Dynamic contracting with limited commitment and the ratchet effect

Gerardi, Dino; Maestri, Lucas Jóver Article Dynamic contracting with limited commitment and the ratchet effect Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Gerardi, Dino; Maestri, Lucas Jóver (2020) : Dynamic contracting with limited commitment and the ratchet effect, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 15, Iss. 2, pp. 583-623, https://doi.org/10.3982/TE2449 This Version is available at: https://hdl.handle.net/10419/253441 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 15 (2020), 583–623 1555-7561/20200583 Dynamic contracting with limited commitment and the ratchet effect Dino Gerardi Collegio Carlo Alberto, Università di Torino Lucas Maestri EPGE, Escola Brasileira de Economia e Finanças We study dynamic contracting with adverse selection and limited commitment. A firm (the principal) and a worker (the agent) interact for potentially infinitely many periods. The worker is privately informed about his productivity and the firm can only commit to short-term contracts. The ratchet effect is in place since the firm has the incentive to change the terms of trade and offer more demanding contracts when it learns that the worker is highly productive. As the parties become arbitrarily patient, the equilibrium outcome takes one of two forms. If the prior probability of the worker being productive is low, the firm offers a pooling contract and no information is ever revealed. In contrast, if this prior probability is high, the firm fires the unproductive worker at the beginning of the relationship. Keywords. Dynamic contracting, limited commitment, ratchet effect. JEL classification. D80, D82, D86. 1. Introduction This paper contributes to the literature on the ratchet effect by analyzing an infinite horizon contracting problem with short-term contracts. We frame the analysis in the context of a labor relationship between a worker and a firm. In each period, the worker can produce a good of quality q∈[01]at a cost that is linear in q. The worker is privately informed about his (persistent) marginal cost, which is low with prior probability p0and high with probability 1−p0. The firm can only commit to short-term contracts, which indicate the payment that the worker receives in the current period if he produces a good of a specified quality. In each period in which the worker is employed, the firm Dino Gerardi: [email protected] Lucas Maestri: [email protected] We are grateful to the three anonymous referees, Luca Anderlini, Paolo Ghirardato, Daniel Gottlieb, Felipe Iachan, Ignacio Monzón, Juan Morales, Cézar Santos, and various seminar audiences for useful comments. The financial support of the European Research Council (Consolidator Grant INFOASYMMARKETS) is gratefully acknowledged. Lucas Maestri acknowledges that this study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES), finance code 001. Lucas Maestri is also grateful for the financial support of CNPq–National Council for Scientific and Technological Development. ©2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE2449 584 Gerardi and Maestri Theoretical Economics 15 (2020) offers a menu with finitely many contracts. The worker can either accept one contract in the menu or reject all contracts and end the relationship. We show that when the discount factor is not too high, the firm is able to extract the worker’s private information independently of the value of the prior. In particular, if the prior p0is high, the firm offers a firing menu in every period. A firing menu contains only one contract, which specifies the efficient quality when the cost is low and yields a payoff equal to 0to the worker. The contract is only accepted by the low-cost worker. Thus, the firm learns the worker’s cost in the first period by firing the high-cost worker. When the prior p0is not too high, the firm employs a sequentially screening procedure. The firm offers two contracts until it discovers the worker’s cost, which occurs in finitely many periods. During the screening procedure, the high-cost worker accepts the first contract while the low-cost worker randomizes between the two contracts. Once the screening process is complete and the firm discovers the cost, the worker delivers the efficient quality and obtains a 0payoff. When the parties are sufficiently patient, the sequentially screening procedure is infeasible. Consider the last period of the screening procedure in which the firm offers a menu that fully separates the two types of worker (in the sense that each type accepts a different contract). The low-cost worker can guarantee a large future payoff by mimicking the high-cost worker. Because of this, it is impossible to design two contracts that simultaneously satisfy the truthtelling constraints of the two types of worker. Only a sufficiently generous contract can prevent the low-cost worker from imitating the highcost worker. But in this case, the high-cost worker has an incentive to adopt the “take the money and run” strategy (i.e., accept the contract designed for the low-cost worker and then quit the relationship).1 The firm could, in principle, adopt more complex dynamic screening strategies. For instance, in the initial phase of the relationship, both types of worker could accept different contracts in the menu with positive, but different, probabilities. Then the firm could use the available information to induce partial or even complete separation (through a firing menu). To investigate the feasibility and optimality of such strategies, we analyze the limiting outcome, as the parties become arbitrarily patient, of all perfect Bayesian equilibria. We show that the limiting equilibrium outcome is unique and takes a very simple form. If the prior is below a certain threshold ˆ p, then, in every period, the firm offers the most profitable contract that the high-cost worker is willing to accept. Both types of worker accept the contract (i.e., they pool) and there is no learning. In contrast, if the prior is above ˆ p, the firm offers the firing menu and the high-cost worker quits the relationship without delay. In both cases, the limiting equilibrium allocation is inefficient. Our results illustrate that when the parties are sufficiently patient, the firm can only screen the worker’s type by firing the high-cost worker. The driving forces behind our findings are similar to those that prevent full separation. When the discount factor is large, it is very costly for the firm to separate the two types of worker and continue the relationship with both of them. A lasting relationship with the high-cost worker provides 1A similar result appears in Laffont and Tirole (1990). Theoretical Economics 15 (2020) Dynamic contracting 585 strong incentives to the low-cost worker to misrepresent his information. Using this fact, we show that the firm would not benefit from engaging in partial screening strategies, even if such policies were feasible. Our benchmark model assumes that the relationship ends when the worker rejects all the contracts in the firm’s menu. This modeling assumption captures situations in which the parties are committed to terminate the relationship upon disagreement. Of course, one can also imagine situations in which the relationship continues even when the parties do not reach an agreement. Therefore, we analyze an extension of the model that allows for rehiring. We study the infinitely repeated game in which, in each period, the firm proposes a menu of contracts from which the worker has to select at most one. We first show that the complete-information version of this game admits a folk theorem. Although the firm has the bargaining power to make offers, the worker can obtain large payoffs by rejecting unfavorable contracts. This is possible because the acceptance of unfavorable contracts by the worker triggers a continuation equilibrium in which the firm implements an efficient allocation that yields a zero payoff to the worker. We use these findings from the complete-information game to show that a version of the folk theorem holds for our model with rehiring.2In particular, when the parties are sufficiently patient, the firm can obtain a payoff arbitrarily close to the payoff of the optimal mechanism with commitment. This paper contributes to the literature on repeated adverse selection with limited commitment pioneered by Freixas et al. (1985), Gibbons (1987), and Laffont and Tirole (1987,1988). In these seminal papers, the parties interact for two periods. One of the main findings is that there is partial separation of the agent’s types in the first period (i.e., the equilibrium is semipooling) and full separation in the second and final period. Therefore, the outcome of two-period environments presents gradual information revelation. In contrast, our paper shows that when the relationship is infinitely repeated and the prior is low, the equilibrium allocation is close to a pooling allocation when the parties are patient. Hart and Tirole (1988) analyze a dynamic model in which the seller makes a rental offer to the buyer in every period. The buyer’s valuation for the good is private information and can take on two values, both of which are larger than the seller’s cost of producing the good. As the parties become sufficiently patient, the equilibrium allocation converges to the efficient allocation, in which both types of buyers consume the good in every period. Note that for high values of the prior, this pooling allocation coincides with the seller’s optimal mechanism under full commitment (i.e., lack of commitment is not detrimental to the seller’s payoff). In a recent paper, Beccuti and Möller (2018)extend Hart and Tirole’s analysis to the case in which the seller is more patient than the buyer. Halac (2012) studies a relational contract model in which the principal is privately informed about his outside option. When the uninformed party has the bargaining power, Coasian forces lead to a pooling outcome when the parties are sufficiently patient. Our work differs from these papers in two respects. First, in our model, the agent’s private 2This finding is reminiscent of earlier contributions to repeated games with incomplete information and simultaneous moves (see Peski 2008 and the references therein). 586 Gerardi and Maestri Theoretical Economics 15 (2020) information is necessary to determine the best course of action and, therefore, pooling allocations are never optimal for the firm under full commitment. Second, we analyze environments in which the ratchet effect leads to inefficiencies.3 Our work is also related to the literature on renegotiation. The seminal paper by Laffont and Tirole (1990) analyzes a two-period model. Recently, Strulovici (2017)and Maestri (2017) study renegotiation in infinite horizon models. These studies find that equilibrium allocations become efficient as the parties become arbitrarily patient. In contrast, in our model the limit allocation is inefficient whenever the firing allocation is not a commitment solution. Bhaskar (2014) studies learning in a dynamic model in which the principal and the agent are ex ante symmetrically informed about the job’s difficulty. When the agent’s effort is unobservable, it is impossible for the principal to design a contract that induces an interior effort level in the first period. Bhaskar and Mailath (2019) consider a related dynamic model and show that inducing high effort becomes prohibitively costly for the principal as the parties become arbitrarily patient. Therefore, the ratchet effect imposes stringent constraints on the learning process of the relationship. In contrast, our paper assumes adverse selection and no exogenous learning, and concludes that the ratchet effect imposes constraints on the amount of private information that is revealed in a dynamic relationship. There is also a connection between our paper and the literature on durable goods monopoly under limited commitment. Ausubel and Deneckere (1989)studyamodel in which the seller posts prices and obtain a folk theorem for the “no gap” case. In our context, a folk theorem holds when rehiring is possible. Skreta (2006,2015)analyzes more general selling mechanisms and shows that posting a price is the seller’s optimal strategy. In these studies, the relationship between the buyer and the seller ends as soon as the durable good is traded, while in our model the parties can make a new transaction every period. Finally, a number of authors have identified situations in which the ratchet effect is mitigated. Kanemoto and MacLeod (1992) argue that competition for secondhand workers guarantees the existence of efficient piece-rate contracts in long-term relationships. Carmichael and MacLeod (2000) show that if entry in a market is difficult, then it is possible to sustain cooperation between an infinitely lived firm and a stream of short lived workers. Our findings suggest that rehiring is another possible remedy to the ratchet effect. The rest of the paper is organized as follows. We present the model in Section 2. In Section 3, we briefly discuss the mechanism design problem with commitment. In Section 4, we show existence of equilibria and provide conditions under which all private information is revealed. Section 5 contains the main result, which completely characterizes the unique equilibrium outcome when the parties are arbitrarily patient. In Section 6, we analyze the extension of the model in which rehiring is possible. Section 7 concludes. Most proofs are relegated to a number of appendices and 3Our work analyzes the relationship between two infinitely lived players. In the context of political economy, several papers study the effects of limited commitment in repeated interactions between one principal and a continuum of privately informed agents (see, among others, Acemoglu et al. 2010,Farhi et al. 2012, and Scheuer and Wolitzky 2016). Theoretical Economics 15 (2020) Dynamic contracting 587 the Supplemental Material, available in a supplementary file on the journal website, http://econtheory.org/supp/2449/supplement.pdf. 2. The model We study a dynamic principal–agent model with adverse selection and short-term contracts, framed in the context of a labor relationship between a firm and a worker. The worker is privately informed about his (persistent) type, which is equal to Lwith prior probability p0∈(01)and is equal to Hwith probability 1−p0. The firm and the worker interact for potentially infinitely many periods. In each period, the worker of type i∈{HL}can produce a good of quality q∈[01]at a cost of θiq,where0< θL<θ H. We refer to the low type L(high type H) as the low- (high-) cost worker. We write θ := θH−θL. The worker bears an additional cost α>0in every period in which he interacts with the firm. The cost αcan be interpreted as the per-period payoff of an outside option available to the worker if he ends the relationship (in Section 7,we discuss the case α=0). The firm’s valuation of a good of quality qis v(q).Thefunctionv:[01]→R+is twice continuously differentiable, increasing, strictly concave, and satisfies v(0)=0.4 Both parties’ preferences are linear in money. When the worker produces a good of quality qand the firm makes a transfer equal to x, the payoff of type i∈{HL}is x−θiq−α, while the firm’s payoff is v(q) −x. We let q∗ i,i∈{HL}, denote the efficient quality produced by type i: q∗ i=argmax q∈[01]v(q) −θiq To make the problem interesting, we assume vq∗ H−θHq∗ H−α>0 This assumption guarantees that the firm prefers hiring the high-cost worker over collecting its outside option, which yields a payoff equal to 0. Moreover, we assume that q∗ H∈(01)and, therefore, q∗ L>q ∗ H.5In this case, the efficient allocation varies with worker’s type. The firm and the worker play the following game. At the beginning of period t= 01, the firm offers a menu mtof contracts to the worker. Each contract is of the form (xtqt)and specifies the transfer xtpaid by the firm and the quality qt∈[01]that the worker must produce. We assume that the quality is verifiable and, thus, each contract is enforceable. After receiving the menu mt, the worker has two options: (i) selecting a 4The concavity of v(·)guarantees that the firm’s screening problem in the proof of Proposition 1 is well behaved. The concavity also allows us to derive a number of useful bounds in the proof of Proposition 2. Finally, the assumption v(0)<∞implies that for large values of the prior, the solution to the mechanism design problem with commitment is to fire the high type (see Section 3). This is used in the proof of the main result. 5In particular, we use this assumption in the proof of Proposition 1 to construct a sequence of separating contracts. 588 Gerardi and Maestri Theoretical Economics 15 (2020) contract from the menu or (ii) rejecting all the contracts and quitting the relationship. In the first case, the game moves to the next period t+1. In the second case, the game ends and both parties obtain a continuation payoff equal to 0. The parties discount future payoffs at the common discount factor δ∈(01). We let M=M j=1(R×[01])jdenote the set of available menus, where M∈{23} is an exogenous upper bound to the number of contracts that a menu can contain. The restriction M≥2guarantees that the menus can contain two contracts (so that it is possible for the firm to separate the two worker types). When the firm offers the menu mt, the set of actions available to the worker is mt∪{∅},where∅denotes the choice of rejecting all the contracts in mtand quitting. We let atdenote the agent’s decision in period t. For every t=12,aperiod-t(nonfinal) public history ht=(m0a0mt−1at−1) consists of all the menus offered by the firm in the previous periods τ=0t −1,as well as all the worker’s decisions, provided that he never chose to quit (i.e., aτ= ∅for every τ=0t −1). We let H0={h0}denote the set containing the empty history h0. We write Htfor the set of all period-tpublic histories. Finally, H=t=01 Htis the set of all (nonfinal) public histories. A behavior strategy σFfor the firm is a sequence {σF t},whereσF tis a function from Htinto (M), mapping the history htinto a (possibly random) menu. A behavior strategy (σHσL)for the worker is a sequence {(σH tσL t)},whereσi t,i∈{HL}, associates to every pair (htmt)∈Ht×Ma probability distribution over the set mt∪{∅}.Wewrite σ=(σFσHσL)for a strategy profile. Finally, we let μ={μ(ht)μ(htmt)}ht∈Hmt∈M denote the firm’s system of beliefs, where μ(ht)and μ(htmt)represent the probability that the firm assigns, at the history htand (htmt), respectively, to the event that the worker’s type is equal to L. Our solution concept is perfect Bayesian equilibrium (PBE or equilibrium henceforth), formally defined below. Definition 1. A PBE of our game is a strategy profile σand a system of beliefs μsuch that the following statements hold: (i) The strategy profile σis sequentially rational given μ. (ii) For every history (htmt)∈Ht×M,μ(htmt)=μ(ht). (iii) For every history (htmt)∈Ht×Mand for every action at∈mt∪{∅},if 1−μhtσH tat|htmt+μhtσL tat|htmt>0 then the belief μ(htmtat)is derived from μ(ht)according to Bayes’ rule: μhtmtat=μhtσL tat|htmt 1−μhtσH tat|htmt+μhtσL tat|htmt In addition to sequential rationality and Bayesian updating whenever possible (i.e., including off-path histories (htmtat)that are reached with positive probability given Theoretical Economics 15 (2020) Dynamic contracting 589 (htmt)), the concept of PBE imposes the “no signaling what you don’t know” condition (Fudenberg and Tirole (1991)) in the sense that the firm does not revise its belief after proposing a menu. Given a strategy profile σand a system of beliefs μ, for each history ht,welet VF(ht;(σμ)) denote the firm’s continuation payoff at ht.WealsoletT∈N∪{∞}denote the random period in which the relationship terminates (we set T=∞if the worker remains employed forever).6Then we have VFht;(σ μ):= E(σμ)(1−δ) T−1  τ=t δτ−tv(qτ)−xτht where E(σμ)[Y|ht]represents the conditional expected value (given ht)oftherandom variable Ygiven the strategy profile σand the system of beliefs μ. Analogously, for every history htwe let Wi(ht;(σμ)) denote the expected continuation payoff at htof the worker of type i∈{HL}.Wehave Wiht;(σ μ):= E(σμ)(1−δ) T−1  τ=t δτ−t(xτ−θiqτ−α) i ht To simplify the notation, we omit the argument (σ μ) and write VF(ht)and Wi(ht) when there is no ambiguity. We also use VF(htmt)and Wi(htmt),i∈{HL},todenote the firm’s and worker’s payoff at the history (htmt). For i∈{HL}and q∈[01],welet πi(q) := v(q) −θiq−α denote the firm’s profits when the quality is q,theworkerisoftypei,andthefirmpays the reservation wage θiq+α. Therefore, πi(q∗ i)represents the highest level of profits that the firm can achieve from the interaction with type i.AsπL(q∗ L)>π H(q∗ H),let ˆ p∈(01) be defined by πH(q∗ H)=ˆ pπL(q∗ L). We conclude this section with a simple result that provides a lower bound to the firm’s payoff under any PBE. Lemma 1. Fix a PBE (σμ). For every history ht∈H, we have VFht;(σ μ)≥maxπHq∗ HμhtπLq∗ L Proof. By contradiction, suppose that there exist a PBE (σμ),ahistoryht,andε>0 such that VFht;(σ μ)<maxπHq∗ HμhtπLq∗ L−ε Suppose that πH(q∗ H)>μ(h t)πL(q∗ L). If the firm offers the menu {(θHq∗ H+α+ ε 2q∗ H)}in every period tt +1 (notice that both types strictly prefer to accept the 6Here, and in what follows, we use N={01}to denote the set of nonnegative integers and adopt the convention that t−1 τ=tδτ−t=0. 590 Gerardi and Maestri Theoretical Economics 15 (2020) contract in the menu rather than quit the relationship), then its continuation payoff is πHq∗ H−ε 2>V Fht;(σ μ) which is a contradiction. Next consider the case πH(q∗ H)≤μ(ht)πL(q∗ L). The firm can guarantee a continuation payoff at least equal to μhtπLq∗ L−ε 2>V Fht;(σ μ) by offering the menu {(θLq∗ L+α+ε 2q∗ L)}in every period tt +1(in equilibrium, the low type must accept the contract in the menu). Intuitively, the following two options are always available to the firm. The first option is to stop learning and offer (θHq∗ H+α q∗ H), the most profitable contract in the class of contracts that are accepted by both types of worker. The second option is to fire the highcost worker and interact only with the low-cost worker. In this case, the most profitable contract is (θLq∗ L+α q∗ L). 3. The commitment allocation It is useful to review the benchmark model in which the firm can fully commit to a sequence of menus (m0m1). This benchmark provides an upper bound to the firm’s profits in the game with limited commitment. It is well known that the solution to the firm’s commitment problem is to replicate the optimal static mechanism (see, for example, Chapter 1 in Laffont and Tirole 1993). The optimal static mechanism is as follows. There exists a critical value pC∈(ˆ p1) such that if p0>p C, the optimal menu (with commitment) is unique and equal to {(θLq∗ L+α q∗ L)}.7The low-cost worker accepts the contract in the menu while the highcost worker rejects it. Thus, the firm’s profits are equal to p0πL(q∗ L). Alternatively, if p0<p C, then the unique optimal menu is xC HqC HxC LqC L=θHqC H+α qC HθLq∗ L+θqC H+α q∗ L (1) for some qC H∈(0q∗ H). The high-cost worker accepts the first contract and obtains a payoff equal to 0. The low-cost worker is indifferent between the two contracts (therefore, he obtains a payoff equal to θqC H) and accepts the second contract. In this case, the firm’s commitment profits are equal to p0vq∗ L−θLq∗ L−θqC H−α+(1−p0)vqC H−θHqC H−α Finally, if p0=pC, then there are two optimal deterministic mechanisms: {(θLq∗ L+ α q∗ L)}and {(xC HqC H)(xC LqC L)}as in (1). In addition, there is a continuum of optimal 7To see why pC>ˆ p,letVC F(p) be the commitment payoff of the firm when the prior is p. The function VC F(·)is strictly increasing. If pC≤ˆ p, then we obtain the contradiction VC F(pC)=pCπL(q∗ L)≤ˆ pπL(q∗ L)= πH(q∗ H)=VC F(0). Theoretical Economics 15 (2020) Dynamic contracting 597 Consider a history htat which the firm offers a menu mtcontaining a contract that leads to a firing region. Lemma 4 provides bounds for the length of the high type’s relationship and for the continuation payoffs conditional on the menu mtbeing offered. The proof of Lemma 4 is given in Appendix B. The driving force behind Lemma 4 is similar to that behind Lemma 3, which establishes that separation with employment cannot occur for large values of δ. The intuition is as follows. Following the acceptance of a contract (xL tqL t)that leads to the firing region, the low type’s continuation payoff is close to 0. Suppose the firm’s relationship with the high type is long lasting. In this case, only a large transfer xL tcan prevent the low type from mimicking the high type. But then it becomes profitable for the high type to accept the contract (xL tqL t)and then quit. We now turn to the inductive step. For every p≥ˆ p,letf(p)∈[0p−ˆ p]be defined by f(p) pπ H(0)+1−f(p) pπHq∗ H=p−f(p)πLq∗ L(2) The function f:[ˆ p1]→[01−ˆ p]is strictly increasing and satisfies f(ˆ p) =0and f(p)<p−ˆ pfor every p> ˆ p.17 Lemma 5. Suppose that the interval [p 1],p∈(ˆ p1), is a firing region. Then [p−f(p) 21] is also a firing region. The rest of the section provides the proof of Lemma 5, which consists of several steps. We outline in detail each step and relegate some technical arguments to Appendix B. The continuation play starting at some history htis an equilibrium of the original game (when the prior is equal to the firm’s belief at ht). It is, therefore, without loss of generality (and convenient in terms of notation) to establish the three properties of a firing region for the initial history h0. We first show that the expected discounted length of the high type’s relationship shrinks to 0(property (i) of Definition 2). This is the main part of the proof of Lemma 5 because, as we show later, the remaining two properties follow from the first. Any equilibrium (σμ) must satisfy the following two conditions. First, the firm’s equilibrium payoff VF(h0)must be at least equal to p0πL(q∗ L). Second, the low type must prefer his strategy to mimicking the high type. Thus, WL(h0)≥WLH(h0),where WLH(h0)denotes the low type’s continuation payoff at h0if he mimics the high type (at every history). We show that if [p1]is a firing region, the prior p0is above p−f(p) 2,and δis close to 1, the two conditions just mentioned can be simultaneously satisfied only if the expected discounted length of the high type’s relationship is close to 0. We proceed in four steps. The payoffs VF(h0),WL(h0),andWLH(h0),andtheexpected length of the relationship are complicated objects since they depend on the entire history of the game. Instead of working directly with these variables, we replace them with some bounds that are easy to express and compare. We do this in the first three steps. In Step 1, we show that it is without loss of generality to restrict attention to a class of simple equilibria. In Step 2, we provide bounds to the continuation payoffs 17Recall that the function πH(·)is concave and, therefore, π H(0)>π  H(0)q∗ H≥πH(q∗ H). 598 Gerardi and Maestri Theoretical Economics 15 (2020) and to the length of the relationship with the high type. These bounds are derived by changing the timing of the transfers in the equilibrium of the game. In Step 3, we introduce an auxiliary game that allows for a simpler expression of these bounds. Finally, in Step 4, we use these bounds to establish property (i) of Definition 2. Step 1: Restriction to a class of simple equilibria. We now show that to establish property (i) of Definition 2 it is without loss of generality to restrict attention to equilibria in which (a) the firm’s strategy in the first period is pure (i.e., the firm does not randomize among different menus at t=0) and (b) the high type’s equilibrium payoff is equal to zero. To see why restriction (a) is without loss, suppose that ((σFσHσL) μ) is a PBE and m0is a menu offered with positive probability by the firm at t=0(σF 0(m0|h0)> 0). Let ˜σFbe the strategy that is identical to σFin every period except the first, in which the firm instead offers the menu m0with certainty ( ˜σF 0(m0|h0)=1). The assessment (( ˜σFσHσL) μ) is also a PBE. Furthermore, the outcome of the equilibrium (( ˜σFσHσL) μ) coincides with the continuation outcome of ((σFσHσL) μ) after the firm proposes the contract m0. Therefore, if it is impossible to establish the first property for arbitrary PBE, then it is also impossible to establish the property for the class of equilibria that satisfy restriction (a). We show next that restriction (b) is also without loss of generality. Suppose that (σ μ) is a PBE in which the firm offers the menu m0(with probability 1) in the first period and that yields a strictly positive payoff WH(h0;(σμ)) to the high type. Then it is possible to construct a new PBE (˜σ ˜μ) that is outcome equivalent to (σμ),exceptforthe fact that the first-period transfers are uniformly decreased by (1−δ)−1WH(h0;(σμ)). In other words, in the first period, the firm replaces every contract (x0q0)in the menu m0with the contract (x0−(1−δ)−1WH(h0;(σμ))q0).18 Finally, notice that the first property of a firing region does not depend on equilibrium transfers. Step 2: Change in the timing of transfers. This step provides bounds to the payoffs and to the length of the relationship that depend only on the quality of the goods delivered by the worker. Let ˜ T∈N∪{∞}denote the random time that stops the play at the first history (h˜ Tm˜ T)at which the menu m˜ Tcontains a contract (x˜ Tq˜ T)accepted with positive probability and for which μ(h˜ Tm˜ T(x˜ Tq˜ T)) ≥p(we set ˜ T=∞if the event does not occur in finite time). We claim that there exist K>0and ¯ δ<1such that for δ> ¯ δ,wehave VFh0≤¯ VFh0:= E(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ Tμh˜ TπLq∗ L+K(1−δ) WLh0≤¯ WLh0:= E(σμ)(1−δ) ˜ T−1  t=0 δtθqt|L+K(1−δ) (3) WLHh0≥WLH h0:= E(σμ)(1−δ) ˜ T−1  t=0 δtθqt|H−K(1−δ) 18In the new equilibrium (˜σ ˜μ), each type of worker accepts the contract (x0−(1− δ)−1WH(h0;(σμ)) q0)with the same probability with which he accepts the contract (x0q0)in the original equilibrium (σμ). Theoretical Economics 15 (2020) Dynamic contracting 599 and E(σμ)(1−δ) T−1  t=0 δt|h0H≤E(σμ)(1−δ) ˜ T−1  t=0 δt|h0H+K(1−δ) (4) Theboundsonthepayoffsprovidedin(3) are derived as follows (the formal derivation is provided in the Appendix). Recall that the high type’s equilibrium payoff WH(h0) is equal to 0.Letm0be the menu offered by the firm at h0and consider any contract (x0q0)∈m0accepted by the high type with positive probability. Let h1denote the history (h0m0(x0q0)).IfWH(h1)=0,thenwehavex0=θHq0+α. If instead WH(h1)>0, then we increase the transfer x0by the amount δ (1−δ) WH(h1). The new transfer is equal to θHq0+α. At the same time, for every menu m1offered at h1,wedecreaseallthetransfers of the contracts in m1by the amount 1 (1−δ) WH(h1m1). These changes leave all parties’ continuation payoffs unchanged. We repeat this procedure in periods 1 ˜ T−1.Thus, for every t=0 ˜ T−1, the new transfer in period tis equal to θHqt+α. Furthermore, it follows from Lemma 4 that at the history (h ˜ Tm˜ T),ifδis close to 1, the firm’s continuation payoff is close to μ(h˜ T)πLq∗ L, while the worker’s payoff is close to 0.19 Combining the expressions of the new transfers with the findings on the continuation payoffs we obtain the bounds in (3). The bound on the expected length of the high type’s relationship (inequality (4)) follows directly from Lemma 4 (property (i)). Step 3: The auxiliary game. We introduce an auxiliary game (a direct mechanism) that replicates the equilibrium outcome from period 0through period ˜ T−1and implements the payoffs ¯ VF(h0),¯ WL(h0),andWLH(h0). In particular, we consider a mechanism in which the worker reveals his type to a designer. The designer selects a history and one of two messages, m0and mp(see below for more details about the mechanism). Let ϒz,forz∈{0p}, denote the expected discounted length of the relationship conditional on the message mz,andlet ˜ qzdenote the expected discounted quality of the good conditional on mz. Using the auxiliary game, we show that ¯ VFh0≤˘ VFh0 := 1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) ¯ WLh0=ϒp˜ qpθ +K(1−δ) WLHh0=1 1−p01−p0 pϒ0˜ q0θ +1−p 1−p0p0 pϒp˜ qpθ −K(1−δ) (5) and E(σμ)(1−δ) T−1  t=0 δt|h0H ≤1 1−p01−p0 pϒ0+1−p 1−p0p0 pϒp+K(1−δ) (6) 19In fact, the high type’s payoff is bounded above by the low type’s payoff, which, in turn, is close to 0(see property (iii) in Lemma 4). 600 Gerardi and Maestri Theoretical Economics 15 (2020) This greatly simplifies our problem since it reduces to four the number of endogenous variables (ϒzand ˜ qz,z∈{0p}) in the expressions of the bounds. We then use the expressions in (5)and(6) to show that the inequalities ˘ VF(h0)≥p0πL(q∗ L)(recall that ˘ VF(h0)≥¯ VF(h0)≥VF(h0))and ¯ WL(h0)≥WLH(h0)can be satisfied only if the right hand side of inequality (6) shrinks to 0(as δgoes to 1), establishing, therefore, the first property of a firing region. We now describe the direct mechanism. The worker reveals his private information to a designer who, in turn, chooses an outcome and reports it to the firm. The outcome consists of a history h˜ Tof the game and a message in {m0mp}. The designer chooses the outcomes in such a way that upon receiving message mz, the firm’s belief is equal to z. Moreover, the likelihoods of the histories reflect those from the equilibrium of the game. Specifically, if the worker announces the low type, then the designer chooses the outcome (h˜ Tmp)with probability Pr(h˜ T|L), the probability of the history h˜ Twhen the worker’s type is low and the parties play the equilibrium (σμ). Alternatively, if the worker announces the high type, then the designer chooses the outcome (h˜ Tmp)with probability Pr(h˜ T|H)μ(h˜ T)(1−p) (1−μ(h˜ T))p and the outcome (h˜ Tm0)with probability Pr(h˜ T|H)[1− μ(h˜ T)(1−p) (1−μ(h˜ T))p ]. We now turn to the payoffs of the firm and the low type. The firm’s payoff depends only on the outcome and not on the worker’s report. Consider an arbitrary history h˜ T= (m0(x0q0)m˜ T(x˜ T−1q˜ T−1)). If the outcome is (h ˜ Tmp), the firm’s payoff is equal to (1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ TpπLq∗ L+K(1−δ) If the outcome is (h ˜ Tm0), the firm’s payoff is equal to (1−δ) ˜ T−1  t=0 δtπH(qt)+K(1−δ) We conclude that if every type reveals his type truthfully, then the firm’s expected payoff is equal to ¯ VF(h0). Consider now the low type. His payoff depends both on the outcome and on his report. First, if the outcome is either (h˜ Tmp)or (h˜ Tm0),withh˜ T=(m0(x0q0)m˜ T (x˜ T−1q˜ T−1)), then the low type obtains a payoff equal to (1−δ) ˜ T−1  t=0 δtθqt In addition, the low type obtains an extra payoff equal to K(1−δ) if he is honest, and equal to −K(1−δ) if he lies to the designer. It follows that the low type’s expected payoff is equal to ¯ WL(h0)if he reveals his type truthfully and is equal to WLH (h0)if he lies to the designer. Finally, we assume that the high type is committed to truthfully reporting his type to the designer. Theoretical Economics 15 (2020) Dynamic contracting 601 It is natural to ask why we introduced the messages m0and mpin the mechanism, given that they do not affect the worker’s payoffs and the firm is a passive player in the auxiliary game. The reason for this is that the additional messages allow us to classify all the histories h˜ Tinto two large classes, depending on whether they are associated with the message m0or the message mp. Recall that when the firm observes the message mz, z∈{0p}, its belief is equal to z. Thus, by the martingale property of the beliefs (see Aumann and Maschler 1995 and Kamenica and Gentzkow 2011), we conclude that the probability of observing the message m0is equal to (1−p0 p), while the probability of observing the message mpis equal p0 p. Below, we use this fact to express the bounds on the continuation payoffs and the length of the relationship in terms of the endogenous variables ϒzand ˜ qz,z∈{0p}. First,observethatwecanrewritethefirm’spayoff ¯ VF(h0)as ¯ VFh0=1−p0 pE(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)m0 +p0 pE(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ TpπLq∗ Lmp+K(1−δ) (7) We now turn to the low type’s payoffs. Fix an arbitrary outcome (h˜ Tmp)and let Pr(h˜ Tmp)denote the ex ante probability of the outcome. Recall that the firm’s belief upon observing the outcome (h˜ Tmp)is equal to p. This immediately implies Prh˜ Tmp=p0 pPrh˜ Tmp|L=1−p0 1−pPrh˜ Tmp|H We conclude that the outcome (h˜ Tmp)is reached with probability p p0Pr(h˜ Tmp) when the worker announces that his type is low, and with probability 1−p 1−p0Pr(h˜ Tmp) when the worker announces that his type is high. Similarly, an outcome (h˜ Tm0)is reached with probability 1 1−p0Pr(h˜ Tm0)if the worker announces the high type and with probability 0if the worker announces the low type (Pr(h˜ Tm0)denotes the ex ante probability of the outcome). Combining these observations, we can rewrite the low type’s payoffs as ¯ WLh0=E(σμ)(1−δ) ˜ T−1  t=0 δtθqtmp+K(1−δ) WLHh0=1 1−p01−p0 pE(σμ)(1−δ) ˜ T−1  t=0 δtθqtm0 +1−p 1−p0p0 pE(σμ)(1−δ) ˜ T−1  t=0 δtθqtmp−K(1−δ) 602 Gerardi and Maestri Theoretical Economics 15 (2020) We next provide the formal definitions of ϒzand ˜ qzfor z∈{0p}: ϒz=E(σμ)(1−δ) ˜ T−1  t=0 δtmz and ˜ qz=1 ϒz E(σμ)(1−δ) ˜ T−1  t=0 δtqtmz if ϒz= 0,and ˜ qz=0otherwise. The definitions of ϒzand ˜ qz,z∈{0p}, allow us to express ¯ WL(h0)and WLH(h0)as in (5). The inequality ¯ VF(h0)≤˘ VF(h0)in (5) follows from (7), the concavity of the function πH(·), and Jensen’s inequality. Finally, using the definitions of ϒ0and ϒpand inequality (4), we are able to bound the length of the high type’s relationship as in (6). Step 4: Bounding the expected length of the high type’s relationship. The following claim establishes the first property of a firing region. Claim 1. Fix K>0and p> ˆ p.ThereexistsK>0such that, for every p0∈[p−f(p) 2p], for every δ,andforevery(ϒz˜ qz)∈[01]2,z∈{0p}, the inequalities 1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) ≥p0πLq∗ L(8) ϒp˜ qpθ +K(1−δ) ≥1 1−p01−p0 pϒ0˜ q0θ +1−p 1−p0p0 pϒp˜ qpθ −K(1−δ) (9) are simultaneously satisfied only if 1 1−p01−p0 pϒ0+1−p 1−p0p0 pϒp+K(1−δ) ≤K(1−δ) (10) The inequalities (8)and(9) capture the constraints ˘ VF(h0)≥p0πL(q∗ L)and ¯ WL(h0)≥ WLH(h0), respectively, while the left hand side of inequality (10) represents the upper bound to the expected discounted length of the high type’s relationship. The proof of Claim 1 is tedious and is relegated to the Appendix. The logic behind this claim is better understood when one considers the problem of maximizing ˘ VF(h0) (with respect to ϒzand ˜ qz,z∈{0p}) subject to the low type’s incentive compatibility constraint: ¯ WL(h0)≥WLH(h0). Clearly, ˘ VF(h0)is maximized by setting ϒ0equal to 1,˜ q0 equal to q∗ H,andϒpequal to 0(recall that p> ˆ pand, therefore, pπL(q∗ L)>π H(q∗ H)≥ πH(˜ qp)for any ˜ qp). However, this would violate the low type’s incentive compatibility constraint. Hence, the following trade-off emerges. To increase the firm’s payoff by increasing ϒ0while also satisfying the incentive compatibility constraint (9), it is also Theoretical Economics 15 (2020) Dynamic contracting 603 necessary to increase ϒp, which decreases the firm’s payoff. Notice that when the prior p0is close to p,(1−p0 p)ϒ0and ˜ q0have a small impact on both the firm’s payoff and the constraint. In contrast, ϒphas a small impact on the constraint and a large (negative) impact on the firm’s payoff. We conclude that for δand p0sufficiently large, the optimal values of (1−p0 p)ϒ0and ϒpmust be close to 0. Therefore, if we could maximize the firm’s payoff, subject to ¯ WL(h0)≥WLH(h0), the solution would be close to a firing allocation, yielding a payoff close to p0πL(q∗ L). For the same reason, any allocation that satisfies ¯ WL(h0)≥WLH(h0)and that is not close to a firing allocation leads to a payoff for the firm smaller than p0πL(q∗ L), hence, violating (8). We have shown that the first property of a firing region holds. In particular, there exist ¯ K>0and ¯ δ<1such that for any δ> ¯ δ, any prior p0above p−f(p) 2,andanyPBE (σ μ),wehaveE(σμ)[(1−δ)T−1 t=0δt|h0H]≤(1−δ) ¯ K. We now turn to the remaining two properties. To verify the second property (the firm’s payoff conditional on type H shrinks to 0weakly faster than 1−δ), notice that VFh0;H≤v(1)E(σμ)(1−δ) T−1  t=0 δth0H≤v(1)¯ K(1−δ) Finally, we use the result above to bound the low type’s continuation payoff WL(h0) (third property). We have p0πLq∗ L≤VFh0≤(1−p0)VFh0;H+p0πLq∗ L−WLh0 which implies WLh0≤1−p0 p0 VFh0;H<1−ˆ p ˆ pVFh0;H≤1−ˆ p ˆ pv(1)¯ K(1−δ) This concludes the proof of Lemma 5. 5.2 Low belief case: p< ˆ p In this section, we characterize the limiting equilibrium outcome when the prior is lower than ˆ p. To do so, we first define the notion of pooling region. Definition 3. The interval [0p]is a pooling region if, for every ε>0,thereexists ¯ δ<1 such that the following statement holds: Fix δ> ¯ δ, an arbitrary PBE (σμ), and a history htat which μ(ht)≤p.Thenwehave (i) E(σμ)[δT]<ε (ii) E(σμ)[(1−δ)T−1 t=0δt|qt−q∗ H|] <ε (iii) For i∈{HL},E(σμ)[(1−δ)T−1 t=0δt(xt−θHq∗ H−α)|i]<ε. When the belief is in a pooling region, the equilibrium allocation converges to the pooling allocation as δgoes to 1. Our next result shows that all beliefs lower than ˆ p belong to a pooling region. 604 Gerardi and Maestri Theoretical Economics 15 (2020) Proposition 3. For every p< ˆ p,theinterval[0p]is a pooling region. Before turning to the proof of Proposition 3, we establish a preliminary result. We show that the equilibrium belief cannot grow too quickly around ˆ pwhen the parties are sufficiently patient. Lemma 6. For every ε>0,thereexists ¯ δ<1such that the following statement holds: Fix δ> ¯ δand a PBE (σμ).Theredoesnotexistahistoryhtwith μ(ht)< ˆ p−εat which the firm offers a menu mtthat contains a contract (xtqt)accepted with positive probability and such that μ(htmt(xtqt)) > ˆ p+ε. The proof of Lemma 6 is given in Appendix B. Recall that for every ε>0, the interval [ˆ p+ε 1]is a firing region. Therefore, it follows from Lemma 4 that if δis close to 1and the belief jumps from μ(ht)< ˆ p−εto μ(htmt(xtqt)) > ˆ p+ε, the firm’s continuation payoff at htmust be close to μ(ht)πL(q∗ L). But then the firm’s payoff would be smaller than πH(q∗ H)(since μ(ht)< ˆ p−εand ˆ pπL(q∗ L)=πH(q∗ H)), contradicting Lemma 1. We now outline the proof of Proposition 3 (see Appendix B for the formal proof). As in the previous section, we simplify the notation and establish the three properties of a pooling region for the initial history h0. The first two properties do not depend on equilibrium transfers. Therefore, it is without loss of generality to establish these properties for the class of equilibria in which (a) the firm’s strategy in the first period is pure; and (b) the high type’s equilibrium payoff is equal to 0.20 We start with the first property and fix p< ˆ p. By contradiction, let us assume that there exists a sequence {δnp0n(σnμn)}∞ n=1such that δnconverges to 1,p0n ∈[0p], (σnμn)is a PBE of the game with discount factor equal to δnand prior equal to p0n,and limn→∞ E(σnμn)[δT n]=ξ>0. We show that for nsufficiently large, it is strictly profitable for the low type to deviate from the equilibrium strategy and mimic the high type. To ease the notation, in what follows we suppress the index nand write δ,p0,and(σμ) to denote an arbitrary element of the sequence. Below, we proceed as follows. First, we replace the low type’s payoffs (from the equilibrium strategy and the deviation) with bounds that depend on the expected discounted time at which the firm’s belief falls for the first time in the firing region. We then compute this expected discounted time conditional on the worker’s type. With this we show that when the parties are sufficiently patient, it is strictly profitable for the low type to deviate and mimic the high type. Fix a small εand now let ˜ T∈N∪{∞}denote the random time that stops the play at the first history (h˜ Tm˜ T)at which the menu m˜ Tcontains a contract (x˜ Tq˜ T)that is accepted with a positive probability and for which μ(h˜ Tm˜ T(x˜ Tq˜ T)) ≥ˆ p+ε. Recall that [ˆ p+ε 1]is a firing region and WH(h0)=0. We proceed similarly as in Section 5.1 by making a change in the timing of the transfers and using Lemma 4 to bound the 20The proof of this claim is identical to the proof provided in Section 5.1 and, therefore, is omitted. Theoretical Economics 15 (2020) Dynamic contracting 605 continuation payoffs at (h˜ Tm˜ T)(see the discussion after inequality (4)). For δclose to 1, the low type’s payoffs WL(h0)and WLH(h0)are bounded as WLh0≤E(σμ)(1−δ) ˜ T−1  t=0 δtθqtL+ε WLHh0≥E(σμ)(1−δ) ˜ T−1  t=0 δtθqtH−ε (11) Also, for δclose to 1,thefirm’spayoffisboundedby VFh0≤E(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ Tμh˜ TπLq∗ L+ε ≤E(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ T(ˆ p+ε)πLq∗ L+ε where the second inequality holds because the belief at the history h˜ Tis bounded above (by definition) by ˆ p+ε. Notice that εcan be arbitrarily small. Therefore, since πH(q∗ H)=ˆ pπL(q∗ L)and q∗ His the unique maximizer of πH(·), the inequality above implies that E(σμ)(1−δ) ˜ T−1  t=0 δtq∗ H−qt≈0(12) for δsufficiently large. If this were not the case, then the firm’s payoff would be strictly smaller than πH(q∗ H)(again, for δsufficiently large). Combining (11)with(12), we obtain that for δclose to 1, the upper bound of WL(h0) is close to θq∗ HE(σμ)(1−δ) ˜ T−1  t=0 δtL=θq∗ H1−E(σμ)δ˜ T|L(13) while the lower bound of WLH (h0)is close to θq∗ HE(σμ)(1−δ) ˜ T−1  t=0 δtH=θq∗ H1−E(σμ)δ˜ T|H(14) We now approximate E(σμ)[δ˜ T|L]and E(σμ)[δ˜ T|H].Wedothisintwosteps.First, we express E(σμ)[δ˜ T|i]as function of E(σμ)[δ˜ T]. Then we approximate E(σμ)[δ˜ T]. Let Pr(h˜ T)denote the (ex ante) probability of reaching the history h˜ T.Thenh˜ Tis reached with a probability μ(h˜ T) p0Pr(h˜ T)if the low type follows his strategy σL, and is 606 Gerardi and Maestri Theoretical Economics 15 (2020) reached with a probability 1−μ(h˜ T) 1−p0Pr(h˜ T)if he mimics the high type and plays the strategy σH. Also, it follows from Lemma 6 that for δclose to 1,μ(h˜ T)must be close to ˆ p. Therefore, for δsufficiently large, we have E(σμ)δ˜ T|L≈ˆ p p0 E(σμ)δ˜ T E(σμ)δ˜ T|H≈1−ˆ p 1−p0 E(σμ)δ˜ T (15) Next, we examine the relationship between E(σμ)[δT]and E(σμ)[δ˜ T]when δis close to 1. First, at the history (h˜ Tm˜ T), the expected discounted length of the relationship with the high type is close to 0(see Lemma 4 and recall that the interval [ˆ p+ε 1]is a firing region). Second, the firm’s belief at h˜ Tmust be close to ˆ p(a value of μ(h˜ T)far away from ˆ pwould contradict Lemma 6). Finally, recall that for δlarge, E(σμ)[δT]is close (by assumption) to ξ. Putting these observations together and using Bayes’ rule, we conclude that E(σμ)[δ˜ T]is close to ξ 1−ˆ pfor δclose to 1. Combining (13), (14), (15), and the last observation, we conclude that for δclose to 1, the upper bound of WL(h0)is close to θq∗ H1−ˆ p p0 ξ 1−ˆ p≤θq∗ H1−ˆ p p ξ 1−ˆ p (where the inequality follows from p0≤p), while the lower bound of WLH (h0)is close to θq∗ H1−1−ˆ p 1−p0 ξ 1−ˆ p≥θq∗ H1−1−ˆ p 1−p ξ 1−ˆ p Hence, since p< ˆ pand ξ>0,wehave θq∗ H1−ˆ p p ξ 1−ˆ p<θq ∗ H1−1−ˆ p 1−p ξ 1−ˆ p which implies the existence of a profitable deviation for values of δclose to 1. Finally, the second and third properties of a pooling region are direct consequences of the first property. Intuitively, if the high type never quits the relationship, the best option for the firm is to implement the best pooling allocation. 6. Rehiring In the model analyzed so far, the worker’s decision to reject all the contracts in the menu is an irreversible action that ends the relationship. In other words, the firm cannot rehire the worker after a period of unemployment. As we argued in the previous two sections, this impairs the firm’s ability to screen the worker. Once the worker reveals his type, his continuation payoff must be equal to 0. The firm can afford to pay the reservation wage, because the worker has no alternative but to end the relationship. This logic does not apply when rehiring is possible. In this case, the worker can credibly threaten the firm to reject offers that pay slightly above the reservation wage, Theoretical Economics 15 (2020) Dynamic contracting 613 We now briefly turn to the case δ> ˆ δand the case p=¯ p=ˆ p(when δ≤ˆ δ). Recall the definitions of Vand in (16)and(17), respectively. For every belief p, we define the setofmenusm(p) as follows. If p< ˆ p,thesetm(p) contains only the menu m(p) = {(θHq∗ H+α q∗ H)}.Ifp> ˆ p,m(p) contains only the menu m(p) ={(θLq∗ L+α q∗ L)}. Finally, the set m(ˆ p) contains both the menu m( ˆ p) ={(θLq∗ L+α q∗ L)}and the menu m(ˆ p) ={(θHq∗ H+α q∗ H)}. The equilibrium strategies and beliefs are defined similarly to the case p<¯ pabove and we omit the details. Appendix B Proof of Lemma 4.FixaPBE(σμ) and a history ht(μ(ht)<p)atwhichthefirmoffers a menu mtwith the properties described in the statement of the lemma. First, notice that if the high type rejects all the contracts in mtwith probability 1 (i.e., the high type quits), then the high type’s length of the relationship, the firm’s payoff (conditional on the high type), and the low type’s payoff are all equal to 0(if the low type’s payoff is strictly positive, the firm’s payoff would fall below μ(ht)πL(q∗ L)). Consider now the case in which the high type accepts a contract in mt,say(xH tqH t), with positive probability. We let ht+1 Hdenote the history (htmt(xH tqH t)).Wealsolet ht+1 Ldenote the history (htmt(xL tqL t)). The fact that type i∈{HL}accepts with positive probability the contract (xi tqi t) implies that (1−δ)xH t−θHqH t−α+δWHht+1 H≥(1−δ)xL t−θHqL t−α+δWHht+1 L and (1−δ)xL t−θLqL t−α+δWLht+1 L≥(1−δ)xH t−θLqH t−α+δWLht+1 H We add the two incentive compatibility constraints and obtain (1−δ)θqL t−qH t+δWLht+1 L−WHht+1 L≥δWLht+1 H−WHht+1 H Recall that μ(ht+1 L)≥pand that [p 1]is a firing region. Therefore, there exist ¯ Kand ¯ δ<1such that WL(ht+1 L)≤¯ K(1−δ) for δ> ¯ δ.Ofcourse,WH(ht+1 L)≥0. This, together with the above inequality, implies (1−δ)θqL t−qH t+¯ K(1−δ) ≥δWLht+1 H−WHht+1 H(21) We now let DH(ht+1 H):= E(σμ)[(1−δ)T−1 τ=t+1δτ−t−1|ht+1 HH]denote the expected discounted time, computed at ht+1 H, until the high type quits. Our next goal is to provide an upper bound to DH(ht+1 H). Thus, without loss, assume that DH(ht+1 H)is strictly positive. We let Qt+1= E(σμ)(1−δ) T−1  τ=t+1 δτ−t−1qτht+1 HH DHht+1 H 614 Gerardi and Maestri Theoretical Economics 15 (2020) denote the expected discounted total quality provided by the high type at the history ht+1 H. Using Jensen’s inequality (recall that the function π(·)is concave), we can bound the firm’s continuation payoff (conditional on type H)as VFht+1 H;H≤E(σμ)(1−δ) T−1  τ=t+1 δτ−t−1π(qτ)ht+1 HH ≤E(σμ)(1−δ) T−1  τ=t+1 δτ−t−1π(Qt+1)ht+1 HH=DHht+1 Hπ(Qt+1) Let ˘ qH∈(0q∗ H)be such that πH(˘ qH)=0and notice that πH(q) < 0for every q<˘ qH. This implies that Qt+1≥˘ qH. In fact, if the last inequality is violated, then VF(ht+1 H;H) is strictly negative and VF(ht+1 H)is strictly less than μ(ht+1 H)πL(q∗ L),contradicting Lemma 1. Notice that one strategy available to the low type is to imitate the high type’s behavior (in every period). Therefore, we conclude that WLht+1 H−WHht+1 H≥θDHht+1 HQt+1≥θDHht+1 H˘ qH Combining the inequality above with inequality (21), we obtain DHht+1 H≤(1−δ)qL t−qH t δ˘ qH +¯ K(1−δ) δθ ˘ qH ≤(1−δ) δ˘ qH1+¯ K θ Hence, for δ>1 2,wehave DHht+1 H≤2(1−δ) ˘ qH1+¯ K θ This, in turn, implies that (for δ>1 2) E(σμ)(1−δ) T−1  τ=t δτ−thtmtH≤(1−δ) +DHht+1 H ≤1+2 ˘ qH +2¯ K θ ˘ qH(1−δ) := ˜ K(1−δ) and establishes part (i). To verify property (ii), notice that the inequality above implies that the firm’s continuation payoff VF(htmt;(σμ)H) is bounded above by v(1)˜ K(1−δ). Finally, we turn to property (iii). The analysis above implies that VFhtmt≤μhtπLq∗ L−WLhtmt+v(1)˜ K(1−δ) (22) Let δbe such that v(1)˜ K1−δ=πHq∗ H 4 Theoretical Economics 15 (2020) Dynamic contracting 615 and notice that for δ≥˜ δ=max{δ¯ δ 1 2}and μ(ht)≤πH(q∗ H) 2πL(q∗ L), μhtπLq∗ L+v(1)˜ K(1−δ) ≤3 4πHq∗ H It follows that if the firm offers the menu mtat the history htand δ≥˜ δ,then μ(ht)> πH(q∗ H) 2πL(q∗ L). Finally, recall that VF(htmt)is bounded below by μ(ht)πL(q∗ L).This and inequality (22)imply WLhtmt≤v(1) μht˜ K(1−δ) ≤2πLq∗ Lv(1)˜ K πHq∗ H(1−δ) This shows that there exists K>0that satisfies the three properties in Lemma 4. Derivation of the inequalities in (3). First, notice that WH(h0)=0implies that the menu m0offered at h0yields a continuation payoff of 0to the high type. Let m0= ((x1 0q1 0)(xk 0qk 0)) be the menu offered at h0.Fori∈{HL},letmi 0denote the set of contracts in m0accepted with positive probability by the type i.Ifthereexistsacontract (xj 0qj 0)in mL 0\mH 0, then ˜ Tcoincides with 0(in fact, the firm’s belief jumps to 1if the worker accepts the contract (xj 0qj 0)) and there is nothing to prove. Therefore, assume that mL 0⊆mH 0and recall that mL 0is nonempty. For every contract (xj 0qj 0)in mH 0,weleth1 j=(h0m0(xj 0qj 0)) denote the history in which the worker accepts the contract (xj 0qj 0)in period 0and we recall that WH(h1 j)≥ 0represents the high type worker’s payoff at h1 j. For every contract (xj 0qj 0)∈mH 0,we replace the payment xj 0with the payment ˜ xj 0=xj 0+δ 1−δWHh1 j which clearly implies ˜ xj 0=θHqj 0+α. To keep the parties’ payoffs unchanged, we also modify the payments in period 1.In particular, for every (xj 0qj 0)∈mH 0, consider the history h1 j.Letm1denote a menu offered at h1 jwith positive probability. We subtract 1 1−δWH(h1 jm1), the high type’s continuation payoff at the moment that the menu m1is offered, from the payment of every contract in m1. Notice that this yields to the high type a continuation payoff (evaluated at the beginning of period 1)equalto0. We recursively apply the procedure outlined above to periods t=1 ˜ T(i.e., we increase the payments in period tand, at the same time, decrease the payments in period t+1). By construction, every contract (xtqt)accepted with positive probability by thehightypeinperiodt=0 ˜ T−1is replaced with the contract (θHqt+α qt), while the payments in every menu offered in period ˜ Tare uniformly decreased by the high type’s continuation payoff. Finally, in every period t=0 ˜ T−1, the set of contracts accepted by the low type is contained in the set of contracts accepted by the high type (this follows from the definition of ˜ T). 616 Gerardi and Maestri Theoretical Economics 15 (2020) The change in the timing of the transfers yields VFh0=E(σμ)(1−δ) ˜ T−1  t=0 δtπH(qt)+I{˜ T<∞}δ˜ TVFh˜ Tm˜ T+WHh˜ Tm˜ Th0 WLh0=E(σμ)(1−δ) ˜ T−1  t=0 δtθqt+I{˜ T<∞}δ˜ TWLh˜ Tm˜ T−WHh˜ Tm˜ Th0L WLHh0≥E(σμ)˜ T−1  t=0 δtθqt−I{˜ T<∞}δ˜ TWHh˜ Tm˜ Th0H The payoff WH(h˜ Tm˜ T)is bounded above by WL(h ˜ Tm˜ T). Therefore, it follows from Lemma 4 and from the fact that [p 1]is a firing region that there exist Kand ¯ δ<1such that for any δ> ¯ δand for any PBE (σμ),thepayoffsVF(h0),WL(h0),andWLH(h0)satisfy the inequalities in (3). Proof of Claim 1. First, assume that ˜ q0≤˘ qH 2and notice that πH(˜ q0)<π H(˘ qH 2)<0 (recall that ˘ qH∈(0q∗ H)satisfies πH(˘ qH)=0). Also notice that ˆ p<p 0<p. If inequality (8)issatisfied,thenwehave 0≤1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) −p0πLq∗ L ≤1−p0 pϒ0πH˘ qH 2+p0 pϒpπHq∗ H−pπLq∗ L+K(1−δ) ≤1−p0 pϒ0πH˘ qH 2+K(1−δ) Putting this and p0<p Ctogether, we obtain 1 1−p01−p0 pϒ0≤− K(1−δ) 1−pCπH˘ qH 2(23) Similarly, we obtain 0≤1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) −p0πLq∗ L ≤1−p0 pϒ0πH(˜ q0)+p0 pϒpπHq∗ H−pπLq∗ L+K(1−δ) ≤p0 pϒpπHq∗ H−pπLq∗ L+K(1−δ) ≤1−f(p) 2pϒpπHq∗ H−pπLq∗ L+K(1−δ) where the last inequality follows from p0∈[p−f(p) 2p]and πH(q∗ H)−pπL(q∗ L)<0. Theoretical Economics 15 (2020) Dynamic contracting 617 Hence, we have 1−p 1−p0p0 pϒp≤ϒp≤K(1−δ) 1−f(p) 2ppπLq∗ L−πHq∗ H (24) For the case ˜ q0≤˘ qH 2, inequalities (23)and(24) imply the result. We now move to the case ˜ q0>˘ qH 2. It follows from the concavity of πH(·)that πH(˜ q0)≤πH(0)+π H(0)˜ q0≤π H(0)˜ q0.Also,πH(˜ qp)<π H(q∗ H)for any ˜ qp= q∗ H.Thus, we have 1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) ≤1−p0 pϒ0π H(0)˜ q0+p0 pϒpπHq∗ H+(1−ϒp)pπLq∗ L+K(1−δ) (25) Suppose that inequality (9) holds. Clearly, the inequality continues to hold if we replace ˜ qpwith 1. This allows us to conclude that ˜ q0,ϒ0,andϒpmust satisfy ϒ0˜ q0≤ϒp+2K(1−δ) θ1 1−p01−p0 p(26) Combining inequalities (25)and(26), we obtain (recall that p0>ˆ p) 1−p0 pϒ0πH(˜ q0)+p0 pϒpπH(˜ qp)+(1−ϒp)pπLq∗ L+K(1−δ) ≤1−p0 pπ H(0)ϒp+p0 pϒpπHq∗ H+(1−ϒp)pπLq∗ L +K1+2π H(0)(1−ˆ p) θ (1−δ) We define K1:= (1+2π H(0)(1−ˆ p) θ ). It follows from inequality (8) and the inequality above that p0πLq∗ L≤1−p0 pπ H(0)ϒp+p0 pϒpπHq∗ H+(1−ϒp)pπLq∗ L+K1(1−δ) which leads to 0≤ϒp1−p0 pπ H(0)+p0 pπHq∗ H−pπLq∗ L+K1(1−δ) ≤ϒp1− p−f(p) 2 pπ H(0)+ p−f(p) 2 pπHq∗ H−pπLq∗ L+K1(1−δ) (27) 618 Gerardi and Maestri Theoretical Economics 15 (2020) The second inequality holds because the expression (1−p0 p)π H(0)+p0 p[πH(q∗ H)− pπL(q∗ L)]is affine in p0, is negative for p0=p,andequalto0for p0=p−f(p)(see the definition of the function f(·)in (2)). Also, recall that p0∈[p−f(p) 2p]. From inequality (27), we obtain ϒp≤K1 p−f(p) 2πLq∗ L−f(p) 2pπ H(0)−1−f(p) 2pπHq∗ H (1−δ) := K2(1−δ) and, thus, 1−p 1−p0p0 pϒp≤ϒp≤K2(1−δ) (28) Finally, using (26)and(28)andp0<p Cwe have 1 1−p01−p0 pϒ0˜ q0 ≤1 1−pC1−p0 pϒp+2K(1−δ) θ ≤1 1−pCϒp+2K(1−δ) θ ≤1 1−pCK2(1−δ) +2K(1−δ) θ  Recall that ˜ q0>˘ qH 2. It follows from the last inequality that 1 1−p01−p0 pϒ0≤2 ˘ qH K2 1−pC+2K θ (1−δ) which coupled with (28) implies the result. Proof of Lemma 6.FixaPBE(σ μ) and a history htas described in the statement of the lemma. The firm’s continuation payoff after offering the menu mtis VFhtmt=1−μhtVFhtmt;H+μhtVFhtmt;L Recall from Proposition 2 that [ˆ p+ε 1]is a firing region. Therefore, it follows from Lemma 4 that there exist ¯ Kand ¯ δ>1−επL(q∗ L) 2¯ Ksuch that δ> ¯ δimplies VF(htmt;H) ≤ ¯ K(1−δ). This, in turn, implies VFhtmt≤(ˆ p−ε)πLq∗ L+¯ K(1−δ) < ˆ p−ε 2πLq∗ L<π Hq∗ H contradicting Lemma 1. Proof of Proposition 3. We start by proving the first property of a pooling region. Without loss of generality, we establish the property at the initial history h0and restrict attention to equilibria in which (a) the firm’s strategy in the first period is pure and (b) the high type’s equilibrium payoff is equal to 0. Theoretical Economics 15 (2020) Dynamic contracting 619 Fix p< ˆ p. By contradiction, suppose that there exists a sequence {δnp0n (σnμn)}∞ n=1such that δnconverges to 1,p0n ∈[0p],(σnμn)isaPBEofthegamewith discount factor equal to δnand prior equal to p0n,and lim n→∞ E(σnμn)δT n=ξ>0(29) Let k:=  2 1−ˆ pand, for k=kk+1,let ˜ Tk≤∞be the random time that stops the play at the first history (h˜ Tm˜ T)at which the menu m˜ Tcontains a contract (x˜ Tq˜ T) accepted with positive probability and for which μ(h˜ Tm˜ T(x˜ Tq˜ T)) ≥ˆ p+1 k.Asusual, we set ˜ Tk=∞if the event does not occur in finite time. It follows from Lemma 4 that for every k≥k,thereexistn1 k∈Nand K1 ksuch that for every n≥n1 k,thePBE(σnμn)satisfies the following property. If the firm offers a menu with a contract that is accepted with positive probability and leads to a belief weakly higher than ˆ p+1 k, then the expected discounted time until the high type quits the relationship is bounded above by K1 k(1−δn). Thus, for n≥n1 k,wehave E(σnμn)δT n−E(σnμn)I{˜ Tk<∞}1−μnh˜ Tkδ˜ Tk n≤K1 k(1−δn) (30) Next, recall that [ˆ p+1 k]is a firing region and Lemma 4 (property (ii)) provides an upper bound to the firm’s continuation payoff when it offers a menu with a contract that leads to a firing region. Therefore, for every k≥k,thereexistn2 k∈Nand K2 ksuch that for every n≥n2 k, the firm’s equilibrium payoff is bounded as22 VFh0;(σnμn)≤E(σnμn)I{˜ Tk<∞}(1−δn) ˜ Tk−1  t=0 δt nπH(qt)+δ˜ Tk nμnh˜ TkπLq∗ L +I{˜ Tk=∞}(1−δn) ˜ Tk−1  t=0 δt nπH(qt)+K2 k(1−δn) Notice that when ˜ Tk<∞, the belief μn(h ˜ Tk)is, by definition, lower than ˆ p+1 kand, therefore, we have μnh˜ TkπLq∗ L<ˆ p+1 kπLq∗ L=πHq∗ H+1 kπLq∗ L Combining the last two inequalities, for every n≥n2 k,weobtain VFh0;(σnμn)≤E(σnμn)I{˜ Tk<∞}(1−δn) ˜ Tk−1  t=0 δt nπH(qt)+δ˜ Tk nπHq∗ H +I{˜ Tk=∞}(1−δn) ˜ Tk−1  t=0 δt nπH(qt)+K2 k(1−δn)+1 kπLq∗ L 22This bound is derived using the same procedure as the one outlined above in the derivation of the inequalities in (3). 620 Gerardi and Maestri Theoretical Economics 15 (2020) =πHq∗ H−E(σnμn)(1−δn) ˜ Tk−1  t=0 δt nπHq∗ H−πH(qt) +K2 k(1−δn)+1 kπLq∗ L This and the fact that the firm’s payoff is bounded below by πH(q∗ H)lead to the result, for every k≥k, lim sup n→∞ E(σnμn)(1−δn) ˜ Tk−1  t=0 δt nπHq∗ H−πH(qt)≤1 kπLq∗ L(31) Inequality (31) implies that for every η>0,thereexistskη∈Nsuch that for every k≥kη,thereexists ˆ nk∈Nsuch that for n≥ˆ nk,wehave E(σnμn)(1−δn) ˜ Tk−1  t=0 δt nq∗ H−qt≤η (32) Furthermore, kηand ˆ nkare such that for every k≥kηand every n≥ˆ nk, ξ 1−ˆ p−η≤E(σnμn)I{˜ Tk<∞}δ˜ Tk n≤ξ 1−ˆ p+η The above result is a consequence of equality (29), inequality (30), and Lemma 6. Fix ε∈(0θq∗ Hξ 4(1−ˆ p)(1+θq∗ H)(ˆ p p−1−ˆ p 1−p)). Recall that for every k, the interval [ˆ p+1 k1]is a firing region and Lemma 4 (property (iii)) provides an upper bound to the low type’s continuation payoff when the firm’s menu contains a contract that leads to a firing region. Finally, recall that if a history htis reached with probability Pr(ht)under (σnμn),then that history is reached with probability μn(ht) p0n Pr(ht)if the worker behaves according to σL nand is reached with probability (1−μn(ht)) (1−p0n)Pr(ht)if the worker behaves according to σH n. Putting together these observations and the last three inequalities, we conclude that there exist ˜ kand ˜ nsuch that for every n≥˜ n, the low type obtains a payoff of at most θq∗ H1−E(σnμn)δ˜ T˜ k n|σL n+ε≤θq∗ H1−ˆ p p0n ξ 1−ˆ p+ε+ε ≤θq∗ H1−ˆ p p ξ 1−ˆ p+ε+ε when he behaves according to σL nand obtains a payoff of at least θq∗ H1−E(σnμn)δ˜ T˜ k n|σH n−ε≥θq∗ H1−1−ˆ p 1−p0n ξ 1−ˆ p−ε−ε ≥θq∗ H1−1−ˆ p 1−p ξ 1−ˆ p−ε−ε Theoretical Economics 15 (2020) Dynamic contracting 621 when he behaves according to σH n.Noticethat θq∗ H1−1−ˆ p 1−p ξ 1−ˆ p−ε−ε−θq∗ H1−ˆ p p ξ 1−ˆ p+ε−ε =θq∗ H ξ 1−ˆ pˆ p p−1−ˆ p 1−p−2ε1+θq∗ H>0 which implies that for nsufficiently large, the low type has an incentive to deviate and follow σH ninstead of the equilibrium strategy σL n. We now turn to the second property of a pooling region. It is again without loss of generality to restrict attention to equilibria in which (a) the firm’s strategy in the first period is pure and (b) the high type’s equilibrium payoff is equal to 0. The second property follows directly from the first property and inequality (32). Finally, we establish the third property. Assume, toward a contradiction, that there exists a sequence {δnp0n(σnμn)}∞ n=1such that δnconverges to 1,p0n ∈[0p],(σnμn) is an arbitrary PBE of the game with discount factor equal to δnand prior equal to p0n, and lim n→∞ E(σnμn)(1−δn) T−1  t=0 δt nxt−θHq∗ H−αi=˜ ξ>0 for some i∈{HL}. Using the first two properties of a pooling region, we conclude that lim sup n→∞ VFh0;(σnμn)≤πHq∗ H−min{p01−p0}˜ ξ<π Hq∗ H which leads to a contradiction and concludes the proof. References Acemoglu, Daron, Mikhail Golosov, and Aleh Tsyvinski (2010), “Dynamic Mirrlees taxation under political economy constraints.” Review of Economic Studies, 77, 841–881. 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