the above unique and non-linear contract does not approach the linear contract as ... of optimal contracts under the double moral-hazard when the agent is risk-.
journal of economic theory 82, 342378 (1998) article no. ET962439
Linear Contracts and the Double Moral-Hazard* Son Ku Kim - and Susheng Wang Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People's Republic of China skkimust.hk; s.wangust.hk Received September 5, 1996; revised April 24, 1998
This paper studies the characteristics of optimal contracts when the agent is riskaverse in the double moral-hazard situation in which the principal also participates in the production process. It is already known that a simple linear contract is one of many optimal contracts under the double moral-hazard when the agent is riskneutral. We find that the agent's optimal incentive scheme in this case is unique and non-linear, but less sensitive to output than would be designed under a single moral-hazard. We also find that the linear contract is not robust in the sense that the above unique and non-linear contract does not approach the linear contract as the agent's risk-aversion approaches zero. Journal of Economic Literature Classification Number: D82. 1998 Academic Press
1. INTRODUCTION A common assumption adopted in the standard moral-hazard literature is that a principal is passive as far as production is concerned. 1 That is, the principal delegates all production decisions to an agent, and designs an incentive contract that is based on common observables correlated with the agent's hidden action choices, such as outputs and realized costs. In many principal-agent relationships, however, the principals do have some choice variables that substantially affect the outcomes. For example, the demand for a product is affected not only by a downstream firm's (agent) sales effort but also by an upstream firm's (principal) manufacturing inputs that determine the quality of the product. The relationship between a franchiser * Correspondence address: Department of Economics, HK University of Science and Technology, Clear Water Bay, Hong Kong, People's Republic of China, Telephone (852) 23587617. Fax: (852) 23582084. E-mail: skkimust.hk. We thank Leonard Cheng, David Hirshleifer, Grant Taylor, and Mike Waldman for their helpful comments. Also, special thanks are given to an associate editor and two anonymous referees. Obviously, all errors are ours. 1 For the standard agency model, see Ross [26], Stiglitz [30], Mirrlees [22], Harris and Raviv [6], Holmstrom [8], and Shavell [28] among others.
342 0022-053198 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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343
and a franchisee is another good example. In her empirical analysis, Lafontaine [16] concludes that a double moral-hazard argument on franchising best explains the data. 2 It is only just recently that the double moral-hazard model has begun to be theoretically discussed in the literature. Romano [25], and Bhattacharyya and Lafontaine [2] show that a simple linear contract in which the principal and the agent proportionally share the output after a certain amount of transfer is made between them implements the second-best outcome when the agent is risk-neutral. When the principal also provides inputs which affect the outcomes, the incentive provision for both the agent's action choice and the principal's own effort level must be taken into account when designing the agent's incentive scheme. There is a strict trade-off between these two incentives, and the full information outcome is not obtainable even if the agent is risk-neutral. Our objective in this paper is two-fold. First, we analyze the characteristics of optimal contracts under the double moral-hazard when the agent is riskaverse. When the agent is risk-averse, risk sharing between the principal and the agent is also to be considered when designing a contract. Being based on the first-order approach, we show that the optimal contract is not generally linear in this case. We also show that the optimal contract under the double moral-hazard is less sensitive to the outcomes than that under the single moral-hazard. Intuition tells us that, since the principal as well as the agent can influence the outcomes, the changes in outcomes convey only partial information about the agent's hidden effort choice. Secondly, we examine whether the simple linear contract, which is one of the many optimal contracts when the agent is risk-neutral, remains as a limiting contract when the agent's risk aversion approaches zero, that is, if linear contracts remain robust with respect to the agent's risk aversion. Romano's or Bhattacharyya and Lafontaine's results seem to suggest that the double moral-hazard framework offers a nice explanation for the prevalence of simple linear contracts. However, there are many other contracts that achieve the same efficiency on the double moral-hazard situation when the agent is risk-neutral. Therefore, we must examine whether and under what conditions the linear contract is not dominated by other forms of contracts before we accept the explanation. It is indeed found that the linear contracts are NOT robust when the agent's risk aversion converges to zero, suggesting that a linear contract is not a good approximation of the optimal contract under double moral-hazard when the agent is almost risk-neutral. There are few other papers which also have examined the optimal contracts under the double moral-hazard. Eswaran and Kotwal [4] examine the 2 Also, see Mathewson and Winter [19] for evidence of moral-hazard on the franchiser's side.
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characteristics of optimal contracts in agriculture under the double moralhazard situation by confining their analysis to linear contracts. Rubin [27] and Mathewson and Winter [19] analyze royalty contracts in franchising under the double moral-hazard. Demski and Sappington [3] show that the full information outcome can be obtained even in the double moral-hazard situation when one party can exercise a buyout option after observing the other party's choice variable. 3 However, none of the above has studied the optimal contracts in double moral-hazard when the agent is risk-averse, and nobody has examined the robustness of linear contracts when the agent's risk aversion converges to zero. Another closely related work was done by Al-Najjar [1]. He considers a situation in which a risk-neutral principal contracts with N risk-averse agents and allocates his own effort inputs among them. Thus, he basically considers a double moral-hazard model with risk-averse agents which is the same as ours. However, one big difference between his and ours is that the number of agents plays a key role in his paper, while it is fixed by one in ours. He shows that when the number of agents increases the situation arbitrarily closely approaches N replications of the single moral-hazard situation. It is because the principal's incentive problem gets smaller as the number of agents increases. The rest of the paper is organized as follows: In Section 2, our basic model is formulated and some results found in the literature assuming the agent is risk-neutral are explained. In Section 3, we characterize the optimal contract under the double moral-hazard when the agent is riskaverse. And in Section 4, we show that the linear contract is not a limiting contract when the agent's risk aversion converges to zero. In Section 5, we provide some numerical examples, and finally, in Section 6, we make some concluding remarks. 2. RISK-NEUTRAL AGENT A risk-neutral principal hires a risk-neutral agent, and they undertake a joint project. After the principal designs a wage contract s for the agent, she provides her effort e # R + , and the agent provides his effort a # R + , noncooperatively (hereafter, we use ``she'' for the principal and ``he'' for the agent). A composite effort is represented by c=C(a, e), where C: R 2+ Ä R is continuous and differentiable. The output, x, which is commonly observable, is a function of the composite effort C(a, e) and the state of nature % # 3, that is, x=X[C(a, e), %]. The randomness of x is suppressed by parameterization, and F(x | c) denotes the output distribution function conditional on the given composite effort, while f (x | c) represents the output density function. Since x is the only observable, the agent's wage scheme s must be based on x, 3
Also, see Mann and Wissink [18].
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i.e., s(x). V(e) and v(a) denote disutilities of the principal's and the agent's efforts, respectively. Assumption 1. (b)
(a)
v$(a) 0, v"(a)>0, v$(0)=0, v$( )= .
V$(e) 0, V"(e)>0, V$(0)=0, V$( )= .
Assumption 2. C a (a, e)>0, C e(a, e)>0, C aa (a, e)0 represents a relative weight placed on the agent's utility in joint benefits. The second constraint implies that the agent's wage contract must exist in a given interval, [0, l]. This constraint is needed to guarantee the existence of an optimal wage contract when the agent is risk-averse. It is already well known that, in a single moral-hazard setting, a lower bound of the wage contract is needed to guarantee the existence of an optimal contract. Otherwise, the principal can attain the result which is arbitrarily close to the full information outcome by severely penalizing the agent with a very small probability. 9 Thus, the principal ends up with no solution. On the other hand, in a double moral-hazard setting, such an existence issue arises not only from the agent's side but also from the principal's side (e.g., penalizing the principal infinitely). Kim and Wang [15] show that, if there is no upper-bound for the wage contract in the double moral-hazard setting, then the double moral-hazard situation arbitrarily closely approaches the single moral-hazard situation. Actually, the principal, by designing a wage contract which is penalizing himself very severely when the outcome is very low, and otherwise is the same as the wage contract which would be optimally designed in the single moral-hazard situation, can effectively provide her own incentive. Since the principal can arbitrarily reduce the outcome range in which she should take the penalty by increasing the penalty amount, she can obtain the result which is arbitrarily close to that in the single moral-hazard situation. Again, the principal thus ends up with no solution in this case. Therefore, for the existence of an optimal contract we need to impose not only the lower bound but also the upper bound on the wage contract. Economically, placing a lower bound can easily be justified by the existence of a limited liability constraint on the agent's side. Thus, to be symmetric (to impose the limited liability constraint on the principal's side), a suitable way to place an upper bound is to set 0 s(x) >0 be the optimal effort choices for (3.3). 11 Eq. (3.6) gives the following solution as a candidate for the optimal wage contract s*(x) of (3.3): *++ 1 C a*( f c f )(x | C*) 1 = , u$[s*(s)] 1++ 2 C e*( f c f ) (x | C*) 10
(3.7)
Given the weight, *, placed on the agent's utility, the agent's utility function u(s) must be (1*) s. Otherwise, we will have a corner solution which is very trivial. 11 The second-best (a*, e*) with the risk-averse agent will generally be different from the second-best (a C, e C ) with the risk-neutral agent.
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where C*#C(a*, e*), C a* #C a (a*, e*) and C e* #C e(a*, e*), provided that 1++ 2 C e( f c f )[x | C(a*, e*)]{0. Lemma 1. For s*(x) satisfying the Euler Eq. (3.6), a second-order necessary condition for (3.3) is H ss[a*, e*, x, *, + 1 , + 2 , s*(x)] 0,
\x.
(3.8)
Conversely, if s* is continuously differentiable, then (3.8) guarantees s* to be an optimal contract. Proposition 1. In Eq. (3.7), it is true that (i) (ii) (iii)
+ 1 >0. + 2 0. + 1 C a*>*+ 2 C e* if l l 0 , where l 0 satisfies l 0 (u(l 0 )&u(0))=*. 12
Note that + 1 and + 2 are the Lagrangian multipliers of the agent's and of the principal's incentive constraints, respectively. Thus, + 1 represents the degree of moral-hazard problem on the agent's side, whereas + 2 represents that on the principal's side. The fact that + 1 >0 implies that the moral-hazard problem on the agent's side is always positive. However, the fact that + 2 0 implies that the moral-hazard problem on the principal's side may be active (+ 2 >0) or may not (+ 2 =0). Let s*(x | e*) be the optimal incentive scheme that would be designed to motivate the agent to take a*, given e* is already taken. Then, it is a single moral-hazard problem, and, from Holmstrom [8], it is straightforward that 1 fc =*++ 1(e*) C a* (x | C*), u$[s*(x | e*)] f for every x for which the above equation has a solution 0 s*(x | e*) l, and otherwise s*(x | e*) is either 0 or l. The fact that + 2 can be zero indicates that when s*(x | e*) is designed for the agent, the residuals for the principal, x&s*(x | e*), may accidentally induce the principal to take e*. In this case, the optimal contract s*(x) in (3.7) will reduce to s*(x | e*). Such asymmetry between + 1 and + 2 arises from the fact that the agent is risk-averse, and the principal is risk-neutral. The incentive provision for a risk-averse party requires that the incentive scheme be a fine tuning on every x, while that for a risk-neutral party does not require that. Now, we draw Fig. 1 to explain the third result in Proposition 1. 12 z(u(z)&u(0)) is strictly increasing in z. By Assumption 6, z(u(z)&u(0)) goes from less than * to + when z goes from 0 to + . Such l 0 thus exists for any * # (0, ).
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FIGURE 1
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Figure 1 depicts (*++ 1 C a*( f c f ))(1++ 2 C e*( f c f )) in two different cases. In there, x c , x 1 , and x 2 denote the output levels satisfying ( fc f )(x c | C*)=0, *++ 1 C a*( f c f )=(x 1 | C*)=0, and 1++ 2 C e*( f c f )(x 2 | C*)=0, respectively. As drawn in Fig. 1, (*++ 1 C a*( f c f ))(1++ 2 C e*( f c f )) is increasing in x almost everywhere (but not monotonic) if + 1 C a*>*+ 2 C e*. Thus, s*(x) in (3.7) is increasing in x almost everywhere as well. The proof of Proposition 1 in the Appendix shows that we heavily rely on our boundary condition assumption such as 0 s(x) l to prove + 1 C a*>*+ 2 C e* . If we use 0 s(x) x+l as a boundary condition, then we cannot rule out the case in which + 1 C a*l 1 , where l 1 is defined in Proposition 2 and ( f c f )(x | C*) is unbounded below,
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FIG. 2.
Optimal contract under double moral hazard, l 0 l l.
there is a unique optimal contract s*(x), which is NOT monotonically increasing, such that
s*(x)=
{
(u$) &1
1++ 2 C e*( f c f )(x | C*)
\*++ C*( f f )(x | C*)+ , 1
0, l,
a
if
x 0 x;
if if
x 3 xl 1 where l 1 is defined in Proposition 2. The proof of Lemma 3 especially shows that l is usually greater than l 1 when l is very large. Thus, as mentioned earlier, we will focus on the case
FIG. 3.
Optimal contract under double moral hazard, l>l 1 .
DOUBLE MORAL-HAZARD
355
in Proposition 3 rather than the one in Proposition 2 to be consistent with our discussion in the next section. Before we go on to the next section, we provide intuitive explanations for Propositions 2 and 3. The range of x can be decomposed by three intervals such as X 3 =(& , x 2 ), X 2 =[x 2 , x 1 ), X 1 =[x 1 , ), where x 1 and x 2 satisfy 1++ 2 C e*( f c f )(x 2 | C*)=0 and *++ 1 C *( a f c f )(x 1 | C*), respectively. Note that x 2 *+ a 2 C e* [Proposition 1(iii)]. Since H ss >0 for x # X 3 , Hamiltonian H is convex in s. Thus, s*(x) in (3.7) is not a maximizing solution but a minimizing solution for x # X 3 . Therefore, from Lemma 1, it follows immediately that s*(x)=0 if lx 3 and s*(x)=l for xl 1 . For x # X 2 , 1++ 2 C e*( f c f )(x | C*) 0 and *++ 1 C a*( f c f )(x | C*)l 1 ), then penalizing the principal with maximum (i.e., s*(x)=l ) and rewarding the agent with maximum when the outcome is very low (i.e., x &
:# , ;
where ;>0;
#=1
if
:=+ .
When : Ä 0, it converges to risk neutrality; and when : Ä + , it converges to the exponential utility function. The advantage of our HARA utility is that the convergencies to the three important cases done by the convergencies of three separate parameters to zero, as shown in (4.2). This point is critical to us, since we will consider : Ä 0. For Merton's HARA utility, however, when : Ä 0, it moves away from the exponential utility. This does not happen to our HARA utility.
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Define the distance of two contracts s 1 and s 2 in S as &s 1 &s 2 &#
inf E/R, m(E)=0
[ sup |s 1(x)&s 2(x)| ], x # Ec
where m( } ) is the Lebesgue measure. When a sequence of contracts [s n ] converges to a contract s 0 based on the above notion of distance between any two contracts, we say that [s n ] converges to s 0 almost surely. Proposition 4. When the HARA utility converges to risk neutrality by any path, i.e., : n Ä 0 + for any sequence [: n ] of :, the optimal contract s*(x) corresponding to risk aversion : n never converges almost surely to any n unbounded contract, including a linear contract, on x # (x, + ), where x satisfies f c[x | C(a, e )]=0 and (a, e ) is the solution of (2.4) and (2.5). 15 In Section 3, we imposed upper-bound l on the wage contract to guarantee the existence of the optimal contract when the agent was risk-averse. However, we did not impose such an upper-bound in Section 2, since the existence of the optimal contract could easily be guaranteed without the upper-bound when the agent was risk-neutral. Therefore, to be consistent with the previous sections, we consider the case in which l is a sufficiently big number such that l>l 1 for any risk aversion of the agent, i.e., for any :. Therefore, to show the non-convergence result, we focus on the contract in Proposition 3 and consider whether or not the right-hand side of the contract, i.e., the part of the contract defined on [x 0 , ), converges to a linear contract. 16 Since Proposition 3 shows that the optimal contract on [x 0 , ) can solely be characterized by (4.4), we see whether s*(x) in (4.4) converges to a linear contract. Furthermore, the proof of Proposition 4 shows that l 1 is bounded as : goes to zero. Thus, from Lemma 3, we can easily see that such l always exists for any :. As we show in the proof, (+ 1 C *)(+ a 2 C e*) is bounded as : Ä 0. Because, as shown in Proposition 3, the right-hand side of the optimal contract is we can easily see that the optimal contract bounded by (u$) &1 ((+ 2 Ce*)(+1 C*)), a does not converge to a linear contract as : Ä 0. Therefore, Proposition 4 indicates that the linear contract s C given by (2.7) is not robust in the sense that a tiny change in the agent's attitude towards risk will result in a dramatic contractual deviation from s C. More strongly, it shows that any utility function in the HARA family does not generate the linear contract as a limiting contract as the agent's risk aversion approaches zero. Thus, We can actually show (a, e )=(a C, e C ). For the non-convergence result, considering the left-hand side of the contract is meaningless because the left-hand side of the contract is characterized by the boundary condition. 15 16
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the linear sharing rule s C is not a good approximation of the optimal contract when the agent's risk aversion is very small. Although we do not explicitly derive the exact form of a limiting contract, the proof in the Appendix in fact shows that the limiting contract varies with the agent's utility function and the stochastic production function f (x | C). Therefore, it is not surprising that there are many optimal contracts, and the linear contract is only one of them when the agent is risk-neutral.
5. NUMERICAL EXAMPLES In this section, we provide some numerical examples of the contracts in Propositions 2 and 3, using exponential utility and distribution functions. We use Mathcad 7 (copyright of Mathsoft) to solve the model, and use PowerPoint 97 (copyright of Microsoft) to draw the contract curve using the data produced by Mathcad 7. We choose 1 u(s)= (1&e &:s ), : 1 f (x | c)= e &(xc), c
:>0, for
x # [0, ),
1 v(a)= a 2, 2 1 V(e)= e 2, 2 c(a, e)=a+e, *=1. Then, we have x&c fc (x | c)= 2 , f c
x 1 =c,
1 c 2 ++ 1(x&c) s(x)= ln 2 : c ++ 2(x&c)
l 0 =0, for
x x 1 .
Note that : denotes the agent's absolute risk aversion. Using the Lagrange function in (3.4), the first-order conditions for (a*, e*) generate
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FIG. 4.
The contract in Proposition 2 with (a) :=1, l=3 and (b) :=0.1, l=3.
+1 =
e*+(a*&e*) [x&s*(x)] f cc dx , 1& u(s*) f cc dx& [x&s*(x)] f cc dx
+2 =
a*&(a*&e*) u(s*) f cc dx . 1& u(s*) f cc dx& [x&s*(x)] f cc dx
We use the above two equations plus the two incentive conditions in (3.3) to determine four variables (a*, e*, + 1 , + 2 ). 17 The Contract in Proposition 2 For the contract in Proposition 2, we choose the initial parameter values :=1 and l=3. We then find a*=0.17,
e*=0.76,
+ 1 =0.73,
+ 2 =0.14,
l 1 =5.15,
and the contract in Fig. 4a. 17
In actual numerical calculations, to reduce the computing time from several hours to about one minute, we need to simplify the four equations using given functions. The derivation of the simplified formulas and the actual MathCAD files of the numerical calculations are available upon request.
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Note that, as calculated above, we have l 1 =5.15>l=3. As drawn in Fig. 4a, the numerical contract shows a pattern which is consistent with the one in Fig. 2. To verify the non-convergence result in Proposition 4, we further choose several very small values of : keeping l=3, and find a clear pattern of nonconvergence. When : approaches zero, the corresponding contract becomes flatter in the region of large x values. For example, for :=0.1 and l=3, we have a*=0.42,
e*=0.55,
+ 1 =0.54,
+ 2 =0.41,
l=5.72,
and the contract in Fig. 4b. Note that the contract with :=0.1 exhibits the same pattern as the one in Fig. 4a, except that the contract is taller. In fact, we can see that the contract in Fig. 4b is flatter than the one in Fig. 4a when x is large. This shows that the contract does not converge to the linear contract in any right-half line of x when the agent's risk aversion approaches zero. The Contract in Proposition 3 For the contract in Proposition 3, we choose the upper bound l=12 which is sufficiently big, keeping :=1. We then find a*=0.24,
e*=0.82,
+ 1 =17.34,
+ 2 =2.45,
l 1 =7.07,
and the contract in Fig. 5a. As calculated above, if l is sufficiently big, we have l>l 1 . This numerically confirms Lemma 3. As drawn in Fig. 5a, the numerical contract shows a pattern which is consistent with the one in Fig. 3. To verify the non-convergence result in this case, we also choose several very small values of : keeping l=12, and find a similar pattern of nonconvergence. For example, for :=0.1 and l=12, we have a*=0.45,
e*=0.54,
+ 1 =1.74,
+ 2 =1.43,
l 1 =7.07,
and the contract in Fig. 5b. Especially, observe that the right-hand side of the contract is far below the upper bound l. This implies that the contract in the right-hand side is not bounded by the upper bound l but internally bounded by (u$) &1 ((+ 2 C e*) (+ 1 C a*)). Thus, this show that our non-convergence result does not hinge upon the upper bound, l.
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FIG. 5.
KIM AND WANG
The contract in Proposition 3 with (a) :=1, l=12 and (b) :=0.1, l=12.
6. CONCLUSION The main focus of this paper is on the double moral-hazard situation in which the principal also participates in the production process. A simple linear contract is one of the optimal contracts when the agent is riskneutral. When the agent is risk-averse, however, there is a unique optimal contract, which is typically non-linear. We show that the agent's rewards in this case are less sensitive to output than that would be designed under single moral-hazard. When the principal also participates in the production process, the changes in output convey only partial information about the agent's hidden action choice, because not only the agent but also the principal is responsible for production. We also examine the robustness of the linear contract by studying the feature of contract when the agent's risk aversion goes to zero. We find that the limiting contract is not a linear contract. Thus, the linear contract is not a good approximation of the optimal contract under double moralhazard when the agent is almost risk-neutral.
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However, we conjecture that, if the agent obtains private information after the contract but before taking action, only the linear contract would achieve the efficient outcome since only the linear contract would make the agent's action choice independent of his private information. However, further research is needed to verify this conjecture.
APPENDIX A: PROOF OF LEMMA 1 If s* is optimal, it must be the solution of max s#S
|
H[a*, e*, x, *, + 1 , + 2 , s(x)] dx.
&
Let h(x) be any Lebesgue measurable function, satisfying h( )=h(& )=0. For any constant t # R, function s t(x)#s*(x)+th(x) will also be admissible and satisfy the initial and terminal conditions s t( )=s*( ) and s t(& ) =s*(& ). Define V(t)#
|
H[a*, e*, x, *, + 1 , + 2 , s t(x)] dx.
&
V reaches the maximum when t=0. The first-order condition is 0=V$(0)=
|
H s[a*, e*, x, *, + 1 , + 2 , s*(x)] h(x) dx.
&
Since h is an arbitrary measurable function taking zero at the two ends, the above implies H s[a*, e*, x, *, + 1 , + 2 , s*(x)]=0,
\x.
The second-order condition is 0 V"(0)=
|
H ss[a*, e*, x, *, + 1 , + 2 , s*(x)] h 2(x) dx,
&
for any measurable function h satisfying h( )=h(& )=0. By a lemma in Kamien and Schwartz [12, p. 39], we then have (3.8). The sufficiency of (3.8) is proven in Kamien and Schwartz [12, p. 38].
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APPENDIX B: PROOF OF PROPOSITION 1 To prove Proposition 1, we need the following lemma as an intermediate step. Let e# (a) solve max R(a, e)&V(e).
(B.1)
e 0
Also, let s*(x | e) be an optimal incentive scheme that would be designed to motivate the agent to take a*, given e is already taken. Then, from Holmstrom [8], it is straightforward that fc 1 =*++ 1(e) C a (a*, e) [x | C(a*, e)], u$[s*(x | e)] f
(B.2)
for every x for which Eq. (B.2) has a solution 0 s*(x | e) l, and otherwise s*(x | e)=0 or s*(x | e)=l. Define
|
SW(s*)# [x&s*(x)] f (x | C*) dx&V(e*) +*
_| u[s*(x)] f (x | C*) dx&v(a*)& ,
and
|
SW c[s*( } | e)]# [x&s*(x | e)] f [x | C(a*, e)] dx&V(e) +*
_| u[s*(x | e)] f [x | C(a*, e)] dx&v(a*)& .
Thus, SW(s*) denotes the joint benefits resulting from s*(x) when the principal's effort is not contractible. In fact, (a*, e*) will be chosen when s*(x) is designed. On the other hand, SW c[s*( } | e)] denotes the joint benefits resulted from s*(x | e) when e is committed (the superscript ``c'' indicates that the principal's effort level is contractible). It is straightforward that SW(s*) SW c[s*( } | e*)],
(B.3)
since the right-hand side is the joint benefits that are obtainable without the principal's incentive constraint, while the left-hand side is the one obtainable with the constraint.
DOUBLE MORAL-HAZARD
Lemma B1. Proof.
365
SW c[s*( } | e)] is strictly increasing in e, for e e# (a*).
Write SW c[s*( } | e)]=R(a*, e)&V(e)&B[s*( } | e)],
where
|
B[s*( } | e)]# [s*(x | e)&*u[s*(x | e)]] f [x | C(a*, e)] dx+*v(a*). Here, B[s*( } | e)] represents the agency cost in motivating the agent to take a* when e is given. Since R e[a*, e# (a*)]&V$[e# (a*)]=0, and by the strict concavity of R(a*, e)&V(e) in e, we have R e(a*, e)&V$(e)>0 for e0 and + 2 0. We will ignore the boundary conditions and show the equivalence of max s # S, a, e 0
| [x&s(x)] f [x | C(a, e)] dx&V(e)
+*
_| u[s(x)] f [x | C(a, e)] dx&v(a)&
s.t. IC 1 : C a IC 2 : C e
| u[s(x)] f [x | C(a, e)] dx v$(a) c
| [x&s(x)] f [x | C(a, e)] dx V$(e), c
(A)
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and
max s # S, a, e 0
| [x&s(x)] f [x | C(a, e)] dx&V(e)
+*
_| u[s(x)] f [x | C(a, e)] dx&v(a)&
(B)
|
s.t. C a u[s(x)] f c[x | C(a, e)] dx=v$(a) Ce
| [x&s(x)] f [x | C(a, e)] dx=V$(e). c
We will have + 1 0 and + 2 0 if (A) and (B) are equivalent. For that, we need to show that the two IC conditions in (A) must be binding for any solution of (A). We first show that IC 1 must be binding. Let (s^ , a^, e^ ) be a solution of (A), with Lagrange multipliers +^ 1 and +^ 2 . We know +^ 1 0 and +^ 2 0. By KuhnTucker Theorem, if IC 1 is not binding, then we must have +^ 1 =0. Define x 2 such that
1++^ 2 C e
fc (x 2 | C )=0, f
18
18
When applying Lemma B1 in the above, we need e^ += e# (a^ ). That is, we need to show e^ C e
| s^ (x) f (x | C) dx.
(E)
c
Consider s(x; + 1 ) s(x; + 1 ) f c(x | c) dx= f c(x | c) dx + 1 + 1
|
|
C a ( f c f )(x | c)
1
| u"[s(x; + )] [*++ C ( f f )(x | c)]
=&
1
1
a
c
2
f c(x | c) dx>0.
Since s(x; 0) f c(x | c) dx=0 and +^ 1 0, we then have
| s^ (x) f (x | C) dx=| s(x; +^ ) f (x | C) ds 0. c
1
c
(E) then implies R e(a^, e^ )&V$(e^ )>0. By R e[a^, e# (a^ )]&V$[e# (a^ )]=0 and the concavity of R(a, e)&V$(e) in e, we must have e^ 0 (it is easy to see that + 1 {0 from the above proof). If IC 2 is not binding, then +^ 2 =0 by the KuhnTucker condition. Then, we have s^ (x)=s^ ( } | e^ ) satisfying fc 1 =*++ 1(e^ ) C a (x | C ). u$[s^ (x | e^ )] f However, by Lemma B1, we can then increase Sw(s^ )=SW[s^ ( } | e^ )] by increasing e^ slightly. That is, by continuity of s^ (x | e) and f c[x | C(a, e)] in e and by Fatou's Lemma, we can find =>0 such that IC 2 still holds strictly: C e(a^, e^ +=) 19
| [x&s^ (x | e^ +=)] f [x | C(a^, e^ +=)] dx>V$(e^ +=), c
Such x 2 may not exist; in that case, we take x 2 =+ .
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while by definition of s^ (x | e), IC$1 still holds for (s^ (x | e^ +=), a^, e^ +=) for any =. Then, by Lemma B1, SW[s^ ( } | e^ +=)]>SW[s^ ( } | e^ )]=SW(s^ ). This contradicts with the fact that (s^ , a^, e^ ) achieves the maximum value under the two IC conditions. 20 Thus, we have proven that + 1 >0 and + 2 0. (ii) + 1 C a*>*+ 2 C e* . If + 2 =0, by the fact that + 1 >0, it is already true that + 1 C *>*+ a 2 C e* . Thus, it suffices to prove under + 2 >0. First, suppose that + 1 C a*=*+ 2 C e*. Then, the optimal contract s* is a fixed wage contract. This is a contradiction since a fixed wage contract gives zero incentive to the agent. Second, suppose + 1 C a*x 2 , by Assumption 4, 1++ 2 C e*
fc (x | C*)>0, f
and since + 1 C *0, (x 2 | C*)>*++ 1 C * a f + 2 C e*
\
+
and *++ 1 C a*( f c f )(x | C*)>0 for x>x 2 . Thus, H ss x 2 , and Eq. (3.7) plus the boundary condition 0 s*(x) l determine an optimal solution for s*(x) when x>x 2 . Since + 1 C **>(+ 1 C *)(+ ((+ 2 C e*) a 2 C e*), we have l>l 0 >(u$) (+ 1 C a*)). 23 Such x 1 may not exist. In that case, we take x 1 =& . 21
369
DOUBLE MORAL-HAZARD
Since + 1 C a*0. Then, from (G.4), we have 1 + 1n C* an f c (x | C *) +A n Gn + 2n C* en f lim = lim =1. nÄ fc nÄ 1 fc (x | C 1++ 2n C*en (x | C *) + *) n n f Gn f 1+A+ 1n C* an
fc (x | C *) n f
(G.5)
Given the fact that G>0, (+ 1n C* an )(+ 2n C* en ) must converge to *. Thus, the limiting contract is internally bounded by (u$) &1 (1*). Therefore, the limiting cannot be a linear contract, which is unbounded.
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