# A More General Utility Function for the Agent

Still keeping quasi-linear utility functions, let  U  = tC(q,θ) now be the agent’s objective  function  with  the  assumptions: Cq > 0, Cθ > 0, Cqq > 0   and Cqqθ > 0. The  generalization  of  the  Spence-Mirrlees  property  used  so  far  is  now C > 0. This latter condition still ensures that the different types of the agent have indif- ference curves which cross each other at most once. It is obviously satisfied in the linear case C(q, θ) = θq  that was analyzed before. Economically, this Spence- Mirrlees property is quite clear; it simply says that a more efficient type is also more efficient at the margin.

The analysis of the set of implementable allocations proceeds closely, as was done previously. Incentive feasible allocations satisfy the following incentive and participation constraints: ### 1. The Optimal Contract

Following the same steps as in section 2.5, the incentive constraint of an efficient type in (2.40) and the participation constraint for the inefficient type in (2.43) are the two relevant constraints for optimization. These constraints rewrite respec- tively as where (with > 0  and > 0  from  the  assumptions made  on  C(·))  and Those constraints are both binding at the second-best optimum, which leads to the following expression of the efficient type’s rent Since > 0, reducing the inefficient agent’s output also reduces, as in Sec- tion 2.6, the efficient agent’s information rent.

With  the  assumptions  made  on  C(·),  one  can  also  check  that  the  principal’s objective function is strictly concave with respect to outputs. The solution of the principal’s program can finally be summarized as follows:

Proposition 2.3: With general preferences satisfying the Spence-Mirrlees property, C > 0,  the  optimal  menu  of  contracts  entails:

• No output distortion with respect to the first-best outcome for the efficient type, qSB = q with A  downward  output  distortion  for  the  inefficient  type, SB < g¯*   with and • Only the efficient type gets a positive information rent given by U SB = .
• The second-best  transfers  are  respectively  given  by  tSB  = C(q, θ) + The first-order conditions (2.47) and (2.49) characterize the optimal solution if the neglected incentive constraint (2.41) is satisfied. For this to be true, we need to have which amounts to We have > 0 from the Spence-Mirrlees property, hence (2.51) is equivalent to SB   qSB.  But  from  our  assumptions  we  easily  derive  that  qSB  = q >  q¯* > SB. So  the  Spence-Mirrlees  property  guarantees  that  only  the  efficient  type’s incentive constraint has to be taken into account.

The critical role played by the Spence-Mirrlees property in simplifying the problem will appear more clearly in models with more than two types.19

Remark: The Spence-Mirrlees property is more generally a constant sign condition  on C. If C  < 0, then Proposition 2.3 is unchanged except that now the inefficient type’s output is distorted upwards SB  > q¯*  >  q.  Indeed,  in  such  a  model,  the  first-best  production  level  of the inefficient type is higher than that level for the efficient type. Moreover,  the  information  rent  of  the  efficient  type  is  still  ,  but  now  an  increase  of  q¯ is  required  to  decrease this  rent  because C  < 0.

### 2. Nonresponsiveness

Coming back to our linear specification of the agent’s cost function, let us assume that the principal’s return from contracting depends also directly on l and is written as  Sjqp lk.  This  is  an  instance  of  a  common  value  model  where  the  agent’s  type directly affects the principal’s utility function. On top of the usual assumptions of a  positive  and  decreasing  marginal  value  of  trade,  we  also  assume  that S  >  1. This latter assumption simply means that the marginal gross value of trade for the principal increases quickly with the agent’s type. For instance, the efficient agent produces a lower quality good than the inefficient one, and the principal prefers a high quality good.

The  first-best  productions  are  now  defined  by Sq(q*, θ) = θ and Sq(q¯*, θ¯) = θ¯.  With  our  assumption  on S,  the  first-best  production  schedule  is  such  that q <  q¯*,  i.e.,  it  does  not  satisfy  the  monotonicity  condition  in  (2.15)  implied  by incentive compatibility.

In this case, there exists a strong conflict between the principal’s desire to have  the  θ¯-type  produce  more  than  the  θ-agent  for  pure  efficiency  reasons  and the monotonicity condition imposed by asymmetric information. This is what Guesnerie and Laffont (1984) call a phenomenon of nonresponsiveness in their general analysis of the principal-agent’s model with a continuum of types. This phenomenon makes screening of types quite difficult. Indeed, the second-best opti- mum  induces  screening  only  when  qSB = q and SB   defined  by satisfy  the  monotonicity  condition  qSB  SB.  However,  when  r is  small  enough, SB  defined in (2.52) is close to the first-best outcome q¯*. Thus, we have SB  > qSB, and the monotonicity condition (2.15) is violated. Nonresponsiveness forces the principal to use a pooling allocation. Figure 2.7 illustrates this nonresponsiveness.

As  in  figure  2.4,  the  pair  of  first-best  contracts  (A, B) is  not  incentive  com- patible. But, contrary to the case of section 2.6.2, the contract C, which makes the θ-type indifferent to whether he tells the truth or takes contract B, is not incentive compatible  for  the  θ¯-type  who  also  strictly  prefers  C  to  B.

One possibility to restore incentive compatibility would be to distort q¯ down to q, which would decrease the θ-type’s information rent to yield contract D while still preserving incentive compatibility for both types. By this action we would obtain a pooling allocation at D. However, the principal can do better by choosing another pooling allocation, which is obtained by moving along the zero isoutility line  of  a  θ¯-type.  Indeed,  the  best  pooling  allocation  solves  problem  (P) below: The harder participation constraint is obviously that of the least efficient type, namely (2.54). Hence, the optimal solution is characterized by and with  qp < q¯*  because S  > 0.

This pooling contract is represented by point E in figure 2.7 (which can be to the left or to the right of D) where the indifference curve (the heavy line through E) of the principal corresponds to the “average” utility function  In summary, when nonresponsiveness occurs, the sharp conflict between the principal’s preferences and the incentive constraints (which reflect the agent’s pref- erences) makes it impossible to use any information transmitted by the agent about his type. Figure 2.7: Nonresponsiveness

### 3. More than Two Goods

Let us now assume that the agent is producing a whole vector of goods q = (q1,…, qn) for the principal. The agent’s cost function becomes C(q, θ) with C(.)  being strictly convex in q. The value for the principal of consuming this whole bundle  is  now  S(q) with  S(·) being  strictly  concave  in  q.

In this multi-output incentive problem, the principal is interested in a whole set of activities carried out simultaneously by the agent. It is straightforward to check  that  the  efficient  agent’s  information  rent  is  now  written  as with .

This leads to second-best optimal outputs. The efficient type produces the first-best vector of outputs qSB = q with The  inefficient  type’s  vector  of  outputs SB  is  instead  characterized  by  the first-order conditions which generalizes the distortion of models with a single good.

Without further specifying the value and cost functions, it is hard to compare the second-best outputs a priori with the first-best outputs defined by the following n first-order conditions: Indeed, it may well be the case that the n first-order conditions (2.58) define a  vector  of  outputs  with  some  components above for  a  subset  of  indices  i.

Turning now to incentive compatibility, summing the incentive constraints for  any  incentive  feasible  contract  yields Obviously, this condition is satisfied if the Spence-Mirrlees property holds  for  each  output  i  and  if  the  monotonicity  conditions for  all  i  are satisfied. Inequality (2.60) is indeed satisfied for the second-best solution (2.58), because  then for  all  i.  However,  the  reverse  is  not  true.  It might  well  be  the  case  that for  some  output  i  and  the  condition (2.60), which is a condition on the whole vector of outputs, nevertheless still holds for  the  second-best  vector  of  outputs  q and SB.

So, in general, the implementability condition (2.60) in a multi-output envi- ronment is more complex than the simple monotonicity condition found in a single-good setting.

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