### 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)* − θ*q *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__^{∗}=

*θ*

__q__^{∗}). Whatever his type, the agent accepts the offer and makes zero profit. The complete information optimal contracts are thus (

__t__^{∗},

__q__^{∗}) if

*θ*=

__and if__

*θ**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*¯

*>*

__, the isoutility curves of the inefficient agent__

*θ**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.