Credible Commitments: The Hostage Model

The simple hostage model serves to illuminate both unilateral and bilateral exchange, permits the concept of specific capital to be extended beyond earlier uses, and clarifies how costs should be described in assessing exchange. While it is primitive and suggestive, rather than refined and definitive, it serves as a paradigmatic wedge by which the importance of private ordering is exposed and is easily made the vehicle for further analysis.

1. Technologies and Costs 

The assessment of alternative contracts will be facilitated by assuming that the product in question can be produced by either of two technologies. One is a general purpose technology; the second is a special purpose technology. The special purpose technology requires greater investment in transaction-specific durable assets and, as described below, is more efficient for servicing steady state demands.

The distinction introduced in Chapter 2 (see especially Figure 2-2) be-tween redeployable and nonsalvageable investments will be employed here. Rather, therefore, than use the convention of fixed and variable costs, the two technologies in question will be described in value realization terms. The value that can be realized by redeploying variable and fixed costs will be given by v. The nonsalvageable value of advance commitments will be denoted by k. The two technologies in question can be described as:

T1: The general purpose technology, all advance commitments of which are salvageable, the redeployable unit operating costs of which are v1.

T2: The special purpose technology, the nonsalvageable value of advance commitments of which are k and the redeployable unit operating costs of which are v2.

2. Contracting 

There are two periods. Orders are placed in the first, and production, if any, occurs in the second. Buyers can either take delivery or refuse it. Demand is stochastic. The gross value to buyers is assumed to be uniformly distributed over the interval 0 to 1, and the quantity demanded at every price will be assumed to be constant, which it will be convenient to set equal to unity. Sunk costs, if any, are incurred in the first period. Inasmuch as sunk costs are incurred for certain while the decision to incur redeployable costs is contingent on the buyer’s decision to confirm or cancel an order, a choice between technologies is interesting only if k + v2 < v1. The demand and cost relations are set out in Figure 7-1.


The criterion by which decisions to take or refuse delivery will be evalu- ated is joint profit maximization. Feasibility and/or bureaucratic disabilities aside, vertical integration assuredly accomplishes the joint profit maximization result. Thus the reference condition for evaluating contracts will be an integrated firm with two divisions, a producing division and a marketing division. The producing division has access to the same two technologies described above, one of which involves specific assets, the other of which does not. Whichever technology is employed, product is transferred between divisions at marginal cost.

FIGURE 7-1. Demand Distribution and Costs of Supply

That k + v2 < v1, does not establish that the special purpose technology (T2) is more efficient. Whether it is or not depends on a net benefit calculation. The expected net benefits of using the general purpose technology (T1) are given by the product of the probability that the integrated firm will decide to produce and the average net benefits that are realized when product is supplied. The integrated firm will decide to produce only if the realized demand price exceeds marginal costs, whence the probability of production under T1 is 1 – v1. The mean net benefits during production periods are (1 — v1 )/2, whence the expected net benefits for technology T1 are:

The expected net benefits for the specific asset technology (T2) are found similarly. Again, the integrated firm will produce whenever realized demand price exceeds marginal costs. Expected net receipts, however, must be reduced by the amount of the earlier investment in specific assets, k, in computing expected net benefits. Thus we have:

where the first term is the expected excess of revenue over out- of-pocket costs.

The specific asset technology will be selected only if b2 > b1,, which requires that


Assume that the inequality in (3) holds and consider the case of autono- mous contracting between a buyer, who services final demand, and a producer, who manufactures the product. Assume that demand and production technologies are as described above. Efficient contracting relations arp-those that replicate the vertical integration result, namely, (1) select the specific asset technology and (2) produce and sell product whenever realized demand price exceeds v2. Assume that both parties are risk-neutral and that the production side of the industry is competitively organized. Whatever contracting relation is described, producers will be willing to supply if a breakeven condition (expressed in expected value terms) can be projected.

Recall that orders are placed in the first period. Specific assets, if any, are committed in the first period in anticipation of second period supply. Whether second period production actually occurs, however, is contingent on demand realizations. Buyers have the option of confirming or canceling orders in the second period. Consider three contracting alternatives:

  1. The buyer purchases specific assets and assigns them to whichever seller submits the lowest bid, p¯.
  2. The producer makes the specific asset investment himself and receives a payment of in the second period if the buyer confirms the order but nothing otherwise.
  3. The producer makes the specific asset investment himself and receives p from the buyer if the buyer confirms the order, is paid a h, 0 ≤ a ≤ 1, if the order is canceled while the buyer pays upon taking delivery and experiences a reduction in wealth of h if second period delivery is canceled.

The third scenario can be thought of as one where the buyer posts a hostage, that he values in amount h, which hostage is delivered to the producer, who values it in amount ah, if the order is canceled.

The producer will break even under contracting relation I if he is compen- sated in amount v2, which is his out-of-pocket cost, for each unit demanded. The low bidder will thus offer to supply product for p¯ = v2. Since the buyer’s net benefits are maximized if he invests in the specific assets, and since product is transferred on marginal cost terms, the contract replicates the vertical integration relation. Contracts of type I are feasible, however, only if the specialized assets are mobile and the specificity is attributable to physical features (e.g. specialized dies). Market procurement can then service the needs of the parties without posing holdup problems by concentrating the ownership of the specific assets on the buyer (who then assigns them to the low bidder). Inasmuch as the buyer can reclaim the dies and, without cost, solicit new bids should contractual difficulties develop, type I contracts yield an efficient result.8

Attention hereafter will be focused on contracts 11 and III, the assump-tion being that asset specificity is of the human, site specific, or dedicated asset kinds. The autonomous buyer will confirm an order under contract II whenever realized demand price exceeds p but not otherwise. The producer will thus break even if , whence

Product will thus be exchanged at a price that exceeds marginal cost under this contracting scenario.9 Plainly if p¯≥ v1, the buyer is better off to scuttle contract II and purchase instead from producers who utilize the (inferior) variable cost technology 7, (and will break even by supplying product on demand for a price of v1).

The buyer will confirm an order under contract III whenever the realized demand price exceeds p‾ h. Let pˆ — h be denoted by m. The seller will then break even when , whence

The case where h = k and a = 1 is one where the buyer gives up wealth in amount of the investment in specific assets in cancellation states, and this is delivered to the producer, who values it in amount k. Under these circumstances, (5) becomes

Since the buyer places an order whenever demand exceeds m = p‾ – h, this yields the result that m = v2, whence orders will be placed whenever demand exceeds v2—which is the efficient (marginal cost) supply criterion.

The buyer’s net benefits under contracting scheme III are

where (1 — m) is the probability of placing an order,-m + (1 — m)/2 is the expected demand price for all orders that are placed, is the payment in demand confirmation states to the producer, and h is the wealth sacrifice in cancellation states (which occur with probability m). Under the assumptions that h = k and a = 1, this reduces to

which is identical to the net benefit calculation for technology T2 under the vertical integration reference condition (see equation [2]).

Accordingly, contracting scheme III accompanied by the stipulations that h = k and a = 1 replicates the efficient investment and supply conditions of vertical integration. Problems arise, however, if h < k or a < 1. The disadvantage, moreover, accrues entirely to the buyer-—since the seller, by assumption, breaks even whatever contracting relation obtains. Thus although after the contract has been made the buyer would prefer to offer a lesser valued hostage and cares not whether the hostage is valued by the producer, at the time of the contract he will wish to assure the producer that a hostage of k for which the producer realizes full value (a = 1) will be transferred in nonexchange states. Failure to make that commitment will result in an increase in the contract price. Thus, whereas producers who are concerned only with ex ante screening can tolerate values of a less than one—see the discussion of ugly princesses in section 4 below—this is not the case at all when ex post opportunism is the concern. If the producer is not indifferent, as between two princesses, each of whom is valued identically by the buyer, the producer’s preferences now need to be taken into account.10

To summarize, therefore, we observe that contract I mimics vertical integration, but only under special asset specificity conditions; contract II is inferior; and contract III yields the vertical integration result if h = k and a = 1. Furthermore, note that an important feature of contract III is that the buyer takes delivery in all demand states for which realized demand exceeds m = pˆ – h. Since the supplier is always paidp upon execution, the buyer sometimes takes delivery when his realized receipts (upon resale of the product) are less than p. This does not, however, signal inefficiency, since orders are never confirmed when realized demand price falls below marginal cost (v2). Indeed, it is precisely because of the hostage feature that efficiency is realized and contract III is superior to contract II.

The above has a bearing on the contracting schema discussed in Chapter 1, to which recurrent reference is made throughout this book. For conve-nience, the basic contracting choices are reproduced (with minor changes) in Figure 7-2. That the (breakeven) prices at nodes A, B, and C should differ was evident at the outset: The technologies differ and, as between nodes B and C, the hazards differ. But there is a further difference between nodes B and C that can be ascertained only by working through the net benefits, as set out in the text: Namely, a node C contract leads to superior asset utilization.

The fact that suppliers are indifferent as between nodes B and C, because an expected breakeven result can be projected under each condition, does not therefore mean that the two outcomes can be regarded with a shrug. On the contrary, both buyers and society have an interest in seeing a node C outcome realized. That applies not merely to intermediate product, which is the focus of this chapter and the next, but also to the organization of labor (Chapter 10) and the supply of capital (Chapter 12).

FIGURE 7-2. A Contracting Schema

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

Leave a Reply

Your email address will not be published. Required fields are marked *