Let us now consider a setting where the agent has to make some initial investment before dealing with the principal. The motivation for this contractual setting is that the agent and the principal can only meet each other after the agent’s investment has been made. Such situations are likely to occur in market environments where the trading relationships are only short term.

### 1. Nonverifiability and Contracting** **

Let us return to the mixed model of section 7.3 with the only difference being that effort is chosen by the agent before contracting, as described by the timing of figure 9.5.

The risk-neutral agent may exert his binary effort *e *at date ** t **= 1 to improve the probability that the good state of nature

__θ__realizes. We denote those probabilities by v(1) = v

_{1}and v(0) = v

_{0}. Δv = v

_{1}— v

_{0}> 0 is the increase in the probability of having an efficient type when the agent makes a high effort.

In the *hold-up *problem, the principal cannot commit to reward the agent *ex ante *for his nonobservable effort. *Ex post *(i.e., at date ** t **= 1), when the principal offers the contract, the agent’s effort has already been sunk. The principal has now lost his role as Stackelberg leader in the design of incentives for effort, and so we must look for a Nash equilibrium between the principal and the agent. The principal offers a contract anticipating a particular choice of effort made by the agent. The agent chooses an effort level anticipating the contract he will receive from the principal.

Figure 9.5: Timing of the Hold-Up Problem

Suppose first that the principal can offer to play a game as in chapter 6. For each value of the effort level *e*ˆ chosen by the agent at date ** t** = 0, the principal can implement the first-best (conditional on

*e*ˆ) in Nash (or subgame perfect, if needed) equilibrium. From section 7.3, we know that the agent always obtains a zero utility level. Anticipating this, the agent exert no effort at date 0.

We will now describe two modelling options that enable the principal to mitigate the hold-up problem. The first option, which has been the focus of most of the literature, is to assume that contracts cannot be signed at date ** t **= 1 and that bargaining between the traders occurs after date

**= 3, when they have both learned the state of nature**

*t**θ*. The second option is to assume that only the agent is informed about the state of nature before date

**= 1, so that asymmetric information exists at date**

*t***= 1, and to maintain the assumption that incentive contracts can be offered at date**

*t***= 1. In both cases, the principal is obliged to give up a rent to the agent, which indirectly provides the agent with some incentives for effort at date**

*t***= 0 and mitigates the hold-up problem.**

*t*### 2. Nonverifiability and Bargaining

Taking the Nash bargaining solution with equal weights to compute their final payoffs, we find that the agent’s *ex ante *expected utility writes as , where, as usual, denote the first-best gains from trade in each state of nature. Hence, the agent invests if and only if

The first-best condition that ensures that exerting an effort is optimal is . The condition (9.39) may no longer hold when . In this case there is underinvestment, and the hold-up problem prevents efficiency.

Of course, this result can be generalized to other allocations of the bargaining power as long as the agent does not have all the bargaining power *ex post*.

**Proposition 9.4: ***Assume that the state of nature is nonverifiable and that the agent has only a limited bargaining power in the negotiation over the ex post gains from trade; then an under-investment may occur.*

The intuition behind this proposition is straightforward. Since the agent only receives half of the *ex post *gains from trade, he has only half of the social incentives to exert effort at the ex ante stage. Underprovision of effort follows.

**Remark: **Simple solutions to this hold-up problem can be found by the contractual partners. First, the *ex post *bargaining power could be fully allocated to the agent, making him residual claimant for the social return to investment. Of course, this solution may not be opti- mal if the principal also has to invest in the relationship19 or if the agent is risk-averse, because he would then bear too much risk. Sec- ond, let us assume that the principal and the agent can agree *ex ante *on an *ex post *allocation (*t*_{0}, *q*_{0}), which stipulates the status quo pay- offs of both the agent and the principal in the *ex post *bargaining taking place when *θ* has realized. This contract is relatively simple to write since it stipulates only one transfer and an output. More- over, we assume that the principal keeps all the bargaining power in the *ex post *bargaining stage. Therefore, he must solve the following problem:

where (9.40) is the agent’s participation constraint, which is obviously binding at the optimum because the principal wants to reduce the agent’s transfer as much as possible.

Since there is complete information *ex post*, the efficient pro- duction levels __q__^{∗} and *q*¯* are chosen by the principal depending upon which state of nature is realized. The agent’s expected payoff is written as when he exerts effort *e*. The agent exerts a positive effort when *q*_{0} is fixed, so that . The status quo output *q*_{0} therefore defines the agent’s marginal incentives to invest. Then, *t*_{0} can be adjusted so that the agent’s expected utility is zero, i.e.,. The principal’s expected payoff becomes , exactly as in a world of complete contracts.

This very nice solution to the hold-up problem is due to Chung (1991). Various other solutions have been found in the incomplete contracts literature. See Tirole (1999) for an exhaustive survey of this litera- ture. Another way to solve the hold-up problem, at least partially, that is well developed in the literature is to allocate property rights. The analysis of this literature is beyond the scope of the present volume, but a few general ideas are worth stressing here. The general framework to analyze the role of prop- erty rights on the assets is due to Grossman and Hart (1986) and Hart and Moore (1990) (see also Hart 1995). Property rights are usually incomplete contracts. Indeed, property rights define what can be obtained by the agents in the status quo of the *ex post *Nash bargaining over the gains from trade. In general, the literature envisions double moral-hazard problems, where both the buyer and the seller must perform a specific investment in the relation- ship and compares the performances of various ownership structures. A major result of this literature is that complementary assets should be owned by the same person (see Hart 1995). Maskin and Tirole (1999) provide conditions such that allocating property rights implements the optimal contract.

### 3. Adverse Selection** **

Let us assume now that the agent is informed about b before date ** t **= 1, when the principal offers him an incentive contract.

Let us denote by v_{e} the conjecture of the principal over the probability that type *θ* realizes. Using the results of chapter 2, the best-response of the principal is thus characterized by an output distortion (for the inefficient type only) q¯(v_{e}), which is defined by

and a positive information rent (for the efficient type only) given by . Note that are both decreasing with v_{e}* *.

Anticipating such a contract, and more specifically the rent * *he will get when he turns out to be efficient, the agent invests in increasing this probability according to the following best response .

Putting together the principal and the agent’s best responses yields the follow- ing characterization of the Nash equilibrium contract and effort in this framework without any commitment.

**Proposition 9.5: ***Assume that the principal cannot commit to a contract before the agent exerts effort. Then, the equilibrium allocation is charac- terized as follows:*

*If**, the agent exerts a positive effort and**is chosen by the principal.*

*If**, the agent does not exert any effort and**is chosen by the principal.**If**, the agent randomizes between exerting**effort or not with respective probabilities ε**and*1 − ε*. We have v*_{e}

When underinvestment occurs with respect to the case with full commitment that we have already discussed in section 7.2.3.21 Again, the logic underlying this result is simple. The agent only receives a share (his information rent) of the overall surplus of production which occurs in period 4; hence he may not have enough incentives to exert effort.

Riordan (1990) used the hold-up model under adverse selection as an ingredient of a theory of vertical integration. Laffont and Tirole (1993, chap. 1) analyzed how cost-reimbursement rules must be adapted to protect the specific investment of regulated firms. Schmidt (1996) used the underinvestment model presented in this section to build a theory of privati- zation. In this theory, the cost of public ownership is the inability of the state to reward a specific investment made at the *ex ante *stage by the public utility.

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