Dynamic costs and moral hazard: A duality-based approach

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1 Available online at ScienceDirect Journal of Economic Theory 166 (2016) Dynamic costs and moral hazard: A duality-based aroach Guy Arie Simon Business School, University of Rochester, Rochester, NY, United States Received 25 November 2014; final version received 8 August 2016; acceted 10 August 2016 Available online 18 August 2016 Abstract The marginal cost of effort often increases as effort is exerted. In a dynamic moral hazard setting, dynamically increasing costs create information asymmetry. This aer characterizes the otimal contract and hels exlain the oular yet thus far uzzling use of non-linear incentives, for examle, in sales-force comensation. The result is obtained by comlementing the standard dynamic rogram with a novel dynamic dual formulation. The dual rogram is monotonic and sub-modular, roviding stronger results, including a roof for the sufficiency of one-shot deviations Elsevier Inc. All rights reserved. JEL classification: C61; D82; D86 Keywords: Dynamic moral hazard; Nonlinear incentives; Private information; Dynamic mechanism design; Duality This aer is based on the first chater of my dissertation at Kellogg. I am indebted to Bill Rogerson, Mark Satterthwaite and esecially Jeroen Swinkels and Michael Whinston for their advice and encouragement throughout this work. I thank the editor, Alessandro Pavan, the associate editor, and referees for extensive and immensely helful comments, many seminar articiants and esecially Paulo Barelli, Doruk Cetemen, Eddie Dekel, Matthias Fahn, Paul Grieco, Jin Li, and Michael Raith rovided helful suggestions. Financial suort from the GM Center at Kellogg is gratefully acknowledged. All remaining errors are my own. address: guy.arie@simon.rochester.edu. htt://dx.doi.org/ /j.jet / 2016 Elsevier Inc. All rights reserved.

2 2 G. Arie / Journal of Economic Theory 166 (2016) Introduction Increasing marginal costs are a standard comonent of economic analysis. In organizational settings, the increase in cost often has a dynamic motivation. A worker icking fruits, for examle, gets tired as the day rogresses. A sales agent tyically deletes the easier sales at the start of the quarter and must exert more effort to generate later sales. Because effort is unobserved, increasing marginal costs generate a rivate-information roblem. After a few eriods, only the agent knows whether he has really worked. If the firm knew the agent s true cost, it would want to increase incentives as effort costs increase. However, the agent may have reviously shirked, and his cost may still be low. As a result, adative olicies decrease the agent s incentive to work early on as he refers to save his effort for the higher rewards. This aer characterizes the otimal mechanism for a dynamic moral hazard setting in which the cost increase deends on the agent s ast, unobservable, effort. The agent starts in an evaluation stage and eventually moves to a comensation stage. In the comensation stage, the agent works for an additional fixed number of eriods that is indeendent of any new outcomes. In the evaluation stage the agent is rewarded only by changes to the exected length of the comensation stage and the number of successes required to enter that stage. In the comensation stage, the ayment er success increases in the costs in all remaining eriods. If the agent does not discount future eriods, he is aid as if all efforts in the comensation stage cost as much as the last effort. If the agent accumulates enough early successes, his ay-er-success later in the quarter will be high. If the agent did not accumulate enough early successes, the contract leads the agent to sto working. The main dearture from existing contracts is that the eventual comensation level here deends on the agent s early erformance. The agent may receive a 5% commission in some realizations and a 15% commission in others. This is difficult to exlain using the existing static or dynamic contracts, 1 yet is consistent with observed contracts, esecially in sales. 2 A growing literature, surveyed below, investigates whether and how inter-temoral agency issues (here, the increasing cost) restrict the otimal contract. The closest dynamic models without inter-temoral issues are DeMarzo and Fishman (2007) and Clementi and Hoenhayn (2006) in which the risk-neutral agent s eriod roduction is bounded from below, the agent can only divert or revent some ositive revenue. In such models, the contract initially rewards the agent only in changes to his continuation utility. If the continuation utility dros below a certain level, the agent is retired. If it reaches the first-best continuation surlus, the firm is sold to the agent and the continuation is fixed at the first-best level. 1 Allowing for rivate information on the agent s side of his initial roductivity can exlain a menu of fixed-wage contracts from which the agent chooses at the outset. Here, as in the contracts discussed by Prendergast (1999), the single contract offers several commission levels. 2 Misra and Nair (2011) document contracts for contact-lens sales agents that ay a commission er sale only if the agent made enough sales in the current quarter. Misra and Nair (2011) note that, according to industry observers, little or no seasonality exists in the underlying demand, and the saleserson has no control over rices. Their estimation confirms that an agent s effort is strongly affected by whether or not the agent exects to meet the threshold for the quarter. Larkin (2014) documents contracts for software sales agents that ay the a commission er sale that increases in the amount the agent sold so far in the quarter. In both aers, the same agent may receive a high commission rate in one quarter and a low (or no) commission in another, based on his results early in the quarter. Joseh and Kalwani (1998) document the extensive use of quota-based contracts in sales force comensation.

3 G. Arie / Journal of Economic Theory 166 (2016) The novel result here is that even when the continuation is fixed, it may not be first best. That is, only a reduced-surlus version of the firm is sold to the agent. Because shirking has the inter-temoral effect of reducing the agent s future costs and thereby increasing the value of the firm to a deviating agent, an agent execting to obtain the firm in the future has a stronger incentive to deviate. The otimal contract s reaction to this concern is to destroy the future value of the firm enough to counteract the agent s increased incentive to shirk. This finding, again, is consistent with the emirical finding of different ay rates to the same agent in different realizations and high volatility of the work decision towards the end of the quarter or fiscal year as found in Oyer (1998). Another imlication of the increasing costs is that the ay rate in the comensation stage, in which the work lan is fixed, is based on the highest effort cost a comlying agent will exert once reaching this stage. 3 Thus, the ay rate may seem significantly excessive comared to the effort cost. This imlication again is consistent with observed, yet so far uzzling, sales contracts (see, e.g., Larkin, 2014). Examles in section 5 show that even in relatively comlicated settings, the otimal contract is simle, consistent with observed contracts, and is significantly more rofitable than the best linear rate or quota contract. Thus, the analysis here shows that increasing marginal costs can rovide micro-economic foundations for the oularity of these schemes. Using dynamic analysis to model increasing marginal cost is concetually different from modeling a long-term relationshi. Here, the dynamics are derived from the fact that roduction is done in stes, and each ste has an observable outcome, not from the actual length of the relationshi. For examle, as in Shearer (2004), the relevant roduction rocess lanting trees may be comleted in a day. What matters is that roduction is comosed of sequential actions (lanting each tree), and that it gets increasingly harder with each action the lanter tires and has to lant in rougher terrain. The focus on the dynamic asect of roduction raises questions such as the distribution of costs across tasks. In some cases, the rincial refers more balanced costs over lower exected costs suggesting that the rincial should consider investing in balancing costs across activities. Moreover, the answer may be reversed if the rincial uses a subotimal linear wage. Other interesting questions consider the contractual effects of slitting a large task into smaller, consecutive ones, without affecting the total first-best surlus. An examle in section 5 shows the imortant role of otimal contracts in such questions. Both the rincial and agent are better off by such slitting if the otimal or quota contract is used, but both are worse off if a iece-rate contract is used. The main model considers a risk-neutral agent deciding every eriod whether to exert costly effort. Period outcome is binary and the robability of success in the eriod increases with current effort. The cost of current effort is a convex function of ast effort. In other words, the marginal cost of effort is increasing and convex. Effort is unobserved and the rincial can commit to a contract at the outset. 4 3 If the agent does not discount future eriods, the ay rate deends only on the highest effort cost. If the agent discounts future eriods, the effect of future, higher effort-cost eriods is discounted. 4 I consider multile ossible actions and outcomes in section 6. Full commitment as here is tyical in the related literature. Deriving the renegotiation-roof contract requires additional constraints on the otimal contract roblem that are beyond the scoe of this aer. In the conclusion, I discuss ossible imlications of renegotiation in light of the current analysis.

4 4 G. Arie / Journal of Economic Theory 166 (2016) 1 50 To see the incentive roblem, suose the robability of success each eriod is 1 2 if the agent exerts effort, and zero otherwise, and the agent s cost for making the n-th effort is n. If both the rincial and the agent consider only current- eriod incentives, a contract aying the agent 2n for success in day n is incentive comatible and rovides the agent zero exected utility clearly first best. However, if the agent considers future ayoffs, this contract is no longer incentive comatible. Shirking in the first eriod and then working whenever asked obtains the agent an exected utility of 1 each eriod. That is, by shirking today, the agent increases his rents from future work. The otimal contract must account for this additional incentive to shirk. To derive the otimal contract, I develo a recursive formulation based on duality. The formulation is tractable and catures the additional friction directly. The main benefit of the dual formulation is a direct roof for the main challenge in analyzing dynamic moral hazard settings with ersistent rivate information the sufficiency of considering only one-shot deviations. This is a result I could not obtain using the standard formulation. The dual methodology develoed here can be alied and extended to other settings. Section 6 illustrates some extensions multile actions and outcomes and different discount factors that require minimal adjustments. 5 The recursive dual formulation in our setting also has structural advantages not common to recursive formulations of dynamic moral hazard roblems. In articular, the recursive roblem is monotonic in addition to convex in its variables, whereas the standard formulation is nonmonotonic. Moreover, although the increasing costs require additional state variables in both formulations, these variables are substitutes only in the dual. The stronger relation between variables allows for more nuanced results (Proosition 4), and more direct roofs for other results (Lemma 5). Finally, the economic interretation of the dual roblem and variables resented below sheds additional light on the economic value of long-term contracting and the rogression of the contract. To understand the recursive dual intuition, suose that before signing the contract, the rincial and agent learn some third arty will have the ower to sto the contract at history h. The rincial must decide, at this early stage, how much to send now to revent this termination. The dual value of history h is the largest amount the rincial would ay. To determine this dual value, we must consider (a) the exected continuation revenue and (b) the exected cost of roviding incentives to the agent to exert the required continuation effort. However, there are additional considerations. If everyone knows ex-ante that the contract will terminate at history h, the utility rovided to the agent starting in h from the otimal contract can no longer rovide incentives (or disincentives) for effort. For examle, in most contracts, the agent is not aid for success in the first eriod. Instead, he is rewarded by a better contract following success than following failure. If the agent knows the third arty will terminate the contract after the first success, giving him zero continuation utility, he may well require ayment for the first success. Thus, the dual value must also consider (c) the effect of the agent s continuation utility on revious histories. The three considerations above are common to all dynamic moral hazard settings. The current setting adds (d) the effect of the agent s current cost on revious histories. That is, if the contract terminates, any gains to the agent from shirking in the ast to obtain a lower cost today are destroyed. If the agent exects the contract to terminate, his incentives to shirk are weaker, reducing the rincial s costs. 5 Other extensions such as an infinite but discrete time horizon and outcome sets require roving additional intermediate results tyically that the infinite sums converge and an interior solution exists (see, e.g., Romeijn et al., 1992; Sharkey, 2010; Ghate, 2015) that are beyond the scoe of this aer.

5 G. Arie / Journal of Economic Theory 166 (2016) The dual value of the history accounts for all four elements (a) (d) above. By contrast, neither the rincial nor agent exect the third arty termination, which instead just aears in history h. In this case, the rincial would ay only u to her exected continuation rofit from history h. The value of avoiding this unexected termination is given by the standard dynamic moral hazard value (cf. Sear and Srivastava, 1987). The dual value therefore has two uses. First, the Duality Theorem requires that the dual value for the first eriod equals the exected rofit from the otimal contract. Economically, this equality must be true because there are no revious eriods to affect. In addition, the interretation above imlies the rincial should commit to terminating the contract if and only if the dual value is negative. Therefore, we can use the dual value to identify the contract s termination histories. The fact that after a ayment has been made, the continuation dual value is highest, whereas the standard value may be negative, best illustrates the economic difference between the standard and dual value. The standard value accounts for the agent s high wages going forward, whereas the dual value accounts for the incentives generated earlier in the contract by the exectation of these high wages. To determine the dual value, the standard romised utility state variable (cf. Sear and Srivastava, 1987) is transformed to the marginal cost to the contract of the agent s utility. This new state variable exactly catures (c) above. A second state variable indicating the marginal cost to the contract of the agent s rivate information catures (d) above. Both of these costs are derived by aggregating the shadow rices (multiliers) on the incentive constraints in the receding histories. Following the literature review, section 2 lays out the dynamic roduction model. Section 3 rovides the dual form and its economic interretation. The otimal contract is formally characterized in section 4. Section 5 uses two examles to illustrate the alication to real-world settings. Section 6 considers the roblem for multile actions and outcomes. Section 7 concludes. All roofs omitted from the text are given in the aendix Literature review Mukoyama and Sahin (2005) study moral hazard with dynamically increasing marginal costs. However, the authors analysis is limited to two eriods and assumes the agent s effort in the first eriod comlements effort in the second eriod. In such cases, the agent s comensation relies more on the second-eriod outcome because second-eriod incentives also induce effort in the first eriod. Consequently, the economic rediction in Mukoyama and Sahin (2005) is that the first-eriod outcome affects the agent s utility less than the second-eriod outcome, and may in fact have no effect. By contrast, increasing marginal costs imly the agent s efforts are substitutes, and thus the rediction here is oosite. 6 Of the recent literature studying inter-temoral agency issues, the closest to mine is the concurrent DeMarzo and Sannikov (2015), where the agent observes rivate information that has a ersistent effect on roductivity. There, too, the contract starts in an evaluation stage and moves to a comensation stage. The analysis shares an imortant economic intuition: as the contract rogresses, the agent accumulates some information rent. In resonse, the contract must limit the 6 Mukoyama and Sahin (2005) also numerically characterize an N-eriod roblem limited to two-eriod ersistence and ex-ost verification of one-shot-deviation sufficiency. Abraham et al. (2011) rovide additional results for a more generalized two-eriod framework.

6 6 G. Arie / Journal of Economic Theory 166 (2016) 1 50 agent s value of the comensation stage. This is otimally accomlished by destroying surlus reducing the value of the firm when sold to the agent. Indeed both in DeMarzo and Sannikov (2015) and here, the novel methodological result is the exlicit characterization of continuation value and incentive constraints using the agent s information rent. The substantive difference between the model here and in DeMarzo and Sannikov (2015) is that the rivate information there is about the exected future revenue, and the agent s marginal cost of affecting the revenue is fixed. 7 This is the exact mirror image of the analysis here, where the exected future revenue is known but the agent s marginal cost is not. As a result, the agent s flow utility in the comensation stage there is constant, whereas while in mine the agent s eriod utility is decreasing (the wage is constant and the costs increase). Methodologically, extending local to global incentive comatibility in DeMarzo and Sannikov (2015) is based on ex-ost verification, whereas here, the roof is direct from the dual analysis. Several other related studies consider more limited forms of ersistent rivate information that revent the need to destroy surlus when selling the firm. Kwon (2015) considers a setting in which the agent has ersistent and exogenously changing rivate information. However, there, the agent s rivate information is limited to whether the relationshi is rofitable in exectation or not, and therefore the continuation after ayment is again first best. Tchistyi (2006) studies a related (otimal insurance) roblem but limits the ersistency of rivate information to one eriod, which restores sufficiency of one-shot-deviations. Sannikov (2014) assumes the agent s effort has a decaying effect on outcomes in the future. However, because the effect there is deterministic, the contracting roblem can be formulated using the net resent value of the effort s roductivity rather than information rent, and so destroying surlus is unnecessary. 8 Another setting in which the otimal tenure mechanism is not first best is the well-known Holmström (1999) career-concerns setting, where the agent s roductivity is unknown. More recently, Garrett and Pavan (2012) study a setting in which only the agent knows his roductivity and this information changes over time. By contrast, here, the agent s roductivity is known when he is aid, yet the contract is still not first best. Thus, different underlying economic friction (uncertain agent ability there vs. otential for intentional cost maniulation here) result in qualitatively different outcomes. In articular, in my setting, even though the agent s true cost and ability are known on-equilibrium, the continuation is not first best. Duality is widely used in economic modeling, dating back to Rockafellar (1970). Vohra (2011) extends the analysis of static adverse selection models by analyzing the dual of the classic adverse selection roblem. Chow (1997) and Marcet and Marimon (2011) (first aearing in 1998) develo a recursive formulation for infinite-horizon roblems using Lagrange multiliers. Duality-based recursive formulations have recently been extended by Messner et al. (2012), Mele (2014), and Messner et al. (2013). 9 7 DeMarzo and Sannikov (2015) consider a continuous-time model with a Brownian Motion outut rocess in which the agent s effort additively increases outut, and therefore many differences exist in the underlying assumtions. DeMarzo and Sannikov (2015) also consider a secial case ( = 0) that is closer to Garrett and Pavan (2012), discussed below. 8 Two related settings are rivate savings by the agent as the intertemoral agency issue (e.g., Fudenberg et al., 1990; Williams, 2011; Edmans et al., 2012 and Williams, 2015) and models in which roduction ends after the first good outcome (e.g., Bergemann et al., 2005; Bonatti and Horner, 2011, and Halac et al., 2016). In both, the contract catures very different trade-offs and takes a different shae. In articular, the firm is never sold to the agent after success. 9 Mele (2014) and Messner et al. (2013) are concurrent with this aer. Messner et al. (2012) considers only a limited set of roblems.

7 G. Arie / Journal of Economic Theory 166 (2016) The aers mentioned above focus on the alicability of the methodology for general infinite-horizon formulations. They rovide conditions under which the recursive rincial-agent roblem formulation has a valid recursive dual formulation based on Lagrange Duality. By contrast, this aer utilizes the dual methodology to obtain stronger results, including a direct roof for alicability of the one-shot-deviation condition. Methodologically, the formal aroach here is different because it derives the dual for the original rather than the recursive roblem, thereby removing differentiability requirements and assumtions relating to the validity of the first-order aroach, which the other aers verify ex-ost. Finally, the model here results in a dual roblem that is a standard minimization roblem with useful monotonicity features, whereas the aers above rovide a saddle-oint (i.e, maxmin) dual roblem and do not attemt to characterize it further. 2. Model 2.1. Setu and rimitives There is a rincial and an agent, both risk neutral. Both have an outside otion normalized to zero. The agent has limited liability, i.e., money can only be transferred to the agent. 10 Time is discrete and the common discount factor is β (0, 1]. In each eriod, the agent chooses an action a A ={0, 1}, where a = 0is interreted as not working in the eriod, and a = 1as working. The agent s work is costly to the agent and unobservable to the rincial. A eriod s roduction outcome is either a success or a failure, denoted by y Y = {0, 1}. Section 6 considers multile actions and outcomes. Both the rincial and the agent observe the outcome of each eriod. 11 The rincial earns a revenue of v from each success (y = 1), and zero from a failure (y = 0). The robability of outcome y given action a is denoted (a, y). To save on notation, set (1, 1) and 0 (0, 1) to denote resectively the robabilities of success with and without effort. Assume 0 0 <<1. To focus on the gains from the agent s effort, we net out the rofit from no effort v 0 for every eriod in which the contract is still active. A ublic history h records the required action (work or not) and observed outcome (y) for each revious eriod. Let H denote the set of all ossible histories (including the null history ). The analysis makes extensive use of histories that began with other histories. The notation h = h 1, h 2 describes the history h 1 and then the history h 2. The set of all histories that begin with h 1 is denoted h h 1 : h h 1 h 2 : h = h 1,h 2. The cost of effort in a eriod deends on the number of receding eriods of work. That is, a sequence c n exists for n 1, 2,... such that in the first eriod of work, the cost of effort is c 1 ; in the second, the cost is c 2, etc. The effort cost deends on actual work. For examle, if the 10 Limited liability is imlemented by restricting eriod ayments to be non-negative. Alternatively, the agent may be aid negative sums in some eriods as long as the total ayment by termination is ositive in all ossible histories. The two restrictions are interchangeable in our setting. Any contract that is feasible and otimal under the latter restriction can be transformed to a contract under the former that rovides the same exected rofits and outcomes for both rincial and agent. See also Remark This assumtion is common to dynamic moral hazard models. Allowing the agent to deviate by shifting one eriod s outcome to a later eriod generates additional interesting questions also for the standard models. Lemma 5 rovides a result for some of the histories. More general results remain for future research.

8 8 G. Arie / Journal of Economic Theory 166 (2016) 1 50 agent shirks in the first eriod, the cost of effort in the second eriod is c 1 and not c 2. We assume that marginal cost is increasing and weakly convex: Assumtion 1. The agent s marginal cost, c n, is strictly increasing and weakly convex in n. Strict convexity is never used in the sequel, and convexity is used to mean weak convexity. That costs are increasing is the motivation of this study; therefore, the first art of the assumtion is natural. 12 In most alications, the total cost of exerted effort at time n is the sum of the receding c n s (i.e., n j=1 c j ) and c n is the marginal cost. The convexity assumtion requires that this marginal cost is increasing and convex. A common examle that fits this assumtion is c n = n, and so the agent s total cost of effort is aroximately n2 2. The convexity assumtion is instrumental to roving the first deviation by the agent is always his most rofitable. Intuitively, at any eriod, shirking decreases the future costs most if the agent never shirked before. However, as I show below, convexity does not imly the first deviation is most rofitable in all feasible contracts. Indeed, a main result of this aer is that this intuition alies to the otimal contract. Convexity has no role in the characterization of the contract limited to single deviations nor in the validity of the dual formulation. The only other role of convexity in the analysis is the imlication that surlus from working becomes negative after enough effort has been exerted creating an endogenous limit on the contract length. The true cost to the agent of working in a eriod is a function of the number of eriods in which the agent actually worked in the ast. Only the agent knows in which eriods he actually did work and in which eriods he shirked. Because only the number of eriods matters, the only ayoff-relevant information in the agent s rivate history is the number of ast shirks: Definition 1. The agent s rivate history (h, s) is the ublic history h and the number of ast shirks s. For any history h, let n(h) denote the number of the next work eriod following history h (i.e. n( ) = 1). Let c(h) c n(h) denote the cost of exerting effort in the eriod following h if the agent comlied with the contract in all revious eriods. Given the imortance of the difference in costs, denote by d n c n c n 1 the n-th increase in effort cost, with d 1 = 0for comleteness. As for costs, let d(h, s) c n(h) s c n(h) s 1 be the decrease in the agent s current eriod effort cost in rivate history (h, s) if the agent would have shirked once more in the ast. Finally, let N FB denote the maximum number of eriods in which consecutive work increases surlus: N FB = max n : c n v ( 0 ) The standard otimal contract roblem This subsection rovides the standard otimal contract roblem formulation and identifies some basic results to be used in the sequel. 12 If costs are fixed, decreasing or exogenous, standard aroaches can be used to characterize the otimal contract.

9 G. Arie / Journal of Economic Theory 166 (2016) The main results will ultimately rely on analyzing the otimal contract roblem as a Linear Program (LP). As is common in analysis based on Linear Programming (see, e.g., Abreu et al., 1991; Vohra, 2011), doing so requires a reformulation of the standard roblem develoed here. A standard contract secifies for each history the required work decision and romised ay schedule. To allow the work requirement to be robabilistic, let Z : H [0, 1] ma from any history h to the robability that the contract will require work (i.e., secifies action a = 1) in the eriod that starts with history h. Let W : H A Y R + denote the non-negative ayment to the agent in the eriod that started with history h if the agent was asked to erform action a A and obtained outcome y Y. The contract in the standard formulation is given by the air of functions Z( ), W ( ). Remark 1. This formulation revents the rincial from ublicly randomizing in history h between two contracts z 1 ( ), W 1 ( ) and z 2 ( ), W 2 ( ) that agree on history h but differ in some continuations. 13 This revention is without loss if the otimal contract can be maed to the solution of a recursive roblem in which randomization between continuation variables is never otimal (cf. Sear and Srivastava, 1987; Clementi and Hoenhayn, 2006). In the analysis here, this is roved in Lemma 3 which shows that the otimal contract is maed to a recursive dual roblem that is a convex minimization roblem, in which randomization between continuations is never otimal. For any contract, let ˆV h and Û h denote resectively the exected continuation rofit for the rincial and utility for the agent starting at history h if the agent always comlies with the contract. Let ˆV a,h, Û a,h denote the corresonding values conditional on the action request (i.e., realization of Z(h)) that was made in the eriod. The following relations are standard: ˆV h = Z(h) ˆV 1,h + (1 Z(h)) ˆV 0,h Û h = Z(h)Û 1,h + (1 Z(h))Û 0,h ˆV a,h = v((a,1) (0, 1)) + [ ] (a, y)(β ˆV h,(a,y) W(h,a,y)) y Û a,h = ac(h) + [ ] (a, y)(w (h, a, y) + βû h,(a,y) ). y (2.1) The otimal contract maximizes ˆV subject to the agent s incentive comatibility (IC) and individual rationality (IR) constraints. For IR, we require that the agent s exected utility at every continuation history is at least as good as his zero-value outside otion: (IR) h : Û h 0. If the agent cannot quit after acceting the contract, a weaker, only ex-ante IR is aroriate (Û0 0). Alternatively, in some roblems considering off-ath IR (i.e., a deviation followed by quitting) might be relevant. Lemma 1 below shows that, because the agent can effectively quit by simly not exerting any effort, IC will imly all of these hold in our roblem. (2.2) 13 Formally, the imlicit assumtion is that the analysis is unaffected by extending the eriod outcome sace to Y ={0, 1} [0, 1] with the second element randomly drawn at the start of the eriod with no effect on costs, success robabilities or revenue.

10 10 G. Arie / Journal of Economic Theory 166 (2016) 1 50 Following Fernandes and Phelan (2000), maintaining incentive comatibility will require monitoring the agent s exected utility conditional on shirking in the ast. We deart slightly from Fernandes and Phelan (2000) here, and instead of secifying the agent s off-ath continuation utility, we secify the off-ath change in utility. This dearture is instrumental in the roofs because it searates the on-ath and off-ath comonents of the incentive roblem. Two things change for a rivate history that deviated from the ublic history. First, the effort cost is lower. Second, the ex-ante robability of arriving to this history is different, because shirking changes the outcome robabilities. Our aroach catures the first change directly in the variable definition and the second change when we integrate the deviation into the agent s value. To cature the difference in effort cost, define ˆD s h as the decrease in exected total cost starting in history (h, s + 1) comared to (h, s), assuming the agent will comly with the contract starting from history h. Because the agent s increase in exected utility from revious shirks is only a result of the cost difference, the continuation utility for an agent at rivate history (h, s) is equal to Û h + s 1 j=0 ˆD j h. As exected, if s = 0, the agent s continuation utility is simly Û h. Following the notation for Û a,h, use the suerscrit a in ˆD s a,h to denote the corresonding values conditional on the action request: ˆD s h = Z(h) ˆD s 1,h + (1 Z(h)) ˆD s 0,h = d(h,s)a + β [ y ˆD a,h s (a,y) ˆD h,(a,y) s ]. (2.3) To construct the IC constraints, we consider only IC if the agent is asked to work, and later verify using Lemma 1 that in the otimal contract, the agent never works if he is not asked to. If the one-shot-deviation rincile were to aly, the IC constraints would require that an agent that never shirked and is considering shirking in the current history, and only in it, is at least as well off by not deviating. As in related treatments, I will refer to these constraints that revent only one-shot deviations as Local Deviation Incentive Constraints (LDIC): (LDIC) h : Û 1,h y [ (0,y) W(h,1,y)+ β )] (Û h,(1,y) + ˆD h,(1,y) 0. (2.4) The only difference from the normal case in which shirking does not affect future costs is the additional ˆD comonent that increases the continuation value after shirking. All LDIC must hold for the contract to be IC. However, as in Fernandes and Phelan (2000), in the current model, the converse is false. 14 LDIC is not sufficient because some contracts may require more work in some eriods that follow a failure than other eriods that follow success. As a result, the rivate information gains (lower costs) from shirking and failing may be too large. The sequel utilizes the following extension of the LDIC: suose an agent that may have shirked in the ast arrives at some rivate history (h, s) and considers one final deviation. The Final Deviation Incentive Constraints (FDIC) require that the agent is at least as well off in this case by following the contract in all later eriods: 14 See Aendix C for an examle contract in which all LDIC hold with equality while the agent s otimal lan is to shirk in the first two eriods.

11 G. Arie / Journal of Economic Theory 166 (2016) s 1 (FDIC) s <n(h) : Û 1,h + y (0,y) [ k=0 ˆD 1,h k W(h,1,y)+ β ( Û h,(1,y) + s k=0 ˆD h,(1,y) k )]. (2.5) Because the agent could have only shirked when asked to work in the ast, the FDIC only consider s smaller than n(h). For any finite contract, satisfying all FDIC imlies IC any sequence of rofitable deviations must have a rofitable final deviation. Theorem 1 establishes that the otimal contract subject to LDIC is finite and satisfies FDIC and thus otimal subject to IC. The otimal contract roblem subject to FDIC is: (P-FDIC) max Z,W ˆV s.t. h : Z(h) [0, 1] W(h,a,y) 0 ˆV,Û, ˆD definitions (2.1) and (2.3) FDIC (2.5) IR (2.2) (2.6) The otimal contract roblem subject to LDIC (i.e., P-LDIC) is the same with the FDIC (2.5) relaced by the LDIC (2.4). The roblem can be transformed to the standard recursive form, using Û and all of ˆD s for the state sace, in an extension of the formulation in Fernandes and Phelan (2000). Instead, I first transform the roblem to an LP. Remark 2. Observing roblem (2.5), it is immediate that infinitely many equivalent ways exist for the otimal contract to ay the agent W dollars in any eriod but the last. For examle, because the discount factor is common, aying a dollar in one eriod is equivalent to aying 1 β dollars in the next. To remove this technical dulication of the otimal contracts, the analysis assumes the otimal contract makes ayments as early as ossible. That is, from all incentivecomatible contracts that secify the same work lan and exected rofits, the otimal contract is the one that ays the most to the agent as early as ossible The LP roblem Neither P-FDIC nor P-LDIC is an LP. For examle, in equations (2.1), both ˆV and Û have Z(h) multilying the next eriod s continuation value, which in turn has its own Z(h). The same alies to the FDIC and LDIC. The technical goal of this section is to remove all the variable multilications from the otimal contract roblems. The main intuition underlying the transformation is that we can consider each history ex ante at the outset of the contract, conditioning on the actual realization. That is, instead of the continuation value ˆV h, we consider the rincial s value for arriving at history h, calculated at the outset of the contract, which we denote V h. Formally, V h is ˆV h multilied by the discounted robability that the contract was not terminated before history h and outcomes were as observed in the history.

12 12 G. Arie / Journal of Economic Theory 166 (2016) 1 50 The following result simlifies the exosition and notation by removing various issues that are suerfluous to identifying an otimal contract. Lemma 1. LDIC imlies IR. In addition, for both P-FDIC and P-LDIC there is an otimal solution (ossibly different for each) in which 1. The required work decision is a stoing decision: if the agent is ever asked not to work, the contract terminates. 2. The agent works for at most N FB eriods. 3. The agent is never aid in a eriod with a failure or without required work. LDIC imlies IR because the agent can always simly shirk. Otimality of a stoing rule follows from the standard inter-changeability between discount factor and robability of termination. Finiteness then follows, because the contract should never require inefficient work. The last result is a direct outcome of risk neutrality and limited liability. Note that Lemma 1 does not rule out randomization of the stoing decision. Lemma 1 imlies it is sufficient to consider finite contracts and, as was assumed initially, the agent always comlies if he is asked not to work. Moreover, once the agent is asked not to work, both layers obtain a continuation value of zero: ˆV 0,h = Û 0,h = ˆD s 0,h = 0. In addition, the following notation simlifications follow from Lemma 1 and will be used in the sequel: 1. The ublic history includes only the sequence of outcomes y, with the imlication that the required action in all revious eriods was a = Define W(h) W(h, 1, 1), recalling that W(h, a, y) > 0 only if a = y = 1. To consider the ex ante values of each history, define an auxiliary function Q(h) that secifies the discounted robability of arriving to history h if the agent fully comlies with the contract. 15 Any contract that satisfies Lemma 1, gives rise to a unique function Q( ), recursively defined by: Q( ) = 1 Q( h, 0 ) = Q(h)βZ(h)(1 ) Q( h, 1 ) = Q(h)βZ(h). (2.7) The LP roblem will use the ex ante arallels of Z(h) and W(h), denoted z(h) and w(h): z(h) Q(h)Z(h) w(h) Q(h)Z(h)W (h). (2.8) Both z(h) and w(h) are uniquely determined by Z(h) and W(h). In articular, both account for the robability the contract reaches history h (i.e. Q(h)), and that effort is requested (i.e., Z(h)). 15 This aroach is similar to that in Bismut (1978), which was recently alied to moral hazard roblems by Williams (2011). The function Q() here is closely related to the function Ɣ() there.

13 G. Arie / Journal of Economic Theory 166 (2016) Using z( ), w( ) to write the ex ante equivalents of the continuation values ˆV, Û, ˆD obtains: V h = z(h) ( 0 ) v w(h) + V h,y = h h [ ] z( h) ( 0 ) v w( h) y U h = w(h) z(h)c(h) + y U h,y) = h h [ ] w( h) z( h)c( h) (2.9) D h s = z(h)d(h, s) + y D h,y s = h h z( h)d( h, s). Defining the FDIC for history (h, s) using the ex ante formulation requires a little care, as we need to adjust the continuation robabilities when shirking: s 1 w(h) z(h)c(h) + z(h)d(h, s) + ( ) s 1 U h,y + D h,y k k=0 y k=0 ( ) ( ) (2.10) 0 w(h) + s 0 U h,1 + D h,1 k + 1 s 0 U h,0 + D h,0 k. 1 k=0 The LHS of the FDIC is the exected continuation utility for an agent that lans to always comly. The RHS catures the four imlications of shirking: The robability of ayment changes from to 0. The cost of effort is avoided. The robabilities of the continuations change: is relaced with 0, and 1 with 1 0. The continuations have one more revious shirk: s 1in the sum of cost savings is relaced with s. Observe that all the continuation values defined in (2.9) are linear in the contract variables z( ), w( ), and therefore so is the FDIC. Finally, if the otimal contract roblem determines z( ), w( ) instead of Z( ), W ( ), a constraint must be added so that any feasible z( ), w( ) can be maed to a feasible Z( ), W ( ). Combining the definitions of z(h) and Q(h) with the constraint that Z(h) [0, 1] imlies that z(h) must satisfy the following Probability Constraint: z( ) 1 z( h, 0 ) z(h)β(1 ) z( h, 1 ) z(h)β. k=0 (2.11) The robability constraint (2.11) binds if and only if Z( h, y ) = 1. If (2.11) is slack, Z( h, y ) equals the ratio between the left and right hand sides of the constraint. When the contract is terminated Z(h) = z(h) = 0. Lemma 2 summarizes the transformation and establishes that z( ), w( ) secify a contract, which will be analyzed in the sequel. Lemma 2. Definitions (2.7) and (2.8) rovide a one-to-one maing between any air Z( ), W ( ) that is feasible for P-FDIC (2.6) and satisfies Lemma 1 and any air z( ), w( ) that is feasible for LP-FDIC (2.12) and satisfies Lemma 1. The objective value from each such

14 14 G. Arie / Journal of Economic Theory 166 (2016) 1 50 z( ), w( ) in LP-FDIC is equal to the objective value in P-FDIC from the corresonding Z( ), W ( ). (LP-FDIC) max z,w 0 V V, U, D definition: (2.9) FDIC: h, s < n(h) : (2.10) Probability Constraint: (2.11) The recursive LP roblem s.t. (2.12) Problem (2.12) can be recast in recursive form. The required state variables will be (n, Q, U, D), all as defined above for every history h. In articular, n denotes the current work eriod to determine the relevant cost c n. Q is the discounted robability the contract has reached this eriod without termination. U is the romised exected continuation utility for the comlying agent. D {D s } n 1 s=0 is a vector of the exected gains for a future comlying agent from one more shirk, conditional on making s shirks in the ast. The vector s length is obvious from the context and omitted from the notation. As before, y denotes the ossible outcomes. For a given state, the otimal contract chooses z and w as above, as well as the standard continuation romises U y. The continuation gains from revious shirks are secified in D y, which has one more element than D as the number of ossible shirks increases by one. Letting F denote the recursive function: F (n, Q, U, D) = max z ( 0) v w (U y,d y,z,w) 0 + F (n + 1,βz,U 1,D 1) + F (n + 1, (1 ) βz,u 0,D 0) subject to Probability (μ) z Q Regeneration U (γ ) U = w zc n + y U y (2.13) Regeneration D s <n (δ s ) D s = zd n s + y D y s The otimal contract solves max F (1, 1,U,D). U 0,D 0 FDIC s <n (λ s ) w zc n s + y 0 w + s 0 (U 1 + k=0 s 1 (U y + D y k ) k=0 D 1 k ) (U 0 + s Dk 0 ) k=0

15 G. Arie / Journal of Economic Theory 166 (2016) Problem (2.13) contains three non-standard elements. First, the state variable Q, imlements the robability constraints (2.11). Second, there is an FDIC for each relevant rivate history s <n, instead of just the one LDIC er history. Finally, the regeneration constraints for the shirking gains (D) are added. Fernandes and Phelan (2000) call the regeneration constraints for D the threat-keeing constraints. The intuition for these additional constraints is as follows: Because the roblem must set some continuation values after deviation in the FDIC, the dynamic roblem must recursively define those values and maintain them just as it does for the original utility in the standard formulation. The recursive LDIC roblem is the same as (2.13) with the FDIC only for s = 0 and the D regeneration constraint only for s = 0. It is rovided in section 4, where it is used to analyze the otimal contract. The dynamic roblem has several difficulties. First, it is not well defined for all values of U and D. For examle, no solution exists if U = 0 and D 0 > 0. The non-existence comlicates roving even standard results, such as concavity. Second, one cannot use this roblem to rove that in the otimal LDIC contract, all FDIC are slack, which is critical for the remainder of the analysis. 16 The dual analysis below rovides a direct roof for sufficiency of the LDIC. 3. Dual form The next subsections develo the economic intuition behind the dual variables, value and roblem. The result is summarized in (3.4) and the corresonding Proosition 1. Partial derivatives will be used throughout this section for intuition only. None of the formal results require differentiability Dual state variables: the effect on receding eriods The recursive dual analysis relaces the continuation utility U and shirking gains D state variables in the recursive formulation (2.13) with dual state variables γ and δ that cature the marginal effect of the continuation variables on the otimal contract. That is, γ catures F( ) U, and δ is a vector with elements δ s that each catures F( ) D s. Two key insights determine γ and δ. To develo both insights, consider a decision to marginally increase the agent s utility in the second eriod after failure in the first (U 0 ). Suose increasing U 0 increases the continuation rofit in the second eriod (after failure) by a small ε, discounted to first-eriod values. However, additional utility after failure decreases the agent s incentive to work. In articular, if the IC in the first eriod was binding, increasing U 0 imlies the IC is now violated. For convenience, a reorganized version of the LDIC from the recursive roblem is reeated here: 16 The standard line of roof would be to show that otimality (or first-order conditions) imlies the LHS of the FDIC for s + 1is larger than the LHS of the FDIC for s. Simlifying, the condition is ( d n s + Ds 1 ) ( D1 s 1 + (1 ) Ds 0 ) ( D0 s 1 0 Ds+1 1 ) ( D1 s + (1 0 ) Ds+1 0 ) D0 s. The technical challenge here is that the Ds+1 0 D0 s term may violate the inequality if the agent works more after failure than after success, because 1 0 is larger than 1. A searate line of roof can be used uniquely for the case that 0 = 0.

16 16 G. Arie / Journal of Economic Theory 166 (2016) 1 50 ( 0 )w zc + U 1 0 U D D (3.1) We see from (3.1) that any mixture of an increase to w or U 1 and a decrease to D 0 or D 1 can comensate for an increase in U 0. The cost to the contract of increasing U 0 therefore deends on the otimal adjustment of all these variables to the change in U 0. Therefore, it may seem that we must at least determine the sensitivity of the otimal contract after success to its state variables, in order to determine the correct marginal cost of increasing U 0. However, if λ is the correct shadow cost of LDIC (3.1), then, by construction, an increase in U 0 decreases the otimal value in the first eriod by exactly λ 0 1 regardless of the secific continuation. Using the IC shadow cost λ relaces the need to know the otimal contract in any other eriod. 17 This intuition extends directly to more eriods and is the first key insight: In any otimal contract, the effect of a change in any of the continuation variables (U, D) on any receding eriod is catured by the shadow cost of that eriod s ICs and only by it. The marginal effect on all receding eriods of a change in a continuation variable is the sum of all the receding shadow costs, using the correct coefficient and robability weights. The second insight is that in any otimal contract, the marginal continuation benefit of increasing the agent s utility or shirking gains must exactly equal the marginal cost imosed by the change on the receding eriods. That is, the marginal effect on F( ) of the continuation utility U (and the same for D) must exactly equal the marginal effect on all receding eriods calculated above. Otherwise, if the former is higher (lower), the contract is imroved by increasing (decreasing) the agent s romised utility. To formalize both insights, consider roblem F(n, Q, U, D) with γ the dual state variable as defined above and let γ y denote the corresonding dual state variable after outcome y. The enveloe theorem 18 F (n,q,u,d) yields γ = U. Alying this equality to the first-order conditions for U 1 and U 0 in (2.13) obtains the law of motion for the dual state variable γ 19 : n 1 γ 1 0 γ + λ s s=0 n 1 = 0; and γ 0 γ s=0 The same rocedure alies for the gains from shirking, D y s, λ s 0 1 = 0. (3.2) Use the IC in all the receding eriods to evaluate the receding-eriods-cost of changing the agent s rivate gains in a eriod (D). Observe that in the otimal contract, any continuation gains from increasing D must exactly equal the cost imlied on receding histories. F (n,q,u,d) D s Aly the enveloe theorem ( = δ s ) to the first-order conditions of (2.13) with resect to Ds y to obtain the laws of motion for γ s, rovided in (3.4). 17 See Chow (1997) chater 2 for a related discussion. 18 Note again that differentiability is not required for the formal derivation in the aendix. 19 The only difference between the LDIC and FDIC roblem is the additional ICs that yield the sum. For the LDIC, the sum has just one element (s = 0).