Analysis of the Risk-Sharing Principal-Agent problem through the Reverse-H{ö}lder inequality
aa r X i v : . [ q -f i n . R M ] D ec Analysis of the Risk-Sharing Principal-Agent problemthrough the Reverse-Hölder inequality
Jessica Martin ∗1 and Anthony Réveillac †11 INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135 Avenue deRangueil 31077 Toulouse Cedex 4 FranceDecember 18, 2019
Abstract
In this paper we provide an alternative framework to tackle the first-best Principal-Agent problem under CARA utilities. This framework leads to both a proof of existenceand uniqueness of the solution to the Risk-Sharing problem under very general assump-tions on the underlying contract space. Our analysis relies on an optimal decompositionof the expected utility of the Principal in terms of the reservation utility of the Agentand works both in a discrete time and continuous time setting. As a by-product thisapproach provides a novel way of characterizing the optimal contract in the CARAsetting, which is as an alternative to the widely used Lagrangian method, and someanalysis of the optimum.
Many economic situations in the areas of optimal contracting or incentive policy designcan be gathered under the so-called Principal-Agent formulation where an Agent is askedto perform an action on behalf of a Principal in exchange for a wage. This formulationhas known a renewed interest since the seminal paper [8] by Holmström and Milgrom in1987 which brought to light a new method to solve a particular type of Principal-Agent ∗ [email protected] † [email protected] Keywords :
Principal Agent problem; First-Best; Risk-Sharing; Borch rule; Reverse Hölder inequality;Optimal Contracting Theory; Existence; Uniqueness. U P and U A are CARA utilities defined as: U P ( x ) = − exp( − γ P x ) and U A ( x ) = − exp( − γ A x ) , where γ P and γ A are two fixed risk aversion coefficients. In a single period setting, in orderto reduce his exposure to uncertainty, the Principal hires the Agent at time t = 0 in atake-it-or-leave-it contract in which (if accepted) the Agent is asked to produce an effort3 ≥ at time t = 0 in return for the payment of a wage W at time t = 1 . The wealth ofthe Agent then becomes X a at time t = 1 with X a = x + a + B, where x is the initial wealth at time t = 0 and B is a square-integrable random variablemodeling the stochastic exposure of the Principal. Note that in this simple model, we as-sume that the Principal fully observes both the outcome X a (at time t = 1 ) and the action a of the Agent, meaning that the Principal actually dictates this action to the Agent. Tomodel the cost of effort for the Agent, we introduce a function κ defined on R + and chosento be strictly convex, continuous and non-decreasing. A simple example of such a functionis the quadratic cost function which, for a fixed constant K > , is defined for any x in R + as κ ( x ) = K | x | . An important remark is that the contract (which from now on will be modeled by thepair (wage, action) = (
W, a ) ) is a take-it-or-leave-it contract that will be accepted by theAgent if a Participation Constraint (PC) condition (or reservation utility constraint) givenbelow is satisfied : E [ U A ( W − κ ( a ))] ≥ U A ( y ) , (1.1)where y ≥ represents the level of requirement for the Agent to accept the contract.In this setting, we analyze the first-best problem which simply writes as : sup ( W,a ) subject to ( . ) E [ U P ( X a − W )] , (1.2)and which can also be written in a continuous time setting. One classical way of provingexistence of a solution to such a problem is to use a variational approach where we finda topology of the underlying contract space that ensures upper semi-continuity, concavityand coercivity of the Principal’s expected utility, whilst rendering the contract space convexand closed. The sticking point here is often coercivity and more particularly coercivity in W . This may be ensured in some cases when B is bounded. However it is difficult to do sounder more general assumptions.In this paper we provide an alternative proof of existence and uniqueness of solutions toProblem (1.2) with next to no further assumptions. To do so, we exploit the properties ofthe exponential function in the utilities to first decompose the Principal’s expected utility A typical example of such a situation is when the Agent is the Principal himself, meaning that as amanager of the firm, the Principal decides his level of work and the salary he pays himself for it. Note thatas the Principal decides of the action of the agent, this action must be a positive effort in the firm as a < would mean that the Agent would sabotage the firm, which of course does not make any sense. Anotherexample is a monopoly type situation.
4n terms of that of the Agent. Using this decomposition and the so-called Reverse HÃűlderinequality, we are able to upper bound the Principal’s expected utility by some constantdepending only on the model’s parameters. We then derive an optimal decomposition, rely-ing on the Reverse HÃűlder equality condition, which provides us with a unique admissiblecontract attaining the upper bound. As a by-product this allows us to shed a new light onthe Borch rule for Risk-Sharing. We thus have an existence and uniqueness proof for theRisk-Sharing problem under CARA utilities along with a characterization of the optimum.We believe that the strength of this method lies in the generality of the settings that itallows us to consider. Indeed we are able to deal with both the discrete and continuoustime case, actions that belong to R + or even to any compact subset of R + , a general effortcost function κ , and more importantly any form on random variable / random process thathas exponential moments (this last assumption is a purely technical as we are optimizingexponential expected utilities). Furthermore, the proof does not rely on any a priori intu-ition on the form of the optimal contract. We thus provide an existence and uniquenessresult for the first-best Principal Agent problem under CARA utilities in more general set-tings that those generally considered in the literature. As a by-product to this approachwe also obtain through our decomposition of the Principal’s expected utility a revisit of thewell-known Borch rule for Risk-Sharing. Finally, we provide some further analysis on theeffect of different parameters on the Principal’s take-home utility. Indeed, we analyze theeffect of enforcing a sub-optimal action as well as study the question of a Principal who hasto choose between two different Agents.We proceed as follows. First in Section 2 we tackle the single-period problem as describedin the introduction. Then in Section 3, we proceed with some contract comparison in thesingle-period setting. Finally in Section 4 we discuss on some extensions, including thecontinuous time problem, before collecting some of the lengthier proofs of our results inSection 5 and concluding in Section 6. In the following we discuss Problem (1.2) as presented in the introduction. To do so weexploit a key result called the Reverse Hölder inequality. As its name suggests it is closelylinked to the more well-known Hölder inequality. We state the result below.
Proposition 2.1 (Reverse Hölder inequality) . Let p ∈ (1 , + ∞ ] . Let F and G be tworandom variables such that G = 0 , P -a.s.. Then :(i) The Reverse Hölder inequality holds, that is, E [ | F × G | ] ≥ E h | F | p i p × E h | G | − p − i − p +1 . ii) In addition, the inequality is an equality, that is, E [ | F × G | ] = E h | F | p i p × E h | G | − p − i − p +1 if and only if there exists some constant (that is non-random) α ≥ such that | F | = α | G | − pp − . In order to proceed with our analysis of Problem (1.2), we define the set of admissiblecontracts for the Risk-Sharing problem C adm as follows : C adm = (cid:8) ( W, a ) ∈ L (Ω) × R + satisfying (1 . (cid:9) , where L (Ω) is the set of square-integrable random variables. Also, for a given a in R + wedefine the set of related admissible wages : W ( a ) = n W ∈ L (Ω) , such that ( W, a ) in C adm o . With these notations in mind, we rewrite Problem (1.2) as : sup ( W,a ) ∈ C adm E [ U P ( X a − W )] , (2.1)or alternatively : sup W ∈W ( a ) sup a ∈ R + E [ U P ( X a − W )] . (2.2)Our aim is to prove existence of a solution to this problem. To do so, we use the inequalityof Proposition 2.1 and the multiplicative nature of the Principal’s utility in order to obtainan attainable upper bound. As a first step, we apply the Reverse Hölder inequality and adecomposition of the Principal’s expected utility. Theorem 2.1.
Let a in R + . We have :(i) For any W in W ( a ) , E [ U P ( X a − W )] ≤ E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA . (2.3) (ii) For W in W ( a ) . The following conditions are equivalent :(ii') E [ U P ( X a − W )]= E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA (2.4)6 ii'') ( a, W ) satisfies the Borch rule, that is, there exists α in R ∗ + such that : U ′ P ( X a − W ) U ′ A ( W − κ ( a )) = α, P − a.s.. (2.5) (ii''') The wage W is of the form : W = γ P γ P + γ A X a + β, with β ∈ R , such that ( W, a ) ∈ C adm . Proof.
See Section 5.1.
Remark 2.1. (i) The Borch rule for Risk-Sharing, derived by Karl Borch in [1] and [2],appears here in (2.5) as a condition for equality between the Princip al’s expected utilityand its decomposition. This sheds a new light onto the rule in the CARA utility case.Indeed, it only allows for contracts that enable an isolation of the effect of the Agent’sexpected utillity in the Principal’s expected utility. We will see further on that theBorch rule is a necessary optimality condition for the Risk-Sharing problem just likeit is when using the Lagrangian method.(ii) When analyzing the first-best problem it seems intuitive that the Principal and theAgent’s utilities should be of opposite effect : the Principal should want to maximizehis utility whilst minimizing that of the Agent. This is encompassed in (2.3) : due tothe negative power − γ P γ A , the upper bound increases as the Agent’s utility decreases. Now that we have a decomposition of the Principal’s expected utility and a condition forit to hold exactly (Borch rule), we turn to further exploiting this decomposition and theappearance of the Agent’s expected utility in order to obtain a bound that is free of W and a . We do this in two stages in the following two Propositions. Proposition 2.2.
Let a in R + . For any W in W ( a ) , E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA ≤ U P ( − y ) × inf ˜ a ∈ R + E (cid:20)(cid:12)(cid:12) U P (cid:0) X ˜ a − κ (˜ a ) (cid:1)(cid:12)(cid:12) γAγA + γP (cid:21) γA + γPγA Proof.
Apply the participation constraint (1.1) to the right hand term and optimize in a inthe left hand term. 7t thus remains to perform the optimization in a . To do so, we introduce the following twonotations : e κ ( p ) := sup x ∈ R + ( px − κ ( x )) , for any p ≥ , and κ ∗ ( p ) := argsup x ∈ R + ( px − κ ( x )) , for any p ≥ . ˜ κ is the Legendre transform of κ , and κ ∗ is its related argument. These are well defined dueto the convexity of κ . We use these two notations in the following Proposition to performthe optimization in a and obtain our upper bound. Proposition 2.3.
For any a in R + it holds that : E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA ≥ E (cid:20)(cid:12)(cid:12)(cid:12) U P (cid:16) X a ∗ − κ ( a ∗ ) (cid:17)(cid:12)(cid:12)(cid:12) γAγA + γP (cid:21) γA + γPγA = exp ( − γ P ( x + ˜ κ (1))) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) γA + γPγA , where a ∗ := κ ∗ (1) . We thus have for any ( W, a ) in C adm : E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA ≤ U P ( x − y + e κ (1)) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) γA + γPγA . Proof.
See Section 5.2.The combination of Theorem 2.1, Proposition 2.2 and Proposition 2.3 allows us to exploita decomposition of the Principal’s expected utility, the Participation Constraint and anoptimization in a in order to upper bound the Principal’s value function. The upper boundthat we obtain is free of W and a and is key for our existence proof. Indeed, we are nowable to show that this upper bound is attained for an admissible contract, which is in factunique. This gives us our main result, which is simultaneously an existence, uniqueness,and characterization result for solutions to the first-best problem and is the object of thefollowing key Theorem. Theorem 2.2 (Existence, Uniqueness and Characterization) . (i) Consider ( W, a ) in anycontract in C adm , then it holds that : E [ U P ( X a − W )] ≤ U P ( x − y + e κ (1)) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) γA + γPγA . ii) Now let ( W ∗ , a ∗ ) be such that : a ∗ = κ ∗ (1) and W ∗ = γ P γ P + γ A X a ∗ + β ∗ ,β ∗ := y + κ ( κ ∗ (1)) − γ P γ A + γ P ( x + κ ∗ (1)) − γ A ln (cid:18) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21)(cid:19) . Then ( W ∗ , a ∗ ) is the only contact verifying (2.5) and saturating the participationconstraint.(iii) Furthermore : E h U P (cid:16) X a ∗ − W ∗ (cid:17)i = U P ( x − y + e κ (1)) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) γA + γPγA . Thus ( W ∗ , a ∗ ) is the unique contract attaining the upper bound. It follows that forany ( W, a ) in C adm : E [ U P ( X a − W )] ≤ E h U P (cid:16) X a ∗ − W ∗ (cid:17)i , and ( W ∗ , a ∗ ) is the optimal contract for the first-best Principal-Agent problem.Proof. See Section 5.3.
Remark 2.2.
Throughout this Section, we suppose that we wish to find an optimal action a amongst the whole of R + . We may in fact also consider optimizations on compact subsetsof R + . For S such a subset, the optimal action is then : a ∗ = argsup a ∈ S a − κ ( a ) We may apply our key Theorem to a more specific Gaussian setting as follows.
Example 2.1.
Consider a Gaussian setting where B ∼ N (0 , , and κ ( x ) := K x forsome K in R + . Then there exists a unique optimal contact for the Risk-Sharing problem.We set : a ∗ := κ ∗ (1) = 1 K ,W ∗ = γ P γ A + γ P X a ∗ + β ∗ , where the optimal β ∗ has the following expression : β ∗ := γ A | γ P | | γ A + γ P | + y + κ ( κ ∗ (1)) − γ P γ A + γ P ( x + κ ∗ (1)) . he contract ( W ∗ , a ∗ ) is the optimal Risk-Sharing contract in this setting. The Principal’soptimal expected utility is worth : E h U P ( X a ∗ − W ∗ ) i = U P ( x − y ) exp (cid:18) γ P (cid:18) − e κ (1) + γ P γ A γ A + γ P ) (cid:19)(cid:19) . Our main Theorem therefore includes the well-known results for the single period Gaussiansetting discussed for example in [5]. We note that it even allows us to go further than [5] andfully specify the intercept β ∗ in the wage rather than leaving it dependent on the Lagrangemultiplier. Remark 2.3.
Theorem 2.2 extends existence of solutions beyond the bounded wealth processsetting. It does so without any assumption on the form of the optimal contract and usingan analytic inequality rather than calculus of variations for which we lack coercivity. Italso provides an important proof of uniqueness. The result therefore completes pre-standingresults on existence and uniqueness by allowing for a general setting (general wealth process,general cost of effort function etc.) in the CARA utility case.
This reasoning discussed in this Section generalizes to continuous time settings and thiswill be a focus of Section 4. In the meantime we go over some further analysis of theRisk-Sharing problem that can be gleaned using the Reverse-Holder inequality.
In this Section we use the Reverse Hölder inequality given in Proposition 2.1 to comparedifferent contracts in the Risk-Sharing setting. Indeed, the multiplicative nature of thedecomposition of the Principal’s expected utility prompts us to compare ratios of utilitiesunder different conditions. This analysis brings to light effects of some choices the Principalmay wish to make.We first study the effect of enforcing another action than the optimal one. The Principalmay indeed wish to over or under work the Agent in some conditions and the followingProposition quantifies the effect of such a choice on the Principal’s own utility. Note thatthe direction of inequality (3.1) may seem counter-intuitive at first but this is due to thenegative sign of the expected utilities.
Proposition 3.1 (Effect of enforcing a sub-optimal a ) . Let a be any positive action and W be any contract in W ( a ) . Let ( W ∗ , a ∗ ) be the optimal contract described in Theorem 2.2.We define the Agent’s action ratio R ( a ) as : R ( a ) := E h | U P ( X a − κ ( a )) | γAγA + γP i E h | U P ( X a ∗ − κ ( a ∗ )) | γAγA + γP i = (cid:18) exp( − γ P ( a − κ ( a )))exp( − γ P ( a ∗ − κ ( a ∗ )) (cid:19) γAγA + γP , nd the Participation Constraint ratio C ( W, a ) as :C ( W, a ) := E [ U A ( W − κ ( a ))] E [ U A ( W ∗ − κ ( a ∗ ))] = E [ U A ( W − κ ( a ))] U A ( y ) . Then it holds that : E [ U P ( X a − W )] E [ U P ( X a ∗ − W ∗ )] ≥ R ( a ) γA + γPγA × C ( W, a ) − γPγA , (3.1) and E [ U P ( X a − W )] ≤ R ( a ) γA + γPγA × E h U P (cid:16) X a ∗ − W ∗ (cid:17)i . (3.2) These inequalities are strict as soon as ( W, a ) = ( W ∗ , a ∗ ) . Furthermore, when ( W, a ) boundsthe Participation Constraint, C ( W, a ) = 1 and we directly obtain 3.2. Finally R ( a ) ≥ . Proof.
The proof of (3.1) is a direct consequence to applying Reverse Holder to E [ U P ( X a − W )] and E (cid:2) U P (cid:0) X a ∗ − W ∗ (cid:1)(cid:3) . Indeed, we obtain : E [ U P ( X a − W )] ≤ E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA , and through optimality for ( W ∗ , a ∗ ) : E h U P (cid:16) X a ∗ − W ∗ (cid:17)i = E (cid:20)(cid:12)(cid:12)(cid:12) U P (cid:16) X a ∗ − κ ( a ∗ ) (cid:17)(cid:12)(cid:12)(cid:12) γAγA + γP (cid:21) γA + γPγA × E [ U A ( W ∗ − κ ( a ∗ ))] − γPγA . Taking the ratio of the two, noting that E [ U P ( X a ∗ − W ∗ )] ≤ which changes the sign of theinequality, we obtain (3.1). To obtain (3.2), note that through the Participation Constraint: E [ U A ( W − κ ( a ))] ≥ E [ U A ( W ∗ − κ ( a ∗ ))] , and thus (as U A is negative) C ( W, a ) ≤ , implying that C ( W, a ) − γPγA ≥ . Finally, weobtain R ( a ) ≥ directly through Proposition 2.3.It is apparent through (3.1) that the ratio of the Principal’s utilities splits into a product oftwo ratios. One of them ( C ( W, a ) ) transfers the effect of the Agent’s participation constraintonto the utility of the Principal : indeed, C ( W, a ) is maximized as soon as ( W, a ) binds (andthus minimizes) the Agent’s utility. The second ratio R ( a ) further restricts the possiblevalues for the Agent’s value function and this no matter the chosen wage. Indeed, we seein (3.2) that a choice of action a may restrict quite consequently the possible attainableutilities for the Principal, no matter whether the associated wage binds the ParticipationConstraint or not. We illustrate this result in the following example.11 xample 3.1. Consider a Principal whose company’s activity temporarily decreases andwho therefore has to underwork by half the Agent. Then the loss in expected utility incurredby the Principal, no matter the wage he pays, is quantified by : R ( a ) γA + γPγA = exp( − γ P ( a ∗ − κ ( a ∗ )))exp( − γ P ( a ∗ − κ ( a ∗ )) Of course, we see that the impact of a choice of action depends in turn on the cost ofaction function κ. For example for a function κ that is convex yet close to linear, choosing anon-optimal action will not have as much as an effect on the Principal as when κ is convexand quadratic. Naturally, we may further analyze the effect of κ and wish to quantify theeffect of two different cost functions. We do so in the following and the analysis is similarto above. Proposition 3.2 (Effect of the action cost function κ ) . Let ( W ∗ , a ∗ ) be the optimumobtained in Theorem 2.2 for the Risk-Sharing problem and a given cost function κ . Let ( W, a ) be some contract in C adm for some cost function ˆ κ and binding the ParticipationConstraint. Then : E [ U P ( X a − W )] E [ U P ( X a ∗ − W ∗ )] ≥ E h | U P ( X a − ˆ κ ( a )) | γAγA + γP i γA + γPγA E h | U P ( X a ∗ − κ ( a ∗ )) | γAγA + γP i γA + γPγA (3.3) In particular if W is of the form γ P γ A + γ P X a + β then the above inequality holds in equality.Thus, if ( W, a ) is the optimum for some particular ˆ κ , one can quantify the effect of ˆ κ onthe obtained optimum relative to that obtained for κ . Indeed, it holds that : E [ U P ( X a − W )] = E h | U P ( X a − ˆ κ ( a )) | γAγA + γP i γA + γPγA E h | U P ( X a ∗ − κ ( a ∗ )) | γAγA + γP i γA + γPγA × E h U P (cid:16) X a ∗ − W ∗ (cid:17)i . (3.4) Proof.
Using similar reasoning to that in the proof of Proposition 3.1, we have that : E [ U P ( X a − W )] E [ U P ( X a ∗ − W ∗ )] ≥ E h | U P ( X a − ˆ κ ( a )) | γAγA + γP i γA + γPγA E h | U P ( X a ∗ − κ ( a ∗ )) | γAγA + γP i γA + γPγA × (cid:18) E [ U A ( W − ˆ κ ( a ))] U A ( y ) (cid:19) − γPγA , (3.5)12nd in particular if ( W, a ) binds the Participation Constraint then (cid:18) E [ U A ( W − ˆ κ ( a ))] U A ( y ) (cid:19) − γPγA = 1 , and using the Reverse-Holder equality condition, we have equality for wages of the form γ P γ A + γ P X a + β saturating the Participation Constraint.This Proposition underlines the effect of a change in action cost function on the Risk-Sharing optimum, encompassing it in the action cost ratio. A corollary to this Propositionextends this result to the situation where a Principal has a choice between two Agents withrespective cost functions κ and ˆ κ , and respective risk aversion coefficients γ A and ˆ γ A , andwishes to choose one and provide him with the related optimal Risk-Sharing contract. Corollary 3.1 (A comparison of two Agents) . Consider a Principal who has a choicebetween two agents characterized by ( γ A , κ, y ) and ( ˆ γ A , ˆ κ, ˆ y ) . Suppose that γ A y = ˆ γ A ˆ y .Let ( W, a ) and ( ˆ W , ˆ a ) be the associated optimal contracts (recalling that a = κ ∗ (1) and ˆ a = ˆ κ ∗ (1) ). Then : E [ U P ( X a − W )] = E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA E (cid:20) | U P ( X ˆ a − ˆ κ (ˆ a )) | ˆ γA ˆ γA + γP (cid:21) ˆ γA + γP ˆ γA × E h U P (cid:16) X ˆ a − ˆ W (cid:17)i . Therefore if E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA E (cid:20) | U P ( X ˆ a − ˆ κ (ˆ a )) | ˆ γA ˆ γA + γP (cid:21) ˆ γA + γP ˆ γA < , the Principal should choose the Agent characterized by ( γ A , κ ) and vice-versa. The choice of Agent therefore depends on a balance between the Agent’s risk aversion andhis action cost function. Note that this result may be generalized beyond the case where γ A y = ˆ γ A ˆ y , one simply has to include the effect of y and ˆ y in the comparison criteria. Example 3.2.
Consider B ∼ N (0 , , κ ( x ) := x and ˆ κ ( x ) := x . Then a = 1 and ˆ a = .The Principal’s expected utility obtained with the first Agent would be : E [ | U P ( X a − κ ( a )) | γAγA + γP ] γA + γPγA = exp (cid:18) − γ P ( x + a − κ ( a )) + γ A γ P γ A + γ P (cid:19) = exp (cid:18) − γ P (cid:18) x + 12 − γ A γ P γ A + γ P (cid:19)(cid:19) . imilarly, his expected utility from the second Agent would be: E [ | U P ( X ˆ a − κ (ˆ a )) | ˆ γA ˆ γA + γP ] ˆ γA + γP ˆ γA = exp (cid:18) − γ P (cid:18) x + 14 − ˆ γ A γ P ˆ γ A + γ P (cid:19)(cid:19) . Therefore in order to maximize his utility, the Principal should employ the first Agent if − γ A γ P γ A + γ P > − ˆ γ A γ P ˆ γ A + γ P , and he should employ the second Agent if not. Note that for more general cost functionsand a Gaussian B , the Principal should employ the first Agent if a − κ ( a ) − γ A γ P γ A + γ P > ˆ a − ˆ κ (ˆ a ) − ˆ γ A γ P ˆ γ A + γ P . Remark 3.1.
In the same vein as the Propositions above, one may also exploit the conse-quences to Reverse-HÃűlder decomposition of the Principal’s utility to compare the optimalRisk-Sharing contract to the optimal Moral Hazard contract, under CARA utilities.
In the following Section, we illustrate the versatility of our framework and our results.Indeed so far we have worked with a single period model with a general production processand a general action cost function. We obtain existence, uniqueness and characterization ofthe Risk-Sharing solution, and some analysis on the model. In the following, we discuss onsome extensions to this setting. One first important setting is the continuous time setting.
We specify the model of interest which is simply a continuous time version of the one stud-ied previously. More precisely we consider one Principal and one Agent. The Principalprovides a single cash flow (wage) W at maturity (denoted T ) to the Agent and requires inexchange an action a = ( a t ) t ∈ [0 ,T ] (that is completely monitored by the Principal) contin-uously in time according to the random fluctuations of the wealth of the firm. A contractwill once again be a pair (wage,action) = ( W, a ) .14e start with the probabilistic structure that is required to define the random fluctuationsof the wealth of the Principal. Let (Ω , F , P ) be a probability space on which a stochasticprocess B := ( B t ) t ∈ [0 ,T ] is defined with its natural and completed filtration F := ( F t ) t ∈ [0 ,T ] .The only requirement on this stochastic process is that sup t ∈ [0 ,T ] E [exp ( qB t )] < + ∞ , ∀ q ∈ R ∗ . This allows for the use of CARA utilities, and is verified for example for Brownian motion.We denote by E [ · ] the expectation with respect to the probability measure P .The Agent will be asked to perform an action a continuously in time, according to theperformances of the firm. Hence we introduce the set P of F -predictable stochastic pro-cesses a = ( a t ) t ∈ [0 ,T ] and the set of actions is given as : H := (cid:26) a = ( a t ) t ∈ [0 ,T ] ∈ P , s.t. E (cid:20) exp (cid:18) q Z T | a t | dt (cid:19)(cid:21) < + ∞ , ∀ q > (cid:27) . As we will work with exponential preferences for the Agent and the Principal, we requireso-called "exponential moments" on the actions and wages. As mentioned previously, thisis a technical assumption. Given a in H , the wealth of the principal at any intermediatetime t between and the maturity T is given by : X at = x + Z t a s ds + B t , t ∈ [0 , T ] , P − a.s., (4.1)where x ∈ R is a fixed real number. For any action a in H , we set F X a := ( F X a t ) t ∈ [0 ,T ] the natural filtration generated by X a . In particular, we are interested in the set of F X a T -measurable random variables which provides the natural set for the wage W paid by thePrincipal to the Agent. More precisely, we set W := (cid:8) F X a T − measurable random variables W, E [exp( q W )] < + ∞ , ∀ q ∈ R ∗ (cid:9) . The fact that we ask for so-called finite exponential moments of any (positive, respectivelynegative) order for the action (respectively for the wage) is purely technical. As we willsee, the optimal contract will satisfy these technical assumptions.The cost of effort for the Agent is once again modeled by a convex, continuous and non-decreasing function κ defined on R + . As explained in the introduction for the single periodproblem, the Agent will accept a contract ( W, a ) in C if and only if the following participa-tion constraint (PC) is satisfied : E (cid:20) U A (cid:18) W − Z T κ ( a t ) dt (cid:19)(cid:21) ≥ U A ( y ) , (4.2) R ∗ := R \ { } y is a given real number, κ : R + → R models the cost of effort for the agent and isas discussed above, and U A ( x ) := − exp( − γ A )( x ) with γ A > the risk aversion parameterfor the Agent. From now on we assume that parameters ( y, γ A ) are fixed. With thesenotations at hand, we can state the Principal’s problem which writes down in term of aclassical first-best problem as follows: sup ( W,a ) ∈ C adm E [ U P ( X aT − W )] , (4.3)where U P ( x ) := − exp( − γ P x ) with γ P > fixed and where C adm := { ( W, a ) ∈ C × H , (4 . is in force } is the set of admissible contracts satisfying the participation constraint (4.2). We notethat this continuous time problem may be dealt with using tools from the field of optimalstochastic control. Indeed, one may exploit Equation (4.2) to obtain a parametrization ofall the wages satisfying the Participation Constraint for a given action process ( a t ) t ∈ [0 ,T ] .This allows us to rewrite Problem (4.3) as a standard optimal control problem and oneway of then proving existence involves using verification results as long as the requiredhypotheses are verified.In the following we provide the continuous time counterpart to Theorem 2.2. It providesexistence, uniqueness and characterization under quite general hypotheses (notably for ageneral process ( B t ) t ∈ [0 ,T ] ). We believe that this theorem brings to light the structure ofthe underlying problem and thus complements possible existence and uniqueness resultsexploiting optimal control. Theorem 4.1 (Existence, uniqueness and characterization) . Consider the contract ( W ∗ , a ∗ ) defined by setting : a ∗ t = κ ∗ (1) for any t in [0 , T ] and W ∗ = γ P γ P + γ A X a ∗ T + β ∗ , where the constant β ∗ is worth : β ∗ := y + T κ ( κ ∗ (1)) − γ P γ A + γ P ( x + T κ ∗ (1)) − γ A ln (cid:18) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A Z T dB t (cid:19)(cid:21)(cid:19) . Then ( W ∗ , a ∗ ) both satisfies and saturates the participation constraint. Furthermore for any ( W, a ) in C adm E [ U P ( X aT − W )] ≤ E h U P (cid:16) X a ∗ T − W ∗ (cid:17)i , ith equality only for ( W ∗ , a ∗ ) . ( W ∗ , a ∗ ) is therefore the unique optimal contract in thecontinuous-time Risk-Sharing problem. Finally, the Principal’s optimal utility is : E h U P (cid:16) X a ∗ T − W ∗ (cid:17)i = U P ( x − y − γ P T e κ (1)) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A Z T dB t (cid:19)(cid:21) γA + γPγA . Proof.
The proof closely mirrors that of Theorem 2.2 with the calculations done in contin-uous time.It is important to note that, as a consequence to this Theorem, the analysis conductedin Section 3 nicely generalizes to this setting and allows once again for the comparison ofcontracts. In this continuous time setting one may even easily analyze the effect of choosinga non-constant action a . Remark 4.1.
In the widely studied case where B is in fact a standard Brownian motion,Theorem 4.1 applies and gives existence and uniqueness. The optimal β ∗ has the expression: β ∗ := T γ A | γ P | | γ A + γ P | + y + T κ ( κ ∗ (1)) − γ P γ A + γ P ( x + T κ ∗ (1)) , and the Principal’s expected utility is : E h U P (cid:16) X a ∗ T − W ∗ (cid:17)i = U P ( x − y ) exp (cid:18) γ P T (cid:18) − e κ (1) + γ P γ A γ A + γ P ) (cid:19)(cid:19) . We therefore complement the characterization work of Muller in [15] by providing an exis-tence and uniqueness proof, as well as providing an alternative characterization method.
Remark 4.2.
The Principal-Agent problem has also been analyzed in the setting where aPrincipal employs several different Agents. Note that for CARA Agents this framework alsoworks.
We now turn to the case of a risk neutral Principal which can be analyzed as a limit of thecase of a CARA Principal, and we do so in the following.
The analysis provided throughout this paper concerns a Principal and an Agent who areboth risk averse with the risk aversion modeled through the CARA utility functions. Infact, the key to this paper is the exponential properties of the CARA utilities. However,an important case in the literature is that of a risk neutral Principal who wishes to employa risk averse Agent. More precisely, if we for example set ourselves in the discrete-timesetting, the Risk-Sharing problem (2.1) becomes :17 up ( W,a ) ∈C adm E [ X a − W ] , (4.4)where we use the same notations as previously (in particular the Participation Constraintis in force for the Agent with utility function U A ( x ) = − exp( − γ A x ) ). Since our ReverseHÃűlder approach relies on the structure of functions U P and U A , we cannot carry it di-rectly in the risk neutral case. However, as it is well-known, the risk neutral frameworkcan be seen as a limit case with formally γ P = 0 by rescaling the mapping U P to become ˜ U P ( x ) := − exp( − γ P x ) − γ P and by letting γ P go to . Hence, we can use our approach with ˜ U P and U A to obtain existence of an optimum and its characterization in the risk-neutralcase.Consider a contract ( W, a ) that satisfies the (PC). Then by Lemma 5.1 (in Section 5.4), E [ X a − W ] = E (cid:20) lim γ P → ˜ U P ( X a − W ) (cid:21) = lim γ P → E h ˜ U P ( X a − W ) i = lim γ P → γ − P ( E [ U P ( X a − W )] + 1) ≤ lim γ P → γ − P (cid:16) E [ U P ( X a ∗ − W ∗ )] + 1 (cid:17) , according to (ii) of Theorem 2.2. Using the explicit computation of the upper bound’svalue, we have that E [ X a − W ] = E (cid:20) lim γ P → ˜ U P ( X a − W ) (cid:21) ≤ lim γ P → γ − P U P ( x − y + e κ (1)) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) γA + γPγA + 1 = x − y + e κ (1) . So we have given the upper bound x − y + e κ (1) to the value problem of the Risk NeutralPrincipal. An explicit computation gives that this upper bound can be attained by choosingthe contract ( W ∗ RN , a ∗ RN ) with a ∗ RN = κ ∗ (1) and W ∗ RN = y + κ ( κ ∗ (1)) , which is formally the optimal contract found in in Theorem 2.2 with γ P = 0 . The optimalparameters have economic meaning : the Principal is neutral to risk and is thus willing togive a fixed wage to his Agent regardless of the performance of the output process. We note18hat in this case, the Risk-Sharing structure of the problem disappears and the Principalcarries all of the risk.We thus have an existence proof for the risk-neutral case, along with a possible charac-terization of the optimum. Of course we may perform the computation of the risk-neutraloptimum more directly without exploiting the risk-averse optimum provided through themethod used in this paper. However, the fact that the risk-neutral case may be seen as alimit of the risk-averse case allows for example for the extension of the results of Section 3to neutral settings. We give such an extension in the following. Proposition 4.1.
Let ( W ∗ , a ∗ ) be the risk-averse optimum for ˆ U P ( x ) = − exp( − ˆ γ P x ) givenin Theorem 2.2 and let ( W, a ) be in C adm . Then : E [ X a − W ] E h ˆ U P ( X a ∗ − W ∗ ) i ≥ lim γ P → γ − P E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA E (cid:20)(cid:12)(cid:12)(cid:12) ˆ U P ( X a ∗ − κ ( a ∗ )) (cid:12)(cid:12)(cid:12) γAγA +ˆ γP (cid:21) γA +ˆ γPγA × U P ( y )ˆ U P ( y ) . Proof.
See Section 5.5.This Proposition is an asymptotic form of Proposition 3.1 for comparison between a risk-neutral and risk-averse Principal : we obtain a decomposition dependent on an action ratioand a participation constraint ratio. Note however that the risk-neutral utility is not signedand this inequality is thus more difficult to exploit. Finally, note that this analysis alsoholds in continuous time.
In this Section we collect the proofs of the technical results we made use of to proceed withour analysis.
Proof.
We fix a in R + and prove each item of the Theorem.(i) For the first result, we express the (expected) utility of the Principal in terms of theone of the Agent. We have : E [ U P ( X a − W )]= E [ U P ( X a − κ ( a )) × exp ( γ P ( W − κ ( a )))]= E h U P ( X a − κ ( a )) × | U A ( W − κ ( a )) | − γPγA i . (5.1)19e wish to extract the Agent’s utility from this expression and obtain at least aninequality. To do so, we need some kind of Hölder inequality. However the classicalHölder inequality cannot be applied for two reasons : first the exponent − γ P γ A of theutility of the Agent is negative; and then the negativity of the mapping U P callsfor the use of a Hölder inequality in the reverse direction. These two features aretaken into account in the so-called Reverse Hölder inequality which can be seenas a counterpart to the classical Hölder inequality and given in Proposition 2.1. Inparticular, we wish to use Item (i). More precisely, let : F := U P ( X a − κ ( a )) , G := | U A ( W − κ ( a )) | − γPγA . (5.2)Note naturally that these two random variables depend on the contract ( W, a ) underinterest.We wish to apply Reverse-Hölder to F and G with some exponent p that we calibrateso that | G | − p − = | U A ( W − κ ( a )) | ; which immediately gives p = 1+ γ P γ A = γ A + γ P γ A > .We thus immediately obtain: E h | F | p i = E h | U P ( X a − κ ( a )) | γAγP + γA i , E h | G | − p − i = − E [ U A ( W − κ ( a ) dt )] . Applying (i) of Proposition 2.1 to F and G with this particular choice of p in Relation(5.1) gives our result : E [ U P ( X aT − W )]= − E [ | F G | ] ≤ E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA . (5.3)(ii) We first prove that (ii ' ) is equivalent to (ii '' ). This involves finding an equality con-dition for (5.3). Through (ii) of Proposition 2.1, Inequality (5.3) is an equality if andonly the contract ( a, W ) is such that there exists a positive constant α such that therandom variables F and G defined in (5.2) enjoys : | F | = α | G | − pp − . By definition of F , G and p = γ A + γ P γ A this condition reads as : | U P ( X a − κ ( a )) || U A ( W − κ ( a ) dt ) | − ( γP + γA ) γA = α | U P ( X aT − W ) || U A ( W − κ ( a )) | = α, which is equivalent to U ′ P ( X aT − W ) U ′ A (cid:16) W − R T κ ( a t ) dt (cid:17) = γ P γ A α. Setting α to γ P γ A α we obtain our result.We now prove that (ii '' ) is equivalent to (ii ''' ). When (ii '' ) holds, ( W, a ) satisfies (4.2)and we have the following series of implications where α is a positive constant : U ′ P ( X a − W ) U ′ A ( W − κ ( a )) = α ⇒ ( γ P + γ A ) W − γ P X a − γ A κ ( a ) = ln (cid:18) α γ A γ P (cid:19) ⇒ W = γ P γ P + γ A X a + γ A γ P + γ A κ ( a ) + ln (cid:18) α γ A γ P (cid:19) , ⇒ W = γ P γ P + γ A X a + β, where β = γ A γ P + γ A κ ( a ) + ln (cid:16) α γ A γ P (cid:17) .Conversely, let us suppose that (ii ''' ) holds. Then ( W, a ) where W = γ P γ P + γ A X a + β satisfies (4.2) and we have that : U ′ P ( X a − W ) U ′ A ( W − κ ( a )) = exp (( γ P + γ A ) β + γ A κ ( a )) ∈ R ∗ + . Let a ∈ R + . We have E h | U P ( X a − κ ( a )) | γAγA + γP i = E h exp ( − γ P ( X a − κ ( a ))) γAγA + γP i E h exp ( − γ P ( x + ( a − κ ( a )) + B )) γAγA + γP i = exp (cid:18) − γ P γ A γ A + γ P x + Φ( a ) (cid:19) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) where c Φ( c ) := − γ P γ A γ A + γ P ( c − κ ( c )) . Note that the mapping Φ is convex on R + , and letting a ∗ := κ ∗ (1) , Φ( c ) ≥ Φ( a ∗ ) = − γ P γ A e κ (1) γ A + γ P , ∀ c ≥ . So, E h | U P ( X a − κ ( a )) | γAγP + γA i ≥ exp (cid:18) − γ P γ A γ A + γ P ( x + ˜ κ (1)) (cid:19) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21) , and we thus deduce our result. We consider the contract ( W ∗ , a ∗ ) defined by setting : a ∗ := κ ∗ (1) and W ∗ = γ P γ P + γ A X a ∗ + β ∗ ,β ∗ := y + κ ( κ ∗ (1)) − γ P γ A + γ P ( x + κ ∗ (1)) − γ A ln (cid:18) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21)(cid:19) . We first study the participation constraint to verify the admissibility of such a contract.We have that : E [ U A ( W ∗ − κ ( a ∗ ))] = E (cid:20) U A (cid:18) γ P γ P + γ A X a ∗ + β ∗ + κ ( κ ∗ (1)) (cid:19)(cid:21) = E (cid:20) U A (cid:18) γ P γ P + γ A ( x + κ ∗ (1) + B ) + β ∗ − κ ( κ ∗ (1)) (cid:19)(cid:21) = U A ( y ) E (cid:20) exp (cid:18) − γ A γ P γ P + γ A B − ln (cid:18) E (cid:20) exp (cid:18) − γ P γ A γ P + γ A B (cid:19)(cid:21)(cid:19)(cid:19)(cid:21) = U A ( y ) . W ∗ belongs to W ( a ∗ ) . According to Item (ii) of Theorem 2.1, the contract satisfiesthe Borch Rule. Furthermore, it is of the form ( W ∗ , κ ∗ (1)) where W ∗ saturates the Partic-ipation Constraint. It follows that the equality conditions to reach the upper bound of thePrincipal’s Expected Utility are verifies and we have that for any a in R + and any W in W ∗ ( a ) , E [ U P ( X a − W )] ≤ E h U P (cid:16) X a ∗ − W ∗ (cid:17)i = U P ( x − y ) exp (cid:18) γ P T (cid:18) − e κ (1) + γ P γ A γ A + γ P ) (cid:19)(cid:19) . We deduce that ( W ∗ , a ∗ ) is the optimal contract for the first-best Principal-Agent problem. Lemma 5.1.
Let ( W, a ) be an admissible contract in C . The sequence of random variables (cid:16) ˜ U P ( X a − W ) (cid:17) <γ P < is uniformly integrable. And so : E [ X a − W ] = E (cid:20) lim γ P → ˜ U P ( X a − W ) (cid:21) = lim γ P → E h ˜ U P ( X a − W ) i . Proof.
The second part of the statement is a consequence of the uniform integrability (UI)and of the fact that the identity mapping is the limit (as γ P goes to ) of ˜ U P . So we focuson the UI property and apply de la Vallée-Poussin criterion. We have : sup <γ P < E h | ˜ U P ( X a − W ) | i = γ − P sup <γ P < E (cid:2) | exp( − γ P ( X a − W )) − | (cid:3) = sup <γ P < E (cid:2) | ¯ X | | exp( − γ P ¯ X ) | (cid:3) , where ¯ X is a random point between and X a − W (using mean value theorem). ByCauchy-Schwarz’s inequality we have, sup <γ P < E h | ˜ U P ( X a − W ) | i ≤ E (cid:2) | ¯ X | (cid:3) / sup <γ P < E (cid:2) exp( − γ P ¯ X ) (cid:3) / . As | ¯ X | ≤ | X a − W | , P -a.s., we have that E (cid:2) | ¯ X | (cid:3) < + ∞ . Regarding the second term, sup <γ P < E (cid:2) exp( − γ P ¯ X ) (cid:3) sup <γ P < (cid:0) P (cid:2) ¯ X ≥ (cid:3) + E (cid:2) exp( − γ P ¯ X ) ¯ X< (cid:3)(cid:1) ≤ E (cid:2) exp( − R ¯ X ) ¯ X< (cid:3) ≤ E [exp(4 R | X a − W | )] < + ∞ . Let ( W ∗ , a ∗ ) be the risk-averse optimum for ˆ U P ( x ) = − exp( − ˆ γ P x ) given in Theorem 2.2and let ( W, a ) be in C adm . Then as the expected utility function is negative, and using theReverse-Holder inequality we obtain : E [ X a − W ] E h ˆ U P ( X a ∗ − W ∗ ) i = lim γ P → γ − P ( E [ U P ( X a − W )] + 1) E h ˆ U P ( X a ∗ − W ∗ ) i ≥ lim γ P → γ − P ( E [ U P ( X a − W )]) E h ˆ U P ( X a ∗ − W ∗ ) i ≥ lim γ P → γ − P E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA ! E (cid:20)(cid:12)(cid:12)(cid:12) ˆ U P ( X a ∗ − κ ( a ∗ )) (cid:12)(cid:12)(cid:12) γAγA +ˆ γP (cid:21) γA +ˆ γPγA × E [ U A ( W ∗ − κ ( a ∗ ))] − ˆ γPγA = lim γ P → γ − P E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × E [ U A ( W − κ ( a ))] − γPγA ! E (cid:20)(cid:12)(cid:12)(cid:12) ˆ U P ( X a ∗ − κ ( a ∗ )) (cid:12)(cid:12)(cid:12) γAγA +ˆ γP (cid:21) γA +ˆ γPγA × ˆ U P ( y ) ≥ lim γ P → γ − P E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA × U P ( y ) ! E (cid:20)(cid:12)(cid:12)(cid:12) ˆ U P ( X a ∗ − κ ( a ∗ )) (cid:12)(cid:12)(cid:12) γAγA +ˆ γP (cid:21) γA +ˆ γPγA × ˆ U P ( y )= lim γ P → γ − P E h | U P ( X a − κ ( a )) | γAγA + γP i γA + γPγA E (cid:20)(cid:12)(cid:12)(cid:12) ˆ U P ( X a ∗ − κ ( a ∗ )) (cid:12)(cid:12)(cid:12) γAγA +ˆ γP (cid:21) γA +ˆ γPγA × U P ( y )ˆ U P ( y ) . Conclusion
This paper uses the Reverse Hölder inequality to derive a new approach to the Risk-SharingPrincipal-Agent problem. Through a specific decomposition of the Principal’s expectedutility (that relies of the multiplicative property of exponential utility functions) we areable to extract the participation constraint in its expectation form. We are then able to toprove existence and uniqueness of the optimal Risk-Sharing plan, whilst also characterizingthe plan whilst and making the Borch rule appear. We note that this analysis allowsfor general hypothesis on the underlying model and works very similarly in both discreteand continuous time settings. It also extends to the risk-neutral case. As a by-productof this work, we are able to analyze the effect of enforcing a sub-optimal action and alsoprovide some insight into the parameters affecting a Principal’s choice between two Agents.Another natural extension to this analysis may be that of choosing a sub-optimal wage.Such a choice may make sense for many reasons such as wanting to enforce limited liability,and is the topic of ongoing research by the authors.
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