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Regulation of a Risk Averse Firm
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
25, 149᎐173 Ž1998.
GA980639
Regulation of a Risk Averse Firm Jean-Jacques Laffont and Jean-Charles Rochet* Institut Uni¨ ersitaire de France, GREMAQ᎐IDEI, Uni¨ ersite´ des Sciences Sociales, F-31042 Toulouse cedex, France Received October 14, 1996
We extend the Laffont᎐Tirole regulation model to the case of risk-averse firms. Our main results are: 1. The impact of risk aversion is to shift the optimal contract toward a cost-plus contract. 2. As compared with the risk-neutral case, distortions are greater and informational rents are smaller. 3. For high levels of risk aversion, the optimal contract involves cost ceilings and the less efficient firms are bunched together. Journal of Economic Literature Classification Numbers: D8, L5. 䊚 1998 Academic Press
Key Words: regulation; risk; bunching; cost-plus contracts
1. INTRODUCTION Principal᎐agent relationships typically involve three types of Žpartially conflicting. objectives: risk sharing must be organized, appropriate effort levels must be induced Ž moral hazard., and truthful information must be elicited Ž ad¨ erse selection.. The literature has in general addressed these issues separately. For example, the moral hazard literature ŽPauly, 1968; Zeckhauser, 1970; Spence and Zeckhauser, 1971; Mirrlees, 1975; Holmstrom ¨ 1979; Shavell, 1979; Grossman and Hart, 1983, among others. has focused on the trade-off between incentives for effort and risk sharing. The main conclusion is that agents must bear more risk than in the first best, to induce a reasonable effort level. Holmstrom ¨ and Milgrom Ž1987. have shown further that, in the case of exponential utility functions and repeated moral hazard situations, the optimal contracts could be taken to be linear. When these assumptions are valid, their model is particularly convenient for the study of risk sharing with moral hazard.1 * E-mail: [email protected]. 1 See, for example, their study of multitasks ŽHolmstrom ¨ and Milgrom, 1987.. 149 0899-8256r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
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On the other hand, the adverse selection literature has studied how a principal should structure his offer of contracts to a privately informed agent, to optimize his own objective under the interim participation constraint of the agent. The optimal contract is in general a nonlinear contract: it can be finely characterized under the so-called Spence᎐Mirrlees conditions, which ensure that only local incentive constraints are binding. An informational rent must be given up by the principal to all types except the less efficient type. Moreover, all agents’ actions are distorted to mitigate the informational rents, except for the most efficient type Žno distortion at the top result; see Mirrlees, 1971; Spence, 1974; Mussa and Rosen, 1978; Green and Laffont, 1979; Baron and Myerson, 1982; Guesnerie and Laffont, 1984, among others.. In the case of risk-neutral agents, Laffont and Tirole Ž1986. have considered simultaneously adverse selection and moral hazard, and have characterized the optimal contract that trades off these two problems. Fixed-price contracts are good for moral hazard but bad for adverse selection. Cost-plus contracts are good for adverse selection and bad for moral hazard. The optimal contract is a nonlinear contract that can sometimes be implemented by a menu of linear sharing rules in which self selection of types is obtained. Moreover, the optimal contract has the same qualitative features as in the pure adverse selection case. What is clearly missing is a simultaneous treatment of the three problems Žrisk sharing, effort inducement, information elicitation ., which must be tackled together in many contractual relationships: sharecropping, managerial incentives, insurance problems, regulation of small firms, etc. The purpose of this paper is to make some progress in this problem by considering a special case that enables us to characterize quite fully the optimal contract. We study a situation with both adverse selection and moral hazard for a risk-averse agent, in which the contract is offered and signed before the agent knows his type. Since the principal is risk-neutral and the agent risk-averse, the optimal contract will involve some insurance Žof the agent by the principal. against this interim risk. Therefore we obtain a novel form of the insurance ᎐incentive trade-off: informational rents become more costly to the principal because they induce variability in the ex post utility of the agent. This is the main intuition behind our results: risk aversion brings the optimal contract closer to a cost-plus contract, with greater distortions and lower rents. Moreover, bunching appears for high levels of risk aversion. Of course, the type of risk we study here is peculiar. It would also be interesting to introduce an ex post risk, i.e., a risk that materializes after
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the effort decision has been made. The basic idea of our modeling is, instead, to maintain a non-noisy relationship between the variables of adverse selection, moral hazard, and the action to be implemented. This allows us to eliminate the moral hazard variable 2 and to reduce technically the problem to an adverse selection problem with risk-averse firms, related to the one studied by Salanie ´ Ž1990.. The model is presented in Section 2. Section 3 provides an analysis of the two-type case. When there is a continuum of types, the problem is much more complex: it is solved for a CARA utility function in Section 4. The special case of the Baron᎐Myerson model is dealt with in Section 5. Examples and concluding comments are gathered in Section 6.
2. THE MODEL Our model is a simple extension of the regulation model of Laffont᎐ Tirole Ž1986., to a situation in which risk is introduced. An utilitarian regulator wishes to realize a public project with social value S. A single firm can realize the project, at a cost C s  y e, where  is an efficiency parameter and e is an effort level with a disutility Ž e . for the management of the firm Žwe assume ⬘ ) 0, ⬙ ) 0, G 0.. This disutility is measured in monetary terms. We assume that the cost C is observable ex post by the regulator Žbut not  and e separately., and we make the accounting convention that it is paid by the regulator. Let t be the net monetary transfer from the regulator to the firm. The firm is supposed to be owned and managed by the same agent, who is risk-averse. Accordingly, this agent obtains Žex post. a utility level: uŽ t y Ž e . . . We assume: u⬘ ) 0, u⬙ F 0. A natural way to introduce some element of risk into this model would be to add a noise term ˜ in the cost, which would become C˜ s  y e q ˜ . Since the cost is the only variable that the regulator observes, we would be in a situation of noisy observation, analogous to the one studied by 2
This follows Laffont᎐Tirole Ž1986.. However, in the risk-neutral case, their results extend immediately to noisy relationships. This is not the case here.
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Caillaud et al. Ž1992. under the assumption of a risk-neutral agent. However, with a risk-averse agent, the simultaneous presence of moral hazard, adverse selection, and noise would make the problem intractable. We have chosen instead to introduce an interim risk about the adverse selection parameter, which brings along two simplifications: the observation of costs remains non-noisy, and the incentive compatibility constraints are unaffected. The timing is as follows: At s 0 the contract is offered to the firm. At s 1, nature chooses  . This parameter is privately observed by the firm, which can then exit the relationship. At s 2, and if it has not exited, the firm chooses the effort e, the regulator observes the cost C s  y e, and the contract is implemented. 䢇 䢇
䢇
Notice that when the contract is offered Žat s 0., the information is symmetric. We also assume that the firm can renege on the contract at s 1, which necessitates the introduction of ex post rational individuality constraints. 3 We are now going to investigate how the introduction of the firm’s risk aversion affects the characteristics of the optimal contract derived by Laffont and Tirole Ž1986.. We start with the simple two-type case.
3. THE OPTIMAL CONTRACT IN THE TWO-TYPE CASE We assume in this section that  can only take two values,  and  Žthe difference  y  is denoted ⌬  .. Ex ante, it is common knowledge that s PrŽ  s  . Žtherefore 1 y s PrŽ  s  ... From the revelation principle, we can restrict ourselves to direct revelation mechanisms  ª Ž t Ž  ., C Ž  .., and we use the following notations: tŽ  . s t,
C Ž  . s C.
tŽ  . s t,
C Ž  . s C.
Similarly, the effort and profit levels are denoted as follows:
3
e s  y C,
s t y Ž e. ,
e s  y C,
s t y Ž e. ,
For simplicity, we assume that these ex post IR constraints imply the ex ante IR constraint that we neglect below.
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The firm’s incentive constraints can be written: t y Ž  y C. G t y  y C ,
Ž 3.1.
t y Ž  y C. G t y Ž  y C..
Ž 3.2.
ž
/
An equivalent way to express these incentive constraints is to write that the difference in profits between one type of firm and the other is at least equal to the difference in costs of effort when the first type ‘‘mimics’’ the second type. Using the notation Ž x . s Ž x . y Ž x y ⌬  ., Ž3.1. and Ž3.2. can thus be rewritten as
G q Ž  y C.,
Ž 3.3.
G y Ž  y C..
Ž 3.4.
The firm’s ex post individual rationality constraints are
G 0,
Ž 3.5.
G 0.
Ž 3.6.
The regulator wishes to maximize expected social welfare under incentive and individual rationality constraints. Let Ž1 q . be the social cost of public funds. Assuming for simplicity that the project is sufficiently valuable to be always realized, consumers’ expected utility is S y Ž 1 q . Ž t q C . q Ž 1 y . Ž t q C . .
Ž 3.7.
The term between square brackets is nothing but the total expected transfer to the firm. We aggregate consumers’ utility with the certainty equivalent of the manager’s profit Ž CE s uy1 w uŽ . q Ž1 y . uŽ .x., to obtain expected social welfare 4 W. Substituting t s q Ž  y C ., we obtain W s S y Ž 1 q . q Ž  y C . q C
ž
/
qŽ 1 y . q Ž  y C . q C
ž
q uy1 Ž u Ž . q Ž 1 y . u Ž . . .
/ Ž 3.8.
4 We consider only deterministic schemes. From Laffont᎐Tirole Ž1993, pp. 119᎐120., we know that stochastic schemes are dominated as long as G 0, and the welfare function W is concave in , , C, C. In our context this is equivalent to saying that the certainty equivalent operator is concave. This is satisfied, for instance, when u is CARA Žsee Section 4..
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The optimal regulation is obtained by maximizing W under the constraints Ž3.3. to Ž3.6.. It is characterized as follows: PROPOSITION 1. When there are only two types, and if the ¨ alue S of the project is large enough, the optimal deterministic regulation entails: 1. An efficient le¨ el of effort for the more efficient type  , i.e., ⬘Ž e . s 1. 2. An incremental rent for the more efficient type Ž i.e., y . equal to Ž . e. 3. An inefficient le¨ el of effort for the less efficient type  , characterized by ⬘Ž e . s 1 y ⬘Ž e . 1q 1y y
⬘ Ž e .
Ž 1 y . Ž 1 q .
1y
u⬘ Ž q Ž e . .
Ž 3.9.
u⬘ CE Ž , e .
where CE Ž , e . s uy1 Ž Ž 1 y . u Ž . q u Ž q Ž e . . . . 4. When 1 q G Ž u⬘Ž Ž e .. q Ž1 y . u⬘Ž0..ru⬘w CEŽ0, e .x, the less efficient firm  has no rent Ž Case 1., as in the risk-neutral case. Otherwise the  firm has a positi¨ e rent Ž Case 2.. Proof. To solve for the maximization of Ž3.8. under the constraints Ž3.3. to Ž3.6., we observe that Ž3.3. and Ž3.6. imply Ž3.5., and we momentarily ignore Ž3.4.. We will check later that it is satisfied by the solution obtained. We are therefore left with two constraints, Ž3.3. and Ž3.6.. Contrary to the case of risk-neutral firms, the individual rationality constraint of the less efficient type, Ž G 0., is not necessarily binding at the optimum. However,
⭸W ⭸
s y Ž 1 q . q
u⬘ Ž . u⬘ Ž CE .
.
Therefore, as ) from Ž3.3., CE - , which implies that ⭸ Wr ⭸ - 0 and that the constraint Ž3.3. is always binding. Substituting Ž3.3. in Ž3.8., we have a new expression of social welfare, which now depends only on three variables, , C, and C: W Ž , C, C . s S y Ž 1 q . q Ž  y C . q Ž  y C . q C
ž
qŽ 1 y . q Ž  y C . q C
ž
/
q uy1 u q Ž  y C . q Ž 1 y . u Ž . ,
ž ž
/
which is to be maximized under the only constraint Ž3.6..
/
/
Ž 3.10.
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We must then distinguish two cases: Case 1. Ž3.6. is binding. Then s 0 and s Ž  y C .. The necessary condition is, at s 0,
⭸W ⭸
F0m
u⬘ Ž . q Ž 1 y . u⬘ Ž . u⬘ Ž CE .
F Ž 1 q . .
Case 2. Ž3.6. is not binding. If this case occurs, then ⭸ Wr⭸ s 0. The value ) 0 is defined by the smallest 5 non-negative solution of ⭸ Wr ⭸ s 0. Maximizing 6 W with respect to C and C, we obtain the desired result. B Remark 1. Contrary to the risk-neutral case, the inefficient firm may have a rent ŽCase 2.. Because of risk aversion, a marginal increase in profit at s 0 may increase the certainty equivalent by more than Ž1 q ., the cost of public funds. Remark 2. The second, more important observation is that the distortion in the inefficient firm’s effort level has an additional term that lowers this effort level even further. Therefore we move toward a scheme even closer to a cost-plus scheme, i.e., a less powerful incentive scheme. The intuition is simply the following. Since the firm is risk-averse, the certainty equivalent of profit CE is lower than the expectation of profit. Therefore it is as if the weight of the firm in the social welfare function were less than 1 Žsay 1 y ⑀ .. Since the cost of transfers to the firm remains Ž1 q ., the weight of the firm’s rent in the social welfare function is yŽ q ⑀ .. So it is as if the rent were socially more costly; hence the optimal regulation entails a greater distortion from efficiency, to decrease the rent further. Remark 3. An interesting example of utility function is the exponential utility: uŽ x . s Ž1 y exp y x .r , which has the merit of neutralizing income effects. In that case, s 0 and u⬘Ž CE . s u⬘Ž Ž e .. q Ž1 y . u⬘Ž0.. The equation giving the inefficient firm’s effort level takes the form
⬘Ž e . s 1 y
1q 1y
= 1y 5 6
⬘Ž e . y
⬘ Ž e .
Ž 1 y . Ž 1 q .
1
q Ž 1 y . exp Ž Ž e . .
.
Because of the certainty equivalent, W is not necessarily concave in . We assume that interior maxima exist Ži.e., with positive effort levels..
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Straightforward differentiation of this expression gives, then, that derd - 0 for an interior maximum, since ⬘ ) 0. As risk aversion increases, the optimal contract moves toward a cost plus contract.7
4. THE OPTIMAL CONTRACT IN THE CONTINUUM CASE We consider the same model as in Section 3, but with a continuum w  ,  x of values for the adverse selection parameter. Let t Ž C . be the net transfer to the firm when the cost equals C s  y e. We define the profit function as
Ž  . s max Ž  , C . C
where Ž  , C . s t Ž C . y Ž  y C . . Ž 4.1.
As explained in Section 3, risk aversion does not affect the incentive compatibility conditions that we obtain after substituting e s  y C. The first-order and second-order incentive compatibility conditions Žwhich are sufficient because ⬙ ) 0 and because the Spence᎐Mirrlees condition ⭸ 2r⭸ ⭸ C ) 0 is fulfilled. are, for all  ,
⬘ Ž  . s y ⬘ Ž e Ž  . . ,
Ž 4.2.
C⬘ Ž  . s 1 y e⬘ Ž  . G 0.
Ž 4.3.
and
By integration of Ž4.2., we obtain
Ž . sŽ. q
H ⬘ Ž e Ž x . . dx.
Ž 4.4.
Therefore the Žex post. individual rationality condition Ž Ž  . G 0 for all  . can be transformed as usual into the simpler condition:
Ž  . G 0. 7
Ž 4.5.
We conjecture that the general equilibrium effect goes in the same direction: if all regulated firms become more risk-averse, the cost of the rents in the social welfare function is likely to be increased, which should also increase the shadow cost of public funds, inducing a further move toward a cost-plus contract.
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Furthermore, from Laffont and Tirole Ž1993., we know that the optimal contract t*Ž C . will be implementable by a menu of linear schedules if t*Ž C . is convex. From Ž4.3. and ⬙ G 0, this condition is satisfied if and only if 8 e⬘Ž  . F 0. Let CEŽ . denote the certainty equivalent of . Taking the same objective function for the regulator as in Section 2, we can write his optimization problem as max W s S y Ž 1 q . E q Ž e . q  y e q CE Ž . under the constraints Ž 4.2. , Ž 4.3. , Ž 4.5. .
Ž 4.6.
To eliminate income effects, we choose the exponential specification ŽCARA utility function. uŽ x . s Ž1 y exp y x .r , which gives CEŽ . s 1 y log E Ž ey .. As in Section 3 Žand see Lemma 1., this implies that the individual rationality constraint of the least efficient firm is always binding at the optimum. LEMMA 1. With a CARA utility function, constraint Ž4.5. is always binding at the optimum. Proof. Suppose by contradiction that Ž  . ) 0 and decrease it by ⑀ . Then by Ž4.4. Ž  . uniformly decreases by ⑀ . Therefore E Ž . and CEŽ . decrease by ⑀ but, because ) 0, the total effect on W is positive. B Under the monotone hazard rate condition Ž drd  Ž F Ž  .rf Ž  .. ) 0., we know that in the case of risk neutrality Ž s 0., the optimal contract is separating or equivalently involves no bunching Žsee Laffont᎐Tirole, 1986.. We show below that it is also true for small enough, in which case we obtain a characterization of the optimal contract.9 PROPOSITION 2. Under the monotone hazard rate assumption and for small enough, an interior solution for the optimal contract is characterized by The first-order condition of the firm’s program is t⬘w C Ž  .x s y ⬘Ž eŽ  ... By differentiation we obtain t⬙ w C Ž  .xC⬘Ž  . s y ⬙ Ž eŽ  .. e⬘Ž  .. Under our assumptions, t⬙ and e⬘ have opposite signs. 9 Technically, the problem turns out to be quite similar to the one studied by Salanie ´ Ž1990.. Salanie ´ studies the case of a principal᎐agent relationship with a risk-averse agent and an ex ante individual rationality condition. Here we integrate in the objective function the analog of his ex ante IR constraint Žour certainty equivalent term., but we also have ex post IR constraints for the agent. 8
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effort le¨ els e*Ž  . and rents *Ž  ., jointly gi¨ en by
⬙ Ž e* Ž  . .
⬘ Ž e* Ž  . . s 1 y
½
fŽ  .
FŽ  . y
H Ž exp y * Ž x . . f Ž x . dx
5
Ž 1 q . E exp y * Ž  . Ž 4.7.
and
*Ž  . s
H ⬘ Ž e* Ž x . . dx.
Ž 4.8.
Furthermore, implementation by a menu of linear schedules is possible. Proof. See Appendix 1. Note that in general, e*Ž⭈. and *Ž⭈. are jointly determined by a system of two integral equations. However, for s 0, *Ž⭈. disappears from Eq. Ž4.7., and we obtain the classical formula
⬘ Ž e* Ž  . . s 1 y
FŽ  .
1 q fŽ  .
⬙ Ž e* Ž  . . .
Ž 4.9.
To see the additional role of risk aversion, Ž4.7. can be rewritten as follows:
⬘ Ž e* Ž  . . s 1 y y
FŽ  .
1 q fŽ  .
⬙ Ž e* Ž  . .
⬙ Ž e* Ž  . . F Ž  . 1q
fŽ  .
½
1y
H exp y * Ž s . f Ž s . ds H exp y * Ž s . f Ž s . ds
5
Ž 4.10. Notice that the additional term vanishes 10 for  s  and  s  . As we show below, risk aversion unambiguously tilts the optimal contract toward cost-plus schemes. 10
The situation is more complex for non exponential utilities. Indeed, let Ž x . s yu⬙ Ž x .ru⬘Ž x . denote the Žlocal. index of risk aversion of any concave VNM utility function. It is not difficult to prove that, in the case of no-bunching, formula Ž4.10. becomes 1 y ⬘ Ž e* Ž  . .
⬙ Ž e* Ž  . .
s
FŽ  .
Ž1 q . f Ž  .
ž
q 1y
H Ž exp y H0 *Ž x . Ž s . ds . f Ž x . dx F Ž  . u⬘ Ž CE Ž * . .
/
.
Ž 4.10bis . When u is not exponential, the second term between brackets does not vanish for  s  .
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As increases, the condition e⬘ F 0 may become violated, and then decentralization by a menu of linear schedules is not possible. Moreover for large enough, the constraint e⬘ F 1 may become binding, and bunching occurs. To fully characterize these phenomena, we pursue the analysis in the case of a uniform distribution on w  ,  x and a quadratic disutility of effort Ž e . s 12 e 2re* Žtherefore, e* represents the first best effort level: ⬘Ž e*. s 1.. In the no-bunching region we can examine the comparative statics of the Žsecond best. optimal effort level. Let us denote by  ª e*Ž  , . this optimal effort function when the index of risk aversion is . PROPOSITION 3. When there is no bunching, effort decreases with the index of risk a¨ ersion, : Ž ⭸ e*r⭸ .Ž  , . F 0. Proof. See Appendix 2. Furthermore, we have: PROPOSITION 4. There exist 1 , 2 with 2 ) 1 ) 0, such that Ža. For F 1 , implementation by a menu of linear schedules is possible: Ž ⭸ e*r⭸ .Ž  , . F 0 for all  . Žb. For F 2 , there is no bunching: Ž ⭸ e*r⭸ .Ž  , . - 1 for all  . Žc. For ) 2 , there is complete bunching on some inter¨ al, w  2 Ž .,  x. Outside this inter¨ al, the solution in¨ ol¨ es no bunching:
⭸ e* ⭸
Ž , . - 1
for  in  ,  2 Ž . w ,
Ž , . s 1
for  in  2 Ž . ,  .
and
⭸ e* ⭸
Proof. See Appendix 3. In the limit case of an infinite risk aversion, we have CE Ž . s Ž  . . Since Ž4.5. remains binding, the solution is the same as with risk neutrality, except that the weight of the profit in the objective function is zero. The solution is then defined by
⬘Ž e Ž  . . s 1 y
FŽ  . fŽ  .
⬙ Ž eŽ  . . .
Ž 4.11.
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FIG. 1. The comparative statics of the second best solution as increases.
All of our results on the comparative statics of the effort level are represented in Fig. 1 Žin the Ž C,  . plane., with C Ž  , . s  y eŽ  , ..
5. THE SPECIAL CASE OF THE BARON᎐MYERSON MODEL We study in this section the extension to a risk-averse firm of the Baron᎐Myerson Ž1982. model. It can be considered a variant 11 of our problem in which there is no effort variable, and the costs are not observable by the regulator. Instead, it is the size of the project Žquantity q produced by the firm. that will be used for inducing self-selection of the 11 In the original formulation ŽBaron᎐Myerson, 1982., the weight of the firm’s profit in the social welfare function is less than one. For homogeneity with the rest of the paper, we examine the variant in which there is a cost of public funds. This allows us to focus on the main difference between the Laffont᎐Tirole approach used in the rest of the paper, and the Baron᎐Myerson approach used in this section, namely that costs are no longer observable.
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different types of firms. The cost function is now C s  q. If P Ž q . is the inverse demand function, we obtain Žas before, Ž  . denotes the profit of the firm. the expression of the social welfare function: W s E V Ž q Ž  . . y Ž 1 q . t Ž  . q CE Ž . ,
Ž 5.1.
where V Ž q . s H0q P Ž s . ds denotes consumers’ surplus, t Ž  . s  q Ž  . q Ž  . is the total transfer to the firm, and CEŽ . is the certainty equivalent of the firm’s profit. W has to be maximized under the constraints
⬘ Ž  . s yq Ž  . q⬘ Ž  . F 0
Ž first-order incentive compatibility .
Ž second-order incentive compatibility .
Ž. G0
Ž individual rationality . .
Ž 5.2. Ž 5.3. Ž 5.4.
Replacing t Ž  . with  q q , and q with y ⬘, we obtain a new expression for W: W Ž . s E V Ž y ⬘ . y Ž 1 q . Ž y ⬘ q . q CE Ž . . Under the CARA specification, EC Ž . s yŽ1r . log EŽexp y ., and Ž5.4. is always binding at the optimum. PROPOSITION 5. When there is no bunching, the second best solution under risk a¨ ersion is characterized by
P Ž q Ž  . . s Ž 1 q .  q
ž
FŽ  . fŽ  .
/
y
H exp y Ž s . dF Ž s . f Ž  . H exp y Ž s . dF Ž s .
.
Ž 5.5. For s 0, we obtain the Baron᎐Myerson formula: P0 Ž q Ž  . . s Ž 1 q .  q
FŽ  . fŽ  .
.
Because of risk a¨ ersion, the distortion is greater when ) 0. Proof. See Appendix 4.
Ž 5.6.
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Still assuming the absence of bunching, Proposition 5 can easily be extended to more general utility functions. Denoting the local index of risk aversion yŽ u⬙ru ⬘.Ž x . by Ž x ., we can write the generalization of formula Ž5.5.: P Ž q Ž  . . s Ž 1 q .  q
ž
FŽ  . fŽ  .
/
H exp y H0 *Ž x . Ž s . ds f Ž x . dx
y
f Ž  . u⬘ Ž CE Ž * . .
.
Using Ž5.6., this can also be written as P Ž q Ž  . . sP0 Ž q Ž  . . q
FŽ  . fŽ  .
1y
H expyH0 *Ž x . Ž s . ds f Ž x . dx F Ž  . u⬘ Ž CE Ž * . .
,
where P0 Ž q Ž  .. is the Baron᎐Myerson price. Therefore the sign of the additional distortion caused by risk aversion is determined by the term between square brackets. In particular, the distortion will always be positive if and only if the function
Ž  . s FŽ  .¨
H␣ Ž s . f Ž s . ds
y

H ¨ Ž ␣ Ž s . . f Ž s . ds
is always positive, where we have introduced the following notations: ¨ Ž x . s u⬘ uy1 Ž x . ,
LEMMA 2.
␣ Ž x . s u *Ž x . .
is always non-negati¨ e m Ž  . G 0.
Proof. ⬘Ž  . s f Ž  .w ¨ Ž H ␣ Ž s . f Ž s . ds . y ¨ Ž ␣ Ž  ..x. Since ¨ and ␣ are decreasing, ⬘ is first positive, then negative. Therefore has an interior maximum. Since Ž  . s 0, is always non-negative if and only if Ž  . G 0. B But now we have that
Ž  . s¨
žH


␣ Ž s . f Ž s . ds y
/
H¨ Ž ␣ Ž s . . f Ž s . ds.
Therefore, by Jensen’s inequality, Ž  . is non-negative only when ¨ is concave Žor linear, as in the exponential specification.. The usual assumption in risk theory Žsee, for instance, Kimball, 1990. is that ¨ is convex.
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This is the case, for instance, when u is DARA, i.e., it exhibits decreasing absolute risk aversion. In that case Ž  . - 0, which means that there exists a threshold  * such that 䢇
For  -  *: risk aversion leads to a higher distortion.
䢇
For  )  *: risk aversion leads to a lower distortion.
6. APPLICATIONS AND CONCLUSION Laffont and Tirole Ž1986. showed that, with risk-neutral agents, noise in the cost observation does not prevent the regulator from implementing the same allocation as in the absence of noise. This result was generalized by Caillaud et al. Ž1992., Demougin Ž1989., and Melumad and Reichelstein Ž1989.. However, when risk aversion is introduced, there is no hope of finding a simple characterization of the optimal contract in the noisy observation case. On the contrary, in the case of non-noisy cost observation, we have shown that even with risk aversion, one can obtain a simple and precise characterization of the optimal contract. This result can be quite useful when the agent is subject to other risks that do not interfere with the observation of his cost. Suppose, for example, that the firm’s activity creates an environmental risk of social cost E with probability , and suppose further that the firm can affect this probability with another effort level d, which interacts with e in the disutility of effort as Ž e q d .. To study the insurance of this environmental risk, risk aversion is crucial, and the results of this paper are an essential building block of such an analysis Žsee Laffont, 1995, for further details.. In conclusion, this paper has provided a solution to a class of principal agent problems in which risk aversion matters, and at the same time both adverse selection and moral hazard affect the outcomes observable by the principal. Risk aversion of the agent, as expected, moves the optimal contract toward cost-plus contracts, i.e., he accepts a higher allocation inefficiency to mitigate the informational rent made more costly by risk aversion. As a consequence, the optimal contract typically involves bunching for inefficient types when risk aversion is strong. This provides a new explanation for the cost ceilings observed in many contracts without having to appeal to special shapes of the hazard rate, as in Laffont and Tirole Ž1993, Chap. 1..
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APPENDIX 1: PROOF OF PROPOSITION 2 Let Ž  . and Ž  . be the Lagrange multipliers associated with constraints Ž4.2. and Ž4.3.. The Lagrangian of the regulator’s optimization program is L s S q CE Ž . y Ž 1 q .
H Ž Ž  . q Ž e Ž  . . q  y e Ž  . . f Ž  . y Ž  . Ž ⬘ Ž  . q ⬘ Ž e Ž  . . . y Ž  . Ž e⬘ Ž  . y 1 . d 
With respect to e, we have a classical calculus of variations problem Žexcept that Ž4.3. is not always binding.. The necessary condition is thus
⬘ Ž  . s Ž ⬘ Ž e Ž  . . y 1. f Ž  . y ⬙ Ž e Ž  . . Ž  . Ž  . Ž e⬘ Ž  . y 1 . s 0
Ž Euler equation . Ž A1.1.
Ž complementarity conditions . . Ž A1.2.
With respect to , this is not a classical problem because of CEŽ .. So we have to compute the derivative of L Žwith respect to . in the direction of an arbitrary differentiable perturbation function hŽ  . satisfying Ž4.5. with equality, i.e., such that hŽ  . s 0.
⭸L ⭸
⭈ h s lim
L Ž q th . y L Ž . t
tª0
s
Eh exp y E exp y
y Ž 1 q .
H h Ž  . f Ž  . y h⬘ Ž  . Ž  . 4 d  ,
or
H
½
exp y Ž  . f Ž  .
Ž 1 q . H exp y Ž x . f Ž x . dx
5
y f Ž  . hŽ  .
q Ž  . h⬘ Ž  . d  s 0 Ž A1.3.
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165
Let BŽ  . s

H
exp y Ž s . f Ž s .
½
Ž 1 q . H exp y Ž x . f Ž x . dx
5
y f Ž s . ds.
Then ŽA1.3. can be rewritten as
Hh Ž  . B⬘ Ž  . d  q H Ž  . h⬘ Ž  . d  s 0. Integrating by parts in the first integral and using the fact that hŽ  . s 0 and B Ž  . s 0, we get
H yB Ž  . q Ž  .
h⬘ Ž  . d  s 0.
Since h⬘Ž  . is arbitrary, we deduce
Ž  . s BŽ  . or
Ž  . s
ey Ž s. f Ž s .

H
Ž 1 q . H ey Ž x . dF Ž x .
ds y F Ž  .
Ž A1.4.
When constraint Ž4.3. is not binding, Ž  . ' 0, and therefore ⬘Ž  . s 0. Substituting ŽA1.4. into ŽA1.1., we get the desired result:
⬘Ž e Ž  . . s 1 y
⬙ Ž eŽ  . . fŽ  .
½
FŽ  . y
H exp y Ž x . f Ž x . dx
Ž 1 q . E exp y Ž  .
5
.
For s 0, Ž drd  . Ž Frf . ) 0 and G 0 ensure that e is strictly decreasing. This will remain true by continuity for small enough. Therefore constraint Ž4.3. is not binding for small enough. Furthermore, since e⬘ - 0, the transfer function is convex for small enough, and implementation with linear schedules is possible.
APPENDIX 2: PROOF OF PROPOSITION 3 In the uniform-quadratic case, the expression of ŽA1.1. becomes, in the no-bunching region Žwhere ⬘Ž  . ' 0.,
ˆe Ž  . s e* q Ž ⌬  . Ž  . ,
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with ⌬  s  y  ,
Ž ⌬  . Ž  . s H


ž
exp y Ž s .
Ž 1 q . ␥
y 1 ds,
/
␥ s E w exp y x .
and
To examine the comparative statics properties of this solution with respect to , we make explicit the dependence with respect to :
ˆe Ž  , . s e* q H


H
with Ž  , . s
ž
exp y Ž s, .
Ž 1 q . ␥ Ž .
y 1 ds
/
ˆe Ž s, . ds e*
and
␥ Ž . s
1
 exp y Ž s, . H ⌬ 
ds.
Hence
⭸ˆ e ⭸
Ž , . s
1
exp y Ž  , . y 1 ) y1, Ž A2.1.
Ž 1 q . ␥ Ž .
and since
⭸ ⭸ ⭸ 2ˆ e ⭸ 2
Ž , . s Ž , . s
yeˆŽ  , . e*
,
ˆ e Ž  , . exp y Ž  , . e* Ž 1 q . ␥ Ž .
Ž A2.2. )0
at any interior solution Žfor which ˆ eŽ  , . ) 0.. The solution is therefore Žstrictly . convex in the no-bunching region. We can eliminate Ž  , . between ŽA2.1. and ŽA2.2.. We obtain a second-order differential equation in ˆ eŽ  , .:
⭸ 2ˆ e ⭸
2
Ž , . s
ˆ eŽ  , . e*
ž
1q
with
ˆe Ž  , . s e*,
⭸ˆ e ⭸
Ž , .
/
Ž A2.3.
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167
and
ˆe Ž  , . s e* y
1q
⌬.
Let uŽ  , . s Ž ⭸ ˆ er⭸ .Ž  , .. We want to prove that u is nonpositive. By differentiation of ŽA2.3. with respect to , we obtain e*
⭸ 2u ⭸
2
Ž  , . s ˆe Ž  , . q u Ž  , . 1 q
ž
q ˆ eŽ  , .
⭸u ⭸
⭸ˆ e ⭸
Ž , .
Ž , .,
/ Ž A2.4.
and u Ž  , . s u Ž  , . s 0. Since u is continuous as a function of  , it has a maximum on w  ,  x, say at  *. Suppose that this maximum is interior. We have, then,
⭸u ⭸
Ž  *, . s 0
⭸ 2u
and
⭸ 2
Ž  *, . F 0.
Therefore by ŽA2.4.,
ˆe Ž  *, . q u Ž  *, . 1 q
ž
⭸ˆ e ⭸
Ž  *, . F 0.
/
Now ŽA2.2. and ŽA2.3. imply that 1 q Ž ⭸ ˆ er⭸ .Ž  *, . ) 0. Therefore Ž . u  *, - 0, which is absurd, since  * is supposed to maximize u and uŽ  , . s uŽ  , . s 0. Notice that we have proved in fact that uŽ  , . - 0 for all  in x  ,  w when there is no bunching. Since uŽ  , . s uŽ  , . s 0, this proves in particular the desired result that u is nonpositive. B
APPENDIX 3: PROOF OF PROPOSITION 4 It is analogous to the proof of Proposition 1 in Salanie ´ Ž1990.. We have just proved that uŽ  , . - 0 for  in x  ,  w and uŽ  , . s 0. Therefore Ž ⭸ ur ⭸ .Ž  , . G 0. But this is also equal to Ž ⭸r⭸ .ŽŽ ⭸ ˆ er⭸ .Ž  , ... Thus Ž ⭸ ˆ er⭸ .Ž  , . is an increasing function of . Parts Ža. and Žb. of Proposition 4 are then immediate consequences of
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LEMMA 3.
⭸ˆ e
lim
⭸
ªq⬁
Ž  , . ) 1.
As a consequence, ᭚ 1 - 2 , such that Ž ⭸ ˆ er⭸ .Ž  , 1 . s 0 Ž⭸ ˆ er⭸ .Ž  , 2 . s 1.
and
Proof of Lemma 3. Suppose by contradiction that ᭙ Ž ⭸ ˆ er⭸ .Ž  , . F 1. By convexity of ˆ e, this implies ᭙ ˆ eŽ  , . G ˆ e Ž  , . y Ž  y  . s e* y
1q
⌬ y Ž  y  ..
Ž A3.1. By integration of ŽA2.3. between  and  , we obtain e*
½
⭸ˆ e
Ž , . y ⭸ s
⭸ˆ e ⭸
Ž , . 1
5 1
Hˆe Ž  , . d  q 2 ˆe Ž  , . y 2 ˆe Ž  , . . 2
2
Ž A3.2.
Since Ž ⭸ ˆ er⭸ .Ž  , . G y1 and Ž ⭸ ˆ er⭸ .Ž  , . F 1, the left-hand side of ŽA3.2. is less than 2 e*r , whereas the right-hand side is Žbecause of ŽA3.1.. greater than a positive term that does not depend on . This is impossible for large enough. B We now come back to Proposition 4: part Žc. remains to be established. For this we need to rewrite the necessary conditions for the case of bunching:
Ž ⌬  . e* ⬘ Ž  . s e Ž  . y e* y Ž ⌬  . Ž  . , Ž  . s Ž  . s 0, Ž  . Ž e⬘ Ž  . y 1 . s 0, ^`_ ^ ` _ G0
Ž Euler . Ž A3.3.
Ž transversality . Ž complementarity .
Ž A3.4. Ž A3.5.
F0
Ž ⌬  . Ž  . s H


ž
exp Ž y Ž s . .
␥ Ž 1 q .
y 1 ds,
/
Ž first-order condition with respect to . , Ž A3.6. where Ž  . s H eŽ s .re* ds; ␥ s EwexpŽy Ž s ..x.
REGULATION OF A RISK-AVERSE FIRM
169
This implies, by differentiating ŽA3.3.,
Ž ⌬  . e* ⬙ s ye* ⬙ y Ž ⌬  . ⬘ s ye* ⬙ q 1 y
exp Ž y .
␥ Ž 1 q .
.
Let us take t s w ␥ Ž1 q .xy1 as a parameter, and define: H Ž . s 2 y t expŽy .. In the no-bunching zone, ⬙ is identically zero, and we have e* ⬙ q 1 s H Ž Ž  . . G 0. However, as indicated by Figure 2, H may take negative values when t ) 2. In that case, e* ⬙ q 1 is identically zero, and we have
Ž ⌬  . e* ⬙ Ž  . s H Ž Ž  . . . We are going to construct the solution to our problem by setting
½ ½
e* ⬙ Ž  . q 1 s Max Ž 0, H Ž Ž  . . . G 0
Ž  . s 0, ⬘ Ž  . s y1
Ž ⌬  . e* ⬙ Ž  . s Min Ž 0, H Ž Ž  . . . F 0
Ž A3.7.
Ž. sŽ. s0
We now prove that the necessary conditions are indeed satisfied. ŽA3.4. is fulfilled by construction; moreover,
concave Ž  . s Ž  . s 0 « ᭙ Ž  . G 0
5
Ž which implies Ž A3.5. . .
FIG. 2. Graph of the function H in the case t ) 0. The value c is equal to Ž1r . logŽ tr2..
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Let us define e Ž  . s ye* ⬘ Ž  . , and 
Ž ⌬  . Ž  . s H Ž t exp Ž y Ž s . . y 1 . ds 
Ž this implies Ž A3.6. . .
We have
Ž 1 y e⬘ . s 1 q e* ⬙ Ž  . G 0, with equality when Ž  . / 0, since then H Ž Ž  .. s 0. Moreover, we have to establish ŽA3.3., by proving that the function d
K Ž  . s Ž ⌬  . e* ⬘ Ž  . y e Ž  . q e* q Ž ⌬  . Ž  . is identically zero. But KŽ  . s 0
¡ ¢ Ž  . s 0
~e Ž  . s ye* ⬘ Ž  . s 0,
since
Ž  . s 0
on  ,  q ⑀ « ⬘ Ž  . s 0
and for all  , K ⬘ Ž  . s Ž ⌬  . e* ⬙ Ž  . q e* ⬙ q Ž ⌬  . ⬘ Ž  . s 0. This proves that K Ž  . is identically zero. Finally, we have two things to do: Ža. Show that bunching occurs on the upper part of the distribution of  . Indeed, this occurs when H w Ž  .x F 0 or Ž  . F c m  G  2 with Ž  2 . s c Žthis is because is decreasing.. Notice that when Ž  . G c there is no bunching. Žb. Show that t can be chosen so that EwexpŽy .x s ␥ s 1rt Ž1 q . . Indeed, if we denote by Ž  , t . the solution of ŽA3.7., it is clear that when t ª 0, e*Ž ⭸ 2r⭸ 2 .Ž  , t . q 1 ª 2 and Ž  , 0. remains bounded, so that lim t ª 0 tEwexpŽy Ž  , t ..x s 0. Similarly, when t ª q⬁, e*Ž ⭸ 2r⭸ 2 .Ž  , t . q 1 ª 0, and Ž  , q⬁. is also bounded, which implies lim t ªq⬁ tEwexpŽy Ž  , t ..x s q⬁. Therefore, by continuity, the solution exists. Finally, in the particular case in which there is no bunching Ž ᭙ H Ž Ž  .. G 0 ., we obtain Eq. ŽA2.3.. Indeed, ŽA3.7. gives e* ⬙ Ž  . q 1 s 2 y t expŽy ..
REGULATION OF A RISK-AVERSE FIRM
171
By differentiating this, we get e* s y
⭸ 2ˆ e ⭸ 2
s t ⬘ exp Ž y . ,
but
¡t expŽ y . s 1 y e* ⬙ s 1 q ⭸ ˆe
~
⭸
¢ ⬘ s y e*ˆ e
Therefore,
⭸ 2ˆ e ⭸
2
s
e*
ž
1q
⭸ˆ e ⭸
/
ˆe,
and we are back to Eq. ŽA2.3..
APPENDIX 4: PROOF OF PROPOSITION 5 The proof is exactly analogous to that of Proposition 2: we differentiate W in the direction of an arbitrary perturbation function h Žsuch that hŽ  . s 0.: W⬘h s lim
W Ž q th . y W Ž .
tª0
t
s E yP Ž q . h⬘ y Ž 1 q . Ž y h⬘ q h . q
h exp y
␥
,
with ␥ s Ewexp y x. Integrating by parts, we obtain a result analogous to the Euler equation,
¡d
f Ž  . Ž yP Ž q Ž  . . q Ž 1 q .  s f Ž  .
~ d ¢P ž q Ž  . / s Ž 1 q .  .
½
exp y
␥
y Ž 1 q .
5
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LAFFONT AND ROCHET
Thus P Ž q Ž  . . s Ž 1 q .  q
FŽ  .
y
fŽ  .
H exp y Ž s . dF Ž s . f Ž  . H exp y Ž s . dF Ž s .
.
def
Let Ž  . s f Ž  .w P0 Ž q Ž  .. y P Ž q Ž  ..x be a measure of the distortion due to risk aversion. We have
Ž  . s Ž  . s 0 ⬘Ž  . s f Ž  . 1 y
and exp y Ž  . H
exp y Ž s . dF Ž s .
.
At  s  , Ž  . s 0, and therefore ⬘Ž  . - 0. If Ž  . was not always non-negative, there would be a ˆ such that Ž ˆ. s 0 and ⬘Ž ˆ. ) 0, i.e., ˆ
F Ž ˆ . s
H exp y Ž s . dF Ž s . H exp y Ž s . dF Ž s .
Ž A4.1.
and
H exp y Ž s .
dF Ž s . ) exp y Ž ˆ . .
Integrating by parts in the right hand side of ŽA4.1., we obtain F Ž ˆ . s
F Ž ˆ . exp y Ž ˆ . H exp y Ž s . dF Ž s .
ˆ
y
H F Ž s . e Ž s . exp y Ž s . ds H exp y Ž s . dF Ž s .
,
which yields a contradiction. B
REFERENCES Baron, D., and Myerson, R. Ž1982.. ‘‘Regulating a Monopolist with Unknown Costs,’’ Econometrica 50, 911᎐930. Caillaud, B., Guesnerie, R., and Rey, P. Ž1992.. ‘‘Noisy Observation in Adverse Selection Models,’’ Re¨ . Econ. Stud. 59, 595᎐615. Demougin, D. Ž1989.. ‘‘A Renegotiation-Proof Mechanism for a Principal-Agent Model with Moral Hazard and Risk Aversion,’’ Rand J. Econ. 20, 256᎐267. Green, J., and Laffont, J. J. Ž1979.. Incenti¨ es in Public Decision Making. Amsterdam: North-Holland. Grossman, S., and Hart, O. Ž1983.. ‘‘An Analysis of the Principal-Agent Problem,’’ Econometrica 51, 7᎐45.
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Guesnerie, R., and Laffont, J. J. Ž1984.. ‘‘Control of Firms under Incomplete Information,’’ J. Public Econ. 25, 329᎐369. Holmstrom, ¨ B. Ž1979.. ‘‘Moral Hazard and Observability,’’ Bell J. Econ. 10, 74᎐91. Holmstrom, ¨ B., and Milgrom, P. Ž1987.. ‘‘Aggregation and Linearity in the Provision of Intertemporal Incentives,’’ Econometrica 55, 303᎐328. Kimball, M. Ž1990.. ‘‘Precautionary Savings in the Small and in the Large,’’ Econometrica 58, 53᎐78. Laffont, J. J. Ž1995.. ‘‘Regulation, Moral Hazard and Insurance of Environmental Risks,’’ J. Public Econ. 58, 319᎐336. Laffont, J. J., and Tirole, J. Ž1986.. ‘‘Using Cost Observation to Regulate Firms,’’ J. Political Econ. 94, 614᎐641. Laffont, J. J., and Tirole, J. Ž1993.. A Theory of Incenti¨ es in Procurement and Regulation Cambridge, MA: MIT Press. Melumad, N., and Reichelstein, S. Ž1989.. ‘‘Value of Communication in Agencies,’’ J. Econ. Theory 47, 334᎐368. Mirrlees, J. Ž1971.. ‘‘An Exploration in the Theory of Optimum Income Taxation,’’ Re¨ . Econ. Stud. 38, 175᎐208. Mirrlees, J. Ž1975.. ‘‘The Theory of Moral Hazard and Unobservable Behavior, Part I,’’ Mimeo, Oxford Univ. Mussa, M., and Rosen, S. Ž1978.. ‘‘Monopoly and Product Quality,’’ J. Econ. Theory 18, 301᎐317. Pauly, M. Ž1968.. ‘‘The Economics of Moral Hazard,’’ Quart. J. Econ. 88, 44᎐62. Salanie, Adverse et Aversion pour le Risque,’’ Ann. Econ. Statist. 18, ´ B. Ž1990.. ‘‘Selection ´ 131᎐150. Shavell, S. Ž1979.. ‘‘On Moral Hazard and Insurance,’’ Quart. J. Econ. 93, 541᎐562. Spence, M. Ž1974.. Market Signalling: Informational Transfer in Hiring and Related Process, Cambridge, MA: Harvard Univ. Press. Spence, M., and Zeckhauser, R. Ž1971.. ‘‘Insurance, Information, and Individual Action,’’ Amer. Econ. Re¨ . 61, 380᎐387. Zeckhauser, R. Ž1970.. ‘‘Medical Insurance: A Case Study of the Trade-Off Between Risk Spreading and Appropriate Incentives,’’ J. Econ. Theory 2, 10᎐26.
25, 149᎐173 Ž1998.
GA980639
Regulation of a Risk Averse Firm Jean-Jacques Laffont and Jean-Charles Rochet* Institut Uni¨ ersitaire de France, GREMAQ᎐IDEI, Uni¨ ersite´ des Sciences Sociales, F-31042 Toulouse cedex, France Received October 14, 1996
We extend the Laffont᎐Tirole regulation model to the case of risk-averse firms. Our main results are: 1. The impact of risk aversion is to shift the optimal contract toward a cost-plus contract. 2. As compared with the risk-neutral case, distortions are greater and informational rents are smaller. 3. For high levels of risk aversion, the optimal contract involves cost ceilings and the less efficient firms are bunched together. Journal of Economic Literature Classification Numbers: D8, L5. 䊚 1998 Academic Press
Key Words: regulation; risk; bunching; cost-plus contracts
1. INTRODUCTION Principal᎐agent relationships typically involve three types of Žpartially conflicting. objectives: risk sharing must be organized, appropriate effort levels must be induced Ž moral hazard., and truthful information must be elicited Ž ad¨ erse selection.. The literature has in general addressed these issues separately. For example, the moral hazard literature ŽPauly, 1968; Zeckhauser, 1970; Spence and Zeckhauser, 1971; Mirrlees, 1975; Holmstrom ¨ 1979; Shavell, 1979; Grossman and Hart, 1983, among others. has focused on the trade-off between incentives for effort and risk sharing. The main conclusion is that agents must bear more risk than in the first best, to induce a reasonable effort level. Holmstrom ¨ and Milgrom Ž1987. have shown further that, in the case of exponential utility functions and repeated moral hazard situations, the optimal contracts could be taken to be linear. When these assumptions are valid, their model is particularly convenient for the study of risk sharing with moral hazard.1 * E-mail: [email protected]. 1 See, for example, their study of multitasks ŽHolmstrom ¨ and Milgrom, 1987.. 149 0899-8256r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
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On the other hand, the adverse selection literature has studied how a principal should structure his offer of contracts to a privately informed agent, to optimize his own objective under the interim participation constraint of the agent. The optimal contract is in general a nonlinear contract: it can be finely characterized under the so-called Spence᎐Mirrlees conditions, which ensure that only local incentive constraints are binding. An informational rent must be given up by the principal to all types except the less efficient type. Moreover, all agents’ actions are distorted to mitigate the informational rents, except for the most efficient type Žno distortion at the top result; see Mirrlees, 1971; Spence, 1974; Mussa and Rosen, 1978; Green and Laffont, 1979; Baron and Myerson, 1982; Guesnerie and Laffont, 1984, among others.. In the case of risk-neutral agents, Laffont and Tirole Ž1986. have considered simultaneously adverse selection and moral hazard, and have characterized the optimal contract that trades off these two problems. Fixed-price contracts are good for moral hazard but bad for adverse selection. Cost-plus contracts are good for adverse selection and bad for moral hazard. The optimal contract is a nonlinear contract that can sometimes be implemented by a menu of linear sharing rules in which self selection of types is obtained. Moreover, the optimal contract has the same qualitative features as in the pure adverse selection case. What is clearly missing is a simultaneous treatment of the three problems Žrisk sharing, effort inducement, information elicitation ., which must be tackled together in many contractual relationships: sharecropping, managerial incentives, insurance problems, regulation of small firms, etc. The purpose of this paper is to make some progress in this problem by considering a special case that enables us to characterize quite fully the optimal contract. We study a situation with both adverse selection and moral hazard for a risk-averse agent, in which the contract is offered and signed before the agent knows his type. Since the principal is risk-neutral and the agent risk-averse, the optimal contract will involve some insurance Žof the agent by the principal. against this interim risk. Therefore we obtain a novel form of the insurance ᎐incentive trade-off: informational rents become more costly to the principal because they induce variability in the ex post utility of the agent. This is the main intuition behind our results: risk aversion brings the optimal contract closer to a cost-plus contract, with greater distortions and lower rents. Moreover, bunching appears for high levels of risk aversion. Of course, the type of risk we study here is peculiar. It would also be interesting to introduce an ex post risk, i.e., a risk that materializes after
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151
the effort decision has been made. The basic idea of our modeling is, instead, to maintain a non-noisy relationship between the variables of adverse selection, moral hazard, and the action to be implemented. This allows us to eliminate the moral hazard variable 2 and to reduce technically the problem to an adverse selection problem with risk-averse firms, related to the one studied by Salanie ´ Ž1990.. The model is presented in Section 2. Section 3 provides an analysis of the two-type case. When there is a continuum of types, the problem is much more complex: it is solved for a CARA utility function in Section 4. The special case of the Baron᎐Myerson model is dealt with in Section 5. Examples and concluding comments are gathered in Section 6.
2. THE MODEL Our model is a simple extension of the regulation model of Laffont᎐ Tirole Ž1986., to a situation in which risk is introduced. An utilitarian regulator wishes to realize a public project with social value S. A single firm can realize the project, at a cost C s  y e, where  is an efficiency parameter and e is an effort level with a disutility Ž e . for the management of the firm Žwe assume ⬘ ) 0, ⬙ ) 0, G 0.. This disutility is measured in monetary terms. We assume that the cost C is observable ex post by the regulator Žbut not  and e separately., and we make the accounting convention that it is paid by the regulator. Let t be the net monetary transfer from the regulator to the firm. The firm is supposed to be owned and managed by the same agent, who is risk-averse. Accordingly, this agent obtains Žex post. a utility level: uŽ t y Ž e . . . We assume: u⬘ ) 0, u⬙ F 0. A natural way to introduce some element of risk into this model would be to add a noise term ˜ in the cost, which would become C˜ s  y e q ˜ . Since the cost is the only variable that the regulator observes, we would be in a situation of noisy observation, analogous to the one studied by 2
This follows Laffont᎐Tirole Ž1986.. However, in the risk-neutral case, their results extend immediately to noisy relationships. This is not the case here.
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Caillaud et al. Ž1992. under the assumption of a risk-neutral agent. However, with a risk-averse agent, the simultaneous presence of moral hazard, adverse selection, and noise would make the problem intractable. We have chosen instead to introduce an interim risk about the adverse selection parameter, which brings along two simplifications: the observation of costs remains non-noisy, and the incentive compatibility constraints are unaffected. The timing is as follows: At s 0 the contract is offered to the firm. At s 1, nature chooses  . This parameter is privately observed by the firm, which can then exit the relationship. At s 2, and if it has not exited, the firm chooses the effort e, the regulator observes the cost C s  y e, and the contract is implemented. 䢇 䢇
䢇
Notice that when the contract is offered Žat s 0., the information is symmetric. We also assume that the firm can renege on the contract at s 1, which necessitates the introduction of ex post rational individuality constraints. 3 We are now going to investigate how the introduction of the firm’s risk aversion affects the characteristics of the optimal contract derived by Laffont and Tirole Ž1986.. We start with the simple two-type case.
3. THE OPTIMAL CONTRACT IN THE TWO-TYPE CASE We assume in this section that  can only take two values,  and  Žthe difference  y  is denoted ⌬  .. Ex ante, it is common knowledge that s PrŽ  s  . Žtherefore 1 y s PrŽ  s  ... From the revelation principle, we can restrict ourselves to direct revelation mechanisms  ª Ž t Ž  ., C Ž  .., and we use the following notations: tŽ  . s t,
C Ž  . s C.
tŽ  . s t,
C Ž  . s C.
Similarly, the effort and profit levels are denoted as follows:
3
e s  y C,
s t y Ž e. ,
e s  y C,
s t y Ž e. ,
For simplicity, we assume that these ex post IR constraints imply the ex ante IR constraint that we neglect below.
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The firm’s incentive constraints can be written: t y Ž  y C. G t y  y C ,
Ž 3.1.
t y Ž  y C. G t y Ž  y C..
Ž 3.2.
ž
/
An equivalent way to express these incentive constraints is to write that the difference in profits between one type of firm and the other is at least equal to the difference in costs of effort when the first type ‘‘mimics’’ the second type. Using the notation Ž x . s Ž x . y Ž x y ⌬  ., Ž3.1. and Ž3.2. can thus be rewritten as
G q Ž  y C.,
Ž 3.3.
G y Ž  y C..
Ž 3.4.
The firm’s ex post individual rationality constraints are
G 0,
Ž 3.5.
G 0.
Ž 3.6.
The regulator wishes to maximize expected social welfare under incentive and individual rationality constraints. Let Ž1 q . be the social cost of public funds. Assuming for simplicity that the project is sufficiently valuable to be always realized, consumers’ expected utility is S y Ž 1 q . Ž t q C . q Ž 1 y . Ž t q C . .
Ž 3.7.
The term between square brackets is nothing but the total expected transfer to the firm. We aggregate consumers’ utility with the certainty equivalent of the manager’s profit Ž CE s uy1 w uŽ . q Ž1 y . uŽ .x., to obtain expected social welfare 4 W. Substituting t s q Ž  y C ., we obtain W s S y Ž 1 q . q Ž  y C . q C
ž
/
qŽ 1 y . q Ž  y C . q C
ž
q uy1 Ž u Ž . q Ž 1 y . u Ž . . .
/ Ž 3.8.
4 We consider only deterministic schemes. From Laffont᎐Tirole Ž1993, pp. 119᎐120., we know that stochastic schemes are dominated as long as G 0, and the welfare function W is concave in , , C, C. In our context this is equivalent to saying that the certainty equivalent operator is concave. This is satisfied, for instance, when u is CARA Žsee Section 4..
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The optimal regulation is obtained by maximizing W under the constraints Ž3.3. to Ž3.6.. It is characterized as follows: PROPOSITION 1. When there are only two types, and if the ¨ alue S of the project is large enough, the optimal deterministic regulation entails: 1. An efficient le¨ el of effort for the more efficient type  , i.e., ⬘Ž e . s 1. 2. An incremental rent for the more efficient type Ž i.e., y . equal to Ž . e. 3. An inefficient le¨ el of effort for the less efficient type  , characterized by ⬘Ž e . s 1 y ⬘Ž e . 1q 1y y
⬘ Ž e .
Ž 1 y . Ž 1 q .
1y
u⬘ Ž q Ž e . .
Ž 3.9.
u⬘ CE Ž , e .
where CE Ž , e . s uy1 Ž Ž 1 y . u Ž . q u Ž q Ž e . . . . 4. When 1 q G Ž u⬘Ž Ž e .. q Ž1 y . u⬘Ž0..ru⬘w CEŽ0, e .x, the less efficient firm  has no rent Ž Case 1., as in the risk-neutral case. Otherwise the  firm has a positi¨ e rent Ž Case 2.. Proof. To solve for the maximization of Ž3.8. under the constraints Ž3.3. to Ž3.6., we observe that Ž3.3. and Ž3.6. imply Ž3.5., and we momentarily ignore Ž3.4.. We will check later that it is satisfied by the solution obtained. We are therefore left with two constraints, Ž3.3. and Ž3.6.. Contrary to the case of risk-neutral firms, the individual rationality constraint of the less efficient type, Ž G 0., is not necessarily binding at the optimum. However,
⭸W ⭸
s y Ž 1 q . q
u⬘ Ž . u⬘ Ž CE .
.
Therefore, as ) from Ž3.3., CE - , which implies that ⭸ Wr ⭸ - 0 and that the constraint Ž3.3. is always binding. Substituting Ž3.3. in Ž3.8., we have a new expression of social welfare, which now depends only on three variables, , C, and C: W Ž , C, C . s S y Ž 1 q . q Ž  y C . q Ž  y C . q C
ž
qŽ 1 y . q Ž  y C . q C
ž
/
q uy1 u q Ž  y C . q Ž 1 y . u Ž . ,
ž ž
/
which is to be maximized under the only constraint Ž3.6..
/
/
Ž 3.10.
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We must then distinguish two cases: Case 1. Ž3.6. is binding. Then s 0 and s Ž  y C .. The necessary condition is, at s 0,
⭸W ⭸
F0m
u⬘ Ž . q Ž 1 y . u⬘ Ž . u⬘ Ž CE .
F Ž 1 q . .
Case 2. Ž3.6. is not binding. If this case occurs, then ⭸ Wr⭸ s 0. The value ) 0 is defined by the smallest 5 non-negative solution of ⭸ Wr ⭸ s 0. Maximizing 6 W with respect to C and C, we obtain the desired result. B Remark 1. Contrary to the risk-neutral case, the inefficient firm may have a rent ŽCase 2.. Because of risk aversion, a marginal increase in profit at s 0 may increase the certainty equivalent by more than Ž1 q ., the cost of public funds. Remark 2. The second, more important observation is that the distortion in the inefficient firm’s effort level has an additional term that lowers this effort level even further. Therefore we move toward a scheme even closer to a cost-plus scheme, i.e., a less powerful incentive scheme. The intuition is simply the following. Since the firm is risk-averse, the certainty equivalent of profit CE is lower than the expectation of profit. Therefore it is as if the weight of the firm in the social welfare function were less than 1 Žsay 1 y ⑀ .. Since the cost of transfers to the firm remains Ž1 q ., the weight of the firm’s rent in the social welfare function is yŽ q ⑀ .. So it is as if the rent were socially more costly; hence the optimal regulation entails a greater distortion from efficiency, to decrease the rent further. Remark 3. An interesting example of utility function is the exponential utility: uŽ x . s Ž1 y exp y x .r , which has the merit of neutralizing income effects. In that case, s 0 and u⬘Ž CE . s u⬘Ž Ž e .. q Ž1 y . u⬘Ž0.. The equation giving the inefficient firm’s effort level takes the form
⬘Ž e . s 1 y
1q 1y
= 1y 5 6
⬘Ž e . y
⬘ Ž e .
Ž 1 y . Ž 1 q .
1
q Ž 1 y . exp Ž Ž e . .
.
Because of the certainty equivalent, W is not necessarily concave in . We assume that interior maxima exist Ži.e., with positive effort levels..
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Straightforward differentiation of this expression gives, then, that derd - 0 for an interior maximum, since ⬘ ) 0. As risk aversion increases, the optimal contract moves toward a cost plus contract.7
4. THE OPTIMAL CONTRACT IN THE CONTINUUM CASE We consider the same model as in Section 3, but with a continuum w  ,  x of values for the adverse selection parameter. Let t Ž C . be the net transfer to the firm when the cost equals C s  y e. We define the profit function as
Ž  . s max Ž  , C . C
where Ž  , C . s t Ž C . y Ž  y C . . Ž 4.1.
As explained in Section 3, risk aversion does not affect the incentive compatibility conditions that we obtain after substituting e s  y C. The first-order and second-order incentive compatibility conditions Žwhich are sufficient because ⬙ ) 0 and because the Spence᎐Mirrlees condition ⭸ 2r⭸ ⭸ C ) 0 is fulfilled. are, for all  ,
⬘ Ž  . s y ⬘ Ž e Ž  . . ,
Ž 4.2.
C⬘ Ž  . s 1 y e⬘ Ž  . G 0.
Ž 4.3.
and
By integration of Ž4.2., we obtain
Ž . sŽ. q
H ⬘ Ž e Ž x . . dx.
Ž 4.4.
Therefore the Žex post. individual rationality condition Ž Ž  . G 0 for all  . can be transformed as usual into the simpler condition:
Ž  . G 0. 7
Ž 4.5.
We conjecture that the general equilibrium effect goes in the same direction: if all regulated firms become more risk-averse, the cost of the rents in the social welfare function is likely to be increased, which should also increase the shadow cost of public funds, inducing a further move toward a cost-plus contract.
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Furthermore, from Laffont and Tirole Ž1993., we know that the optimal contract t*Ž C . will be implementable by a menu of linear schedules if t*Ž C . is convex. From Ž4.3. and ⬙ G 0, this condition is satisfied if and only if 8 e⬘Ž  . F 0. Let CEŽ . denote the certainty equivalent of . Taking the same objective function for the regulator as in Section 2, we can write his optimization problem as max W s S y Ž 1 q . E q Ž e . q  y e q CE Ž . under the constraints Ž 4.2. , Ž 4.3. , Ž 4.5. .
Ž 4.6.
To eliminate income effects, we choose the exponential specification ŽCARA utility function. uŽ x . s Ž1 y exp y x .r , which gives CEŽ . s 1 y log E Ž ey .. As in Section 3 Žand see Lemma 1., this implies that the individual rationality constraint of the least efficient firm is always binding at the optimum. LEMMA 1. With a CARA utility function, constraint Ž4.5. is always binding at the optimum. Proof. Suppose by contradiction that Ž  . ) 0 and decrease it by ⑀ . Then by Ž4.4. Ž  . uniformly decreases by ⑀ . Therefore E Ž . and CEŽ . decrease by ⑀ but, because ) 0, the total effect on W is positive. B Under the monotone hazard rate condition Ž drd  Ž F Ž  .rf Ž  .. ) 0., we know that in the case of risk neutrality Ž s 0., the optimal contract is separating or equivalently involves no bunching Žsee Laffont᎐Tirole, 1986.. We show below that it is also true for small enough, in which case we obtain a characterization of the optimal contract.9 PROPOSITION 2. Under the monotone hazard rate assumption and for small enough, an interior solution for the optimal contract is characterized by The first-order condition of the firm’s program is t⬘w C Ž  .x s y ⬘Ž eŽ  ... By differentiation we obtain t⬙ w C Ž  .xC⬘Ž  . s y ⬙ Ž eŽ  .. e⬘Ž  .. Under our assumptions, t⬙ and e⬘ have opposite signs. 9 Technically, the problem turns out to be quite similar to the one studied by Salanie ´ Ž1990.. Salanie ´ studies the case of a principal᎐agent relationship with a risk-averse agent and an ex ante individual rationality condition. Here we integrate in the objective function the analog of his ex ante IR constraint Žour certainty equivalent term., but we also have ex post IR constraints for the agent. 8
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effort le¨ els e*Ž  . and rents *Ž  ., jointly gi¨ en by
⬙ Ž e* Ž  . .
⬘ Ž e* Ž  . . s 1 y
½
fŽ  .
FŽ  . y
H Ž exp y * Ž x . . f Ž x . dx
5
Ž 1 q . E exp y * Ž  . Ž 4.7.
and
*Ž  . s
H ⬘ Ž e* Ž x . . dx.
Ž 4.8.
Furthermore, implementation by a menu of linear schedules is possible. Proof. See Appendix 1. Note that in general, e*Ž⭈. and *Ž⭈. are jointly determined by a system of two integral equations. However, for s 0, *Ž⭈. disappears from Eq. Ž4.7., and we obtain the classical formula
⬘ Ž e* Ž  . . s 1 y
FŽ  .
1 q fŽ  .
⬙ Ž e* Ž  . . .
Ž 4.9.
To see the additional role of risk aversion, Ž4.7. can be rewritten as follows:
⬘ Ž e* Ž  . . s 1 y y
FŽ  .
1 q fŽ  .
⬙ Ž e* Ž  . .
⬙ Ž e* Ž  . . F Ž  . 1q
fŽ  .
½
1y
H exp y * Ž s . f Ž s . ds H exp y * Ž s . f Ž s . ds
5
Ž 4.10. Notice that the additional term vanishes 10 for  s  and  s  . As we show below, risk aversion unambiguously tilts the optimal contract toward cost-plus schemes. 10
The situation is more complex for non exponential utilities. Indeed, let Ž x . s yu⬙ Ž x .ru⬘Ž x . denote the Žlocal. index of risk aversion of any concave VNM utility function. It is not difficult to prove that, in the case of no-bunching, formula Ž4.10. becomes 1 y ⬘ Ž e* Ž  . .
⬙ Ž e* Ž  . .
s
FŽ  .
Ž1 q . f Ž  .
ž
q 1y
H Ž exp y H0 *Ž x . Ž s . ds . f Ž x . dx F Ž  . u⬘ Ž CE Ž * . .
/
.
Ž 4.10bis . When u is not exponential, the second term between brackets does not vanish for  s  .
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159
As increases, the condition e⬘ F 0 may become violated, and then decentralization by a menu of linear schedules is not possible. Moreover for large enough, the constraint e⬘ F 1 may become binding, and bunching occurs. To fully characterize these phenomena, we pursue the analysis in the case of a uniform distribution on w  ,  x and a quadratic disutility of effort Ž e . s 12 e 2re* Žtherefore, e* represents the first best effort level: ⬘Ž e*. s 1.. In the no-bunching region we can examine the comparative statics of the Žsecond best. optimal effort level. Let us denote by  ª e*Ž  , . this optimal effort function when the index of risk aversion is . PROPOSITION 3. When there is no bunching, effort decreases with the index of risk a¨ ersion, : Ž ⭸ e*r⭸ .Ž  , . F 0. Proof. See Appendix 2. Furthermore, we have: PROPOSITION 4. There exist 1 , 2 with 2 ) 1 ) 0, such that Ža. For F 1 , implementation by a menu of linear schedules is possible: Ž ⭸ e*r⭸ .Ž  , . F 0 for all  . Žb. For F 2 , there is no bunching: Ž ⭸ e*r⭸ .Ž  , . - 1 for all  . Žc. For ) 2 , there is complete bunching on some inter¨ al, w  2 Ž .,  x. Outside this inter¨ al, the solution in¨ ol¨ es no bunching:
⭸ e* ⭸
Ž , . - 1
for  in  ,  2 Ž . w ,
Ž , . s 1
for  in  2 Ž . ,  .
and
⭸ e* ⭸
Proof. See Appendix 3. In the limit case of an infinite risk aversion, we have CE Ž . s Ž  . . Since Ž4.5. remains binding, the solution is the same as with risk neutrality, except that the weight of the profit in the objective function is zero. The solution is then defined by
⬘Ž e Ž  . . s 1 y
FŽ  . fŽ  .
⬙ Ž eŽ  . . .
Ž 4.11.
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FIG. 1. The comparative statics of the second best solution as increases.
All of our results on the comparative statics of the effort level are represented in Fig. 1 Žin the Ž C,  . plane., with C Ž  , . s  y eŽ  , ..
5. THE SPECIAL CASE OF THE BARON᎐MYERSON MODEL We study in this section the extension to a risk-averse firm of the Baron᎐Myerson Ž1982. model. It can be considered a variant 11 of our problem in which there is no effort variable, and the costs are not observable by the regulator. Instead, it is the size of the project Žquantity q produced by the firm. that will be used for inducing self-selection of the 11 In the original formulation ŽBaron᎐Myerson, 1982., the weight of the firm’s profit in the social welfare function is less than one. For homogeneity with the rest of the paper, we examine the variant in which there is a cost of public funds. This allows us to focus on the main difference between the Laffont᎐Tirole approach used in the rest of the paper, and the Baron᎐Myerson approach used in this section, namely that costs are no longer observable.
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different types of firms. The cost function is now C s  q. If P Ž q . is the inverse demand function, we obtain Žas before, Ž  . denotes the profit of the firm. the expression of the social welfare function: W s E V Ž q Ž  . . y Ž 1 q . t Ž  . q CE Ž . ,
Ž 5.1.
where V Ž q . s H0q P Ž s . ds denotes consumers’ surplus, t Ž  . s  q Ž  . q Ž  . is the total transfer to the firm, and CEŽ . is the certainty equivalent of the firm’s profit. W has to be maximized under the constraints
⬘ Ž  . s yq Ž  . q⬘ Ž  . F 0
Ž first-order incentive compatibility .
Ž second-order incentive compatibility .
Ž. G0
Ž individual rationality . .
Ž 5.2. Ž 5.3. Ž 5.4.
Replacing t Ž  . with  q q , and q with y ⬘, we obtain a new expression for W: W Ž . s E V Ž y ⬘ . y Ž 1 q . Ž y ⬘ q . q CE Ž . . Under the CARA specification, EC Ž . s yŽ1r . log EŽexp y ., and Ž5.4. is always binding at the optimum. PROPOSITION 5. When there is no bunching, the second best solution under risk a¨ ersion is characterized by
P Ž q Ž  . . s Ž 1 q .  q
ž
FŽ  . fŽ  .
/
y
H exp y Ž s . dF Ž s . f Ž  . H exp y Ž s . dF Ž s .
.
Ž 5.5. For s 0, we obtain the Baron᎐Myerson formula: P0 Ž q Ž  . . s Ž 1 q .  q
FŽ  . fŽ  .
.
Because of risk a¨ ersion, the distortion is greater when ) 0. Proof. See Appendix 4.
Ž 5.6.
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Still assuming the absence of bunching, Proposition 5 can easily be extended to more general utility functions. Denoting the local index of risk aversion yŽ u⬙ru ⬘.Ž x . by Ž x ., we can write the generalization of formula Ž5.5.: P Ž q Ž  . . s Ž 1 q .  q
ž
FŽ  . fŽ  .
/
H exp y H0 *Ž x . Ž s . ds f Ž x . dx
y
f Ž  . u⬘ Ž CE Ž * . .
.
Using Ž5.6., this can also be written as P Ž q Ž  . . sP0 Ž q Ž  . . q
FŽ  . fŽ  .
1y
H expyH0 *Ž x . Ž s . ds f Ž x . dx F Ž  . u⬘ Ž CE Ž * . .
,
where P0 Ž q Ž  .. is the Baron᎐Myerson price. Therefore the sign of the additional distortion caused by risk aversion is determined by the term between square brackets. In particular, the distortion will always be positive if and only if the function
Ž  . s FŽ  .¨
H␣ Ž s . f Ž s . ds
y

H ¨ Ž ␣ Ž s . . f Ž s . ds
is always positive, where we have introduced the following notations: ¨ Ž x . s u⬘ uy1 Ž x . ,
LEMMA 2.
␣ Ž x . s u *Ž x . .
is always non-negati¨ e m Ž  . G 0.
Proof. ⬘Ž  . s f Ž  .w ¨ Ž H ␣ Ž s . f Ž s . ds . y ¨ Ž ␣ Ž  ..x. Since ¨ and ␣ are decreasing, ⬘ is first positive, then negative. Therefore has an interior maximum. Since Ž  . s 0, is always non-negative if and only if Ž  . G 0. B But now we have that
Ž  . s¨
žH


␣ Ž s . f Ž s . ds y
/
H¨ Ž ␣ Ž s . . f Ž s . ds.
Therefore, by Jensen’s inequality, Ž  . is non-negative only when ¨ is concave Žor linear, as in the exponential specification.. The usual assumption in risk theory Žsee, for instance, Kimball, 1990. is that ¨ is convex.
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This is the case, for instance, when u is DARA, i.e., it exhibits decreasing absolute risk aversion. In that case Ž  . - 0, which means that there exists a threshold  * such that 䢇
For  -  *: risk aversion leads to a higher distortion.
䢇
For  )  *: risk aversion leads to a lower distortion.
6. APPLICATIONS AND CONCLUSION Laffont and Tirole Ž1986. showed that, with risk-neutral agents, noise in the cost observation does not prevent the regulator from implementing the same allocation as in the absence of noise. This result was generalized by Caillaud et al. Ž1992., Demougin Ž1989., and Melumad and Reichelstein Ž1989.. However, when risk aversion is introduced, there is no hope of finding a simple characterization of the optimal contract in the noisy observation case. On the contrary, in the case of non-noisy cost observation, we have shown that even with risk aversion, one can obtain a simple and precise characterization of the optimal contract. This result can be quite useful when the agent is subject to other risks that do not interfere with the observation of his cost. Suppose, for example, that the firm’s activity creates an environmental risk of social cost E with probability , and suppose further that the firm can affect this probability with another effort level d, which interacts with e in the disutility of effort as Ž e q d .. To study the insurance of this environmental risk, risk aversion is crucial, and the results of this paper are an essential building block of such an analysis Žsee Laffont, 1995, for further details.. In conclusion, this paper has provided a solution to a class of principal agent problems in which risk aversion matters, and at the same time both adverse selection and moral hazard affect the outcomes observable by the principal. Risk aversion of the agent, as expected, moves the optimal contract toward cost-plus contracts, i.e., he accepts a higher allocation inefficiency to mitigate the informational rent made more costly by risk aversion. As a consequence, the optimal contract typically involves bunching for inefficient types when risk aversion is strong. This provides a new explanation for the cost ceilings observed in many contracts without having to appeal to special shapes of the hazard rate, as in Laffont and Tirole Ž1993, Chap. 1..
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APPENDIX 1: PROOF OF PROPOSITION 2 Let Ž  . and Ž  . be the Lagrange multipliers associated with constraints Ž4.2. and Ž4.3.. The Lagrangian of the regulator’s optimization program is L s S q CE Ž . y Ž 1 q .
H Ž Ž  . q Ž e Ž  . . q  y e Ž  . . f Ž  . y Ž  . Ž ⬘ Ž  . q ⬘ Ž e Ž  . . . y Ž  . Ž e⬘ Ž  . y 1 . d 
With respect to e, we have a classical calculus of variations problem Žexcept that Ž4.3. is not always binding.. The necessary condition is thus
⬘ Ž  . s Ž ⬘ Ž e Ž  . . y 1. f Ž  . y ⬙ Ž e Ž  . . Ž  . Ž  . Ž e⬘ Ž  . y 1 . s 0
Ž Euler equation . Ž A1.1.
Ž complementarity conditions . . Ž A1.2.
With respect to , this is not a classical problem because of CEŽ .. So we have to compute the derivative of L Žwith respect to . in the direction of an arbitrary differentiable perturbation function hŽ  . satisfying Ž4.5. with equality, i.e., such that hŽ  . s 0.
⭸L ⭸
⭈ h s lim
L Ž q th . y L Ž . t
tª0
s
Eh exp y E exp y
y Ž 1 q .
H h Ž  . f Ž  . y h⬘ Ž  . Ž  . 4 d  ,
or
H
½
exp y Ž  . f Ž  .
Ž 1 q . H exp y Ž x . f Ž x . dx
5
y f Ž  . hŽ  .
q Ž  . h⬘ Ž  . d  s 0 Ž A1.3.
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Let BŽ  . s

H
exp y Ž s . f Ž s .
½
Ž 1 q . H exp y Ž x . f Ž x . dx
5
y f Ž s . ds.
Then ŽA1.3. can be rewritten as
Hh Ž  . B⬘ Ž  . d  q H Ž  . h⬘ Ž  . d  s 0. Integrating by parts in the first integral and using the fact that hŽ  . s 0 and B Ž  . s 0, we get
H yB Ž  . q Ž  .
h⬘ Ž  . d  s 0.
Since h⬘Ž  . is arbitrary, we deduce
Ž  . s BŽ  . or
Ž  . s
ey Ž s. f Ž s .

H
Ž 1 q . H ey Ž x . dF Ž x .
ds y F Ž  .
Ž A1.4.
When constraint Ž4.3. is not binding, Ž  . ' 0, and therefore ⬘Ž  . s 0. Substituting ŽA1.4. into ŽA1.1., we get the desired result:
⬘Ž e Ž  . . s 1 y
⬙ Ž eŽ  . . fŽ  .
½
FŽ  . y
H exp y Ž x . f Ž x . dx
Ž 1 q . E exp y Ž  .
5
.
For s 0, Ž drd  . Ž Frf . ) 0 and G 0 ensure that e is strictly decreasing. This will remain true by continuity for small enough. Therefore constraint Ž4.3. is not binding for small enough. Furthermore, since e⬘ - 0, the transfer function is convex for small enough, and implementation with linear schedules is possible.
APPENDIX 2: PROOF OF PROPOSITION 3 In the uniform-quadratic case, the expression of ŽA1.1. becomes, in the no-bunching region Žwhere ⬘Ž  . ' 0.,
ˆe Ž  . s e* q Ž ⌬  . Ž  . ,
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with ⌬  s  y  ,
Ž ⌬  . Ž  . s H


ž
exp y Ž s .
Ž 1 q . ␥
y 1 ds,
/
␥ s E w exp y x .
and
To examine the comparative statics properties of this solution with respect to , we make explicit the dependence with respect to :
ˆe Ž  , . s e* q H


H
with Ž  , . s
ž
exp y Ž s, .
Ž 1 q . ␥ Ž .
y 1 ds
/
ˆe Ž s, . ds e*
and
␥ Ž . s
1
 exp y Ž s, . H ⌬ 
ds.
Hence
⭸ˆ e ⭸
Ž , . s
1
exp y Ž  , . y 1 ) y1, Ž A2.1.
Ž 1 q . ␥ Ž .
and since
⭸ ⭸ ⭸ 2ˆ e ⭸ 2
Ž , . s Ž , . s
yeˆŽ  , . e*
,
ˆ e Ž  , . exp y Ž  , . e* Ž 1 q . ␥ Ž .
Ž A2.2. )0
at any interior solution Žfor which ˆ eŽ  , . ) 0.. The solution is therefore Žstrictly . convex in the no-bunching region. We can eliminate Ž  , . between ŽA2.1. and ŽA2.2.. We obtain a second-order differential equation in ˆ eŽ  , .:
⭸ 2ˆ e ⭸
2
Ž , . s
ˆ eŽ  , . e*
ž
1q
with
ˆe Ž  , . s e*,
⭸ˆ e ⭸
Ž , .
/
Ž A2.3.
REGULATION OF A RISK-AVERSE FIRM
167
and
ˆe Ž  , . s e* y
1q
⌬.
Let uŽ  , . s Ž ⭸ ˆ er⭸ .Ž  , .. We want to prove that u is nonpositive. By differentiation of ŽA2.3. with respect to , we obtain e*
⭸ 2u ⭸
2
Ž  , . s ˆe Ž  , . q u Ž  , . 1 q
ž
q ˆ eŽ  , .
⭸u ⭸
⭸ˆ e ⭸
Ž , .
Ž , .,
/ Ž A2.4.
and u Ž  , . s u Ž  , . s 0. Since u is continuous as a function of  , it has a maximum on w  ,  x, say at  *. Suppose that this maximum is interior. We have, then,
⭸u ⭸
Ž  *, . s 0
⭸ 2u
and
⭸ 2
Ž  *, . F 0.
Therefore by ŽA2.4.,
ˆe Ž  *, . q u Ž  *, . 1 q
ž
⭸ˆ e ⭸
Ž  *, . F 0.
/
Now ŽA2.2. and ŽA2.3. imply that 1 q Ž ⭸ ˆ er⭸ .Ž  *, . ) 0. Therefore Ž . u  *, - 0, which is absurd, since  * is supposed to maximize u and uŽ  , . s uŽ  , . s 0. Notice that we have proved in fact that uŽ  , . - 0 for all  in x  ,  w when there is no bunching. Since uŽ  , . s uŽ  , . s 0, this proves in particular the desired result that u is nonpositive. B
APPENDIX 3: PROOF OF PROPOSITION 4 It is analogous to the proof of Proposition 1 in Salanie ´ Ž1990.. We have just proved that uŽ  , . - 0 for  in x  ,  w and uŽ  , . s 0. Therefore Ž ⭸ ur ⭸ .Ž  , . G 0. But this is also equal to Ž ⭸r⭸ .ŽŽ ⭸ ˆ er⭸ .Ž  , ... Thus Ž ⭸ ˆ er⭸ .Ž  , . is an increasing function of . Parts Ža. and Žb. of Proposition 4 are then immediate consequences of
168
LAFFONT AND ROCHET
LEMMA 3.
⭸ˆ e
lim
⭸
ªq⬁
Ž  , . ) 1.
As a consequence, ᭚ 1 - 2 , such that Ž ⭸ ˆ er⭸ .Ž  , 1 . s 0 Ž⭸ ˆ er⭸ .Ž  , 2 . s 1.
and
Proof of Lemma 3. Suppose by contradiction that ᭙ Ž ⭸ ˆ er⭸ .Ž  , . F 1. By convexity of ˆ e, this implies ᭙ ˆ eŽ  , . G ˆ e Ž  , . y Ž  y  . s e* y
1q
⌬ y Ž  y  ..
Ž A3.1. By integration of ŽA2.3. between  and  , we obtain e*
½
⭸ˆ e
Ž , . y ⭸ s
⭸ˆ e ⭸
Ž , . 1
5 1
Hˆe Ž  , . d  q 2 ˆe Ž  , . y 2 ˆe Ž  , . . 2
2
Ž A3.2.
Since Ž ⭸ ˆ er⭸ .Ž  , . G y1 and Ž ⭸ ˆ er⭸ .Ž  , . F 1, the left-hand side of ŽA3.2. is less than 2 e*r , whereas the right-hand side is Žbecause of ŽA3.1.. greater than a positive term that does not depend on . This is impossible for large enough. B We now come back to Proposition 4: part Žc. remains to be established. For this we need to rewrite the necessary conditions for the case of bunching:
Ž ⌬  . e* ⬘ Ž  . s e Ž  . y e* y Ž ⌬  . Ž  . , Ž  . s Ž  . s 0, Ž  . Ž e⬘ Ž  . y 1 . s 0, ^`_ ^ ` _ G0
Ž Euler . Ž A3.3.
Ž transversality . Ž complementarity .
Ž A3.4. Ž A3.5.
F0
Ž ⌬  . Ž  . s H


ž
exp Ž y Ž s . .
␥ Ž 1 q .
y 1 ds,
/
Ž first-order condition with respect to . , Ž A3.6. where Ž  . s H eŽ s .re* ds; ␥ s EwexpŽy Ž s ..x.
REGULATION OF A RISK-AVERSE FIRM
169
This implies, by differentiating ŽA3.3.,
Ž ⌬  . e* ⬙ s ye* ⬙ y Ž ⌬  . ⬘ s ye* ⬙ q 1 y
exp Ž y .
␥ Ž 1 q .
.
Let us take t s w ␥ Ž1 q .xy1 as a parameter, and define: H Ž . s 2 y t expŽy .. In the no-bunching zone, ⬙ is identically zero, and we have e* ⬙ q 1 s H Ž Ž  . . G 0. However, as indicated by Figure 2, H may take negative values when t ) 2. In that case, e* ⬙ q 1 is identically zero, and we have
Ž ⌬  . e* ⬙ Ž  . s H Ž Ž  . . . We are going to construct the solution to our problem by setting
½ ½
e* ⬙ Ž  . q 1 s Max Ž 0, H Ž Ž  . . . G 0
Ž  . s 0, ⬘ Ž  . s y1
Ž ⌬  . e* ⬙ Ž  . s Min Ž 0, H Ž Ž  . . . F 0
Ž A3.7.
Ž. sŽ. s0
We now prove that the necessary conditions are indeed satisfied. ŽA3.4. is fulfilled by construction; moreover,
concave Ž  . s Ž  . s 0 « ᭙ Ž  . G 0
5
Ž which implies Ž A3.5. . .
FIG. 2. Graph of the function H in the case t ) 0. The value c is equal to Ž1r . logŽ tr2..
170
LAFFONT AND ROCHET
Let us define e Ž  . s ye* ⬘ Ž  . , and 
Ž ⌬  . Ž  . s H Ž t exp Ž y Ž s . . y 1 . ds 
Ž this implies Ž A3.6. . .
We have
Ž 1 y e⬘ . s 1 q e* ⬙ Ž  . G 0, with equality when Ž  . / 0, since then H Ž Ž  .. s 0. Moreover, we have to establish ŽA3.3., by proving that the function d
K Ž  . s Ž ⌬  . e* ⬘ Ž  . y e Ž  . q e* q Ž ⌬  . Ž  . is identically zero. But KŽ  . s 0
¡ ¢ Ž  . s 0
~e Ž  . s ye* ⬘ Ž  . s 0,
since
Ž  . s 0
on  ,  q ⑀ « ⬘ Ž  . s 0
and for all  , K ⬘ Ž  . s Ž ⌬  . e* ⬙ Ž  . q e* ⬙ q Ž ⌬  . ⬘ Ž  . s 0. This proves that K Ž  . is identically zero. Finally, we have two things to do: Ža. Show that bunching occurs on the upper part of the distribution of  . Indeed, this occurs when H w Ž  .x F 0 or Ž  . F c m  G  2 with Ž  2 . s c Žthis is because is decreasing.. Notice that when Ž  . G c there is no bunching. Žb. Show that t can be chosen so that EwexpŽy .x s ␥ s 1rt Ž1 q . . Indeed, if we denote by Ž  , t . the solution of ŽA3.7., it is clear that when t ª 0, e*Ž ⭸ 2r⭸ 2 .Ž  , t . q 1 ª 2 and Ž  , 0. remains bounded, so that lim t ª 0 tEwexpŽy Ž  , t ..x s 0. Similarly, when t ª q⬁, e*Ž ⭸ 2r⭸ 2 .Ž  , t . q 1 ª 0, and Ž  , q⬁. is also bounded, which implies lim t ªq⬁ tEwexpŽy Ž  , t ..x s q⬁. Therefore, by continuity, the solution exists. Finally, in the particular case in which there is no bunching Ž ᭙ H Ž Ž  .. G 0 ., we obtain Eq. ŽA2.3.. Indeed, ŽA3.7. gives e* ⬙ Ž  . q 1 s 2 y t expŽy ..
REGULATION OF A RISK-AVERSE FIRM
171
By differentiating this, we get e* s y
⭸ 2ˆ e ⭸ 2
s t ⬘ exp Ž y . ,
but
¡t expŽ y . s 1 y e* ⬙ s 1 q ⭸ ˆe
~
⭸
¢ ⬘ s y e*ˆ e
Therefore,
⭸ 2ˆ e ⭸
2
s
e*
ž
1q
⭸ˆ e ⭸
/
ˆe,
and we are back to Eq. ŽA2.3..
APPENDIX 4: PROOF OF PROPOSITION 5 The proof is exactly analogous to that of Proposition 2: we differentiate W in the direction of an arbitrary perturbation function h Žsuch that hŽ  . s 0.: W⬘h s lim
W Ž q th . y W Ž .
tª0
t
s E yP Ž q . h⬘ y Ž 1 q . Ž y h⬘ q h . q
h exp y
␥
,
with ␥ s Ewexp y x. Integrating by parts, we obtain a result analogous to the Euler equation,
¡d
f Ž  . Ž yP Ž q Ž  . . q Ž 1 q .  s f Ž  .
~ d ¢P ž q Ž  . / s Ž 1 q .  .
½
exp y
␥
y Ž 1 q .
5
172
LAFFONT AND ROCHET
Thus P Ž q Ž  . . s Ž 1 q .  q
FŽ  .
y
fŽ  .
H exp y Ž s . dF Ž s . f Ž  . H exp y Ž s . dF Ž s .
.
def
Let Ž  . s f Ž  .w P0 Ž q Ž  .. y P Ž q Ž  ..x be a measure of the distortion due to risk aversion. We have
Ž  . s Ž  . s 0 ⬘Ž  . s f Ž  . 1 y
and exp y Ž  . H
exp y Ž s . dF Ž s .
.
At  s  , Ž  . s 0, and therefore ⬘Ž  . - 0. If Ž  . was not always non-negative, there would be a ˆ such that Ž ˆ. s 0 and ⬘Ž ˆ. ) 0, i.e., ˆ
F Ž ˆ . s
H exp y Ž s . dF Ž s . H exp y Ž s . dF Ž s .
Ž A4.1.
and
H exp y Ž s .
dF Ž s . ) exp y Ž ˆ . .
Integrating by parts in the right hand side of ŽA4.1., we obtain F Ž ˆ . s
F Ž ˆ . exp y Ž ˆ . H exp y Ž s . dF Ž s .
ˆ
y
H F Ž s . e Ž s . exp y Ž s . ds H exp y Ž s . dF Ž s .
,
which yields a contradiction. B
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