A Class of Explicit optimal contracts in the face of shutdown
AA Class of Explicit optimal contracts in the face of shutdown
Jessica Martin ∗1 and Stéphane Villeneuve †21 INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135 Avenue deRangueil 31077 Toulouse Cedex 4 France Université Toulouse 1 Capitole, (TSE-TSMR), 2 Rue du Doyen Gabriel Marty,31000 ToulouseFebruary 2, 2021
Abstract
What type of delegation contract should be offered when facing a risk of the magnitude ofthe pandemic we are currently experiencing and how does the likelihood of an exogenous earlytermination of the relationship modify the terms of a full-commitment contract? We studythese questions by considering a dynamic principal-agent model that naturally extends theclassical Holmström-Milgrom setting to include a risk of default whose origin is independent ofthe inherent agency problem. We obtain an explicit characterization of the optimal wage alongwith the optimal action provided by the agent. The optimal contract is linear by offering botha fixed share of the output which is similar to the standard shutdown-free Holmström-Milgrommodel and a linear prevention mechanism that is proportional to the random lifetime of thecontract. We then tweak the model to add a possibility for risk mitigation through investmentand study its optimality.
Keywords : Principal-Agent problems, default risk, Hamilton-Jacobi Bellman equations.
Without seeking to oppose public health and economic growth, there is no doubt that the manage-ment of the Covid crisis had serious consequences on entire sectors of the economy. The first fewmonths of 2020 will go down in world history as a period of time characterized by massive layoffs,forced closures of non-essential companies, disruption of cross-border transportation whilst popu-lations were subject to lockdown and/or social distancing measures and hospitals and the medicalworld struggled to get a grasp on the Sars-Cov-2 pandemic. Whilst the immediate priority wassaving lives, decongesting hospitals and preventing the spread of the disease, many extraordinary ∗ [email protected] † [email protected] authors acknowledge funding from ANR PACMAN and from ANR under grant ANR-17-EUR-0010 (Investisse-ments d’Avenir program). The second author gratefully thanks the FdR-SCOR “Chaire Marché des risques et créationde valeurs". a r X i v : . [ m a t h . O C ] J a n conomic support measures were taken to help businesses and individuals stay afloat during theseunprecedented times and in the hope of tempering the economic crisis that would follow. Althoughthe world has lived through many crises over the past centuries, from several Panics in the 1800s andthe Great Depression of the 1930s to the more recent Financial Crisis of 2008, never before has theglobal economy as a whole come to such a standstill due to an external event. Such large shutdownrisks do not only materialize during pandemics but throughout many major other large events. Themassive bushfires that affected Australia towards the end of 2019, temporarily halting agriculture,construction activity and tourism in some areas of the country are another recent example. As webegin to see a glimpse of hope for a way out through vaccination, the focus is turning to buildingthe world of tomorrow with the idea that we must learn to live with such risks. This paper triesto make its contribution by focusing on a simple microeconomic issue. In a world subject to moralhazard, how can we agree to an incentive contract whose obligations could be made impossible orat least very difficult because of the occurrence of a risk of the nature of the Covid19 pandemic?Including such a shutdown risk-sharing in contracts seems crucial going forward for at least tworeasons. First, it is not certain that public authority will be able to continue to take significanteconomic support measures to insure the partners of a contract if the frequency of such global riskswere to increase. On the other hand, the private insurance market does not offer protection againstthe risk of a pandemic which makes pooling too difficult. It therefore seems likely that we will haveto turn to an organized form of risk sharing between the contractors.Economic theory has a well-developed set of tools to analyze incentive and risk-sharing problemsusing expected-utility theory. Most of the now abundant literature related to dynamic contractingthrough a Principal-Agent model has, so far, mostly been based on continuously governed (eg.Brownian motion) output-processes. This was the case of the foundational work of Holmström andMilgrom [12] and many of its many extensions such as those of Schattler and Sung in [20] and [21],and the more recent contributions of Sannikov in [19], and Cvitanic et al [5] and [6]. However somerelatively recent works have introduced jump processes into continuous time contracting. Biaiset al. were the first to do so in [3] where they study optimal contracting between an insurancecompany and a manager whose effort can reduce the occurence of an underlying accident. In asimilar vein, work by Capponi and Frei [4] also used a jump-diffusion driven outcome process inorder to include the possibility for accidents to negatively affect revenue. Here, we extend theclassical Holmström and Milgrom [12] framework to include a shutdown risk. We do not claimthat this model with CARA preferences is general enough to come up with robust economic facts,but it has the remarkable advantage of being explicitly creditworthy, which allows us to find anexplicit optimal contract that disentangles the incentives from external risk-sharing and allows usto understand the sensitivity of the optimal contract to the different exogenous parameters of themodel. Our work uses a jump-diffusion process too and presents some structural similarities with[4] : both models consider a risk averse principal and agent with exponential utility and reach anexplicit characterization of the optimal wage. However, Capponi and Frei combine the continuouspart of the diffusion and the accident jump process additively and they are able to allow preventionthrough intensity control of the jump process. This makes sense as many accidents are preventablethrough known measures. Our framework uses a different form of jump diffusion to enable the shut-down event to completely stop revenue generation in a continuous-time setting. This is done bybuilding on a standard continuous Brownian-motion based output process. The main novelty is themultiplicative effect of the jump risk : upon the arrival of the risk, the whole of the output processcomes to a halt. As an extension, we allow the halt to no longer be a complete fatality : production2ay continue at a degraded level through an investment by the principal. From a methodolog-ical viewpoint, our reasoning uses a now standard method in dynamic contracting based on [19]and [6] which consists in transforming both the first-best and second-best problems into classicalMarkovian control problems. The solution to these control problems can be characterized througha Hamilton-Jacobi-Bellman equation. Quite remarkably, this equation has, in our context, an ex-plicit solution that is closely linked to a so-called Bernouilli ODE which facilitates many extensions.To the best of our knowledge, this paper is the first to explicitly introduce a default in a dynamicPrincipal-agent framework, in both a first-best (also called full Risk-Sharing) and second-best (alsocalled Moral Hazard) setting. A key feature of our study is that the shape of the optimal contract islinear. More precisely, the agent’s compensation is the sum of two functions: the first is linear withrespect to the output and serves to give the incentives, while the second is linear with respect to theeffective duration of the contract and serves to share the default risk. While the linear incentive partof the contract is in line with the existing literature on continuous-time Principal-Agent problemswithout default under exponential utilities, the risk-sharing part deserves some clarification. Thecontract exposes both agents to a risk of exogenous interruption but it has two different regimesthat are determined by an explicit relation between the risk-aversions and the agent’s effort cost.Under the first regime, the agent is more sensitive to the risk of default than the principal. Inthis case, the principal deposits on the date a positive amount onto an escrow account whosebalance will then decrease over time at a constant rate. It is crucial to observe that the later thedefault arrives, the more the amount in the escrow account decreases to a point where it may evenbecome negative. If the default occurs, the principal transfers the remaining balance to the agent.Under the second regime, the principal is more sensitive to the risk of default. In this case, theprincipal deposits a negative amount into the escrow account, which now grows at a constant rateand symmetrical reasoning applies. This linearity contrasts with the optimum obtained in [4] asthe additive contribution of their jump process to revenue generation leads to a sub-linear wage.This result is coherent with the paper by Hoffman and Pfeil [11] which proves that, in line with theempirical studies by Bertrand and Mullainathan [2], the agent must be rewarded or punished for arisk that is beyond his control.Finally, this paper also explicitly characterizes the optimal contract when a possibility for shutdownrisk mitigation exists at a cost. Such a possibility is coherent with agency-free external risk: preven-tion is not possible, at least on a short-term or medium-term time scale. At best the principal maybe able to invest to mitigate its effects. Crucially we find that in many circumstances, investing isnot optimal for the principal. When it is, it is only optimal up until some cutoff time related to abalance between the cost of investment, the agent’s rents and possible remaining gain.The rest of the document is structured as follows. In Section 2, we present the model and thePrincipal-Agent problems that we consider. In Section 3, we analyse the first-best case where theprincipal observes the agent’s effort. Then in Section 4, we give our main results and analysis. InSection 5, we extend our model to include a possibility for mitigation upon a halt. The model is inherited from the classical work of Holmström and Milgrom [12]. A principal contractswith an agent to manage a project she owns. The agent influences the project’s profitability by3xerting an unobservable effort. For a fixed effort policy, the output process is still random and theidiosyncratic uncertainty is modeled by a Brownian motion.We assume that the contract matures at time
T > and both principal and agent are risk-aversewith CARA preferences. The departure from the classical model is as follows: we assume the projectis facing some external risk that could partially or totally interrupt the production at some randomtime τ . The probability distribution of τ is assumed to be independent of the Brownian motion thatdrives the uncertainty of the output process and also independent of the agent’s actions. Finally,we assume that the contract offers a transfer W at time T from the principal to the agent that isa functional of the output process. Let
T > be some fixed time horizon. The key to modeling our Principal-Agent problems underan agency-free external risk of default is the simultaneous presence over the interval [0 , T ] of acontinuous random process and a jump process as well as the ability to extend the standard math-ematical techniques used for dynamic contracting to this mixed setting. Thus, we shall deal withtwo kinds of information : the information from the output process, denoted as F = ( F t ) t ≥ andthe information from the default time, i.e. the knowledge of the time where the default occurredin the past, if the default has appeared. This construction is not new and occurs frequently inmathematical finance .The complete probability space that we consider will be denoted as (Ω , G , P ) , with two independentstochastic processes :• B a standard one-dimensional F -Brownian motion,• N the right-continuous single-jump process defined as N t = τ ≤ t , t in [0 , T ] where τ is somepositive random variable independent of B that models the default time. N will also be referred to as the default indicator process. We therefore use the standard approachof progressive enlargement of filtration by considering G = {G t , t ≥ } the smallest complete right-continuous extension of F that makes τ a G -stopping time. Because τ is independent of B , B is a G -Brownian motion under P according to Proposition 1.21 p 11 in [1]. We also suppose that thereexists a bounded deterministic compensator of N , Λ t = (cid:82) t λ ( s ) ds for some bounded function λ ( . ) called the intensity implying that: M t = N t − (cid:90) t λ ( s )(1 − N s ) ds, t ∈ [0 , T ] is a G -compensated martingale. Note that through knowledge of the function λ, the principal andagent can compute at time 0 the probability of default happening over the contracting period [0 , T ] .Indeed : P ( τ ≤ T ) = 1 − exp( − Λ T ) . We first suppose for computational ease that the intensity λ is a constant. We will see in Section4.3 that our results may easily be lifted to more general deterministic compensators. We refer the curious reader to the two important references [1] and [9]. emark 2.1. Here we will suppose that the compensator of N is common knowledge to both thePrincipal and the Agent. We could imagine settings where the Principal and Agent’s beliefs regardingthe risk of default may differ : this natural extension of our work would call for analysis of thedynamic contracting problem under hidden information which is left for future research. We suppose that the agent agrees to work for the principal over a time period [0 , T ] and provide upto the default time a costly action ( a t ) t ∈ [0 ,T ] belonging to A , where A denotes the set of admissible F -predictable strategies that will be specified later on. The Principal-Agent problem models therealistic setting where the principal cannot observe the agent’s effort. As such the agent chooseshis action in order to maximize his own utility. The principal must offer a wage based on theinformation driven by the output process up to the default time that incentivizes the agent to workefficiently and contribute positively to the output process. Mathematically, the unobservability ofthe agent’s behaviour is modeled through a change of measure. Under P , we assume that theproject’s profitability evolves as X t := x + (cid:90) t (1 − N s ) dB s . Thus, P corresponds to the probability distribution of the profitability when the agent makes noeffort over [0 , T ] . When the agent makes an effort a = ( a t ) t , we shall assume that the project’sprofitability evolves as X t := x + (cid:90) t a s (1 − N s ) ds + (cid:90) t (1 − N s ) dB as , where B a is a F -Brownian motion under a measure P a . The agent fully observes the decompositionof the production process under a measure P a whilst the principal only observes the realization of X t . In order for the model to be consistent, the probabilities P and P a must be equivalent for all ( a t ) t ∈ [0 ,T ∧ τ ] belonging to A . Therefore, we introduce the following set of actions B = (cid:110) a = ( a t ) t : F -predictable and taking values in [ − A, A ] for some A > (cid:111) . The action process in B are uniformly bounded by some fixed constant A > that will be assumedas large as necessary. For a ∈ B , we define P a as d P a d P |G T = exp (cid:32)(cid:90) T a s (1 − N s ) dB s − (cid:90) T | a s | (1 − N s ) ds (cid:33) := L T . Because E ( L T ) = 1 , ( B at ) t ∈ [0 ,T ] with B at = B t − (cid:82) t a s (1 − N s ) ds, t ∈ [0 , T ] is a G -Brownian motionunder P a according to Proposition 3.6 c) p 55 in [1]. It is key to note that if halt occurs, i.e. if τ ≤ T , then the production process is halted before T meaning that : X at ∧ τ = X at , t ∈ [0 , T ] . Letus then observe that an action a = ( a t ) t of B can be extended to a G -predictable process (˜ a t ) t ∈ [0 ,T ] by setting ˜ a t = a t t ≤ τ .The cost of effort for the agent is modeled through a quadratic cost function : κ ( a ) := κ a , for κ > some fixed parameter. As a reward for the agent’s effort, the principal pays him a wage W
5t time T . W is assumed to be a G T ∧ τ random variable which means that the payment at time T in case of an early default is known at time τ . The principal and the agent are considered to berisk averse and risk aversion is modeled through two CARA utility functions : U P ( x ) := − exp( − γ P x ) and U A ( x ) := − exp( − γ A x ) , where γ P > and γ A > are two fixed constants modeling the principal’s and the agent’s riskaversion.In this setting and for any given wage W , the agent maximizes his own utility and solves : V A ( W ) = sup a ∈B E a (cid:34) U A (cid:32) W − (cid:90) T κ ( a s (1 − N s )) ds (cid:33)(cid:35) . (2.1)A wage W is said to be incentive compatible if there exists an action policy a ∗ ( W ) ∈ B thatmaximises (2.1) and thus satisfies V A ( W ) = E a ∗ ( W ) (cid:34) U A (cid:32) W − (cid:90) T κ ( a ∗ s ( W )(1 − N s )) ds (cid:33)(cid:35) . When the principal is able to offer an incentive compatible wage W , she knows what the agent’s bestreply will be. As such the principal establishes a set A ∗ ( W ) ⊂ B of best replies for the agent for anyincentive compatible W . Therefore, the first task is to characterize the set of incentive-compatiblewages W IC . Only then may the principal consider maximizing his own utility by solving : sup W ∈W IC sup a ∗ ∈A ∗ ( W ) E a ∗ ( W ) (cid:104) U P (cid:16) X a ∗ ( W ) T − W (cid:17)(cid:105) (2.2)under the participation constraint E a ∗ ( W ) (cid:34) U A (cid:32) W − (cid:90) T κ ( a ∗ s ( W )(1 − N s )) ds (cid:33)(cid:35) ≥ U A ( y P C ) , (2.3)where y P C is a monetary reservation utility for the agent.
Remark 2.2.
Problem (2.2) has been thoroughly analyzed in a setting where the output processmay not default (see the pioneer papers [12], [20] ). Setting κ = 1 for simplicity, the optimal actionis constant and given by : a ∗ = γ P + 1 γ P + γ A + 1 , and the optimal wage is linear in the output: W = y P C + a ∗ X T + (cid:18) γ A −
12 ( a ∗ ) (cid:19) T. We may naturally expect to encounter an extension of these results in this setting. Optimal First-best contracting
We begin with analysis of the first-best benchmark (the full Risk-Sharing problem) which leads to asimple optimal sharing rule. Of course this problem is not the most realistic when it comes to mod-eling dynamic contracting situations. However it provides a benchmark to which we can comparethe more realistic Moral Hazard situation. Indeed, the principal’s utility in the full Risk-Sharingproblem is the best that the principal will ever be able to obtain in a contracting situation as hemay observe (and it is thus assumed that he may dictate) the agent’s action.To write the first-best problem, we assume that both the principal and the agent observe thevariations of the same production process ( X at ) t ∈ [0 ,T ] under P : X at := x + (cid:90) t a s (1 − N s ) ds + (cid:90) t (1 − N s ) dB s . t ∈ [0 , T ] (3.1)The agent is guaranteed a minimum value of expected utility through the participation constraint : E (cid:34) U A (cid:32) W − (cid:90) T κ ( a s (1 − N s )) ds (cid:33)(cid:35) ≥ U A ( y P C ) , (3.2)but has no further say on the wage or action. Consider the admissible set : A P C := { ( W, a ) such that W is G T ∧ τ measurable with E [exp( − γ A W )] < + ∞ , ( a t ) t ∈ B , and (3 . is satisfied } . The full Risk-Sharing problem involves maximizing the principal’s utility across A P C : sup ( W,a ) ∈A PC E [ U P ( X aT − W )] . (3.3) A first step to optimal contracting in this first-best setting involves answering the following question:can we characterize the set A P C ? Following the standard route, we will first establish a necessarycondition. For a given pair ( W, a ) ∈ A P C , let us introduce the agent’s continuation utility ( U ( W,a ) t ) t as follows: U ( W,a ) t := E t (cid:34) U A (cid:32) W − (cid:90) Tt κ ( a s (1 − N s )) ds (cid:33)(cid:35) , where we use the shorthand notation : E t [ . ] := E [ . |G t ] . We may write the Agent’s continuation valueprocess as the product : U ( W,a ) t = M ( W,a ) t D ( W,a ) t , where : M ( W,a ) t := E t (cid:34) U A (cid:32) W − (cid:90) T κ ( a s (1 − N s )) ds (cid:33)(cid:35) and D ( W,a ) t := exp (cid:18) − γ A (cid:90) t κ ( a s (1 − N s )) ds (cid:19) . Observe that for any admissible pair ( W, a ) ∈ A P C , the process M = ( M ( W,a ) t ) t is a G -squareintegrable martingale. According to the Martingale Representation Theorem for G -martingales7see [1], Theorem 3.12 p. 60), there exists some predictable pair ( z s , l s ) in H × H , where H isthe set of F -predictable processes Z with E (cid:104)(cid:82) T | Z t | dt (cid:105) < + ∞ , such that : M ( W,a ) t := M ( W,a )0 + (cid:90) t z s (1 − N s ) dB s + (cid:90) t l s (1 − N s ) dM s . Integration by parts yields the dynamic of U , noting that D has finite variation : dU ( W,a ) t = − γ A κ ( a t (1 − N s )) U ( W,a ) t dt + D ( W,a ) t z t (1 − N s ) dB t + D ( W,a ) t l t (1 − N s ) dM t . Setting Z ( W,a ) t := D ( W,a ) t z t ∈ H and K ( W,a ) t := D ( W,a ) t l t ∈ H , we obtain: dU ( W,a ) t = − γ A κ ( a t (1 − N s )) U ( W,a ) t dt + Z ( W,a ) t (1 − N s ) dB t + K ( W,a ) t (1 − N s ) dM t . By construction, we have that U ( W,a ) T = U A ( W ) . It follows that (cid:16) U ( W,a ) t , Z ( W,a ) t , K ( W,a ) t (cid:17) is asolution to the BSDE: − dU ( W,a ) t = − Z ( W,a ) t (1 − N s ) dB t − K ( W,a ) t (1 − N s ) dM t + γ A κ ( a t (1 − N s )) U ( W,a ) t dt, (3.4)with U ( W,a ) T = U A ( W ) . Therefore, (3.2) is satisfied if and only if U ( W,a )0 ≥ U A ( y P C ) . Remark 3.1.
Let S be the set of G − adapted RCLL processes U such that E [ sup ≤ t ≤ T | U t | ] < + ∞ . Through Proposition 2.6 of [7], the solution to (3.4) is unique in ( S × H × H ) . Indeed, the driver g ( ω, U ) = γ A κ ( a t (1 − N t )) U is uniformly Lipschitz in U because ( a t ) t is bounded and the terminalcondition is in L . To sum up, we have the following necessary condition for admissibility.
Lemma 3.1. If ( W, a ) ∈ A P C then there exists a unique solution (cid:16) U ( W,a ) t , Z ( W,a ) t , K ( W,a ) t (cid:17) in ( S × H × H ) to the BSDE (3.4) such that U ( W,a )0 ≥ U A ( y P C ) . To obtain a sufficient condition, we introduce, for π = ( y , a, β, H ) ∈ R × B × H × H , the wageprocess ( W πt ) t defined as W πt := y + (cid:90) t β s (1 − N s ) dB s + (cid:90) t H s (1 − N s ) dM s + (cid:90) t (cid:110) γ A β s (1 − N s ) + κ ( a s (1 − N s )) λγ A [exp( − γ A H s ) − γ A H s ](1 − N s ) (cid:27) ds, (3.5)and consider the set Γ := (cid:8) ( y , a, β, H ) ∈ R × B × H × H such that y ≥ y P C and E [exp( − γ A W πT )] < + ∞ . (cid:9) . We have the following result. 8 emma 3.2.
For any π ∈ Γ , the pair ( W πT , a ) belongs to A P C .Proof.
We apply Itô’s formula to the process Y πt = U A ( W πt ) to obtain dY πt = − γ A Y πt β t (1 − N t ) dB t + Y πt (cid:0) e − γ A H t − (cid:1) (1 − N t ) dM t − γ A κ ( a t (1 − N t )) Y πt dt. Moreover, because π ∈ Γ , Y πT = U A ( W πT ) is square-integrable. Remark 3.1 yields the triplet (cid:0) Y πt , − γ A Y πt β t , Y πt ( e − γ A H t − (cid:1) is the unique solution in ( S × H × H ) to BSDE (3.4) withterminal condition U A ( W πT ) when π ∈ Γ . Therefore, Y π = U A ( y ) = E (cid:34) U A (cid:32) W πT − (cid:90) T κ ( a s (1 − N s )) ds (cid:33)(cid:35) ≥ U A ( y P C ) , and thus (3.2) is satisfied. Remark 3.2.
The admissible contracts are essentially the terminal values of the controlled processes (3.5) for π ∈ Γ . The difficulty is that we do not know how to characterize the β and H processesthat guarantee that π belongs to Γ . Nevertheless, it is easy to check by a standard application of theGronwall lemma that if β and H are bounded then π ∈ Γ . This last observation will prove to becrucial in the explicit resolution of our problem. Using Lemma 3.2, the full Risk-Sharing problem under default writes as the Markovian controlproblem : V F BP := sup π =( y ,a,Z,K ) ∈ Γ E (cid:104) U P (cid:16) X ( x ,a ) T − W πT (cid:17)(cid:105) , (3.6)where X ( x ,a ) t is given by : dX ( x ,a ) s = a s (1 − N s ) ds + (1 − N s ) dB s , with X ( x ,a )0 = x and the wage process is given by : dW πs = Z s (1 − N s ) dB s + K s (1 − N s ) dM s + (cid:26) γ A Z s (1 − N s ) + κ ( a s (1 − N s )) + λγ A [exp( − γ A K s ) − γ A K s ](1 − N s ) (cid:27) ds, with W π = y .We have the following key theorem for the first-best problem. Theorem 3.1.
Let a ∗ t = 1 κ , Z ∗ t = γ P γ P + γ A , and let : K ∗ t = 1 γ P + γ A log(Φ ( t )) , where : Φ ( t ) := (cid:18) c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c (cid:19) γP + γAγA , (3.7)9 ith : c := γ P γ A γ P + γ A ) − γ P κ − λ γ P + γ A γ A and c := λ γ P + γ A γ A . Then π ∗ = ( y P C , a ∗ , Z ∗ , K ∗ ) ∈ Γ parameterizes the optimal contract ( W π ∗ T , a ∗ ) for the first-bestproblem. The rest of this subsection is dedicated to the proof of this Theorem. We first make the followingobservation. As X remains constant after τ , the principal has no further decision to make after thedefault time. Thus, its value function is constant and equal to U P ( x − y ) on the interval [ τ, T ] .We now focus on the control part of the problem (i.e. computation of the optimal control triplet ˜ π = ( a, Z, K ) for a given pair ( x , y ) ). To do so, we follow the dynamic programming approachdeveloped in [17], Section 4 to define the value function V (0 , x , y ) = sup ˜ π ∈ ˜Γ E (cid:34) U P ( X aT − W ˜ πT )(1 − N T ) + (cid:90) T U P ( X at − W ˜ πt ) λe − λt dt (cid:35) , (3.8)where ˜Γ = (cid:8) ˜ π ∈ B × H × H (cid:9) , Because Γ ⊂ R × ˜Γ , we have V F BP ≤ sup y ≥ y PC V (0 , x , y ) . According to stochastic control theory, the Hamilton-Jacobi-Bellman equation associated to thestochastic control problem (3.8) is the following (see [16]): ∂ t v ( t, x, y ) + sup a,Z,K (cid:26) ∂ x v ( t, x, y ) a + ∂ y v ( t, x, y ) (cid:20) γ A Z + κ ( a ) + λγ A [exp( − γ A K ) − (cid:21) + λ [ U P ( x − y − K ) − v ( t, x, y )] + ∂ yy v ( t, x, y ) Z ∂ xx v ( t, x, y ) + ∂ xy v ( t, x, y ) Z (cid:27) = 0 , (3.9)with the boundary condition : v ( T, x, y ) = U P ( x − y ) . It happens that the HJB equation (3.9) is explicitly solvable by exploiting the separability propertyof the exponential utility function.
Lemma 3.3.
The function v ( t, x, y ) = U P ( x − y )Φ ( t ) with : Φ ( t ) = (cid:18) c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c (cid:19) γP + γAγA , where : c = γ P γ A γ P + γ A ) − γ P κ − λ γ P + γ A γ A and c = λ γ P + γ A γ A , solves (in the classical sense) the HJB partial differential equation (3.9).Furthermore a ∗ t = 1 κ , Z ∗ t = γ P γ P + γ A and K ∗ t = 1 γ P + γ A log(Φ ( t )) are the optimal controls. roof. We search for a solution to Equation (3.9) for a v of the form : v ( t, x, y ) = U P ( x − y )Φ ( t ) , with Φ a positive mapping. Such a v satisfies (3.9) if and only if Φ ( t ) solves the PDE : Φ (cid:48) ( t ) + inf a,Z,K (cid:26) − γ P Φ ( t ) a + γ P Φ ( t ) (cid:18) γ A Z + κ ( a ) + λγ A { exp( − γ A K ) − } (cid:19) + γ P Φ ( t ) Z γ P ( t ) − γ P Φ ( t ) Z + λ (exp( γ P K ) − Φ ( t )) (cid:27) = 0 , with the boundary condition Φ ( T ) = 1 . As Φ is a positive mapping, the infimum is well defined.We derive the following first order conditions that must be satisfied by the optimal controls : γ P Φ ( t ) = γ P κa Φ ( t ) γ P Φ ( t ) Z ( γ A + γ P ) = γ P Φ ( t ) γ P Φ ( t ) λ exp( − γ A K ) = γ P λ exp( γ P K ) , equating to : a ∗ = 1 κ , Z ∗ = γ P γ P + γ A , K ∗ = log(Φ ( t )) γ P + γ A . It follows that : inf a,Z,K (cid:26) − γ P Φ ( t ) a + γ P Φ ( t ) (cid:18) γ A Z + κ ( a ) + λγ A { exp( − γ A K ) − } (cid:19) + γ P Φ ( t ) Z γ P ( t ) − γ P Φ ( t ) Z + λ (exp( γ P K ) − Φ ( t )) (cid:27) = − γ P Φ ( t ) a ∗ + γ P Φ ( t ) (cid:18) γ A Z ∗ + κ ( a ∗ ) + λγ A { exp( − γ A K ∗ ) − } (cid:19) + γ P Φ ( t ) Z ∗ γ P ( t ) − γ P Φ ( t ) Z ∗ + λ (exp( γ P K ∗ ) − Φ ( t ))= Φ ( t ) γ P γ A γ P + γ A ) (cid:124) (cid:123)(cid:122) (cid:125) terms with Z ∗ − Φ ( t ) γ P κ (cid:124) (cid:123)(cid:122) (cid:125) terms with a ∗ − λ γ P + γ A γ A Φ ( t ) + λ γ P + γ A γ A Φ ( t ) γPγP + γA . (cid:124) (cid:123)(cid:122) (cid:125) terms with K ∗ We may inject this expression back into the PDE on Φ . Doing so yields the following Bernoulliequation : Φ (cid:48) ( t ) + c Φ ( t ) + c Φ ( t ) γPγP + γA = 0 , Φ ( T ) = 1 , where c = γ P γ A γ P + γ A ) − γ P κ − λ γ P + γ A γ A and c = λ γ P + γ A γ A . The unique solution to this equation is (see for instance [22]) : Φ ( t ) = (cid:18) c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c (cid:19) γP + γAγA , and the result follows. 11 roof of Theorem 3.1 . The value function v ( t, x, y ) = U P ( x − y )Φ ( t ) is a classical solutionto the HJB equation (3.9). A standard verification theorem yields that v = V . Through Lemma3.3, the optimal controls for the full Risk-Sharing problem are : a ∗ t = 1 κ , Z ∗ t = γ P γ P + γ A and K ∗ t = 1 γ P + γ A log(Φ ( t )) , with Φ as defined in Lemma 3.3. These controls are free of y and it follows that : V (0 , x , y ) = E (cid:104) U P (cid:16) X ( x ,a ∗ ) T − W ( y ,a ∗ ,Z ∗ ,K ∗ ) T (cid:17)(cid:105) , is a decreasing function of y . Thus we obtain sup y ≥ y PC V (0 , x , y ) = E (cid:104) U P (cid:16) X ( x ,a ∗ ) T − W ( y PC ,a ∗ ,Z ∗ ,K ∗ ) T (cid:17)(cid:105) . Finally, we observe that the optimal controls are bounded and thus Remark (3.2) yields π ∗ =( y P C , a ∗ , Z ∗ , K ∗ ) ∈ Γ . As a consequence, sup y ≥ y PC V (0 , x , y ) = E (cid:104) U P (cid:16) X ( x ,a ) T − W ( y PC ,a ∗ ,Z ∗ ,K ∗ ) T (cid:17)(cid:105) ≤ V F BP . Because the reverse inequality holds, the final result follows.
The following is dedicated to our main result for the Moral Hazard problem. We shall state ourmain theorem with the explicit optimal contract before turning to some analysis of the effect of theshutdown on dynamic contracting. In the case of moral hazard, one is forced to make a strongerassumption about the nature of a contract. This stronger hypothesis will naturally appear tojustify the martingale optimality principle. In our setting, a contract is a G T ∧ τ measurable randomvariable W such that for every β ∈ R , we have E [exp( βW )] < + ∞ . A first step to optimal contracting involves answering the preliminary question: can we characterizeincentive compatible wages and if so what is the related optimal action for the agent? The charac-terization of incentive compatible contracts relies on the martingale optimality principle (see [13]and [18]) that we recall below.
Lemma 4.1 (Martingale Optimality Principle) . Given a contract W , consider a family of stochasticprocesses R a ( W ) := ( R at ) t ∈ [0 ,T ] indexed by a in B that satisfies :1. R aT = U A ( W − (cid:82) T κ ( a s (1 − N s )) ds ) for any a in B R a. is a P a -supermartingale for any a in B . R a is independent of a .4. There exists a ∗ in B such that R a ∗ is a P a ∗ -martingale.Then, R a ∗ = E a ∗ (cid:34) U A ( W − (cid:90) T κ ( a ∗ s ) ds ) (cid:35) ≥ E a (cid:34) U A ( W − (cid:90) T κ ( a s (1 − N s )) ds ) (cid:35) , meaning that a ∗ is the optimal agent’s action in response to the contract W . We will construct such a family following the standard route. Consider a given contract W , wedefine the family R a ( W ) := ( R at ) t ∈ [0 ,T ] by R at := − exp (cid:18) − γ A (cid:18) Y t ( W ) − (cid:90) t κ ( a s (1 − N s )) ds (cid:19)(cid:19) , where ( Y ( W ) , Z ( W ) , K ( W )) in ( S × H × H ) is the unique solution of the following BSDE under P Y t ( W ) = W − (cid:90) Tt f ( Z s ( W ) , K s ( W ))(1 − N s ) ds − (cid:90) Tt Z s ( W )(1 − N s ) dB s − (cid:90) Tt K s ( W )(1 − N s ) dM s , (4.1)with f ( z, k ) := 12 γ A z + λk + λγ A ( e − γ A k −
1) + inf a ∈B { κ ( a ) − az } . Remark 4.1.
The theoretical justification of the well-posedness of the BSDE (4.1) deserves somecomments. The first results were obtained in [14] and [8] when the contract W is assumed to bebounded. The necessary extension in our model when W admits an exponential moment has beentreated recently in the paper [15]. By construction, R aT = U A ( W − (cid:82) T κ ( a s (1 − N s )) ds ) for any a in B . Moreover, R a = Y ( W ) isindependent of the agent’s action a . We compute the variations of R a and obtain : = − γ A R as Z s (1 − N s ) dB s + R as ( e − γ A K s − − N s ) dM s + R as γ A (cid:26) γ A Z s − f ( Z s , K s ) + κ ( a s (1 − N s )) + λK s + λγ A ( e − γ A K s − (cid:27) (1 − N s ) ds. = − γ A R as Z s (1 − N s ) dB as + R as ( e − γ A K s − − N s ) dM s + R as γ A (cid:26) γ A Z s − f ( Z s , K s ) + κ ( a s (1 − N s )) + λK s + λγ A ( e − γ A K s − − a s Z s (cid:27) (1 − N s ) ds. Thus R a is a P a -super-martingale for every a in B , the function a ∗ ( z ) = − A z ≤− κA + zκ − κA ≤ z ≤ κA + A z ≥ κA is a unique minimizer for f and R a ∗ is a P a ∗ -martingale. As a consequence, every contract W is incentive compatible which a unique best reply a ∗ ( Z ( W )) . Finally, a contract W satisfies theparticipation constraint if and only if Y ( W ) ≥ y P C .13elying on the idea of Sannikov [19] and its recent theoretical justification by Cvitanic, Possamaiand Touzi [6], we will consider the agent promised wage Y ( W ) as a state variable to embed theprincipal’s problem into the class of Markovian problems, by considering the sensitivities of theagent’s promised wage Z ( W ) and K ( W ) as control variables. For π = ( y , Z, K ) ∈ [ y P C ; + ∞ ) × H × H , we define under P , the control process called the agent continuation value W ( y ,Z,K ) t = y + (cid:90) t Z s (1 − N s ) dB s + (cid:90) t K s (1 − N s ) dM s + (cid:90) t f ( Z s , K s )(1 − N s ) ds. (4.2)Under P ∗ := P ( a ∗ ( Z )) , we thus have W ( y ,Z,K ) t = y + (cid:90) t Z s (1 − N s ) dB ∗ s + (cid:90) t K s (1 − N s ) dM s (4.3) + (cid:90) t (cid:26) γ A Z s + κ ( a ∗ ( Z s )) + λγ A [exp( − γ A K s ) − γ A K s ] (cid:27) (1 − N s ) ds = y + (cid:90) t Z s (1 − N s ) dB ∗ s + (cid:90) t K s (1 − N s ) dN s + (cid:90) t (cid:26) γ A Z s + κ ( a ∗ ( Z s )) + λγ A [exp( − γ A K s ) − (cid:27) (1 − N s ) ds Now, we consider the set ζ = (cid:110) π = ( Z, K ) ∈ H × H such that ∀ β ∈ R , E (cid:104) exp( βW ( y,Z,K ) T ) (cid:105) < + ∞ for y ∈ R (cid:111) . By construction, W ( y,π ) T is a contract that satisfies the participation constraint for every π ∈ ζ and y ≥ y P C . Moreover, by the well-posedness of the BSDE (4.1) , every contract W that satisfiesthe participation constraint can be written W ( Y ( W ) ,Z ( W ) ,K ( W )) T with π ( W ) = ( Z ( W ) , K ( W )) ∈ ζ .Therefore, the problem of the principal can now be rewritten as the following optimisation problem V P := sup y ≥ y pc v (0 , x, y ) , where v (0 , x, y ) = sup π ∈ ζ E ∗ [ U P ( X T ∧ τ − W πT ∧ τ )] (4.4)To characterize the optimal contract, we will proceed analogously as in the full risk sharing caseby constructing a smooth solution to the HJB equation associated to the Markov control problem(4.4) given by ∂ t v ( t, x, y ) + inf Z,K (cid:26) ∂ x v ( t, x, y ) Zκ + ∂ y v ( t, x, y ) (cid:20) γ A Z + κ ( a ∗ ( Z )) + λγ A [exp( − γ A K ) − (cid:21) + λ [ U P ( x − y − K ) − v ( t, x, y )] + ∂ yy v ( t, x, y ) Z ∂ xx v ( t, x, y ) + ∂ xy v ( t, x, y ) Z (cid:27) , (4.5)14 emma 4.2. Assume the constant A in the definition of the set of admissible efforts B satisfies A > γ P + κ − κ ( γ P + γ A ) + 1 . Then, the function U P ( x − y )Φ ( t ) , with Φ ( t ) = (cid:18) c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c (cid:19) γP + γAγA , where c = γ P γ A γ P + γ A + κ − ) − γ P κ − ( γ P + κ − )2( γ P + γ A + κ − ) − λ γ P + γ A γ A and c = λ γ P + γ A γ A . solves in the classical sense the HJB equation (5.3). In particular Z ∗ t = γ P + κ − γ P + γ A + κ − and K ∗ t =1 γ P + γ A log(Φ ( t )) , Proof.
Because the assumption on A implies a ∗ ( z ) = z/κ , the proof of this lemma is a directadaptation of the proof of Lemma 3.3 to which we refer the reader.We are in a position to prove the main result of this section Theorem 4.1.
We have the following explicit characterizations of the optimal contracts. Let A asin the Lemma 4.2 and let Z ∗ t = γ P + κ − γ P + γ A + κ − and K ∗ t = γ P + γ A log(Φ ( t )) , where Φ is definedas in (3.7) with the constants : c := γ P γ A γ P + γ A + κ − ) − γ P κ − ( γ P + κ − )2( γ P + γ A + κ − ) − λ γ P + γ A γ A and c := λ γ P + γ A γ A . Then ( y P C , Z ∗ , K ∗ ) parametrizes the optimal wage for the Moral Hazard problem. The Agent per-forms the optimal action Z ∗ κ .Proof. Because the function U P ( x − y )Φ ( t ) is a classical solution to the HJB equation (5.3) and theoptimal controls are bounded and free of y , we proceed analogously as in the proof of Theorem 3.1.Finally, we have to prove that the optimal wage W ∗ = Y ( y PC ,Z ∗ ,K ∗ ) T admits exponential momentsto close the loop. According to (4.3), we have W ∗ = y P C + Z ∗ B ∗ T ∧ τ + 12 (cid:18) γ A + 1 κ (cid:19) ( Z ∗ ) ( T ∧ τ )+ K ∗ τ − τ ≤ T + (cid:90) T λγ A [exp( − γ A K ∗ s ) − − N s ) ds. Because ( B ∗ t ) t is a Brownian motion and K ∗ t is deterministic, it is straightforward to check that W ∗ admits exponential moments. 15 .2 Model analysis The optimal contract includes two components. One is linear in the output with an incentivizingslope that is similar to the classical optimal contract found in [12]. This is necessary to implementa desirable level of effort. The other is unrelated to the incentives but linked to the shutdown risksharing. It is key to observe that this second term is nonzero even if the shutdown risk does notmaterialize before the termination of the contract.The characterization of the optimal contracts in Theorem 4.1 sparks an immediate observation: thetwo parties only need to be committed to the contracting agreement up until T ∧ τ . Therefore inthis simple model, using an expected-utility related reasoning and without considering mechanismssuch as employment law, the occurence of the agency-free external risk, halting production, leadsto early contract terminations. This is in line with what actually happened during the Covidpandemic. Indeed in the USA and in eight weeks of the pandemic, 36.5 million people applied forunemployment insurance. In more protective economies, mass redundancies were only preventedthrough the instauration of furlough type schemes allowing private employees’ wages to temporarilybe paid by gouvernements. This phenomena makes fundamental sense : a principal whose outputprocess is completely halted cannot enforce the agent to work hard because she has no revenue toprovide the incentives. Let’s focus on the second term: K ∗ τ − τ ≤ T + (cid:90) T λγ A [exp( − γ A K ∗ s ) − − N s ) ds, (4.6)Understanding the effect of these extra terms is crucial to fully understand the sharing of theagency-free shutdown risk. First, we show that the sign of the control K ∗ is constant. Lemma 4.3.
Let c and c be the relevant constants given in Theorem 4.1 then the optimal control ( K ∗ t ) t ∈ [0 ,T ] can be expressed as : K ∗ t = 1 γ A log (cid:18) E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )(( T − t ) ∧ τ ) (cid:19)(cid:21)(cid:19) t ∈ [0 , T ] . (4.7) Proof.
We have that : K ∗ t = 1 γ P + γ A log(Φ ( t )) , with Φ ( t ) = (cid:18) c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c (cid:19) γP + γAγA . The aim here is to link this expression for Φ to that of an expected value. As such, we considerthe following expected value that decomposes as shown : E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )(( T − t ) ∧ τ ) (cid:19)(cid:21) = E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )( T − t ) (cid:19) τ>T − t (cid:21) + E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c ) τ (cid:19) τ ≤ T − t (cid:21) . c = γ P + γ A γ A λ , the first term of the expected value rewrites as follows : E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )( T − t ) (cid:19) τ>T − t (cid:21) = exp (cid:18) γ A γ P + γ A ( c + c )( T − t ) (cid:19) exp ( − λ ( T − t ))= exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) . It remains to compute the second term. We obtain : E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c ) τ (cid:19) τ ≤ T − t (cid:21) = (cid:90) T − t λ exp (cid:18) γ A γ P + γ A ( c + c ) s (cid:19) exp ( − λs ) ds = (cid:90) T − t λ exp (cid:18) γ A γ P + γ A ( c + c ) s − λs (cid:19) ds = (cid:90) T − t λ exp (cid:18) γ A γ P + γ A c s (cid:19) ds = (cid:20) λc γ P + γ A γ A exp (cid:18) γ A γ P + γ A c s (cid:19)(cid:21) T − t = (cid:20) c c exp (cid:18) γ A γ P + γ A c s (cid:19)(cid:21) T − t = c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c . Combining both terms we reach the final expression : E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )(( T − t ) ∧ τ ) (cid:19)(cid:21) = c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c . Therefore we identify that : Φ ( t ) = (cid:18) E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )(( T − t ) ∧ τ ) (cid:19)(cid:21)(cid:19) γP + γAγA . As a consequence, we may also rewrite K ∗ t . Indeed : K ∗ t = 1 γ P + γ A log(Φ ( t )) , and with the new expression for Φ we obtain the result : K ∗ t = 1 γ A log (cid:18) E (cid:20) exp (cid:18) γ A γ P + γ A ( c + c )(( T − t ) ∧ τ ) (cid:19)(cid:21)(cid:19) . Remark 4.2.
We have the same expression for the optimal control K ∗ in the first-best case, usingfor c and c the relevant constants given in Theorem 3.1.
17s a consequence, this alternative form for K ∗ leads to easy analysis of the sign of the control,given in the following lemma. Lemma 4.4.
The sign of K ∗ over the contracting period [0 , T ] is constant and entirely determinedby the model’s risk aversions γ P and γ A , and the Agent’s effort cost κ. Indeed, the sign of K ∗ is equal to the sign of γ P γ A − γ P κ − − ( κ − ) . Moreover, K ∗ t varies monotonously in time, with K ∗ T = 0 . Proof.
From the expression (4.7), we easily deduce that :• If c + c = 0 then K ∗ t = 0 for every t ∈ [0 , T ] ,• if c + c > then K ∗ t > for t ≤ τ and the function t → K ∗ t decreases,• if c + c < then K ∗ t < for t ≤ τ and the function t → K ∗ t increases.Replacing c and c by their relevant expressions in each case leads to the result.Finally, we will show that the risk-sharing component of the contract is in fact linear with respectto the default time. This is a strong result of our study for which we had no ex-ante intuition. Corollary 4.1.
The shutdown risk-sharing component of the optimal wage is linear in the defaulttime. More precisely, the optimal wage is W ∗ = y P C + Z ∗ B ∗ T ∧ τ + 12 (cid:18) γ A + 1 κ (cid:19) ( Z ∗ ) ( T ∧ τ ) + K ∗ − (cid:18) c γ P + γ A + λγ A (cid:19) ( T ∧ τ ) . Proof.
Because K ∗ T = 0 , the optimal wage can be written W ∗ = y P C + Z ∗ B ∗ T ∧ τ + 12 (cid:18) γ A + 1 κ (cid:19) ( Z ∗ ) ( T ∧ τ ) + f ( T ∧ τ ) , with f ( t ) = K ∗ t + (cid:90) t λγ A (exp( − γ A K ∗ s ) − ds, t ∈ [0 , T ] . Let us define g ( t ) = c + c c exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − c c . We have g (cid:48) ( t ) = − ( c + c ) γ A γ P + γ A exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) . Therefore, ∂∂t K ∗ t = 1 γ A g (cid:48) ( t ) g ( t )= 1 γ P + γ A − ( c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) c + c c exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c c = c γ P + γ A − ( c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) ( c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c . ∂∂t (cid:90) t λγ A (exp( − γ A K ∗ s ) − ds = λγ A (exp( − γ A K ∗ t ) − and so : ∂∂t (cid:90) t λγ A (exp( − γ A K ∗ s ) − ds = λγ A (exp( − γ A K ∗ t ) − λγ A (cid:18) g ( t ) − (cid:19) well-defined as g ( t ) > on [0 , T ]= λγ A c + c c exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c c − = c λγ A c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c − λγ A Finally, we have : f (cid:48) ( t ) = ∂∂t K ∗ t + ∂∂t (cid:90) t λγ A (exp( − γ A K ∗ s ) − ds = c γ P + γ A − ( c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) ( c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c + c λγ A c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c − λγ A = − c γ P + γ A c + c ) exp (cid:16) c γ A γ P + γ A ( T − t ) (cid:17) − c (cid:26) ( c + c ) exp (cid:18) c γ A γ P + γ A ( T − t ) (cid:19) − λ γ A γ P + γ A (cid:27) − λγ A = − c γ P + γ A − λγ A as c = λγ A γ P + γ A = − (cid:18) c γ P + γ A + λγ A (cid:19) . We may note that the default related part of the wage paid under full-Risk-Sharing also writesunder this form, where we take the relevant values for K ∗ , c and c . Remark 4.3.
A little algebra gives f ( t ) = K ∗ − (cid:18) c γ P + γ A + λγ A (cid:19) t = K ∗ − ( c + c ) tγ P + γ A he slope of f is of opposite sign to the sign of c + c . As a consequence, • If c + c = 0 then f ( t ) = 0 for every t ∈ [0 , T ] , • if c + c > then f (0) = K ∗ > for t ≤ τ and the function t → f ( t ) decreases, • if c + c < then f (0) = K ∗ < for t ≤ τ and the function t → f ( t ) increases. K ∗ > is the extra compensation asked at the signature of the contract by an agent who is moresensitive to the shutdown risk than the principal. We can easily visualize the sign of the control K ∗ as a function of γ P , γ A and κ . Below we plot the sign depending on the risk-aversions and fixing κ = 1 and κ = 2 . The first two plots (Figure 9 and 2) correspond to the full Risk-Sharing casewhilst Figures 3 and 4 show the sign of K ∗ in the Moral Hazard case. The x -axis holds the valuesof γ A and the y -axis the values of γ P . Both risk-aversion constants are valued between and and the origin is in the bottom left corner. Blue encodes a negative sign and green a positive sign.Figure 1: Sign of K ∗ dependingon γ P and γ A for κ = 1 . Figure 2: Sign of K ∗ dependingon γ P and γ A for κ = 2 .Figure 3: Sign of K ∗ dependingon γ P and γ A for κ = 1 . Figure 4: Sign of K ∗ dependingon γ P and γ A for κ = 2 .We observe that in most situations, the sign of K ∗ is positive. A negative sign occurs when eitherthe principal or the agent are close to being risk-neutral (symmetrically so in the full Risk-Sharingcase but asymmetrically so in the Moral Hazard case : the sign switches from negative to positiveat a much lower level of risk-aversion for the principal than the agent). Also note that increasingthe agent’s effort coefficient κ decreases the level of risk-aversion for which K ∗ goes from positiveto negative. Note that it is known by both the Principal and the Agent at time whether the20ontract will fall into either regime.Our key result shows that the risk of shutdown adds an extra linear term to the optimal compen-sation that, up to the underlying constants, has the same structure under both full Risk Sharingand under Moral Hazard. Even if the optimal contract separates the role of incentives from that ofshutdown risk sharing, the amount of the insurance deposit K ∗ depends strongly on the magnitudeof the moral hazard problem. Therefore, we may naturally quantify the effect of Moral Hazard onthe risk-sharing part. With this question in mind we compare the values of K ∗ under Risk-Sharingand Moral Hazard for different parameter values. This may be observed in the Figures 5 to 9below where we represent the values of K ∗ under Moral Hazard (red) and Risk-Sharing (blue) as afunction of one of the underlying parameters ( γ P , γ A , λ, κ, T ) whilst fixing the remaining 4.These figures lead to a crucial observation : under any given set of parameters, the value of K ∗ under Moral Hazard is always greater than the value of K ∗ under Risk Sharing. Note that in somesettings, given some parameters the signs of the two cases are not always the same as the sign isthat of c + c which depends on the values of γ P , γ A and κ in a different manner in Risk-Sharingand Moral Hazard. Of course though, as soon as K ∗ for Risk-Sharing is positive, K ∗ for MoralHazard is positive too. We may note that although λ and T do not impact the sign, variations intheir values have an impact on the magnitude | K ∗ | : the higher the risk of early termination of thecontract, the lower the amount of insurance requested by the agent and the longer the duration ofthe contract, the higher the amount of insurance requested by the agent.Figure 5: Values of K ∗ depending on γ P γ A = 0 . , λ = 1 , κ = 1 , T = 1 γ A = 1 . , λ = 1 , κ = 1 , T = 1 γ A = 5 , λ = 1 , κ = 1 , T = 1 Figure 6: Values of K ∗ depending on γ A γ P = 0 . , λ = 1 , κ = 1 , T = 1 γ P = 1 , λ = 1 , κ = 1 , T = 1 γ P = 3 , λ = 1 , κ = 1 , T = 1 K ∗ depending on λ γ P = 4 , γ A = 4 , κ = 1 , T = 1 γ P = 1 , γ A = 1 , κ = 1 , T = 1 Figure 8: Values of K ∗ depending on κ γ P = 1 , γ A = 1 , λ = 1 , T = 1 γ P = 1 , γ A = 3 , λ = 1 , T = 1 γ P = 3 , γ A = 3 , λ = 1 , T = 1 Figure 9: Values of K ∗ depending on T γ P = 1 , γ A = 1 , λ = 1 , κ = 1 γ P = 1 , γ A = 3 , λ = 1 , κ = 1 γ P = 3 , γ A = 3 , λ = 1 , κ = 1 λ does have a quantifiable effect of the wage as it affects its expectedvalue : E ( f ( T ∧ τ )) = K ∗ − c + c γ P + γ A (cid:18) − e − λT T (cid:19) . This expected value is represented below as a function of γ A and γ P for different values of λ (with T = 1 and κ = 1 fixed). The first three figures (Figures 10, 11 and 12 ) concern the full Risk-Sharingcase whilst the second set of figures (Figures 13, 14 and 15) concern the Moral Hazard setting.Figure 10: Expected value de-pending on γ P and γ A for λ =0 . . Figure 11: Expected value de-pending on γ P and γ A for λ =1 . Figure 12: Expected value de-pending on γ P and γ A for λ =5 .Figure 13: Expected value de-pending on γ P and γ A for λ =0 . . Figure 14: Expected value de-pending on γ P and γ A for λ =1 . Figure 15: Expected value de-pending on γ P and γ A for λ =5 .The value seems to be in many cases very close to 0. As such, the agent earns, on average, avery similar wage to a "stopped" Holmstrom-Milgrom wage. However when the principal and theagent are particularly risk-averse, the expected value increases quite notably and the agent gainsslightly more on average. This is in line with the papers by Hoffman and Pfeil [11] and Bertrandand Mullainathan [2] which show the agent must be rewarded for a risk that is beyond his control.Note that the simulations show that for fixed levels of risk-aversion the expected value increases23s λ increases. For example for γ P = γ A = 7 , in Figure 13 the expected value is approximatelyworth . and in Figure 14 it is approximately equal to . . As such the risk-averse agent isincreasingly rewarded for the uncontrollable risk.Finally, we can make a few further comments related to the underlying expected utilities in thisnew contracting setting. First, the agent’s expected utility is the same under both full Risk-Sharingand Moral Hazard. Indeed he walks away with his participation constraint : E (cid:34) U A (cid:32) W ∗ − (cid:90) T κ ( a ∗ s ) ds (cid:33)(cid:35) = U A ( y P C ) , where ( W ∗ , a ∗ ) designates the Risk-Sharing or Moral Hazard optimal contract under shutdownrisk. Such a result also holds in the same setting without shutdown risk : the agent tracts averagethe same expected utility under full Risk-Sharing or Moral Hazard, with or without an underlyingagency-free external risk.When it comes to the principal we may wonder how his expected utility may be affected by thepossibility of a halt in production. In particular we may question what the principal loses innot being able to observe the agent’s action under a likelihood of agency-free external risk ? Toanswer this we denote as V RS (0 , x, y ) the Principal’s expected utility under full Risk-Sharing and V MH (0 , x, y ) under Moral Hazard. We have that : V RS (0 , x, y ) = U P ( x − y )Φ RS (0) and V MH (0 , x, y ) = U P ( x − y )Φ MH (0) where Φ RS and Φ MH are the related functions from Theorem 3.1 and Theorem 4.1. With theseexpressions we have that : V MH (0 , x, y ) V RS (0 , x, y ) = Φ MH (0)Φ RS (0) . The Moral Hazard problem involves optimizing across a more restricted set of contracts. Thereforewe know that : V MH (0 , x, y ) ≤ V RS (0 , x, y ) , and as U P ( x − y ) < : Φ RS (0) ≤ Φ MH (0) . We may question whether this inequality leads to a big gap between the expected utilities and weanswer this by plotting the ratio Φ MH (0)Φ RS (0) for different values of λ and with T = 1 and κ = 1 fixed.We first observe a standard result : for low levels of risk-aversion the ratio is close to 1 and theprincipal does not lose much by not observing the agent’s actions. As the values of risk-aversionincrease the principal loses out more and more by not being in a first best setting. This classicalresult comes with an observation that is specific to the presence of shutdown risk : as λ increases(and therefore as the chance of shutdown risk occurring before T increases), the ratio stays closeto 1 for higher and higher levels of risk-aversion. For example in Figure 16 when γ P = γ A = 6 we observe that Φ MH (0)Φ RS (0) ≈ yet in Figure 18 for the same levels of risk-aversion we have that Φ MH (0)Φ RS (0) ≈ . So a high possibility of some production halt occurring reduces the gap between thefull Risk-Sharing contract and the Moral Hazard contract. Such a phenomena may be due to thefact that a high possibility of a production halt at some point in the time interval means that thewage process evolves on a time period that is on average shorter before stopping. There is thus less24igure 16: Ratio depending on γ P and γ A for λ = 0 . . Figure 17: Ratio depending on γ P and γ A for λ = 1 . Figure 18: Ratio depending on γ P and γ A for λ = 5 .time for a significant gap to appear between the full Risk-Sharing case and the Moral Hazard case.As this analysis comes to a close we finish this section by discussing a possibility for extension withmore general deterministic compensators. (Λ t ) t ∈ [0 ,T ] Throughout this paper, we have considered a constant compensator λ for the jump process. Thischoice allows for clearer calculations but it is key to note that our results extend to the case where λ is no longer constant such as : Λ t = (cid:90) t λ s ds, with ( λ s ) s ∈ [0 ,T ] some deterministic positive mapping such that Λ T < + ∞ . The proofs for theoptimal contracting are simply a direct extension of the proofs of the previous sections. Of coursedue to the independence between B and N , a time dependent compensator does not induce anychange to the Holmstrom-Milgrom part of the wages. Only the part related to K ∗ is affected. Weprovide the details in the following. The full Risk-Sharing problem
The optimal wage for the Risk-Sharing problem in such a setting is of the form : W t = y ∗ + (cid:90) t Z ∗ s (1 − N s ) dB s + (cid:90) t K ∗ s (1 − N s ) dM s + (cid:90) t (cid:26) γ A Z ∗ s + κ ( a ∗ s (1 − N s )) + λ s γ A [exp( − γ A K ∗ s ) − γ A K ∗ s ] (cid:27) (1 − N s ) ds, where : y ∗ = y P C , a ∗ t = 1 κ , Z ∗ t = γ P γ P + γ A and K ∗ t = 1 γ P + γ A log(Φ ( t )) , with Φ ( t ) solution to the Bernouilli equation : 25 (cid:48) ( t ) + c ( t )Φ ( t ) + c ( t )Φ ( t ) γPγP + γA = 0 , Φ ( T ) = 1 , where c ( t ) = γ P γ A γ P + γ A ) − γ P κ − λ t γ P + γ A γ A and c ( t ) = λ t γ P + γ A γ A . The Moral Hazard problem
The optimal wage in the Moral-Hazard problem is again of the form : W t = y ∗ + (cid:90) t Z ∗ s (1 − N s ) dB ∗ s + (cid:90) t K ∗ s (1 − N s ) dM s + (cid:90) t (cid:26) γ A Z ∗ s − Z ∗ s κ + λ s K ∗ s + λ s γ A ( e − γ A K ∗ s − (cid:27) (1 − N s ) ds, where : y ∗ = y P C , Z ∗ t = γ P + κ − γ P + γ A + κ − and K ∗ t = 1 γ P + γ A log(Φ ( t )) , with Φ ( t ) solution to the Bernouilli equation : Φ (cid:48) ( t ) + c ( t )Φ ( t ) + c ( t )Φ ( t ) γPγP + γA = 0 , Φ ( T ) = 1 , with : c ( t ) = γ P γ A γ P + γ A + κ − ) − γ P κ − ( γ P + κ − )2( γ P + γ A + κ − ) − λ t γ P + γ A γ A and c ( t ) = λ t γ P + γ A γ A . This paper has so far modeled the occurence of a halt as a complete fatality suffered by bothparties in the contracting agreement. Yet the recent crisis has highlighted the ability of humansand businesses to react and adapt when faced with adversity. We now include such phenomena inthe contracting setting by allowing the principal to invest upon a halt in order to continue some formof (possibly disrupted) production. This is quite a natural and realistic variant on our initial model.Indeed when faced with a period of lockdown, companies may for example invest in teleworkinginfrastructure so that a number of employees whose jobs are doable remotely can continue to work.Similarly, jobs that require some form of presence could continue if companies invest in protectiveequipment and adapt their organization. We may wonder how such a mechanism may affect optimalcontracting.
Mathematically, we consider that the production process evolves as previously up until τ ∧ T . Ifa halt happens at some time τ ≤ T , we allow the principal to invest an amount i > to continueproduction at a degraded level θ ∈ (0 , . It is assumed that the investment decision is at theprincipal’s convenience. It is modeled by a control D which is a G τ -measurable random variable26ith values in { , } . The θ parameter is firm-specific and reflects the effectiveness of the post-shutdown reorganization.Under the initial probability measure P , the output process X t evolves as X t = x + (cid:90) t ((1 − N s ) + N s D =1 ) dB s . We recall from [10] Lemma 4.4. the decomposition of a G -adapted process φ . There exist a F -adapted process φ and a family of processes ( φ t ( u ) , u ≤ t ≤ T ) that are F t ⊗ B ( R + ) measurablesuch that φ t = φ t t<τ + φ t ( τ )11 t ≥ τ . Contract:
A contract W is a G T -measurable random variable satisfying E (exp( − γ A W )) < + ∞ ofthe form W = W T <τ + ( W , T ( τ )11 D =1 + W , T ( τ )11 D =0 )11 τ ≤ T . where W is F T -mesurable and W , t ( u ) and W , t ( u ) are F t ⊗ B ( R + ) measurable. We will assumethat W , T ( τ ) = W , T ∧ τ ( τ ) since in the absence of investment, it is no longer necessary to give incen-tives after τ . Effort process:
In this setting, the agent will adapt his effort to the occurence of the shutdown risk.This is mathematically modeled by a G -adapted process ( a t ) t in the form a t = a t t<τ + a t ( τ )11 t ≥ τ where a and a are respectively F -adapted and F t ⊗ B ( R + ) measurable. Furthermore, we assumethat the effort processes are bounded by some constant A . We then define P a as d P a d P |G T = L θT , with L θT = exp (cid:32)(cid:90) T a s (1 − N s ) + θa s ( τ ) N s D =1 dB s − (cid:90) T ( a s ) (1 − N s ) + θ ( a s ( τ )) N s D =1 ds (cid:33) Because the processes a and a are bounded, ( B at ) t ∈ [0 ,T ] with B at = B t − (cid:90) t ( a s (1 − N s ) + θa s ( τ ) N s D =1 ) ds, t ∈ [0 , T ] is a G -Brownian motion under P a . Under P a , the output process evolves as X t = x + (cid:90) t ( a s (1 − N s ) + θa s ( τ ) N s D =1 ) ds + (cid:90) t ((1 − N s ) + N s D =1 ) dB s . We first make the following observation. After τ , if the default time occurs before the maturity ofthe contract, the principal has a binary decision to take. If she decides to not invest, she gets thevalue V , ( x, y ) = U P ( x − y ) where x is the level of input and y is the agent continuation value.On the other hand, if she decides to invest, she will face for t ≥ τ the moral hazard problem ofHolmstrom and Milgrom for which we know the optimal contract and the associated value function V , ( t, x, y ) = U P ( x − y )Φ ( t, θ ) Φ ( t, θ ) = exp( − γ P C inv ( T − t )) and C inv := (cid:16) γ P + θ κ (cid:17) (cid:0) γ P + γ A + θ κ (cid:1) − γ P . Because the principal has to pay a sunk cost i > to invest, she will decide optimally to invest ifand only if at τ for a given ( x, y ) , she observes V , ( τ, x, y ) ≥ V , ( x, y ) , or equivalently C inv ( T − τ ) > i . Hence, if C inv > , the optimal control will be D ∗ = 11 { τ Lemma 5.1. 1. Investment for mitigation is never optimal upon a halt if : C inv < or i > T C inv . 2. Now suppose that : C inv > and i < T C inv . Mitigation is optimal up until the cutoff time t max defined as : t max := T − iC inv . Note that i < T C inv guarantees that t max ≥ . We are in a position to solve the before-default principal problem. Proceeding analogously as inSection 4, the before-default value function is given by the Markovian control problem V P = sup y ≥ y PC V (0 , x, y ) , with V (0 , x , y ) = sup π =( Z,K ) ∈ ζ E (cid:34) U P ( X πT − W πT )(1 − N T ) + (cid:90) T max( V , ( X πt , W πt ) , V , ( t, X πt , W πt )) λe − λt dt (cid:35) , (5.1)and dX t = a ∗ ( Z t )(1 − N t ) dt + (1 − N t ) dB ∗ t ,dW πt = Z s (1 − N s ) dB ∗ s + K s (1 − N s ) dM s (5.2) + (cid:26) γ A Z s + κ ( a ∗ ( Z s )) + λγ A [exp( − γ A K s ) − γ A K s ] (cid:27) (1 − N s ) ds. Remark 5.1. To be perfectly complete, we develop in the appendix the martingale optimality prin-ciple which makes it possible to obtain the dynamics (5.2) . Theorem 5.1. We have the following explicit characterizations of the optimal contracts. Assumethe constant A in the definition of the set of admissible efforts B satisfies A > γ P + κ − κ ( γ P + γ A ) + 1 . et Z ∗ t = γ P + κ − γ P + γ A + κ − and K ∗ t = 1 γ P + γ A log Φ ( t )min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) with Φ as defined above with : c := γ P γ A γ P + γ A + κ − ) − γ P κ − ( γ P + κ − )2( γ P + γ A + κ − ) − λ γ P + γ A γ A and c ( t ) = λ γ P + γ A γ A min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) γAγP + γA Then ( y P C , Z ∗ , K ∗ ) parametrizes the optimal wage for the Moral Hazard problem with a possibilityfor mitigation. The Agent performs the optimal action Z ∗ κ before τ and θZ ∗ κ after τ when τ Assume A > γ P + κ − γ P + γ A + κ − . The function v ( t, x, y ) = U P ( x − y )Φ ( t ) , with Φ ( t ) := exp( − c t ) (cid:40) exp( c γ A γ P + γ A T ) + γ A γ P + γ A (cid:90) Tt c ( s ) exp( γ A γ P + γ A c s ) ds (cid:41) γP + γAA where c = γ P γ A γ P + γ A + κ − ) − γ P κ − ( γ P + κ − )2( γ P + γ A + κ − ) − λ γ P + γ A γ A and c ( t ) = λ γ P + γ A γ A min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) γAγP + γA . olves in the classical sense the HJB equation (5.3). In particular Z ∗ t = γ P + κ − γ P + γ A + κ − and K ∗ t = γ P + γ A log Φ ( t )min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) .Proof. The proof of this lemma is a direct adaptation of the proof of Lemma 4.2 to which we referthe reader.The proof of the final result relies on the regularity of v and a standard verification result. Becausethe controls are free of y , we deduce that V P = V (0 , x, y P C ) . The main change brought about by investment involves the halt related control K ∗ . Indeed theoptimal Z ∗ in the Moral Hazard case are simply the optimal "Holmström-Milgrom" controls for therelated production process. At first glance, the optimal control K ∗ seems to be quite different fromthat of Theorem 4.1. However one may verify that when we are in a setting where investment isnever optimal (through the criteria of Lemma 5.1), the expression for K ∗ simplifies to exactly thatof Theorem 4.1. The key to deduce this is that in such a setting, min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) = 1 .We may therefore focus our analysis on the effects on investment when investing may be optimal(i.e. when C inv > and i < T C inv ). In such a setting, K ∗ has two phases :- before t max , K ∗ is adjusted to account for the possibility of risk mitigation- after t max , K ∗ has the same values as without mitigation. Indeed : min (cid:110) , exp( γ P i )Φ ( t, θ ) (cid:111) = 1 for t ≥ t max . We are able to analyze the effect of different parameters and to do so represent the deterministicpart of K ∗ as a function of time in the following figures.We fix parameters γ P = κ = T = 1 , γ A = 0 . : again this allows for mitigation to be optimal beforesome t max .Figure 19: λ = 0 . , i = 0 . , θ = 0 . Figure 20: λ = 1 , i = 0 . , θ =0 . . Figure 21: λ = 5 , i = 0 . , θ =0 . .We immediately observe that with mitigation, the value of K ∗ before t max and is higher thanwithout mitigation : the possibility for mitigation shrinks the opportunities for speculation (seeFigures 19 to 21) and increasingly so as the probability of a halt increases. In fact the sign of K ∗ may now change over the duration of the contracting period : see Figure 21. Quite naturally, t max i = 0 . , λ = 1 , θ = 0 . Figure 23: i = 0 . , λ = 1 , θ =0 . . Figure 24: i = 0 . , λ = 1 , θ = 0 . .Figure 25: i = 0 . , λ = 1 , θ =0 . Figure 26: i = 0 . , λ = 1 , θ =0 . .varies with θ and i . Indeed it decreases as i increases or θ decreases : as the cost of investmentincreases and/or the level of degradation in continued production increases, more time is neededfor investment for continued production to be worth it.31 Appendix We sketch the martingale optimality principle arising from the Agent’s problem in the investmentsetting. We set H G is the set of G − adapted processes Z with E [ (cid:82) T Z s ds ] < + ∞ and S G is the setof G − predictable processes Y with cadlag paths such that Z with E [sup t ∈ [0 ,T ] Y t ] < + ∞ . For a G τ − measurable random variable D with values in { , } that models the investment decision, weconsider a contract W as a G T measurable r.v. which can be decomposed under the form : W = W T <τ + (cid:16) W , T ( τ ) D =1 + W , T ( τ ) D =0 (cid:17) τ ≤ T , where W is F T -measurable, and W , t ( u ) and W , t ( u ) are F t ⊗B ( R + ) measurable, with in particular W , T ( τ ) = W , T ∧ τ ( τ ) . Given a contract W , the agent faces the following control problem, sup a ∈B E P a E (cid:34) U A (cid:32) W − (cid:90) T κ ( a s ) ds (cid:33)(cid:35) . Remember that an effort process is now a G − adapted process ( a t ) t ∈ [0 ,T ] and consequently has theform : a t = a t t<τ + a t ( τ ) t ≥ τ where a is F − adapted and a is F t ⊗ B ( R + ) measurable and where both are assumed bounded bysome constant A . By convention, we still denote by B the set of such effort. Lemma 6.1. Suppose that there exists some unique triplet ( Y, Z, K ) in S G × H G × H G such that : Y t = W − (cid:90) Tt Z s ((1 − N s ) + N s D =1 ) dB s − (cid:90) Tt K s (1 − N s ) dM s − (cid:90) Tt f ( s, Z s , K s ) ds, where f ( s, Z s , K s ) = (cid:18) λK s + λγ A ( e − γ A K s − (cid:19) (1 − N s )+ 12 γ A Z s ((1 − N s ) + N s D =1 ) + inf a ∈B { κ ( a s ) − a s Z s (1 − N s ) − θa s Z s τ ≤ s,D =1 } , then R at = U A (cid:18) Y t − (cid:90) t κ ( a s ) ds (cid:19) satisfies a Martingale Optimality Principle for the Agent’s problem in this setting.Proof. By construction, R aT = U A (cid:16) W − (cid:82) T κ ( a s ) ds (cid:17) and R a is independent of the Agent’s action32 . As in Section 4, we compute the variations of R a to obtain: dR as = − γ A R as Z s ((1 − N s ) + N s D =1 ) dB as + R as ( e − γ A K s − − N s ) dM s + R as γ A (cid:18) γ A Z s ((1 − N s ) + N s D =1 ) − f ( s, Z s , K s ) + κ ( a s ) + ( λK s + λγ A ( e − γ A K s − − N s ) (cid:19) + R as γ A ( − a s Z s (1 − N s ) + θa s N s D =1 )= − γ A R as Z s ((1 − N s ) + N s D =1 ) dB as + R as ( e − γ A K s − − N s ) dM s + R as γ A (cid:18) γ A Z s ((1 − N s ) + N s D =1 ) − f ( s, Z s , K s ) + κ ( a s (1 − N s )) + κ ( θa s N s D =1 ) + ( λK s + λγ A ( e − γ A K s − − N s ) (cid:19) + R as γ A (cid:0) − a s Z s (1 − N s ) + θa s N s D =1 (cid:1) and therefore R a is a super-martingale for every a in B . Setting : a s ∗ ( z ) = − A zκ < − A + A zκ >A + zκ − A ≤ zκ ≤ A and a s ∗ ( z ) = − A θzκ ≤− A + A θzκ >A + θzκ − A ≤ θzκ ≤ A , then a ∗ t = a t ∗ t<τ + a t ∗ t ≥ τ . We get that R a ∗ is a P a ∗ -martingale and the Agent’s response given W is then a ∗ . It remains to show that there actually exists a unique solution to ( Y, Z, K ) in S G × H G × H G to : Y t = W − (cid:90) Tt Z s ((1 − N s ) + N s D =1 ) dB s − (cid:90) Tt K s (1 − N s ) dM s − (cid:90) Tt f ( s, Z s , K s ) ds, where f ( s, Z s , K s ) = (cid:18) λK s + λγ A ( e − γ A K s − (cid:19) (1 − N s )+ 12 γ A Z s ((1 − N s ) + N s D =1 ) + inf a ∈B { κ ( a s ) − a s Z s (1 − N s ) − θa s Z s τ ≤ s,D =1 } . To to this, first note for any s in [0 , T ] fixed, and for any t ∈ [ s, T ] there exists a unique pair ( Y i , Z i ) ∈ S G × H G solution to the BSDE : Y it ( s ) = W , T ( s ) D =1 − (cid:90) Tt f ( Z is ( s )) ds − (cid:90) Tt Z is ( s ) dB s , (6.1)where f ( z ) = γ A z +inf a ∈B ( κ ( a ) − θaz ) and where the notation ( Y i ( s ) , Z i ( s )) is used to emphasizethe dependency in s of the terminal condition and its effect on the solution. This existence resultsimply follows from the fact that for each s , (6.1) is now simply a Brownian BSDE that fits intothe classical quadratic setting of Briand and Hu. We may then set : ˜ W = Y iτ ( τ ) τ ≤ T D =1 + W , T ∧ τ ( τ ) τ ≤ T D =0 + W T <τ , G T ∧ τ measurable random-variable. We set : f ( z, k ) = 12 γ A z + λk + λγ A ( e − γ A k − 1) + inf a ∈B ( κ ( a ) − aZ ) . This fits right into the setting of the recent work [15] on a default BSDE for Principal Agentproblems. In particular, there exists a unique triplet ( ˜ Y , ˜ Z, ˜ K ) in S G × H G × H G such that : ˜ Y t = ˜ W − (cid:90) T ∧ τt ∧ τ ˜ Z s dB s − (cid:90) T ∧ τt ∧ τ ˜ K s dM s − (cid:90) T ∧ τt ∧ τ f ( ˜ Z s , ˜ K s ) ds. 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