A holdup situation leading to *underinvestment *occurs when relationship-specific investments are
introduced in a contracting situation. This means that the supplier's default payoff increases less
than the gain from trading with the buyer: the supplier's marginal return to his investment is no
longer the same independent of whether he sells to the buyer or on the spot market. Thus the first-

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best investment level is not automatically achieved. The formal presentation of this problem is central to this section.

Bolton and Dewatripont (2004 pp. 560–562) present the holdup problem by a simple model as
follows. There are two contracting parties, a prospective buyer and a prospective supplier, who can
enter a relationship in which they can end up trading a quantity , - at a price . At this point
it is important to note, that in order to apply the model in the context of nuclear reactor
procurement, *quantity *must be interpreted as *quality. *From now on, the notation is referred to as
*level of trade. *Their utility from the trade depends on the buyer’s valuation and the supplier’s
production cost . These utilities are uncertain at the time of contracting and can be influenced by
specific investments made by each party at an earlier date. The following assumptions are made:

* + ( )

where the buyer’s cost of making an investment is ( ), and

* + ( )

where the supplier’s cost of making an investment is ( ). The investment functions are increasing and convex, and the investments are sunk whatever the ex-post level of trade. The ex-post payoffs are thus

( )

for the buyer, and

( )

for the supplier. The timing is as follows. First, the parties contract; second, they simultaneously choose their investment levels and ; third, the both learn the state of nature, ( ) affecting the buyer’s valuation and the supplier’s cost, and fourth, the contract is executed.

For simplicity, it is assumed that . Under this assumption, the ex-post efficient level of trade is if ( ) and 0 otherwise. Since the parties are assumed to be risk neutral, the ex-ante efficiency is equivalent to investment efficiency, i.e. and must result from

* ( ) ( ) ( )+

The first-order conditions yield the optimal investment levels and :

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( ) ( )

and

( ) ( )

In the standard contracting problem the state of nature ( ) and the investment levels and are not contractible, although the state of nature is observable to both parties ex post. If ex-post spot contracting is possible, and if the gains are evenly divided between buyer and supplier, there will be underinvestment in equilibrium after is realized and investments and are sunk. Figure 3:

Underinvestment with holdup problem below illustrates this. The black curves represent the optimal investment functions and , and the grey curves represent the best response functions under spot contracting, given by

( ) ( ^{ })

and

( ) ( ^{ })

The difficulty faced by the contracting parties in a dynamic setting with uncertainty is how to formulate an optimal long-term contract that is independent of the state of nature, which mitigates this underinvestment problem, moving the investment levels from the second-best equilibrium toward the first-best equilibrium.

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**Figure 3: Underinvestment with holdup problem **

The following subsection presents one approach to mitigating the underinvestment problem. Bolton and Dewatripont expand on the model above, and present a solution where underinvestment is avoided by awarding one of the two contracting parties full bargaining power.

**5.1.1 Mitigating underinvestment with bargaining power **

Bolton and Dewatripont (2004, pp. 563–564) present a positivistic solution to mitigating underinvestment. In the model, it is assumed that the parties can specify default options whenever trade is possible. The level of trade, ̃, is defined such that

̃( ) ( )

where ( ) is the supplier’s payoff. The following contractual mechanism can be considered: Once the state of the world has been realized, the parties play the following game: in stage 1, the buyer can make an offer ( ) to the supplier; in stage 2, the supplier accepts the offer and trade takes

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place at these terms, or rejects it, in which case ̃ is traded at a prespecified price ̃ designed to share ex-ante surplus according to initial bargaining shares. The buyer has full bargaining power in this first version of the two-stage game; he will therefore offer to trade the ex-post efficient quantity while leaving the supplier indifferent between trade and his default option. This is sufficient to guarantee ex-post efficiency. However, as described in section 5.1.1 this is not a sufficient condition for the investments to be efficient. The supplier anticipates obtaining his default option payoff whatever the ex-post level of trade, so he solves

* ̃ ̃ ( ) ̃ ( )+

Given the interpretation of ̃, investment level is the supplier’s optimal choice, whatever the buyer’s investment. Finally, as the buyer has full bargaining power, he is the residual claimant on her investment and solves

* ( ) , ̃ ̃ ( ) ̃ ( )-

The buyer thus maximizes total surplus minus the payoff of the supplier, which does not depend on the investment of the buyer, and minus his own investment. Consequently, the buyer chooses and the supplier chooses .

The mechanism described above induces efficient bilateral investments. For the buyer, the efficient investment level is achieved by making him the residual claimant. The supplier, in turn, is incentivized into efficient investment behavior despite having no bargaining power at all. The supplier’s incentive to invest stems from the default option, whose attractiveness rises when his production cost decreases. The existence of the default option thus makes the supplier sensitive to his investment. In the context of nuclear power reactor procurement, however, the existence of a default option is an unrealistic assumption – a half-built nuclear power plant is hardly of any value.

Therefore it is straightforward to assume that a default option alone is not a sufficient incentive for the supplier to make first-best investment decisions. The following section focuses on a variety of other contractual solutions.