Optimal Contract for Machine Repair and Maintenance

Transcription

1 Optimal Contract for Machine Repair an Maintenance Feng Tian University of Michigan, Peng Sn Dke University, Izak Denyas University of Michigan, A principal hires an agent to repair a machine when it is own an maintain it when it is p, an earns a flow revene when the machine is p. Both the p an own times follow exponential istribtions. If the agent exerts effort, the owntime is shortene, an ptime is prolonge. Effort, however, is costly to the agent an nobservable to the principal. We sty optimal ynamic contracts that always ince the agent to exert effort while maximizing the principal s profits. We formlate the contract esign problem as a stochastic optimal control moel with incentive constraints in continos time over an infinite horizon. Althogh we consier the contract space that allows payments an potential contract termination time to take general forms, the optimal contracts emonstrate simple an intitive strctres, making them easy to escribe an implement in practice. Key wors : ynamic, moral hazar, optimal control, jmp process, maintenance 1. Introction In this paper, we sty a ynamic contract esign problem over an infinite horizon, in which a risk netral principal hires a risk netral agent to more efficiently operate a proction process ( machine ), which changes between two states: p an own. The state of the machine is pblic information. The p state yiels a constant flow of revene to the principal. The machine is sbject to ranom shocks which cases it to go own. When it is own, the machine can be repaire to be p again. Withot the agent, the machine stays in the p an own states for exponentially istribte ranom time perios with certain baseline rates. The agent has the expertise to improve maintenance an repair proceres by recing the instantaneos rate for breaking own, an increasing the instantaneos rate to recover from the own state, if the agent exerts effort. Exerting effort is costly to the agent, an the effort cost may be ifferent for repairing or maintaining the machine. Whether an when the agent pts in effort is the agent s private information. The principal wol like to ince the agent s effort, an is able to commit to a long term contract, which involves payments an potential termination contingent on pblic information. We 1

2 2 Athor: allow general forms of payments, incling both instantaneos payments an flow payments. The principal is allowe to terminate the contract any time, incling terminating the contract with a probability less than one when the machine changes state. We also assme that the agent has limite liability. That is, the agent can ecie to qit an never owes money to the principal. Althogh there is a wie literatre on maintenance an repair, the majority of this literatre focses on optimal maintenance an repair concte by a central ecision maker an has largely ignore the isses case by agency. In most settings, maintenance an repair is concte by an agent, an the principal that obtains revenes from the eqipment is not able to observe what effort the agent is exerting to keep his eqipment working. Ths, this paper makes a contribtion to the maintenance/repair literatre by moeling agency isses that have not been central to the literatre. The paper also contribtes to the ynamic contract esign literatre by consiering an environment with two machine states, which yiels ynamics that o not appear to arise in traitional settings withot sch a mlti-state environment. We formlate the ynamic contract esign problem as a continos time stochastic optimal control problem with incentive compatibility constraints, an characterize the optimal contract that inces fll effort from the agent. It is a stanar reslt in the ynamic moral hazar literatre that the agent s contination tility (also referre to as promise tility, see Spear an Srivastava (1987)) constittes the state space of the optimal control moel. Therefore, the optimal contract keeps track of this contination tility in aition to the machine state. The corresponing principal s total isconte tility at each machine state is a fnction of the agent s contination tility. There is a large literatre on maintenance scheling, althogh the classical maintenance literatre focses on the optimal scheling of maintenance activities in a centralize context (see, for example, Pierskalla an Voelker 1976, Paz an Leigh 1994, McCall 1965, Barlow an Proschan 1965). There are several papers consiering maintenance otsorcing contracts involving a maintenance agent an a cstomer. In particlar, Mrthy an Asgharizaeh (1998) sties a gametheoretic moel in which an agent offers several options of contracts to a cstomer, incling charging a fee for each repair ring a given ration of time, or charging a lmp sm fee for repairing the machine whenever it breaks own. The cstomer ecies whether to hire the agent epening on the propose contract. Mrthy an Asgharizaeh (1999) extens the moel to incle mltiple cstomers. Asgharizaeh an Mrthy (2) frther allows the agent to choose the nmber of cstomers an the nmber of service channels besies the pricing strategy. Following this line of work, Tarakci et al. (26) evelop incentive contracts to achieve channel coorination. A clear istinction of or paper is that we consier time epenent ynamic contracts while the aforementione papers either consier static settings or repeate single-perio settings.

3 Athor: 3 Plambeck an Zenios (2) is the first paper to consier a ynamic principal-agent moel of maintenance contract esign in a iscrete-time setting with a finite time horizon. In each perio, if the machine is own, the manager (agent) col choose between high an low effort levels, which frther etermine the probability the machine comes back p in the following perio. There is no moral hazar isse when the machine is p. We consier an infinite horizon continos time moel. More fnamentally, in or setting the agent nees to choose the effort level in both machine states, which yiels argably richer strctres in the optimal contract. The origin of the continos time ynamic contract literatre is often creite to the seminal paper Sannikov (28), which consiers a principal hiring an agent to control the rift of a Brownian motion. Several papers have applie similar techniqes in ifferent settings with applications mostly in corporate finance (see, for example, DeMarzo an Sannikov 26, Biais et al. 27, F 215, to name a few). Instea of controlling the rift of a Brownian motion, in or moel, the agent exerts effort to change the arrival rates of Poisson processes. Previos literatre has stie one-sie problems, i.e., either ecreasing or increasing the arrival rate of a Poisson process. Biais et al. (21), for example, consiers a firm (principal) hiring a manager (agent) to exert private effort to ecrease the arrival rate of large losses, moele as a Poisson process, when the two players have ifferent time iscont rates. Myerson (215) sties essentially the same moel as in Biais et al. (21), except that the two players share the same time iscont rate. In contrast, Sn an Tian (217) consier the case of increasing the arrival rate of a Poisson process by the agent s private effort. Varas (217), Shan (217), an Green an Taylor (216) sty similar moels with a finite nmber of arrivals an aitional featres, sch as averse selection isses or mltiple players. Becase of limite liability, the optimal contract strctres are ifferent for ecreasing verss increasing arrival rates. The common theme between the two one-sie cases is that the optimal ynamic contracts often involve letting the promise tility to take a constant jmp pon an arrival, which is pwar for the case of increasing the arrival rate, an ownwar for ecreasing the arrival rate. Or paper generalizes the previos literatre by stying contracts that ince the agent to alternatively increase an ecrease two ifferent arrival rates over time. The combine control problem is more complex, an the optimal soltion more intricate. In particlar, the ynamics of the contination tility following or optimal control policy is not a mere combination of one-sie control policies. Even thogh a change of machine state also inces pwar or ownwar jmps in the promise tility, the jmps may not be constant anymore. Specifically, following or optimal contract, whenever the machine breaks own, the agent nees to be penalize by at least an amont, enote as β, so that the agent has the incentive to prolong the p state an elay the penalty. De to limite liability, the principal cannot charge

4 4 Athor: the agent money. Therefore, the penalty takes the form of a rection (ownwar jmp) of the promise tility. Similarly, whenever the machine recovers from a own state, the agent nees to be reware with a bons, at least β, to motivate his efforts to shorten the own perio. The bons may be pai either monetarily, or as an increase (pwar jmp) in the contination tility. The exact optimal contract strctre iffers between the cases of β β an β < β. If β β, it is optimal to set the pwar an ownwar jmps in the promise tility exactly at β an β, respectively. If β < β, on the other han, the optimal contract is mch more complex, which appears to possess two featres new in the literatre. First, the incentive compatibility constraints are not always bining ner the optimal contract, which appears to contrast sharply with previos literatre (see, for example, Sannikov 28, Biais et al. 21, Shan 217, Sn an Tian 217). Secon, when the machine recovers from the own state, an pwar jmp of β may not be able to carry the promise tility to be above the level β. In this case, the principal nees to ranomize the promise tility between zero an a level at or above β. Sch a ranomization between an β wol not be srprising given or nerstaning of Myerson (215). What may appear srprising is that the higher en of the ranomization col be strictly above β. The rest of this paper is organize as follows: We introce the moel an erive the incentive compatibility constraints in Section 2. In Section 3 an 4, we characterize the optimal incentive compatible contract ner the conition β β an β < β, respectively. Finally, in Section 5, we provie a sfficient conition ner which the optimal incentive compatible contracts are optimal even if we allow shirking. Althogh the sfficient conition oes always hol, in the e-companion of this paper, we propose alternative simple contracts that allow shirking which may otperform the optimal incentive compatible contracts when the sfficient conitions o not hol. 2. Moel Consier a principal operating a process (e.g. a machine ) in a continos time setting. At any time t, the state of the machine, θ t, is either p or own, enote as or, respectively. The principal receives a revene at a positive rate R per nit of time when the machine is p. When the machine is own, the revene is zero. When the machine is p, the machine is sbject to ranom shocks that arrive exponentially with rate µ > that reslt in the machine breaking own. Once the machine is own, it takes an an exponentially istribte time with rate µ > to repair it back to state. There are many settings in which the above escribe sitation arises (for example, factories proce proct which can be sol for revene when the eqipment in the factories are working, similarly, airlines can only generate revene when their planes are in fnctioning state). The principal can hire an agent to improve the process (e.g,, better maintain an repair the machine (the plane etc.) so that it fails less often an when it fails, it is repaire an is available

5 Athor: 5 again to generate revenes faster). In particlar, whenever the agent exerts effort, the instantaneos rate of breaking own from state is rece to µ (, µ ) e to the better processes the agent implements with his efforts. Similarly, at state, the agent s effort increases the instantaneos rate of recovering to µ > µ. The agent s effort is not observable to the principal, an costs the agent at a rate c at state an c at state per time nit to maintain an repair the machine, respectively. At any point in time t, the pblic information incles all the time epochs the machines changes state by time t. Formally, we enote a left-continos conting process N t to represent the total nmber of pblic events, i.e., change of machine states, p to time t. Let F N be the filtration generate by the initial state θ an conting process N = {N t }. Frther enote an F N -preictable ν = {ν t } to represent the agent s effort process, sch that ν t {, 1} for any time t. Specifically, ν t = 1 an ν t = represent that the agent exerts effort an shirks at time t, respectively. Therefore, at any point in time t, the arrival rate of process N is µ(θ t, ν t ) := [µ ν t + µ (1 ν t )] θ t = + [µ ν t + µ (1 ν t )] θ t =. (1) The effort cost rate at time t is c(θ t, ν t ) := ν t (c θt = + c θt =). (2) We assme that the principal has the commitment power to a long-term contract base on pblic information. Overall, a ynamic contract Γ = (L, τ, q) incles a payment process L, a contract termination time τ, an a stochastic termination process q. Specifically, enote an F N -aapte process L = {L t } t to represent the cmlative payment from the principal to the agent p to time t. The payment can be an instantaneos one, I t, or a flow with rate l t, sch that L t = I t + l t t. We assme the agent has limite liability. That is, the agent oes not pay the principal, or, I t an l t. The contract not only incles payments, bt also the possibility of terminating the agent at a ranom time τ. We consier two ways of contract termination. First, at any point in time t when the machine changes state (i.e., N t = 1), we allow the principal to terminate the contract ranomly, with probability q t [, 1], where the probability q t is a fnction of Ft N, which represents all of the information on machine state changes ntil time t. Therefore, the contract also contains an F N - aapte process q = {q t } t for ranom contract termination. Secon, we also allow the principal to terminate the agent even at times of no machine state change an withot ranomization as a fnction of the whole history Ft N. As will be clear later in the paper, allowing ranom termination is crcial to constrct optimal contracts for certain moel parameter settings. The principal an

6 6 Athor: the agent are both risk-netral an iscont ftre cash flows at rate r. Given contract Γ an effort process ν, the agent s expecte total isconte tility is the cmlative payments mins the effort cost, expresse as the following τ (Γ, ν, θ ) = E e rt (L t c(θ t, ν t )t) θ. (3) In orer to express the principal s profit ner a contract Γ, we first nee to express the principal s baseline revene, v τ, after terminating the agent. It is clear that the vale of v τ epens on the state of machine at termination time τ, where v τ = v if the state of the machine is p when the agent is terminate, an v τ = v if the state of the machine is own when the agent is terminate. It can be verifie that withot the agent 1, the principal s total isconte ftre profit takes the following vales for states an, respectively, r + µ R µ v :=, an v := R. (4) r r + µ + µ r r + µ + µ Overall, the principal s expecte total isconte profit ner a contract Γ an effort process ν is efine as 2 τ U(Γ, ν, θ ) = E e rt (R θt =t L t ) + e rτ v τ θ. (5) Therefore, for initial states θ = or θ =, we can efine a game between the two players, in which the principal esigns the optimal contract Γ that maximizes tility U(Γ, ν, θ ), anticipating the agent s effort process ν that maximizes (Γ, ν, θ ). Throghot the paper, we focs on stying contracts that ince the agent to always exert effort (so calle incentive compatible contracts). 3 Later in Section 5 we provie a sfficient conition on moel parameters sch that it is inee optimal for the principal to only focs on incentive compatible contracts. Incentive Compatibility A contract Γ is incentive compatible (IC) if in eqilibrim, the agent has the incentive to always exert effort (to exert effort to better maintain the machine at state so that the machine s failre rate rops to µ an to exert effort to better an faster repair the machine so that it comes back p at rate µ at state ), i.e. ν := {ν t = 1} t [,τ]. That is, the contract is incentive compatible if 4 (Γ, ν, θ ) (Γ, ν, θ ), F N -preictable effort process ν, θ {, }. (IC) In this paper we focs on the class of incentive compatible contracts that always ince effort. We will characterize the incentive compatible contracts in a recrsive manner, so that the contract esign problem may be formlate as a stochastic optimal control problem. Therefore, we will first introce the agent s contination tility at time t following cotract Γ an effort process ν as, τ W t (Γ, ν) = E e r(s t) (L s c(ν s, θ s )s) F t {t τ}. (6) t

7 Athor: 7 It is clear that W (Γ, ν) = (Γ, ν, θ ) for θ consistent with F. We assme that the agent can ecie to qit if ftre payments o not compensate for effort costs. That is, the contination tility nees to always be non-negative (also calle the inivial rationality (IR) conition), i.e., W t, t. (IR) The following Lemma provies the evoltion of the agent s contination tility W t ner a contract Γ, which is the so-calle promise keeping (PK) conition in the ynamic contract literatre. It also provies an eqivalent conition for (IC) in terms of the contination tility W t. Lemma 1. For any contract Γ, there exists an F N -preictable process H t sch that for t [, τ], W t =rw t t + c(ν t, θ t )t L t [(1 X t )H t + X t W t ] N t + [(1 q t )H t + q t W t ] µ(ν t, θ t )t, (PK) in which Bernolli ranom variable X t takes vale 1 with probability q t. Frthermore, contract Γ satisfies (IC) if an only if q t W t + (1 q t )H t β for θ t =, an q t W t + (1 q t )H t β for θ t =, (7) for all t [, τ], where β := Finally, we have H t W t for all t, which garantees (IR). c µ µ an β := c µ µ. (8) The (PK) conition is a stanar reslt for the ynamics of the agent s contination tilities over time. To facilitate nerstaning, it is helpfl to consier its heristic erivation base on iscrete time approximation. Consier a small time interval [t, t + δ). Assme that the agent s contination tility W t evolves continosly to W t+ in this interval, nless there is a change of machine state, with probability µ(θ t, ν t )δ. With a change of state, the contination tility takes a jmp either to W t H t, with probability 1 q t, or to (termination), with probability q t. We note that H t col be positive or negative, inicating the agent s contination tility can increase or ecrease after the change of the machine s state. Also taking into consieration the effort cost c(θ t, ν t )δ, payment l t δ, an time iscont rate r (for the moment, ignore the instantaneos payment I t ), the above escription of the iscrete time approximation of the contination tility implies the following, W t = c(θ t, ν t )δ + l t δ + µ(θ t, ν t )δ [q t + (1 q t )(W t H t )] + [1 (µ(θ t, ν t ) + r)δ] W t+. (9)

8 8 Athor: As δ approaches, replace it with t. Frther sbtract W t+ from both sies an note that as W t+ approaches W t, e to continity, we obtain W t+ W t = rw t + c(θ t, ν t ) l t + [(1 q t )H t + q t W t ] µ(θ t, ν t ) t, an, therefore, the terms involving t in (PK). The change of machine state (N t = 1) reslts in a jmp of either W t or H t, epening on the otcome of the ranom variable X t, which is captre in the term [(1 X t )H t + X t W t ] N t in (PK). Finally, the instantaneos payment I t also inces a jmp in W t. Following stanar IC conitions in Sannikov (28) an Biais et al. (21), one wol only obtain the reslt that the magnite of H t is larger than β or β. Or (IC) conition in Lemma 1, however, generalizes the stanar form so that the contination tility W t takes a ownwar jmp of either H t, or W t to, which represents contract termination. In Section 4, we show that the probability of ranom termination, q t, col inee be positive ner the the optimal contract. The vales β an β efine in eqation (8) reflect the ratios between effort cost an improvement in the repair or failre rates, which reveal the intition behin the (IC) conition. Specifically, assme that when the machine is own, the principal wol pay the agent β pon its recovery. The agent can then weigh the effort cost to repair the machine over the next δ time nits, c δ, against the higher expecte bons, β (µ µ )δ. Or efinition of β ensres that these two terms are the same (i.e. β (µ µ )δ = c δ), an, therefore, the agent is inifferent between shirking an exerting effort. The term β has a similar meaning when the machine is p. Combining β an β, it is intitive that to incentivize the agent to always exert effort, we nee to satisfy conition (7). That is, when the machine is p, the agent s contination tility wol rop by β on average when it breaks own, an, when the machine is own, the contination tility wol jmp p by β on average when it recovers. Also, we will show that the strctre of the optimal contracts are ifferent in the case β β an the case β < β. Frther, becase the agent s contination tility mst be non-negative at all times, an it mst be rece by an amont H t when the state changes at time t accoring to (PK), we mst have W t H t. Hence, conition q t W t +(1 q t )H t (Γ, ν ) β in (7) implies that any incentive compatible contract mst satisfy the following conition, W t β when θ t =. (1) This is becase a ownwar jmp of β at machine breakown has the potential to lea to a negative contination vale, while the agent can secre a zero contination vale by simply walking away. Therefore, when the machine is p an W t is lower than a threshol β, the principal

9 Athor: 9 nees to ranomize the contination tility to either (terminating the contract), or back to the threshol. This is why the ranomize termination process q t is reqire for the optimal contract. Interestingly, as we will show in the next two sections, the ranom termination only occrs if β < β. If β β, on the other han, the optimal contract always garantees W t β when the machine is p. In the next two sections, we sty an characterize in etail the optimal contracts that ince the agent to always exert effort before termination. 3. The Case β β In this section we consier the case β β. The strctre of the optimal contract in this section, althogh new, may not appear srprising to reaers alreay familiar with the continos time contracting literatre (Biais et al. 21, Sn an Tian 217). However, this section provies a gentle preparation to the more complex an elicate strctre in the optimal contract for the case β < β in the next Section. In this section, we first introce a simple incentive compatible contract in Section 3.1, which helps s to lay the fonation of the optimal contract ner a reglarity conition, to be presente in Section 3.2. Section 3.3 frther provies the principal s vale fnctions ner the optimal contract an the proof of optimality. When the reglarity conition oes not hol, we sty the optimal contracts in the e-companion A simple incentive compatible contract Before we introce the optimal contract, it is worth looking at a simple incentive compatible contract that never terminates the agent. This contract elivers constant contination tilities w an w when the machine s state is an, respectively. To achieve incentive compatibility, we jst let the constraints (7) to be bining an q t =. Hence, the contination tility rops by β when the machine breaks own. So the contination tilities shol satisfy w = w + β. (11) On the other han, when the machine recovers, the contination tility nees to take an pwar jmp of β from w. However, becase the contination tility at state is w, an β β, the principal pays the ifference β β as an instantaneos payment to the agent at the moment the machine recovers. To make this work, we still nee a flow payment with rate l when the agent is maintaining the machine at state. Hence, the instantaneos payment provies the incentive for the agent to repair the machine faster, an the flow payment incentivizes the agent to maintain the machine sch that it stays in the p state longer.

10 1 Athor: With the escriptions above, in the following, we erive the expressions for l, w an w throgh iscrete time approximation. When the machine is p, we consier a small time interval [t, t + δ], in which the agent gets a flow payment l δ. With probability µ δ, the machine breaks own an payment stops. Otherwise, with probability 1 µ δ, the machine contines p an rnning. On the other han, when the machine is own, in a small time interval [t, t + δ], with probability µ δ, the machine recovers an the agent gets an instantaneos payment β β. Otherwise, with probability 1 µ δ, the machine remains own. Hence, the contination tilities w δ an w δ in states an, respectively, have the following relationship: w δ = c δ + µ δ(β β + γ w δ ) + (1 µ δ)γ w δ, w δ = c δ + l δ + µ δγ w δ + (1 µ δ)γ w δ. (12) Together with eqation (11) an lim w δ = w, δ lim w δ = w, δ we are able to solve for w, w an l, an obtain w = µ β c, w = w + β an l = r w + µ β + c. (13) r It can be verifie that the corresponing societal vale (smmation of the principal an the agent s tilities) at states an are, respectively, v = µ (R c ) (r + µ )c r(r + µ + µ ) an v = (r + µ )(R c ) µ c r(r + µ + µ ). (14) In the seqel, we refer to this simple contract as Γ. Here we focs on stying the optimal contract ner the conition that the society is better off when the agent exerts fll effort, an there is no termination, i.e., v v, (15) which is eqivalent to µ c + (r + µ ) c R h := r + µ + µ. (16) µ µ + (r + µ ) µ Also, ner β β, conition (15) implies that v v. (17) Conition (15) an (17) also state that in the centralize problem, exerting fll effort to repair an maintain the machine is better than not exerting effort at all when the machine starts in states an, respectively. Later in e-companion, we analyze the case when conition (15) is violate.

11 Athor: 11 Althogh this simple contract Γ satisfies all the incentive compatibility constraints, it is actally not optimal. This is becase the simple contract only ses the carrot of payments withot the stick of termination. At the en of Section 3, Proposition 4 shows that this simple contract is actally the optimal incentive compatible contract if terminating the agent is not allowe. The reason that we introce this simple contract is becase accoring to the optimal contract, it is possible that the contination tilities eventally become w an w. After that point, the optimal contract is ientical to the simple contract, an the agent is never terminate. However, following the optimal contract, it is also possible that the contination tilities never reach w an w before the agent is terminate Optimal contract when v v In this sbsection, we will evelop a contract Γ 1, an we will prove that this contract is optimal for the principal in the next sbsection. The contract keeps track of the agent s contination tility. Figre 1 epicts two sample trajectories of the agent s contination tility in the propose contract where the machine starts at state θ =. Figre 1 Two sample trajectories of contination tility with µ = 6, µ = 9, µ = 5, µ = 2, c =.8, c = 1, r =.9, R = 7.5. In this case, w =.74, w = 1.1 an β =.27 < β =.33. The policy starts from W = w =.26. The two ashe horizontal lines represent the level of w an w, respectively. The pwar jmp level when the machine is repaire is β an the ownwar rop level when the machine breaks own is β. The contination tility starts from an initial contination tility W = w (, w ). While repairing the machine, this tility keeps ecreasing (the exact form to be specifie later) ntil either the machine is repaire or the tility reaches. If the machine has not recovere before the

12 12 Athor: tility W t reaches, the principal terminates the agent. The otte crve in Figre 1 represents this sitation, where the contination tility ecreases to zero at time τ. On the other han, if the machine recovers at time t with W t >, the tility W t takes an pwar jmp of level min{β, w W t } an the agent is pai (W t +β w ) + instantaneosly. See the soli crve in Figre 1, which represents another sample trajectory. Between [, t 1 ], the contination tility is ecreasing over time. At t 1, it jmps p by β becase W t1 < w β. The corresponing instantaneos payment is. Then the contract contines with the agent maintaining the machine in the p state, while the contination tility keeps increasing ntil either it reaches w, or the machine breaks own. Between [t 1, t 2 ], the contination tility is increasing over time. At time t 2, the machine breaks own an the contination tility rops by β. Again, between [t 2, t 3 ], the agent is repairing the machine with the contination tility ecreasing over time. After t 3, the machine oes not break own before the contination tility reaches w at time ˆt 3, at which point the flow payment l starts. After time ˆt 3, the agent s contination tility is jmping back an forth between w an w. The contract becomes exactly the same as the simple contract we introce in the previos sbsection. Henceforth, when the machine is p, the agent maintains the machine with the flow payment l ntil it breaks own; when the machine is own, the agent repairs it an receives an instantaneos payment β β at the moment the machine recovers. Note that when the machine is own, the ynamics of W t follow: W t = r(w t w )t + min{ w W t, β }N t. (18) This is aligne with the sample trajectories epicte in (PK). The iffsion term jst represents that if W t is below the bon w, it keeps ecreasing before the machine recovers. Whenever it reaches w, W t is maintaine at w before the state of the machine changes. The secon term represents the pwar jmp that W t takes when the machine recovers. When the machine is p, we have W t = rw t + µ β + c W t < w t β N t. (19) Again, the iffsion term represents that if W t is below w, it keeps increasing. If W t = w at some t, W t is kept at w before the machine breaks own. The last term β N t represents that every time the machine breaks own, the agent s contination tility rops by β. Combining eqations (18) an (19), we provie a formal efinition of the propose optimal contract in the following. Definition 1. For a machine starting from state θ {, }, efine contract Γ 1(w) = (L, q, τ ) as the following, where w [β, w ] if the initial state is, an w [, w ] if the initial state is.

13 Athor: 13 i. The ynamics of the agent s contination tility W t follows W t = r(w t w )t + min{ w W t, β }N t θ t = + (rw t + µ β + c ) Wt < w t β N t θ t =, (DW1) from the initial contination tility W = w. ii. The payment to the agent follows L t = l W t = w θ t =t + (W t + β w ) + θ t =N t. iii. The ranom termination probability is qt =, (i.e. there is no ranom termination) an the termination time is τ = min{t : W t = }. It is worth noting that the ynamics of W t in the propose optimal contract follows (PK), with H t = β θt = + β θt =, L t = L t an q t = qt. Also, in the propose optimal contract, the incentive compatible constraints (7) are bining, an the principal never ranomly terminates the contract. It is only possible to terminate the agent when the machine is own (note that we o not terminate the agent exactly at the point when the machine goes own bt when the contination tility reaches zero, e.g., after a long enogh own perio). On the other han, when the machine is p, the agent s contination tility is always greater than β. This is becase if the initial state of the machine is p, the initial contination tility wol be at least β an keeps going p ntil the first break own; after the agent has finishe repair once, the contination tility wol always jmp to a level above β β to start the p state Vale fnctions an proof of optimality when v v In this section, we first heristically erive the ynamics of the principal s tility, as a fnction of the agent s contination tility ner the propose optimal contract Γ 1 efine in Definition 1. Later, in Proposition 2, we prove that or erive vale fnction is the actal optimal vale fnction of the principal. Specifically, let J (w) an J (w) represent the principal s total tility at time t when the agent s contination tility is w if the machine s state is an, respectively. If the machine s crrent state is, consier a small time interval [t, t + δ]. With probability µ δ the machine recovers an changes to state, the principal pays the agent (w + β w ) +, an, corresponingly, the contination tility jmps p to min{w + β, w }. With probability 1 µ δ, on the other han, the machine stays in, an the contination tility evolves to w + r(w w )δ. Therefore, we have J (w) = e µ rδ δ (w + β w ) + + J (min{w + β, w }) + (1 µ δ)j (w + r(w w )δ) + o(δ). Sbtracting J (w) an iviing δ on both sies, then letting δ approach, we obtain (µ + r)j (w) = r(w w )J (w) + µ J (min{w + β, w }) µ (w + β w ) +. (2)

14 14 Athor: Similarly, consier the machine s crrent state at, an a small time interval [t, t + δ], when the principal collects revene Rδ an the agent s contination tility w β. With probability µ δ, the machine breaks own an changes to state, an the contination tility rops to w β. With probability 1 µ δ, on the other han, the machine stays in, the contination tility evolves to w + (rw + µ β + c )δ if w < w, an the principal pays the agent l δ if w = w contination tility stays at w. Therefore, while the J (w) =Rδ + e rδ µ δj (w β ) + (1 µ δ) J (w + (rw + µ β + c )δ w< w ) l w= w + o(δ). Following similar steps as before, we obtain that for w [β, w ], (µ + r)j (w) = (rw + µ β + c ) w< w J (w) + R + µ J (w β ) l w= w. (21) The bonary conitions are J () = v an J () = v, (22) reflecting that the principal receives baseline revenes v an v (as efine in (4)), pon terminating the agent in states an, respectively. For the interval [, β ], we simply exten the fnction J (w) to be linear, that is, J (w) = J () + J (β ) J () β w, for w [, β ]. (23) As we have emonstrate, the agent s contination tility never falls into the interior of this interval if we follow the optimal contract accoring to Definition 1. However, having an extene efinition of J (w), for that interval will be sefl as we erive the proof of the optimality of the contract in Definition 1. It is helpfl to consier the societal vale fnctions, efine below as the smmation of the principal an the agent s tilities, V (w) = J (w) + w an V (w) = J (w) + w. (24) Following (2)-(23), we obtain the following system of ifferential eqations for V an V, (µ + r)v (w) = µ V (min{w + β, w }) c r( w w)v (w), w [, w ], (25) (µ + r)v (w) = c + R + µ V (w β ) + (rw + µ β + c ) w< w V (w), w [β, w ], (26) V (w) = V () + V (β ) V () w, β (27) V () = v an V () = v. (28) Frthermore, as soon as the contination tility reaches w at state, contract Γ 1 becomes ientical to the simple contract stie in Section 3.1. This implies the following bonary conitions in which v an v are efine in (14). V ( w ) = v an V ( w ) = v, (29)

15 Athor: 15 Proposition 1. The system of ifferential eqations (25) an (26) with bonary conitions (27), (28) an (29) has a niqe soltion: the pair of fnctions V (w) on [, w ] an V (w) on [, w ], both of which are increasing an strictly concave. Following (24), we also have niqely efine fnctions J (w) an J (w) that satisfy eqations (2) an (21) with the corresponing bonary conitions; both fnctions are concave with maximizers w an w on [, w ] an [, w ], respectively. Next, we show that fnctions J (w) an J (w) are inee the vale fnctions of the principal ner contract Γ 1(w), which starts with a contination tility of w at the initial states θ = an θ =, respectively. Proposition 2. For any state θ {, } an contination tility w [, w θ ], we have U(Γ 1(w), ν, θ) = J θ (w). That is, fnctions J (w) an J (w) are eqal to the principal s total isconte tilities of following contract Γ 1 when the initial state of the machine is an, respectively. Figre 2 provies a nmerical example of societal vale fnctions V an V an the principal s vale fnctions J an J. To implement the contract, the principal nees to esignate the initial contination tility W. The initial contination tility shol be w if the machine starts at state θ = an shol be w if the machine starts at state θ =. Note that e to concavity, if J (β ) J (), then w β. Otherwise, the optimal initial contination tility w =, an in this case, it is better not to hire the agent if the initial state of the machine is. Finally, to show that the contract Γ 1 efine in Definition 1 is inee optimal, in the next proposition, we will first show that fnctions J an J are pper bons for the principal s tility ner any incentive compatible contract Γ, if the machine starts at states an, respectively. Proposition 3. For any incentive compatible contract Γ an any initial state θ {, }, we have J θ (Γ, ν, θ) U(Γ, ν, θ), in which we exten the fnction J θ (w) = J θ ( w θ ) (w w θ ) for w > w θ. Therefore, we know that for any incentive compatible contract Γ an initial state θ, U(Γ, ν, θ) J θ ((Γ, ν, θ)) J θ (wθ) = U (Γ 1(wθ), ν, θ), where the first ineqality follows from Proposition 3, the secon ineqality follows from the fact that wθ is the maximizer of J θ, an the thir eqality follows from Proposition 2. This implies the following main reslt of this section. Theorem 1. The optimal incentive contract is Γ 1(wθ) if β β an the machine starts from state θ {, }. That is, U Γ 1(wθ), ν, θ U(Γ, ν, θ) for any incentive compatible contract Γ an state θ.

16 16 Athor: (a) Societal Vale fnctions (b) Principal s Vale fnctions Figre 2 Vale fnctions with µ = 6, µ = 9, µ = 5, µ = 2, c =.8, c = 1, r =.9, R = 7.5. In this case, w =.74, w = 1.1 an β =.27 < β =.33. We have shown that if R is sfficiently large ((15) or eqivalently (16) is satisfie), then Γ 1(w θ) is the optimal incentive contract if β β. The next reslt states that if R is not large enogh, it is also better for the principal to not hire the agent than motivating effort. Theorem 2. If R < r + µ + µ β, (3) we have v θ U(Γ, ν, θ) for any incentive compatible contract Γ an state θ {, }. One can easily verify that the right han sie of (16) is greater or eqal to the right han sie of (3) an they are eqal when β = β. Hence, when β > β, we characterize moel parameters into three regions where the three regions may be easily characterize by focsing on the revene rate R when the machine is p, fixing other moel parameters. When R is sfficiently large, the incentive compatible constraints in eqation (7) are always bining, an the ynamic contract emonstrates the richest strctre compare with other parameter settings. In this case the simple contract stie in Section 3.1 is an absorbing state. If R is sfficiently small, in comparison, no incentive compatible contract (incling hiring an agent only to maintain or repair analyze in the Online Appenix) performs better for the principal than not hiring the agent at all. As we will emonstrate in Section 5, for these moel parameters, not hiring the agent is the best strategy for the principal, even among contracts that allow shirking. Finally, if R is in between the two aforementione regions (both (16) an (3) are violate), we sty the optimal contracts in the e-companion.

17 Athor: 17 Finally, if we o not allow contract termination, the following Proposition shows that the simple contract Γ introce in Section 3.1 is optimal. Proposition 4. We have U( Γ, ν, θ) U(Γ, ν, θ) for any state θ {, } an incentive compatible contracts Γ sch that τ =. 4. The case β < β If β < β, the contract in Definition 1 is no longer incentive compatible. To see this, consier the sitation that the machine recovers at time t with W t < β β. If the contination tility still jmps p by β, then W t+ < β. Following conition (1), the principal cannot incentivize the agent if his contination tility W t < β while maintaining the machine. As we will show in the following, the optimal contract nees to involve ranom termination when the agent s contination tility is low. Frthermore, when the contination tility is high, the optimal contract involves a region where one of the incentive compatible constraints in (7) is not bining. This is qite pecliar, becase, as far as we know, IC constraints are always bining in optimal contracts stie in the continos time moral hazar literatre (see, for example, Sannikov 28, Biais et al. 21, Shan 217, Sn an Tian 217). The strctre of this section mirrors Section 3. Similar to Section 3.1, we first propose a simple incentive compatible contract in Section 4.1, which alreay involves non-bining (IC) constraints. In Sections 4.2 an 4.3, we first sty the optimal contract ner conition (17), which, parallel to (16), is eqivalent to µ c + (r + µ )c R h := r + µ + µ, (31) µ µ + (r + µ ) µ an implies (15) when β < β. This optimal contract involves ranom termination. Finally, in e-companion, we sty optimal contracts when conition (17) is violate A simple incentive compatible contract Before we introce the optimal contract, we introce a simple incentive compatible contract that never terminates the agent. This contract elivers constant contination tilities ŵ an ŵ when the machine s state is an, respectively. In orer to satisfy the incentive compatible constraints (7), we mst have ŵ ŵ = β, (32) so that when the machine breaks own, the ownwar jmp in the contination tility eqals β. However, this implies that the pwar jmp in the contination tility when the machine

18 18 Athor: recovers is also β, which is higher than β as is reqire in (7). Therefore, the incentive compatible constraints (7) are not bining in state. Following the same iscrete time approximation approach as in Section 3.1, we obtain two eqations for ŵ, ŵ an the flow payment l at state, similar to (12). Together with (32), we obtain ŵ = µ β c, ŵ = ŵ + β, an l = rŵ + µ β + c. (33) r Becase β < β, it is clear that ŵ an ŵ are higher than w an w as efine in (13), respectively. In this simple contract, the agent is never terminate. Therefore, the societal tility in states an remains v an v as efine in (14), respectively. In the seqel, we refer to this simple contract as ˆΓ. Later in Section 4.2, we show that the agent s contination tility has a chance to eventally become ŵ an ŵ following the optimal contract. After reaching that point, the optimal contract becomes ientical to the simple contract, an the agent is never terminate, similar to the role of the simple contract Γ of Section 3.1 in the last section. At the en of Section 4, similar to Proposition 4, we present Proposition 8, which shows that this simple contract is actally the optimal incentive compatible contract if terminating the agent is not allowe. Uner conition (31), contract ˆΓ achieves social efficiency. At this point, it is worth explaining the intitive reason why following the optimal contract, the social efficiency shol be achieve at some point. First, becase the principal an agent has the same time iscont rate, they have the same total isconte valation for any payments. Therefore, maximizing the principal s vale fnction is eqivalent to maximizing the societal vale fnction (total tility between the principal an the agent as a fnction of the contination tility). Hence, becase social efficiency is achievable when the agent s contination tility is high enogh following contract ˆΓ, it mst be achievable at the same contination tility level ner the optimal contract Optimal Contract when v v In the following, we illstrate the strctre of the optimal contract sing Figre 3 before formally efining the optimal contract. Once again, the contract keeps track of the agent s contination tility W t over time. The ynamics of W t, however, are more complicate than the optimal contract in Section 3.2. In particlar, if W t (, w ) in state, then the contination tility keeps ecreasing ntil either the machine is repaire, or the contination tility reaches an the agent is terminate. If W t [ w, ŵ ] in state, on the other han, the contination tility remains a constant ntil the machine is repaire. If, pon recovery to state, the contination tility is below β, however, conition (1) implies that the machine cannot stay in state at the crrent contination tility level. Instea, the principal ranomly terminates the contract or resets W t to be at or above β.

19 Athor: 19 Figre 3 epicts two sample trajectories following the propose contract starting at state θ = from an initial contination tility W = w (, ŵ ). First, focs on the soli crve. The contination tility ecreases over time while the agent provies effort to get the machine p, ntil time t 1, when the machine recovers. At this point, the contination tility jmps p by β an the agent starts maintaining the machine at state. The contination tility keeps increasing ntil time t 2, when the machine breaks own. In the time interval [t 2, t 4 ], the contination tility behaves the same way as it oes in [, t 2 ], with the machine recovering at t 3. When the machine breaks own again at time t 4, however, the contination tility is alreay so high that it will still be above w after a ownwar jmp of β. Becase W t w at state, the contination tility is kept at this level as a constant, ntil the machine recovers at time t 5. At this point in time the contination tility takes an pwar jmp (rw t5 + c )/µ > β, or, the IC constraint (7) at state is not bining. After time t 5, the machine stays in state while the contination tility increases to reach ŵ at time t 5, at which point the contract follows ˆΓ as efine in Section 4.1. Note that following this sample trajectory, the strctre of the optimal contract after time t 4 behaves ifferently from the optimal contract Γ 1 efine in Section 3 (becase the contination tility remains constant even thogh the machine is own). Now we focs on the other sample trajectory in Figre 3, the otte crve. The machine is in state ring time intervals [, ˆt 1 ] an [ˆt 2, ˆt 3 ], an in state ring [ˆt 1, ˆt 2 ]. The contination tility ecreases in state an increases in state. Right before the machine recovers for the secon time, at ˆt 3, the contination tility is below β β. Therefore, even an pwar jmp of β cannot raise the contination tility above β. In light of the iscssion in the beginning of this section, the agent is terminate with probability q ˆt 3 = (β Wˆt 3 +)/β. On the other han, with probability 1 q ˆt 3, the agent s contination tility is reset to β. It is clear that ranomization nees to occr at state if the contination tility is below the threshol β. In fact, the threshol below which the ranom termination occrs oes not have to be exactly β. It can be at a more general level of ˆβ β. That is, as long as the contination tility W t is below ˆβ in state, the agent is ranomly terminate with probability qt = ( ˆβ W t )/ ˆβ. If termination oes not happen, the contination tility is reset to ˆβ. In the contract epicte in Figre 3, we have ˆβ = β, bt this eqality oes not necessarily always hol an we may have ˆβ β. Formally, we efine the following contract, Γ ˆβ, an later show that the optimal contract follows this efinition with an appropriately chosen vale of ˆβ β. Definition 2. For any ˆβ [β, w ), efine contract Γ ˆβ(w) = (L, q, τ ) for w [, ŵ θ ] if the initial state of the machine is θ {, }.

20 2 Athor: Figre 3 Two sample trajectories of contination tility with moel parameters µ = 2, µ = 1, µ = 6, µ = 2, c = 1, c = 1.2, r =.8, R = 2. In this case, w = 3, ŵ = 6, ŵ = 7 an β = 1 > β =.6. The policy starts from w =.95. The soli crve represents a sample trajectories which brings the agent to the point of never terminate. The otte crve represents another sample trajectory in which the agent is terminate e to a ranom raw at a point when the machine recovers. i. The ynamics of the agent s contination tility W t, follows rw t + c W t = r(w t w ) Wt < w t + W t ( w,ŵ ] + µ Wt ( ˆβ β, w ] β (1 X t )( ˆβ W t ) X t W t N t + Wt <β ˆβ θ t = + [(rw t + µ β + c )t Wt < w β N t ] θ t =, (DW2) from an initial contination tility W = w. ii. The payment to the agent follows L t = l θ t = W t =ŵ t. iii. The ranom termination probability is q t = ˆq(W t ) Wt +β < ˆβ θ t =, in which ˆq(w) = ˆβ (w + β ). (34) ˆβ It is worth noting that in this contract, constraint (7) is not always bining. Specifically, if W t > w, following Γ ˆβ we have q t = an H t = (rw t + c )/µ < β. Before we rigorosly prove the optimality of the contract, here we first explain the intition why constraint (7) is not always

21 Athor: 21 bining in the case β < β. If we ha to force incentive compatible constraints to be bining, the pwar jmp in the contination tility is β, which is smaller than the ownwar jmp β. Therefore, no matter where the contination tility starts from, a ownwar jmp of β cannot be flly compensate by an pwar jmp of β. As a reslt, regarless of where the contination tility starts from, a sample trajectory (however nlikely) with a seqence of very freqent state switches eventally rives the contination tility own to. The existence of sch sample trajectories implies that social efficiency wol not be acheivable. This contraicts the argments at the en of Section 4.1 that the optimal contract shol be able to achieve social efficiency. This contraiction implies that we cannot enforce IC constraints to be bining all the time Vale fnctions an optimality when v v There are some important istinctions in the approach to etermine the principal s vale fnctions, in the case of β < β, compare with the one in Section 3.3, becase here we nee to specify the threshol ˆβ that efines when/if the agent will be ranomly terminate. First, let J (w) an J (w) represent the principal s vale fnctions for states an, respectively. Following Definition 2 an similar heristic erivation steps as in Section 3.3, we obtain the following system of ifferential eqations. In particlar, for state, eqation (2) in Section 3.3 becomes the following three eqations (µ + r)j (w) = µ J w + rw + c, µ w [ w, ŵ ], (35) r(w w )J (w) = (µ + r)j (w) µ J (w + β ), w [ ˆβ β, w ], an (36) r(w w )J (w) = (µ + r)j (w) µ ˆq(w)J () + 1 ˆq(w) J (β), w [, ˆβ β ]. (37) For state, the ifferential eqation is similar to (21) for w [ ˆβ, ŵ ]. That is, (rw + µ β + c ) w<ŵ J (w) = (µ + r)j (w) R µ J (w β ) + l w=ŵ, w [ ˆβ, ŵ ] (38) De to ranomization, we may frther exten fnction J (w) to the interval [, ˆβ] as a linear fnction with a slope a, that is, J (w) = J () + aw, w [, ˆβ]. (39) The principal receives baseline revenes v an v, as efine in (4), pon termination in states an, respectively, which implies the following bonary conitions J () = v an J () = v. (4) We first present the following reslt regaring general soltions to the aforementione ifferential eqations.

22 22 Athor: Lemma 2. For any a > 1, there exists a niqe pair of fnctions J a ˆβ an J, respectively, that satisfy (35)-(4). Frthermore, fnctions J a ˆβ continosly ifferentiable, except for J a ˆβ (w) at w = ˆβ. an J a ˆβ, in place of J a ˆβ (w) an J (w) are twice It is straightforwar to show that it is sfficient to focs only on the case a > 1, becase the slope a which represents how mch the tility of the principal changes as the agent s contination tility increases can never be less than 1 (otherwise, this wol mean that we are rewaring the agent for estroying vale). Next, we etermine the threshol ˆβ for a given parameter a. The iea is to set ˆβ sch that fnction J a ˆβ (w) is ifferentiable at ˆβ if possible, so that we achieve smooth pasting 5 between (38) an (39). We efine the following fnction for ˆβ [β, w ), f a ( ˆβ) := J a ˆβ ( ˆβ ) J a ˆβ ( ˆβ+ ) (r ˆβ + µ β + c ) (41) Clearly, we can achieve smooth pasting if there exists a ˆβ sch that f a ( ˆβ) =. Lemma 3. For any a > 1, fnction f a ( ˆβ) is increasing in ˆβ on [β, w ), an lim β w f( ˆβ) >. Therefore, the following qantity β a is well efine, β, f a (β ), β a := fa 1 (), f a (β ) <, (42) in which f 1 a is the invervse fnction of f a. Frthermore, as soon as the contination tility reaches ŵ in state, the contract Γ ˆβ becomes ientical to ˆΓ, an the agent will no longer be terminate. This implies the following bonary conitions J āβ ā (ŵ ) = v ŵ an J āβ ā (ŵ ) = v ŵ, (43) in which v an v are the societal vale fnction when the agent is never terminate, as efine in (14). Now we are reay to niqely etermine the vale a in eqation (39) for the vale fnction. Proposition 5. There exists a niqe ā > sch that lim J āβ ā w ŵ (w) = J āβ ā (ŵ ) = v ŵ an lim J āβ ā w ŵ where threshol βā is efine accoring to (42). Frthermore, fnctions J āβ ā both strictly concave, an, lim w ŵ w J āβ ā (w) = lim w ŵ w J āβ ā (w) = 1. (w) = J āβ ā (ŵ ) = v ŵ, (44) (w) an J āβ ā (w) are

23 Athor: 23 Figre 4 (a) Societal s Vale fnctions (b) Principal s Vale fnctions Vale fnctions with µ = 2, µ = 1, µ = 6, µ = 2, c = 1, c = 1.2, r =.9, R = 2. In this case, w = 3, ŵ = 6 an ŵ = 7 an β = 1 > β =.6. Similar to Proposition 2, the following reslt shows that J āβ ā (w) an J āβ ā (w) specifie in Proposition 5 are inee the principal s total isconte tility ner contract Γ βā(w), as state in the next reslt. Proposition 6. For any state θ {, } an contination tility w [, w θ ], we have U(Γ βā, ν, θ) = J āβ ā θ (w). That is, vales J āβ ā (w) an J āβ ā (w) are eqal to the principal s total isconte tilities of following contract Γ βā state of the machine is an, respectively. Figres 4 an 5 epict the principal s vale fnctions J āβ ā from the initial contination tility w when the initial (w) an J āβ ā (w), as well as societal vale fnctions V āβ ā θ (w) = J āβ ā θ (w) + w, similar to Figre 2 of Section 3. In particlar, Figre 4 epicts a case where the threshol βā = β, while Figre 5 epicts a case with βā > β with the se of smooth pasting. Now we are reay to show that the contract Γ βā is inee optimal. Similar to Proposition 3 an Theorem 4, we have the following main reslt of this section. Theorem 3. For any incentive compatible contract Γ an initial state θ {, }, we have J āβ ā θ Γ, ν, θ U(Γ, ν, θ), in which we exten the fnction J āβ ā θ (w) = J āβ ā θ ( w θ ) (w w θ ) for w > w θ. Therefore, enoting w θ to represent a maximizer of fnction J āβ ā θ, we have U Γ βā(wθ), θ U(Γ, ν, θ) for any incentive compatible contract Γ an state θ. That is, the optimal incentive compatible contract is Γ βā(w θ), if β > β an the machine starts from state θ {, }.

24 24 Athor: Figre 5 (a) Societal s Vale fnctions (b) Principal s Vale fnctions Vale fnctions with µ = 14, µ = 4, µ = 6, µ = 5, c = 4, c = 3, r =.9, R = In this case, ŵ = 3.33, ŵ = 4.33 an β = 1 > β =.6. ā =.668 an βā = 1.52, w =.4 Frthermore, as we can see from Figre 5, where the threshol βā > β, the fnction J āβ ā (w) is monotonically ecreasing, or, the maximizer w =. That is, if the initial state of the machine is, it is better for the principal not to hire the agent than to motivate the agent s fll effort. This is generally tre, as confirme in the following reslt. Proposition 7. If βā > β, then we have the slope ā <. In other wors, if it is optimal to hire the agent at the initial state, then the threshol βā in contract Γ βā mst be eqal to β. On the other han, in Figre 5, we have w >. Therefore, even if βā > β an it is better not to hire the agent for the initial state, it may still be optimal to hire the agent if the initial state is. We have shown that if R is sfficiently large ((17) or eqivalently (31) is satisfie), then Γ ˆβ(w) is the optimal incentive contract if β < β. Similar to Theorem 2 in Section 3.3, the following reslt inicates that, it is also better for the principal to not hire the agent than motivating effort if R is not large enogh. Theorem 4. If R r + µ + µ β, (45) we have v θ U(Γ, ν, θ) for any incentive compatible contract Γ an state θ {, }. Similar to Section 3, when β β, we also characterize moel parameters into three regions by revene R. When R is sfficiently large, the optimal contract follows Γ ˆβ(w), where the incentive

25 Athor: 25 compatible constraints in eqation (7) may not be always bining. When R is small enogh, on the other han, no incentive compatible contract otperforms not hiring the agent at all for the principal. Finally, when R takes moerate vales between the high an low vale cases mentione above, we sty the optimal contracts in e-companion. Finally, if we o not allow contract termination, the following Proposition shows that the simple contract ˆΓ introce in Section 4.1 is optimal. Proposition 8. We have U(ˆΓ, ν, θ) U(Γ, ν, θ) for any state θ {, } an incentive compatible contracts Γ sch that τ =. 5. Incentive compatibility So far, we have focse on analyzing optimal contracts that ince fll effort from the agent before termination. Reslts in the last two sections inicate that for a set of given moel parameters, it is fairly easy to obtain optimal incentive compatible contracts an the corresponing vale fnctions J an J. In this section, we first provie sfficient conitions base on compte J an J, which can be se to verify if the optimal incentive compatible contracts that obtain fll effort from the agent are, in fact, optimal, even if we allow shirking. When the conition is not satisfie, it may be preferable for the principal to hire the agent jst to maintain or jst to repair an we analyze these contracts in the e-companion. Following the optimality conition presente in Lemma 5 in the Appenix, we obtain the following sfficient conition for optimality of maintaining incentive compatibility. Proposition 9. It is optimal to always ince effort from the agent before contract termination if fnction J (w) an J (w) efine by (2)-(23) in Section 3.3; or (35)-(4) in Section 4.3, satisfy the following two conitions, an ϕ (w) := rj (w) + µ J (w) rwj (w) µ max h w {hj (w) + J (w h)}, for w >, (46) ϕ (w) := rj (w) + µ J (w) R rwj (w) µ max h w {hj (w) + J (w h)}, for w β. (47) It is worth noting that Proposition 9 is a parallel reslt to conition (54) in Biais et al. (21), Proposition 8 in DeMarzo an Sannikov (26) an Proposition 6 in Varas (217). However, or conitions are more complex than the corresponing conitions in the literatre, involving solving a single imensional maximization problem in both (46) an (47). This complexity is e to the key ifference between or paper an the aforementione continos time ynamic contracting papers: in all the other papers, the agent is only responsible for one task whereas in ors, the agent

26 26 Athor: is responsible for two tasks. This inces complexity becase the principal s vale fnction will frther epen on the machine s states an. Specifically, imagine, for the moment, that we replace the term J (w h) in (46) to J (w h), so that there wol be only one state. (the own state) It is easy to verify that in this case, concavity of the vale fnction J (w h) implies that the optimal h in this maximization problem shol be. (The intitive interpretation is that there is no change in the agent s contination tility associate with arrivals ring the perio when the agent is allowe to shirk.) Conseqently, the expression ϕ (w) wol be greatly simplifie to be a monotone fnction, which yiels a sfficient conition only involving evalating the vale fnction at its bonaries. In or case, however, concavity of fnctions J (w) an J (w) o not garantee that the optimal h takes vale. (That is, in general contracts allowing shirking, the agent s contination tility still nees to incle jmps as the machine changes states when the agent shirks.) This exactly explains the reason why or verification conitions are more complex than those in the aforementione literatre, an highlights the istinct featre of or set-p with two machine states. Fortnately, the principal s vale fnctions J (w) an J (w) efine in the previos sections are, in fact, qite easy to compte. Therefore, conitions (46) an (47) can be easily verifie nmerically for any moel parameter settings. From Sections 3 an 4, we learn that it is better not hiring the agent if R is too low. The following reslt inicates that in this region, sfficient conitions (46) an (47) are garantee to hol. Corollary 1. If β β an conition (3) hols, or, if β < β an conition (45) hols, then conitions (46) an (47) hol. Corollary 1 implies that if R is low enogh, then not hiring the agent is not only the optimal incentive compatible contract, bt also the best strategy among all contracts. In this case the principal s vale fnction is a linear fnction with slope 1, which allows s to easily verify conitions (46) an (47). Note that when revene R is in the mile range, we show in the e-companion that the optimal incentive compatible contract ictates the principal to hire the agent only if the machine starts in the favore state, an to terminate the agent as soon as the state changes. If we allow shirking instea, the principal may benefit from hiring the agent to exert effort when the machine is in the favore state, while allowing the agent to shirk when the machine is in the other state an wait for the favore state to come back. In the e-companion of this paper, we provie the optimal contracts when the agent is hire to exert effort only when the machine is p (resp. own), an call them one-sie contracts. When sfficient conitions (46) an (47) o not hol, it is possible that one-sie contracts improve pon

27 Athor: 27 Figre 6 Moel parameters: µ = 1, µ = 2, µ = 4, µ = 2, r = 1, R = 35. optimal incentive compatible contracts. In Figre 6, we provie sch a comparison for ifferent moel parameter settings. In Figre 6, we vary effort costs c an c, while keeping all other parameters the same. For each choice of moel parameters, we test if sfficient conitions (46) an (47) hol. The white region in the figre epicts moel parameters sch that these sfficient conitions hol. In particlar, there is a white sqare region on the pper right corner, when both effort costs are high. The moel parameters in this region correspon to the sitation that the revene rate R is in the lowest interval, which is captre in Corollary 1. The grey an black regions captre moel parameters ner which at least one of the sfficient conitions (46) an (47) oes not hol, i.e., incentive compatible contracts may not be optimal if shirking is allowe. In particlar, the grey area in the figre represent parameters ner which optimal one-sie contracts otperform incentive compatible contracts. One explanation is that one of the costs is relatively high in the grey area where it is reasonable to ince one-sie effort. Black is the only remaining area in which onesie contracts o not improve pon optimal incentive compatible contracts. The black area is relatively small an in this region shirking may not improve pon incentive compatible contracts, since (46) an (47) are only sfficient conitions. Frthermore, the non-monotone bonaries of white regions in Figre 6 emonstrates that the verification fnctions ϕ (w) an ϕ (w) are not globally monotone in c or c. Actally, nmerical tests inicate that fnctions ϕ (w) an ϕ (w) are not monotone in any of the moel parameters. Optimal ynamic contracts that allow shirking when the sfficient conition in Proposition 9 is violate are generally har to obtain, an may involve complex strctres that reners them