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Specific investments, flexible adaptation, and requirement contracts.

I. INTRODUCTION

In bilateral procurement relationships, ex post opportunism may cause underinvestment in relationship-specific investments, as is shown by Klein et al. (1978) and Williamson (1985). In such situations, long-term contracts can help to mitigate this so-called hold-up problem. In a world of positive transaction costs, long-term contracts are necessarily incomplete. (1) An inflexible incomplete contract may lead to ex post maladaptation in the presence of uncertainty, as in Williamson (1985). Therefore, a transaction cost-economizing contract needs to provide both incentives for efficient relationship-specific investments and flexibility for efficient adaptation. (2)

In a simple hold-up model with contractible cost-related variables, I show that efficient adaptation and investments can be achieved through a requirement contract, which gives the buyer the right to decide on how much to buy ex post and contains a price provision with a two-part tariff structure, including a fixed component and a unit charge that may vary with quantities and cost-related variables. This result Provides a formal explanation of contractual arrangements observed in vertical procurement relationships.

Formally, my result can be viewed as an extension of Williamson's (1983) hostage model in which only the seller makes investments, the procurement quantity is fixed, and there is no cost uncertainty. My result is also related to previous formal studies on the take-or-pay contracts as in Hubbard and Weiner (1986), Crocker and Masten (1988), and Masten (1988). In a simpler model without cost uncertainty and specific investments, they show that a unilateral options contract with a take-or-pay feature gives the buyer the right incentives to make efficient quantity decisions. Particularly, Crocker and Masten (1988) and Masten (1988) emphasize the take-or-pay contract's ability to induce efficient adaptation. Another related paper is by Bos (1996), who shows that when the production cost is contractible, a target-cost pricing contract that requires the seller to share a portion of the cost overrun can achieve the first best outcome.

The rest of the article is organized as follows. Section II provides the background of the issue studied. Section III lays out the model and characterizes the first best benchmark. Section IV derives the optimal contract. Section V concludes.

II. BACKGROUND

A growing recent literature has attempted to find contractual solutions to the hold-up problem. Most of the papers in the literature use Hart and Moore's (1988) classical model (or its variation). In that model, there are two risk-neutral firms: a buyer and a seller of an intermediate good. Each firm can, before production, make a relationship-specific investment to enhance the future value of the good (in the case of the buyer) or to reduce the production cost of the good (in the case of the seller). Neither the investments nor the value nor the cost of the intermediate good are contractible ex ante. In the absence of a contract, there will generally be underinvestment.

Some of the contractual solutions to the underinvestment problem rely on complex direct-revelation mechanisms that utilize private or unverifiable information to implement efficient investments as in, for example, Rogerson (1992) and Maskin and Tirole (1999), but they are not robust to renegotiation. Several articles have focused on contracts that make explicit use of ex post renegotiation to achieve efficiency. Particularly, Aghion et al. (1994), Chung (1991), and Edlin and Reichelstein (1996) show that simple fixed-price-fixed-quantity contracts may suffice to achieve the first best outcome. Such contracts are, in general, not efficient ex post. But as long as mutually beneficial renegotiation of the initial contract is allowed, both ex ante and ex post efficiency can be achieved either by a properly designed renegotiation game (as in Aghion et al. and Chung) or by court-imposed standard breach remedies (as in Edlin and Reichelstein). Noldeke and Schmidt (1995) consider a more sophisticated contract. They s how that the first best can be achieved by an option contract that stipulates a price schedule depending on whether trade takes place or not and gives the seller the option to decide whether to trade or not. Again, the initial contract is generally renegotiated to achieve ex post efficiency.

In reality, however, complex direct-revelation mechanisms are rarely observed. Simple fixed-price contracts are also not commonly used in bilateral vertical procurement. Moreover, the optimal simple contracts identified in the literature are almost always renegotiated. In practice, contract renegotiation does not occur as often as theory predicts.

Empirical studies show that long-term contracts in vertical procurement relationships with specific investments typically have the following features. First, prices are generally contingent on some standard costs or some variables that determine the actual costs. Fixed-price contracts are rarely used. (3) In contrast to the assumption used in the aforementioned theoretical literature, cost-related variables are often contractible to a certain degree. Joskow (1985) observes that indexed-price contracts are common in the relationship between coal mines and electricity-generating plants. In an indexed contract, the price is broken down into components (labor, materials and supplies, depreciation, profit, taxes, etc.), and then each component escalates according to an appropriate input price (which accounts for inflation) and productivity index.

Second, transacting quantities are often determined ex post by one party (typically the buyer) such as in a requirement contract. Joskow (1985) reports that almost all of the coal contracts in his sample are full requirement contracts, which specify that quantities of coal delivered should meet the requirements of the power plant. In other words, because requirements vary, the power plant decides how much coal to buy ex post. Crocker and Masten (1991) note that transactors are more willing to adopt mechanical pricing arrangements where contracts allow flexibility in other dimensions, such as quantity. Goldberg and Erickson (1987) observe that when a coker is a sole supplier of petroleum coke to a calciner, it is almost invariably the calciner who decides on the quantity.

Third, nonlinear or two-part pricing, such as the minimum take-or-pay provision, is frequently used to protect the seller's specific investments, as is shown by Masten and Crocker (1985) and Masten (1988). Joskow (1985) also finds that 5 out of 21 contracts m his sample had nonlinear price schedules; most of the contracts have a minimum take-or-pay provision; and one had a two-part tariff. (4)

One must note that the objective of the theoretical literature is not to provide formal explanations of the contractual forms actually used in the real world but to show whether in the absence of complete contracting there exist in theory efficient contractual solutions to the hold-up problem, which is a theoretically important question.

The objective of the present article is rather modest. Instead of looking for a new type of efficient contracts in a more general setup, this study, by analyzing a special case of the standard model, aims to derive contracts that have characteristics similar to those observed in empirical studies.

III. THE MODEL

The model has two periods and two risk-neutral firms: an upstream firm (the seller) and a downstream firm (the buyer). At date 1 (ex ante), both firms can make relationship-specific investments, [beta] for the buyer and [delta] for the seller. Let h([beta]) and g([delta]) denote the strictly increasing, convex, and differentiable cost functions of the investments. Production takes place at date 2 (ex post). Both the value and the variable production cost of the good are random variables, the values of which are realized and observed by both parties at the beginning of date 2. The value of the procurement to the buyer, denoted as V(q, [beta], [omega]), is a function of quantity bought, the investments made by the buyer, and the state of nature [omega] [member of] [OMEGA], which may reflect demand uncertainty. The variable cost of production, denoted as C(q, [delta], [theta]), is a function of the quantity produced, the investments made by the seller, and the state of nature [theta] [member of] [THETA]. Both V( q, [beta], [omega]) and C(q, [delta], [theta]) are positive, differentiable, and increasing in q. Furthermore, V(q, [beta], [omega]) is increasing in [beta] while C(q, [delta], [theta]) is decreasing in [delta]. In other words, the buyer's ex ante investments are of a value-enhancing type while the seller's investments are of a cost-reducing type.

Both [omega] and [theta] are realized at the beginning of date 2, that is, after investment but before production. At date 1, both parties are uncertain but have common beliefs about [omega] and [theta], which are represented by the cumulative distribution functions F([omega]) and G([theta]) respectively. At the beginning of date 1, the buyer and the seller can negotiate a contract. But we assume that the investments, [beta] and [delta], the realized value V and cost C, and [omega] are observable but not verifiable in the sense that they cannot be contracted on. (5) Departing from the literature, however, we assume, based on the empirical observations cited in section II, that the cost-related random factor [theta] is contractible.

Efficiency requires that given [beta], [delta], [omega], and [theta], the quantity q maximizes

(1) V(q, [beta], [omega]) - C(q, [delta], [theta]).

For simplicity, it is assumed that there exists a unique interior maximizer, denoted as [q.sup.*]([beta], [delta], [omega], [theta]).

We define

(2) S([beta], [delta], [omega], [theta] [equivalent to] V([q.sup.*]([beta], [delta], [omega], [theta]), [beta], [omega]) - C([q.sup.*]([beta], [delta], [omega], [theta]), [delta], [theta]),

which is the ex post total surplus.

Efficiency also requires that [beta] and [delta] maximize the expected total surplus,

(3) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] S([beta], [delta], [omega], [theta]) dF([omega]) dG([theta]) - h([beta]) - g([delta]).

Again, we assume that there exists a pair of unique interior solutions ([[beta].sup.*], [[delta].sup.*]) to the above maximization problem, and that they are defined by the following first-order conditions:

(4) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [partial]S([[beta].sup.*], [[delta].sup.*], [omega], [theta])/[partial][beta] dF([omega]) dG([theta]) - [partial]h([[beta].sup.*])/[partial][beta] = 0, and

(5) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [partial]S([[beta].sup.*], [[delta].sup.*], [omega], [theta])/[partial][delta] dF([omega]) dG([theta]) - [partial]g([[delta].sup.*])/[partial][delta] = 0.

By the envelop theorem, the marginal conditions are equivalent to

(6) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [partial]V([[q.sup.*]([[beta].sup.*], [[delta].sup.*], [omega], [theta]), [[beta].sup.*], [omega])/[partial][beta] dF([omega]) dG([theta]) - [partial]h([[beta].sup.*])/[partial][beta] = 0, and

(7) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [partial]C([q.sup.*]([[beta].sup.*], [[delta].sup.*], [omega], [theta]), [[delta].sup.*], [theta])/[partial][delta] dF([omega]) dG([theta]) + [partial]g([[delta].sup.*])/[partial][delta] = 0.

IV. OPTIMAL CONTRACT

At the beginning of date 1, the parties can sign a procurement contract. I consider the following requirement contract, which gives the buyer the right to decide at date 2 on the procurement quantity q and specifies a price schedule,

(8) P(q,[theta])=C(q, [[delta].sup.*], [theta]) + T.

This schedule has a natural interpretation. It can be thought of as having a two-part tariff structure with a fixed component T and a unit charge p(q, [theta]) = C(q, [[delta].sup.*], [theta])/q for q > 0. This unit price is nonlinear and can be interpreted as the average standard cost, which is contingent on the cost-related state of nature and may vary with quantity. The fixed component is determined by the relative bargaining power of the two firms. In the case where the buyer can make a take-it-or-leave it offer, T can be interpreted as a compensation for the seller's specific investments. It is called the "hostage" in Williamson's (1983) famous article, whereas Hubbard and Weiner (1986) interpret it as a take-or-pay provision.

I follow the backward induction principle to analyze the game between the two firms. At date 2, after investments (i.e., [beta] and [delta]) are sunk and the state of nature (i.e., [omega] and [theta]) is realized, the buyer would choose q to maximize

(9) V(q, [beta], [omega]) - P(q, [theta]).

Suppose for now that there is no renegotiation at date 2. (I will allow possible renegotiation later.) Substituting (8) into (9), the buyer's choice of q maximizes

(10) V(q, [beta], [omega]) - C(q, [[delta].sup.*], [theta])- T.

This yields a unique solution [q.sup.*]([beta], [[delta].sup.*], [omega], [theta]). At date 1, the buyer chooses [beta] to maximize

(11) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [V ([q.sup.*] ([beta], [[delta].sup.*], [omega], [theta]), [beta], [omega]) - C ([q.sup.*] ([beta], [[delta].sup.*], [omega], [theta]), [[delta].sup.*], [theta])]

x dF ([omega]) dG([theta]) - T - h([beta]).

By assumption, this program has a unique solution [[beta].sup.*],

Given the buyer's choice of [[beta].sup.*] and [q.sup.*] ([[beta].sup.*], [[delta].sup.*], [omega], [theta]), the seller chooses [delta] to maximize

(12) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [C([q.sup.*] ([[beta].sup.*], [[delta].sup.*], [omega], [theta]), [[delta].sup.*], [theta])

- C([q.sup.*] ([[beta].sup.*], [[delta].sup.*], [omega], [theta]), [delta], [theta])]

x dF([omega]) dG([theta]) + T - g([delta]).

Because the first item in the double integral is independent of [delta], the maximizer [delta] satisfies the following first-order condition:

(13) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] [partial]C ([q.sup.*] ([[beta].sup.*], [[delta].sup.*], [omega], [theta]), [delta], [theta])

/[partial][delta] dF([omega]) dG([theta]) + [partial]g([delta])/[partial][delta] = 0.

From (7) we know that [delta] = [[delta].sup.*] is a solution to (13). I claim that it is the unique solution that maximizes (12), and the seller's expected payoff is T - g([[delta].sup.*]). To see this, suppose there exists an investment level [delta] [not equal to] [[delta].sup.*] that solves (13) and maximizes (12). Then the seller's expected payoff must be greater or equal to T - g([[delta].sup.*]). Note from (11) that the buyer's expected payoff does not depend on the seller's choice of [delta] and it is

(14) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] S([[beta].sup.*], [[delta].sup.*], [omega], [theta]) dF ([theta]) dG([theta]) - T - h([[beta].sup.*]).

Therefore, if [delta] [not equal to] [[delta].sup.*] is the seller's optimal investment choice, then the total expected surplus should be greater or equal to

(15) [[integral].sub.[THETA]] [[integral].sub.[OMEGA]] S([[beta].sup.*], [[delta].sup.*], [omega], [theta]) dF([omega]) dG([theta])

- h([[beta].sup.*]) - g([[theta].sup.*]) - h([[beta].sup.*]) - g([[delta].sup.*]),

which is the first-best level of the expected total surplus. But this contradicts the assumption that [[beta].sup.*] and [[delta].sup.*] are the unique first-best investment choices.

To summarize, under the specified contract without renegotiation, the efficient investment and quantity choices ([[beta].sup.*], [[delta].sup.*], [q.sup.*]) are the unique subgame perfect equilibrium. I call this equilibrium the efficient equilibrium.

In deriving the efficient equilibrium, I assumed that there is no renegotiation at date 2. While in that equilibrium, renegotiation, even if allowed, would not take place, allowing renegotiation in the off-equilibrium path may potentially lead to different outcomes. If the seller's ex ante investment differs from the efficient level, that is, [delta] [not equal to] [[delta].sup.*], then, without renegotiation, the buyer's ex post quantity choice would be inefficient, and hence there would be room for mutually beneficial renegotiation. The anticipation of possible ex post renegotiation would in turn affect the ex ante choice of investments. In the following, I show that given the specified contract, ([[beta].sup.*], [[delta].sup.*], [q.sup.*]) is still an equilibrium even if we allow for the possibility of ex post renegotiation.

First note that given the seller's choice of [[delta].sup.*], the buyer's quantity choice [q.sup.*] ([beta], [[delta].sup.*], [omega], [theta]) is ex post efficient. Thus there will be no renegotiation, and [[beta].sup.*] is the buyer's best strategy.

Next we show that [[delta].sup.*] is the seller's best strategy given the buyer's choice of [[beta].sup.*]. Suppose the seller chooses [delta] = [delta]' [not equal to] [[delta].sup.*]. Then, with efficient renegotiation the ex post quantity choice is [q.sup.*]([[beta].sup.*], [delta]', [omega], [theta]) rather than [q.sup.*]([beta], [[delta].sup.*], [omega], [theta]). The ex post total surplus is maximized and equal to S([[beta].sup.*], [delta]', [omega], [theta]). Note that the buyer can guarantee him- or herself an expected postrenegotiation payoff that is no less than the amount given by (14).

Therefore, regardless of how the renegotiation surplus is divided between the two firms, the seller's expected postrenegotiation payoff is no greater than

(16) [[integral].sub.[THETA]][[integral].sub.[OMEGA]] [S([[beta].sup.*], [delta]', [omega], [theta])

- S([[beta].sup.*], [[delta].sup.*], [omega], [theta])] dF([omega]) dG([theta])

- [h([[beta].sup.*]) + g([delta]')] + [T + h([[beta].sup.*])].

Given the uniqueness of the efficient outcome, we have

(17) [[integral].sub.[THETA]][[integral].sub.[OMEGA]] S([[beta].sup.*], [delta]', [omega], [theta]) dF([omega]) dG([theta])

- [h([[beta].sup.*]) + g([delta]')]

< [[integral].sub.[THETA]][[integral].sub.[OMEGA]] S([[beta].sup.*], [delta]', [omega], [theta]) dF([omega]) dG([theta])

- [h([[beta].sup.*]) + g([[delta].sup.*])].

Thus, the value of (16) is less than T - g([[delta].sup.*]), which is the seller's expected payoff from choosing [[delta].sup.*]. In other words, given the buyer's choice of [[beta].sup.*], the seller cannot gain from ex post renegotiation by choosing an inefficient investment level.

To conclude, I have shown that given a requirement contract with a price schedule specified in (8), ([[beta].sup.*], [[delta].sup.*], [q.sup.*]) is an equilibrium whether or not one allows for possible ex post renegotiation; in this equilibrium, the contract is not and need not be renegotiated even if it is allowed. There may or may not exist other equilibriums if renegotiation is allowed. Because the identified equilibrium is efficient, the two firms would be able to coordinate with each other and play this efficient equilibrium if other less efficient equilibriums do exist.

V. CONCLUSION

The optimal contract is designed such that each firm is made the residual claimant to the benefits from their own investments and hence has the right incentives to make those investments efficiently. The contract also provides sufficient flexibility for ex post adaptation without the need of renegotiation by giving the buyer the unilateral rights to decide on the quantity, and the price schedule is chosen such that the buyer's unilateral decision is ex post efficient given that the ex ante investments are efficient.

One lesson from this exercise is that sometimes predictions based on a general model do not always conform to the reality better than those derived from a more specific model, particularly when the choice variable (contractual forms in this case) is of a discrete nature.

The simple model employed in this article cannot, of course, capture the richness of the real-world contracts. It is restricted, for example, to situations where the standard cost is contractible. Nevertheless, my result sheds light on some of the crucial aspects of vertical procurement contracting.

(1.) See Williamson (1985), Grossman and Hart (1986), and Hart and Holmstrom (1987).

(2.) I ignore integration, which has transaction costs of its own, given that the optimal contract in my model achieves efficiencies both in ex ante investments and in ex post adaptation.

(3.) As Goldberg and Erickson (1987) noted, a fixed-price contract may induce wasteful precontractual information acquisition. In their study of petroleum coke contracts, Goldberg and Erickson argue that a principal reason to include price adjustment mechanisms in longterm contracts is to reduce precontractual oversearch of information. Zhu (2000) shows formally how the possibility of acquiring information before signing a contract may make a fixed price contract undesirable and, as a result, cause underinvestment.

(4.) Also see Williamson (1983) and Goldberg and Erickson (1987).

(5.) Here, I follow the standard assumptions. It should be clear later that if ex post renegotiation is not allowed, my result holds even if these variables are not observable ex post.

REFERENCES

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Masten, S., and K. Crocker. "Efficient Adaptation in Long-Term Contracts: Take-or-Pay Provisions for Natural Gas." American Economic Review, 75(5), 1985, 1083-93.

Noldeke, G., and K. Schmidt. "Option Contracts and Renegotiation: A Solution to the Hold-Up Problem." Rand Journal of Economics, 26(2), 1995, 163-79.

Rogerson, W "Contractual Solutions to the Hold-Up Problem." Review of Economic Studies, 59(4), 1992, 777-93.

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-----. The Economic Institutions of Capitalism. New York: Free Press, 1985.

Zhu, T. "Holdups, Simple Contracts, and Information Acquisition." Journal of Economic Behavior and Organization, 42(4), 2000, 549-60.

TIAN ZHU *

* I wish to thank an anonymous referee for helpful comments.

Zhu: Associate Professor, Division of Social Science, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Phone 852-2358-7834, Fax 852-2335.0014, E-mail sotimzhu@ust.hk
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