Moral Hazard: More than Two Levels of Effort

1. A Discrete Model

 Let us extend the model of chapter 4 by allowing more than two levels of effort. Consider the more general case, with n levels of production q₁ < q₂ < ··· < q_n and K levels of effort with 0 = e₀ < e₁ < ··· < e_k-1 and the disutilities of effort Ψ(e_k) = Ψ_k for all k in {0,… , K − 1}. We still make the normalization Ψ₀ = 0 and assume that Ψ_k  is increasing in k. Let π_ik  for i in {1, . . ., n} also denote the prob- ability  of  producing q_i   when  the  effort  level  is e_k.  Let  S_i = S(q_i ) denote  the  value for the principal of production q_i  for i in {1, . . ., n}. The agent still has a separable utility  function  over  monetary  transfer  and  effort  U  = u(t) − ψ(e),  where  u(·) is increasing and concave (u‘ > 0 and u“ ≤ 0) with the normalization u(0) = 0. In such an environment, a contract is a set of transfers {(t₁,… tn)} corresponding to all possible output levels. As in chapter 4, we proceed in two steps. First, we compute the second-best cost of inducing effort e_k for the principal. We denote this cost by C_k^SB. Second, we find the optimal level of effort from the principal’s point of view, taking into account both the costs and benefits of each action e_k .

Let us thus denote by (P_k ) the cost minimization problem of a principal willing to implement effort e_k. Using our by now standard change of variables, the important  variables  are  the  utility  levels  in  each  state  of  nature,  i.e.,  u_i   = u(t_i ) or  alternatively  t_i = h(u_i ) where  h  = u⁻¹  is  increasing  and  convex  (h’ >  0  and h” > 0).  (P_k) is  now  a  convex  problem  that  is  written  as:

(5.1) is the incentive constraint preventing the agent from exerting effort e_k’ , for k’ # k when the principal wants to implement effort e_k. There are K − 1 such constraints. (5.2) is the agent’s participation constraint when he exerts effort e_k. We denote the multiplier of (5.1) by and still, as in chapter 4, the multiplier of (5.2.) by μ. The value of this problem is the second-best cost of implementation C_k^SB for effort e_k.

It should be immediately clear that the second-best cost of implementing effort  e_k   is  such  that  C_k^SB  ≥ C_k^FB  = h(ψ_k),  where  C_k^FB  denotes  the  first-best  cost  of implementing effort e_k. This is so because the presence of incentive constraints in problem (P_k) implies that the value of this problem is necessarily not less than under complete information. Note that the inequality above is strict whenever one of  the  incentive  constraints  in  (5.1)  is  binding  at  the  optimum  of  (P_k).

The necessary and sufficient first-order conditions for the optimization of pro- gram (P_k) are thus written as

where t_ik is the transfer given to the agent in state i when the principal wants to implement effort e_k.

The new difficulty coming from more than two levels of effort is that there may be several incentive constraints binding, i.e., several multipliers that may be different from zero. Looking only at local incentive constraints may not be enough  to  characterize  the  solution  to  (P_k),  and  the  optimal  payments  are  then the solutions of a complex system of nonlinear equations. However, if the only binding incentive constraint is the local downward incentive constraint, the first- order condition for problem (P_k) is written simply as

When the cumulative distribution function of production is a convex function of the level of effort, and when MLRP holds, the local approach described above can be validated, as has been shown by Grossman and Hart (1983). We prove this proposition in section 5.1.3 for the case of a continuum of effort levels and a continuum of performances, but the same kind of proof also applies in this discrete framework.

Even if describing the behavior of the second-best cost of implementation C_k^SB is in general a difficult task, one may try to get some insights on how the principal chooses the second-best level of effort. The optimal second-best effort is indeed defined as

Finding this second-best effort e^SB is a rather difficult problem and there is, a priori, no reason to be sure that it is below its first-best value. Under- as well as over-provision of effort may now be obtained at the second-best effort.

Under-provision was obtained in chapter 4. To understand how over-provision may also arise, let us consider the following example with three possible levels of effort  e₀, e₁,  and  e₂,  and  two  possible  outcomes  yielding  S¯ and  S  to  the  principal.

The  probabilities  that  S¯ is  realized  are,  respectively, π₀, π₁  and  π₂,  with  π₀  < π₁   < π₂    and  the  corresponding  disutilities  of  effort  being  ψ₀  = 0  < ψ₁   < ψ₂. Under complete information, the intermediate effort is chosen when

The first inequality means that effort e₁ is preferred to e₀. The second inequal- ity means that e₁ is also preferred to e₂.

Under moral hazard, let us first observe that the first-best effort e₁ may no longer  be  implementable.  Indeed,  let  us  denote  by  u¯  and  u  the  levels  of  utility offered  to  the  agent  when  S¯ and  S  realize.  Incentive  compatibility  requires  that

so that the agent prefers exerting effort e₁ rather than e₀. Similarly, we must also have

to ensure that the agent prefers exerting effort e₁ than e₂. However, when , the set of payoffs (u¯, u), such that (5.6) and (5.7) are both satisfied, is empty. Hence, e₁ can no longer be implemented.

Effort e₀ remains obviously implementable with a null payment in each state of nature. Finally, effort e₂ is implemented when the incentive constraint

which ensures that effort e₂ is preferred to both e₁ and e₀, and the participation constraint

are both satisfied.

When , the maximand on the right-hand side of (5.8) is .

Hence, the second-best cost of implementing effort e₂ can easily be computed as

Therefore, effort e₂ is second-best optimal when

It  is  easy  to  check  that  one  can  find  values  of  S¯− S,  so  that  (5.5)  and  (5.11) both hold simultaneously. Our conclusion is in proposition 5.1.

Proposition 5.1: With more than two levels of effort, the second-best effort level may be greater than the first-best level.

2. Two Outcomes with a Continuum of Effort Levels 

To reduce the cumbersome difficulty of the discrete case, modellers have often preferred to allow for a continuum of effort levels. With more than two states of nature, one meets the soon-important technical problems analyzed in section 5.1.3. With only two possible levels of performance, the analysis nevertheless remains very tractable as we will demonstrate in this section. To make some progress in this  direction,  we  reparameterize  the  model  by  assuming  that π(e) = e,  for  all  e in  [0, 1].  Hence,  the  agent’s  effort  level  equals  the  probability  of  a  high  perfor- mance. The disutility of effort function ψ(e) is increasing and convex in e (ψ‘ > 0 and ψ” > 0) with ψ(0) = 0. Moreover, to ensure interior solutions, we assume that the Inada conditions ψ'(0) = 0 and ψ'(1) = +∞ both hold. Let us finally consider a risk-neutral agent with zero initial wealth who is protected by the limited liability constraints

and

Faced with an incentive contract , this agent chooses an effort e, such that

By strict concavity of the agent’s objective function, the incentive constraint rewrites with the following necessary and sufficient first-order condition:

The principal’s program becomes

As in the model of section 4.3, the limited liability constraint in (5.12) (resp.(5.13))  is  again  binding  (resp.  slack).  Replacing  t¯ with ψ'(e)  in  the  principal’s objective function, the principal’s reduced program (P’) is written as

When ψ”’ > 0 the principal’s objective function is strictly concave in e, and direct optimization leads to the following equation defining the second-best level of effort e^SB:

This second-best effort is obviously lower than the first-best effort e^∗, which is defined by

The first-best effort is such that the marginal benefit ΔS of increasing effort by  a  small  amount  de  is  just  equal  to  the  marginal  disutility  of  doing  so ψ(e^∗)de.

Under moral hazard, the marginal benefit ΔSde must be equal to the marginal cost ψ'(e^SB)de  plus  the  marginal  cost  of  the  agent’s  limited  liability  rent  e^SBψ”(e^SB)de. Indeed, with moral hazard the limited liability rent of the agent is strictly positive  because  this  rent  is  also  rewritten  as , where the right-hand side inequality is derived from the convexity of ψ(·), ψ(0) = 0, and the fact that e^SB > 0. Reducing this rent, which is costly from the principal’s point view, calls for decreasing effort below the first-best.

Remark: The model with a risk-neutral agent protected by limited lia- bility bears some strong resemblance to the adverse selection model of chapter 2. Indeed, in both models the principal reduces the expected volume of trade with the agent to reduce the latter’s information rent. Distortions in effort are now replacing distortions in output to reduce this information rent.

3. The First-Order Approach 

Let us now consider the case where the risk-averse agent may exert a continuous level of effort e in a compact interval [0, e¯] and by doing so incurs a disutility efeg which is increasing and convex (e^r > 0 and e^rr > 0) with ψ'(0) = 0. To avoid  corner  solutions,  we  will  also  assume  that  the  Inada  conditions ψ'(0) = 0 and ψ'(e¯) = +∞ are  satisfied.

The agent’s performance q˜ may take any possible value in the compact inter- val Q = [q, q¯] with the conditional distribution F(q|e) and the everywhere positive density  function  f(q|e).  We  assume  that  F(·|e) is  twice  differentiable  with  respect to e and that those distributions all have the same full support Q.

Formally, a contract {t(q˜)} that implements a given level of effort e must now satisfy the following incentive constraints:

and the participation constraint

The principal’s problem is thus

We  denote  by  {(t^SB(·), e^SB)} the  solution  to  (P).  The  first  difficulty  with  this problem is to ensure that such an optimum exists within the class of all admissible functions  t(·).  For  instance,  Mirrlees  (1999)  has  shown  that  the  problem  may sometimes have no optimal solution in the class of unbounded sharing rules. The difficulty here comes from the lack of compacity of the set of incentive feasible contracts.6 We leave aside these technicalities to focus on what we think is the main difficulty of problem fP g—simplifying the infinite number of global incentive constraints in (5.18) and replacing those constraints by the simpler local incentive constraint:

which is a necessary condition if the optimal effort level is positive.

This constraint simply means that the agent is indifferent about his choice between effort e and increasing (or decreasing) his effort by an amount de when he receives the compensation schedule {t(q˜)}.

Let  us  thus  define  (P^R) as  the  relaxed  problem  of  the  principal,  where  the infinite number of constraints (5.18) have now been replaced by (5.20):

We denote by {(t^R(·), e^R)} the solution to this relaxed problem. We will first characterize this solution. Then we will find sufficient conditions under which the  solution  of  the  relaxed  problem  (P^R) satisfies  the  constraints  of  the  original problem  (P).  Hence,  we  will  have  obtained  a  characterization  of  the  solution  for problem  (P).

Let  us  first  characterize  the  solution  to  the  relaxed  problem  (P^R).  Denoting the multiplier of (5.20) by λ and the multiplier of (5.19) by μ, we can write the Lagrangian L of this problem as

Pointwise optimization with respect to t(q) yields

The left-hand side of (5.22) is increasing with respect to t^R(q), because u” < 0. Provided that λ > 0, MLRP (here in its strict version for simplicity),

guarantees that the right-hand side of (5.22) is also increasing in q. Hence, under MLRP,  t^R(q) is  strictly  increasing  with  respect  to  q.

Remark: Note that the probability that the realized output is greater than a given q  when effort e is exerted is 1 − F(q|e). Let us check that increasing e raises this probability when MLRP holds. We have

where is the  derivative  of the  log-likelihood of f (·). But, by MLRP, α(q,e) is increasing in q. α(q,e) cannot  be  everywhere  negative,  because  by  definition . Hence, there exists q^∗ such that: α(q, e) ≤ 0 if and  only  if  q  ≤ q^∗.

F_e(q|e) is  decreasing  in  q  (resp.  increasing)  on , we necessarily have F_e(q|e) ≤ 0 for any q  in . Hence, when the agent exerts an effort e which is greater than e’, the distribution of output with e dominates the distribution of output with e^r in the sense of first-order stochastic dominance.

We now show that indeed λ > 0. Let us first denote by e^R the effort level solution  of  (P^R).  Multiplying  (5.22)  by  f(q|e^R) and  integrating  over  yields

since .  The  expectation  operator  with  respect  to  the  distribution of output induced by effort e^R  is denoted by. Because u’ > 0, we have μ > 0 and the participation constraint in (5.19) is binding.

Using (5.22) again, we also find

Multiplying both sides of (5.25) by and integrating over yields

where cov(.) is the covariance operator.

Using  the  slackness  condition  associated  with  (5.20),  namely ,  we  finally  get

Since u(·) and u’(·) covary in opposite directions, we necessarily have λ ≥ 0. Moreover,  the  only  case  where  this  covariance  is  exactly  zero  is  when  t^Rfqg is a constant for all q. But then the incentive constraint (5.20) can no longer be satisfied at a positive level of effort. Having proved that λ > 0, we derive the following proposition from above.

Proposition  5.2:  Under MLRP, the solution t^R(q) to the relaxed problem (P^R) is  increasing  in  q.

We can rewrite the agent’s expected utility when he receives the scheme {t^R(q)} and exerts an effort e as

where the second line is obtained simply by integrating by parts, and the third line uses  the  fact  that  F(q|e) = 0  and  F(q¯|e) = 1  for  all  e.

Since  ψ” >  0,  ψ‘ >  0  and  t^R(q) is  increasing  when  MLRP  holds,  U(e) is concave  in  e  as  soon  as  F_ee(q|e) > 0  for  all  (q, e).  This  last  property  is  called  the convexity of the distribution function condition (CDFC).

Remark 1: Joined to MLRP, CDFC captures the idea that increasing the agent’s effort also increases, at a decreasing rate, the probability 1 − F(q|e) that  the  realized  output  is  greater  than  q.

Remark 2: Note that CDFC may not always hold. Let us assume, for instance, that production is linked to effort as follows: q = e + ∈ where ∈ is  distributed  on  ]−∞, +∞[ with  a  cumulative  distribution  function G(·). Then, CDFC implies that the distribution of s has an increasing density, a rather stringent assumption.

Remark 3: It may seem surprising that such stringent assumptions are needed to prove the simple and intuitive result that the agent’s reward increases as his performance increases. However, remember that the dependence  of  t^R(·) on  q  (which  is  bad  from  the  insurance  point  of view) is interesting only to the extent that it creates incentives for effort. For a higher q to be a signal of a high effort, it must be that an increase of effort increases production unambiguously,7 but also that the informativeness of q about e increases with q (this is ensured by MLRP).

Since U(e) is concave for a solution of the relaxed problem, the first-order condition (5.20) is sufficient to characterize the incentive constraints. Accordingly, the  first-order  conditions  of  problem  (P )  are  the  same  as  those  of  problem  (P^R). Proposition 5.3 provides a summary.

Proposition 5.3: Assume that both MLRP and CDFC both hold. If the optimal effort level is positive, it is characterized by the solution of a relaxed  problem  (P^R) using  the  first-order  approach  in  (5.20).  We  have {(t^SB(·), e^SB)} = {(t^R(·), e^R)}.

This solution {t^SB(·), e^SB} is then characterized by the binding participation constraint (5.19), the incentive constraint (5.20), and the two first-order conditions of the principal’s problem, namely (5.22) and

Given the highly restrictive assumptions imposed to prove proposition 5.3, the validity of the first-order approach is somewhat limited. Furthermore, when the first-order approach is not valid, using it can be very misleading. The true solution may not even be one among the multiple solutions of the first-order conditions for the relaxed problem.8 As a consequence, most of the applied moral hazard literature  adopts  the  discrete  {0, 1} formalization  outlined  in  chapter.

An  example:     Take   S(q)  =  q   and   the   probability   distribution   of   production .

Note that . This distribution satisfies the MLRP property, since .

Moreover, we have 0, i.e., an increase of effort increases production in the first-order stochastic

dominance sense. Finally, ,  so   that   CDFC   is   not satisfied

The agent has utility function  is immediately writ-ten as

For a high enough status quo utility level U₀ of the agent, one can solve the binding participation constraint of the agent, the first order condition (5.20) of the agent’s problem with respect to e, and the first order condition (5.29) of the principal’s problem with respect to e, and obtain the optimal values of the parameters λ^SB, μ^SB, and e^SB of the transfer function, which becomes

Note that the agent’s objective function is strictly concave despite the fact that CDFC is not satisfied. Indeed,

Designing by t^FB the sure transfer associated with the first-best, figure 5.1 summarizes the analysis.

With respect to the first-best, the agent is punished for q ≤ q^∗ and rewarded for q > q^∗. The agent bears some risk that induces him to exert some effort e^SB, which depends on U .

The first-order approach has been one of the most debated issues in contract theory in the late seventies and early eighties. Mirrlees (1975) was the first to point out the limits of this approach and argued that it is valid only when the agent’s problem has a unique maximizer.10i11 Later he offered a proof for the use of this approach when the conditions MLRP and CDFC both hold. This proof was corrected by Rogerson (1985a). Finally, Jewitt (1988) offered a direct proof that the multiplier of the local incentive constraint was positive.12 Jewitt also showed that CDFC can be relaxed, pro- vided that the agent’s utility function satisfies further fine properties. Brown, Chiang, Ghosh, and Wolfstetter (1986) provided a condition validating the first-order approach independently of restrictions on the information struc- tures. Aloi (1997) extended the first-order approach to nonseparable utility functions for the agent. Sinclair-Desgagné (1994) generalized the first-order approach to the case where the principal observes several dimensions of the agent’s performance. Grossman and Hart (1983) gave an exhaustive char- acterization of the agent’s incentive scheme. Their approach is based on a complete description of the incentive and participation constraints when the performances take n ≥ 2 values and the agent’s effort belongs to a finite set. Araujo and Moreira (2001) showed how to solve the moral hazard problem in the absence of a first-order approach.

Figure 5.1: First-Best and Second-Best Transfers

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.