The Complete Information Optimal Contract

1. First-Best Production Levels

First suppose that there is no asymmetry of information between the principal and the agent. The efficient production levels are obtained by equating the principal’s marginal value and the agent’s marginal cost. Hence, first-best outputs are given by the following first-order conditions

and

The  complete  information  efficient  production  levels  q and  q¯*  should  be  both carried  out  if  their  social  values,  respectively   and  W¯*  = , are non-negative. The social value of production when the agent is  efficient,  W ,  is  greater  than  when  he  is  inefficient,  namely  W¯*.  Indeed,  we have  by  definition  of  q,  which  maximizes  S(q)θq and   because . For trade to be always carried out, it is thus enough that production be socially valuable for the least efficient type, i.e., the following condition must be satisfied

a hypothesis that we will maintain throughout this chapter. As the fixed cost F plays no role other than justifying the existence of a single agent, it is set to zero from now on in order to simplify notations.

Note that, since the principal’s marginal value of output is decreasing, the optimal  production  levels  defined  by  (2.4)  and  (2.5)  are  such  that  q >  q¯*,  i.e., the optimal production of an efficient agent is greater than that of an inefficient agent.

2. Implementation of the First-Best

For a successful delegation of the task, the principal must offer the agent a utility level that is at least as high as the utility level that the agent obtains outside the relationship (for each value of the cost parameter). We refer to these constraints as the agent’s participation constraints. If we normalize to zero the agent’s out- side opportunity utility level (sometimes called his status quo utility level), these participation constraints are written as

To implement the first-best production levels, the principal can make the following take-it-or-leave-it offers to the agent: If θ = θ¯ (resp. θ), the principal offers the  transfer  t¯*  (resp.  t)  for  the  production  level  q¯*  (resp.  q)  with  (resp. t = θq).  Whatever  his  type,  the  agent  accepts  the  offer  and  makes  zero  profit. The complete information optimal contracts are thus (t, q) if θ = θ and if  θ = θ¯.

Importantly, under complete information delegation is costless for the princi- pal, who achieves the same utility level that he would get if he was carrying out the task himself (with the same cost function as the agent).

3. A Graphical Representation of the Complete Information Optimal Contract

In  figure  2.2,  we  draw  the  indifference  curves  of  a  θ-agent  (heavy  curves)  and  of a θ¯-agent (light curves) in the (q, t) space. The isoutility curves of both types cor-respond to increasing levels of utility when one moves in the northwest direction. These indifference curves are straight lines with a slope l corresponding to the agent’s  type.  Since  θ¯ >  θ,  the  isoutility  curves  of  the  inefficient  agent  θ¯ have  a greater slope than those of the efficient agent. Thus, the isoutility curves for dif- ferent types cross only once. Throughout this chapter and the next one we will come back several times to this important property called the single-crossing or Spence-Mirrlees property.

Figure 2.2: Indifference Curves of Both Types

The complete information optimal contract is finally represented in figure 2.3 by  the  pair  of  points  (A, B).  For  each  of  those  two  points,  the  strictly  concave indifference curve of the principal is tangent to the zero rent isoutility curve of the corresponding type. Note that the isoutility curves of the principal correspond to increasing levels of utility when one moves in the southeast direction. Thus the principal reaches a higher profit when dealing with the efficient type. We denote by V¯*  (resp. V ) the principal’s level of utility when he faces the θ¯- (resp. θ-) type. Because the principal has all the bargaining power in designing the contract, we have  V¯*  = W¯ *  (resp.  V = W )  under  complete  information.

Remark:  In Figure 2.3, the payment t is greater than t¯*, but we note that  t can  be  greater  or  smaller  than  t¯*  depending  on  the  curvature of  the  function  S(·),  as  it  can  be  easily  seen  graphically.

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