we compute the cost of the junior agent as 1 or isnotrelevantforthecomparisonofprofits in the two strategies, since the firm pays the same cost under both, the old and the new contracts. Therefore, we can develop the analysis of the (stationary) equilibria of our model by focusing on the profits firms make in one period, provided that we consider the cost of the junior agent as being generated the previous period, that is, as long as we associate a cost of 1 , instead of , to the junior agent. From now on, we will refer to this level of profits as “a firm’s one-period profits” and we will denote e=−1 +(−∆)−.Afirm hasincentivestoswitchfromcontractto contract 0if and only if e()e(0). 16 4 Long-term contracts in equilibrium Consider a firm that owns a project whose additional value in case of success is ∈£ ¤ and that signs LT contracts with junior workers. At each period ,thefirm runs the project withthejunioragentthatithiresatand with the senior worker that it hired at period −1. The senior agent has ability with probability and ability with probability 1−, as his ability was unknown at −1. As previously said, the ability of the agent is publicly known before he starts working as a senior; hence, the LT contract signed at −1may have payments contingent on the ability of the agent when senior.12 All workers are ex-ante identical and there are more junior workers than positions to fill. Therefore, at any period there are unemployed junior agents ready to accept any LT contract that provides them with an expected utility equal to their (two-period) outside utility +. Hence, the participation constraint (PC ) specifies that the total expected utility the worker obtains in the relationship be at least equal to +. Following the discussion of the previous section, a candidate LT contract for equilibrium ( ∆ ∆)maximizes the firm’s one-period profits, also taking into account the ICCs and the limited liability constraints (LLC), that is, it solves max (∆∆)−1 +((−∆)−)+(1−)((−∆)−) s.t. +£(∆+−()2)+(1−)(∆+−()2)¤≥+ =1 22∆,=1 22∆ ≥ ≥ ≥. If the contract does not satisfy the previous program, then the firm can deviate by offering adifferent acceptable LT agreement to junior agents and obtain larger discounted profits. We state the characteristics of the candidate LT contract in Proposition 2, where we denote e≡q2 +(1−)2 ,  1≡2 er1 (−)+−and  2≡4 er1 (−)+−. 12As will be clear later, this flexibility has no effect on the optimal contract. Therefore, at the candidate equilibrium contract, no third party needs to verify the ability of the agent. 17 Proposition 2 If firm is in the set R ,thenitoffers the following LT contract: Region  :If  1,then 13  ()=µ =  = =1 (−)+−1 422e2∆ =∆ =¶ Region  :If∈£ 1  2¤then  ()=Ã =  = = ∆ =∆ =2 er1 (−)+−! Region  :If  2then  ()=µ =  = = ∆ =∆ = 2¶ We now explain the main characteristics of the LT contract  (). Despite the absence of risk aversion, the moral hazard problem of the senior agent induces an inefficiency due to the presence of limited liability that restricts the capacity of the firm to induce theseniorworkertoexertahigheffort. Therefore, the firm is interested in relaxing the senior agent’s limited liability constraint, which explains why it concentrates as much as possible the agent’s payments in his second period of life (i.e., the firmpaystoayoung worker the minimum possible wage:  =.) Young agents accept contracts with a low payoffbecause of the credible promise to be “well” paid when they are senior. The limited liability constraints also explain why, unless is very low, workers are paid the minimum salary if the outcome turns out to be a failure:  = =. The impact of limited liability on bonuses and on payoffs obtained by agents and firms differs depending on the profitability of the project (as well as on the level of agents’ reservation utility +,costofeffort , and “average” probability of success e). Some characteristics are shown in Figure 2. For high values of (Region  ),theoptimalbonusdependsonlyonthevalueof the project. The firm shares half of the value in the event of success because it maximizes profits when the senior agent supplies effort  =1 42 for = . Given this bonus, the worker ends up with a utility larger than +(i.e., he obtains informational rents). 13In this region, there are other contracts that are also candidates for equilibrium. Any combination of ,and that satifies +(+(1−))+1 422e2=+andsuchthateach variable is higher than , is also a candidate as it would give the same profits to the firm. 18 R1LT R (PC) does not bind (LLC) binds 1/2 of FB efforts (PC) binds (LLC) binds efforts increase in UJ and US (PC) binds (LLC) does not bind FB efforts R R R/2       R2LT Figure 2: Incentives in the optimal LT contracts For intermediate values of (Region  ), the equilibrium payment scheme also depends on  ,and , as the participation constraint (together with the limited liability constraint) binds. Given that the firm needs to provide a level of utility of +, it gives it in terms of bonuses, which lead to a senior agent’s effort of  = q1 (−)+− for = . Finally, firmswithlow-valuedprojects(Region ) give all the project’s returns to the worker (they set ∆=)inexchangeforafixed payment (a franchise-type contract). Therefore, agents obtain their total outside utility +and they provide, when senior, the first-best level of effort  =1 22 for = . Next corollary provides the expression of the firm’s one-period profits for  (). Corollary 1 The firm’s one-period profits under  ()are: Region  :If  1,then e ()=−1 −+1 422e2 Region  :If∈£ 1  2¤then e ()=+1 er1 (−)+−−1 [2−(1 + )]−2 Region  :If  2then e ()=+2e2 82−(1 + )  19 The profit function e ()is continuously differentiable and convex in . 5 Short-term contracts in equilibrium All firms signing ST contracts hire similar young workers, as they are indistinguishable ex-ante. Concerning senior workers, they can decide to hire high-ability or low-ability workers. Consider an equilibrium where some firms sign ST contracts. A fraction of those firms offer contracts to high-ability senior agents. Denote by the (minimum) level of utility that this type of agent obtains at the equilibrium.14 Similarly, denote by the (minimum) level of utility received by low-ability senior workers. Both and need to be higher than or equal to Additionally, given the limited liability constraint and the competition among firms, and, possibly, can be strictly higher than Therefore, a junior agent is ready to sign an ST contract that provides a utility level lower than  as long as the reduction is not higher than the expected extra utility he will obtain when senior. Formally, the salary that the junior agent is ready to accept must satisfy: +[+(1−)]≥+, where we denote and the expected utility of a highand a low-ability worker. For example, if all the low-ability workers obtain the same in all the possible jobs, then =. The candidate equilibrium contract of firm in R(R) to a high- (low-) ability senior agent must be the optimal one-period contract for this agent, taking into account that it must grant him a level of utility of at least (); that is, it solves max (∆)+(−∆)− s.t. ∆+−()2≥ =1 22∆ ≥ 14Given the limited liability constraint, similar senior agents might obtain different utility levels at equilibrium. A firm with a very high ends up providing its senior agent a utility level higher than as its participation constraint will not be binding (see also, Alonso-Pauli and Pérez-Castrillo, forthcoming). 20 for = . Next proposition provides the candidate equilibrium contract for those firms, where we use the notation  1()≡2 p−and  2()≡4 p−. Proposition 3 If firm is in the set Rwith ∈{ },thenitoffers the following ST contract to a senior agent: Region  ():If  1(),then  ( )=µ =−1 422 2∆ =¶ Region  ():If∈£ 1()  2()¤then  ( )=µ = ∆ =2 p−¶ Region  ():If  2()then  ( )=µ = ∆ = 2¶ In Region  (), senior agent’s effort is the first-best level  =1 22 while in Region  ()his effort is lower than the first-best level:  =1 √− In these two regions, the agent’s expected utility is Finally, in Region  ()where the project is very valuable, the senior agent’s effort is  =1 42 for =  and he receives an informational rent. His expected utility in this region is +1 1622 2  Corollary 2 provides the expression of the firm’s one-period profits under  ( ), denoting  the equilibrium salary paid to junior agents. Corollary 2 Afirm in the set Rwith ∈{ }obtains the following one-period profits with  ( ) Region  ():If  1(),thene ¡   ¢=+1 422 2−−1   Region  ():If∈£ 1()  2()¤then e ¡   ¢=−2++1 √−−1   Region  ():If  2()then e (  )=−+1 822 2−1   The profit function e ¡   ¢is continuously differentiable and convex in . 21 6 Equilibrium matching and equilibrium contracts The previous sections identify the equilibrium contracts once we know the type of agreements firms offer (that is, once the sets R ,Rand Rare determined) and the levels of utility and that they must guarantee to lowand high-ability agents. In the present section, we characterize equilibria where at least some firms offer ST contracts. Therefore, we identify the distribution of firms in R ,Rand R, the levels and and the minimum salary that firms must offer to juniors under ST contracts. We look for equilibria where =. Low-ability workers do not have special skills and the firms will not compete for them.15 On the other hand, the level of will be determined by the equilibrium conditions, that is, by the (marginal) firm’s willingness to pay to attract a high-ability worker instead of either attracting a low-ability one, or signing an LT contract. We develop the analysis for markets where high-ability workers are not abundant but they make a difference for the firm they work for. That is, we consider environments with many “normal” workers and some “stars”. Assumption 1 reflects this idea, together with the reasonable hypothesis that the outside reservation utility of a senior agent is larger or equal to that of a junior worker (part (i)). Assumption 1 (ii) states that the proportion of high-ability agents is small enough. Finally, Assumption 1 (iii) reproduces the idea that the difference among the two types of agent is large enough. Assumption 1 The parameters satisfy the following conditions: (i) ≥, (ii)   1+2, (iii) ³ ´21+ 1   Why may some firms be interested in LT relationships while others prefer to secure high-ability agents through ST contracts? Even more, why would a firm choose a strategy that implies contracting low-ability agents through ST contracts, instead of offering LT contracts and, sometimes, benefiting from high-ability senior agents? The two main equilibrium variables that make firms prefer one or another type of contract are the 15However, at equilibrium the measure of senior workers with low ability is the same as the measure of firms looking for them. Therefore, other equilibria may exist where  for all low-ability players. 22 salary of a young worker  (or rather, the comparison between  and )andthe difference between the cost of a highversus a low-ability senior agent, that is, − The firms that obtain large profits in the event of success, that is, firms with a high , are ready to pay a high price to always hire a good senior agent given his added value in terms of increased probability of success. Therefore, firms at the right end of the interval £ ¤must be those most interested in signing ST contracts to hire high-ability senior agents. Similarly, firms that do not care much about agents’ effort, i.e., firms with a low , pay more attention to the potential savings they can make in a junior’s contract if they offer him an ST contract than to the gains obtained through an LT contract, or by securing a high-ability agent. Therefore, firms at the left end of £ ¤are the likely candidates to sign ST contracts to hire low-ability senior agents. Lemma1providesafirst confirmation of the previous intuitions. It compares the slopes, in terms of , of the profits obtained from the different types of contract. Lemma 1 Under Assumption 1, the slopes of the profit functions satisfy the following relations:16 (a)    ( )  (),forall ; (b)    ( )   (  ),forall ,andforall≥;and (c)   ()   (  ),forall ,andforall≥. Afirm’s ST profits increase with the value of success when it hires a low-ability worker. However, this increase is smaller than that of a firm’s profits under the optimal LT contract (part (a)). It is also smaller than the rate at which its profits increase if it hires high-ability workers through ST contracts (part (b)). A higher implies a larger interest in securing the services of a high-ability worker, which explains the previous relations. A similar argument gives the intuition of part (c) in the lemma. Letusdenotebythe value that would “balance” the set of firms if all the firms with  would hire low-ability workers while all the firms with ≥would hire high-ability workers, that is, is characterized by () 1−()≡1−  16Lemma 1 (a) and 1 (b) do not depend on Assumption 1. However, if Assumption 1 does not hold, then Lemma 1 (c) may fail if  ≡h2 +(1−)2  2 i£1 (−)+−¤+. 23 Also, we denote b the value that makes the firm indifferent between using LT contracts and hiring low-ability senior workers through ST contracts, when the junior salary is  =,thatis, b is characterized by e (b )=e (b  ). As we check in Claim 1in the proof of Theorem 1, under Assumption 1 firm b lies in regions  and  ()Therefore, we can easily calculate b :b ≡2 (2 −2 )p−. We first consider the case where b ∈[ )that is, some of the firms in the market have a low-valued project, but there is a relatively high number of firms with valuable projects. Theorem 1 Suppose ≤b  , and denote  the firm such that ³b ´= (1 −)(). Then, under Assumption 1, an equilibrium exists where (i) firms with ≤b offer ST contracts: to junior workers and  ( )to lowability senior workers, (ii) firms with ∈³b  ´offer the LT contracts  (), (iii) firms with ≥ offer ST contracts: to junior workers and  (  )to high-ability senior workers, where  is such that e ()=e ( ), (iv) junior workers accept both LT contracts that guarantee them +and ST contracts with  =,and (v) senior workers accept contracts that guarantee them .17 When is high enough, that is, the population of firms is not concentrated on low levels of  then, at equilibrium, firms are divided according to three hiring strategies. Firms with low-valued projects use ST contracts and only hire low-ability seniors; firms with a high also use ST contracts but they only hire high-ability seniors; and firms with intermediary s use LT contracts. The rationale behind Theorem 1 is the following. Firms with more profitable projects give more importance to hiring the high-ability worker, and they offer more to attract them. This increases the expected utility of a junior worker when he accepts the ST contract: if he turns out to be of high ability he will obtain a large utility level. The 17At equilibrium, high-ability workers receive a level of utility of, at least,   .However,outof equilibrium, they should be ready to accept lower offers, as long as they guarantee . 24 R ˆR Ro o )( ~RE LT   S ST LUwRE ,, ~  R o R   oo H ST HHwRE ,, ~  ShortTer m low-ability workers LongTer m ShortTer m high-ability workers Figure 3: Profit functions at equilibrium expectation of this potential reward leads workers to accept a wage =when junior which is under their reservation utility because they will be compensated in the future (in expected terms) for this sacrifice. Firms with low take advantage of this reduction inthewagethatcanbeoffered to junior workers who sign ST contracts: their value of the project is low enough so that the reduction in the wage of junior agents more than compensates the fact that they always end up hiring low-ability senior workers. Given the difference in equilibrium salaries between highand low-ability senior workers, firms with intermediary do not perceive a large difference between hiring one type or another. Therefore, it is better for them to profit from the additional improvement in efficiency due to the commitment they make through LT contracts. Figure 3draws the LT and ST profits, as a function of , for the equilibrium values for salaries and utility . As shown in Lemma 1, the slope of e (  )is always higher than that of e ()which in turn is higher than the slope of e ( ). At equilibrium, the market price that a firm has to pay in order to attract a high-ability worker ( ), is such that the three profit functions cross as shown in Figure 3. Itisworthnotingthateventhoughalljuniorworkersareidenticalwhentheysigntheir equilibrium contracts and they perform identical jobs, their expected utility is different 25 up receiving high remuneration when ST contract are in place which, in turn, allows the reduction of the payment to juniors, who foresee the prospects of a very high wage when seniors. Consequently, some firms having less lucrative ventures20 may not be able to retain the high-talented workers, but they indirectly profit from the existence of such workersasitallowsthemtohirejuniorsatamuchlowercost. At equilibrium, we often find that two types of firms use short-term contracts: firms in which the success of the project depends very much on the senior’s effort, which always end up hiring high-ability senior workers; and firms whose profits do not depend too much on the effort, which hire low-ability senior workers. Intermediate firms may use long-term or short-term contracts, depending on several market characteristics. We show that coexistence of both types of contract is more likely when there is a relevant fraction of firms with profitable projects, when the reservation utility of young workers is low and the minimum wage is high, when the discount rate is small, when there is a large difference between the productivity of highand low-ability workers, and when the agents’ effort is not too costly. In addition to the equilibrium with short-term contracts that often exists, there always exists an equilibrium where all firms choose a long-term contracts (see Proposition 1). However, we argue that, in our environment, whenever the equilibrium with short-term contracts and the one with only LT contracts coexist, the former is more “robust” or “sensible” as the latter is a “knife-edge” result. The full long-term outcome is sustained by the fact that, since no other firm is choosing a short-term contract, no firm can profit from the enhanced flexibility that short-term contracts offer. A small amount of firms with low-valued projects and another with high-valued projects have incentives to switch from LT to ST agreements to obtain higher profits. Appendix A Proof of Proposition 1 Proof. We firstnotethat,inasituationwhereallfirms sign LT contracts, if a firm follows the strategy of offering ST contracts to its workers, it necessarily hires as senior 20Firms where the role of the senior is less important for the outcome. 32 agentatperiodthe same agent that it hired as a junior at period −1.Also,theonly alternative occupation for the senior agent is to get out of the market, since no other firm is interested in hiring him, independent on his ability. Then, any sequence of ST contracts can be replicated as an LT contract. Therefore, the optimal ST contracts cannot give higher profits than the optimal LT contracts. B Proof of Proposition 2 Proof. Substituting and by their value and multiplying the objective function by ,thefirm’s program can be rewritten as: max (∆∆) + µ1 222 ∆(−∆)−¶+(1 −)µ1 222 ∆(−∆)−¶− s.t. + µ1 422 ∆2 +¶+(1 −)µ1 222 ∆2 +¶≥+(1) ≥ ≥ ≥. Let  ,and be the Lagrange multipliers corresponding to the constraints. The Kuhn-Tucker (first-order) conditions of the above maximization problem include the constraints, and the non-negativity of the multipliers: ≥0,≥0,≥0,≥0The derivatives of the Lagrangian with respect to ∆and ∆are:  1 222 (−2∆)+ 1 222 ∆=0 (2) (1 −)1 222 (−2∆)+ (1 −)1 222 ∆=0(3) which imply that ∆=∆, which we denote ∆in the rest of the proof. The derivatives of the Lagrangian with respect to ,and are: −1++=0 − + +=0(4) −(1 −)+(1 −)+=0 which imply =1−,= (1 −)and =(1 −)(1−); therefore, either the three constraints are binding or none is. The last Kuhn-Tucker conditions are: ∙+∙µ+1 422 ∆2 ¶+(1−)µ+1 422 ∆2 ¶¸−(+)¸=0 33 (−)=0 (−)=0 (−)=0 From (2) and (4) we can deduce that: =2− ∆and = µ ∆−1¶ We study the different regions where the Kuhn-Tucker conditions may be satisfied: Case /1:0,0,0,0Payments when young and in case of failure are ===and the bonus in case of success is ∆=2 q1 [(+)−(1 + )]. Finally, this is a candidate only if ∈[01], i.e., ∈"2 er1 [(+)−(1 + )]4 er1 [(+)−(1 + )]# Case 2 :=00 0,0.Then===,and∆= 2.In this case the participation constraint holds only if ≥4 q1 [(+)−(1 + )] (The candidate at the lower bound of this case coincides with the candidate at the higher bound of Case 1.) Case 3:===0.Then=1and ∆=. We write the participation constraint as +(+(1−))+1 422e2=+ Any combination of ,and that satisfies the previous constraint and such that the three values are larger or equal to constitutes an optimal solution (in particular, the values proposed in the proposition). This can be the case only if +(+(1−))≥ (1 + ),thatis≤2 q1 [(+)−(1 + )]. Theuniquecandidateforeachvalueofis the optimal solution of the firm’s maximization program. From the optimal contract in each case, it is immediate to compute agent’s effort(s) and utility, and firm’s profits. Additionally, easy calculations show that the function  ()is continuously differentiable in . 34 C Proof of Proposition 3 Proof. Substituting by its value in the firm’s program, we can rewrite it as max (∆)+1 222 ∆(−∆)− s.t. 1 422 ∆2 +≥ ≥. Let   be the Lagrange multipliers corresponding to the constraints. The Kuhn-Tucker (first-order) conditions of the above maximization problem include the constraints, and the non-negativity of the multipliers: ≥0,≥0The derivatives of the Lagrangian with respect to and ∆are −1++=0(5) 1 222 (−2∆)+1 222 ∆=0(6) From (5) and (6) we can deduce that: =2− ∆ and = ∆−1 We study the different regions. Case /1:0,0Payment are =and ∆=2 √−. This is a candidate only if ≥0and ≥0,i.e.,∈h2 √− 4 √−i Case 2 :=00.Then=,and∆= 2. In this case the participation constraint holds only if ≥4 √− Case 3:=0.Then∆=, which implies =10. The participation constraint is 1 422 2+=Therefore, =−1 422 2≥if and only if ≤2 √− Theuniquecandidateforeachvalueofis the optimal solution of the firm’s maximization program. From the optimal contract in each case, it is immediate to compute agent’s effort(s) and utility. D Proof of Lemma 1 Proof. We highlight that the three derivatives that we consider in the lemma,    ( ),   (),and   (  ), have a similar shape: they are first linear in until they 35 reach some 1(either  1(),or 1,or 1()), then they are constant until they reach a second threshold 2and, from 2on, they are linear in again. The proof of the three parts in the lemma is similar. We write a complete proof of part (a) and we point out the main elements of parts (b) and (c). (a) First, notice that if lies in both regions  ()and  ,   =1 222  1 22e2=  The same comparison holds if lies in both regions  ()and  . Additionally, if lies in both regions  ()and  ,   =1 p−  1 eq1 (−)+−=   Second, if  1≥ 1()(and  2≥ 2()), then   is increasing in a larger region of parameters than    before becoming constant (at a higher level than    in region  ()). Finally, even if    starts increasing again (i.e., it reaches region  ())before  (because  2()≤ 2), it is always lower than the latter, since it is lower even when = 2, given that we have seen that      for any which lies in both regions  ()and  . Third, suppose  1  1()(and  2  2()). Given that    is smaller than   when    reaches the region where it becomes constant, and that it is certainly also smaller when it starts increasing again (because   has reached this region before), it is not possible that the two derivatives cross. Therefore,      for any 0. (b) If lies in both regions  ()and  (),   =1 222  1 222 =    The same comparison holds in regions  ()and  (). Also, if lies in both regions  ()and  (),   =1 p−  1 √−=    Therestoftheproofisidenticaltotheoneinpart(a). (c) For in both regions  and  ()(and similarly in  and  ()),   =1 22e2 1 222 =   .Iflies in both regions  and  (), then   =1 eq1 (−)+−  1 √−=   if and only if  2 2 £1 (−)+−¤+. If this inequality holds, the rest of the proof of Lemma (c) is identical to the one in part (a). A sufficient condition is ≥e2 2 ∙1 (−)+−¸+(7) which, given Assumption 1 (i), is implied by (2 −e2)e2,i.e.,(1 −)(2 −2 ) 2 +(1−)2 ,or,((1 −)−)2  2  (1+)(1−)Assumption 1 (ii) implies that (1− 36 )−0Therefore, given Assumption 1 (iii), the inequality holds if ((1 −)−)³1+ 1  ´≥ (1 + )(1 −),i.e.,(1 −)−≥0which closes the proofs. E Proof of Theorem 1 Proof. We do the proof through a series of claims. Claim 1:If =,then b   1()and b   1. ProofofClaim1:Ifthevalueb that satisfies e (b )=e (b  )lies in both regions  and  ()(i.e., b   1()and b   1), then b =2 (2 −2 )p− Moreover, it is easy to check that each of the inequalities b   1()and b   1is equivalent to the following: 2 (−)¡2 −2 ¢(−)(8) Given Assumption 1 (i), (8) is implied by Assumption 1 (ii). Claim 2: Consider the value b such that e ³b   ´=e ³b    =b ´. If junior workers anticipate that they will obtain at least b when senior if they turn out to be high-ability, then they are ready to accept  =. ProofofClaim2: We proceed as follows. We conjecture that b is such that b   1³b ´=2 qb − we will compute the corresponding b in this region, and then we will show that it is indeed the case that b   1³b ´. Therefore, b is defined by −1  −+³ 2´2b 2=−1  −b +³ 2´2b 2 i.e., b =+¡1 2¢2(2 −2 )b 2or b =+1  (−)For this value,  1³b ´= 2 q+¡1 2¢2(2 −2 )b 2−.Therefore,b   1³b ´holds if and only if b 2 ³2 ´2h+¡1 2¢2(2 −2 )b 2−i, i.e., 2 b 2(2)2(−), which is equivalent to (8). Finally, given b , and taking into account that ≥b and ≥, a junior worker is ready to accept an ST contract with  whenever  ++  ¡1 2¢2(2 −2 )b 2≥+, that is, when  ≥. Claim 3:e ¡  =¢e ¡  = ¢for any ≥b and for any  b . 37 ProofofClaim3:Giventhedefinition of b in Claim 3,e ³b   =´≥ e ³b   = ´for any ≥b . Then, the claim follows after Lemma 1 (b). Claim 4:e ¡  =¢max ©e ¡  = ¢e ()ªfor any  b . ProofofClaim4:Thefirst inequality follows after Claim 3,alsotakingintoaccountthat  b implies  b . The second inequality follows the definition of b and Lemma 1(a). Claim 5:e ()≥max ©e ¡  = ¢e ¡  =¢ªfor any ∈hb  i. ProofofClaim5:Thefirst part of the inequality follows after the characterization of   in part (vi) of the theorem, by the property that  b and Lemma 1 (c). The second part follows the definition of b and Lemma 1 (a). Claim 6:e ¡  = ¢max ©e ()e ¡  =¢ªfor any  . ProofofClaim6: By the same argument as in Claim 5, the maximum of the two terms inside the maximization is e (). Then, the inequality is implied by the characterization of  in part (vi) of the theorem, by the property that  b and Lemma 1 (c). FProofofTheorem2 Proof. Given that the behavior of the workers is optimal by construction, we prove the theorem if we show that firms’ strategies are optimal. We do it through a series of claims. Claim 1: ≤− ¡ 2¢2(2 −2 ) ProofofClaim1.Giventhat≥ and ≥,−(+(1−)−)≤ − ¡ 2¢2(2 −2 )Moreover ≤− ¡ 2¢2(2 −2 )because this inequality is equivalent to ≤b . Claim 2:≤ 1( ) ProofofClaim2:≤2 q+¡ 2¢2(2 −2 )−if and only if ≤2 p−=  1()which is implied by the fact that ≤b and b ≤ 1()(by Claim 1in the proof of Theorem 1). Claim 3: ¡   ¢= ¡  ¢≥ () 38 ProofofClaim3.Giventhat≤ 1()and ≤ 1( )the first equality comes directly from the definition of  To prove the inequality, we notice that ≤ 1 because ≤b and b ≤ 1(by Claim 1in the proof of Theorem 1). Given that ≤ 1and ≤ 1()the inequality can be written as − +¡ 2¢2≥−− 1 (−)+³ 2´2 By Claim 1,asufficient condition is −³− ¡ 2¢2(2 −2 )´+ ¡ 2¢2≥−−1 (−)+³ 2´2This inequality holds because it is equivalent to ≤b  Claim 4:e ¡  ¢max ©e (   )e ()ªfor any   ProofofClaim4: It follows from Claim 3and Lemma 1 (a) and (b). Claim 5:e (   )≥max ©e ¡  ¢e ()ªfor any ≥ ProofofClaim5: It follows from Claim 3and Lemma 1 (b) and (c). G Proof of Theorem 3 Proof. We recall that b is characterized by e (b )=e (b  ).Ifb ,then b for all ∈£ ¤. Therefore, Lemma 1 (b) implies e ()e ( ) for all ∈£ ¤. It easily follows that e ()e (  )for all ∈£ ¤, ≥and ≥. Therefore, at equilibrium, no ST contract can be signed, since it would imply that some firms choose the strategy of keeping low-ability senior workers, which is dominated by the strategy of always offering LT contracts. H Proof of Theorem 4 Proof. The proofs of theorems 1 and 2 and that of Lemma 1 only use Assumption 1 to show that the inequalities (7) and (8) hold. Therefore, we prove theorem 4 if we show that Assumption 2 also imply (7) and (8). We write Assumption 2 as  ¡2 −2 ¢(−) 2 (−) and (1 −)¡2 −2 ¢(−)e2(−). The first inequality corresponds to (8). Moreover, it is easy to check that the second inequality also corresponds to (7) (with strict inequality). 39 References [1] Alonso-Paulí, E., Pérez-Castrillo, D. (forthcoming), Codes of Best Practice in Competitive Markets for Managers. 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