Avoiding the Cost of a Bad Hire: On the Optimality of Pay to Quit Programs

Transcription

1 Avoiding the Cost of a Bad Hire: On the Optimality of Pay to Quit Programs Dana Foarta Stanford GSB Takuo Sugaya Stanford GSB November 18, 2017 Abstract Contracts that compensate workers if they choose to leave an organization or Pay to Quit programs are becoming increasingly prevalent in both established and new companies. In this paper, we address the puzzle of why such employment contracts are offered. We propose a model to study the optimal employment contract when firms face both an adverse selection problem they need to find a good fit for the project and a moral hazard problem they need to incentize employee effort. Moreover, hiring the bad fit is costly for the firm because taking on a worker requires the firm to relinquish an outside option, coming for instance form being able to search for a new candidate. We fully characterize the optimal employment contract in this environment and derive the conditions under which offering a payment for quitting is optimal. Foarta: Stanford Graduate School of Business, 655 Knight Way, Stanford, CA, 94305, tel: 650) , ofoarta@stanford.edu. Sugaya: Stanford Graduate School of Business, 655 Knight Way, Stanford, CA, 94305, tel: 650) , tsugaya@stanford.edu. 1

2 1 Introduction An increasingly common practice among both small and large companies has been to incentivize the early removal of potentially bad fits by offering contracts that compensate employees who decide to quit called Pay to Quit programs. The online retailer Zappos famously introduced such contracts in the 2000s. 1 As described by the Harvard Business Review, Zappos wants to learn if there s a bad fit between what makes the organization tick and what makes individual employees tick and it s willing to pay to learn sooner rather than later. 2 Similar Pay to Quit programs were subsequently adopted by Amazon in 2014 and by some Silicon Valley start-ups afterwards. 3 A motivation often offered for this phenomenon is that organizations are structured in ways that are increasingly reliant on employee cultural fit. A misfit employee is very costly for the organization. He does not only cost the organization in terms of productivity the usual concern of supplying too little effort, but also in terms of the effectiveness of the organizational structure a problem of selecting the right type of person for the team. Moreover, in many cases, organizations cannot immediately compensate for a bad hire by expanding the team and hiring other employees who are good fits. Organizational charts and team sizes cannot be easily expanded, so an employee must in many cases leave for a new one to be hired. Even when teams can be expanded, organizations face high fixed, up-front costs of training and onboarding employees. A misfit employee who is fired shortly into his tenure can be particularly costly given these training costs. Finally, the process of removing a misfit hire is in itself diffi cult and costly, 4 even unfeasible in some cases, 5 and a protracted 1 The offer was $2000 and the take-up rate about 3% accessed on August 8, 2017). 3 The monetary compensation was up to $5000 in the case of Amazon and 10% of salary in the case of startup Mavens. See In one survey of 6000 hiring professionals worldwide, more than half mentioned that they have been affected by the costs of a bad hire More Than Half of Companies in the Top Ten World Economies Have Been Affected By a Bad Hire, CareerBuilder, 2013) 5 For example, union-protected jobs or public servants. 2

3 firing process can negatively impact morale in the organization more broadly. Given the high costs of bad hires, a key concern for both firms and public institutions is how to optimally structure contracts that incentivize employee effort the moral hazard problem and at the same time incentivize low ability employees to select out in order to free up the spot for another candidate the adverse selection problem. Devising such contracts is not an easy task, as a candidate s ability is usually better known to the candidate than to the organization, especially when this refers to what makes the employee tick. Moreover, the incentives offered for the low ability candidates to select out must not be so tempting that high ability candidates also choose to take them. In this paper, we study this fundamental problem. We derive the optimal contracts that firms should offer workers in a setting in which i) the workers ability is unobservable to the firm there is adverse selection), ii) the worker s input effort) is not observable, but it is correlated with the project s success there is moral hazard), and iii) if and only if a worker selects out of working on the project, the firm can exercise an outside option for example, it can offer the contract to a new candidate. 6 Our model considers a firm that has one project and exactly one worker can work on that project. The pool of available workers consists of both low ability and high ability types. The high ability types have a lower cost of supplying effort. This matters for the firm because the worker s effort increases the payoff from the project, and yet, the amount of effort supplied cannot be observed by the firm. The firm is matched with exactly one candidate from the pool of available workers. Based on the contract he is offered, the candidate either works on the project or selects out. If the candidate stays to work on the project, the outcome of the project is revealed at the end of the period. If the candidate selects out, he performs no work, and the firm exercises an outside option. The firm can offer contracts that specify a monetary reward based on the observable project outcome, a probability with which the worker stays to work on the project, and any fixed payment for the worker who does not stay on the project. 6 This outside option can be seen as the baseline productivity achieved without hiring a new employee in the case of a bad fit having a negative effect. 3

4 We characterize the optimal contract for the firm in this environment. In designing this contract, the firm faces three challenges. First, in order to respond the moral hazard problem, the firm would like to offer high powered incentives, that induce the most effort from the worker. This implies offering higher monetary rewards for better outcomes. Second, in order to respond to adverse selection, the firm would like for the low ability type to select out of the project. This would require offering a contract that retains the low ability worker less often. The relative value to the firm of solving the moral hazard problem incentivizing effort and solving the adverse selection problem retaining the high ability type dictates the balance between monetary rewards and retention probabilities in the optimal contract. There is, however, a third challenge for the firm, namely that it must offer contracts that lead each type of worker to select the contract designed for his type the incentive compatibility problem. The interaction of this third challenge with the previous two determines whether payments to select out are optimal. The main result of the paper identifies the condition which makes it optimal for the employment contract to offer the worker a payment when selecting out of the project. We start from the case in which, with full information about worker types, the firm would prefer to employ only the high ability worker and to offer this worker a higher wage than it would the low type worker. Yet, such a contract would not be incentive compatible when worker types are unknown. Hence, the firm must offer one contract with a lower wage and higher retention probability, and another contract with higher wage and lower retention probability. The condition for the fixed payment is that there must be misalignment between the agent and principal in how they rank these two contracts. Such misalignment emerges if the principal is more willing to reduce the wage give up solving moral hazard) in exchange for increasing the retention probability solving adverse selection) when she faces the high ability worker, while the high ability worker is more willing to give up retention probability in exchange for higher wage than the low ability worker, or vice-versa. This misalignment between the relative values placed on these two instruments wage and retention probability) by the firms versus the workers prevents the contract without a 4

5 fixed payment from being incentive compatible for the workers and profitable to the firm. The only way to obtain both incentive compatibility and profitability is by offering the low ability type a fixed payment in case of no retention. Since the low ability worker has a higher marginal cost of effort than the high ability worker, a fixed payment is relatively more valuable for the low ability type. Theoretically, our contribution to analyzing this multidimensional mechanism design problem is to derive the relevant condition for the misalignment between firms and workers, and by doing so, to provide a proper analogue of the single-crossing condition of singledimensional problems. We show that the misalignment condition corresponds to the logsubmodularity/ supermodularity of the net total gain from retaining the worker as a function of wage and type the net gain from retention for the firm and the worker together and the log-submodularity/ supermodularity of the worker s value as a function of wage and type. If the net total gain is log-supermodular log-submodular) in wage and type, then the principal is more less) willing to reduce wage in exchange for higher retention probability when she faces the high type; and if the agent s value is log-supermodular log-submodular) in wage and type, then the high type agent is more less) willing to accept a low wage in exchange for a high retention probability. When the two values are both log-submodular or both log-supermodular, there is alignment between the relative benefit to the firm and to the high ability worker. Conversely, misalignment emerges when one of the values is log-supermodular while the other is log-submodular. Both moral hazard and adverse selection are essential to obtaining this key result. With only moral hazard, a fixed payment would never be used, since it does not incentivize effort. With only adverse selection, no production is the least informative outcome, hence making the use of a fixed payment ineffi cient. Therefore, studying an environment with both moral hazard and adverse selection is essential for understanding the optimality of paying workers to quit. Moreover, we highlight the importance of differentiating between organization specific fit and genuine productivity differences. We provide an extension to the model that adds type-dependent outside values for the workers, to reflect genuine productivity differences, 5

6 and show that, in such cases, offering a payment to select out may backfire and result in the high ability types selecting out. The model has wide policy implications for contracts both in the private sector and in public organizations. The results imply that contracts with Pay to Quit provisions are desirable in organizations for which organization-specific fit is of key importance solving adverse selection dominates), while the "best fit" employees value compensation relatively more than fit the moral hazard problem dominates). This misalignment between what the firm values most and what the employee values most is suggested in the motivating examples of Amazon and Zappos. If such misalignment is present, then Pay to Quit programs can improve outcomes, by selecting out low ability types without being too tempting for the high ability types. Also, we discuss how other types of programs can be viewed as similar in scope to the Pay to Quit programs. Examples include early tracking programs, which are more prevalent in public bureaucracies, or early retirement packages offered, for example, in higher education. Related Literature Agency theory has extensively explored the optimal design of contracts with either moral hazard or adverse selection; 7 however, the case in which both moral hazard and adverse selection are present is less well understood. Chade and Swinkels 2016) analyze the general principal-agent problem with both moral hazard and adverse selection, and provide a set of suffi cient conditions under which we can decouple the problem into adverse selection and moral hazard. There are two main differences between their paper and ours. First, our main focus is on the case when the principal has an outside option. Second, our principal targets both effort level as in their model as well as the retention probability for each type, which adds another dimension to the mechanism design problem. Gottlieb and Moreira 2014) also analyze a principal-agent problem in which two types of agents who have private information about their type and about the distribution of outcomes conditional on whether effort was exerted. In this general framework, Gottlieb and Moreira 7 Starting with the seminal papers of Mirrlees 1975), Hölmstrom 1979), Myerson 1981), Grossman and Hart 1983), and Maskin and Riley 1984). 6

7 2014) show that with risk neutral types and limited liability for the agents, all agents are offered a single contract. We consider this same setting, but add another component to the principal s problem: the existence of an outside option if the current worker is not retained. With this outside option, the optimal contract can in fact be differentiated for each type and feature a payment when the agent is not retained. This happens because the firm s ability to remove a bad hire from the project in order to free up its outside option may be suffi cient to compensate for any losses from not retaining the current candidate. The focus of our paper is on optimal contracts in environments in which it is valuable to select out low ability types rather than employ them and learn about their type the firm can be matched with a different candidate if the current one is not retained. This relates our work to the literature on screening in the presence of both moral hazard and adverse selection Chiu and Karni, 1998; De Meza and Webb, 2001; Jullien et al., 2007). While this literature considers a similar framework, we add to it the outside option for the firm if it does not retain the current candidate. Our focus on the selection of workers given their organization-specific fit links our paper to the literature on optimal contracts when workers also have intrinsic motivations Murdock, 2002; Benabou and Tirole, 2003; Besley and Ghatak, 2005). This literature assumes that workers derive utility directly from being in the organization for which they are a fit. We instead examine the case in which the organization-specific fit only affects the worker s cost of effort. Besley and Ghatak 2005) examine the case with both adverse selection in terms of intrinsic motivation and unverifiable worker effort, and they obtain the result that all worker types are offered the same contract. We show that offering different probabilities of retention or a fixed non-contingent payment are necessary for differentiated contracts to be optimal. The use of non-contingent payments is also examined in Benabou and Tirole 2003) in a model with intrinsically motivated agents. In their model, such payments are used by the principal as a signalling device to the agent when the principal has private information about the agent s ability. In our paper, non-contingent payments are used in order to select out the low ability types when the principal does not have information about types. Our paper also 7

8 relates to the literature on incentives provision for downsizing the public sector. Jeon and Laffont 1999) consider the optimal contracts for inducing low ability public employees to voluntarily leave a public sector job. While their model only considers the adverse selection problem, we examine optimal contracts with both adverse selection and moral hazard. Finally, there is a rich and growing literature on dynamic contracts with both moral hazard and adverse selection Strulovici, 2011; Garrett and Pavan, 2012; Halac et al., 2016); while the focus in this literature is on the evolution of the relationship between the firm and the employee once hiring happens, our paper focuses instead on the problem of selecting out a bad type employee ex ante, when the cost of a bad hire is high. To focus on ex ante selection before production, we analyze the environment with a one-period production. The rest of the paper is organized as follows. Section 2 describes the environment and the structure of the game. It also introduces the proper analogue to the standard single crossing condition in this environment, namely log-submodularity/supermodularity. Section 3 studies the optimal contract for a firm with an exogenous outside option, and Section 4 illustrates it by numerical examples. Section 5 presents several extensions that show the robustness of the optimal contract to introducing outside options for agents, multiple project outcomes, endogenizing the firm s outside option, and multiple agent types. Section 6 discuss empirical and policy implications of the model and Section 7 concludes. The Appendix contains the proofs and further extensions of the model. 2 The Model We consider an environment with two players: a principal and an agent. The agent has two possible types, θ {H, L}, with µ θ representing the prior that the agent is type θ. The agent is hired by the principal to work on a project, the outcome of which depends on the agent s private effort. The type θ agent has a cost of effort c θ, e) for e R +, which satisfies c e, c ee, c eee 0, c θ 0, 1) 8

9 where the subscripts denote derivatives with respect to the stated parameter. Moreover, the H type has lower marginal cost of effort than the L type: c eθ 0 for each e R +. The set of possible outcomes is denoted by Y = {0, 1}, where y = 1 denotes a project success and y = 0 denotes a failure. The probability of a success is given by qe θ ), where qe θ ) = q 0 + q 1 e θ, with q 0, q 1 R. 8 By the revelation principle, we can focus on the contract in which, upon arrival, the agent truthfully declares his type. Depending on the declared type θ, the principal offers the contract C θ. In particular, C θ specifies a probability of retention p θ, a wage w θ y) contingent of the realized output y Y, if the agent is retained on the project, and a payment w θ ) in case of no retention. If the agent is not retained, he is removed right away and paid w θ ), so he does not exert any effort on the project. For notational convenience, we also define w θ w θ y = 1) w θ y = 0), the relative reward received by a retained agent θ in case of a success rather than a failure, and v θ p θ w θ y = 0) + 1 p θ ) w θ ), the expected base payment if nothing is produced either because the project fails or because the agent is not retained. This is what we call a fixed payment the payment not contingent on a good outcome such as a life-stile track, or the payment upon quitting. We assume that the agent s type θ is not observable adverse selection), and effort e θ is not also observable moral hazard). We also assume limited liability: w θ y) 0 and 8 Note that, as long as q is concave, we can always re-measure the effort such that the linearity is obtained for convex cost c. 9

10 w θ ) 0 θ, y. 9 Timing of Actions. The game can be summarized in the following timing of actions: 1. An agent arrives and declares his type ˆθ. On the equilibrium path, he reports his true type: ˆθ = θ. 2a. With probability p θ, the principal retains the agent for the project. After retention, the agent chooses private effort e θ and is paid w θ y) given the observable outcome y. This is the end of the game. 2b. With probability 1 p θ, the principal does not retain the agent. She pays w θ ) to the agent. The principal obtains an exogenous outside option with value W. Payoffs. C θ is given by Given the risk neutrality, the total expected payoff for agent θ from contract p θ V θ, w θ ) + v θ, where V θ, w θ ) qe θ, w θ )) w θ c θ, e θ, w θ )) denotes the payoff given the optimal effort with reward w θ. The equilibrium effort e θ, w θ ) for type θ is determined by e θ, w θ ) = arg max qe) w θ c θ, e). 2) e We assume that type H brings strictly higher profit to the principal than type L for the same wage: Assumption 1 The cost c θ, e) is continuously differentiable with effort e [ 0, 1 q 0 q 1 ], lim e 0 c e L, e) = 0, lim e 1 q 0 q 1 c e H, e) q 1, and c e H, e) c e L, e) < 0 for each e > 0. 9 Otherwise, selling a project with a price accepted only by the high type will be optimal. 10

11 Assumption 1 provides that necessary conditions such that e H, w) > e L, w) and e H, w H ) < 1, since it is suboptimal for the principal to offer w θ 1. Assumption 2 We assume that W > max w π L, w). Assumption 2 allows us to focus on the case in which selecting the right fit is of high importance for the principal. Specifically, with full information over the agents types, the principal would prefer the outside option over hiring type L The Principal s Problem Upon retaining an agent of type θ, the principal obtains expected payoff π θ, w θ ) q e θ, w θ )) 1 w θ ). Defining the social welfare function as S θ, w θ ) q e θ, w θ )) c θ, e θ, w θ )), we can write π θ, w θ ) = S θ, w θ ) V θ, w θ ). 3) We analyze the optimal contract offered by the principal in order to maximize her expected profit. The principal offers these contracts under the constraint that the ex-ante value for each agent θ from contract C θ is no less than Cˆθ for each ˆθ θ. Denoting by J the equilibrium value of the principal, her problem is characterized by the following program: J W ) = max {p H,p L,w H,w L,v H,v L } µ H [p H π H, w H ) + 1 p H ) W v H ] +µ L [p L π L, w L ) + 1 p L ) W v L ], 4) 10 In the Appendix, we provide the analysis of the case in which Assumption 2 does not hold. 11

12 subject to 11 p H V H, w H ) + v H p L V H, w L ) + v L ; IC H ) p L V L, w L ) + v L p H V L, w H ) + v H. IC L ) The principal maximizes her payoff J W ) subject to the two incentive compatibility constraints. Constraint IC H ensures that the type H agent prefers the contract p H, w H, v H ) to the contract designed for the type L agent. Similarly, constraint IC L ensures that the type L agent prefers the contract p L, w L, v L ) to the contract designed for the type H agent, p H, w H, v H ). We present the problem and results by assuming for now that W is exogenous, W 0, W MAX), where W MAX = max wh πh, w H ). 12 In Section 5.3, we present results endogenizing W. Throughout the paper, we analyze the non-trivial case in which the principal hires at least one of the agents with a positive probability, so max {p H, p L } > 0. The guess is verified if the optimal value with this guess is greater than W. Otherwise, p H = p L = v H = v L = 0 is optimal. 2.2 A Useful Condition As mentioned in the introduction, the key condition for v L > 0 is the alignment between the principal s and agent s preferences. To characterize when preferences are aligned, we make use of the following property. 13 Definition 1 A function F θ, w) is log-supermodular log-submodular) if F θ, w) F θ, w ) F θ, w) F θ, w ) )0 if and only if θ θ ) w w ) )0. 5) 11 The definition of V θ, w) and πθ, w) implies 2). 12 If W W MAX, not hiring any candidate is trivially optimal. 13 See Athey 2002) for other economic applications of log-supermodularity/log-submodularity. 12

13 If log F θ, w) is twice-differentiable, then it is log-supermodular log-submodular) if d 2 )0. We say a function is regular if, given θ, θ {L, H}, it is globally log-supermodular or log-submodular for each w, w [0, 1] with F θ, w), F θ, w ), F θ, w), F θ, w ) 0. dθdw log F θ, w) Intuitively, log-supermodularity implies that an increase in w is relatively more valuable when θ is higher. 14 Applied to V θ, w), log-supermodularity means that, all else constant, an increase in the reward w leads to a relatively higher increase in type H s payoff. If S θ, w) is also log-supermodular, then an increase in w increases the social welfare upon retention relatively more when the agent is of type H. To show why the property of log-supermodularity/submodularity will be useful in analyzing our problem, we provide the following intuition. Consider the case in which no fixed payment is offered, so v H = v L = 0. In this case, the principal offers a contract that features a reward w and a probability of retention p. We can therefore represent each contract on the two dimensional space of p, w). For any arbitrary contract C H, consider the set of contracts C that would make type θ indifferent between C H and C. On the indifference curve obtained in this way, the marginal rate of substitution between p and w satisfies V θ, w) dp + pv w θ, w) dw = 0 dp dw = pv w θ, w) V θ, w). The property of log-submodularity of V θ, w) implies that dp/dw is decreasing in θ. Type H is willing to sacrifice more monetary rewards in exchange for the increase in retention probability he values higher retention probabilities relative to higher rewards compared to type L. This is illustrated in Figure 1. The intersection of the indifference curves is C H and the shaded area is the set of C L that satisfies IC H ) and IC L ) with v L = 0. We can perform a similar analysis the net social welfare created by type θ, which is given by p S θ, w) W ). Since utility is transferrable, the principal maximizes the excess social welfare minus the rent to the agent. When IC θ is binding, the rent to the agent is fixed 14 In percentage terms, F θ, w + ) F θ, w)) /F θ, w). 13

14 Figure 1: Log-submodular V according to contract C θ offered to type θ θ. Hence, the log-submodularity of S θ, w) W implies that the principal is more willing to give up offering high rewards in exchange for increasing retention, i.e., she values higher retention probabilities relative to higher rewards when faced with type H, compared to the case when faced with type L. ln order to streamline the analysis and highlight the intuition behind the main result, we present the results for the case in which V θ, w) is globally log-submodular and S θ, w) W is regular. In Appendix F, we provide the complementary analysis. Remark 1 There exist cost functions c θ, e) such that V θ, w) is log-submodular and S θ, w) W is regular. An example of a cost function that leads to V θ, w) log-submodular and S θ, w) W is regular is given in Section 4. Assumption 3 The cost function c θ, e) has a form that leads to V θ, w) log-submodular and S θ, w) W regular. Under this assumption, we proceed to analyze the optimal employment contract. 14

15 3 The Optimal Contract 3.1 Preliminaries We begin by deriving several properties of objective 4), that allow us to characterize the optimal contract. First, Assumption 2 implies that constraint IC L holds with equality, since otherwise the principal would like to reduce p L. Hence, the principal faces the threat of type L imitating type H, and she must therefore adjust the menu of contracts to dissuade type L from choosing the contract designed for type H. Next, we also show that the type H agent is offered no fixed base payment, and at least one type of agent is always retained for the project. Lemma 1 The optimal contract has the property that no fixed payment is offered to the type H agent v H = 0), and one of the agent types is always retained either p L = 1 or p H = 1). Proof. In Appendix A.2. The principal only rewards the H type in case of the success of the project. Intuitively, increasing the reward increases the effort provided by the agent. Since the agent is riskneutral, the principal can incentivize the highest effort from type H by loading the entire reward on the positive outcome. Moreover, this change makes type L less willing to pretend to be of type H since he has to pay a higher effort cost for a good outcome. Also, at least one of the agents is always retained. Otherwise, increasing p θ and v θ proportionally would proportionally increase the principal s payoff relative to the outside option W given the guess that max {p H, p L } > 0). Next, we establish that the incentive compatibility constraint for the type H agent must hold with equality whenever v L > 0. Lemma 2 In the optimal contract, if the type L agent is offered a fixed payment v L > 0), then constraints IC L and IC H bind. Proof. In Appendix A.3. 15

16 The fixed payment v L > 0 amounts to a cost for the principal without any benefit in terms of output. With constraint IC L binding, the principal can reduce v L and increase w L, keeping the payoff to type L constant. This is feasible as long as type H still prefers the contract designed for his type, i.e., up until constraint IC H binds. Finally, we show that, even with a fixed payment, the principal cannot offer type L both a lower reward and a lower retention probability. Lemma 3 It is the case that w H w L and p H p L or w H w L and p H p L. Proof. In Appendix A.4. Without a fixed payment, offering type L both a lower reward and a lower retention probability is not incentive compatible. With a fixed payment, the previous lemma implies constraints IC H and IC L bind, which narrows down the possible combination of wages and retention probabilities. 3.2 Contract where type L is never retained Given the above preliminaries, we proceed to solve for the optimal contract. First, if type L is very unproductive, then the principal never retains this type: Proposition 1 Retention of type H only) If W S L, 1), then only the type H agent is retained in the optimal contract: v L > 0, p L = 0, p H = 1. Conversely, if W < S L, 1), then type L is retained with positive probability p L > 0). Proof. In Appendix A.5. If W S L, 1), then type L is never retained. He is instead offered a fixed payment that makes his contract as appealing as taking the contract designed for type H. By not retaining type L, the principal loses at most S L, 1) and she gains W from the outside option. Conversely, if W < S L, 1), then the principal can gain by retaining type L. To see this, consider starting from p L = 0 and w L = 1 and increasing p L by a small ε > 0. The 16

17 principal can then reduce v L by εv L, 1) = εsl, 1) and keep type L indifferent 15. When retained, the type L agent creates the social surplus S L, 1), and the principal captures ε SL, 1) W ). Therefore, this would be a gainful deviation for the principal. 3.3 Contract where both types may be retained The following proposition characterizes when the principal puts more weight on solving the moral hazard by offering w H w L ) and when she puts more weight on solving the adverse selection by offering p H p L ). This trade-off emerges given Lemma 3. The result also shows when the fixed payment is needed, which stands as the paper s main focus: Proposition 2 When W < S L, 1), the optimal contract has the following properties: Suppose V θ, w) is log-submodular. 1. Principal-Agent Alignment) If Sθ, w) W is also log-submodular, then v L = 0, w L w H, and 1 = p H p L > Principal-Agent Misalignment) If Sθ, w) W is log-supermodular, then v L 0, w H w L, and 1 = p L p H > 0. Moreover, v L > 0 if and only if w H w L. Proof. In Appendix A.7. The result emerges from two forces, captured by the log-supermodularity log-submodularity) of Sθ, w) W and the log-submodularity of V θ, w). First, whether S θ, w) W is logsubmodular log-supermodular) determines whether w L w H w H w L ). The principal has two possible incentive compatible deviations that she could undertake: offer all agents the contract designed for type H, or offer all agents the contract designed for type L. If the principal offers all agents the contract designed for type H, then the principal gains over the original set of contracts if this new contract offers her a higher payoff from type L, i.e., p L S L, w L ) V L, w L )) + 1 p L ) W v L < p H S L, w H ) V L, w H )) + 1 p H ) W. 15 Type H s incentive compatibility is satisfied for a small ε. 17

18 Given constraint IC L, this implies p L S L, w L ) W ) < p H S L, w H ) W ). 6) Similarly, if the principal offers both types the contract designed for the type L agent, then this deviation is gainful only if p L S H, w L ) W ) > p H S H, w H ) W ). 7) Combining 6) and 7) delivers the result that the log-submodularity of S θ, w) W is equivalent to w L w H, and the log-supermodularity of S θ, w) W is equivalent to w H w L. Intuitively, log-submodularity of S θ, w) W means that an increase in w granted to type L is relatively more valuable for the excess social welfare than an increase in w granted to type H. That is, the log-submodularity implies that the principal puts more weight on solving the adverse selection problem, by offering w H w L and p H p L. Symmetrically, the log-supermodularity implies that the principal puts more weight on solving the moral hazard problem by offering w H w L and p H p L. The second force driving the result comes from the log-submodularity log-supermodularity) of V θ, w). This determines whether a fixed base payment will be used in the optimal contract. Consider the case in which S θ, w) W is log-submodular, so that w L w H. Acting towards solving the adverse selection problem leads the principal to choose p H p L. This implies that type H is compensated for the reduced reward w H by being offered a higher probability of retention p H. If V θ, w) is log-submodular, then such an increase in p H is relatively more valuable to type H than to type L. Thus, in this case, the principal s actions towards solving adverse selection also act towards solving incentive compatibility. Therefore, no fixed payment v L 0 is necessary. In the case in which S θ, w) W is log-supermodular, the principal puts more weight on the moral hazard problem, so w H w L and p H is decreased below p L. With V θ, w) log-submodular, the decrease in p H would be relatively more beneficial for type L than for type H. Then, without a fixed payment, type L would 18

19 prefer to mimic type H. Hence, the fixed payment v L 0 is necessary in order to ensure incentive compatibility. To summarize, if both S θ, w) W and V θ, w) are log-submodular, then the incentives of the principal and those of the agent are aligned; the principal values solving the adverse selection problem by increasing p H relatively more than solving the moral hazard problem, and the type H agent values retention the action that solves adverse selection over reward the action that the solves moral hazard problem relatively more compared to type L s valuation of these two components. When S θ, w) W is log-supermodular, the principal values solving the moral hazard problem by increasing w H relatively more than solving the adverse selection problem. With V θ, w) log-submodular, this means that the principal s incentives are misaligned with the agent s, requiring a fixed base payment v L 0 in order to correct for this misalignment and ensure incentive compatibility of the optimal contract. When Misalignment is Likely to Occur Recall that log-submodularity or supermodularity) of F is equivalent to the sign of d2 log F θ,w) dwdθ being negative or positive). By the envelope theorem, the sign of d2 log V θ,w) dwdθ is determined by q 1 e θ θ, w) V θ, w) [q 0 + q 1 e θ, w)] V θ θ, w) and that of d2 logsθ,w) W ) dwdθ is determined by q 1 1 w) e wθ θ, w) [S θ, w) W ] q 1 1 w) e θ θ, w) S θ θ, w). Log-submodular of V θ, w) emerges if e θ θ, w) is small, so that the effort level is similar between types, but V θ θ, w) is large, so the absolute level of the cost is lower for type H. On the other hand, log-supermodular of S θ, w) W emerges if e wθ θ, w) is large, meaning that the reaction of the high-powered incentive is different between types, but e θ θ, w) and S θ θ, w) are small. Among other differences, notice that log-submodularity of V θ, w) depends on the comparison of the effort level between types, while that of S θ, w) W 19

20 depends on the comparison of the reaction to the incentives. This intuition is that when we increase w, the agent s value V θ, w) increases proportionally to the probability of the good outcome directly related to the effort level. The net social value S θ, w) W, however, increases only due to higher effort, since moving money from the principal to the agent does not have a direct effect on the welfare. Characterization of the optimal contract in the alignment case We now provide a more detailed characterization of the optimal contract. First, consider the case in which both V θ, w) and Sθ, w) W are log-submodular. Given Lemma 1 and Proposition 2, it follows that p H = 1 and v H = v L = 0. Substituting constraint IC L into 4) we obtain max µ V L, w H ) H π H, w H ) W ) + µ L w H,w L,w L w H 0 V L, w L ) π L, w L) W ). 8) Proposition 3 Alignment case) If both V θ, w) and Sθ, w) W are log-submodular, then the optimal contract solves 8). In Section 4, we provide an example in which the optimal contract features w H < w L and p H > p L. This result is markedly different from the results obtained by Gottlieb and Moreira 2014), who study a similar problem, but without an outside option for the principal. In that environment, all agents are retained and offered the same contract. This highlights the difference brought about by the existence of the outside option. 16 The fact that the principal has an outside option creates a non-trivial cost of adverse selection retaining the agent implies giving up the outside option W. Hence the principal can benefit from lowering p L. Characterization of the optimal contract in the misalignment case Next, consider the case in which Sθ, w) W is log-supermodular. Since the preferences of the principal and 16 Although Gottlieb and Moreira 2014) only consider the deterministic contract, we can show that allowing stochastic replacement without outside options does not change the result that all agents are always retained and offered the same contract. Details are available upon request. 20

21 agent are misaligned, both IC H ) and IC L ) hold with equality. Substituting constraints IC H ) and IC L ) into 4) we obtain max w H,w L,w H w L 0 µ H V H, w L ) V L, w L ) V H, w H ) V L, w H ) π H, w H) W ) + µ L π L, w L ) W V L, w H ) + V L, w L )). 9) Proposition 4 Misalignment case) If V θ, w) is log-submodular and Sθ, w) W is logsupermodular, then the optimal contract solves 9). Moreover, we have v L > 0 if and only if w H > w L. See Section 4 for the example with w H > w L and v L > 0. Again, the comparison with Gottlieb and Moreira 2014) highlights the difference brought about by the existence of the outside option. Lowering p H is not so costly if the outside option is suffi ciently attractive, since the principal can at least obtain the outside option. Without outside option, lowering p H would be too costly, and the optimal contract would be the same for both types. Decoupling and first order approach One may wonder if it would be possible to solve the problem by the methods previously suggested in the literature: decoupling and first order approach. Since we have the multi-dimensional problem, decoupling suggested by Chade and Swinkels 2016) does not work as it is. For the first order approach, the diffi culty is to control double deviation, that is, the case when type θ reports his type as θ and takes an effort level not corresponding to either type θ s or type θ s equilibrium effort. To avoid this problem, we can assume additive separability of the effect of types and efforts on the outcomes. For example, we can assume the probability of the good outcome is equal to q e) + χ θ) and the cost of effort is c e). Then, since the type does not affect the first order condition for the optimal effort, given the contract C, each type supplies the same effort. 17 However, this additive separability makes both V θ, w) and S θ, w) W log-submodular, which is only one of the cases we examine, 17 This specification is similar to Garrett and Pavan 2012) 21

22 and the case in which the fixed payment is not used. 3.4 Comparison to the social welfare maximizing contract Finally, let us present the type of contracts that would maximize social welfare, and compare it with the contracts to maximize the principal s profit. Consider the problem for a social planner who offers employment contracts p θ, w θ, v θ ) in order to maximize max E θ [p θ S θ, w θ ) + 1 p θ ) W ], subject to the incentive compatibility constraints for the type H agent and the type L agent, respectively: p H V H, w H ) + v H p L V H, w L ) + v L ; p L V L, w L ) + v L p H V L, w H ) + v H. If the social planner retains the agent, she obtains S θ, w θ ), while if she does not retain the agent, she obtains the exogenous continuation value W. Proposition 5 The social planner s solution can take one of two forms: 1. Retention of type H only) If W > S L, 1), then w H = 1, p H = 1, v H = 0, and p L = 0, w L = 0, v L = V L, 1). 2. Retention of both agents) If W S L, 1), then w θ = 1, p θ = 1, v θ = 0 for θ = H, L Proof. In Appendix A.10. Since utility is transferrable, the social planner cares only about incentivizing the maximum effort and hiring the right type. For the first purpose, we obtain the well-known results that she sells the project to the worker in order to generate the maximum effort. For the second, since the exogenous continuation payoff satisfies W < S H, 1), the social planner 22

23 always benefits from keeping the type H agent; however, keeping the type L agent is optimal only if the continuation value W is smaller than the social value of retaining the type L agent, S L, 1). When comparing this contract to the one that maximizes the principal s profit, notice that the difference in results comes from the fact that the profit-maximizing principal also cares about leaving enough profit to herself. Hence, the solution is in general different; however, the condition for p L = 0 is the same. If p L is very close to 0, however high w L is, IC H ) is not binding when she sets v L to satisfy IC L ). Since IC L ) binds, the principal maximizes the social welfare from the low type without worrying about the effect on IC H ) recall that her profit is the social welfare minus the rent for the agent, and the latter is fixed given IC L ) binding). Hence, her incentive is the same as the social planner s. 4 Numerical Illustration In what follows, we provide numerical examples for both the alignment and the misalignment cases. We assume that the good outcome takes value y = 10. This re-scaling allows us to minimize computational errors without qualitatively changing the results. 18 Alignment case Consider the case in which µ H =.7 and the functions q e) and c θ, e) take the following functional forms: q e) = q 1 e, ) 1 c H, e) = 1 e e 1 /80, ) 1 c L, e) = 1 e 1 / With y = 1, everything is the same by also rescaling cθ, e)/10. 23

24 Figure 2: Optimal Contrant Form as a Function of the Outside Option Under these specifications, S θ, w) W is log-submodular for each W with S θ, w) W and V θ, w) is log-submodular. Hence Propositions 1 and 3 characterize the optimal contract. Figure 2 shows the form of the optimal contract when q 1 =.1 and W varies between 10 2 and As q 1 and W are very low, both types are ineffi cient, and the outside option is too low to be worth eliminating any of them. Since both types have similarly low productivity, the principal does not find it optimal to offer separate contracts. As W increases, the adverse selection problem increases, and offering different contracts becomes optimal. If W = 0.75, for q 1 =.1, the value of q 1 is very low, and so the L type is very ineffi cient. Hence, the principal would like to set p H > p L, which can be achieved in an incentive compatible contract if w H < w L. For W suffi ciently large, not hiring any agent and receiving the outside option is optimal. We can also examine the effect of increasing q 1. For q 1.2, the principal offers the same contract to both types, since now the low type is suffi ciently productive. Finally, we can provide an example where the optimal contract retains only the type H agent and provides a payment for the type L agent to select out. Setting µ H =.99, q 1 =.8, and W = SL, 1) Since we start with a very low value of W, Assumption 2 is not always satisfied. In this case, it is possible that IC L ) does not bind. As long as it binds, Propositions 1 and 3 are still valid characterizations of the optimal contract. See Appendix G. 24

25 delivers this case. The intuition behind the result is that the probability of encountering a type L is small and W is greater than S L, 1), so never retaining this type is almost costless for the principal. Misalignment case For the misalignment case, consider the case in which µ H =.5, W = 3.41 and the functions q e) and c θ, e) take the following functional forms: 20 q e) = e, c H, e) =.5e + max {e.5, 0} +.4 max {e.8, 0}, c L, e) = e max {e.5, 0}. The cost functions are piecewise linear. It immediately follows that the optimal wages are in the set { 0, 5, 10, 19, } , corresponding to the marginal cost divided by q1 =.3. If we restrict our attention to those wages, we can show that S θ, w) W is log-supermodular and V θ, w) is log-submodular. Intuitively, the cost function ensures that the high type reacts to the incentive much more than the low type, which makes S θ, w) W is log-supermodular. Moreover, π θ, w) is concave in w characterizes the optimal contract. 22 In addition, S L, 1) > W = Hence, Proposition We can show that, if the principal is forced to offer the same wage, then she offers w H = w L = 10 3 and the expected profit is When the principal offers differentiated contracts by setting w H = 19 3 and w L = 10 3 to induce more effort by type H solving the moral hazard problem), we can show that the principal s payoff is increased. By Proposition 3, we have v L > 0 at the optimal contract. Note that W is close to the profit coming from the same wage contract. Hence, the principal does not bare a high cost of reducing p H in 20 One may notice that the example below does not satisfy 1). We can make the cost function continuously differentiable by modifying it slightly around the kink to satisfy 1). We keep linearity to simplify the algebra. 21 We use c eee 0 only to show the concavity of π θ, w) in w; hence, piecewise linearity does not change the statement of the lemmas or propositions. 22 Lemma 5 in Appendix ensures that IC L ) binds in this problem. 25

26 order to solve the moral hazard problem. 5 Robustness and Extensions 5.1 Outside option for agents In the following extension, we consider the case with exogenous outside options for both the principal and the agents. As before, the firm has outside option W. The agent of type θ has an outside option V O θ. We assume that V O H option. In particular, V O H > V O L V L O, that is, type H has a higher outside implies that the more productive type in the current match is genuinely more productive and so obtains higher value in the outside option. On the other hand, V O H = V O L The principal s problem is implies that the type of the current match is just the match-specific fit. max µ H p H π H, w H ) W ) + µ L [p L π L, w L ) W ) v L ] p H,w H,p L,w L,v L subject to constraints [ ] p θ V θ, wθ ) Vθ O + vθ max { pˆθ [ ] V θ, wˆθ) Vθ O + vˆθ, 0 }. 10) where constraint 10) denote the incentive compatibility and participation constraint for type θ {H, L}. The following is the counterpart of Assumption 2: Assumption 4 The following conditions hold: W > arg max w subject to V L,w) V O L π L, w). The results of Lemma 2 are readily generalized for the non-trivial contract: Lemma 4 The optimal contract v H = 0, and p L > 0 together with v L > 0 imply IC H ) is binding. 26

27 Proof. Analogous to the proof of Lemma 2. Since Assumption 4 implies that the principal always wants to decrease p L, if the L type leaves voluntarily due to the participation constraint, it is beneficial to the principal. Hence, for the L type, constraint 10) is equivalent to p L [ V L, wl ) V O L ] + vl = p H [ V L, wh ) V O L ] 0. 11) A main difference from the baseline case emerges in the condition of when to replace the L type completely: 23 Proposition 6 If VH O = V L O, then not retaining the type L agent p L = 0) is optimal if and only if S L, 1) VL O W 0. If V H O > V L O, then not retaining the type L agent p L = 0) is optimal only if S L, 1) VL O W 0. Proof. In Appendix B.1. When the agents have the same outside option, Proposition 6 shows that not retaining type L is equivalent to the net social social gain from keeping this type being negative. The intuition is that, if p L were to ever be positive, then the principal could increase her objective by decreasing p L and increasing v L so as to keep type L indifferent. Moreover, the change would be incentive compatible for type H, since this type obtains a higher value from the retention than type L. Similarly, if S L, 1) VL O W > 0 but p L = 0, then since IC H ) is redundant for suffi ciently small p L, the principal could obtain a higher payoff by increasing p L with w L = 1 and reducing v L, keeping the rent to type L constant. When the outside options of the two agents differ VH O > V L O ), the problem for the principal changes if S L, 1) VL O W 0. In this case, the principal cannot reduce p L if the incentive compatibility constraint for type H is binding. Intuitively, since the H type has a higher outside option, if the L type contract allows the agent to capture the outside option with a higher probability due to the lower probability of retention, then such a contract may be more attractive to the H type. On the other hand, if S L, 1) VL O W > 0, 23 See Appendices B.2 and B.3 for the full characterization. 27