- Email: [email protected]
Price competition in noncooperative joint ventures
International
Journal
of Industrial
Organization
Price competition ventures
12 (1994) 53-69.
North-Holland
in noncooperative
joint
Ian Gale* Department of Economics, University of Wwconsin, Madison. WI 53706, USA Final version
accepted
November
1992
Firms that participate in a cotenancy Joint venture are able to utilize capacity that is left unused by other firms. This paper considers a cotenancy involving two or more firms. In equilibrium, the firms quote prices that are socially optimal given the level of aggregate capacity. When there is open entry, the resulting (common) equilibrium price converges to the Ramsey optimal price. A cotenancy may be of use as an alternative to direct regulation of a natural monopoly or as an antitrust remedy, as it combines the benefits of single plant production with competition at the marketing stage. 1.
Introduction
Many production joint ventures have the feature that participants in the joint venture are direct competitors at the marketing stage. Natural gas pipelines provide one example. Within this class of ‘noncooperative’ joint ventures are joint ventures known as cotenancies.’ In a ‘cost center’ cotenancy, production is overseen by an independent management company. Output and pricing decisions are made independently by the co-owners, who essentially place orders for output with the management company. Whenever economies of scale exist, the most obvious way to capture them is to have a single firm produce. One option is to have an unregulated monopolist, which will lead to an inefficiently low level of output. Another option is to impose direct regulation, which will lead to inefftciencies because of incentive problems [Baron (1988)]. Yet another option is to auction off the exclusive right to serve the market [Demsetz (1968)]. However, the existence of economies of scale does not require that there be a monopoly seller. Joint ventures provide one means of combining the benefits of single plant production with competition at the marketing stage. This paper Correspondence to: Ian Gale, Department of Economics, University of Wisconsm, WI 53706, USA. *I am grateful to seminar participants at the University of Wisconsm, the Antitrust of the U.S. Department of Justice, and the 1992 Econometric Society Meetings in New The comments of Buz Brock, Robert Masson (the editor), Stan Reynolds and Marius have improved this paper. ‘The terms ‘competitively-ruled cotenancies’ and ‘competitive rules joint ventures’ used. 0167-7187/94/$07S!O 0 1996Elscvier SSDI 0167-7187(93)00383-Y
Science B.V. All rights reserved
Madison, Division Orleans. Schwartz are also
54
f. Gate, Price com~ririon in noneoo~rati~e joint ventures
examines the performance of cost center cotenancies in the presence of economies of scale. There is growing support for the use of cotenancies as an alternative to direct regulation of natural monopolies and as an antitrust remedy. The original reference is Lewis and Reynolds (1979). Reynolds (1990) WarrenBoulton and Woodbury (1990), and Alger (1991) all provide in-depth discussions. Alger and Braman (1991) examine a model that is similar in spirit to the one studied here. Rassenti et al. (1993) evaluate cotenancies in an experimental setting. The contribution of the current paper is that it provides a complete analysis of a price-setting game that captures the essence of a cost center cotenancy. The performance of cotenancies as an alternative to direct regulation and as an antitrust remedy can then be examined. The setting for the analysis is the following. Two or more firms jointly own a facility that operates as a cost center cotenancy. For concreteness, one can think of firms that own a natural gas pipeline. Each firm has the right individually to pump an amount of gas equal to its ownership share times the pipeline’s capacity. This absolute quantity is the owner’s ‘dedicated capacity’. The firms are subject to competitive rules, which are intended to enhance rivalry among the firms. These rules are based on rules in force in an actual cotenancy. An aluminum rolling mill that was built by the Atlantic Richfield Company (Arco) operates as a cost center cotenancy. In 1984, Alcan Aluminum agreed to buy Arco’s aluminum operations, but the purchase was challenged by the U.S. Department of Justice. A consent decree was ultimately accepted. It called for the rolling mill to be jointly owned by Alcan and Arco, and to be operated by an independent management company. The most notable of the com~titive rules contained in the consent decree was the following: ‘Each party to the joint venture may utilize any unused portion of the other party’s capacity by assuming the variable costs, but not the fixed costs, attributable to the added production.‘* The game analyzed here captures the impact of such a ‘use-or-lose’ provision. A firm that still faces demand after using its dedicated capacity can then use the other firms’ excess capacity. The analysis is divided into two parts, depending on whether capacity is already in place. When capacity is sunk, imposing a use-or-lose provision leads to prices that are socially optimal given the level of aggregate capacity. When capacity is not sunk, the results are still positive, although not unambiguously so. Consider a two-firm cotenancy and a standard duopoly with separate plants. Suppose that there are economies of scale (i.e. declining long-run average cost). When these economies are small, the cotenants may install less aggregate capacity and generate less total surplus than the ‘U.S. II. Alcan Aluminum, 605
F. Supp.619(1985),p. 625.
I. Gale, Price competition
in noncooperative
55
joint ventures
duopolists. With large economies of scale, the cotenancy will dominate in terms of capacity and total surplus. The results are strongest when there is open entry into the cotenancy. As the number of co-owners increases, the common equilibrium price converges to long-run average cost. That is, there will be Ramsey optimal pricing. 3 Moreover, total cost will be minimized because there is no duplication of facilities. The results suggest that there may be a role for cotenancies as a complement, if not an alternative, to traditional direct regulation. With existing or new capacity, the combination of a use-or-lose provision and open entry leads to a constrained Pareto optimum. The case for cotenancies as an antitrust remedy is somewhat less strong. The difference arises because open entry may not be a reasonable requirement when a cotenancy is used as an antitrust remedy. As noted above, the performance of cotenancies is weaker when open entry is not feasible. The basic price-setting game is described in section 2, and the equilibrium is determined. Section 3 examines the two-stage game in which firms choose capacities first, and then play the price-setting game. These results are used in section 4 to evaluate cotenancies in different regulatory and antitrust settings. Extensions are contained in section 5. 2. The price-setting game There are N firms that participate in a cost center cotenancy that produces a single homogeneous good. The facility has aggregate capacity X=x1 + and firm j has dedicated capacity x,. Firm j has first claim on xz+...+xN, xj units of output, in a sense to be made precise below. There is a (short-run) marginal cost of c up to the capacity of the facility and a fixed cost of production F. An independent management company oversees production. There is a continuum of risk-neutral consumers indexed by t E [0,11. The total measure of consumers is equal to one. A consumer of type t has a reservation price P(t) for a single unit of the good, where P( .) is a continuous, decreasing function. This implies that there is a measure q of consumers with reservation prices of P(q) or above. The ‘market-clearing’ price (i.e. the price at which all of the output can be sold when all firms produce up to capacity) is P(X). Demand is denoted by D( .). The firms announce prices at r =O. Whenever demand exceeds supply, the consumers with the highest reservation prices are served first. This is the efficient rationing rule. Under this rule output is allocated efficiently so there is no possibility of mutually beneficial trade among consumers. (Section 5 contains a discussion of alternative rationing rules.) If two or more firms select the same price, consumers who attempt to purchase from these firms patronize the firms in proportion to their capacities. %nce
there is only a single market,
Ramsey
optimal
pricing
entails average
cost pricing.
56
1. Gale, Price competition in noncooperative joint ventwes
The use-or-lose provision in the Alcan consent decree gave each firm the opportunity to use the other firm’s excess capacity. This is modeled here as follows. There are two sales periods: z= 1 and T=2. At t = 1, the consumers whose reservation prices exceed the lowest price attempt to purchase from the firm (or firms) offering that price.4 That firm fills as many orders as it can, up to the limit imposed by its dedicated capacity, and incurs a cost of c per unit of output. All remaining unserved consumers then decide individually whether to attempt to purchase from the firm with the second-lowest price. (Consumers whose reservation prices exceed the second-lowest price will not buy at z = 1 if they will be able to buy from the low-price firm at z =2.) That firm then fills as many orders as it can. This process continues until the remaining firm with the lowest price makes no sales or else all firms have made sales. The same process takes place at x=2, with one difference.’ Now the firms use the capacity that was not used at t= 1. A firm incurs the marginal cost of c per unit of additional output when it uses the excess capacity. It is assumed throughout that a firm must serve all consumers that it is able to serve. (A firm that has the option of limiting output after announcing its price would never have an incentive to do so in equilibrium.) Subgame perfect Nash equilibrium is the equilibrium concept used here. Strategies must be mutual best responses at each stage of the game. The firms price-setting strategies must be mutual best responses given the strategies of the consumers, while the consumers’ strategies must be mutual best responses given the prices and capacities of the firms. In particular, consumers correctly assess the probability that their orders will be filled, and they behave optimally given these assessments. The first step in characterizing the equilibrium is to determine outputs for all combinations of prices. When all prices are above the market-clearing price only the firm with the lowest price will make sales. Consumers will attempt to purchase from the low-price firm first. Since there is enough capacity in aggregate to serve all who wish to purchase at the lowest price, it is not optimal to purchase from any firm with a strictly higher price. Those consumers who are unable to purchase from the low-price firm at z= 1 will all wait until z =2, at which point the low-price firm has access to the other firms capacity. When some prices are below the market-clearing price, all firms with prices &An earlier version of the paper had consumers submit irrevocable orders to the firms. The firms served as many consumers as they could using their dedicated capacity. Those firms with waiting lists were able to use the other firms’ excess capacity at the next stage, beginning with the firm with the lowest price. The ~uilib~um prices and sales are identical in the two extensive forms. ‘Another interpretation of the sequence of events is that orders are contirmed at z= 1 and T= 2, and then productIon and delivery take place at 7 = 3.
I. Gale, Price competition
in noncooperative
joint
57
ventures
below P(X) make sales while those with prices above P(X) do not. Suppose that N=2 and pixt consumers with reservation prices exceeding p2. These latter consumers will all attempt to purchase from firm 2, so there will be no excess capacity available for use at r=2. Any member of this group who does not attempt to purchase from firm 2 at r= 1 will not be able to purchase at all. In this case firm i’s output is xi. Now suppose that p1
output for
7 =
1 is
firmi is
Qi=
I
$)‘D(pi),
i=1,2 ,..., j,
0,
i=j+l,j-t-2
(1) ,..., N.
When prices satisfy
output for firm i is
1
Xi+&(X-x(k)),
Qi=
Proof.
i=1,2 ,..., j,
Xi,
i=j+l,
0,
i=k+l,k+2
1
The proof is in appendix A.
q
j+2 ,..., k, ,..., N.
(2)
I. Gale, Price competition in noncooperative joint ventures
58
Relabelling the firms gives the outputs for all permutations of these prices. A firm charging a price above the market-clearing price Z’(X) only makes sales if all other prices are at least as high. But if all firms charge the same price, and it exceeds the market-clearing price, then each firm has an incentive to cut price unilaterally if P(X) > c. If P(X) SC, then the incentive to undercut ends when at least two prices are equal to c. A consequence is that all firms will select the price that is socially optimal given the level of aggregate capacity.6 Proposition 1.
(i=1,2,...,
When
P(X) >c,
the
equilibrium
prices
are
pi=P(X)-E
N). When P(X) 5 c, the equilibrium prices satisfy pi 2 c, and at least
two firms quote pi = c. Proof.
The proof is in appendix B.
0
When the market-clearing price exceeds marginal cost, the firms all quote the market-clearing price. When the market-clearing price is below marginal cost, there is marginal cost pricing. No cost or demand information is required by the regulator, nor are any subsidies required. All that is needed for this result is that the use-or-lose provision be imposed. In a multi-period setting there will be socially optimal pricing in each period. Note also that the result does not depend on the number of firms so open entry (whereby the incumbents are required to sell capacity to entrants at a set price) would have no impact. One recent paper uses a similar model and predicts the same prices. In Alger and Braman (1991) firms can make price offers at any time. Consumers submit orders, which cannot be withdrawn, although consumers can accept a lower price if one is subsequently offered. While the authors claim existence and uniqueness of equilibrium, the extensive form that they examine is not presented explicitly. 3. The capacity-choice
game
The previous sections have examined cotenancies in which firms’ dedicated capacities are sunk. This section takes a step back and looks at capacity choice by firms that form a cotenancy prior to building capacity. This situation could arise as the result of regulatory or antitrust decisions to challenge a proposed cooperative joint venture, for example.’ The outcome ‘%rictly speaking, the firms will quote a price that is infinitesimally below P(X) when P(X)>,. Suppose that there is a smallest unit of currency. The unique pure-strategy equilibrium has all the firms quoting the largest price strictly below P(X) in this case. In what follows, this distinction will be ignored. ‘Most proposed joint ventures above a certain size must undergo prior antitrust review by the Department of Justice or the Federal Trade Commission [Kwoka (1989, p. 49)].
I. Gale, Price competition
in noncooperative
jotnt ventures
59
of the two-stage cotenancy game will be determined first. For purposes of comparison, the equilibrium of the analogous two-stage game with separate facilities will also be determined.
The firms choose their individual investments in capacity at time t= - 1, and then select prices at r=O. The total cost of building x units of capacity is @P(X).It is assumed that the average cost of capacity, @(x)/x, is strictly decreasing, and that the long-run average cost, cf [@(x)/~], cuts P(x) from below exactly once. A rule for allocating the cost of capacity among the firms must also be specified. The rule in the Alcan consent decree governing the allocation of costs of new capacity will be used here. Shares of the capacity costs are equal to the firms’ ownership shares8 In other words, firm i pays the average cost of capacity for its xi units of capacity. A strategy for firm j now consists of a rule for selecting x,, and a rule for selecting pj given the capacities chosen, Consumer behavior is not changed by adding the capacity-choice stage, so the outputs are unchanged for given prices and capacities. Strategies for the price-setting game must be best responses given the capacities chosen, and the capacities chosen must be best responses given the equilibrium of the ensuing price-setting game. The results of section 2 demonstrate that either all firms produce up to their dedicated capacity or else there is (short-run) marginal cost pricing. The latter outcome would give the firms negative profits in the two-stage game, so firms will choose capacities such that they all produce up to capacity.’ Firm 1 selects
when x,=Xj for j # 1. The first-order condition is
8U.S. O. Alcan Aluminum, 605 F. Supp. 619 (198.5). p. 625. ‘It will be assumed that the tixed costs of productlon equal zero. A fixed cost of capacity ~111 have the same effect, smce fixed costs of production are allocated according to ownership shares, not output shares.
JIO
C
60
I. Gale, Price competition
where w~x/(x+~jN_~ order condition is
in noncooperatwe
joint ventures
Zj) is firm l’s ownership share. The symmetric lirst-
o=P’(Nx)x+P(Nx)-c-
N - 1 @(NT) ;@‘(Nx)-N ~NZ
(5)
(If long-run average cost does not decline too quickly, (5) characterizes the unique equilibrium. The focus will be on symmetric equilibria throughout.) When firm 1 expands its investment in capacity it pays only a fraction of the true marginal cost of capacity since it pays only a fraction of the total capacity cost. However, as its ownership share rises, the firm must pay an increased share of the cost of the inframarginal capacity. The net effect is that in equilibrium each firm faces an effective marginal cost of capacity that is a weighted average of the true marginal cost and the true average cost of capacity. The larger N is, the closer the effective marginal cost is to the true average cost.
3.2. The standard duopoly game The standard duopoly game in which two firms have separate plants and are not subject to a use-or-lose provision serves as a benchmark for a twofirm cotenancy. This section characterizes the equilibrium of the duopoly game that is analogous to the cotenancy just examined. The firms quote prices at r =O. At r = 1, consumers purchase. They first attempt to purchase from the low-price firm, as before. The difference is that there is only one round of purchases here, so neither firm has the opportunity to use the other firm’s excess capacity. Suppose that pi
P(2cz) -c - g(3).
(6)
I. Gale,
Price
competrtion
in noncooperative
joint
ventures
61
4. The prospects for cotenancy This section uses the analysis of the previous two sections to evaluate the prospects for employing cotenancies as an alternative to traditional direct regulation and as an antitrust remedy. The different contexts have different reference points for measuring performance and, as a consequence, they will have different rules governing entry. In a regulatory setting, the appropriate standard is Ramsey optima1 pricing since a regulator with complete cost and revenue information could induce a monopolist to price at long-run average cost. In this case it would be reasonable to impose open entry on the cotenancy. In an antitrust setting, the reference point is the pre-merger level of competition. Open entry would not be a reasonable requirement in this case, and only the parties to the proposed merger would typically be involved [Warren-Boulton and Woodbury (1990)]. The case of sunk capacity is examined first. 4.1. Existing
capacit?
When capacity is sunk, forming a cotenancy simply means imposing a use-or-lose provision here. It was shown in section 2 that this action enhances competition and leads to socially optimal pricing, with or without open entry. A cotenancy arrangement would therefore appear to offer benefits as an alternative to traditional direct regulation of natural monopolies. This analysis is also relevant to the opening and closing of duplicate plants. Suppose that two companies wish to merge and close one of their two pipelines. This will reduce the total fixed costs of production but will also eliminate rivalry at the marketing stage. Imposing a use-or-lose provision on the remaining pipeline would maintain rivalry. Since this case involves a net reduction in aggregate capacity, there may be a reduction in consumers’ surplus, which must be weighed against the reduction in fixed costs. At the same time, consumers’ surplus could actually rise if the pipeline that is closed down is sufficiently small. Now consider the case of an incumbent monopolist and a potential entrant. Suppose that the incumbent currently has excess capacity. The incumbent and the entrant would both prefer to merge rather than have the entrant build new capacity and compete as a duopolist. However, a cotenancy could be welfare-superior to the duopoly outcome, in addition to being superior to the merger. In particular, if the economies of scale are sufficiently great, cotenancy will generate strictly more total surplus than will the duopoly. lo “‘Another possibility, which is not treated here, is that economies of scope are realized the operations are merged. Kwoka and Warren-Boulton (1986) discuss this issue.
when
62
I. Gale, Price competition in noncooperative joint ventures
4.2. New capacity
A cotenancy could be formed prior to the construction of capacity, either in a regulatory setting or in an antitrust setting. For example, two firms might propose a cooperative joint venture. Antitrust authorities have several options. They could approve the joint venture, they could challenge it but suggest that the firms submit independent proposals instead, or they could challenge it but suggest that the firms form a cotenancy. (In an antitrust setting, open entry would typically not be an appropriate remedy for a proposed cooperative joint venture involving two firms.) The monopoly and duopoly equilibria will then be used as benchmarks for a two-firm cotenancy here. It could also happen that two firms would submit independent proposals to a regulator to build plants in a new market. The regulator could approve one or both of the firms’ proposals, or she could turn down both but suggest that the firms form a cotenancy instead. I1 The regulator could also impose open entry on the cotenancy, which would allow new firms to enter by paying the average cost of capacity for their capacity. The case without open entry is examined first. The cotenancy dominates monopoly, even when the cotenants face a higher effective marginal cost of capacity. The first-order condition for the cotenants in the capacity-choice game is
which can be rewritten as o=[p’(2x)2x+P(2x)-c-q2x)]+P(2x)-c-
@(2X) x.
(8)
The last three terms, take together, equal price minus long-run average cost. This must clearly be positive in the cotenancy equilibrium, so the term in square brackets must be negative. This means that when aggregate capacity is at the level where the cotenancy first-order condition is satisfied, the monopolist’s profit would be declining in capacity. Hence, capacity and total surplus are greater under cotenancy. While a two-firm cotenancy dominates monopoly, the following example demonstrates that total surplus can be lower in the cotenancy than in a duopoly. Suppose that demand is weakly concave and that the cost of capacity is Q(x) = &, + $x for x 5 x* and Q(x) = $*x for x > x*, where 4* > 4 “The idea that a regulator might suggest that proposals be combined mto a joint venture in order to capture the economies of scale goes back at least to the Northern Natural Gas case of 1968. See Kahn (1988, p. 160) for a dlscussion.
I. Gale, Price competition
in noncooperatl
d joEnt ventures
63
and #*=x*(#* - 4). The average cost of capacity is continuous, strictly decreasing for x < x*, and constant thereafter. Now suppose that the duopolists have an interior maximum right at x* in the capacity-choice game. The associated first-order condition is 0 = P’f2x.*)x* + P(2X*) - c- rfi_
(9
The first-order condition in the cotenancy (presuming that aggregate capacity is at least x*) is 0 = p’(23i + P(2.Z)- c - 4*_
(10)
Since @+> Ip, the equilibrium aggregate capacity will be larger in the duopoly. Consumers’ surplus will also be larger under duopoly, as will total surplus, since the duopolists are producing at the minimum of long-run average cost. It is then straightforward to construct an example in which total surplus is lower under cotenancy, even though long-run average cost is strictly decreasing everywhere. Aggregate capacity and consumers’ surplus can be lower in the cotenancy even when the true marginal cost of capacity is decreasing. For example: if G(x)=& +4x for all x, then the cotenants still face a higher effective marginaf cost of capacity than the duopolists do since the true average cost af capacity always exceeds the true marginal cost. Capacity would again be lower under cotenancy. Examples can be constructed in which the cotenants face a higher effective marginal cost of capacity, even though the true marginal cost is itself strictly decreasing. Total surplus can be higher in the cotenancy, even though consumers’ surplus is lower, since the duopolists do not produce at the minimum of long-run average cost here. To this point some extreme examples have shown that a cotenancy need not generate greater total surplus than a duopofy. When the economies of scale in plant construction are great, the effective marginal cost of capacity in the cotenancy will be below the corresponding marginal cost in the duopoly. In this case aggregate capacity will be larger under cotenancy, so total surplus will necessarily be larger as well.” When open entry is feasible, another method of generating greater total surplus is to alfow open entry. Consider the first-order condition in the capacity-choice game with N firms. As N increases, the equilibrium dedicated capacity of the individual cotenants converges to zero, and the first-order condition (5) converges to the follo~ng expression:
‘?here may be multiple equilibria of the twa-stage duopoly game when the marginal cost of capacity is strictly decreasing. However, any equilibrium of the two-stage game in which the firms use pure strategtes in seiecting quantities is ako a pure-strategy Cournot equilibrium [Osborne and Pitch& [19&Q].
64
I. Gale, Price ci~~petition in noncoo~rar~ve joint ventures
O=P(NZ)--c-F.
(11)
(The dependence of X on N is implicit.) Since long-run average cost cuts the demand curve from below exactly once, there is a unique level of aggregate capacity at which (11) is satisfied. As the number of cotenants grows, the equilibrium price converges to the Ramsey optimal price, i.e. long-run average cost. The regulator does not need to know the cost of capacity or the demand curve. All that is required is that the use-or-lose provision be imposed along with open entry. The most important result is that there is always room for more firms in a cotenancy, in the following sense. The use-or-lose provision, in conjunction with open entry, increases output, but not at the expense of economies of scale. Entry brings the benefit of increased competition at the marketing stage, without forsaking economies of scale in production or plant construction.
5. Extensions This section examines various assumptions form game.
made in the basic extensive
5.1. Different rationing rules
Consider the impact of different rationing rules. Suppose first that all prices strictly exceed P(X), in which case there is enough capacity in aggregate to serve all who wish to purchase at the lowest price. Under any rationing rule, consumers will only purchase from the firm(s) with the lowest price. It follows that whenever P(X)sc, at least two firms quote a price equal to marginal cost and all purchases will be made at that price. Likewise, whenever P(X) > c, there cannot be an equilibrium in which all prices strictly exceed P(X). There is always at least one firm with an incentive to cut price unilaterally and make all of the sales, under any rationing rule. Consider random rationing in particular and suppose that P(X)>c. It is equilibrium behavior here as well for all firms to quote pi= P(X). Suppose that all firms select the market-clearing price. No firm has an incentive to raise its price unilaterally, since its sales would drop to zero. Now suppose that firm j deviates unilaterally and quotes a price pj X - xj consumers remaining with reservation
I. Gale, Price competition
in noncooperative
joint
ventures
65
prices above P(X). Since X--xi is the amount of capacity remaining, there are more of these consumers remaining than there is capacity. It is equilibrium behavior for all those remaining consumers with reservation prices above P(X) to attempt to purchase from the firms charging P(X). Suppose that an individual consumer with a reservation price above P(X) were to deviate and not attempt to purchase at r= 1. This consumer would not be able to purchase at z =2 since the others will all attempt to purchase at r = 1. There will be no excess capacity available for use at r =2, so it is optimal for all consumers with reservation prices above P(X) to attempt to purchase at the price P(X) at z= 1, if they do not purchase at the price p, first. Since firm j’s output remains at xj the deviation is not profitable, so it is an equilibrium for all firms to quote P(X). 5.2. The role of dedicated capacity The firms’ property rights in their dedicated capacity are crucial for the results here. Suppose that a firm with a strictly lower price could use all of the common capacity immediately, For example, if firm 1 had a strictly lower price, it would have the opportunity to serve all consumers who wished to purchase, before firm 2 had the opportunity to serve anyone. All consumers with reservation prices above pi would attempt to purchase from firm 1, and firm 1 would make all of the sales. Fixing capacities, the only difference between this latter rule and the original use-or-lose provision occurs when p1 < pz < P(x, +x2). Under this latter rule, only firm 1 would make any sales: Qi =x1 +x2 and Q2 =O. The equilibrium prices are always pi =p2 =c in this case, so it can not be optimal for both firms to invest in capacity (with probability one). 5.3. A dynamic price-setting game The price-setting game in section 2 was a simultaneous move game in which firms chose their prices once and for all. Consider the following alternative price-setting game. There are two firms. Firm 1 moves in oddnumbered periods, firm 2 in even-numbered periods. When it is a firm’s turn to move, the firm must quote a price or pass. If the firms pass on successive moves the price-setting game ends. The remainder of the game is exactly the same as in section 2. This game has a structure that would appear to foster competition since a firm always has the opportunity to move after its opponent does. In particular, the price-setting game does not end until neither firm wishes to lower its price any further. In fact, it is equilibrium behavior for firm 1 to quote the monopoly price, for firm 2 to match it, and then for each firm to pass on its next move. The firms earn the monopoly profit in aggregate in
66
1. Gale, Price co~~titt~n
in nonc~~~rative joint ventures
this case. If demand is concave, for example, this outcome is supported by off-the-equilibrium-path strategies that have firms match lower prices that exceed c, and pass otherwise. Now suppose that the price-setting game has a finite horizon. If firm 2 has the last move, and P(X)>c, the results of Lemma 1 indicate that firm 2 should either undercut firm 1 or quote P(X) --E. Knowing this, firm 1 will want to offer a best price of P(X)--&, so the outcome will be the same as in the original game. When P(X)sc there are multiple equilibria. Firm 1 is indifferent among all prices above c since firm 2 will undercut any such price in the last period of a subgame perfect Nash equilibrium. 6. Conclusion
This paper has examined the performance of joint ventures in which firms have the opportunity to use other firms’ excess capacity. The use-or-lose provision was shown to induce socially optimal pricing. Cotenancies may therefore be a useful alternative to direct regulation of natural monopolies. Cotenancies may also be useful as an antitrust remedy, especially when there are large economies of scale in production or in construction of a facility. The benefits of a cotenancy are greatest when open entry can be imposed. As the number of firms that build capacity in a cotenancy grows, the (common) equilibrium price converges to the Ramsey optimal price. Industries that would be good candidates for cotenancies include telecommunications, electricity transmission and natural gas pipeline transportation. But capacity need not refer to physical capital. A use-or-lose provision could be imposed on an industry in which licenses are used to implement quantitative restrictions. Import quota licenses provide one example. A useor-lose provision would counter the hoarding of licenses in such a case. Airport takeoff and landing slots (as well as airport gates) represent additional possibilities. The analysis here has presented a model that puts cotenancies in a favorable light, and some caveats are in order. For example, there could be problems in measuring the costs of production. That is, a regulator may not be able to discern whether marginal cost is being paid when one firm uses another firm’s excess capacity. Analogous points hold for the cost of capacity. It bears repeating, however, that no demand information or subsidies have been used by the regulator here. Appendix A Proof of Lmzma I. Suppose that all prices exceed P(X):
I. Gale. Puce compet~tron
in noncooperatwe
joint
rentures
67
Any consumer with a reservation price of p1 or above is assured of being able to purchase, either initially or after the excess capacity of the higherpriced firms is used, since D(p,)5X. These consumers will all attempt to purchase from the j firms with the lowest price, and those firms will have aggregate output of min ix(j), D(p,)) at z= 1. The remaining consumers, if any, with reservation prices above p1 will be served by those firms at r=2. Since sales are proportional to capacities when firms select the same price, this gives the desired result (1). Now suppose that some prices are strictly below P(X): Pi =??I= . ..=pj
Aggregate output will equal X. If not, some consumers with reservation prices above P(X) do not buy. These consumers would be strictly better off purchasing at r=2 from one of the j firms with the lowest price. The li firms with the lowest prices must all sell output equal to their dedicated capacity at T = 1. Suppose, to the contrary, that firm m 5 k sold less than x, at z= 1. There will exist a positive measure of consumers with reservation prices greater than pm who are certain never to be able to purchase in equilibrium since L&p,)> X. If an individual consumer in this group attempts to purchase from firm m at z= 1, he will be successful and will get a strictly positive surplus. Hence. the first k firms must have output equal to capacity at T = 1. The X consumers with the highest reservation prices will purchase in equilibrium, and consumers with higher reservation prices will purchase at (weakly) lower prices. If not, the consumers with the higher reservation prices could mimic the strategies of the consumers with the lower reservation prices who are served at the lower price. The consumers with the higher reservation prices will then be served first, given the rationing rule. Since the first k firms sell up to capacity at z= 1, the excess capacity available at r=2 is no larger than X--x(k). The best possible outcome for the consumers who will purchase in equilibrium would be for the first consumers to purchase from the first j firms and for the X-C(k)+C(j) remaining consumers of type r 5 X to purchase from firms j+ 1 through k, in order. In fact, this is the equilibrium outcome. The x(j) consumers with the highest reservation prices purchase from the first j firms at T= 1. The X-C(k) consumers with the next highest reservation prices do not purchase at z= 1. Rather, they purchase from the first j firms at T = 2. The next xj+ , consumers purchase from firm j+ 1 at r= 1, and this process continues until km k is reached. Since all consumers of type t>X have reservation prices strictly below P(X), none of them will be able to purchase. Any consumer of type t SX who deviates can only end up paying a higher price or not buying at all. This gives the desired result (2). /-J
68
I. Gale, Price competition in n~~co~~erat~~ joint Dentures
Appendix B Proof oj” Proposition 1. Suppose that P(X)>c. The supremum of the prices on which the firms place strictly positive density must be at least equal to P(X). Raising a price that is strictly below P(X) increases profit, when all other prices are also strictly below P(X), since Qi=xt for all such prices. Either all firms have a supremum of P(X), or some have a supremum strictly above P(X). But a firm whose supremum was strictly greater than the larger of another firm’s supremum and P(X) would, with probability one, make no sales when it charges prices in the interval between its supremum and the larger of these two prices. Hence, the firms must all have the same supremum. Now suppose that the common supremum is strictly greater than P(X) and that no firm places point mass on the common supremum. When a firm selects this price it will get no profit since, with probability one, its sales will be zero. (Likewise, if it selects a price that is arbitrary close to the supremum, its expected profit will be arbitrarily close to zero.) It follows that there must be mass points for all firms at the top of the support. However, a price strictly above P(X) cannot be a mass point for all firms. Moving the mass just below that price gives a firm a discontinuous increase in expected profit. The common supremum must therefore equal P(X). However, if all firms select pi=P(X), then an individual firm has an incentive to undercut. It follows that when P(X)>c, the unique equilibrium has each firm selecting a price just below P(X). It is clearly suboptimal for a firm to make sales at a price below marginal cost. Repeating the arguments above indicates that at least two firms price at marginal cost when P(X) 5 c. 171
Afger, Daniel, 1991, Competitive joint ventures for natural gas pipelines, Mimeo. Alger, Daniel and Susan Braman, 1991, Competitive joint ventures and the regulation of natural monopolies, Mimeo. Baron, David, 1988, The design of regulatory mechanisms and Institutions, in: R. Schmalensee and R. Willig, eds., Handbook of industrial organization (North-Holland, Amsterdam). Demsetz, Harold, 1968, Why regulate utilities?, Journal of Law and Economics I 1, 55-6.5. Kahn. Alfred. 1988. The economics of regulation. Vol. I1 (MIT Press, Cambridge, MA). Krepi, David and jose Scheinkman, 19g3, Quantity pre~ommitment and Bertrand com~tition yield Cournot outcomes, Bell Journal of Economics 14, 326-337. Kwoka, John E., Jr., 1989, International joint venture: General Motors and Toyota, in: John E. Kwoka, Jr. and Lawrence J. White, eds., The antitrust revolution (Scott, Foreman and Company, Glenview, IL). Kwoka, John E., Jr. and Frederick Warren-Boulton, 1986, Efficiencies. failing firms, and alternatives to merger: A policy synthesis, Antitrust Bulletin 31, 431-450. Lewis, Lucinda and Robert ReynoIds, 1979, Appraising alternatives to regulation for natural monopolies, in: Edward J. Mitchell, ed., Oil pipelines and public policy: Analysis of proposals for industry reform and reorganization (American Enterprise Institute, Washington, DC).
1. Gale, Price competition in noncooperative joint ventures
69
Osborne, Martin and Carolyn Pitchik, 1986, Price competition in a capactty-constrained duopoly, Journal of Economic Theory 38,238260. Rassentt, Stephen, Stanley Reynolds and Vernon Smith, 1993, Cotenancy and competitton in an experimental double auction market for natural gas pipeline networks, Economic Theory (forthcommg). Reynolds, Stanley, 1990, Competition and cost sharing among joint venture participants, Mimeo. Warren-Boulton, Frederick and John Woodbury, 1990, The design and evaluation of competittve rules joint ventures for mergers and natural monopolies, Paper presented at the Amencan Enterprrse Institute Conference on Policy Approaches to the Deregulation of Network Industries, l&l 1 Oct.
Journal
of Industrial
Organization
Price competition ventures
12 (1994) 53-69.
North-Holland
in noncooperative
joint
Ian Gale* Department of Economics, University of Wwconsin, Madison. WI 53706, USA Final version
accepted
November
1992
Firms that participate in a cotenancy Joint venture are able to utilize capacity that is left unused by other firms. This paper considers a cotenancy involving two or more firms. In equilibrium, the firms quote prices that are socially optimal given the level of aggregate capacity. When there is open entry, the resulting (common) equilibrium price converges to the Ramsey optimal price. A cotenancy may be of use as an alternative to direct regulation of a natural monopoly or as an antitrust remedy, as it combines the benefits of single plant production with competition at the marketing stage. 1.
Introduction
Many production joint ventures have the feature that participants in the joint venture are direct competitors at the marketing stage. Natural gas pipelines provide one example. Within this class of ‘noncooperative’ joint ventures are joint ventures known as cotenancies.’ In a ‘cost center’ cotenancy, production is overseen by an independent management company. Output and pricing decisions are made independently by the co-owners, who essentially place orders for output with the management company. Whenever economies of scale exist, the most obvious way to capture them is to have a single firm produce. One option is to have an unregulated monopolist, which will lead to an inefficiently low level of output. Another option is to impose direct regulation, which will lead to inefftciencies because of incentive problems [Baron (1988)]. Yet another option is to auction off the exclusive right to serve the market [Demsetz (1968)]. However, the existence of economies of scale does not require that there be a monopoly seller. Joint ventures provide one means of combining the benefits of single plant production with competition at the marketing stage. This paper Correspondence to: Ian Gale, Department of Economics, University of Wisconsm, WI 53706, USA. *I am grateful to seminar participants at the University of Wisconsm, the Antitrust of the U.S. Department of Justice, and the 1992 Econometric Society Meetings in New The comments of Buz Brock, Robert Masson (the editor), Stan Reynolds and Marius have improved this paper. ‘The terms ‘competitively-ruled cotenancies’ and ‘competitive rules joint ventures’ used. 0167-7187/94/$07S!O 0 1996Elscvier SSDI 0167-7187(93)00383-Y
Science B.V. All rights reserved
Madison, Division Orleans. Schwartz are also
54
f. Gate, Price com~ririon in noneoo~rati~e joint ventures
examines the performance of cost center cotenancies in the presence of economies of scale. There is growing support for the use of cotenancies as an alternative to direct regulation of natural monopolies and as an antitrust remedy. The original reference is Lewis and Reynolds (1979). Reynolds (1990) WarrenBoulton and Woodbury (1990), and Alger (1991) all provide in-depth discussions. Alger and Braman (1991) examine a model that is similar in spirit to the one studied here. Rassenti et al. (1993) evaluate cotenancies in an experimental setting. The contribution of the current paper is that it provides a complete analysis of a price-setting game that captures the essence of a cost center cotenancy. The performance of cotenancies as an alternative to direct regulation and as an antitrust remedy can then be examined. The setting for the analysis is the following. Two or more firms jointly own a facility that operates as a cost center cotenancy. For concreteness, one can think of firms that own a natural gas pipeline. Each firm has the right individually to pump an amount of gas equal to its ownership share times the pipeline’s capacity. This absolute quantity is the owner’s ‘dedicated capacity’. The firms are subject to competitive rules, which are intended to enhance rivalry among the firms. These rules are based on rules in force in an actual cotenancy. An aluminum rolling mill that was built by the Atlantic Richfield Company (Arco) operates as a cost center cotenancy. In 1984, Alcan Aluminum agreed to buy Arco’s aluminum operations, but the purchase was challenged by the U.S. Department of Justice. A consent decree was ultimately accepted. It called for the rolling mill to be jointly owned by Alcan and Arco, and to be operated by an independent management company. The most notable of the com~titive rules contained in the consent decree was the following: ‘Each party to the joint venture may utilize any unused portion of the other party’s capacity by assuming the variable costs, but not the fixed costs, attributable to the added production.‘* The game analyzed here captures the impact of such a ‘use-or-lose’ provision. A firm that still faces demand after using its dedicated capacity can then use the other firms’ excess capacity. The analysis is divided into two parts, depending on whether capacity is already in place. When capacity is sunk, imposing a use-or-lose provision leads to prices that are socially optimal given the level of aggregate capacity. When capacity is not sunk, the results are still positive, although not unambiguously so. Consider a two-firm cotenancy and a standard duopoly with separate plants. Suppose that there are economies of scale (i.e. declining long-run average cost). When these economies are small, the cotenants may install less aggregate capacity and generate less total surplus than the ‘U.S. II. Alcan Aluminum, 605
F. Supp.619(1985),p. 625.
I. Gale, Price competition
in noncooperative
55
joint ventures
duopolists. With large economies of scale, the cotenancy will dominate in terms of capacity and total surplus. The results are strongest when there is open entry into the cotenancy. As the number of co-owners increases, the common equilibrium price converges to long-run average cost. That is, there will be Ramsey optimal pricing. 3 Moreover, total cost will be minimized because there is no duplication of facilities. The results suggest that there may be a role for cotenancies as a complement, if not an alternative, to traditional direct regulation. With existing or new capacity, the combination of a use-or-lose provision and open entry leads to a constrained Pareto optimum. The case for cotenancies as an antitrust remedy is somewhat less strong. The difference arises because open entry may not be a reasonable requirement when a cotenancy is used as an antitrust remedy. As noted above, the performance of cotenancies is weaker when open entry is not feasible. The basic price-setting game is described in section 2, and the equilibrium is determined. Section 3 examines the two-stage game in which firms choose capacities first, and then play the price-setting game. These results are used in section 4 to evaluate cotenancies in different regulatory and antitrust settings. Extensions are contained in section 5. 2. The price-setting game There are N firms that participate in a cost center cotenancy that produces a single homogeneous good. The facility has aggregate capacity X=x1 + and firm j has dedicated capacity x,. Firm j has first claim on xz+...+xN, xj units of output, in a sense to be made precise below. There is a (short-run) marginal cost of c up to the capacity of the facility and a fixed cost of production F. An independent management company oversees production. There is a continuum of risk-neutral consumers indexed by t E [0,11. The total measure of consumers is equal to one. A consumer of type t has a reservation price P(t) for a single unit of the good, where P( .) is a continuous, decreasing function. This implies that there is a measure q of consumers with reservation prices of P(q) or above. The ‘market-clearing’ price (i.e. the price at which all of the output can be sold when all firms produce up to capacity) is P(X). Demand is denoted by D( .). The firms announce prices at r =O. Whenever demand exceeds supply, the consumers with the highest reservation prices are served first. This is the efficient rationing rule. Under this rule output is allocated efficiently so there is no possibility of mutually beneficial trade among consumers. (Section 5 contains a discussion of alternative rationing rules.) If two or more firms select the same price, consumers who attempt to purchase from these firms patronize the firms in proportion to their capacities. %nce
there is only a single market,
Ramsey
optimal
pricing
entails average
cost pricing.
56
1. Gale, Price competition in noncooperative joint ventwes
The use-or-lose provision in the Alcan consent decree gave each firm the opportunity to use the other firm’s excess capacity. This is modeled here as follows. There are two sales periods: z= 1 and T=2. At t = 1, the consumers whose reservation prices exceed the lowest price attempt to purchase from the firm (or firms) offering that price.4 That firm fills as many orders as it can, up to the limit imposed by its dedicated capacity, and incurs a cost of c per unit of output. All remaining unserved consumers then decide individually whether to attempt to purchase from the firm with the second-lowest price. (Consumers whose reservation prices exceed the second-lowest price will not buy at z = 1 if they will be able to buy from the low-price firm at z =2.) That firm then fills as many orders as it can. This process continues until the remaining firm with the lowest price makes no sales or else all firms have made sales. The same process takes place at x=2, with one difference.’ Now the firms use the capacity that was not used at t= 1. A firm incurs the marginal cost of c per unit of additional output when it uses the excess capacity. It is assumed throughout that a firm must serve all consumers that it is able to serve. (A firm that has the option of limiting output after announcing its price would never have an incentive to do so in equilibrium.) Subgame perfect Nash equilibrium is the equilibrium concept used here. Strategies must be mutual best responses at each stage of the game. The firms price-setting strategies must be mutual best responses given the strategies of the consumers, while the consumers’ strategies must be mutual best responses given the prices and capacities of the firms. In particular, consumers correctly assess the probability that their orders will be filled, and they behave optimally given these assessments. The first step in characterizing the equilibrium is to determine outputs for all combinations of prices. When all prices are above the market-clearing price only the firm with the lowest price will make sales. Consumers will attempt to purchase from the low-price firm first. Since there is enough capacity in aggregate to serve all who wish to purchase at the lowest price, it is not optimal to purchase from any firm with a strictly higher price. Those consumers who are unable to purchase from the low-price firm at z= 1 will all wait until z =2, at which point the low-price firm has access to the other firms capacity. When some prices are below the market-clearing price, all firms with prices &An earlier version of the paper had consumers submit irrevocable orders to the firms. The firms served as many consumers as they could using their dedicated capacity. Those firms with waiting lists were able to use the other firms’ excess capacity at the next stage, beginning with the firm with the lowest price. The ~uilib~um prices and sales are identical in the two extensive forms. ‘Another interpretation of the sequence of events is that orders are contirmed at z= 1 and T= 2, and then productIon and delivery take place at 7 = 3.
I. Gale, Price competition
in noncooperative
joint
57
ventures
below P(X) make sales while those with prices above P(X) do not. Suppose that N=2 and pi
output for
7 =
1 is
firmi is
Qi=
I
$)‘D(pi),
i=1,2 ,..., j,
0,
i=j+l,j-t-2
(1) ,..., N.
When prices satisfy
output for firm i is
1
Xi+&(X-x(k)),
Qi=
Proof.
i=1,2 ,..., j,
Xi,
i=j+l,
0,
i=k+l,k+2
1
The proof is in appendix A.
q
j+2 ,..., k, ,..., N.
(2)
I. Gale, Price competition in noncooperative joint ventures
58
Relabelling the firms gives the outputs for all permutations of these prices. A firm charging a price above the market-clearing price Z’(X) only makes sales if all other prices are at least as high. But if all firms charge the same price, and it exceeds the market-clearing price, then each firm has an incentive to cut price unilaterally if P(X) > c. If P(X) SC, then the incentive to undercut ends when at least two prices are equal to c. A consequence is that all firms will select the price that is socially optimal given the level of aggregate capacity.6 Proposition 1.
(i=1,2,...,
When
P(X) >c,
the
equilibrium
prices
are
pi=P(X)-E
N). When P(X) 5 c, the equilibrium prices satisfy pi 2 c, and at least
two firms quote pi = c. Proof.
The proof is in appendix B.
0
When the market-clearing price exceeds marginal cost, the firms all quote the market-clearing price. When the market-clearing price is below marginal cost, there is marginal cost pricing. No cost or demand information is required by the regulator, nor are any subsidies required. All that is needed for this result is that the use-or-lose provision be imposed. In a multi-period setting there will be socially optimal pricing in each period. Note also that the result does not depend on the number of firms so open entry (whereby the incumbents are required to sell capacity to entrants at a set price) would have no impact. One recent paper uses a similar model and predicts the same prices. In Alger and Braman (1991) firms can make price offers at any time. Consumers submit orders, which cannot be withdrawn, although consumers can accept a lower price if one is subsequently offered. While the authors claim existence and uniqueness of equilibrium, the extensive form that they examine is not presented explicitly. 3. The capacity-choice
game
The previous sections have examined cotenancies in which firms’ dedicated capacities are sunk. This section takes a step back and looks at capacity choice by firms that form a cotenancy prior to building capacity. This situation could arise as the result of regulatory or antitrust decisions to challenge a proposed cooperative joint venture, for example.’ The outcome ‘%rictly speaking, the firms will quote a price that is infinitesimally below P(X) when P(X)>,. Suppose that there is a smallest unit of currency. The unique pure-strategy equilibrium has all the firms quoting the largest price strictly below P(X) in this case. In what follows, this distinction will be ignored. ‘Most proposed joint ventures above a certain size must undergo prior antitrust review by the Department of Justice or the Federal Trade Commission [Kwoka (1989, p. 49)].
I. Gale, Price competition
in noncooperative
jotnt ventures
59
of the two-stage cotenancy game will be determined first. For purposes of comparison, the equilibrium of the analogous two-stage game with separate facilities will also be determined.
The firms choose their individual investments in capacity at time t= - 1, and then select prices at r=O. The total cost of building x units of capacity is @P(X).It is assumed that the average cost of capacity, @(x)/x, is strictly decreasing, and that the long-run average cost, cf [@(x)/~], cuts P(x) from below exactly once. A rule for allocating the cost of capacity among the firms must also be specified. The rule in the Alcan consent decree governing the allocation of costs of new capacity will be used here. Shares of the capacity costs are equal to the firms’ ownership shares8 In other words, firm i pays the average cost of capacity for its xi units of capacity. A strategy for firm j now consists of a rule for selecting x,, and a rule for selecting pj given the capacities chosen, Consumer behavior is not changed by adding the capacity-choice stage, so the outputs are unchanged for given prices and capacities. Strategies for the price-setting game must be best responses given the capacities chosen, and the capacities chosen must be best responses given the equilibrium of the ensuing price-setting game. The results of section 2 demonstrate that either all firms produce up to their dedicated capacity or else there is (short-run) marginal cost pricing. The latter outcome would give the firms negative profits in the two-stage game, so firms will choose capacities such that they all produce up to capacity.’ Firm 1 selects
when x,=Xj for j # 1. The first-order condition is
8U.S. O. Alcan Aluminum, 605 F. Supp. 619 (198.5). p. 625. ‘It will be assumed that the tixed costs of productlon equal zero. A fixed cost of capacity ~111 have the same effect, smce fixed costs of production are allocated according to ownership shares, not output shares.
JIO
C
60
I. Gale, Price competition
where w~x/(x+~jN_~ order condition is
in noncooperatwe
joint ventures
Zj) is firm l’s ownership share. The symmetric lirst-
o=P’(Nx)x+P(Nx)-c-
N - 1 @(NT) ;@‘(Nx)-N ~NZ
(5)
(If long-run average cost does not decline too quickly, (5) characterizes the unique equilibrium. The focus will be on symmetric equilibria throughout.) When firm 1 expands its investment in capacity it pays only a fraction of the true marginal cost of capacity since it pays only a fraction of the total capacity cost. However, as its ownership share rises, the firm must pay an increased share of the cost of the inframarginal capacity. The net effect is that in equilibrium each firm faces an effective marginal cost of capacity that is a weighted average of the true marginal cost and the true average cost of capacity. The larger N is, the closer the effective marginal cost is to the true average cost.
3.2. The standard duopoly game The standard duopoly game in which two firms have separate plants and are not subject to a use-or-lose provision serves as a benchmark for a twofirm cotenancy. This section characterizes the equilibrium of the duopoly game that is analogous to the cotenancy just examined. The firms quote prices at r =O. At r = 1, consumers purchase. They first attempt to purchase from the low-price firm, as before. The difference is that there is only one round of purchases here, so neither firm has the opportunity to use the other firm’s excess capacity. Suppose that pi
P(2cz) -c - g(3).
(6)
I. Gale,
Price
competrtion
in noncooperative
joint
ventures
61
4. The prospects for cotenancy This section uses the analysis of the previous two sections to evaluate the prospects for employing cotenancies as an alternative to traditional direct regulation and as an antitrust remedy. The different contexts have different reference points for measuring performance and, as a consequence, they will have different rules governing entry. In a regulatory setting, the appropriate standard is Ramsey optima1 pricing since a regulator with complete cost and revenue information could induce a monopolist to price at long-run average cost. In this case it would be reasonable to impose open entry on the cotenancy. In an antitrust setting, the reference point is the pre-merger level of competition. Open entry would not be a reasonable requirement in this case, and only the parties to the proposed merger would typically be involved [Warren-Boulton and Woodbury (1990)]. The case of sunk capacity is examined first. 4.1. Existing
capacit?
When capacity is sunk, forming a cotenancy simply means imposing a use-or-lose provision here. It was shown in section 2 that this action enhances competition and leads to socially optimal pricing, with or without open entry. A cotenancy arrangement would therefore appear to offer benefits as an alternative to traditional direct regulation of natural monopolies. This analysis is also relevant to the opening and closing of duplicate plants. Suppose that two companies wish to merge and close one of their two pipelines. This will reduce the total fixed costs of production but will also eliminate rivalry at the marketing stage. Imposing a use-or-lose provision on the remaining pipeline would maintain rivalry. Since this case involves a net reduction in aggregate capacity, there may be a reduction in consumers’ surplus, which must be weighed against the reduction in fixed costs. At the same time, consumers’ surplus could actually rise if the pipeline that is closed down is sufficiently small. Now consider the case of an incumbent monopolist and a potential entrant. Suppose that the incumbent currently has excess capacity. The incumbent and the entrant would both prefer to merge rather than have the entrant build new capacity and compete as a duopolist. However, a cotenancy could be welfare-superior to the duopoly outcome, in addition to being superior to the merger. In particular, if the economies of scale are sufficiently great, cotenancy will generate strictly more total surplus than will the duopoly. lo “‘Another possibility, which is not treated here, is that economies of scope are realized the operations are merged. Kwoka and Warren-Boulton (1986) discuss this issue.
when
62
I. Gale, Price competition in noncooperative joint ventures
4.2. New capacity
A cotenancy could be formed prior to the construction of capacity, either in a regulatory setting or in an antitrust setting. For example, two firms might propose a cooperative joint venture. Antitrust authorities have several options. They could approve the joint venture, they could challenge it but suggest that the firms submit independent proposals instead, or they could challenge it but suggest that the firms form a cotenancy. (In an antitrust setting, open entry would typically not be an appropriate remedy for a proposed cooperative joint venture involving two firms.) The monopoly and duopoly equilibria will then be used as benchmarks for a two-firm cotenancy here. It could also happen that two firms would submit independent proposals to a regulator to build plants in a new market. The regulator could approve one or both of the firms’ proposals, or she could turn down both but suggest that the firms form a cotenancy instead. I1 The regulator could also impose open entry on the cotenancy, which would allow new firms to enter by paying the average cost of capacity for their capacity. The case without open entry is examined first. The cotenancy dominates monopoly, even when the cotenants face a higher effective marginal cost of capacity. The first-order condition for the cotenants in the capacity-choice game is
which can be rewritten as o=[p’(2x)2x+P(2x)-c-q2x)]+P(2x)-c-
@(2X) x.
(8)
The last three terms, take together, equal price minus long-run average cost. This must clearly be positive in the cotenancy equilibrium, so the term in square brackets must be negative. This means that when aggregate capacity is at the level where the cotenancy first-order condition is satisfied, the monopolist’s profit would be declining in capacity. Hence, capacity and total surplus are greater under cotenancy. While a two-firm cotenancy dominates monopoly, the following example demonstrates that total surplus can be lower in the cotenancy than in a duopoly. Suppose that demand is weakly concave and that the cost of capacity is Q(x) = &, + $x for x 5 x* and Q(x) = $*x for x > x*, where 4* > 4 “The idea that a regulator might suggest that proposals be combined mto a joint venture in order to capture the economies of scale goes back at least to the Northern Natural Gas case of 1968. See Kahn (1988, p. 160) for a dlscussion.
I. Gale, Price competition
in noncooperatl
d joEnt ventures
63
and #*=x*(#* - 4). The average cost of capacity is continuous, strictly decreasing for x < x*, and constant thereafter. Now suppose that the duopolists have an interior maximum right at x* in the capacity-choice game. The associated first-order condition is 0 = P’f2x.*)x* + P(2X*) - c- rfi_
(9
The first-order condition in the cotenancy (presuming that aggregate capacity is at least x*) is 0 = p’(23i + P(2.Z)- c - 4*_
(10)
Since @+> Ip, the equilibrium aggregate capacity will be larger in the duopoly. Consumers’ surplus will also be larger under duopoly, as will total surplus, since the duopolists are producing at the minimum of long-run average cost. It is then straightforward to construct an example in which total surplus is lower under cotenancy, even though long-run average cost is strictly decreasing everywhere. Aggregate capacity and consumers’ surplus can be lower in the cotenancy even when the true marginal cost of capacity is decreasing. For example: if G(x)=& +4x for all x, then the cotenants still face a higher effective marginaf cost of capacity than the duopolists do since the true average cost af capacity always exceeds the true marginal cost. Capacity would again be lower under cotenancy. Examples can be constructed in which the cotenants face a higher effective marginal cost of capacity, even though the true marginal cost is itself strictly decreasing. Total surplus can be higher in the cotenancy, even though consumers’ surplus is lower, since the duopolists do not produce at the minimum of long-run average cost here. To this point some extreme examples have shown that a cotenancy need not generate greater total surplus than a duopofy. When the economies of scale in plant construction are great, the effective marginal cost of capacity in the cotenancy will be below the corresponding marginal cost in the duopoly. In this case aggregate capacity will be larger under cotenancy, so total surplus will necessarily be larger as well.” When open entry is feasible, another method of generating greater total surplus is to alfow open entry. Consider the first-order condition in the capacity-choice game with N firms. As N increases, the equilibrium dedicated capacity of the individual cotenants converges to zero, and the first-order condition (5) converges to the follo~ng expression:
‘?here may be multiple equilibria of the twa-stage duopoly game when the marginal cost of capacity is strictly decreasing. However, any equilibrium of the two-stage game in which the firms use pure strategtes in seiecting quantities is ako a pure-strategy Cournot equilibrium [Osborne and Pitch& [19&Q].
64
I. Gale, Price ci~~petition in noncoo~rar~ve joint ventures
O=P(NZ)--c-F.
(11)
(The dependence of X on N is implicit.) Since long-run average cost cuts the demand curve from below exactly once, there is a unique level of aggregate capacity at which (11) is satisfied. As the number of cotenants grows, the equilibrium price converges to the Ramsey optimal price, i.e. long-run average cost. The regulator does not need to know the cost of capacity or the demand curve. All that is required is that the use-or-lose provision be imposed along with open entry. The most important result is that there is always room for more firms in a cotenancy, in the following sense. The use-or-lose provision, in conjunction with open entry, increases output, but not at the expense of economies of scale. Entry brings the benefit of increased competition at the marketing stage, without forsaking economies of scale in production or plant construction.
5. Extensions This section examines various assumptions form game.
made in the basic extensive
5.1. Different rationing rules
Consider the impact of different rationing rules. Suppose first that all prices strictly exceed P(X), in which case there is enough capacity in aggregate to serve all who wish to purchase at the lowest price. Under any rationing rule, consumers will only purchase from the firm(s) with the lowest price. It follows that whenever P(X)sc, at least two firms quote a price equal to marginal cost and all purchases will be made at that price. Likewise, whenever P(X) > c, there cannot be an equilibrium in which all prices strictly exceed P(X). There is always at least one firm with an incentive to cut price unilaterally and make all of the sales, under any rationing rule. Consider random rationing in particular and suppose that P(X)>c. It is equilibrium behavior here as well for all firms to quote pi= P(X). Suppose that all firms select the market-clearing price. No firm has an incentive to raise its price unilaterally, since its sales would drop to zero. Now suppose that firm j deviates unilaterally and quotes a price pj
I. Gale, Price competition
in noncooperative
joint
ventures
65
prices above P(X). Since X--xi is the amount of capacity remaining, there are more of these consumers remaining than there is capacity. It is equilibrium behavior for all those remaining consumers with reservation prices above P(X) to attempt to purchase from the firms charging P(X). Suppose that an individual consumer with a reservation price above P(X) were to deviate and not attempt to purchase at r= 1. This consumer would not be able to purchase at z =2 since the others will all attempt to purchase at r = 1. There will be no excess capacity available for use at r =2, so it is optimal for all consumers with reservation prices above P(X) to attempt to purchase at the price P(X) at z= 1, if they do not purchase at the price p, first. Since firm j’s output remains at xj the deviation is not profitable, so it is an equilibrium for all firms to quote P(X). 5.2. The role of dedicated capacity The firms’ property rights in their dedicated capacity are crucial for the results here. Suppose that a firm with a strictly lower price could use all of the common capacity immediately, For example, if firm 1 had a strictly lower price, it would have the opportunity to serve all consumers who wished to purchase, before firm 2 had the opportunity to serve anyone. All consumers with reservation prices above pi would attempt to purchase from firm 1, and firm 1 would make all of the sales. Fixing capacities, the only difference between this latter rule and the original use-or-lose provision occurs when p1 < pz < P(x, +x2). Under this latter rule, only firm 1 would make any sales: Qi =x1 +x2 and Q2 =O. The equilibrium prices are always pi =p2 =c in this case, so it can not be optimal for both firms to invest in capacity (with probability one). 5.3. A dynamic price-setting game The price-setting game in section 2 was a simultaneous move game in which firms chose their prices once and for all. Consider the following alternative price-setting game. There are two firms. Firm 1 moves in oddnumbered periods, firm 2 in even-numbered periods. When it is a firm’s turn to move, the firm must quote a price or pass. If the firms pass on successive moves the price-setting game ends. The remainder of the game is exactly the same as in section 2. This game has a structure that would appear to foster competition since a firm always has the opportunity to move after its opponent does. In particular, the price-setting game does not end until neither firm wishes to lower its price any further. In fact, it is equilibrium behavior for firm 1 to quote the monopoly price, for firm 2 to match it, and then for each firm to pass on its next move. The firms earn the monopoly profit in aggregate in
66
1. Gale, Price co~~titt~n
in nonc~~~rative joint ventures
this case. If demand is concave, for example, this outcome is supported by off-the-equilibrium-path strategies that have firms match lower prices that exceed c, and pass otherwise. Now suppose that the price-setting game has a finite horizon. If firm 2 has the last move, and P(X)>c, the results of Lemma 1 indicate that firm 2 should either undercut firm 1 or quote P(X) --E. Knowing this, firm 1 will want to offer a best price of P(X)--&, so the outcome will be the same as in the original game. When P(X)sc there are multiple equilibria. Firm 1 is indifferent among all prices above c since firm 2 will undercut any such price in the last period of a subgame perfect Nash equilibrium. 6. Conclusion
This paper has examined the performance of joint ventures in which firms have the opportunity to use other firms’ excess capacity. The use-or-lose provision was shown to induce socially optimal pricing. Cotenancies may therefore be a useful alternative to direct regulation of natural monopolies. Cotenancies may also be useful as an antitrust remedy, especially when there are large economies of scale in production or in construction of a facility. The benefits of a cotenancy are greatest when open entry can be imposed. As the number of firms that build capacity in a cotenancy grows, the (common) equilibrium price converges to the Ramsey optimal price. Industries that would be good candidates for cotenancies include telecommunications, electricity transmission and natural gas pipeline transportation. But capacity need not refer to physical capital. A use-or-lose provision could be imposed on an industry in which licenses are used to implement quantitative restrictions. Import quota licenses provide one example. A useor-lose provision would counter the hoarding of licenses in such a case. Airport takeoff and landing slots (as well as airport gates) represent additional possibilities. The analysis here has presented a model that puts cotenancies in a favorable light, and some caveats are in order. For example, there could be problems in measuring the costs of production. That is, a regulator may not be able to discern whether marginal cost is being paid when one firm uses another firm’s excess capacity. Analogous points hold for the cost of capacity. It bears repeating, however, that no demand information or subsidies have been used by the regulator here. Appendix A Proof of Lmzma I. Suppose that all prices exceed P(X):
I. Gale. Puce compet~tron
in noncooperatwe
joint
rentures
67
Any consumer with a reservation price of p1 or above is assured of being able to purchase, either initially or after the excess capacity of the higherpriced firms is used, since D(p,)5X. These consumers will all attempt to purchase from the j firms with the lowest price, and those firms will have aggregate output of min ix(j), D(p,)) at z= 1. The remaining consumers, if any, with reservation prices above p1 will be served by those firms at r=2. Since sales are proportional to capacities when firms select the same price, this gives the desired result (1). Now suppose that some prices are strictly below P(X): Pi =??I= . ..=pj
Aggregate output will equal X. If not, some consumers with reservation prices above P(X) do not buy. These consumers would be strictly better off purchasing at r=2 from one of the j firms with the lowest price. The li firms with the lowest prices must all sell output equal to their dedicated capacity at T = 1. Suppose, to the contrary, that firm m 5 k sold less than x, at z= 1. There will exist a positive measure of consumers with reservation prices greater than pm who are certain never to be able to purchase in equilibrium since L&p,)> X. If an individual consumer in this group attempts to purchase from firm m at z= 1, he will be successful and will get a strictly positive surplus. Hence. the first k firms must have output equal to capacity at T = 1. The X consumers with the highest reservation prices will purchase in equilibrium, and consumers with higher reservation prices will purchase at (weakly) lower prices. If not, the consumers with the higher reservation prices could mimic the strategies of the consumers with the lower reservation prices who are served at the lower price. The consumers with the higher reservation prices will then be served first, given the rationing rule. Since the first k firms sell up to capacity at z= 1, the excess capacity available at r=2 is no larger than X--x(k). The best possible outcome for the consumers who will purchase in equilibrium would be for the first consumers to purchase from the first j firms and for the X-C(k)+C(j) remaining consumers of type r 5 X to purchase from firms j+ 1 through k, in order. In fact, this is the equilibrium outcome. The x(j) consumers with the highest reservation prices purchase from the first j firms at T= 1. The X-C(k) consumers with the next highest reservation prices do not purchase at z= 1. Rather, they purchase from the first j firms at T = 2. The next xj+ , consumers purchase from firm j+ 1 at r= 1, and this process continues until km k is reached. Since all consumers of type t>X have reservation prices strictly below P(X), none of them will be able to purchase. Any consumer of type t SX who deviates can only end up paying a higher price or not buying at all. This gives the desired result (2). /-J
68
I. Gale, Price competition in n~~co~~erat~~ joint Dentures
Appendix B Proof oj” Proposition 1. Suppose that P(X)>c. The supremum of the prices on which the firms place strictly positive density must be at least equal to P(X). Raising a price that is strictly below P(X) increases profit, when all other prices are also strictly below P(X), since Qi=xt for all such prices. Either all firms have a supremum of P(X), or some have a supremum strictly above P(X). But a firm whose supremum was strictly greater than the larger of another firm’s supremum and P(X) would, with probability one, make no sales when it charges prices in the interval between its supremum and the larger of these two prices. Hence, the firms must all have the same supremum. Now suppose that the common supremum is strictly greater than P(X) and that no firm places point mass on the common supremum. When a firm selects this price it will get no profit since, with probability one, its sales will be zero. (Likewise, if it selects a price that is arbitrary close to the supremum, its expected profit will be arbitrarily close to zero.) It follows that there must be mass points for all firms at the top of the support. However, a price strictly above P(X) cannot be a mass point for all firms. Moving the mass just below that price gives a firm a discontinuous increase in expected profit. The common supremum must therefore equal P(X). However, if all firms select pi=P(X), then an individual firm has an incentive to undercut. It follows that when P(X)>c, the unique equilibrium has each firm selecting a price just below P(X). It is clearly suboptimal for a firm to make sales at a price below marginal cost. Repeating the arguments above indicates that at least two firms price at marginal cost when P(X) 5 c. 171
Afger, Daniel, 1991, Competitive joint ventures for natural gas pipelines, Mimeo. Alger, Daniel and Susan Braman, 1991, Competitive joint ventures and the regulation of natural monopolies, Mimeo. Baron, David, 1988, The design of regulatory mechanisms and Institutions, in: R. Schmalensee and R. Willig, eds., Handbook of industrial organization (North-Holland, Amsterdam). Demsetz, Harold, 1968, Why regulate utilities?, Journal of Law and Economics I 1, 55-6.5. Kahn. Alfred. 1988. The economics of regulation. Vol. I1 (MIT Press, Cambridge, MA). Krepi, David and jose Scheinkman, 19g3, Quantity pre~ommitment and Bertrand com~tition yield Cournot outcomes, Bell Journal of Economics 14, 326-337. Kwoka, John E., Jr., 1989, International joint venture: General Motors and Toyota, in: John E. Kwoka, Jr. and Lawrence J. White, eds., The antitrust revolution (Scott, Foreman and Company, Glenview, IL). Kwoka, John E., Jr. and Frederick Warren-Boulton, 1986, Efficiencies. failing firms, and alternatives to merger: A policy synthesis, Antitrust Bulletin 31, 431-450. Lewis, Lucinda and Robert ReynoIds, 1979, Appraising alternatives to regulation for natural monopolies, in: Edward J. Mitchell, ed., Oil pipelines and public policy: Analysis of proposals for industry reform and reorganization (American Enterprise Institute, Washington, DC).
1. Gale, Price competition in noncooperative joint ventures
69
Osborne, Martin and Carolyn Pitchik, 1986, Price competition in a capactty-constrained duopoly, Journal of Economic Theory 38,238260. Rassentt, Stephen, Stanley Reynolds and Vernon Smith, 1993, Cotenancy and competitton in an experimental double auction market for natural gas pipeline networks, Economic Theory (forthcommg). Reynolds, Stanley, 1990, Competition and cost sharing among joint venture participants, Mimeo. Warren-Boulton, Frederick and John Woodbury, 1990, The design and evaluation of competittve rules joint ventures for mergers and natural monopolies, Paper presented at the Amencan Enterprrse Institute Conference on Policy Approaches to the Deregulation of Network Industries, l&l 1 Oct.