With the same model as in chapter 2, we now assume that the parameter θ is unknown at the contracting date (date t = 0) but becomes common knowledge between the two parties, the principal and the agent, later on (at date t = 1). First we examine the case where no initial contract has been signed at date t = 0.
At date t = 1, the principal is informed about θ and can make a take-it-or- leave-it offer to the agent at date t = 2 under complete information. See figure 6.1 for the timing of the game.
These take-it-or-leave-it offers implement the first-best volumes of trade and obviously leave no rent to the agent since the principal has all the bargaining power at date t = 2. For instance, when the agent is efficient, his output q∗ satisfies S’(q∗) = θ, and the transfer t∗ he receives from the principal is t∗ = θq∗. Similarly, the inefficient agent produces q¯* such that , and the transfer ¯t∗ he receives just covers his cost: .
Figure 6.1: Timing with No Ex Ante Contract
So far we have assumed that the principal was endowed with all the bargain- ing power both at the ex ante and at the interim stages, i.e., before the agent has learned the piece of information θ or just after. In the nonverifiability paradigm, it is frequent to analyze a more even distribution of the bargaining power at date t = 2. For example, the principal may have performed an (unmodeled) invest- ment specific to the relationship, so that he finds himself in a position of bilateral monopoly vis-à-vis the agent, thereby justifying a nonzero bargaining power ex post for the agent.
Obviously, changing the principal and the agent’s bargaining powers at date t = 2 does not affect allocative efficiency. To see that, let us assume that the principal and the agent bargain ex post, i.e., at date t = 2, over the entire gains from trade. See figure 6.2.
To model this bargaining, we use the cooperative Nash bargaining solution with the principal and the agent having now equal weights in the negotiation. In state θ, they agree on a transfer t and production q, which are solutions to the following problem:
We easily find that the Nash bargaining solution consists of the first-best out- put q∗(θ) and a transfer tNB(θ), which satisfy
Figure 6.2: Timing with No Ex Ante Contract and with Ex Post Bargaining
As a result, both the principal and the agent receive an equal share of the first-best gains from trade. Denoting the principal and the agent’s shares of the surplus respectively by V NB(θ) and UNB(θ), we have thus
where W ∗(θ) is the first-best surplus in state θ.
Remark: Similar results would also hold with any kind of cooperative or noncooperative bargaining solution, for instance, the Rubinstein (1982) alternative offers bargaining game. The particular way of split- ting the ex post surplus has no allocative impact. The volume of trade remains always at its first-best value.
More generally, the higher the agent’s bargaining power in the negotiation over the surplus at date t = 2, the lower (resp. the higher) the principal’s (resp. the agent’s) gains from trade. As a corollary, if the principal expects to be stuck in a bilateral relationship at date t = 2 with a lower bargaining power than at the ex ante stage, he will prefer to contract with the agent at the ex ante stage (date t = 0). Next, we study what can then be achieved with such ex ante contracting.
Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.