Existence and non-existence in the moral hazard problem
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ScienceDirect Journal of Economic Theory 150 (2014) 668–682 www.elsevier.com/locate/jet
Existence and non-existence in the moral hazard problem ✩ Sofia Moroni b,∗ , Jeroen Swinkels a a Kellogg School of Management, Northwestern University, Evanston, IL, United States b Department of Economics, Yale University, New Haven, CT, United States
Received 6 May 2013; final version received 9 September 2013; accepted 17 September 2013 Available online 1 October 2013
Abstract We provide a new class of counter-examples to existence in a simple moral hazard problem in which the first-order approach is valid. In contrast to the Mirrlees example, unbounded likelihood ratios on the signal technology are not central. Rather, our examples center around the behavior of the utility function as utility diverges to negative infinity. For any utility function, such as ln(w), in which utility diverges to negative infinity at a finite wealth level, existence will fail for some specifications of the agent’s cost of effort. When utility diverges to negative infinity only as wealth does as well, existence holds for all specifications of the agent’s cost of effort if and only if the agent continues to dislike risk as wealth diverges to negative infinity. When there is a finite lower bound on utility, existence is assured. For those cases where existence fails, we characterize the limit of near optimal contracts. © 2013 Elsevier Inc. All rights reserved. JEL classification: D86 Keywords: Moral hazard; Principal–agent models; Optimal contract; Existence; Mirrlees non-existence example
1. Introduction Mirrlees [5] provides a classic example of a moral hazard problem with no optimal solution. In the example, which relies on an unbounded likelihood ratio, there is a sequence of contracts ✩
We thank Dirk Bergemann, Johannes Hörner, Ohad Kadan, Larry Samuelson, the Editor, the Associate Editor and an anonymous referee for helpful comments. We would also like to thank the seminar audience at Yale University. * Corresponding author. Fax: +1 203 436 9023. E-mail addresses: [email protected] (S. Moroni), [email protected] (J. Swinkels). 0022-0531/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jet.2013.09.011
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that have cost converging to the full information first best, but no contract that achieves the first best. Jewitt, Kadan, and Swinkels [4] (henceforth JKS) provide a proof of existence when likelihood ratios are bounded. We point to an implicit, and not always correct, assumption in [4] by exhibiting a new class of examples in which there is non-existence of the optimal contract even if the likelihood ratio is bounded. The problem in these examples is that we know that an optimal contract must be of the form given by Holmström [2]. But, in some settings, the principal can easily run out of room to provide adequate incentives by contracts of this form. In particular, we show conditions under which any contract of the required form that gives the right utility provides inadequate incentives. When utility diverges at a finite consumption level the existence problem is severe – for any given utility function and information structure one can find a specification of the agent’s cost of effort and reservation utility such that an optimal contract does not exist. Thus for example, existence is not in fact guaranteed in the workhorse example of log utility. When utility diverges to −∞ only as consumption does as well, we show a necessary and sufficient condition for the missing step for the proof in [4] to hold, and hence for existence. The condition is that, in a sense to be made precise, the agent remains risk averse as consumption diverges to −∞. Under a technical regularity condition, the condition fails only if u has bounded slope, and so becomes essentially linear as consumption diverges. Hence, the existence problem is not severe for this case. For settings where the utility is bounded below, the original JKS proof goes through without problem. Because our setting has bounded likelihood ratios, we show that unlike in the Mirrlees example we cannot approach the first best.1 But, we provide a characterization of both the limit to which any sequence of contracts with costs approaching the second best must converge, and the limit cost. We also show that one can approach the second best using contracts that bound utility strictly above −∞.2 The limit contract has an intuitive form. It can be thought of as starting from a contract of the standard form that provides both too much utility and too little incentives, and then modifying it by providing very low utility on a small interval near the lowest signal, which both eliminates the extra utility the base contract provides and relaxes the incentive constraint so as to restore feasibility. Since the problem is too much utility and too little incentives, and since the trade-off of extra incentives per unit of utility taken away is most favorable at the lowest signal, the “right” way to do this is to concentrate the modification arbitrarily close to the lowest signal, and it is this that leads to non-existence. 2. Model The model is standard. A risk neutral principal employs a risk averse agent. The agent’s utility function over final wealth is u : D → R where u is in C 2 with u > 0 and u < 0. To focus attention, we assume D ⊆ R is an interval with upper bound ∞, and lower bound d, where d may 1 In the case of unbounded likelihood ratios, one can concentrate punishments at outcomes which are very unlikely at the desired effort level, but the probability of which rises fast with a deviation. Here, the only way to provide incentives is to face the agent with risk at the desired action. 2 The case where utility has a lower bound above −∞ is one for which JKS’s existence proof is valid, and for which contracts have a simple characterization as truncated versions of standard Holmström [2] contracts.
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be finite or −∞, and where d may or may not be in D. We assume u(d) ≡ limw↓d u(w) = −∞, limw↑∞ u(w) = ∞, and limw↑∞ u (w) = 0. Write also u (d) for limw↓d u (w).3 The agent chooses an effort level e ∈ [0, e] ¯ which is unobservable to the principal. The effort cost is given by the function c(e) ∈ C 2 with c (e) > 0 for e > 0 and c (e) > 0. The agent’s utility is additively separable and equal to u(w) − c(e). An outcome x ∈ X = [0, 1] is realized according to F (x|e) ∈ C 3 . The density of F is f , with f (x|e) > 0 for all x and e. As f is continuous, f (x|e) is uniformly bounded from zero and ∂ fe (x|e) so fe (x|e)/f (x|e) is also bounded. We assume that ∂x f (x|e) is strictly positive, so that the monotone likelihood ratio property (MLRP) holds. The principal’s gross benefit of e is given by B(e) = xf (x|e) dx. The agent is compensated according to π(x) : (0, 1] → D.4 Let U (π, e) ≡ u π(x) f (x|e) dx. The agent’s net utility is given by U (π, e) − c(e). Let C(π, e) ≡ π(x)f (x|e) dx be the expected cost to the principal for contract π and effort e. The participation constraint is U (π, e) − c(e) u0 ,
(IR)
where u0 is the agent’s outside option. The incentive compatibility constraint is e ∈ arg max U (π, e) ˆ − c(e) ˆ . e∈[0, ˆ e] ¯
Following Rogerson [6], we work with the (doubly) relaxed incentive constraint Ue (π, e) c (e).
(IC)
Various primitives for the validity of this exist. The simplest is that Convexity of the Distribution Function (CDFC) is satisfied so that Fee (x|e) 0 for all x and e.5 For any given e > 0, the relaxed cost minimization problem is min C(π, e) π
(P)
s.t. (IC), (IR). 3 For the case d > −∞, it follows from u(d) = −∞ that u (d) = ∞. When d = −∞, it is possible that u (d) > −∞. 1. Assume for example that for w < −1, u(w) = w + w 4 We write things this way to allow for the possibility that at the zero measure point x = 0 the principal may wish to pay d, where (recall) d may not be in D. Since u(d) = −∞, it cannot be that the principal wants to pay d on a positive measure set of signals in any contract satisfying the agent’s individual rationality constraint. 5 One could also apply the less stringent conditions from Jewitt [3], except at one point later in the paper at which we approach the second-best optimum by contracts which are subject to a minimum payment constraint.
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Let b =
1 u (d) .
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For any given e, μ > 0, and b b, let
δ(x|e) =
fe (x|e) fe (0|e) − f (x|e) f (0|e)
and define the contract πb,μ,e (·) implicitly on [0, 1] by 1 = b + μδ(x|e), u (πb,μ,e (x))
(1)
where we note that by MLRP, for x > 0, πb,μ,e (x) ∈ D. Note that this is just a re-parametrization (0|e) . of the standard Holmström [2] (λ, μ)-contract, where in particular, b = λ + μ ffe(0|e) Lemma 1. A necessary condition for π to solve (P) is that π = πb,μ,e (·) for some b b and μ > 0 where U (πb,μ,e , e) = u0 + c(e) and Ue (πb,μ,e , e) = c (e). Proof. This is essentially standard (see Holmström [2]). But, given that one point of this paper is to argue that there is some lack of clarity in previous existence proofs, we provide an elementary proof for completeness. Let π : (0, 1] → D be an optimal contract. Choose disjoint Borel (X2 |e) (X1 |e) > FFe(X and sets X1 , X2 , X3 and X4 of (0, 1] with positive F (·|e)-measure where FFe(X 2 |e) 1 |e) Fe (X4 |e) F (X4 |e)
(X3 |e) > FFe(X . Consider raising utility by F (Xz2 |e) on X2 , and lowering it by F (Xz1 |e) on X1 .6 3 |e) This leaves the expected utility of the agent unchanged, and changes his incentives at rate
Fe (X2 |e) Fe (X1 |e) − , F (X2 |e) F (X1 |e) while changing the expected cost to the principal at z = 0 at rate 1 1 1 1 f (x|e) dx − f (x|e) dx F (X2 |e) u (π(x)) F (X1 |e) u (π(x)) X2 X1 1 1 X2 − E X 1 . =E u (π(x)) u (π(x)) So, if E
1 u (π(x)) X2 Fe (X2 |e) F (X2 |e)
−
1 u (π(x)) X1 Fe (X1 |e) F (X1 |e)
−E
0 follows since otherwise Ue (π, e) 0. That 1 > b. That (IR) binds follows b b follows since π(x) ∈ D for all x ∈ (0, 1], and hence u (π(x)) from the standard thought experiment of removing a small constant from utility everywhere. That (IC) binds is also standard. If not, then pick disjoint intervals X1 and X2 where X2 is strictly to 6 Here and in many places that follow, we will allow ourselves to consider variational arguments in which the variation creates a non-monotonicity in the contract. This is consistent with the logic of the relaxed problem, for which the monotonicity of an optimal contract is a result, not an assumption. One could build each variation to be monotone, but only at considerable extra notational cost.
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the right of X1 . Lower payments on X2 by z, and raise them on X1 by z to restore (IR). Since (IC) is slack, this contract is feasible for small z, and since payments on X2 are strictly larger than those on X1 and the agent is risk averse, the principal saves money when z is small. 2 3. Non-existence: a class of simple examples Let us turn to the new class of non-existence examples. In this section, we work with u(w) = ln w. In the next section, we work with more general utility functions. Since we have assumed that ffe is bounded, our class includes a set of cases for which Jewitt, Kadan, and Swinkels [4] assert existence. Lemma 2. Let u(w) = ln w. Then, Ue (π0,1,e , e) is finite, and for any c(·) with c (e) > Ue (π0,1,e , e), there exists no optimal contract implementing e. The idea of the proof is to show that Ue (π0,1,e , e) is the strongest incentive that can be provided with a contract of the form required by Lemma 1. Example 1. Let
1 f (x|e) = 1 + (e − 1) x − 2
be the standard FGM linear copula. Then, it is straightforward that δ(x|1) = x and so 1 Ue (π0,1,e , e) =
ln δ(x|1) fe (x|1) dx
0
1 =
1 1 (ln x) x − dx = . 2 4
0
Hence, for any c(·) with c (1) > 14 , there is no optimal contract implementing e = 1. Proof of Lemma 2. That Ue (π0,1,e , e) is finite will be shown in Lemma 4. Because u(w) = ln w, 1 we have u (w) = w, and so 1 Ue (πb,μ,e , e) =
ln b + μδ(x|e) fe (x|e) dx,
0
from which ∂ Ue (πb,μ,e , e) = ∂b
1 0
where by MLRP,
1 b+μδ(x|e)
1 fe (x|e) dx, b + μδ(x|e)
is positive and decreasing in x. Hence, since fe (·|e) single crosses 0 1 from below, and so, since 0 fe (x|e) = 0, ∂ Ue (πb,μ,e ) 0. ∂b
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But, ∂ Ue (πb,μ,e ) = ∂μ
1 0
δ(x|e) fe (x|e) dx 0, b + μδ(x|e)
with equality if and only if b = 0, since the fraction is constant in x for b = 0, and strictly increasing otherwise. It follows that for each μ, Ue (πb,μ,e ) is maximized by taking b = 0, and that Ue (π0,μ,e ) is independent of μ. Hence for any c(·) and e for which c (e) > Ue (π0,1,e ), no contract of the form πb,μ,e satisfies (IC), and so by Lemma 1, there is in fact no optimal contract. 2 The difficulty for the proof in [4] is that it implicitly assumes that as b → b, so that utility at the worst outcome diverges to −∞, expected utility and incentives diverge. As Example 1 illustrates, this can fail. 4. A necessary and sufficient condition for existence We now present a necessary and sufficient condition for existence of the optimal contract for each specification of costs and the outside option. We will see that the condition always fails when d > −∞, and so the counterexample from the previous section is in fact very general. But, we will also see that when d = −∞, the condition is extremely mild, so that except in somewhat tortured examples, existence is guaranteed. We also note that the problem we identify in the proof by JKS is not an issue when u(d) > −∞. Hence, existence is essentially fine except in the case where utility goes to −∞ at some finite pay level, but is in doubt in any such case. 4.1. Asymptotic risk aversion Say that utility function u has Asymptotic Risk Aversion (ARA) if lim
w↓d
−u(w) + w = −∞. u (w)
(2)
When d is finite, then, since limw↓d −u(w) u (w) 0, ARA must fail. On the other hand, when d = −∞, then ARA is very weak. In particular, the following lemma shows that only if u converges to a finite limit, so that the agent becomes effectively risk neutral as income tends to −∞, can ARA fail.7 Lemma 3. Assume that d = −∞ and that sufficient for ARA is that u (d) = ∞.
−u(w) u (w)
has a well defined limit as w → −∞. Then,
7 When u does converge to a finite limit, ARA may or may not be satisfied. It fails when u(·) has the form u(w) = 1 for w < −2, but is satisfied when u(w) = w + ln(−w) for w < −2. w+ w
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Proof. If limw↓−∞ −u(w) u (w) = 0, then (2) is immediate. So, assume that limw↓−∞ u(w) u(w)u (w) ∂ ∂w − u (w) + w = (u (w))2 , we need 0 lim
w↓−∞
−u(w) u (w)
> 0. Since
−u(w) −u (w) dw = ∞, u (w) u (w)
w
and so given that limw↓−∞ 0
−u(w) u (w)
> 0 it is sufficient that limw↓−∞
0
−u (w) w u (w)
dw = ∞. But,
−u (w) dw = ln u (w) − ln u (0) u (w)
w
which diverges since limw↓−∞ u (w) = ∞.
2
The key implication of ARA is contained in the following lemma. Lemma 4. Fix e > 0 and μ > 0. Then, lim U (πb,μ,e , e) = −∞
(3)
b↓b
if and only if u has ARA, and lim Ue (πb,μ,e , e) = ∞
(4)
b↓b
if and only if u has ARA.8 ∂ fe (x|e) Proof. Fix e > 0 and μ > 0. Since F ∈ C 3 it follows from our assumptions that ∂x f (x|e) is bounded away from 0 and ∞. The implication is that h(·|e), defined as the density of δ(x|e), is also bounded from below by some h > 0, and above by some h¯ < ∞ on the support [0, δ(1|e)]. Let ω(t), defined by 1 u (ω(t))
= t,
(5)
be the amount the agent is paid when
1 u
= t . Then, changing variables,
1 lim U (πb,μ,e , e) = lim b↓b
u(πb,μ,e )f (x|e) dx
b↓b
0 δ(1|e)
u ω(b + μz) h(z|e) dz,
= lim b↓b
0
and, 8 As will be seen in the proof, both limits are guaranteed to exist in the extended reals.
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1 lim Ue (πb,μ,e , e) = lim b↓b
u(πb,μ,e )
b↓b
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fe (x|e) f (x|e) dx f (x|e)
0
fe (0|e) u ω(b + μz) z + h(z|e) dz. f (0|e)
δ(1|e)
= lim b↓b
0
Choose zˆ > 0 and bˆ > b such that zˆ +
fe (0|e) f (0|e)
< 0 and u(ω(bˆ + μˆz)) < 0. Then, both
δ(1|e)
u ω(b + μz) h(z|e) dz
lim b↓b
zˆ
and fe (0|e) u ω(b + μz) z + h(z|e) dz f (0|e)
δ(1|e)
lim b↓b
zˆ
are well defined and finite by the dominated convergence theorem, since in particular, for b ∈ ˆ and z ∈ [ˆz, δ(1|e)] [b, b], −∞ < u ω(b + μˆz) u ω(b + μz) u ω bˆ + μδ(1|e) < ∞ (0|e) is bounded as well since δ(1|e) < ∞. and since z + ffe(0|e) ¯ while But, on [0, zˆ ], h is bounded by h and h, fe (0|e) ¯ fe (0|e) fe (0|e) −∞ < h z+ h(z|e) zˆ + h < 0. f (0|e) f (0|e) f (0|e) It follows that the limits in (3) and (4) are well-defined, and that both (3) and (4) are equivalent to zˆ lim b↓b
u ω(b + μz) dz = −∞
0
or, changing variables and discarding a constant, to b+μˆ z
u ω(z) dz = −∞.
lim b↓b
(6)
b
Integrating by parts, b+μˆ z
b+μˆz u ω(z) dz = u ω(z) zb −
b
b+μˆ z
u ω(z) ω (z)z dz
b
= u(b + μˆz)(b + μˆz) − u ω(b) b −
b+μˆ z
ω (z) dz
b
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= u(b + μˆz)(b + μˆz) − u ω(b) b − ω(b + μˆz) + ω(b) u(ω(b)) = u(b + μˆz)(b + μˆz) − − ω(b + μˆz) + ω(b) u (ω(b)) where both the simplification within the integral and the last equality use (5). But, lim u(b + μˆz)(b + μˆz) − ω(b + μˆz) = u(b + μˆz)(b + μˆz) − ω(b + μˆz) b↓b
and thus is finite and so (6) is equivalent to −u(w) lim + w = −∞, w↓d u (w) as claimed.
2
Before we move on, we make two notes. First, an issue similar to what we have identified can also arise if limw↑∞ u (w) > 0. Then, unless an analog to ARA holds, one can have a case where, (x|e) = holding fixed μ, utility does not diverge as we increase b towards the value where b + μ ffe(x|e) 1 limw↑∞ u (w) .
We rule out this case both because we find it less economically interesting, and because the technical point is already made in the case we examine. Second, Carlier and Dana [1] do not, as far as we can tell, have the same problem as JKS. The key is that in their environment, the principal is also strictly risk averse with utility v(·), where in particular there is α ∈ (0, 1), such that as w ↑ ∞, v(w) w α → 0. This implies that as large amounts are transferred from the agent to the principal, there remains enough curvature in the system as to avoid the discontinuity issue discussed here. In particular note that since v(w) w α → 0, limw↑∞ v (w) = 0. 4.2. The result and proof We begin with a small lemma. For the balance of the paper, assume that some e > 0 has been fixed. Lemma 5. Let u not have ARA. Then, there is a unique μˆ such that if we let πˆ = πb,μ,e ˆ , then U (π, ˆ e) = u0 + c(e). Proof. This follows immediately from the intermediate value theorem, noting that by Lemma 4 U (πb,μ,e , e) is finite and continuous in μ for all μ > 0, and that for x > 0, πb,μ,e (x) is increasing in μ with limμ↓0 πb,μ,e (x) = −∞ and limμ↑∞ πb,μ,e (x) = ∞ and so by the monotone convergence theorem, limμ↓0 U (πb,μ,e , e) = −∞ and limμ↑∞ U (πb,μ,e , e) = ∞. 2 Proposition 1. An optimal contract implementing e exists if and only if either u has ARA or u does not have ARA, but Ue (π, ˆ e) c (e). Thus, when ARA holds, existence holds regardless of c(·) and u0 . But, when ARA fails, then existence will hold for some specifications of c(·) and u0 , but one can always find a specification of c(·) and u0 where existence fails. ˜ μ) Proof of necessity. Assume Ue (πˆ , e) < c (e), and, given Lemma 1 let (b, ˜ = (b, μ) ˆ be such that π˜ = πb, is optimal, where U ( π ˜ , e) = U ( π ˆ , e) = u + c(e). Then, by Lemma 5, b˜ > b, 0 ˜ μ,e ˜
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and so, since U (π˜ , e) = U (π, ˆ e), it must be that μ˜ < μ, ˆ and that π˜ single crosses πˆ from above. But then, since 1 U (πˆ , e) − U (π˜ , e) =
u(πˆ ) − u(π˜ ) f (x|e) dx = 0,
0
it follows from MLRP that 1 Ue (πˆ , e) − Ue (π˜ , e) =
fe (x|e) u(πˆ ) − u(π˜ ) f (x|e) dx > 0, f (x|e)
0
and so, as πˆ fails (IC), so does π˜ .
2
Proof of sufficiency. The point here is simply that the JKS construction works once either (3) holds or Ue (πˆ , e) c (e). In particular, assume (3). Then, for any given c, u0 , e, and μ 0, there is, by the intermediate value theorem, b(μ) > b such that (IR) holds with equality at πb(μ),μ,e . Because b(μ) > b, payoffs and incentives are continuous in b and μ at (b(μ), μ), and so, as in [4], b(·) is continuous and hence Ue (πb(μ),μ,e , e) is continuous in μ. But, Ue (πb(0),0,e , e) = 0 and JKS argue that when μ is large enough, (IC) is slack.9 But then, again by the intermediate value theorem, there is a μ where Ue (πb(μ),μ,e , e) = c (e), and, as JKS show, this is sufficient ˆ to establish optimality. Similarly assume ARA fails, but Ue (πˆ , e) c (e). Then, for μ < μ, ˆ and so on the domain [0, μ], ˆ b(μ) and Ue (πb(μ),μ,e , e) are continub(μ) > b by definition of μ, ous in μ, and we are done as before by the intermediate value theorem. 2 5. Characterizing near optimal contracts Assume that there is no optimal contract. It still must be the case that over the set of feasible solutions implementing e in the (doubly relaxed) moral hazard problem there is an infimum C ∗ (e) of expected costs. In this section, we characterize nearly optimal contracts. This gives us a tight expression for C ∗ (e), gives insight into how existence fails, and allows us to show that, in contrast to the Mirrlees counter-example, in this example, C ∗ (e) is strictly higher than the full information first best. We begin with a small lemma. Lemma 6. Assume there is no optimal contract implementing e. Then, there exists a unique pair μ∗ > 0 and τ > 0 such that if we let π ∗ = πb,μ∗ ,e , then U π ∗ , e = c(e) + u0 + τ (7) and fe (0|e) Ue π ∗ , e = c (e) + τ. f (0|e)
(8)
9 The argument is particularly simple here. Let x ∗ be such that f (x ∗ |e) = 0. Consider the contract π which pays z e z for x x ∗ , and pays a constant k(z) for x < x ∗ , where k(z) > d is chosen so that πz satisfies (IR) with equality. Choose some zˆ large enough that (IC) is slack at πzˆ . Now, note that for μ sufficiently large, the contract πb(μ),μ,e will pay less than πzˆ (0) = k(ˆz) at 0 (as μ → ∞, it must be that b(μ) → b) but, since b + μδ(x ∗ |e) diverges, will for μ large enough, pay more than zˆ at x ∗ . Thus, πb(μ),μ,e single crosses πzˆ from below, and so provides stronger incentives than it by Lemma 5 of [4].
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Proof. Define ν(μ) = U (πb,μ,e , e) − c(e) − u0 , and K(μ) = Ue (πb,μ,e , e) − c (e) − ν(μ)
fe (0|e) , f (0|e)
and let μˆ be as defined in Lemma 5, so that ν(μ) ˆ = 0 and hence any μ < μ(e) ˆ has ν(μ) < 0. By Proposition 1 and Lemma 5, ν(μ) ˆ = 0 and K(μ) ˆ < 0. But, as μ ↑ ∞, U (πb,μ,e , e), and hence ν(μ), diverges to ∞, while Ue (πb,μ,e , e) remains weakly positive (since πb,μ,e is monotone and (0|e) < 0, there is at least since MLRP implies first order stochastic dominance). Hence, since ffe(0|e) ∗ ∗ ∗ ∗ one μ > μˆ with K(μ ) = 0. By construction, μ and τ = ν(μ ) satisfy the requisite conditions. To see that μ∗ is unique, note that ∂ fe (0|e) ∂ K(μ) = Ue (πb,μ,e , e) − U (πb,μ,e , e) ∂μ ∂μ f (0|e)
1 ∂ = u ω b + μδ(x|e) δ(x|e)f (x|e) dx ∂μ 0
1 =
u ω b + μδ(x|e) ω b + μδ(x|e) δ 2 (x|e)f (x|e) dx > 0.
2
0
Remark 1. π ∗ is the unique optimal solution to the auxiliary problem in which costs are c(·) ˆ (0|e) τ. where c(e) ˆ = c(e) + τ , and cˆ (e) = c (e) + ffe(0|e) This follows directly from [4, Proposition 1]. It turns out that in a very strong sense, π ∗ is also the basis for a characterization of near optimal solutions to (P). In particular, by construction, π ∗ ∗ e (0|e) clears (IR) by τ , but fails (IC) by −f f (0|e) τ . Choose a small interval near 0, and reduce π on that interval so as to reduce overall utility by τ . When the interval is small, this improves incentives ∗ e (0|e) by approximately −f f (0|e) τ , and results in a contract with cost approximately C(π , e) − τ b. The content of the next proposition is not only that this construction approximates the best that can be done, but that any sequence of near optimal contracts must look very much like this construction. Proposition 2. Assume there is no optimal contract implementing e. Then, 1. C ∗ (e) = C(π ∗ , e) − bτ , 2. C ∗ (e) can be approximated by contracts each of which have a lower bound d > d on payments, and, 3. if we let πn (x) be any sequence of contracts satisfying (IR) and (IC) for e and with C(πn , e) → C ∗ , then, for each ε > 0, Pr πn (x) − π ∗ (x) > ε → 0. 2
Example 2. Return to the setting of Example 1. Assume c(e) = e2 and that u0 = −10. Then, effort levels e e˜ ≈ 0.31 can be implemented by optimal contracts, but higher effort levels cannot. Using Proposition 2, one can (numerically, but not in closed form) find the supremum of payoffs to the principal for any given e > e. ˜ In this example, it turns out that e = 1 maximizes
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this supremum. Hence, the primal problem of finding an optimal e and π does not have a solution even though the dual problem does for some effort levels. To prove Proposition 2, we begin with a lemma. Lemma 7. Let π be any contract. For d > d, define πd (x) = max(π(x), d). Then, lim u πd (x) f (x|e) dx = u π(x) f (x|e) dx, d↓d
lim
d↓d
and
u πd (x) fe (x|e) dx =
lim
d↓d
u π(x) fe (x|e) dx
(9) (10)
πd (x)f (x|e) dx =
π(x)f (x|e) dx.
(11)
Proof. As d ↓ d, u(πd (x)) and πd (x) decrease monotonically, and converge to u(π(x)) and π(x). Hence, (9) and (11) follow by Lebesgue’s monotone convergence theorem. To see (10), let (x|e) and note that M be a bound on ffe(x|e) u πd (x) fe (x|e) dx − u π(x) fe (x|e) dx fe (x|e) f (x|e) dx = u πd (x) − u π(x) f (x|e) M u πd (x) − u π(x) f (x|e) dx which converges to 0 by (9).
2
Proof of Proposition 2. Let us first show that C ∗ C(π ∗ , e) − τ b, and that C(π ∗ , e) − τ b can be approached by finite contracts. For ε > 0 and d > d, define πε,d by adding ε to π ∗ (x) where fe (x|e) > 0 and by replacing any payment below d by d. By Lemma 7, for any ε > 0, d > d can be chosen small enough that U (πε,d , e) > u0 + c(e) + τ, fe (0|e) τ, f (0|e) C(πε,d , e) < C π ∗ , e + ε. Ue (πε,d , e) > c (e) +
and
τ For any y ∈ (0, 1], define the contract π˜ from πε,d by reducing utility by F (y|e) utils on (0, y]. τ Since u(s) = u(d) − F (y|e) has solution s > d, π˜ remains bounded, with U (π˜ , e) > u0 + c(e) (0|e) τ , for y small enough, and C(π˜ , e) < C(π ∗ , e) + ε − τ b. Since Ue (πε,d , e) > c (e) + ffe(0|e) ∗ Ue (π˜ , e) > c (e). Hence, π˜ implements e at cost at most C(π , e) + ε − τ b. Since ε > 0 was arbitrary, C ∗ C(π ∗ , e) − τ b, and C(π ∗ , e) − τ b can be approached by finite contracts.
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Assume C ∗ < C(π ∗ , e) − τ b. Then, there are π and κ > 0 with U (π, e) u0 + c(e), Ue (π, e) c (e), and C(π, e) = C π ∗ , e − τ b − κ. Fix y > 0 such that u (π1∗ (y)) < b + κτ (by definition of π ∗ limy↓0 u (π1∗ (y)) = b) and for z > 0, z define πz (x) from π ∗ by setting u(πz (x)) = u(π ∗ ) + F (y|e) on (0, y], so that U (πz , e) = u0 + c(e) + τ + z. Note that fe (0|e) Ue (πz , e) c (e) + (τ + z). f (0|e) Define π˜ z implicitly on (0, 1] by τ z u π˜ z (x) = u πz (x) + u π(x) . τ +z τ +z Then, τ z U (π˜ z , e) = U (πz , e) + U (π, e) τ +z τ +z z τ c(e) + u0 + τ + z + c(e) + u0 τ +z τ +z = c(e) + u0 + τ, and τ z Ue (πz , e) + Ue (π, e) τ +z τ +z τ fe (0|e) z c (e) + (τ + z) + c (e) τ +z f (0|e) τ +z fe (0|e) = c (e) + τ , f (0|e)
Ue (π˜ z , e) =
and so π˜ z is a feasible solution to the auxiliary problem. Now by (12), ∂ π˜ z (x) 1 τ ∂u(πz (x)) ∂ z = + u π(x) − u πz (x) , ∂z u (π˜ z (x)) τ + z ∂z ∂z τ + z and so ∗ ∂ π˜ z (x) 1 1 ∂u(πz (x)) = + u π(x) − u π (x) , ∂z z=0 u (π ∗ (x)) ∂z τ z=0 from which y ∂ π˜ z (x) 1 1 = f (x|e) f (x|e) dx ∗ ∂z u (π (x)) F (y|e) z=0 0 ∗ 1 1 u π(x) − u π (x) f (x|e) dx + τ u (π ∗ (x)) 1 1 π(x) − π ∗ (x) f (x|e) dx ∗ + u (π (y)) τ
(12)
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=
1 u (π ∗ (y))
−b−
681
κ τ
0, Pr πn (x) − π ∗ (x) > ε → 0. Assume by way of contradiction that for some ε > 0, δ > 0, and along a subsequence, Pr πn (x) − π ∗ (x) > ε > δ.10 Choosing xˆ > 0 so that F (x|e) ˆ < 2δ , and letting εˆ = εu (π ∗ (1) + ε) > 0, δ Pr {x > x} ˆ ∩ u πn (x) − u π ∗ (x) > εˆ > . 2 Let
(13)
ˆ + u−1 (u) ˜ u−1 (u) ˆ + u˜ −1 u ρ(u, ˆ u) ˜ = −u 2 2
be the amount saved by replacing a 50/50 lottery between uˆ and u˜ by its certainty equivalent. Then, ρ(u, ˆ u) ˜ is continuous, and by convexity of u−1 , strictly positive when uˆ = u. ˜ For uˆ = u, ˜ ρ is decreasing in its smaller argument and increasing in its larger argument. Hence, since [u(π ∗ (x)), ˆ u(π ∗ (1))] is compact, r=
min
∗ (x)),u(π ∗ (1))],|u− u∈[u(π ˆ ˆ ˜ u|ˆ ˆ ε
ρ(u, ˆ u) ˜ > 0.
(14)
Let πˆ n (x) be defined for all x by u(πˆ n (x)) = 12 u(π ∗ (x)) + 12 u(πn (x)). Then, τ U (πˆ n , e) u0 + c(e) + , 2 τ fe (0|e) Ue (πˆ n , e) c (e) + . 2 f (0|e)
(15) (16)
Now, simply paying according to πn half the time and π ∗ half the time costs 12 C(πn , e) + 1 ∗ ∗ 2 C(π , e). But, (13) puts a lower bound on the probability that u(πn ) and u(π ) are significantly different, with x > x, ˆ while (14) puts a lower bound on what is saved by replacing the relevant lottery with its certainty equivalent, and hence, δ 1 1 C(πˆ n , e) C(πn , e) + C π ∗ , e − r 2 2 2 1 1 ∗ 1 δ = C(πn , e) + C + bτ − r. 2 2 2 2 Using that C(πn , e) → C ∗ , choose some n˜ such that 1 1 C(πˆ n˜ , e) C ∗ + bτ − δr. 2 3 10 The proof extends in a simple manner to the case where contract π involves randomization. n
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1 Form π˜ by increasing πˆ n˜ by 12 δr where fe > 0. Then, C(π˜ , e) C ∗ + 12 bτ − 14 δr, and (0|e) τ Ue (π˜ , e) > c (e)+ τ2 ffe(0|e) . Hence, forming π˜ y by reducing u(π˜ ) by 12 F (y|e) on (0, y] and choosing y small enough,
U (π˜ y , e) u0 + c(e), Ue (π˜ y , e) c (e), and 1 C(π˜ y , e) < C ∗ − δr, 4 contradicting that C ∗ was the infimum of costs for contracts implementing e.
2
In the Mirrlees example, non-existence is shown by exhibiting a sequence of contracts that implement e, but approach the cost u−1 (u0 + c(e)) of a full information setting. Since μ∗ > 0, it follows immediately that in our setting C ∗ (e) is strictly higher than the first best. In particular, we have C π ∗ , e > u−1 u0 + c(e) + τ , and so
C π ∗ , e − bτ > u−1 u0 + c(e) + τ − bτ u−1 u0 + c(e) .
It also follows from Proposition 2, and from [4, Proposition 1] that if one lets bd and μd be such that the optimal contract implementing e subject to paying no less than d pays the larger 1 of d and the solution to u (π(x)) = bd + μd δ(x|e) then (bd , μd ) → (b, μ∗ ). References [1] G. Carlier, R.A. Dana, Existence and monotonicity of solutions to moral hazard problems, J. Math. Econ. 41 (2004) 826–843. [2] Bengt Holmström, Moral hazard and observability, Bell J. Econ. 10 (1979) 74–91. [3] Ian Jewitt, Justifying the first order approach to principal–agent problems, Econometrica 56 (1988) 1177–1190. [4] Ian Jewitt, Ohad Kadan, Jeroen M. Swinkels, Moral hazard with bounded payments, J. Econ. Theory 143 (2008) 59–82. [5] J. Mirrlees, The theory of moral hazard and unobservable behavior: part I, Rev. Econ. Stud. 66 (1999) 3–21. [6] William P. Rogerson, The first-order approach to principal–agent problems, Econometrica 53 (1985) 1357–1367.
ScienceDirect Journal of Economic Theory 150 (2014) 668–682 www.elsevier.com/locate/jet
Existence and non-existence in the moral hazard problem ✩ Sofia Moroni b,∗ , Jeroen Swinkels a a Kellogg School of Management, Northwestern University, Evanston, IL, United States b Department of Economics, Yale University, New Haven, CT, United States
Received 6 May 2013; final version received 9 September 2013; accepted 17 September 2013 Available online 1 October 2013
Abstract We provide a new class of counter-examples to existence in a simple moral hazard problem in which the first-order approach is valid. In contrast to the Mirrlees example, unbounded likelihood ratios on the signal technology are not central. Rather, our examples center around the behavior of the utility function as utility diverges to negative infinity. For any utility function, such as ln(w), in which utility diverges to negative infinity at a finite wealth level, existence will fail for some specifications of the agent’s cost of effort. When utility diverges to negative infinity only as wealth does as well, existence holds for all specifications of the agent’s cost of effort if and only if the agent continues to dislike risk as wealth diverges to negative infinity. When there is a finite lower bound on utility, existence is assured. For those cases where existence fails, we characterize the limit of near optimal contracts. © 2013 Elsevier Inc. All rights reserved. JEL classification: D86 Keywords: Moral hazard; Principal–agent models; Optimal contract; Existence; Mirrlees non-existence example
1. Introduction Mirrlees [5] provides a classic example of a moral hazard problem with no optimal solution. In the example, which relies on an unbounded likelihood ratio, there is a sequence of contracts ✩
We thank Dirk Bergemann, Johannes Hörner, Ohad Kadan, Larry Samuelson, the Editor, the Associate Editor and an anonymous referee for helpful comments. We would also like to thank the seminar audience at Yale University. * Corresponding author. Fax: +1 203 436 9023. E-mail addresses: [email protected] (S. Moroni), [email protected] (J. Swinkels). 0022-0531/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jet.2013.09.011
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that have cost converging to the full information first best, but no contract that achieves the first best. Jewitt, Kadan, and Swinkels [4] (henceforth JKS) provide a proof of existence when likelihood ratios are bounded. We point to an implicit, and not always correct, assumption in [4] by exhibiting a new class of examples in which there is non-existence of the optimal contract even if the likelihood ratio is bounded. The problem in these examples is that we know that an optimal contract must be of the form given by Holmström [2]. But, in some settings, the principal can easily run out of room to provide adequate incentives by contracts of this form. In particular, we show conditions under which any contract of the required form that gives the right utility provides inadequate incentives. When utility diverges at a finite consumption level the existence problem is severe – for any given utility function and information structure one can find a specification of the agent’s cost of effort and reservation utility such that an optimal contract does not exist. Thus for example, existence is not in fact guaranteed in the workhorse example of log utility. When utility diverges to −∞ only as consumption does as well, we show a necessary and sufficient condition for the missing step for the proof in [4] to hold, and hence for existence. The condition is that, in a sense to be made precise, the agent remains risk averse as consumption diverges to −∞. Under a technical regularity condition, the condition fails only if u has bounded slope, and so becomes essentially linear as consumption diverges. Hence, the existence problem is not severe for this case. For settings where the utility is bounded below, the original JKS proof goes through without problem. Because our setting has bounded likelihood ratios, we show that unlike in the Mirrlees example we cannot approach the first best.1 But, we provide a characterization of both the limit to which any sequence of contracts with costs approaching the second best must converge, and the limit cost. We also show that one can approach the second best using contracts that bound utility strictly above −∞.2 The limit contract has an intuitive form. It can be thought of as starting from a contract of the standard form that provides both too much utility and too little incentives, and then modifying it by providing very low utility on a small interval near the lowest signal, which both eliminates the extra utility the base contract provides and relaxes the incentive constraint so as to restore feasibility. Since the problem is too much utility and too little incentives, and since the trade-off of extra incentives per unit of utility taken away is most favorable at the lowest signal, the “right” way to do this is to concentrate the modification arbitrarily close to the lowest signal, and it is this that leads to non-existence. 2. Model The model is standard. A risk neutral principal employs a risk averse agent. The agent’s utility function over final wealth is u : D → R where u is in C 2 with u > 0 and u < 0. To focus attention, we assume D ⊆ R is an interval with upper bound ∞, and lower bound d, where d may 1 In the case of unbounded likelihood ratios, one can concentrate punishments at outcomes which are very unlikely at the desired effort level, but the probability of which rises fast with a deviation. Here, the only way to provide incentives is to face the agent with risk at the desired action. 2 The case where utility has a lower bound above −∞ is one for which JKS’s existence proof is valid, and for which contracts have a simple characterization as truncated versions of standard Holmström [2] contracts.
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be finite or −∞, and where d may or may not be in D. We assume u(d) ≡ limw↓d u(w) = −∞, limw↑∞ u(w) = ∞, and limw↑∞ u (w) = 0. Write also u (d) for limw↓d u (w).3 The agent chooses an effort level e ∈ [0, e] ¯ which is unobservable to the principal. The effort cost is given by the function c(e) ∈ C 2 with c (e) > 0 for e > 0 and c (e) > 0. The agent’s utility is additively separable and equal to u(w) − c(e). An outcome x ∈ X = [0, 1] is realized according to F (x|e) ∈ C 3 . The density of F is f , with f (x|e) > 0 for all x and e. As f is continuous, f (x|e) is uniformly bounded from zero and ∂ fe (x|e) so fe (x|e)/f (x|e) is also bounded. We assume that ∂x f (x|e) is strictly positive, so that the monotone likelihood ratio property (MLRP) holds. The principal’s gross benefit of e is given by B(e) = xf (x|e) dx. The agent is compensated according to π(x) : (0, 1] → D.4 Let U (π, e) ≡ u π(x) f (x|e) dx. The agent’s net utility is given by U (π, e) − c(e). Let C(π, e) ≡ π(x)f (x|e) dx be the expected cost to the principal for contract π and effort e. The participation constraint is U (π, e) − c(e) u0 ,
(IR)
where u0 is the agent’s outside option. The incentive compatibility constraint is e ∈ arg max U (π, e) ˆ − c(e) ˆ . e∈[0, ˆ e] ¯
Following Rogerson [6], we work with the (doubly) relaxed incentive constraint Ue (π, e) c (e).
(IC)
Various primitives for the validity of this exist. The simplest is that Convexity of the Distribution Function (CDFC) is satisfied so that Fee (x|e) 0 for all x and e.5 For any given e > 0, the relaxed cost minimization problem is min C(π, e) π
(P)
s.t. (IC), (IR). 3 For the case d > −∞, it follows from u(d) = −∞ that u (d) = ∞. When d = −∞, it is possible that u (d) > −∞. 1. Assume for example that for w < −1, u(w) = w + w 4 We write things this way to allow for the possibility that at the zero measure point x = 0 the principal may wish to pay d, where (recall) d may not be in D. Since u(d) = −∞, it cannot be that the principal wants to pay d on a positive measure set of signals in any contract satisfying the agent’s individual rationality constraint. 5 One could also apply the less stringent conditions from Jewitt [3], except at one point later in the paper at which we approach the second-best optimum by contracts which are subject to a minimum payment constraint.
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Let b =
1 u (d) .
671
For any given e, μ > 0, and b b, let
δ(x|e) =
fe (x|e) fe (0|e) − f (x|e) f (0|e)
and define the contract πb,μ,e (·) implicitly on [0, 1] by 1 = b + μδ(x|e), u (πb,μ,e (x))
(1)
where we note that by MLRP, for x > 0, πb,μ,e (x) ∈ D. Note that this is just a re-parametrization (0|e) . of the standard Holmström [2] (λ, μ)-contract, where in particular, b = λ + μ ffe(0|e) Lemma 1. A necessary condition for π to solve (P) is that π = πb,μ,e (·) for some b b and μ > 0 where U (πb,μ,e , e) = u0 + c(e) and Ue (πb,μ,e , e) = c (e). Proof. This is essentially standard (see Holmström [2]). But, given that one point of this paper is to argue that there is some lack of clarity in previous existence proofs, we provide an elementary proof for completeness. Let π : (0, 1] → D be an optimal contract. Choose disjoint Borel (X2 |e) (X1 |e) > FFe(X and sets X1 , X2 , X3 and X4 of (0, 1] with positive F (·|e)-measure where FFe(X 2 |e) 1 |e) Fe (X4 |e) F (X4 |e)
(X3 |e) > FFe(X . Consider raising utility by F (Xz2 |e) on X2 , and lowering it by F (Xz1 |e) on X1 .6 3 |e) This leaves the expected utility of the agent unchanged, and changes his incentives at rate
Fe (X2 |e) Fe (X1 |e) − , F (X2 |e) F (X1 |e) while changing the expected cost to the principal at z = 0 at rate 1 1 1 1 f (x|e) dx − f (x|e) dx F (X2 |e) u (π(x)) F (X1 |e) u (π(x)) X2 X1 1 1 X2 − E X 1 . =E u (π(x)) u (π(x)) So, if E
1 u (π(x)) X2 Fe (X2 |e) F (X2 |e)
−
1 u (π(x)) X1 Fe (X1 |e) F (X1 |e)
−E
0 follows since otherwise Ue (π, e) 0. That 1 > b. That (IR) binds follows b b follows since π(x) ∈ D for all x ∈ (0, 1], and hence u (π(x)) from the standard thought experiment of removing a small constant from utility everywhere. That (IC) binds is also standard. If not, then pick disjoint intervals X1 and X2 where X2 is strictly to 6 Here and in many places that follow, we will allow ourselves to consider variational arguments in which the variation creates a non-monotonicity in the contract. This is consistent with the logic of the relaxed problem, for which the monotonicity of an optimal contract is a result, not an assumption. One could build each variation to be monotone, but only at considerable extra notational cost.
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the right of X1 . Lower payments on X2 by z, and raise them on X1 by z to restore (IR). Since (IC) is slack, this contract is feasible for small z, and since payments on X2 are strictly larger than those on X1 and the agent is risk averse, the principal saves money when z is small. 2 3. Non-existence: a class of simple examples Let us turn to the new class of non-existence examples. In this section, we work with u(w) = ln w. In the next section, we work with more general utility functions. Since we have assumed that ffe is bounded, our class includes a set of cases for which Jewitt, Kadan, and Swinkels [4] assert existence. Lemma 2. Let u(w) = ln w. Then, Ue (π0,1,e , e) is finite, and for any c(·) with c (e) > Ue (π0,1,e , e), there exists no optimal contract implementing e. The idea of the proof is to show that Ue (π0,1,e , e) is the strongest incentive that can be provided with a contract of the form required by Lemma 1. Example 1. Let
1 f (x|e) = 1 + (e − 1) x − 2
be the standard FGM linear copula. Then, it is straightforward that δ(x|1) = x and so 1 Ue (π0,1,e , e) =
ln δ(x|1) fe (x|1) dx
0
1 =
1 1 (ln x) x − dx = . 2 4
0
Hence, for any c(·) with c (1) > 14 , there is no optimal contract implementing e = 1. Proof of Lemma 2. That Ue (π0,1,e , e) is finite will be shown in Lemma 4. Because u(w) = ln w, 1 we have u (w) = w, and so 1 Ue (πb,μ,e , e) =
ln b + μδ(x|e) fe (x|e) dx,
0
from which ∂ Ue (πb,μ,e , e) = ∂b
1 0
where by MLRP,
1 b+μδ(x|e)
1 fe (x|e) dx, b + μδ(x|e)
is positive and decreasing in x. Hence, since fe (·|e) single crosses 0 1 from below, and so, since 0 fe (x|e) = 0, ∂ Ue (πb,μ,e ) 0. ∂b
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But, ∂ Ue (πb,μ,e ) = ∂μ
1 0
δ(x|e) fe (x|e) dx 0, b + μδ(x|e)
with equality if and only if b = 0, since the fraction is constant in x for b = 0, and strictly increasing otherwise. It follows that for each μ, Ue (πb,μ,e ) is maximized by taking b = 0, and that Ue (π0,μ,e ) is independent of μ. Hence for any c(·) and e for which c (e) > Ue (π0,1,e ), no contract of the form πb,μ,e satisfies (IC), and so by Lemma 1, there is in fact no optimal contract. 2 The difficulty for the proof in [4] is that it implicitly assumes that as b → b, so that utility at the worst outcome diverges to −∞, expected utility and incentives diverge. As Example 1 illustrates, this can fail. 4. A necessary and sufficient condition for existence We now present a necessary and sufficient condition for existence of the optimal contract for each specification of costs and the outside option. We will see that the condition always fails when d > −∞, and so the counterexample from the previous section is in fact very general. But, we will also see that when d = −∞, the condition is extremely mild, so that except in somewhat tortured examples, existence is guaranteed. We also note that the problem we identify in the proof by JKS is not an issue when u(d) > −∞. Hence, existence is essentially fine except in the case where utility goes to −∞ at some finite pay level, but is in doubt in any such case. 4.1. Asymptotic risk aversion Say that utility function u has Asymptotic Risk Aversion (ARA) if lim
w↓d
−u(w) + w = −∞. u (w)
(2)
When d is finite, then, since limw↓d −u(w) u (w) 0, ARA must fail. On the other hand, when d = −∞, then ARA is very weak. In particular, the following lemma shows that only if u converges to a finite limit, so that the agent becomes effectively risk neutral as income tends to −∞, can ARA fail.7 Lemma 3. Assume that d = −∞ and that sufficient for ARA is that u (d) = ∞.
−u(w) u (w)
has a well defined limit as w → −∞. Then,
7 When u does converge to a finite limit, ARA may or may not be satisfied. It fails when u(·) has the form u(w) = 1 for w < −2, but is satisfied when u(w) = w + ln(−w) for w < −2. w+ w
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Proof. If limw↓−∞ −u(w) u (w) = 0, then (2) is immediate. So, assume that limw↓−∞ u(w) u(w)u (w) ∂ ∂w − u (w) + w = (u (w))2 , we need 0 lim
w↓−∞
−u(w) u (w)
> 0. Since
−u(w) −u (w) dw = ∞, u (w) u (w)
w
and so given that limw↓−∞ 0
−u(w) u (w)
> 0 it is sufficient that limw↓−∞
0
−u (w) w u (w)
dw = ∞. But,
−u (w) dw = ln u (w) − ln u (0) u (w)
w
which diverges since limw↓−∞ u (w) = ∞.
2
The key implication of ARA is contained in the following lemma. Lemma 4. Fix e > 0 and μ > 0. Then, lim U (πb,μ,e , e) = −∞
(3)
b↓b
if and only if u has ARA, and lim Ue (πb,μ,e , e) = ∞
(4)
b↓b
if and only if u has ARA.8 ∂ fe (x|e) Proof. Fix e > 0 and μ > 0. Since F ∈ C 3 it follows from our assumptions that ∂x f (x|e) is bounded away from 0 and ∞. The implication is that h(·|e), defined as the density of δ(x|e), is also bounded from below by some h > 0, and above by some h¯ < ∞ on the support [0, δ(1|e)]. Let ω(t), defined by 1 u (ω(t))
= t,
(5)
be the amount the agent is paid when
1 u
= t . Then, changing variables,
1 lim U (πb,μ,e , e) = lim b↓b
u(πb,μ,e )f (x|e) dx
b↓b
0 δ(1|e)
u ω(b + μz) h(z|e) dz,
= lim b↓b
0
and, 8 As will be seen in the proof, both limits are guaranteed to exist in the extended reals.
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1 lim Ue (πb,μ,e , e) = lim b↓b
u(πb,μ,e )
b↓b
675
fe (x|e) f (x|e) dx f (x|e)
0
fe (0|e) u ω(b + μz) z + h(z|e) dz. f (0|e)
δ(1|e)
= lim b↓b
0
Choose zˆ > 0 and bˆ > b such that zˆ +
fe (0|e) f (0|e)
< 0 and u(ω(bˆ + μˆz)) < 0. Then, both
δ(1|e)
u ω(b + μz) h(z|e) dz
lim b↓b
zˆ
and fe (0|e) u ω(b + μz) z + h(z|e) dz f (0|e)
δ(1|e)
lim b↓b
zˆ
are well defined and finite by the dominated convergence theorem, since in particular, for b ∈ ˆ and z ∈ [ˆz, δ(1|e)] [b, b], −∞ < u ω(b + μˆz) u ω(b + μz) u ω bˆ + μδ(1|e) < ∞ (0|e) is bounded as well since δ(1|e) < ∞. and since z + ffe(0|e) ¯ while But, on [0, zˆ ], h is bounded by h and h, fe (0|e) ¯ fe (0|e) fe (0|e) −∞ < h z+ h(z|e) zˆ + h < 0. f (0|e) f (0|e) f (0|e) It follows that the limits in (3) and (4) are well-defined, and that both (3) and (4) are equivalent to zˆ lim b↓b
u ω(b + μz) dz = −∞
0
or, changing variables and discarding a constant, to b+μˆ z
u ω(z) dz = −∞.
lim b↓b
(6)
b
Integrating by parts, b+μˆ z
b+μˆz u ω(z) dz = u ω(z) zb −
b
b+μˆ z
u ω(z) ω (z)z dz
b
= u(b + μˆz)(b + μˆz) − u ω(b) b −
b+μˆ z
ω (z) dz
b
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= u(b + μˆz)(b + μˆz) − u ω(b) b − ω(b + μˆz) + ω(b) u(ω(b)) = u(b + μˆz)(b + μˆz) − − ω(b + μˆz) + ω(b) u (ω(b)) where both the simplification within the integral and the last equality use (5). But, lim u(b + μˆz)(b + μˆz) − ω(b + μˆz) = u(b + μˆz)(b + μˆz) − ω(b + μˆz) b↓b
and thus is finite and so (6) is equivalent to −u(w) lim + w = −∞, w↓d u (w) as claimed.
2
Before we move on, we make two notes. First, an issue similar to what we have identified can also arise if limw↑∞ u (w) > 0. Then, unless an analog to ARA holds, one can have a case where, (x|e) = holding fixed μ, utility does not diverge as we increase b towards the value where b + μ ffe(x|e) 1 limw↑∞ u (w) .
We rule out this case both because we find it less economically interesting, and because the technical point is already made in the case we examine. Second, Carlier and Dana [1] do not, as far as we can tell, have the same problem as JKS. The key is that in their environment, the principal is also strictly risk averse with utility v(·), where in particular there is α ∈ (0, 1), such that as w ↑ ∞, v(w) w α → 0. This implies that as large amounts are transferred from the agent to the principal, there remains enough curvature in the system as to avoid the discontinuity issue discussed here. In particular note that since v(w) w α → 0, limw↑∞ v (w) = 0. 4.2. The result and proof We begin with a small lemma. For the balance of the paper, assume that some e > 0 has been fixed. Lemma 5. Let u not have ARA. Then, there is a unique μˆ such that if we let πˆ = πb,μ,e ˆ , then U (π, ˆ e) = u0 + c(e). Proof. This follows immediately from the intermediate value theorem, noting that by Lemma 4 U (πb,μ,e , e) is finite and continuous in μ for all μ > 0, and that for x > 0, πb,μ,e (x) is increasing in μ with limμ↓0 πb,μ,e (x) = −∞ and limμ↑∞ πb,μ,e (x) = ∞ and so by the monotone convergence theorem, limμ↓0 U (πb,μ,e , e) = −∞ and limμ↑∞ U (πb,μ,e , e) = ∞. 2 Proposition 1. An optimal contract implementing e exists if and only if either u has ARA or u does not have ARA, but Ue (π, ˆ e) c (e). Thus, when ARA holds, existence holds regardless of c(·) and u0 . But, when ARA fails, then existence will hold for some specifications of c(·) and u0 , but one can always find a specification of c(·) and u0 where existence fails. ˜ μ) Proof of necessity. Assume Ue (πˆ , e) < c (e), and, given Lemma 1 let (b, ˜ = (b, μ) ˆ be such that π˜ = πb, is optimal, where U ( π ˜ , e) = U ( π ˆ , e) = u + c(e). Then, by Lemma 5, b˜ > b, 0 ˜ μ,e ˜
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677
and so, since U (π˜ , e) = U (π, ˆ e), it must be that μ˜ < μ, ˆ and that π˜ single crosses πˆ from above. But then, since 1 U (πˆ , e) − U (π˜ , e) =
u(πˆ ) − u(π˜ ) f (x|e) dx = 0,
0
it follows from MLRP that 1 Ue (πˆ , e) − Ue (π˜ , e) =
fe (x|e) u(πˆ ) − u(π˜ ) f (x|e) dx > 0, f (x|e)
0
and so, as πˆ fails (IC), so does π˜ .
2
Proof of sufficiency. The point here is simply that the JKS construction works once either (3) holds or Ue (πˆ , e) c (e). In particular, assume (3). Then, for any given c, u0 , e, and μ 0, there is, by the intermediate value theorem, b(μ) > b such that (IR) holds with equality at πb(μ),μ,e . Because b(μ) > b, payoffs and incentives are continuous in b and μ at (b(μ), μ), and so, as in [4], b(·) is continuous and hence Ue (πb(μ),μ,e , e) is continuous in μ. But, Ue (πb(0),0,e , e) = 0 and JKS argue that when μ is large enough, (IC) is slack.9 But then, again by the intermediate value theorem, there is a μ where Ue (πb(μ),μ,e , e) = c (e), and, as JKS show, this is sufficient ˆ to establish optimality. Similarly assume ARA fails, but Ue (πˆ , e) c (e). Then, for μ < μ, ˆ and so on the domain [0, μ], ˆ b(μ) and Ue (πb(μ),μ,e , e) are continub(μ) > b by definition of μ, ous in μ, and we are done as before by the intermediate value theorem. 2 5. Characterizing near optimal contracts Assume that there is no optimal contract. It still must be the case that over the set of feasible solutions implementing e in the (doubly relaxed) moral hazard problem there is an infimum C ∗ (e) of expected costs. In this section, we characterize nearly optimal contracts. This gives us a tight expression for C ∗ (e), gives insight into how existence fails, and allows us to show that, in contrast to the Mirrlees counter-example, in this example, C ∗ (e) is strictly higher than the full information first best. We begin with a small lemma. Lemma 6. Assume there is no optimal contract implementing e. Then, there exists a unique pair μ∗ > 0 and τ > 0 such that if we let π ∗ = πb,μ∗ ,e , then U π ∗ , e = c(e) + u0 + τ (7) and fe (0|e) Ue π ∗ , e = c (e) + τ. f (0|e)
(8)
9 The argument is particularly simple here. Let x ∗ be such that f (x ∗ |e) = 0. Consider the contract π which pays z e z for x x ∗ , and pays a constant k(z) for x < x ∗ , where k(z) > d is chosen so that πz satisfies (IR) with equality. Choose some zˆ large enough that (IC) is slack at πzˆ . Now, note that for μ sufficiently large, the contract πb(μ),μ,e will pay less than πzˆ (0) = k(ˆz) at 0 (as μ → ∞, it must be that b(μ) → b) but, since b + μδ(x ∗ |e) diverges, will for μ large enough, pay more than zˆ at x ∗ . Thus, πb(μ),μ,e single crosses πzˆ from below, and so provides stronger incentives than it by Lemma 5 of [4].
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Proof. Define ν(μ) = U (πb,μ,e , e) − c(e) − u0 , and K(μ) = Ue (πb,μ,e , e) − c (e) − ν(μ)
fe (0|e) , f (0|e)
and let μˆ be as defined in Lemma 5, so that ν(μ) ˆ = 0 and hence any μ < μ(e) ˆ has ν(μ) < 0. By Proposition 1 and Lemma 5, ν(μ) ˆ = 0 and K(μ) ˆ < 0. But, as μ ↑ ∞, U (πb,μ,e , e), and hence ν(μ), diverges to ∞, while Ue (πb,μ,e , e) remains weakly positive (since πb,μ,e is monotone and (0|e) < 0, there is at least since MLRP implies first order stochastic dominance). Hence, since ffe(0|e) ∗ ∗ ∗ ∗ one μ > μˆ with K(μ ) = 0. By construction, μ and τ = ν(μ ) satisfy the requisite conditions. To see that μ∗ is unique, note that ∂ fe (0|e) ∂ K(μ) = Ue (πb,μ,e , e) − U (πb,μ,e , e) ∂μ ∂μ f (0|e)
1 ∂ = u ω b + μδ(x|e) δ(x|e)f (x|e) dx ∂μ 0
1 =
u ω b + μδ(x|e) ω b + μδ(x|e) δ 2 (x|e)f (x|e) dx > 0.
2
0
Remark 1. π ∗ is the unique optimal solution to the auxiliary problem in which costs are c(·) ˆ (0|e) τ. where c(e) ˆ = c(e) + τ , and cˆ (e) = c (e) + ffe(0|e) This follows directly from [4, Proposition 1]. It turns out that in a very strong sense, π ∗ is also the basis for a characterization of near optimal solutions to (P). In particular, by construction, π ∗ ∗ e (0|e) clears (IR) by τ , but fails (IC) by −f f (0|e) τ . Choose a small interval near 0, and reduce π on that interval so as to reduce overall utility by τ . When the interval is small, this improves incentives ∗ e (0|e) by approximately −f f (0|e) τ , and results in a contract with cost approximately C(π , e) − τ b. The content of the next proposition is not only that this construction approximates the best that can be done, but that any sequence of near optimal contracts must look very much like this construction. Proposition 2. Assume there is no optimal contract implementing e. Then, 1. C ∗ (e) = C(π ∗ , e) − bτ , 2. C ∗ (e) can be approximated by contracts each of which have a lower bound d > d on payments, and, 3. if we let πn (x) be any sequence of contracts satisfying (IR) and (IC) for e and with C(πn , e) → C ∗ , then, for each ε > 0, Pr πn (x) − π ∗ (x) > ε → 0. 2
Example 2. Return to the setting of Example 1. Assume c(e) = e2 and that u0 = −10. Then, effort levels e e˜ ≈ 0.31 can be implemented by optimal contracts, but higher effort levels cannot. Using Proposition 2, one can (numerically, but not in closed form) find the supremum of payoffs to the principal for any given e > e. ˜ In this example, it turns out that e = 1 maximizes
S. Moroni, J. Swinkels / Journal of Economic Theory 150 (2014) 668–682
679
this supremum. Hence, the primal problem of finding an optimal e and π does not have a solution even though the dual problem does for some effort levels. To prove Proposition 2, we begin with a lemma. Lemma 7. Let π be any contract. For d > d, define πd (x) = max(π(x), d). Then, lim u πd (x) f (x|e) dx = u π(x) f (x|e) dx, d↓d
lim
d↓d
and
u πd (x) fe (x|e) dx =
lim
d↓d
u π(x) fe (x|e) dx
(9) (10)
πd (x)f (x|e) dx =
π(x)f (x|e) dx.
(11)
Proof. As d ↓ d, u(πd (x)) and πd (x) decrease monotonically, and converge to u(π(x)) and π(x). Hence, (9) and (11) follow by Lebesgue’s monotone convergence theorem. To see (10), let (x|e) and note that M be a bound on ffe(x|e) u πd (x) fe (x|e) dx − u π(x) fe (x|e) dx fe (x|e) f (x|e) dx = u πd (x) − u π(x) f (x|e) M u πd (x) − u π(x) f (x|e) dx which converges to 0 by (9).
2
Proof of Proposition 2. Let us first show that C ∗ C(π ∗ , e) − τ b, and that C(π ∗ , e) − τ b can be approached by finite contracts. For ε > 0 and d > d, define πε,d by adding ε to π ∗ (x) where fe (x|e) > 0 and by replacing any payment below d by d. By Lemma 7, for any ε > 0, d > d can be chosen small enough that U (πε,d , e) > u0 + c(e) + τ, fe (0|e) τ, f (0|e) C(πε,d , e) < C π ∗ , e + ε. Ue (πε,d , e) > c (e) +
and
τ For any y ∈ (0, 1], define the contract π˜ from πε,d by reducing utility by F (y|e) utils on (0, y]. τ Since u(s) = u(d) − F (y|e) has solution s > d, π˜ remains bounded, with U (π˜ , e) > u0 + c(e) (0|e) τ , for y small enough, and C(π˜ , e) < C(π ∗ , e) + ε − τ b. Since Ue (πε,d , e) > c (e) + ffe(0|e) ∗ Ue (π˜ , e) > c (e). Hence, π˜ implements e at cost at most C(π , e) + ε − τ b. Since ε > 0 was arbitrary, C ∗ C(π ∗ , e) − τ b, and C(π ∗ , e) − τ b can be approached by finite contracts.
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Assume C ∗ < C(π ∗ , e) − τ b. Then, there are π and κ > 0 with U (π, e) u0 + c(e), Ue (π, e) c (e), and C(π, e) = C π ∗ , e − τ b − κ. Fix y > 0 such that u (π1∗ (y)) < b + κτ (by definition of π ∗ limy↓0 u (π1∗ (y)) = b) and for z > 0, z define πz (x) from π ∗ by setting u(πz (x)) = u(π ∗ ) + F (y|e) on (0, y], so that U (πz , e) = u0 + c(e) + τ + z. Note that fe (0|e) Ue (πz , e) c (e) + (τ + z). f (0|e) Define π˜ z implicitly on (0, 1] by τ z u π˜ z (x) = u πz (x) + u π(x) . τ +z τ +z Then, τ z U (π˜ z , e) = U (πz , e) + U (π, e) τ +z τ +z z τ c(e) + u0 + τ + z + c(e) + u0 τ +z τ +z = c(e) + u0 + τ, and τ z Ue (πz , e) + Ue (π, e) τ +z τ +z τ fe (0|e) z c (e) + (τ + z) + c (e) τ +z f (0|e) τ +z fe (0|e) = c (e) + τ , f (0|e)
Ue (π˜ z , e) =
and so π˜ z is a feasible solution to the auxiliary problem. Now by (12), ∂ π˜ z (x) 1 τ ∂u(πz (x)) ∂ z = + u π(x) − u πz (x) , ∂z u (π˜ z (x)) τ + z ∂z ∂z τ + z and so ∗ ∂ π˜ z (x) 1 1 ∂u(πz (x)) = + u π(x) − u π (x) , ∂z z=0 u (π ∗ (x)) ∂z τ z=0 from which y ∂ π˜ z (x) 1 1 = f (x|e) f (x|e) dx ∗ ∂z u (π (x)) F (y|e) z=0 0 ∗ 1 1 u π(x) − u π (x) f (x|e) dx + τ u (π ∗ (x)) 1 1 π(x) − π ∗ (x) f (x|e) dx ∗ + u (π (y)) τ
(12)
S. Moroni, J. Swinkels / Journal of Economic Theory 150 (2014) 668–682
=
1 u (π ∗ (y))
−b−
681
κ τ
0, Pr πn (x) − π ∗ (x) > ε → 0. Assume by way of contradiction that for some ε > 0, δ > 0, and along a subsequence, Pr πn (x) − π ∗ (x) > ε > δ.10 Choosing xˆ > 0 so that F (x|e) ˆ < 2δ , and letting εˆ = εu (π ∗ (1) + ε) > 0, δ Pr {x > x} ˆ ∩ u πn (x) − u π ∗ (x) > εˆ > . 2 Let
(13)
ˆ + u−1 (u) ˜ u−1 (u) ˆ + u˜ −1 u ρ(u, ˆ u) ˜ = −u 2 2
be the amount saved by replacing a 50/50 lottery between uˆ and u˜ by its certainty equivalent. Then, ρ(u, ˆ u) ˜ is continuous, and by convexity of u−1 , strictly positive when uˆ = u. ˜ For uˆ = u, ˜ ρ is decreasing in its smaller argument and increasing in its larger argument. Hence, since [u(π ∗ (x)), ˆ u(π ∗ (1))] is compact, r=
min
∗ (x)),u(π ∗ (1))],|u− u∈[u(π ˆ ˆ ˜ u|ˆ ˆ ε
ρ(u, ˆ u) ˜ > 0.
(14)
Let πˆ n (x) be defined for all x by u(πˆ n (x)) = 12 u(π ∗ (x)) + 12 u(πn (x)). Then, τ U (πˆ n , e) u0 + c(e) + , 2 τ fe (0|e) Ue (πˆ n , e) c (e) + . 2 f (0|e)
(15) (16)
Now, simply paying according to πn half the time and π ∗ half the time costs 12 C(πn , e) + 1 ∗ ∗ 2 C(π , e). But, (13) puts a lower bound on the probability that u(πn ) and u(π ) are significantly different, with x > x, ˆ while (14) puts a lower bound on what is saved by replacing the relevant lottery with its certainty equivalent, and hence, δ 1 1 C(πˆ n , e) C(πn , e) + C π ∗ , e − r 2 2 2 1 1 ∗ 1 δ = C(πn , e) + C + bτ − r. 2 2 2 2 Using that C(πn , e) → C ∗ , choose some n˜ such that 1 1 C(πˆ n˜ , e) C ∗ + bτ − δr. 2 3 10 The proof extends in a simple manner to the case where contract π involves randomization. n
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1 Form π˜ by increasing πˆ n˜ by 12 δr where fe > 0. Then, C(π˜ , e) C ∗ + 12 bτ − 14 δr, and (0|e) τ Ue (π˜ , e) > c (e)+ τ2 ffe(0|e) . Hence, forming π˜ y by reducing u(π˜ ) by 12 F (y|e) on (0, y] and choosing y small enough,
U (π˜ y , e) u0 + c(e), Ue (π˜ y , e) c (e), and 1 C(π˜ y , e) < C ∗ − δr, 4 contradicting that C ∗ was the infimum of costs for contracts implementing e.
2
In the Mirrlees example, non-existence is shown by exhibiting a sequence of contracts that implement e, but approach the cost u−1 (u0 + c(e)) of a full information setting. Since μ∗ > 0, it follows immediately that in our setting C ∗ (e) is strictly higher than the first best. In particular, we have C π ∗ , e > u−1 u0 + c(e) + τ , and so
C π ∗ , e − bτ > u−1 u0 + c(e) + τ − bτ u−1 u0 + c(e) .
It also follows from Proposition 2, and from [4, Proposition 1] that if one lets bd and μd be such that the optimal contract implementing e subject to paying no less than d pays the larger 1 of d and the solution to u (π(x)) = bd + μd δ(x|e) then (bd , μd ) → (b, μ∗ ). References [1] G. Carlier, R.A. Dana, Existence and monotonicity of solutions to moral hazard problems, J. Math. Econ. 41 (2004) 826–843. [2] Bengt Holmström, Moral hazard and observability, Bell J. Econ. 10 (1979) 74–91. [3] Ian Jewitt, Justifying the first order approach to principal–agent problems, Econometrica 56 (1988) 1177–1190. [4] Ian Jewitt, Ohad Kadan, Jeroen M. Swinkels, Moral hazard with bounded payments, J. Econ. Theory 143 (2008) 59–82. [5] J. Mirrlees, The theory of moral hazard and unobservable behavior: part I, Rev. Econ. Stud. 66 (1999) 3–21. [6] William P. Rogerson, The first-order approach to principal–agent problems, Econometrica 53 (1985) 1357–1367.
