Strategic Capacity Investment Under Hold‐Up Threats: The Role of Contract Length and Width

We analyze the impact of the length of incomplete contracts on investment and surplus sharing. In the bilateral relationship explored, the seller controls the input and the buyer invests. With two‐part tariffs, the length of the contract is irrelevant: the surplus is maximal and goes to the seller. In linear contracts, the seller prefers the shortest contract and the buyer the longest one. Further, the commitment period concentrates the incentives, whereas afterwards there is rent extraction. The socially efficient contract is as short as possible; yet, long contracts can be promoted because of the surplus they allocate to the buyer.


INTRODUCTION
Some equipment is very long-lived, and yet its capacity is determined upfront and cannot be readjusted. When making this long-term decision, the investor needs to anticipate the possible evolution of the prices of equipment-related inputs and outputs. The investor often seeks to obtain guarantees about future prices, but the equipment might last longer than any reasonable contract's duration. This duration might be a concern when the equipment uses an input produced by a firm with market power. If the commitment is limited, then investors might fear a hold-up, that is, unilateral price increases that will extract rent from the installed capital (Williamson, 1971). The energy sector at large offers a number of instances in which trade-offs must be found via contracts.
We examine investments for which the installed capacity acts as a cap on subsequent purchases. Once the investment has been made, buyers are locked into the chosen technology or standard, but they keep the flexibility to purchase less than nominal capacity. The obvious examples are oil and gas pipelines and most large industrial fuel-consuming appliances. Other important sectors exhibit similar features. For instance, there are those firms that invest in specific production lines using patented technologies. These firms dimension their Our model shows a simple and realistic context in which the shortest contract is the best. After the contract expires, prices are designed for rent extraction. Contracts must therefore be structured so that the commitment period concentrates the incentives to invest, and the seller should concede temporary advantageous conditions. When the commitment period is long, the committed price inevitably mixes two objectives: incentives and rent extraction. But these objectives are antagonistic, and a longer contract is less efficient at inducing profitable investment. The negative impact on the social surplus of a longer contract is very unequally shared between the parties. One could think they both somehow lose. In fact, the seller is a net loser, whereas the buyer produces less surplus but is still a net winner. 1 A longer contract actually protects expropriable investors (their profit) rather than the investment itself (social surplus).
We also argue that simpler contracts that are less efficient in general, such as contracts with purely linear prices, are robust, and more importantly, guarantee buyers a share of the surplus. Linear pricing as well as the duration of the contract could be seen as general rules that parties can agree on beforehand to prevent contract failure, or that can be imposed by regulators in the interest of buyers, while letting the parties negotiate about contractual details. In our view, contract length and width (price structure) can be viewed as key features of the ' "constitution" governing the ongoing relationship' (Goldberg, 1976, p. 428). This view motivates our comparative statics analysis of the contract length in relation to the restrictions on the price structure.
The paper is organized as follows. Section 2 sets up the model. We analyze various scenarios with two-part tariffs in Section 3. Linear contracts are justified and characterized in Section 4. The proofs are relegated to the Appendix.

Trade and Payoffs
The game involves two players: the seller and the buyer. The model is in continuous time with infinite horizon. At instant t, the seller sells to the buyer a quantity qt of a commodity produced at a constant unit cost c; he receives in exchange a payment τt, expressed in units of the numéraire good.
The instantaneous utility of the buyer is quasi-linear. It can be written (up to some irrelevant constant) u(qt) − τt. We will use the iso-elastic utility function 1 Castaneda (2006) also finds a negative effect of longer contracts on investment, but in his model both parties are worse off. The contract in his analysis is such that the buyer pays a lump sum upfront and that the seller invests after acceptance. The buyer thus is reluctant to accept long contracts that could establish extortion for a long time, which in turn means that the seller's return on them is poor that depresses investment.
where d is a positive scale parameter and where ε is the price elasticity of demand. We assume that ε > 1. The seller sets the tariffs {τt(·), t ≥ 0} where the argument of τt(·) is qt. Only linear and two-part tariffs are used. The general shape of the total payment at date t to the seller for quantity qt is therefore τt(qt) = mt + ptqt, where mt is a per-time-unit fixed fee and pt is the marginal price.
Thus, the instantaneous buyer demand Q(·) exhibits constant price elasticity: The inverse demand function Q −1 is denoted P.
The consumption of the commodity requires an equipment with an indefinite service life, installed at date t = 0, whose size A determines maximum consumption once and for all. For example if the equipment is a pipeline, imports cannot exceed the pipeline's capacity: Unless otherwise specified, the investment cost (k per unit) is assumed to be borne by the buyer, who chooses non-cooperatively the investment size A (the case where the seller is the investor is also discussed further on). The future is discounted at rate r.
The intertemporal surplus of the buyer is S u q q e dt kA The intertemporal profit of the seller is The First Best. The social optimum requires u′(qt) = c + rk, which means constant consumption: ∀t, qt = Q(c + rk). The term rk is the unit amortization cost of the durable investment. The efficient investment is thus

Market Power and Limited Commitment
Trade takes place only after the equipment has been built; however, trading terms can be determined beforehand by means of an agreed tariff. T is the commitment duration.
The Manchester School 1. At date 0, the seller makes a take-it-or-leave-it offer to the buyer consisting of tariffs {τt(·), t ∈ [0, T)}, valid until T. The buyer accepts this offer if it leaves him with a non-negative surplus. 2. The buyer invests A. 3. From t = 0 to T, the buyer purchases qt at each date t. 4. At date T, the contract expires. At each date t ≥ T, the seller makes a take-it-or-leave-it offer to the buyer consisting of tariff τt(·). The buyer accepts this offer if it leaves him with a non-negative instantaneous surplus, and consumes the corresponding qt.
Without loss of generality, the seller's pricing strategies can be restricted to constant marginal prices over each period: p0 during the first period (0 ≤ t < T) and pT during the second period (for t ≥ T). The way the fixed payments are staggered over time within the first period does not matter in the sense that paying mt at all dates t ∈ [0, T) is equivalent to an upfront payment 2 The contract can be summarized by (M0, p0).
In the fourth step, the buyer cannot change his investment, thus whether the seller offers a durable contract or (as we assume) a short-term contract does not make a difference. Indeed, any optimal contract has to implement the same successive identical offers.
The following sections will discuss performance of the market with respect to the first-best allocation. The analysis will focus on a bilateral relationship where the investor is the buyer. The case where the buyer invests before any tariff is proposed is discussed in Subsection 3.3. The case where the seller invests is discussed in Subsection 3.5.

Unlimited Commitment (T = + ∞)
Implementation of the first-best is straightforward. If p0 = c and the fixed fee is reasonable (i.e. such that the buyer participates), the buyer invests A*. To extract the surplus, the seller must choose which is the present value of the perpetual flow u(A*) − (c + rk)A*. This is the generalization of the well-known static result. Note that implementation of the first best via unlimited two-part tariffs does not depend on the demand function.
2 Liquidity constraints on either player could limit this freedom, which we do not consider.

Limited Commitment (0 < T < + ∞)
At date T the investment A is sunk and nothing can prevent the seller from exerting hold-up on the buyer by setting at each instant a tariff that captures the entire surplus from the relationship. For t ≥ T, any tariff scheme (mt, pT) such that pT ≤ P(A) (to avoid under-consumption) and mt + (pT − c)A = u(A) (to ensure participation) will do. A simple example is pT = c combined with a fixed fee mt = u(A), ∀t ≥ T. The buyer anticipates that he will obtain no surplus from T on. Assume the price during the first period is constant and denoted by p0. After rearrangement, the buyer solves: where the expression reveals that, because of expropriation ex post, the investment A has to be amortized during the first period. The equivalent flow cost of investment is thus rk e rT 1− − per unit of equipment. The buyer will invest under the condition that M0 is small enough to ensure participation. The seller anticipates that he can obtain the total surplus from T on, whose present value is e r u A cA Clearly, the seller will choose the largest possible fixed fee and the corresponding profit-maximizing price.
Proposition 1: When the seller can commit to a two-part tariff (M0, p0) for a limited duration T, he will set a marginal price below his marginal cost: p c e e rk rT rT 0

The Manchester School
This leads the buyer to undertake the optimal investment A* = Q(c + rk). The fixed fee allows the seller to capture the entire first-stage surplus: Then the following tariff allows him to capture the entire second-stage surplus: An interesting feature of this game is that the consumption good is subsidized during the contract: it is priced below its marginal cost, the subsidy being the amortization of capital from date T on, i.e. it is an advance compensation for the hold-up period. The shorter the contract duration and the larger the investment cost, the more generous the first-stage unit price reduction must be in order to stimulate investment. Remark that p0 is negative if T is sufficiently short.
The two instruments of the contract play different roles: p0 is used to give the right investment incentives to the buyer, and M0 transfers the surplus to the seller. The seller is willing to set a below-cost unit price in order to encourage investment because this does not prevent him from capturing the entire surplus from the relationship, through the fixed fee in the first stage, then by exerting hold-up after expiry of the contract.

No Commitment
Suppose now the seller cannot commit to a tariff (M0, p0). Strategically speaking, this corresponds to the scenario where the buyer invests before any tariff is proposed. This is a situation of pure hold-up: once the investment is made, the seller will repeatedly set the same tariff (m, p) that captures the entire ex post surplus for each period, which means that the buyer cannot recoup his investment costs. As a consequence, the buyer's surplus if he invests is negative. Therefore, there is no equilibrium with positive investment: no commitment and powerful tools is the worst-case scenario.

Endogenous Duration
The first-best investment level can be attained whatever T > 0. The interest of the alternative scenarios with respect to T lies in the timing of transfers and the structure of tariffs. In all cases, the whole surplus goes into the seller's hands. In the analysis, the contract duration T is treated as exogenous. Now suppose the seller or the buyer has the power to determine the duration of the contract. All durations T > 0 are equivalent for both players, while no commitment is inefficient, thus the choice will be any strictly positive duration.

Investment by the Seller
Suppose the investment is made by the seller. Fixed fees allow a monopolistic seller to capture the entire surplus both during and after the contract; therefore, it is optimal for him to choose the first-best investment level A* and to always set the variable price p = c + rk to induce a consumption equal to A*. In fact, whether the investment is undertaken by the seller or the buyer, two-part tariffs yield the efficient level of investment and allocate the surplus to the seller.

LINEAR TARIFFS WITH EXPIRY DATE
Before studying the structure of the equilibrium with linear tariffs, we give two theoretical reasons why they are important in practice.
For the first argument, there is a basic result on two-part tariffs that must be recalled. Take the static standard monopoly. Suppose that the fixed part of a two-part tariff is capped by regulation, and let's see what happens when the cap goes down.
• As long as the cap is high enough, the constraint does not matter. The seller sets a marginal price equal to the marginal cost and extracts the entire surplus through the fixed fee. This is the best case for the seller and it is socially optimal. • When the cap on the fixed part is binding and goes down, the seller inevitably decreases the fixed fee and raises the marginal price above marginal cost. The buyer's consumption decreases and his surplus stays at zero. • As the cap on the fixed part falls further down, the seller fixes the price at the standard monopoly level before being forced to a pure linear price. He continues to set the highest permissible fee.
The pure linear price is a limit case of the latter regime with the cap at zero. In that important regime, capped two-part tariffs all yield the same marginal price, quantity and investment level as linear tariffs. Only the surplus sharing is affected via the cap-dependent fixed fee (the buyer's surplus increases as the cap decreases). Therefore studying linear prices is not as restrictive as it may seem as the predictions encompass the case of two-part tariffs with relatively tight caps on fixed fees.
This being recalled, a defense of linear pricing relies on its distributive role: it guarantees the buyer a share of the surplus. Linear pricing may emerge as a general rule for a wide set of contexts imposed in the buyer's interest. In the absence of information on future costs or preferences, it is extremely robust as a protection against full rent extraction.

A General Characterization
We assume now that, at all dates t, the buyer has to pay a linear price pt, and that prices are guaranteed until date T > 0 only.

The Manchester School
A priori, during the first stage t ∈ [0, T], different prices could be used at different dates. In fact, the following lemma proves that this is not necessary: Lemma 1: If the seller uses linear prices at each date, it is always optimal for him to restrict himself to a constant price during the commitment period.
In the first stage (from t = 0 to T) the price is p0 and consumption is q0, while in the second stage (from t = T to +∞) the price is pT and consumption is qT. The game is sequential, with successive decisions p0, A, q0, pT, qT. As usual, it will be solved by backward induction, but a second lemma will eliminate further large classes of dominated strategies.
The buyer can either passively invest the capacity corresponding to the monopoly quantity, or use his investment choice to influence the outcome. In addition, the seller can either passively set p0 at the anticipated marginal willingness to pay of the buyer, or use p0 strategically to influence his investment behavior, typically to stimulate investment.
This defines the following alternative behaviors. We shall see what they imply and which prevail in equilibrium.
Definition 1: The buyer is said to be active when his investment choice A induces a response pT from the seller that differs from the unconstrained monopoly price ε ε c −1 . Otherwise he is said to be passive.

Definition 2:
The seller is said to be active when his price choice p0 induces a response A from the buyer that differs from A = Q(p0). Otherwise he is said to be passive.
To begin with, we will prove that the investment is never oversized in the second stage, and that the second-stage price equates demand with capacity. Otherwise, the seller could slightly adjust his first-period price without affecting the second period; setting the price p0 closer to the unconstrained monopoly price would enhance profits. The next lemma also proves that the investment can never be oversized in the first stage either.
2. When the buyer is passive, p p P A c , and the seller is also passive. 3. At equilibrium, A ≤ Q(p0).
The lemmas enable us to summarize the four candidate regimes in Table 1.

Contract Length and Width
Theorem 1 (equilibrium prices in the general case): As we are mostly interested in the situation where both players use their opportunities of strategically influencing the other party's behavior, we will concentrate the analysis in the text on the case where the seller and the buyer are active. For the other cases, Fig. 1 illustrates how changing T can make the equilibrium prices cross all three regimes (details are given in Appendix A.5).
From the theorem, we see that a necessary and sufficient condition for both players to be active for all T is Note that for the right hand-side to be positive, the elasticity of demand must exceed 1 5 2 + , the golden ratio (≃1.6). Equilibrium prices as a function of T are given in Fig. 2.

The Value of Commitment
One could think that longer commitment periods are necessarily better for the seller: any price profile that can be implemented with a short

Corresponds to the No-Commitment Scenario (Subsection 4.3)
Contract Length and Width 11 commitment duration (say T1) can be implemented with a longer commitment duration (say T2). But from this, it doesn't follow that the (so to speak) T2-buyer will replicate the behavior of the (so to speak)

T1-buyer.
Assume that the T1-buyer and the T2-buyer face the same price profile and that they consider investing A. How do they value a marginal increase in A? Identically from t = 0 to t = T1: price is fixed anyway. Identically also from t = T2 on: the post-commitment price is a function of capacity only. Now, an increase in capacity will yield a price reduction between T1 and T2 for the T1-buyer, whereas the price remains constant in that interval for the T2-buyer. This means that the value of increasing A is necessarily less for the T2-buyer than for the T1-buyer. There is no ambiguity: facing the same price profile, the T2-buyer (whose contract is longer) invests less than the T1-buyer. This shows that a mere replication does not work.
As A = Q(pT) is a decreasing function of pT, which (from Theorem 1) is an increasing function of T in all regimes, the following paradoxical result obtains.
Proposition 2 (investment): The equilibrium investment level decreases with respect to the contract duration for all T > 0.
As the seller cannot commit to refrain from extracting rent after contract expiry, his only means to stimulate investment is to offer at t = 0, before A is chosen, the guarantee of a low price until T. The smaller T, the more generous the bargain must be: p0 can even be negative, i.e. consumption is subsidized during the contract when the contract duration is short. For example, when T is close to zero, but still strictly positive, p0 becomes infinitely negative in the case where c rk < − − ε ε 1 1 (see Fig. 2).
Remark. Intuitively, one would think that the equilibrium prices are increasing functions of the investment cost, as a higher k tends to dampen investment and consumption. This is true for the hold-up price pT, but not necessarily for the contract price. See the analysis in Appendix A.6 and Fig. 4 therein.
As social welfare increases with respect to the investment level, Proposition 2 implies that welfare decreases with respect to the contract duration. With linear prices, more commitment through longer contracts is detrimental to the seller and has the effect of depressing investment. In a nutshell, the first price p0 serves to push investment whereas pT serves to extract rents: the two successive prices approach a two-part tariff, the first price providing the marginal incentives (investment) and the second price the inframarginal ones (participation). Although the performance of these linear prices is imperfect, this view clarifies the effects of T across agents, and for the economy, as we explain now.
At the beginning of Section 4, we recalled how a cap on the fixed fee benefits the buyer. Longer durations are similar to lower (i.e. more Contract Length and Width 13 advantageous) caps: the longer the commitment, the smaller the impact of the hold-up period is. In compensation, as the second price loses impact when T increases, the seller has to mix in the first price two objectives (incentives and rent extraction), which is the typical inefficiency of plain static linear prices. What about the buyer? As T increases, the buyer's surplus increases even though the surplus to be shared is smaller.

No Commitment
An alternative scenario which does not fit in the previous structure, has to be considered: no commitment at all. The timing is simplified: 1. The buyer invests A at a unit cost k.
2. At each date t, the seller makes a take-it-or-leave-it offer pt to the buyer. If the buyer accepts, he buys qt.
In effect, this case differs, as we shall see, from the case T → 0 in the previous game. Indeed, whenever T > 0, and whatever small T may be, there are a price before and a price after A is chosen.
Clearly the same price and quantities (denoted respectively by p and q) will be chosen at all dates. The buyer anticipates that he will suffer hold-up as soon as the investment is realized, but extortion is imperfect as linear prices are used.
The following proposition summarizes the results.
Proposition 4 (no commitment at all): The equilibrium price and investment level are As in both cases the capacity is fully used [q = A = Q(p)], the outcome is ex post efficient; however, ex ante efficiency is not achieved. Indeed, the equilibrium price is always higher than the social optimum p = c + rk, which implies that the equilibrium investment level is always suboptimal.
No commitment and 'some' commitment (T > 0) are qualitatively different. In other terms, to encourage investment, the smallest contract is the best (Proposition 2), no contract at all is the worst case. See NC (no commitment) prices, profits and welfare in Figs 2 and 3. In particular, note that p NC is larger than any p0 or pT.

Investment by the Seller
An alternative assumption could be that the investment is undertaken by the seller. Let us examine the following game: 1. The seller chooses the investment size A and sets price p.

The buyer purchases q indefinitely.
Commitment is not relevant here, because the optimal price for the seller does not change over time. There is no need to create incentives and extract rents in two separate episodes. This means that contract duration does not matter.
The seller solves the program of a monopolist whose marginal cost is c + rk: the equilibrium is characterized by This game is essentially equivalent to the previous game with T = + ∞ with buyer investment: the outcomes (investment level, profit and surplus of the parties) are the same. The seller simply increases the commodity price by rk to pass on his investment cost.
Does this solve the seller's problem? In fact, the seller would prefer to let the buyer invest. The reasoning is as follows. If the seller invests, he can invest at the level that he wishes, but the linear prices limit his ability to extract the rent thus generated. As the investment cost is difficult to recoup, his incentive to invest in the first place is limited. Conversely, if the buyer invests, the linear prices protect his effort (he is able to keep a fraction of the surplus generated). So the incentives that the seller gives to the buyer are leveraged by the buyer's own incentives, which makes the seller's contribution to investment (through rebates) relatively small. This allows the seller to extract a bigger rent through the second price pT.

CONCLUSION
In this paper, we have examined a situation where contracts are incomplete in several respects: both the pricing structure and the duration of commitment are restricted. All of the pricing schemes that we have analyzed share the following feature: some commitment is always better than none. The reason is intuitive: the seller can offer a significant rebate that gives the buyer perfect or at least fairly good incentives to invest. The commitment period secures the indispensable thrust.
A longer contract never gives higher investment incentives. Tariffs with a sufficient 'width', such as two-part tariffs, are powerful enough to make the contract length irrelevant. With linear tariffs, the investment level is a strictly decreasing function of the contract length. When the contract duration is very short, the seller chooses to subsidize investment heavily to maximize the surplus that will subsequently be extracted. A longer commitment postpones the hold-up period, and the seller prefers to extract rent during the contract period by setting higher prices, with a negative effect on investment incentives. Social welfare, the seller's profit, and the buyer's surplus do not vary in parallel when the contract length changes. Contrary to the seller's profit and social welfare, the buyer's surplus increases with length. A long commitment period means that there is a cap on the rent extracted by the seller, which benefits the buyer.
The link between contract duration and investment incentives in a noncompetitive environment has not been thoroughly established by economic theory. One reason is that duration is not just another parameter in a contract, complete or incomplete. Its interaction with other dimensions can be quite counterintuitive. Our model offers a plausible theoretical interpretation for empirical results that link contract duration and investment incentives. In an empirical study of coal contracts, Joskow (1987) finds evidence of a positive relationship between asset 'specificity'-as measured by different proxies-and contract length (see also Neumann and von Hirschhausen, 2008). This theory is not developed beyond the reasoning that specificity is a weakness and that duration is a protection. Specificity is weaker if the seller can use the productive capacity to deliver the product to several buyers. In this case, our model suggests that the seller's market power can be used to seek the shortest possible contracts. Conversely, if the asset is very specific, for example, if there is a single buyer, market power is rebalanced in favor of the buyer, and the seller must concede a longer contract.

A.1 Proof of Proposition 1
Proof in the text. ■

A.2 Proof of Lemma 1
Consider a contract of length T, such that the producer commits to a price profile {pt}0≤t≤T. We assume that for all real number x, sets of dates like {t|pt ≤ x} are Lebesgue-measurable (a measure denoted by d), which is not restrictive in our context and it is technically more rigorous for the argument. The reasoning will be developed in three steps.
Step  The buyer will equate qt with A for t ∈ I only, and will be strictly unconstrained at other dates, so that his program can be rewritten, for the only part that depends on A: At equilibrium the buyer's best choice is If he anticipates this reaction, the seller wants to approach as much as possible the interior solution of the corresponding profit maximizing program. This interior solution is In any case, the initial pricing choice appears not to be optimal. This proves step 1 by contradiction.
Step 2. Then we prove that consumption cannot be lower than chosen capacity during the contract period, in other terms that q0 = A. Indeed, suppose that for some t in [0, T], qt < A, which means that Q(pt) < A.
A reduction in pt is beneficial for the producer in the short run, because the current price will be brought closer to the monopoly price so that profit will increase, and it will have no detrimental effect in the long run, because investment will not be affected. Indeed, as long as capacity is not binding at time t, the current price has no other effect than on current consumption. As a consequence, any contract price profile offered by the producer is necessarily such that the buyer will consume A for all t in [0, T].
Step 3. Finally, we prove that the seller can never be worse off by restricting himself to a constant contract price.  3. When the seller is passive, by definition A = Q(p0). When the buyer is passive, from Lemma 2 (point 2), A = Q(p0). When both the buyer and the seller are active, A = Q(pT) ≠ Q(p0). We will now prove by contradiction that A > Q(p0) cannot be an equilibrium. Indeed, it implies p p c But it is clearly profitable for the buyer to reduce p0: in the first period where q0 = Q(p0) he would get closer to his static monopoly profit, and in addition, this would induce the buyer to invest more, bringing second-period consumption closer to the seller's preferred level A. Therefore, in an equilibrium with active buyer and active seller, necessarily A < Q(p0). ■

A.4 Proof of Theorem 1
Whenever the buyer is active, he solves: where the first term is the present buyer utility from the contract, the second term is the present utility for the subsequent period, and the third is investment cost. The fact that the buyer is active enables us to replace pT with P(A); Lemma 2 (1.) enables us to replace consumption in the first period with A.
Since the seller is active, p0 < pT: the seller offers a lower price in the first period so that the buyer is incited to invest more, then once the capacity is fixed he sets a higher price. To choose p0, the seller solves To complete the proof of (c) in the theorem, we use equation (36) For case (b), we have both p0 = pT and equation (40). Thus the price is re rT εk. From Lemma 2 (1. and 2.), we can deduce that a necessary and sufficient condition for the buyer to be active is p c T > − ε ε 1 . stages offsets the first-stage loss. In this case only, the contract price and the hold-up price differ. The contract price can even be a decreasing function of k, and become negative when k is high, if the contract length is short enough: the seller accepts a loss that will be compensated by larger sales volumes after contract expiry.

A.7 Proof of Proposition 2
Proof in the text. ■

A.8 Proof of Proposition 3
Profit of the seller. The seller's profit can be expressed as a function of the contract duration: T