A New Class of Non Existence Examples for the Moral Hazard Problem Sofia Moroni and Jeroen Swinkels April, 23 Abstract We provide a class of counter-examples to existence in a simple moral hazard problem in which the first-order approach is valid. These examples involve utility functions such as lnw in which utility diverges to negative infinity at a finite wealth level. In contrast to the Mirrlees example, unbounded likelihood ratios are not needed, and no sequence of contract can approach the full information first best. The examples expose a hidden and not always correct assumption in the existence proof of Jewitt, Kadan, and Swinkels 28. We show that their result and proof method is valid under very mild conditions outside of this class of utility functions. In particular, if utility only goes to negative infinity when wealth does as well, then any utility function where the agent continues to dislike risk as wealth diverges to negative infinity will suffice. Introduction Mirrlees 999 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 the cost of which converges to the full information first best, but no contract that exactly achieves the first best. Jewitt, Kadan, and Swinkels 28, henceforth JKS provide a proof of existence when likelihood ratios are bounded. We point to an implicit, and not always correct, assumption in JKS 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 on the one hand, we know that an optimal contract must be of the form given by Hölmstrom 979. But, in some settings, the principal can easily run out of room to provide adequate incentives by contracts of the required 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. When utility diverges to only as consumption does as well, we show a necessary and sufficient condition for the missing step in JKS to hold, and hence for existence. The condition is that, in a sense to be made We thank Larry Samuelson and Ohad Kadan for helpful comments.
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. As is intuitive given bounded likelihood ratios, we show that unlike in the Mirrlees example we cannot approach the first best. But, we show that one can approach the second best using contracts that bound utility strictly above, and provide a characterization of both the it to which these contracts converge, and the it cost. The it contract has an intuitive form. It can be thought of as beginning 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 einates the extra utility the base contract provided and relaxes the incentive constraint so as to restore feasibility. Since the problem is too much utility and too little incentives, the right way to do this is to concentrate the modification closer and closer to the lowest signal because it is here that the trade-off of providing extra incentives per unit of utility taken away is most favorable, 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 > and u <. To focus attention, we assume D R is an interval with upper bound, and lower bound d, where d may be finite or, and where d may or may not be in D. We assume u d w d u w =, w u w =, and w u w =. Write also u d for w d u w. 2 The agent chooses an effort level e [, ē] which is unobservable to the principal. The effort cost is given by the function ce C 2 with c e > for e > and c e >. The agent s utility is additively separable and equal to uw ce. An outcome x X = [, ] is realized according to Fx e C 3. The density of F is f, with f x e > for all x and e. As f is continuous, f x e is uniformly bounded from zero and so f e x e/ f x e is also bounded. We assume that x fe x e is strictly positive, so that the monotone likelihood ratio property MLRP, holds. f x e The principal s gross benefit of e is given by Be = x f x e dx The agent is compensated according to πx :, ] D. 3 Let Uπ, e uπx f x edx. These are cases for which JKS s existence proof is valid, and for which contracts have a simple characterization as truncated versions of standard Hölmstrom 979 contracts. 2 For the case d >, it follows from u d = that u d =. When d =, it is possible that u d >. Assume for example that for w <, u w = w + w. 3 We write things this way to allow for the possibility that at the zero measure point x = 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. 2
The agent s net utility is given by Uπ, e ce. Let Cπ, e πx f x edx be the expected cost to the principal of contract π and effort e. The participation constraint is Uπ, e ce u, where u is the outside option. The incentive compatibility constraint is IR e arg max Uπ, ê cê. ê [,ē] Following Rogerson 985, let the doubly relaxed incentive constraint be U e π, e c e, IC and assume Convexity of the Distribution Function CDFC so that F ee x e for all x and e. 4 Under CDFC the First Order Approach is valid. Write b for u. For any given e > the relaxed cost minimization problem is d s.t. min Cπ, e π IC, IR. For any given e, µ >, and b b, write δ x e = f ex e f x e f e e f e and define the contract π b,µ,e implicitly on [, ] by u = b + µδ x e, π b,µ,e x where we note that by MLRP, for x >, π b,µ,e x D, and where we follow the convention that π b,µ,e = d. Note that this is just a re-parametrization of the standard Hölmstrom 979 λ, µ-contract, where in particular, b = λ + µ f e e f e. Lemma A necessary condition for optimality in the doubly relaxed problem is that the contract is of the form π b,µ,e for some b b and µ > where U π b,µ,e, e = u + c e and U e π b,µ,e, e = c e. Proof This is essentially standard see Hölmstrom 979. 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 :, ] D be an optimal contract. Choose disjoint Borel sets X, X 2, X 3 and X 4 of, ] with positive F e-measure where F ex 2 e FX 2 e > F ex e FX e and F ex 4 e FX 4 e > 4 We could also impose the less stringent conditions from Jewitt 988, except in the section in which we approach the second-best by constrained optimal contracts. 3
F e X 3 e FX 3 e z. Consider raising utility by FX 2 e on X 2, and lowering it by expected utility of the agent unchanged, and changes his incentives at rate F e X 2 e F X 2 e F e X e F X e, while changing the expected cost to the principal at z = at rate So, if F X 2 e = E E X 2 u π x X 2 u πx X 2 u π x f x e dx F X e E. E F e X 2 e FX 2 e F ex e FX e u πx X u π x X < E u πx X 4 z FX e on X. This leaves the X u π x E F e X 4 e FX 4 e F ex 3 e FX 3 e f x e dx u πx X 3 then the principal can improve on π by increasing payments on X 2 and X 3, and decreasing them on X and X 4 so as to restore the original incentives. It follows that a necessary condition for optimality is that the contract is of the form π b,µ,e. That µ > follows since otherwise U e π, e. That b b follows since π x D for all x, ], and hence u > b. That IR binds follows πx 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 X and X 2 where X 2 is strictly to the right of X. Lower payments on X 2 by z, and raise them on X by z to restore IR. Since IC is slack, this contract is feasible for small z, and since payments on X 2 are strictly larger than on X and the agent is risk averse, the principal saves money when z is small., 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. Note in what follows that since we have assumed that f e f is bounded, our class includes a set of cases for which Jewitt, Kadan, and Swinkels 28 assert existence. Lemma 2 Let u w = ln w. Then, U e π,,e, e is finite, and for any c with U e π,,e, e < c e, there exists no optimal contract implementing e. The point of the proof is to show that U e π,,e, e is in fact that strongest incentive that can be provided with a contract of the form required by Lemma. Example Let f x e = + e x 2 4
be the standard FGM linear copula. Then, it is straightforward that δ x = x and so U e π,,e, e = = ln δ x f e x dx ln x x dx = 2 4. Hence, for any c with c > 4, there is no optimal contract implementing e =. Proof of Lemma 2 That U e π,,e, e is finite will be shown in Lemma 4. Because u w = ln w, we have u = w, and so w from which U e π b,µ,e, e = ln b + µδ x e f e x e dx, b U e π b,µ,e, e = b + µδ x e f e x e dx, where by MLRP, b+µδx e is positive and decreasing in x. Hence, since f e e single crosses from below, and so, since f e x e =, b U e π b,µ,e. But, µ U δ x e e π b,µ,e = b + µδ x e f e x e dx, with equality if and only if b =, since the fraction is constant in x for b =, and strictly increasing otherwise. It follows that for each µ, U e π b,µ,e is maximized by taking b =, and that U e π,µ,e is independent of µ. Hence for any c and e for which c e > U e π,,e, no contract of the form π b,µ,e satisfies IC, and so by Lemma, there is in fact no optimal contract. The difficulty for the proof in JKS 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 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 note also 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. 5
4. Asymptotic risk aversion Say that utility function u has Asymptotic Risk Aversion ARA if u w w d u w + w =. When d is finite, then, since w d uw u w, ARA must fail. On the other hand, when d =, then ARA is very weak. In particular, we have the following lemma which shows only if u converges to a finite it, so that the agent becomes effectively risk neutral as income tends to, can ARA fail. 5 Lemma 3 Assume that d =. Assume also that wu w uw w. Then, sufficient for ARA is that u d =. Proof We want to prove that and u w u w have well defined its as uw w u + w =. 2 w The derivative of this expression is given by uwu w u w 2 and therefore we need uwu w w w u w 2 =. For sufficiency, note that by the convergence test for integrals the previous expression holds if w uwu w u w 2 w = w uw u ww u w u w w 2 }{{} = A >. 3 uw If w u ww =, then 2 holds. If w uw u >, then 3 holds because A diverges. ww This observation also follows by the convergence test since the integral of the numerator of A diverges given our assumption that w u = and given u u = lnu w, while dw is w finite. 2 The key implication of ARA is contained in the following lemma. Lemma 4 Fix e > and µ >. Then, if and only if u has ARA, and U π b,µ,e, e = 4 U e π b,µ,e, e = 5 5 When u does converge to a finite it, ARA may or may not be satsified. It fails when u has the form u w = w + w for w < 2, but is satisfied when u w = w + ln w for w < 2. 6
if and only if u has ARA. 6,7 Proof Fix e > and µ >. Since F C 3 it follows from our assumptions that x fe x e f x e bounded away from and. The implication is that h e defined as the density of δ x e, is also bounded from below by some h >, and above by some h < on support [, δ e]. Let ω t, defined by is u = t, 6 ω t be the amount the agent is paid when u = t. Then, changing variables, U π b,µ,e, e = = δ e uπ b,µ,e f x e dx u ω b + µz h z e dz, and, U e π b,µ,e, e = = uπ b,µ,e f e x e f x e dx f x e u ω b + µz δ e z + f e e f e h z e dz. Choose ẑ > and ˆb > b such that ẑ + f e e f e < and uωˆb + µẑ <. Then, both and δ e u ω b + µz h z e dz ẑ δ e u ω b + µz ẑ z + f e e f e h z e dz are well defined and finite by the dominated convergence theorem, since in particular, for b [b, ˆb], and z [ẑ, δ e] < u ω b + µẑ u ω b + µz uωˆb + µδ e < and since z + f e e is bounded as well since δ e <. f e But, on [, ẑ], h is bounded by h and h, while < f e e f e h z e h z e z + f e e f e ẑ + f e e f e h <. 6 As will be seen in the proof, both its are guaranteed to exist in the extended reals. 7 Carlier and Dana 24 also prove existence of an optimal contract in a principal agent context. They assume that both the principal and agent are strictly risk averse, and because they restrict utility functions to have full domain, our results do not immediately apply to their setting. But, for utility functions of full domain that fail ARA, we would again speculate that when c is sufficiently large, the principal will run out of room to provide sufficient incentives with contracts of the standard form. 7
It follows that the its in 4 and 5 are well-defined, and that both 4 and 5 are equivalent to ẑ u ω b + µz dz = or, changing variables and discarding a constant, to Integrating by parts, b+µẑ b u ω z dz = u ω z z b+µẑ b b+µẑ b u ω z dz =. 7 b+µẑ u ω z ω z zdz b = ub + µẑb + µẑ u ω b b b+µẑ b ω z dz = ub + µẑ b + µẑ u ω b b ωb + µẑ + ω b u ω b = ub + µẑ b + µẑ u ωb + µẑ + ω b ω b where both the simplification within the integral and the last equality use 6. But, ub + µẑ b + µẑ ωb + µẑ = ub + µẑ b + µẑ ωb + µẑ and thus is finite and so 7 is equivalent to as claimed. u w w d u w + w =, Before we move on, we make two notes. First, an issue similar to what we have identified can also arise if w u w >. Then, unless an analog to ARA holds, one can have a case where, holding fixed µ, utility does not diverge as we increase b towards the value where b + µ f ex e w u w f x e =. We rule this case out 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 25 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,, such that as w, vw w α. This implies that as large amounts are transfered from the agent to the principal, there remains enough curvature in the system as to avoid the discontinuity issue here. In particular note that since vw w α, w v w =. 4.2 The Result and Proof We begin with a small lemma. For the balance of the paper, assume that some e > has been fixed. 8
Lemma 5 Let u not have ARA. Then, there is a unique ˆµ such that if we let ˆπ = π b, ˆµ,e, then U ˆπ, e = u + c e. Proof This follows immediately from the intermediate value theorem, noting that by Lemma 4 U π b,µ,e, e is finite and continuous in µ for all µ >, and that for x >, π b,µ,e x is increasing in µ with µ π b,µ,e x = and µ π b,µ,e x = and so by the monotone convergence theorem, µ U π b,µ,e, e = and µ U π b,µ,e, e =. Proposition 2 An optimal contact implementing e exists if either u has ARA or u does not have ARA, but U e ˆπ, e c e. An optimal contract does not exist if u does not have ARA and U e ˆπ, e < c e Thus, when ARA holds, existence holds regardless of c and u. But, when ARA fails, then existence will hold for some specifications of c and u, but one can always find a specification of c and u where existence fails. Proof of Necessity Assume U e ˆπ, e < c e, and, given Lemma let b, µ = b, ˆµ be such that π = π b, µ,e is optimal, where U π, e = U ˆπ, e = u + c e. Then, by Lemma 5, b > b, and so, since U π, e = U ˆπ, e, it must be that µ < ˆµ, and that π single crosses ˆπ from above. But then, since it follows from MLRP that U ˆπ, e U π, e = U e ˆπ, e U e π, e = and so, as ˆπ fails IC, so does π. u ˆπ u π f x e dx =, u ˆπ u π f e x e f x e dx >, f x e Proof of Sufficiency The point here is simply that the JKS construction works once either 4 holds or U e ˆπ, e c e. In particular, assume 4. Then, for any given c, u, e, and µ, there is, by the intermediate value theorem, b µ > b such that IR holds with equality at π bµ,µ,e. Because b µ > b, payoffs and incentives are continuous in b and µ at b µ, µ, and so, as in JKS, b is continuous and hence U e π bµ,µ,e, e is continuous in µ. But, U e π b,,e, e = and JKS argue that when µ is large enough, IC is slack. 8 But then, again by the intermediate value theorem, there is a µ where U e π bµ,µ,e, e = c e, and, as JKS show, this is sufficient to establish optimality. Similarly assume ARA fails, but U e ˆπ, e c e. Then, for µ < ˆµ, b µ > b by definition of ˆµ, and so on the domain [, ˆµ], b µ and U e π bµ,µ,e, e are continuous in µ, and we are done as before by the intermediate value theorem. 8 The argument is particularly simple here. Let x be such that f e x e =. Consider the contract π z which pays z for x x, and pays a constant k z for x < x, where k z > d is chosen so that π z satisfies IR with equality. Choose some ẑ large enough that IC is slack at πẑ. Now, note that for µ sufficiently large, the contract π bµ,µ,e will pay less than πẑ = k ẑ at as µ, it must be that b µ b but, since b + µδ x e diverges, will for µ large enough, pays more than ẑ at x. Thus, π bµ,µ,e single crosses πẑ from below, and so provides stronger incentives than it by Lemma 5 of JKS. 9
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 of expected costs. In this section, we characterize nearly optimal contracts. This gives us a tight expression for C, gives insight into how existence fails, and allows us to show that, in contrast to the Mirrlees counter-example, in this example, C is strictly higher than the full information first best. Proposition 3 Assume there is no optimal contract implementing e. Then, there is one and only one pair µ > and τ > such that if we let π = π b,µ,e, then and Furthermore, U π, e = c e + u + τ U e π, e = c e + f e e f e τ. C = C π, e bτ, and C e can be approximated by contracts each of which have a lower bound d > d on payments. Note that it follows from JKS that π is the unique optimal solution to the auxiliary problem in which costs are ĉ where ĉ e = c e + τ, and ĉ e = c e + f e e f e τ. We begin with a Lemma. Lemma 6 Let π be any feasible contract. For d > d, define π d x = max π x, d. This is trivially also measurable. Then, u π d x f x e dx = u π x f x e dx, 8 d d and d d u π d x f e x e dx = u π x f e x e dx 9 d d π d x f x e dx = π x f x e dx. Proof As d d, u π d x and π d x decrease monotonically, and converge to u π x and π x. Hence, 8 and follow by Lebesgue s monotone convergence theorem. To see 9, note that u π d x f e x edx u π x f e x e dx = u π d x u π d x f ex e f x e dx f x e M u π d x u π d x f x e dx which converges to by 8.
Proof of Proposition 3 Define ν µ = U π b,µ,e, e c e u, and K µ = U e π b,µ,e, e c e ν µ f e e f e. Any µ < ˆµ e has ν µ <. By Lemma 5 and Proposition 2, ν ˆµ = and K ˆµ <. But, as µ, U π b,µ,e, e, and hence ν µ diverge to, while U e π b,µ,e, e remains weakly positive. Hence, since f e e <, there is at least one µ f e > ˆµ with K µ =. By construction, µ and τ = ν µ satisfy the conditions of the proposition. To see that µ is unique, note that µ K µ = µ U e π b,µ,e, e f e e f e U π b,µ,e, e = u ω b + µδ x e δ x e f x e dx µ = u ω b + µδ x e ω b + µδ x e δ 2 x e f x e dx >. Fix ε >, and define π ε,d by adding ε to π x where f e x e > and by replacing any payment below d by d. By Lemma 6, for any ε >, d > d can be chosen small enough that U π ε,d, e > u + c e + τ, U e π ε,d, e > c e + f e e τ, and f e C π ε,d, e < C π, e + ε. For any y, ], define the contract π from π ε,d by reducing utility by u s = u d τ utils on, y]. Since Fy e τ Fy e has solution s > d, π remains bounded, with U π, e > u + c e and C π, e < C π, e + ε bτ. Since U e π ε,d, e > c e + f e e f e τ, for y small enough, U e π, e > c e. Thus, since ε > was arbitrary, C C π, e τb, and C can be approached by finite contracts. Assume C < C π, e τb. Then, there is π with U π, e u + c e, U e π, e c e, and C π, e = C π, e τb κ for some κ >. Fix y > such that u π y < b + τ κ and for z >, define π z x from π by setting u π z x = u π + z Fy e on, y], so that U π z, e = u + c e + τ + z. Note that and U e π z, e c e + f e e τ + z, f e z C π z, e z= u π y.
Then, and Let α z = z τ+z,and define π z implicitly on, ] by u π z x = α z u π z x + α z u π x. U π z, e = α z U π z, e + α z U π, e α z c e + u + τ + z + α z c e + u = c e + u + τ, U e ˆπ z, e = α z U e π z, e + α z U e π, e α z c e + f e e f e τ + f e e f e z + α z c e = c e + τ f e e f e, and so ˆπ z is a feasible solution to the auxiliary problem. Now by, and so ˆπ z x z = ˆπ z x z u α z u π z x + α z u π x u π z x, ˆπ z x z = z= u πz x u π + x z τ u π x u π x, from which ˆπz x y f x e z = z= u π x F y e f x e dx + τ u π x u π x u π x f x e dx u π y + π x π x f x e dx τ = u π y b κ τ < by choice of y, which contradicts that π is the optimal solution in the auxiliary problem. In the Mirrlees example, non-existence is shown by exhibiting a sequence of contracts that implement e, but approach the cost u u + c e of a full information setting. Since µ >, it follows immediately that in our setting the second best cost is strictly higher than the the first best. In particular, we have C π, e > u u + ce + τ, and so C π, e bτ > u u + ce + τ bτ u u + ce. It also follows from Proposition 3 that if one lets b d and µ d be such that the optimal contract implementing e subject to paying no less than d pays the larger of d and the solution to u πx = b d + µ d δ x e then b d, µ d b, µ d. 2
References [] Carlier, G., and R. A. Dana, 24, Existence and Monotonicity of Solutions to Moral Hazard Problems, Journal of Mathematical Economics, 4, 826-843. [2] Holmström, Bengt, 979; Moral Hazard and Observability, Bell Journal of Economics,, 74-9. [3] Jewitt, Ian 988; Justifying the First Order Approach to Principal-Agent Problems, Econometrica, 56, 77-9. [4] Jewitt, Ian, Kadan, Ohad, and Swinkels, Jeroen M. 28; Moral Hazard with Bounded Payments, Journal of Economic Theory, 43, 59-82. [5] Mirrlees, J., 999; The Theory of Moral Hazard and Unobservable Behavior: Part I, Review of Economic Studies, 66, 3-2. 3