# Credible Commitments: The Hostage Model Extended

Assume that the two firms are engaged in tied bilateral trade and that both have made specific asset investments of k in support of each other. Assume further that each firm incurs redeployable costs of production of v2 and that is the price at which product is traded. In deciding whether to take delivery or cancel an order, a firm needs to consider not merely the net gain from procurement but also the net gain from supply. Let the net gain from buying and selling product be given by bB and bS, respectively. The combined gain from observing reciprocity is then given by bR = bB + bS. Net benefits upon taking delivery in the purchase market will be given by

while net benefits from the simultaneous sale of product (given that specific assets in amount k have already been sunk) are given by

The net benefits of noncancellation—that is, of continuing reciprocal trade (given that one s trading counterpart does not renege)—are then

which will be positive so long as demand realization in the market for which product is purchased exceeds the marginal cost of own production.

Although the specific asset term, k, appears nowhere in these expressions, that does not mean it is irrelevant. Thus assume, as before, that demand in both markets is uniformly distributed over the interval 0 to 1. Then the expected net benefits of reciprocity will be positive only if the probability of trade under the reciprocal trading criterion (namely, 1 – v2) times the expected gain from remunerative exchange (1 – v2)/2 exceeds the value of nonsalvageable assets, k. Thus the inequality (1 – v2) 2/2 — k > 0 must be satisfied.

More significant is the fact that only if specific assets are committed in support of the exchange will the benefits from the sale of product be given by bs= – v2. If, for example, one of the parties to the exchange were to employ the general purpose technology T1, instead, the net benefits from supplying product for which is received would be b’s = pˆ – v1. The criterion for assessing whether to cancel or not would then be b’R = p — v1, which would call for cancellation in demand states where p < v1. One party to the bilateral exchange would thus find cancellation attractive under circumstances where the other, because it has made specific asset investments, would want product to be traded. The symmetrical exposure of specific assets avoids that result.

Alternatively, the issues can be presented as follows. As above, let bB = p – be the buyer’s net benefits from unilateral trade if he decides to take delivery and bS = v2 be the net gain to the seller from making delivery. Assume that = 2 and v2 = 1 and that there are three possible state realizations: p = 4, p = 2 — ∈, and p = 0. It is the buyer’s decision to accept delivery or refuse it. The payoffs associated with each decision, demand realization pair are:

The efficient choice is for the seller to deliver in the two more favorable demand realization states and to shut down, thereby saving variable costs, if p = 0, however, the buyer consults only his own profits in deciding on whether to accept or refuse delivery, the buyer will take delivery only in the most favorable demand state and will refuse it if p = 2-∈ or if p = 0, since bB < 0 in both instances.

Enlarging the transaction from one of unilateral to bilateral trade changes the payoffs in such a way as to eliminate this inefficiency. Thus now the payoff to both parties is given by bR = p – v2. The corresponding payoffs faced by both parties are now identical and are given by:

Product will thus be traded in the demand states, but production will be shut down if p = 0, which is the efficient result. Reciprocity can thus be regarded as a reaction to the inherent strains (and resulting inefficiencies) that would occur under a unilateral trading regime. Public policy insistence that anything other than unilateral trading is “unnatural” and presumptively antisocial is simply mistaken.

Source: Williamson Oliver E. (1998), The Economic Institutions of Capitalism, Free Press; Illustrated edition.