A Theory of Bilateral Oligopoly

A THEORY OF BILATERAL OLIGOPOLY

KENNETH HENDRICKS and R. PRESTON MCAFEE∗

In horizontal mergers, concentration is often measured with the Hirschman– Herfindahl Index (HHI). This index yields the price–cost margins in Cournot competition. In many modern merger cases, both buyers and sellers have market power, and indeed, the buyers and sellers may be the same set of firms. In such cases, the HHI is inapplicable. We develop an alternative theory that has similar data requirements as the HHI, applies to intermediate good industries with arbitrary numbers of firms on both sides, and specializes to the HHI when buyers have no market power. The more inelastic is the downstream demand, the more captive production and consumption (not traded in the intermediate market) affects price–cost margins. The analysis is applied to the merger of the gasoline refining and retail assets of Exxon and Mobil in the western United States. (JEL L13, L41)

I. INTRODUCTION in a bilateral oligopoly market in evaluating the competitiveness of the market? Merging a net The seven largest refiners of gasoline on buyer with a net seller produces a more bal- the west coast of the United States account anced firm, bringing what was formerly traded for over 95% of the production of California in the intermediate good market inside the firm. Air Resources Board (CARB) certified gasoline Will this vertical integration reduce the exercise sold in the region. The seven largest brands of of market power and produce a more compet- gasoline also account for over 97% of retail sales itive upstream market? Or will the vertically of gasoline. Thus, the wholesale gasoline market integrated firm restrict supply to other nonin- on the west coast is composed of a number tegrated buyers, particularly if they are rivals in of large sellers and large buyers who compete the downstream market? against each other in the downstream retail There is a voluminous theoretical literature market. What will be the effect of a merger of that addresses these questions. Most of the vertically integrated firms on the wholesale and literature considers situations in which one or retail markets? This question has relevance with two sellers supply one or two buyers who the mergers of Chevron and Texaco, Conoco and compete in a downstream market and models Phillips, Exxon and Mobil, and BP/Amoco and their interactions as a bargaining game.1 Sellers Arco, all of which have been completed in the negotiate secret contracts with buyers specifying past decade. a quantity to be purchased and transfers to When monopsony or oligopsony faces an be paid by the buyer. The bilateral bargaining oligopoly, most analysts consider that the need in these models is efficient, so there is no for protecting buyers from the exercise of mar- distortion in the wholesale market. Gans (2007) ket power is mitigated by the market power of uses the model of vertical contracting to derive the buyers and vice versa. Thus, even when the buyers and sellers are separate firms, an analysis 1. See Rey and Tirole (2008) for a survey of this based on dispersed buyers or dispersed sellers literature. Several of the main papers in this literature is likely to err. How should antitrust authori- are Hart and Tirole (1990), McAfee and Schwartz (1994), ties account for the power of buyers and sellers O’Brien and Shaffer (1992), Segal (1999), and de Fontenay and Gans (2005). *We thank Jeremy Bulow and Paul Klemperer for their useful remarks and for encouraging us to explore downstream concentration. ABBREVIATIONS Hendricks: Department of Economics, University of Texas CARB: California Air Resources Board at Austin, Austin TX 78712. HHI: Hirschman–Herfindahl Index McAfee: Yahoo! Research, 3333 Empire Blvd., Burbank, CA MHI: Modified Hirschman–Herfindahl Index 91504. Phone 818-524-3290, Fax 818-524-3102, Email [email protected]

391 Economic Inquiry doi:10.1111/j.1465-7295.2009.00241.x (ISSN 0095-2583) Online Early publication November 12, 2009 Vol. 48, No. 2, April 2010, 391–414 © 2009 Western Economic Association International 392 ECONOMIC INQUIRY a concentration index that measures the amount Moreover, our model will suffer from the same of distortion in the vertical chain as a result of flaws as the Cournot model in its application both horizontal concentration among buyers and to antitrust analysis. Elasticities are treated as sellers and the degree of vertical integration. constants when they are not, and the relevant However, the vertical contracting models do elasticities are taken as known. However, the not describe intermediate good markets like the analysis can be applied to markets with arbitrary wholesale gasoline market in the western United numbers of sellers and buyers, who individually States. The market consists of more than two have the power to influence price, and buyers sellers and two buyers, and trades occur at a who may compete against each other in a down- fixed, and observable, price. Other papers study stream market. The analysis is simple to apply, vertical mergers by assigning the market power and permits the calculation of antitrust effects in either to buyers or to sellers, but not both.2 a practical way. These models are excellent for assessing some Our approach is based on the Klemperer and economic questions, including the incentive to Meyer (1989) market game. In their model, sell- raise rival’s cost, the effects of contact in several ers submit supply functions and behave strate- markets, or the consequences of refusals-to-deal. gically, buyers are passive and report their true But they do not address the implications of demand curves, and price is set to clear the mar- bilateral market power that we wish to study ket. We allow the buyers to behave strategically in this paper. in submitting their demand functions, and apply Traditional antitrust analysis presumes dis- a similar concept of equilibrium as Klemperer persed buyers. Given such an environment, the and Meyer. As is well known, supply function Cournot model (quantity competition) suggests models have multiple equilibria. Klemperer and that the Hirschman–Herfindahl Index (HHI), Meyer (1989) reduce the multiplicity by intro- which is the sum of the squared market shares of ducing stochastic demand, and they show that, the firms, is proportional to the price–cost mar- if the support is unbounded, then the equilib- gin, which is the proportion of the price that rium is unique. More recently, Holmberg (2004, is a markup over marginal cost. Specifically, 2005) has shown that capacity constraints and the HHI divided by the elasticity of demand a price cap can lead to uniqueness. A number equals the price–cost margin. The HHI is zero of authors (e.g., Turnbull 1981; Green 1996, for perfect competition and one for monopoly. 1999; and Akgun 2005) obtain uniqueness by The HHI has the major advantage of simplicity restricting the supply schedules to be linear. Our and low data requirements. In spite of well- approach is similar but we do not require lin- publicized flaws, the HHI continues to be the earity. In our model, sellers can select from workhorse of concentration analysis and is used a one-parameter family of nonlinear schedules by both the U.S. Department of Justice and the indexed by production capacity, and buyers can Federal Trade Commission. The HHI is inap- select from a one-parameter family of nonlinear plicable, however, to markets where the buyers schedules indexed by consumption or retailing are concentrated, particularly if they compete in capacity. Thus, sellers can exaggerate their costs a downstream market. by reporting a capacity that is less than it in fact Our objective in this paper is to offer an is, and buyers can understate demand. The main alternative to the HHI analysis that applies to advantage of restricting the selection of sched- homogenous good markets with linear pricing ules is that it allows us to study the strategic where buyers are concentrated and with (i) sim- interaction between sellers and buyers. ilar informational requirements, (ii) the Cournot In a traditional assessment of concentration model as a special case, and (iii) an underly- according to the U.S. Department of Justice ing game as plausible as the Cournot model. Merger guidelines, the firms’ market shares The model we offer suffers from the same are intended, where possible, to be shares of flaws as the Cournot model. It is highly styl- capacity. This is surprising in light of the fact ized and static. It uses a “black box” pricing that the Cournot model does not suggest the use mechanism motivated by the Cournot analysis. of capacity shares in the HHI, but rather the share of sales in quantity units (not revenue). 2. See, for example, Hart and Tirole (1990), Ordover, Like the Cournot model, the present study Saloner, and Salop (1990), Salinger (1988), Salop and suggests using the sales data, rather than the Scheffman (1987), Bernheim and Whinston (1990). An alternative to assigning the market power to one side of capacity data, as the measure of market share. the market is Salinger’s sequential model. Capacity plays a role in our theory, and indeed HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 393 a potential test of the theory is to check that demand, so markups are lower in our model than actual capacities, where observed, are close to in the Cournot model. Furthermore, the rivals’ the capacities consistent with the theory. responses are determined by their marginal The merger guidelines assess the effect of the costs, so markups depend upon cost elasticities merger by summing the market shares of the as well as the demand elasticity. Larger firms merging parties.3 Such a procedure provides a have higher markups, and markups are higher in useful approximation, but is inconsistent with markets where marginal costs are steep. Struc- the theory (either Cournot or our theory), since tural models of vertically related markets typ- the theory suggests that, if the merging parties’ ically estimate markups under the assumption shares do not change, then the prices are unlikely that sellers post prices that buyers take as fixed to change as well. We advocate a more com- in a sequential vertical-pricing game.5 In our putationally intensive approach, which involves model, sellers and buyers move simultaneously estimating the capacities of the merging parties and the division of rents from market power from the pre-merger market share data. Given depends upon cost and demand functions. We those capacities, we then estimate the effect of investigate the conditions under which the first- the merger on the industry, taking into account order conditions of our model can be used to the incentive of the merged firm to restrict out- estimate demand and cost parameters. put (or demand, in the case of buyers). Horizon- Our model also provides an interesting alter- tal mergers among sellers in intermediate input native for studying spot electricity markets. The market, where buyers are manufacturing firms, operation of these markets closely resembles are more likely to raise price and be profitable our market game: generating firms submit sup- in our model than in the Cournot model because ply schedules, buyers report the demands of capacity reports of sellers are typically strategic their retail customers who face regulated prices, complements, not strategic substitutes. In whole- and an independent system operator chooses the sale markets, the buyers are retailers, and they spot price to equate reported supply to market typically respond to enhanced seller power by demand. Green and Newberry (1992) was the reducing their reported demand, thereby miti- first study of wholesale markets for electricity gating the effects of the merger and complicat- as a game in which firms’ strategies are their ing its impact on prices and profits. The model choices of supply functions.6 Empirical stud- treats horizontal mergers by buyers symmetri- ies of these markets have applied supply func- cally. Vertical mergers in our model generate tion models or Cournot models to predict the large efficiency gains because they eliminate potential for generating firms to exercise market two “wedges,” the markup by the seller and the power in spot electricity markets and to estimate markdown by the buyer. Foreclosure effects are their markups.7 Our model generates a markup important when the merging firms are large. equation that combines the advantages of the Structural models of homogenous good mar- supply function model with the simplicity of the kets with dispersed buyers use an ad hoc mod- Cournot model. ification of the Cournot first-order conditions to The second section presents a model of inter- estimate seller markups. They find that markups mediate good markets and derives the equilib- 4 are typically much lower than Cournot markups rium price/cost margins and the value/price mar- and attribute this finding to the Cournot model’s gins, which is the equivalent for buyers, for failure to account for a firm’s expectations of vertically separated markets and for vertically how rivals will respond to its output choices. In our model, each firm understands that reductions in its supply will be partially offset by increases 5. Several recent studies include Goldberg and Verboven (2001), Villas-Boas and Zhao (2005), and Villas-Boas and in the outputs of its rivals. Their response Hellerstein (2006). means that the elasticity of each firm’s resid- 6. Other studies include Anderson and Philpott (2002), ual demand function exceeds the elasticity of Anderson and Xu (2002, 2005), and Baldick et al. (2004). 7. The studies of markups include Borenstein and Bushnell (1999), Borenstein, Bushnell, and Wolak (2003), 3. Farrell and Shapiro (1990) and McAfee and Williams and Wolak (2003) on California; Hortacsu and Puller (2008) (1992) independently criticize the Cournot model while on Texas; Mansur (2007), Bushnell, Mansur, and Saravia using a Cournot model to address the issue. (2008) on California, New England, and Pennsylvania, New 4. See Bresnahan (1989) for a description of the method- Jersey, Maryland (PJM); Green and Newbery (1992), Green ology and a survey of a number of empirical studies. More (1996), Brunekreeft (2001), Sweeting (2007), and Wolfram recent studies include Genesove and Mullin (1998) and Clay (1999) on England and New Wales; and Wolak (2000, 2003, and Troesken (2003). 2007) on Australia. 394 ECONOMIC INQUIRY integrated markets. The third section extends oligopoly. A market with no vertically integrated the model to spot markets in electricity and firms is called a vertically separated market. wholesale markets in which buyers compete in a These markets can be further decomposed into downstream market. The fourth section analyzes oligopoly markets, in which sellers have market horizontal and vertical mergers. The fifth section power and buyers do not, and oligopsony mar- examines identification issues that would arise in kets, in which buyers have market power but trying to apply our model to market data. The sellers do not. A market with one or more ver- sixth section applies the model to the merger tically integrated firms is a vertically integrated of the west coast assets of Exxon and Mobil to market. illustrate the plausibility and applicability of the In what follows, we will need to distinguish theory. The final section concludes. between two kinds of intermediate good markets based on the type of buyer. Markets in which buyers are manufacturing firms are called II. INTERMEDIATE GOOD MARKETS intermediate input markets. In these markets, We begin with a standard model of a market a buyer j combines the intermediate input qj for a homogenous intermediate good Q.There with capacity kj using a constant returns to scale are n firms, indexed by i from 1 to n. Each seller technology F(qj ,kj ) to produce a good yj that 10 i produces output xi using a constant returns to it sells at price r. Thus, its revenue function scale production function with fixed capacity γi . can be expressed as Thus, seller i’s production costs take the form = qi x V(qj ,kj ) rki f . C(x , γ ) = γ c i , ki (1) i i i γ i A manufacturing firm that has twice the capacity where c(·) is convex and strictly increasing.8 of another firm can produce twice as much at the Each buyer j consumes intermediate output same average productivity. qj and values that consumption according to Markets in which buyers are retail firms are a function V(qj ,kj ) where kj is buyer j’s called wholesale markets. In these markets, a capacity for processing the intermediate output. buyer j purchases qj to resell to final consumers = We assume that V is homogenous of degree one at price r.Hereyj qj and firm j’s valuation so that it can be expressed as of qj is given by qj = − qj (2) V(qj ,kj ) = kj v , V(qj ,kj ) rqj kj w k kj j where v(·) is concave and strictly increasing.9 qj qj = kj r − w A firm may be both a seller and a buyer, that kj kj is, it may produce the intermediate good and also consume it. Such firms are called vertically where w represents unit selling costs. A retailer integrated, although they may be net sellers or with twice as much selling capacity (e.g., num- net buyers. Letting p denote the market-clearing ber of stores) can sell twice as much at the same price in the intermediate good market, the profits unit cost. to a vertically integrated firm i are given by In the Cournot model of oligopoly markets, sellers submit quantities and the market chooses (3) price to equate reported supply to demand. q x The equilibrium price is equal to each buyer’s π = p (x − q ) + k v i − γ c i . i i i i k i γ marginal willingness to pay, but exceeds each i i seller’s marginal cost of supply. In the standard The profits to a firm if it is either a pure seller or oligopsony model, buyers submit quantities and a pure buyer can be obtained by setting qi = 0 the market chooses price to equate reported sup- or xi = 0, respectively. ply and demand. The equilibrium price is equal Markets in which both sellers and buy- to each seller’s marginal cost, but exceeds each ers exercise market power are called bilateral buyer’s true willingness to pay. In each of these

8. In addition, we assume that c (z) −→ ∞ as z −→ ∞ . 10. The production function could include other inputs 9. In addition, we assume v (z) −→ 0asz −→ ∞ . provided their quantities are proportional to qi . HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 395 models, one side of the market is passive and balance equation, the other side behaves strategically, anticipat- n n ing the market-clearing mechanism in order to = = = manipulate prices. Our interest, however, is in a (4) Q qi xi X market where both buyers and sellers recognize i=1 i=1 their ability to unilaterally influence the price and the marginal conditions, and behave strategically. In order to model this type of market, we extend the Klemperer and (5) Meyer model, in which sellers behave strategi- q x cally in submitting supply schedules, to allow j = = i = v p c γ ,i,j 1, .., n. buyers to behave strategically in submitting their kj i demand schedules. In adopting this model, however, we impose Note that, if everyone tells the truth, then the some restrictions on the schedules that the firms equilibrium outcome is efficient. Our model can can report. Sellers have to submit cost schedules be viewed as turning the market into a black which come in the form γc(x/γ), and buyers box, as in fact happens in the Cournot model, have to submit valuation functions which come where the price formation process is not mod- in the form kv(q/k). In a mechanism design eled explicitly. Given this black box approach, framework, agents can lie about their type, but it seems appropriate to permit the market to be they cannot invent an impossible type. The efficient when agents do not, in fact, exercise admissible types in our model satisfy (1) and unilateral power. Such considerations dictate the (2), and agents are assumed to be bound by competitive solution, given the reported types. this message space. For sellers, the message Any other assumption would impose inefficiencies in the market mechanism, rather than having space is a one-parameter family of schedules inefficiencies arise as the consequence of the indexed by production capacity, and for buyers, rational exercise of market power by firms with the message space is a one-parameter family significant market presence. of schedules indexed by consumption capacity. Each firm anticipates the market mecha- Therefore, a seller’s type is a capacity γ,a nism’s decision rule in submitting its reports. buyer’s type is a capacity k, and if the firm From Equation (4), it follows that is vertically integrated, its type is a pair of capacities (γ,k). Agents (simultaneously) report ki Q γiQ their types to the market mechanism, but in (6) qi = ,xi = , doing so, they do not have to tell the truth. K A seller can exaggerate its costs by reporting a where capacity γ that is less than what it is in reality, n n and a buyer can understate its willingness to (7) K = k ,= γ . pay by reporting a capacity k that is less than i i = = what it is in reality.11 Values and costs are not i 1 i 1 well specified at zero capacity. However, the Thus, given the firms’ reports, market output solution can be calculated for arbitrarily small Q(, K) solves the equation but positive capacities and zero capacity handled as a limit. Firms with zero capacity would then Q Q (8) v = c , report zero capacity. K Given the agents’ reports, the market mechanism chooses price p to equate reported supply which depends only on the aggregate production and reported demand, and allocates the output and consumption capacity reports. Market price efficiently. The solution is characterized by the p(, K) is obtained by substituting Q(, K) into the marginal conditions of Equation (5), and the output is allocated to sellers and buyers using the market share Equation (6). 11. A buyer’s marginal willingness to pay for another The firms’ actual types are common knowl- unit of input at q units of input is measured by v q .Since k edge to the firms. Thus, in choosing their reports, v is concave, this derivative is increasing in k. Therefore, a buyer understates its willingness to pay by underreporting firms know the true types of other firms. Then its capacity. the payoff to a vertically integrated firm i from 396 ECONOMIC INQUIRY

submitting reports (γi, ki) is as firm i’s equilibrium marginal cost and marginal valuation. π = ki Q(, K) (9) i ki v THEOREM 1. Suppose markets are vertically kiK separated. Then γ iQ(, K) − σ −γ c p ci i i γ (14) = , i ε + η − σ p (1 i ) ki γi and −p(, K)Q(, K) − . K v − p s i = i . (15) ε − + η If firm i is a seller with no consumption capac- p (1 si) = ≡ ity, then ki ki 0; similarly, if firm i is a γ γ buyer with no production capacity, then γ = COROLLARY 1. (i) / is less than 1 and de- i creasing in γ; (ii) k/k is less than 1 and decreas- γi ≡ 0. The Nash equilibrium to the market game consists of a profile of reports with the ing in k. property that (i) each firm correctly guesses the reports of other firms and (ii) no firm has an The exercise of market power by sellers and incentive to submit a different, feasible report. buyers creates a double markup problem. Sellers report less than their true capacity, thereby overstating their marginal cost. Since the market A. Equilibrium mechanism equates price to reported marginal costs, it exceeds each seller’s actual marginal We first derive and discuss equilibrium cost. Buyers report less than their true capacity, markups in vertically separated markets. We thereby understating their true willingness to consider several special cases including the pay. As a result, price is less than each buyer’s Cournot model. We then derive and discuss actual marginal willingness to pay. The corollary the equilibrium markups in vertically integrated establishes that, on both sides of the market, markets. the distortion is larger for firms with larger Before stating the theorems, we require some capacities. additional notation. The market demand function As in the standard Cournot model, seller Qd is given by v (Qd/K) = p, so the market markups are constrained by the elasticity of elasticity of demand is demand. If demand is elastic, then ε is large and sellers’ profit margins are small. However, (10) d − the seller’s margins also depend upon the elas- ε =− p dQ = v (Q/K) . ticity of supply. In the Cournot model, if a Q dp (Q/K)v (Q/K) seller restricts output, market supply falls by the Similarly, the market supply function Qs is same amount, and the price response depends given by c(Qs/)= p, so the market elasticity only upon the elasticity of demand. In a sup- of supply is ply function model such as ours, if a seller tries to restrict output by reporting a higher marginal p dQs c(Q/ ) (11) η = = . cost schedule, the reported market supply shifts Q dp (Q/ )c(Q/ ) to the left causing price to rise, but other sellers σ move up their reported supply curves, expand- Let i and si denote firm i’s market share ing their output. Thus, the fall in market supply in production and consumption, respectively. is less than the reduction in the seller’s output. Given any profile of reports, the market shares The price response depends upon the slope of are equal to reported capacity shares, that is, the reported supply curves. If marginal costs are roughly constant (i.e., c 0),thenη is very γi ki (12) σi = ,si = . large, individual sellers cannot raise price sig- K nificantly by constricting their supply schedule, Finally, define and the Bertrand outcome arises. On the other hand, if marginal cost curves are steeply sloped σ Q s Q c ≡ c i ,v ≡ v i . (i.e., c /c −→ ∞ ), then η approaches 0, and (13) i γ i i ki the Cournot outcome arises. Since our model HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 397 treats buyers and sellers symmetrically, the same analysis is motivated by the Cournot model. reasoning applies to buyer markups. Theorem 3 gives the equivalent of the HHI for We turn next to vertically integrated markets. the present model. We will refer to it as the modified Hirschman–Herfindahl index (MHI). THEOREM 2. Suppose markets are vertically It has the same useful features—it depends only integrated. Then, in any interior equilibrium, on market shares and elasticities—but there are = vi ci and two important differences. First, it suggests that analysts also need to consider the elasticity of (16) supply in evaluating the competitiveness of the v − p c − p s − σ i = i = i i . market. Second, it generalizes the analysis to p p ε(1 − si) + η(1 − σi ) vertically integrated markets and suggests that analysts use the firms’ net positions to measure There are two immediate observations. First, the effects of market power. As noted above, each vertically integrated firm is technically effi- zero net demand causes no inefficiency. Thus, cient about its production; that is, its marginal an intermediate good market in which each firm cost is equal to its marginal value. Thus, the is vertically integrated and supplies only itself is firm cannot, in the equilibrium allocation, gain perfectly efficient. However, with even a small from secretly producing more and consuming but nonzero net demand or supply, size exacer- that output. This is not to say that the firm could bates the inefficiency. not gain from the ability to secretly produce In this framework, the shares are of pro- and consume, for the firm might gain from this duction or consumption, and not capacity. The ability by altering its reports appropriately. For U.S. Department of Justice Merger Guidelines example, if the firm is a net seller, it will try generally calls for evaluation shares of capac- to raise price by restricting supply and overstat- ity. While our analysis begins with capacities, ing demand. It accomplishes the first by report- the shares are actual shares of production (σ ) γ i ing a production capacity that is less than its or consumption (si ), rather than the capacity actual capacity γ, and the second by reporting a for production and consumption, respectively. consumption capacity k that exceeds its actual Firms may have the same capacity in produc- capacity of k. A net buyer does the opposite, tion and consumption but nevertheless choose reporting higher production capacity and lower to be a net seller or a net buyer depending upon consumption capacity than its actual capacities. market conditions. The use of actual consump- Second, net buyers value the good more than tion and production is an advantageous feature the price, and net sellers value the good less of the theory, since these values tend to be read- than price. Thus, net buyers restrict their demand ily observed, while capacities are not. Finally, below that which would arise in perfect competi- the shares are shares of the total quantity and tion, and net sellers restrict their supply. In both not revenue shares. However, like the Cournot cases, the gain arises because of price effects. model, our model is not designed to handle industries with differentiated products, which is THEOREM 3. The (quantity weighted) aver- the situation where a debate about revenue ver- age difference between marginal valuations and sus quantity shares arises. marginal costs satisfies: n n 1 B. Intermediate Input Markets with Constant (17) siv − σi c p i i Elasticities i=1 i=1 n An important special case of our model is one (s − σ )2 = i i . in which value and cost elasticities are constant ε(1 − si) + η(1 − σi ) i=1 and buyers are manufacturing firms. Suppose marginal cost is given by In evaluating proposed horizontal mergers in vertically separated markets, antitrust agencies c(z) = z1/η, (and courts) focus primarily on demand elasticity, the concentration levels in the industry where η > 0, and marginal willingness to pay is prior to the merger and the predicted change in given by concentration levels due to the merger, where = −1/ε concentration is measured using the HHI. This v (z) z , 398 ECONOMIC INQUIRY where ε > 1.12 Given any vector of capacity significantly in the intermediate good market. reports, the market clearing conditions yield Indeed, the size of the misrepresentation is pro- closed form solutions for output portional to the discrepancy between price and marginal value or cost, as given by Theorem 2, ε ε+η η ε+η Q(, K) = /( )K /( ) adjusted for the demand elasticity. This is hardly surprising, since the constant demand and sup- and price ply elasticities insure that marginal values can be = −1/(ε+η) 1/(ε+η) converted to misrepresentations in a log-linear p(, K) K , fashion. which facilitates a quantitative assessment of Equation (19) provides the formula for lost firm misrepresentations and the cost of those trades. Here, there are two effects. Net buyers misrepresentations. If elasticities vary, the for- underrepresent their demand, but overrepresent mulae derived from the constant elasticity case their supply. On balance, net buyers under- apply approximately, with the error determined represent their net demands, which is why by the amount of variation in the elasticities. the quantity-weighted average marginal value exceeds the quantity-weighted average marginal Let Qf represent the first best quantity, which arises when all firms are sincere in their behav- cost. Equation (19) provides a straightforward ior, and p be the associated price. Then: means of calculating the extent to which a f market is functioning inefficiently, both before and after a merger, at least in the case where the With constant elasticities, the THEOREM 4. elasticities are approximately constant. size of the firms’ misrepresentations is given by Equation (20) gives the effect of strategic −ε k s − σ behavior in the model on price. Note that the i = + i i price can be larger, or smaller, than the efficient (18) 1 ε − + η − σ ki (1 si) (1 i ) full-information price. Market power on the η γi si − σi buyer’s side (high values of si) tends to decrease = 1 + . the price, with buyers exercising market power. γi ε(1 − si) + η(1 − σi ) Similarly, as σi increases, the price tends to Moreover, rise.

(19) III. EXTENSIONS η ε+η n ε /( ) Qf si − σi = si 1 + Q ε(1 − si ) + η(1 − σi ) In this section we consider two extensions i=1 of the model. The ﬁrst is to markets in which ε ε+η n −η /( ) si − σi the buyers compete in a downstream market. × σi 1 + ε(1 − si ) + η(1 − σi ) The second is to spot markets such as electricity i=1 markets in which ﬁrms are net traders. and

(20) ⎡ ⎤ A. Downstream Competition −η 1/(ε+η) n s − σ ⎢ σ 1 + i i ⎥ In many, perhaps, even most, applications, = i pf ⎢ i 1 ε(1 − si ) + η(1 − σi ) ⎥ = ⎢ ε ⎥ . the assumption that a buyer in the intermedi- p ⎣ n s − σ ⎦ s 1 + i i ate good market can safely ignore the behavior = i ε − + η − σ i 1 (1 si ) (1 i ) of other firms in calculating the value of consumption is unfounded. This is particularly true Equation (18) confirms the intuition that the when the buyers are retail firms. In this section, misrepresentation is largest for the largest net we extend the model to wholesale markets in traders, and small for those not participating which retail firms compete in quantities in the downstream market. 12. The associated cost and valuation functions are Recall that the value of consumption to a retail firm is given by η η+ η ε ε− ε c(z) = z( 1)/ ,v(z)= z( 1)/ . η + 1 ε − 1 qi V(qi,ki) = r(Q)qi − kiw . ki HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 399 where r(Q) is the downstream inverse demand the downstream market has perfectly elastic and w represents unit selling costs. Firm prof- demand. As a result, r is a constant, and (25) its are: readily reduces to Theorem 3. Note, however, that Theorem 4 does not apply since, in the qi (21) π (γ,k)= r(Q)q − k w wholesale market, the market-clearing condi- i i i k i tions fail to yield closed form solutions for output and prices.13 As we shall see in the next −γ xi − − ic p(qi xi). section, this distinction between the intermedi- γi ate input and wholesale market also matters for As before, we calculate the efficient solution, merger analysis. which satisfies: When B = 0, there is a constant retailing cost w. This case is analogous to Cournot, in that all (22) p = c (Q/ ) = c (xi/γi ) firms are equally efficient at selling, although and the firms vary in their efficiency at producing. = + = + In this case, (25) reduces to (23) r(Q) p w (Q/K) p w (qi /ki). n ACσ2 Let α be the elasticity of downstream demand, | = i β MHI B=0 and be the elasticity of the selling cost w.Let A(1 − σi ) + C i=1 θ be ratio of the intermediate good price p to the final good price r. The observables of the n θσ2 i analysis will be the market shares (both produc- = . η(1 − σi ) + θα tion, σi , and retail, si), the elasticity of final good i=1 demand, α, of selling cost, β, of production cost, η θ = The Herfindahl index reflects the effect of the , and the price ratio p/r. It will turn out wholesale market through the elasticity of sup- that the elasticities enter in a particular way, and ply η.Ifη = 0, the Cournot HHI arises. For thus it is useful to define: positive η, the possibility of resale increases the (24) A = α−1,B= (1 − θ)β−1, price/cost margin. This increase arises because a firm with a large capacity now has an alter- −1 and C = θη . native to selling that capacity on the market. We replicate the analysis of Section 3 in A firm with a large capacity can sell some of the Appendix for this more general model. The its Cournot level of capacity to firms with a structure is to use the efficiency equations to smaller capacity. The advantage of such sales construct the value to firm i of reports of to the large firm is the reduction in desire of the smaller firms to produce more, which helps ki and γi. The first-order conditions provide necessary conditions for a Nash equilibrium to increase the retail price. In essence, the larger the reporting game. These first-order conditions firms buy off the smaller firms via sales in the are used to compute the price/cost margin, intermediate good market, thereby reducing the weighted by the firm shares. In particular, we incentive of the smaller firms to increase their look for an MHI given by: production. Equation (25) can be decomposed into n 1 Herfindahl-type indices for three separate mar- MHI = [(r(Q)−p−w)s + (p−c )σ ], r i i i i kets: transactions, production, and consumption. i−1 Note, where wi = w(qi /γi). The main theorem characterizes the MHI for 13. In wholesale markets, Q(, K) is defined implicitly an interior solution. by the equilibrium condition η β Q 1/ Q 1/ THEOREM 5. In an interior equilibrium, = r − , K (25) whereas in intermediate good markets, market clearing n − σ 2 + 2 − σ + σ2 − BC(si i ) ABs (1 i ) AC (1 si ) implies = i i . MHI − − σ + − σ + − A(1 si )(1 i ) B(1 i ) C(1 si ) η − ε i=1 Q 1/ Q 1/ = , While complex in general, this formula K has several important special cases. If A = 0, which yields an explicit solubion for Q(, K). 400 ECONOMIC INQUIRY

n − σ + − − σ 2 = B(1 i ) C(1 si) (si i ) MHI −1 −1 A(1 − si )(1 − σi ) + B(1 − σi ) + C(1 − si) C (1 − σi ) + B (1 − si ) i=1 n A(1 − σ )(1 − s ) Bs2 Cσ2 + i i i + i . A(1 − si)(1 − σi ) + B(1 − σi ) + C(1 − si ) (1 − si) (1 − σi ) i=1

The MHI is an average of three separate indices. approximate the description of very elastic mar- The first index corresponds to the transactions ket demand.15 However, the case of A = 0also in the intermediate good market. In form, this corresponds to the case where the buyers do not term looks like the expression in Theorem 3, compete in a downstream market, and may thus adjusted to express the elasticities in terms of have alternative applications. the final output prices. The second expression In the Appendix, we provide the formulae is an average of the indices associated with governing the special case of constant elas- production and consumption of the intermediate ticities. It is straightforward to compute the good. These two indices ignore the fact that reduction in quantity that arises from a confirms consume some of their own production. centrated market, as a proportion of the fully When the downstream market is very elas- efficient, first-best quantity. Moreover, we pro- tic, as we have already noted, A is near zero. In vide programs which take market shares as this case, the MHI reduces to that of Theorem 3, inputs and compute the capital shares of the because elastic demand in the downstream mar- firms, the quantity reduction, and the effects of a ket eliminates downstream effects, so that the merger.16 only effects arise in the intermediate good market. In contrast, when the downstream market is relatively inelastic, downstream effects dom- B. Electricity Markets inate, and the MHI is approximately an average Theoretical and empirical studies of whole- of the Herfindahl indices for the upstream and sale electricity markets have applied supply downstream markets, viewed as separate mar- function models or Cournot models to predict kets. the potential for generating firms to exercise In a sense, these limiting cases provide a res- market power in spot electricity markets and olution of the question of how to treat captive to estimate markups. Our model generates a consumption. When demand is very inelastic, markup equation that combines the advantages as with gasoline in California, then the issue of the supply function models with the simplic- of captive consumption can be ignored without ity of the Cournot model. major loss; it is gross production and consump- In day-ahead or real-time balancing mar- tion that matters. In this case, it is appropriate kets, generating firms submit supply schedules, to view the upstream and downstream markets buyers report the amounts that they need for as separate markets and ignore the fact that their retail customers, and an independent sys- the same firms may be involved in both. In tem operator chooses price to equate reported particular, a merger of a pure producer and a pure retailer should raise minimal concerns. On supply to market demand. An important fac- the other hand, when demand is very elastic tor determining the generating firms bidding (A near zero), gross consumption and gross pro- behavior in these markets is their contract posi- duction can be safely ignored, and the market tions. With the exception of firms in the Cal- treated as if the producers and consumers of the ifornia market, generating firms typically sign intermediate good were separate firms, with net forward contracts with buyers in which they trades in the intermediate good the only issue agree to deliver a fixed amount of electricity that arises.14 Few real-world cases are likely to at a predetermined price. Generating firms that

14. However, the denominator still depends on gross 15. When market demand is very elastic, it is likely that production and consumption, rather than net production and there are substitutes that have been ignored. It would usually consumption. This can matter when mergers dramatically be preferable to account for such substitutes in the market, change market shares, and even the merger of a pure rather than ignoring them. producer and pure consumer can have an effect. 16. This program is available on McAfee’s Web site. HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 401 have signed such vertical contracts are essen- THEOREM 6. In an interior equilibrium tially vertically integrated, and they may be p − c σ − s net sellers or net buyers in the spot market. i = i i . Green (1999) and Wolak (2000) discuss the the- p η(1 − σi ) + α oretical implications of forward contracts and show that they make the spot market more The theorem states that the firm’s reported competitive. capacity exceeds its true capacity (i.e., marginal Let qi denote firm i’s forward contract quan- cost exceeds reported marginal cost) when the tity and let r denote the contract price. Gener- firm produces less than its contract quantity, and ating firms typically take short positions in the the opposite is true when it produces more than forward market, in which case qi is positive. its contract quantity. It reports truthfully when Firm i’s profit from supplying xi at price p is it is balanced. (If demand is perfectly inelastic, given by then α is equal to zero in the above formula.) The intuition is that, in former case, the firm is xi πi = p(xi − qi ) − vic + rqi. in a net buy position and wants to lower price, vi whereas in the latter case, it is in a net sell posi- The firm’s revenues consist of two components: tion and wants to raise price. Markups in our the amount it earns from its contract position model will vary across firms depending upon and the payment it receives from either reducing their contract positions, and with demand con- its supply below q or from increasing its supply ditions if the elasticity of costs is not constant. i Since marginal cost functions in the electricity above qi. When it reduces its supply, it is in a net buyer position, buying the reduction in supply markets are approximately L-shaped, our model at price p from the spot market and selling predicts that markups are essentially zero in low this amount to its customers at the contract demand periods and higher during high demand price of r. When it increases its supply, firm periods, particularly for large net sellers. This is consistent with the evidence presented in Bush- i is in a net seller position, selling the increase 18 at price p to retailers. The profit function can nell, Mansur, and Saravia (BMS) (2008). They also be interpreted as the profits of a vertically find that prices are very close to marginal costs integrated firm that sells electricity to its retail during off-peak hours and higher during peak customers at a regulated price of r. hours. We assume that the firms face a downward- Our markup equation is closely related to the sloping inverse demand function p(X).17 The markup equations that have been estimated in operator knows the marginal cost schedules but the literature. Wolak (2000, 2003) assumes that does not know the capacity that firms have avail- each firm i faces a stochastic residual demand, able or at least cannot force them to make all RDi (p, ), and submits a bid schedule that is ex of their capacity available. Firms are asked to post optimal. That is, for each realization of the report their available generating capacities. Con- random variable ,xi (p) is a best reply to the ex tract positions are common knowledge among post residual demand and satisfies the optimality the firms. On the basis of these reports, the condition operator equates demand to supply and allo- − σ − p ci i si cates output across firms by equating reported = . p α (p, ) marginal costs so in equilibrium, i α ε x where i (p, ) is the elasticity of the residual p(X) = c i ν demand facing firm i. It incorporates both the elasticity of demand and the aggregate elasticity and for i = 1, .., n. Note that the allocation of supply bid by firm i’s rivals and can be esti- rule does not depend upon the firms’ contract mated from data on the bid schedules of firm positions, so firms report only their production capacity. Let α(p) be the elasticity of demand = 18. Bushnell, Mansur, and Saravia (2008) study markups at price p and define si qi /X. in California, New England, and the Pennsylvania, New Jer- sey, Maryland (PJM) electricity markets using the Cournot model to predict the potential for generating firms to 17. Market demand is downward-sloping in some elec- exercise market power. Borenstein and Bushnell (1999) and tricity markets because it is equal to the fixed retail demand Borenstein, Bushnell, and Wolak (2002) also use the Cournot less a competitive import supply. model to study markups in the California electricity market. 402 ECONOMIC INQUIRY i’s rivals. In his study of Australian electric- firms do not change their capacity reports.19 ity markets, Wolak has data on a firm’s con- However, as Farrell and Shapiro (1990) have tract positions, and he develops a procedure for observed, this rule ignores the fact that post- recovering the firm’s cost function from the ex merger behavior is likely to be different from post optimality condition. Hortacsu and Puller pre-merger behavior since the merging firms (2008) show that ex post optimal bid functions will internalize the negative externality that their are a Bayesian equilibrium when the firms’ con- pre-merger actions imposed on each other’s tract positions are private information and bid profits. An equilibrium analysis is necessary and strategies are additively separable in the pri- Farrell and Shapiro provide such an analysis vate information. The additivity assumption also for Cournot oligopoly markets. They investigate implies that the elasticity of the residual demand the relationship between HHI and consumer function does not depend upon . They exploit and social welfare, and provide necessary and the availability of data on the firms’ marginal sufficient conditions under which a merger raises cost and bid schedules in the Texas electric- price. They also provide sufficient conditions ity market to infer the firms’ contract positions under which profitable mergers raise welfare. and then use the markup equations to test the Mergers without cost synergies generally raise ex post optimality conditions. Sweeting (2007) price but are often not profitable in the Cournot also uses the ex post optimality condition in his model, since the merging firms reduce output study of the Wales electricity market to test the and rival firms respond by expanding their hypothesis of optimal bidding behavior under output. McAfee and Williams (1992) provide various assumptions about the firms’ contract conditions under which mergers are profitable positions. for the special case of quadratic costs and linear demand. We begin our equilibrium analysis of hori- IV. MERGERS zontal merger by examining firms’ best replies. Substituting equilibrium output and price into In this section we examine the equilibrium 20 effects of horizontal and vertical mergers. The the profit function of seller i and taking logs , we obtain constant returns to scale assumption facilitates the study of mergers. It implies that the merger η + 1 π = Q(, K) − of two firms i and j produces a firm with con- log i η log log sumption capacity ki + kj and production capac- γ + γ η − η η+ η ity i j , and thereby is subject to the same + log ν − ν 1/ ν( 1)/ . analysis. In what follows, we focus primarily on i η + 1 i i mergers in vertically separated markets for two reasons. First, previous merger studies typically Differentiating yields firm i’s best reply, which make this assumption and we wish to compare solves the results of our analysis to their results. Sec- (26) ond, the vertically separated market provides a − 1/η polar case in which qualitative results can be vi − ∂Q = 1 (vi/v) 1 1/η . obtained under the assumption of constant elas- ∂ Q ((η + 1)/η) − (vi/vi ) ticities. The analysis illustrates the economic forces at work in vertically integrated markets, The right-hand side of Equation (26) is decreas- where the impact of mergers depends upon the ing in vi. The two terms on the left-hand side values of the elasticity parameters and hence requires a more quantitative analysis. We will 19. Applying this rule to a merger of firms 1 and 2 using assume throughout this section that elasticities, our index of market power yields including that of downstream demand, are con- = − ρ σ2 + − ρ σ2 + σ σ stant. MHI (1 1) 1 (1 2) 2 2 1 2, where

ε + (1 − σ1 − σ2) A. Horizontal Mergers ρi = < 1. ε + (1 − σi )η The DOJ’s Merger Guidelines estimate the Therefore, MHI exceeds HHI (although MHI < HHI). impact of a horizontal merger in oligopoly 20. Note that maximizing log πi is the same as maxi- markets under the assumption that the merged mizing πi . HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 403 of the equation capture the impact of capacity on the same side of the market will respond, reports of other firms. The first term is firm and their responses are mutually reinforcing. i’s market share, which falls with reports by This leads to the following predictions about other sellers, and the second involves the pro- the equilibrium impact of horizontal mergers. duction capacity output elasticity. An analogous A merger among sellers reduces reported supply equation determines the buyer’s best reply. The but does not affect reported demand. Hence key issue that determines whether reports of price increases and output falls. Similarly, a other firms are strategic substitutes or comple- merger among buyers reduces reported demand ments is their impact on the production capacity but does not affect reported supply, so both price output elasticity (or consumption capacity out- and output fall. The merging firms profit from put elasticity in the case of buyers). This impact a merger in two ways. First, it gives them more will depend upon the type of market. market power to reduce capacity and raise price, In intermediate input markets in which buy- and second, other firms on the same side of ers face a constant downstream price, it is the market will do the same. The latter effect easily verified that the output elasticities are reflects the key difference between our model constants.21 and the Cournot model. In our model, best replies are strategic complements, whereas in LEMMA 1. Consider a vertically separated, the Cournot model, they are strategic substitutes. intermediate input market with constant cost and Akgun (2005) obtains similar results in a supply value elasticities and a fixed downstream price. function model of oligopoly markets in which Then (i) the capacity reports by a seller and a the sellers are restricted to reporting linear buyer are independent of each other and (ii) the supply schedules. capacity reports of any pair of sellers or buyers In wholesale markets, the equilibrium output are strategic complements. elasticities are functions of the aggregate production and retailing capacity.23 This introduces Lemma 1 implies that intermediate input mar- two new effects into the analysis of a horizon- kets with no vertically integrated firms are not tal merger, which complicates the analysis of only structurally separate, they are also strategi- a merger. First, the market is no longer strate- cally separate. It also implies that the reporting gically separate. If two sellers merge, buyers game is a log supermodular game.22 will respond. Tedious calculations reveal that the capacity reports of a buyer and a seller are strate- THEOREM 7. Consider a vertically sepa- gic complements as long as downstream demand rated, intermediate input market with constant is not too inelastic.24 Thus, when the merg- cost and value elasticities and a fixed down- ing sellers try to reduce their reported capac- stream price. A horizontal merger among sellers ity, buyers will reduce their capacity reports, reduces reported production capacity, increases thereby lowering reported demand. In this way, price, and decreases output. A horizontal merger the enhanced market power of sellers is miti- among buyers reduces reported consumption gated by buyers exercising buyer power. Prices capacity and decreases price and output. Hor- are lower, mergers are less profitable, and inef- izontal mergers are always profitable for the ficiency costs increase. In fact, the strategic merging firms. complementarity between the buyer and seller We show in the Appendix that the best reply reports can lead to nonexistence of an interior of merging firms to pre-merger reports of other solution. For example, if the merger among sell- firms is always to report a capacity that is less ers creates a monopoly, and there is only one than the sum of their pre-merger reports. It buyer who sells at a fixed price of r, it can then follows from Lemma 1 that only firms be shown that the only intersection point of the best replies is (0,0). Clearly, in this case, 21. Recall that 23. In this case, Q(, K) is defined implicitly by Q(, K) = ε/(ε+η)Kη/(ε+η). η ε Q 1/ Q 1/ + − Q1/α = 0. K 22. Substituting the expression for Q(, K) given in the previous footnote, it is easily verified that each seller’s (buyer’s) profit function is log supermodular in the capacity 24. A sufficient condition for strategic complementarity reports of other sellers (buyers) and independent of the is ε ≥ α. However, if α is sufficiently small, the second-order capacity reports of buyers (sellers). conditions will be violated. 404 ECONOMIC INQUIRY a merger would not be profitable. Second, the only the buyer in the market merges with a seller reports may no longer be strategic com- seller, reported demand does not change, but plements. An increase in the reports of other reported supply increases, lowering price and sellers reduces the production capacity output increasing output. Thus, a vertical merger in elasticity (assuming downstream demand is not monopoly or monopsony markets always leads too inelastic). This effect dominates the market to foreclosure, with the magnitude of the effect share effect when sellers have a lot of market depending upon the elasticities of supply and power (i.e., η is small). demand and upon market shares. In markets with multiple buyers and sellers, the vertically integrated firm increases both capacity reports, B. Vertical Mergers which causes rivals on both sides of the mar- The voluminous literature on vertical merg- ket to increase their capacity reports. Hence, ers is primarily concerned with two issues: effi- reported supply and demand increase, output ciency and foreclosure. Vertical mergers can increases, but the impact on price is ambiguous. generate substantial efficiency gains by eliminat- Intuitively, the foreclosure effect is important ing the double markup problem that arises from when the vertically integrated firm is either a sellers exercising market power in the upstream large net seller or a large net buyer. market and buyers exercising market power in The assumption that the market is verti- downstream markets. This is the reason why cally separated is crucial to the merger analysis. antitrust agencies are less likely to contest a A vertical merger in a vertically integrated mar- vertical merger than a horizontal merger. The ket can lead to a decrease in reported demand primary concern that the agencies have about a or supply. The reason is that reported pro- vertical merger is the risk that the vertically inte- duction capacity and consumption capacity are grated firm will foreclose the intermediate input strategic substitutes for a vertically integrated market to other buyers with whom it may be firm. A similar issue arises with vertical merg- competing or, more generally, raise their costs ers in wholesale markets. Even if the market is by increasing input prices. Analogous effects vertically separated, increases in reported con- arise when the vertically integrated firm prevents sumption capacity reduce the reported capacity other sellers from selling input into the interme- of sellers, and increases in reported production diate market or forcing them to accept a lower capacity reduce the reported capacity of buyers. price. Thus, in these cases, the impact of a vertical The standard models assign market power merger on price and output needs to be com- either to buyers or to sellers, but not both. puted on a case–case basis. In contrast, in our model, sellers misrepresent their costs and buyers misrepresent their willingness to pay. Thus, in equilibrium, the “wedge” V. IDENTIFICATION between marginal cost and marginal value is Suppose a researcher has data on prices and the sum of the seller markup and buyer mark- quantities in a market for t = 1, .., T periods down. A vertical merger eliminates this “wedge” and wants to use our model to estimate cost and between the merging seller and buyer. Con- demand parameters, under what conditions is the sequently, vertical mergers in our model have model identified? stronger efficiency effects than in the standard In addressing this question, it is useful to models. begin with the standard model that has been esti- What about foreclosure effects? To obtain mated in numerous empirical studies of market some insight into this issue, we consider inter- power (e.g., Porter 1983; Genesove and Mullin mediate input markets with a single seller (or 1998; Clay and Troesken 2003). The model buyer) and a constant downstream price. In this assumes that the market is vertically separated case, it can be shown that a vertical merger and that buyers are price-takers. Market demand does not change the seller’s production capacity is given by report, but it does cause the merged firm to over- state its consumption capacity. It then follows pt = P(Qt ,Kt ,Zt ,udt ; δ), from Lemma 1 that reported demand increases substantially as other buyers respond by increas- where Zt are observed demand shifters, udt rep- ing their reported consumption capacity. Hence, resents unobserved (to the researcher) demand both output and price increase. Similarly, when shocks that are independent over time, and δ are HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 405 the unknown demand parameters. The pricing cost and value elasticities. More precisely, sup- equation for sellers is specified as pose = φ 1/η −1/η pt = c (Qt ,Wt ,ust; φ) c (Qt ,Wt ,ust) Wt Qt t ust ∂P(Q ,K ,Z ; δ) and −λQ t t t , t ∂Q = δ −1/ε 1/ε v (Qt ,Zt ,udt ) Zt Qt Kt udt , where Wt are observed factors that shift marginal where (ust,udt) are distributed multivariate log- cost, ust represents unobserved supply shocks, normal with mean zero and covariance . and φ are the unknown cost parameters. Here This is the model that Porter estimates under we have assumed that only the aggregate quan- the assumption that buyers are price-takers, tity data are available, so c is the marginal cost which, in terms of our model, means that they of the average firm. The parameter λ is known are reporting their true capacity. As we have as the “conduct” parameter and interpreted as observed previously, the elasticities of reported the average of the firms’ conjectures on how demand and supply are constant in this model, aggregate supply will change with an increase in independent of the capacity reports. Further- their output. In the Cournot model, rivals cannot more, the argument given for Lemma 1 also react so λ is equal to 1 but, in empirical work, implies that capacity reports of sellers and it is often useful to allow markups to vary from buyers are independent of the observed and the Cournot markups. As is well known (see unobserved factors shifting demand and sup- Bresnahan 1989), the above model is identified ply. Thus, as long as actual capacities are con- if instruments are available for the endogenous stant over time, and K are constants, as are variables in the two equations. Shifts in marginal the firms’ market shares. This in turn implies cost can be used to identify the demand param- that λ is a constant and that the derivative of eters, and shifts in demand and in slope of the reported demand schedule does not vary demand can be used to identify the cost and with reported buyer capacity. The variation in conduct parameters. In the various empirical markups over time is coming from exogenous studies surveyed by Bresnahan (1989), estimates variation in the observed and unobserved factors of λ range from 0.05 to 0.65. More recently, but not from the endogenous variables, and K. Genesove and Mullin report estimates of λ for As a result, changes in W shift the reported sup- the sugar industry at the turn of the century rang- ply but not the reported demand, thereby identi- ing from 0.038 to 0.10, with the latter computed fying the demand parameters; changes in Z shift directly from the data on prices and marginal the reported demand but not the reported supply, costs. Clay and Troesken report similarly low thereby identifying the cost parameters.25 estimates for λ in the whiskey industry at the Our model is not identified if elasticities turn of the century. These estimates suggest that (and/or slopes of reported demand and supply) dynamic considerations do matter and lead to depend upon reported capacities and these differ lower markups. from actual capacities due to the exercise of In our model, market power. This will typically be the case in ε λ = , 25. Solving for equilibrium and taking logs, the structural ε + (1 − σ)η model is given by σ εδη φεη ε where denotes the market share of the aver- log Qt = log Zt − log Wt + log age firm. It is not a free parameter but in general ε + η ε + η ε + η η εη u depends upon the elasticities of reported demand + K + dt ε + η log ε + η log and supply evaluated at the equilibrium mar- ust ket quantity. Note that λ is bounded between εδ φη ε P = Z + W − 0 and 1, with the upper bound achieved when log t ε + η log t ε + η log t ε + η log η = 0 (i.e., the Cournot case). Hence, our model 1 ε u + log K + log dt provides a potential explanation for why esti- ε + η ε + η u mated markups are typically lower than Cournot st markups. It is straightforward to show that the structural parameters (ε, η, δ, φ) can be recovered from parameter estimates of the Our model is identified in vertically sepa- reduced form regressions of log Qt and log Pt on a constant, rated, intermediate input markets with constant log Zt and log Wt . 406 ECONOMIC INQUIRY

TABLE 1 Approximate Market Shares

Reﬁning Market Reﬁning Capital Retail Market Retail Capital Company i Share (σi ) Share Share (si ) Share

Chevron 1 26.4 (26.6) 29.5 (29.5) 19.2 (19.5) 19.0 (19.0) Tosco 2 21.5 (21.7) 21.7 (21.7) 17.8 (18.0) 17.8 (17.8) Equilon 3 16.6 (16.7) 16.1 (16.1) 16.0 (16.2) 16.0 (16.0) Arco 4 13.8 (13.9) 13.0 (13.0) 20.4 (20.7) 22.0 (22.0) Mobil 5 7.0 (13.3) 6.2 (12.4) 9.7 (17.5) 9.3 (17.8) Exxon 6 7.0 (0.0) 6.2 (0.0) 8.9 (0.0) 8.5 (0.0) Ultramar 7 5.4 (5.4) 4.7 (4.7) 6.8 (6.9) 6.4 (6.4) Paramount 8 2.3 (2.3) 2.0 (2.0) 0.0 (0.0) 0.0 (0.0) Kern 9 0.0 (0.0) 0.0 (0.0) 0.3 (0.3) 0.27 (0.27) Koch 10 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18) Vitol 11 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18) Tesoro 12 0.0 (0.0) 0.0 (0.0) 0.2 (0.2) 0.18 (0.18) PetroDiamond 13 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09) Time 14 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09) Glencoe 15 0.0 (0.0) 0.0 (0.0) 0.1 (0.1) 0.09 (0.09)

Note: West coast CARB gasoline post-merger numbers are given in parentheses. wholesale markets and in vertically integrated of the Exxon–Mobil merger. Nevertheless, we markets. In these markets, markups will be a have tried to use the best available data for function of K and , which are likely to depend the analysis. In Table 1, we provide a list upon the unobserved shocks affecting demand of market shares, along with our estimates and supply. As a result, λ and P will vary over of the underlying capital shares and the post- time and the variation will be correlated with merger market shares, which will be discussed the unobserved shocks. One could try to find below. The data come from Leffler and Pulliam instruments for K and but data on reported (1999). capacities are typically not available. From Table 1, it is clear that there is a significant market in the intermediate good of bulk (unbranded) gasoline, prior to branding and VI. APPLICATION: THE EXXON–MOBIL MERGER the addition of proprietary additives. However, Our second application is to a merger of the actual size of the intermediate good market Exxon and Mobil’s gasoline refining and retail- is larger than one might conclude from Table 1, ing assets in the western United States. The west because firms engage in swaps. Swaps trade coast gasoline market is relatively isolated from gasoline in one region for gasoline in another. the rest of the nation, both because of transporta- Since swaps are balanced, they will not affect tion costs26 and because of the requirement of the numbers in Table 1. gasoline reformulated for lower emission, a type It is well known that the demand for gasoline of gasoline known as CARB. is very inelastic. We consider a base case of an Available market share data are generally elasticity of demand, α,of1/3. We estimate θ to imperfect, because of variations due to shut- be 0.7, an estimate derived from an average of downs and measurement error, and the present 60.1 cents spot price for refined CARB gasoline, analysis should be viewed as an illustration out of an average of 85.5 (net of taxes) at the of the theory rather than a formal analysis pump in the year 2000.27 We believe the selling

26. There is currently no pipeline permitting transfer of 27. We will use all prices net of taxes. As a consequence, Texas or Louisiana reﬁned gasoline to California, and the the elasticity of demand builds in the effect of taxes, so Panama Canal cannot handle large tankers, and in any case that a 10% retail price increase (before taxes) corresponds is expensive. Nevertheless, when prices are high enough, approximately to a 17% increase in the after tax price. Thus, CARB gasoline has been brought from the Hess reﬁnery in the elasticity of 1/3 corresponds to an actual elasticity of the Caribbean. closer to 0.2. HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 407

TABLE 2 Analysis of Exxon–Mobil Merger

Cases Markup as a Percent of Retail Price Balanced Pre-merger Post-merger αβηSymmetric Asymmetric Markup Markup Reﬁnery Sale Retail Sale

1 1/3 5 /2 6.9 (98.4) 18.4 (95.3) 20.0 (94.6) 21.3 (94.3) 20.1 (94.6) 21.2 (94.3) 1 1/5 5 /2 7.9 (98.7) 21.6 (96.0) 23.6 (95.4) 25.2 (95.2) 23.7 (95.4) 25.2 (95.2) 1 1/3 3 /2 7.0 (98.4) 18.7 (95.3) 20.3 (94.6) 21.7 (94.3) 20.5 (94.6) 21.6 (94.4) 1/3 5 1/3 8.7 (98.2) 23.0 (94.6) 25.1 (93.8) 26.7 (93.5) 25.2 (93.8) 26.7 (93.5)

Note: Quantity as a percent of fully efficient quantity is given in parentheses. cost to be fairly elastic, with a best estimate the sale of retailing, we combine the refining of β = 5. Similarly, by all accounts refining capacity. costs are quite inelastic; we use η = 1/2as The estimated shares of capital are presented the base case. We will consider the robustness in Table 1. These capital shares reflect the to parameters below, with α = 1/5, β = 3, and incentives of large net sellers in the intermediate η = 1/3. market to reduce their sales in order to increase Table 2 presents our summary of the Exxon/ the price, and the incentive of large net buyers Mobil merger. The first three columns provide to decrease their demand to reduce the price. the assumptions on elasticities that define the Equilon, the firm resulting from the Shell- four rows of calculations. The fourth column Texaco merger, is almost exactly balanced and provides the markups that would prevail under thus its capital shares are relatively close to a fully symmetric and balanced industry, that is, its market shares. In contrast, a net seller in one comprised of 15 equal sized firms. This is the intermediate market like Chevron refines the best outcome that can arise in the model, significantly less than its capital share, but retails given the constraint of 15 firms, and can be close to its retail capital share. Arco, a net buyer used as a benchmark. The fifth column consid- of unbranded gasoline, sells less than its share ers a world without refined gasoline exchange, from its retail stores, but refines more to its in which all 15 companies are balanced, and share of refinery capacity. The estimates also is created by averaging production and con- reflect the incentives of all parties to reduce sumption shares for each firm. This calculation their downstream sales to increase the price, provides an alternative benchmark for compari- that is, the larger an incentive the larger is the son, to assess the inefficiency of the intermediate retailer. good exchange. The next four columns use the The sixth column of Table 2 provides the existing market shares, reported in Table 1, as an pre-merger markup, or MHI, and is a direct input, and then compute the price–cost margin calculation from Equation (24) using the mar- and quantity reduction, pre-merger, post-merger, ket shares of Table 1. The seventh, eighth, and with a refinery sale, or with a sale of retail out- ninth columns combine Exxon and Mobil’s capi- lets, respectively. tal assets in various ways. The seventh combines Table 2 does not use the naive approach of both retail and refining capital. The eighth com- combining Exxon and Mobil’s market shares, an bines retail capital, but leaves Exxon’s Benicia approach employed in the Department of Justice refinery at the hands of an alternative supplier Merger Guidelines. In contrast to the merger not listed in the table. This corresponds to a sale guidelines approach, we first estimate the capital of the Exxon refinery. The ninth and last column held by the firms, then combine this capital considers the alternative of a sale of Exxon’s in the merger, then compute the equilibrium retail outlets. (It has been announced that Exxon given the post-merger allocation of capital. The will sell both its refining and retailing operations estimates are not dramatically different from in California.) those that arise using the naive approach of the Our analysis suggests that without divestiture merger guidelines. To model the divestiture of the merger will, under the hypotheses of the refining capacity, we combine only the retailing theory, have a small effect on the retail price. capital of Exxon and Mobil; similarly, to model In the base case, the markup increases from 408 ECONOMIC INQUIRY

20% to 21%, and the retail price increases 1%.28 TABLE 3 Moreover, a sale of a refinery eliminates most of Analysis of Exxon–Mobil Merger the price increase; the predicted price increase is less than a mil. Unless retailing costs are Expected Percentage Quantity Decrease much less elastic than we believe, a sale of retail Cases outlets accomplishes very little. The predicted Full Refinery Retail changes in prices, as a percent of the pretax αβη Merger Sale Sale retail price, are summarized in Table 2. The 1 1/3 5 /2 0.31 (0.94) 0.03 (0.09) 0.30 (0.90) unimportance of retailing is not supported by 1 1/5 5 /2 0.27 (1.36) 0.02 (0.11) 0.25 (1.29) Hastings (2004). 1 1/3 3 /2 0.32 (0.97) 0.05 (0.15) 0.30 (0.89) The predicted quantity, as a percentage of the 1/3 5 1/3 0.35 (1.06) 0.03 (0.08) 0.34 (1.03) fully efficient quantity, is presented in Table 3, in parentheses. The first three columns present the Note: Percentage price increase is given in parentheses. prevailing parameters. The next three columns correspond to the conceptual experiments discussed above. The symmetric column considers VII. CONCLUSION fifteen equal-sized firms. The balanced asym- This paper presents a method for measur- metric column uses the data of Table 1, but aver- ing industry concentration in intermediate good ages the refining and retail market shares to yield markets. It is especially relevant when firms a no-trade initial solution. The pre-merger col- have captive consumption, that is, some of the umn corresponds to Table 1; post-merger com- producers of the intermediate good use some bines Exxon and Mobil. Finally, the last two or all of their own production for downstream columns consider a divestiture of a refinery and sales. retail assets, respectively. We see the effects of The major advantages of the theory are its the merger through a small quantity reduction. applicability to a wide variety of industry struc- Again, we see that a refinery sale eliminates tures, its low informational requirements, and nearly all of the quantity reduction. its relatively simple formulae. The major dis- The analysis used the computed market advantages are the special structure assumed in shares rather than the approach espoused by the the theory and the static nature of the analysis. U.S. Department of Justice Merger Guidelines. The special structure mirrors Cournot, and thus Our approach is completely consistent with the is subject to the same criticisms as the Cournot theory, unlike the merger guidelines approach, model. For all its defects, the Cournot model which sets the post-merger share of the merg- remains the standard model for antitrust anal- ing firms to the sum of their pre-merger shares. ysis; the present theory extends Cournot-type This is inconsistent with the theory because analysis to a new realm. the merger will have an impact on all firms’ We considered the application of the the- shares. In Table 1, we provide our estimate ory to wholesale electricity markets and to the of the post-merger shares alongside the pre- merger of Exxon and Mobil assets in the west- merger shares. Exxon and Mobil were respon- ern United States. Several reasonable predic- sible for 18.6% of the refining, and we esti- tions emerge. In wholesale electricity markets, mate that the merger will cause them to contract firm markups are approximately zero during low to 17.4%. The other firms increase their share, demand periods, high during high demand peri- though not enough to offset the combined firm’s ods, and vary depending upon the firm’s net contraction. position and market share. In west coast gaso- There is little to be gained by using the naive line markets, the industry produces around 95% merger guidelines market shares, because the of the efficient quantity and the merger reduces analysis is sufficiently complicated to require quantity by a small amount, around 0.3%. The machine-based computation. However, we repli- price–cost margin is on the order of 20% and cated the analysis using the naive market shares, rises by a percentage point or two. A sale of and the outcomes are virtually identical. Thus, it Exxon’s refinery eliminates nearly all of the pre- appears that the naive approach gives the right dicted price increase. This last prediction arises answer in this application. because retailing costs are relatively elastic, so that firms are fairly competitive downstream. 28. The percentage increase in the retail price can be Thus, the effects of industry concentration arise computed by noting that p = q−A. primarily from refining, rather than retail. Hence HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 409

the sale of a refinery (Exxon and Mobil have one Q ε−1 = (v − p) (1 − s ) + s refinery each) cures most of the problem associ- K i i i ε−1 + η−1 ated with the merger. The naive approach based ε−1 (ηε)−1 +(p−c ) σ −p(s −σ ) . purely on market shares gives answers similar i i ε−1 + η−1 i i ε−1 + η−1 to the more sophisticated approach of first computing the capital levels, combining the capital Similarly, of the merging parties, then computing the new equilibrium market shares. Finally, it is worth ∂πi si Q ∂Q noting that the computations associated with the = v − p si ∂γˆ i ki ∂ present analysis are straightforward, and run in a few seconds on a modern PC. σ Q 1 γˆ ∂Q + p − c i Q − i + σ 2 i As with Cournot analysis, the static nature γi ∂ of the theory is problematic. In some industries, −1 ∂p Q η entry of new capacity is sufficiently easy that −Q(si − σi ) = (v − p) si ∂ i ε−1 + η−1 entry would undercut any exercise of market η−1 power. The present theory does not accommo- +(p − c ) 1 − σ + σ date entry, and thus any analysis would need i i i ε−1 + η−1 a separate consideration of the feasibility and (ηε)−1 29 +p(s − σ ) . likelihood of timely entry. When entry is i i ε−1 + η−1 an important consideration, the present analysis provides an upper bound on the ill-effects Thus, merger. π π K ∂ i + ∂ i = − Another limitation of the theory is the restric- vi ci . Q ∂kˆ Q ∂γˆ i tion to homogenous good markets. An extension i of the theoretical approach to markets in which − = In an interior equilibrium, then, vi ci 0. Either of sellers offer differentiated good would be chal- the first-order conditions yields (16). lenging but worth exploring. TheCaseof si = 0, σi > 0. If σi > 0, the first-order condition onγ î holds with equality. Consequently, using si = 0,

APPENDIX Q η−1 0 = (p − c ) 1 − σ + σ Proof of Theorems 1 and 2 i i i ε−1 + η−1 − Before proceeding, note that differentiating the equilib- (ηε) 1 +p(0 − σ ) . rium condition in Equation (8) implies that i ε−1 + η−1 K ∂Q ε−1 ∂Q η−1 = , = This yields Q ∂K ε−1 + η−1 Q ∂ ε−1 + η−1 − σ p ci i K ∂P ∂P (ηε)−1 = , and =− = . p ε + η(1 − σi ) P ∂K P ∂ ε−1 + η−1 a formula that respects (15) (for s = 0). In addition, we Differentiating the firm’s profit function in Equation (9) i have and substituting the relations given above yields the following first-order conditions: π ∂ i 0 , ∂kˆ = i si 0 ∂πi si Q 1 kî ∂Q = v − p Q − + si ∂kˆ k K K2 ∂K i i which gives σi Q ∂Q ∂p + p − c σ − Q(s − σ ) − −σ γ i i i v (0) p i i ∂K ∂K . p ε + η(1 − σi )

29. McAfee, Simons and Williams (1992) present a Summarizing, Cournot-based merger evaluation theory that explicitly accommodates entry in the analysis. Ghosh and Morita − − σ vi p si i (2007) study entry in Salinger-type model (with downtream , with equality if si > 0. ﬁrms as price-takers). p ε(1 − si ) + η(1 − σi ) 410 ECONOMIC INQUIRY and Rewrite (18) to obtain

− − σ − σ ε p ci si i si i , with equality if σi > 0. ki = kˆi 1 + . p ε(1 − si ) + η(1 − σi ) ε(1 − si ) + η(1 − σi )

Suppose ki = 0, then kv(q/k) = [(v(q/k))/(1/k)] ≈ Thus qv (q/k) −−→ 0. Thus, an agent with ki = 0 will report k→0 kˆ = 0. Similarly, an agent with γ = 0 produces zero. This ε i i k s − σ yields (14) and (15). i = s 1 + i i . n ˆ i ε − + η − σ j=1 kj (1 si ) (1 i )

Proof of Corollary 1 Thus

Supposeγ ˆ i > γˆ j . Then Equation (6) implies that xi > ε x and hence that σ > σ . It then follows from the first- n k n s − σ j i j i=1 i = + i i n si 1 . order conditions of firms i and j that kˆ ε(1 − si ) + η(1 − σi ) i=1 i i=1 xi xj xi xj γî γˆj c γj . A similar calculation gives γi γj γi γj γi γj

−η n γ n s − σ The reasoning for buyers is similar. i=1 i = σ + i i n i 1 . = γˆi ε(1 − si ) + η(1 − σi ) Proof of Theorem 3 : It is readily checked that the i 1 i=1 following substitutions hold, even when a share is zero.

Note that, with constant elasticity, actual quantity is n n n v si − c σi = (v − p)si i i i n η/(ε+η) n ε/(ε+η) i=1 i=1 i=1 Q = kˆi γˆ i , n n n i= i= + − σ = − + − σ 1 1 (p ci ) i (vi p)si (p vi ) i i=1 i=1 i=1 n n and = (v − p)s − (v − p)σ i i i i i=1 i=1 n η/(ε+η) n ε/(ε+η) n Qf = ki γi . = − − σ = = (vi p)(si i ) i 1 i 1 i=1 n p(s − σ )2 Substitution gives (19). One obtains (20) from = i i . ε(1 − si ) + η(1 − σi ) i=1

p v Q/ n kˆ = i=1 i Proof of Theorem 4 n pf v Qf/ i=1 ki Applying (8), (16) and (18): −1/ε 1/ε Q n k Q n kˆ = i=1 i = f i=1 i n n Qf kˆ Q = ki i=1 i i 1 −1/ε ⎡ ⎤ si Q si Q η ε+η 1/ε = v n /( ) n ε/(ε+η) n k γ kˆ ki ki = ⎣ i=1 i i=1 i i=1 i ⎦ n ˆ n γ n − σ i=1 ki i=1 î i=1 ki = + si i p 1 ε(1 − si ) + η(1 − σi ) 1/(ε+η) n γ n kˆ = i=1 i i=1 i n n Q si − σi γ = v 1 + . i=1 ˆ i i=1 ki n ε − + η − σ = kî (1 si ) (1 i ) i 1 ⎡ ⎤ 1 −σ −η ε+η n σ + si i ⎢ i=1 i 1 ε(1−s )+η(1−σ ) ⎥ = ⎣ i i ⎦ −σ ε . This readily gives the first part of (18); the second half n si i = si 1 + is symmetric. i 1 ε(1−si )+η(1−σi ) HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 411

Proof of Theorem 5 K dQ = (w − w ) 1 − s + s i i i Q dK Using the market calculations (22) and (23), rewrite K dQ proﬁts to obtain +(c − c )σ − β−1ws i i Q dK i

−1 −1 K dQ π = − +(β w si + η c σi ) i r(Q)qi ki w(qi /ki ) Q dK −γ c(x /γ ) − p(q − x ) i i i i i = − − + K dQ (w wi ) 1 si si = − − Q dK (r(Q) p)qi ki w(qi /ki ) K dQ K dQ +px − γ c(x /γ ) +(c − c )σ − r Bs − (Bs + Cσ ) i i i i i i Q dK i i i Q dK = w (Q/K)qi − ki w(qi /ki ) = − − + K dQ (w wi ) 1 si si +c (Q/ )xi − c(xi /γi ) Q dK ˆ ˆ + − σ K dQ ki ki Q (c ci ) i = w (Q/K) Q − ki w Q dK K K ki − − K dQ − σ K dQ γˆ i γˆi Q r Bsi 1 C i . +c (Q/ ) Q − γi c . Q dK Q dK γi

Similarly, and symmetrically, The equilibrium quantity is given by

∂π dQ = i = − r(Q)− w (Q/K) − c (Q/ ) = 0. 0 (w wi )si Q ∂γˆi Q d + − − σ + σ dQ (c ci ) 1 i i From this equation, and applying (24), it is a routine Q d computation to show: dQ dQ −r Cσ 1 − − Bs . i Q d i Q d − 2 K dQ = K (w )Q/K Q dK Q r − w /K − c / These equations can be expressed, substituting the elas- − β−1 ticities with respect to capacity, as = (r p) rα−1 + (r − p)β−1 + pη−1 B = . A + B + C A + B + C − si (A + C) Bσi Csi A + B + C − σi (A + B) Similarly, − + − σ × (w wi )/r = Bsi (A C) BC i − σ + − . (c ci )/r c i (A B) BCsi dQ C = . Q d A + B + C The determinant of the left-hand-side matrix is given by

Differentiating πi , and using the analogous notation for

DET = [A + B + C − si (A + C)] π × + + − σ + − σ = K ∂ i = − − + K dQ [A B C i (A B)] BCsi i 0 (w wi ) 1 si si Q ∂kˆi Q dK = (A + B + C)[(A + B + C)(1 − si )(1 − σi ) si Q K dQ −w + (c − c )σ + − σ + − σ K i i Q dK Bsi (1 i ) C(1 si ) i ] Q Q K dQ = (A + B + C)[A(1 − si )(1 − σi ) + ws + cσ i K i Q dK +B(1 − σi ) + C(1 − si )]. 412 ECONOMIC INQUIRY

Thus,

− 1 + + − σ + − σ + − σ (w wi )/r = A B C i (A B) B i Bsi (A C) BC i − − + + − + σ + − (c ci )/r DET Csi A B C si (A C) c i (A B) BCsi + + − σ + + − σ − σ σ + − = 1 (A B C i (A B))(Bsi (A C) BC i ) B i (c i (A B) BCsi ) DET −Csi (Bsi (A + C) − BCσi ) + (A + B + C − si (A + C))(cσi(A + B) − BCsi ) A + B + C (Bs (A + C) − BCσ ) − ABs σ A + B + C B[C(s − σ ) + As (1 − σ )] = i i i i = i i i i . DET (Cσi (A + B) − BCsi ) − ACsi σi DET C[B(σi − si ) + Aσi (1 − si )]

Thus, The efﬁcient solution satisﬁes r − w Qf − ki n − − + − σ Qf (r(Q) p wi )si (p ci ) i c γ = 0, or MHI = i = r i 1 − β 1/β n 1/ − /α Qf −β n (w − w )s + (c − c )σ 0 = Q 1 − r s [1 − θ − ψ ] = i i i i f Q i i r i=1 i=1 − η 1/η n 1/ n + + Qf −η A B C − r σi [θ − χi ] . = (si B[C(si − σi ) Q DET i=1 i=1 = −1/α +Asi (1 − σi )] + σi C[B(σi − si ) Thus, substituting and dividing by r Q − α β −1/β +Aσ (1 − s )]) Q 1/ Q 1/ n i i 0 = f − f s [1 − θ − ψ ]−β Q Q i i i=1 n A + B + C = − σ 2 − η [BC(si i ) 1/η n 1/ DET Qf −η i=1 − σ [θ − χ ] , Q i i i=1 +ABs2(1 − σ ) + ACσ2(1 − s )] i i i i or n + β −1/β BC(s −σ )2+ABs2(1−σ )+ACσ2(1−s ) A (1/ ) n = i i i i i i . = Qf − θ − ψ −β A(1−si )(1−σi )+B(1−σi )+C(1−si ) 1 si [1 i ] = Q i 1 i=1 + η −1/η The Constant Elasticity Case Q A (1/ ) n + f σ [θ − χ ]−η , Q i i i=1 ⎛ ⎞ B[C(si −σi )+Asi (1−σi )] or − −σ + −σ + − w − w A(1 si )(1 i ) B(1 i ) C(1 si ) − /β i = r ⎝ ⎠ −(A+(1/β)) n 1 − σ − + σ − Q −β c ci C[B( i si ) A i (1 si )] 1 = s [1 − θ − ψ ] − −σ + −σ + − i i A(1 si )(1 i ) B(1 i ) C(1 si ) Qf i=1 ψ − η ≡ r i . −(A+(1/η)) n 1/ χ Q −η i + σ [θ − χ ] . Q i i f i=1 Then 1/β This equation can be solved for the ratio Qf/Q,which si Q = = − ψ = − θ − ψ yields the underproduction. wi w r i r(1 i ), ki or Proof of Theorem 6

−β ki = si Q[r(1 − θ − ψi )] . First, we need to define a couple of elasticities. Differ- entiating the equilibrium condition, Similarly, 1/η 1/α Q −η Q = , γi = σi Q[r(θ − χi )] . HENDRICKS & MCAFEE: THEORY OF BILATERAL OLIGOPOLY 413 with respect to yields the supply elasticity SIAM Journal on Control and Optimization, 41(4), 2002, 1212–28. ∂Q α = . Anderson, E. J., and H. Xu. “Supply Function Equilibrium Q ∂ η + α in Electricity Spot Markets with Contracts and Price Caps.” Journal of Optimization Theory and Applica- Similarly, define the equilibrium price elasticity tions, 124, 2005, 257–83. ∂p −1 Baldick, R., R. Grant, and E. Kahn. “Theory and Applica- = . p ∂ η + α tion of Linear Supply Function Equilibrium in Elec- tricity Markets.” Journal of Regulatory Economics, 25, Differentiating firm i’s profits with respect to its capacity 2004, 143–67. report and substituting the above elasticities into the first- Bernheim, Douglas, and Michael Whinston. “Multimarket order condition, we obtain Contact and Collusive Behavior.” Rand Journal of Economics, 21(1), 1990, 1–26. Borenstein, Severin, and James Bushnell. “An Empirical ∂p ∂Q 1 v Analysis of the Potential for Market Power in Cali- 0 = [Qσ − q ] + p σ + pQ − i ∂ i i ∂ i fornia’s Electricity Market.” Journal of Industrial Eco- nomics, 47(3), 1999, 285–323. 1 v ∂Q Borenstein, Severin, James Bushnell, and Frank Wolak. −c − i X + σ i i ∂ “Measuring Market Inefficiencies in California’s Restructured Wholesale Electricity Market.” American ∂p p − c ∂Q Economic Review, 92(5), 2003, 1376–1405. = [σ − s ] + i σ + (1 − σ ) ∂ p i i p ∂ Q i i Bresnahan, Timothy F. “Empirical Studies of Industries with Market Power,” in Handbook of Industrial −(σ − s ) (p − c ) η(1 − σ ) + α Organization, Vol. II, edited by R. Schmalensee and = i i . + i i . η + α p η + α R. D. Willig. Amsterdam: North-Holland, 1989. Brunekreeft, G. “A Multiple-Unit, Multiple-Period Auction Proof of Theorem 7 in the British Electricity Market.” Energy Economics, 23(1), 2001, 99–118. ∗ Bushnell, James B., Erin T. Mansur, and Celeste Saravia. Let vi denote the equilibrium pre-merger reports and n “Vertical Arrangements, Market Structure, and Com- ∗ = ∗ let −12 vi denote the aggregate capacity reported by petition: An Analysis of Restructured U.S. Electricity i=2 = Markets.” American Economic Review, 98(1), 2008, sellers 2 through n.Definez12 v12/v1 and assume, without 237–66. ≥ loss of generality, that v1 v2. Fixing its rival reports to Clay, Karen, and Werner Troesken. “Further Tests of Static their pre-merger levels, the merged firm’s equilibrium best Oligopoly Models: Whiskey, 1882-1898.” Journal of reply solves Industrial Economics, 51(2), 2003, 151–66. de Fontenay, Catherine C., and Joshua Gans. “Vertical εη + αη z12 Integration in the Presence of Upstream Competition.” ν + −1 ∗ + εη + αη + αε Rand Journal of Economics ( 1 v2) −12 z12 , 36(2), 2005, 544–72. Farrell, Joseph, and Carl Shapiro. “Horizontal Mergers: An − 1/η Equilibrium Analysis.” American Economic Review, 1 z12 = . 80, 1990, 107–26. η + η − 1/η (( 1)/ ) z12 Gans, Joshua. “Concentration-Based Merger Tests and Ver- tical Market Structure.” Journal of Law & Economics, The fact that 50(4), 2007, 661–80. Genesove, David, and Wallace P. Mullin. “Testing Static ν + −1 ∗ −1 ∗ + ∗ ( 1 v2) −12 Industrial Organization, 52(4), 2004, 535–46. ical Economy, 100, 1992, 929–53. Anderson, E. J., and A. B. Philpott. “Using Supply Func- Hart, Oliver, and Jean Tirole. “Vertical Integration and tions for Offering Generation into an Electricity Mar- Market Foreclosure.” Brookings Papers on Eco- ket.” Operations Research, 50(3), 2002, 477–89. nomic Activity: Microeconomics, Special Issue 1990, Anderson, E. J., and H. Xu. “Necessary and Sufficient Con- 205–76. ditions for Optimal Offers in Electricity Markets.” Hastings, J. “Vertical Relationships and Competition in Retail Gasoline Markets: Empirical Evidence from 414 ECONOMIC INQUIRY

Contract Changes in Southern California.” American Salinger, Michael. “Vertical Mergers and Market Foreclo- Economic Review, 94(1), 2004, 317–28. sure.” Quarterly Journal of Economics, 103(2), 1988, Holmberg, P. “Unique Supply Function Equilibrium with 345–56. Capacity Constraints.” Working Paper, University of Salop, Steven, and David Scheffman. “Cost Raising Strate- Uppsala, 2004. gies.” Journal of Industrial Economics, 36(1), 1987, Holmberg, P. “Asymmetric Supply Function Equilibrium 19–34. with Constant Marginal Costs”, Working Paper, Uni- Segal, Igal. “Contracting with Externalities.” Quarterly versity of Uppsala, 2005. Journal of Economics, 114(2), 1999, 337–88. Hortacsu, Ali, and Steven L. Puller. “Understanding Strate- Sweeting, Andrew. “Market Power in the England and gic Bidding in Restructured Electricity Markets: Wales Wholesale Electricity Market 1995-2000.” Eco- A Case Study of ERCOT.” Rand Journal of Eco- nomic Journal, 117(520), 2007, 654–85. nomics, 39(1), 2008, 86–114. Turnbull, S. “Choosing Duopoly Solutions by Consistent Klemperer, Paul, and Margaret Meyer. “Supply Function Conjectures and by Uncertainty.” Economics Letters, Equilibria in Oligopoly Under Uncertainty.” Econo- 7, 1981, 144–48. metrica, 57(6), 1989, 1243–77. Villa-Boas, S. B. “Vertical Contracts Between Manu- Leffler, Keith, and Barry Pulliam. “Preliminary Report to facturers and Retailers: Inference with Limited the California Attorney General Regarding California Data.” Review of Economic Studies, 74(2), 2007, Gasoline Prices.” November 22, 1999. 625–52. Mansur, Erin T. “Upstream Competition and Vertical Inte- Villa-Boas, S. B., and R. Hellerstein. “Identification of Sup- gration in Electricity Markets.” Journal of Law & Eco- ply Models of Retailer and Manufacturer Oligopoly nomics, 50(4), 2007, 125–56. Pricing.” Economics Letters, 90(1), 2006, 132–40. McAfee, R. Preston, and M. Schwartz. “Opportunism in Villa-Boas, J. M., and Y. Zhao. “Retailers, Manufacturers, Multilateral Vertical Contracting: Nondiscrimination, and Individual Consumers: Modeling the Supply Exclusivity, and Uniformity.” American Economic Side in Ketchup Marketplace.” Journal of Marketing Review, 84(1), 1994, 210–30. Research, 42, 2005, 83–95. McAfee, R. Preston, and Michael Williams. “Horizontal Wolak, Frank. “An Empirical Analysis of the Impact of Mergers and Antitrust Policy.” Journal of Industrial Hedge Contracts on Bidding Behavior in a Com- Economics, XL, 1992, 181–87. petitive Electricity Market.” International Economic McAfee, R. Preston, Joseph Simons, and Michael Williams. Review, 14(2), 2000, 1–39. “Horizontal Mergers in Spatially Differentiated Nonco- Wolak, Frank. “Measuring Unilateral Market Power in operative Markets.” Journal of Industrial Economics, Wholesale Electricity Markets: The California Market, XL, 1992, 349–57. 1998-2000.” American Economic Review, May 2003, Mortimer, J. H. Forthcoming. “Vertical Contracts in the 425–430. Video Rental Industry.” Review of Economic Studies, Wolak, Frank. “Identification and Estimation of Cost forthcoming. Functions Using Observed Bid Data: An Application O’Brien, D. P., and Greg Schaffer. “Vertical Control with to Electricity Markets,” in Advances in Economics and Bilaternal Contracts.” Rand Journal of Economics, Econometrics: Theory and Applications, Eighth World 23(3), 1992, 299–308. Congress, Vol. II, edited by M. Dewatripont, L. P. Ordover, Janusz, Garth Saloner, and Steven Salop. “Equi- Hansen, and S. J. Turnovsky. New York: Cambridge librium Vertical Foreclosure.” American Economic University Press 2003, 133–69. Review, 80(1), 1990, 127–42. Wolak, Frank. “Quantifying the Supply-Side Benefits Porter, R. H. “A Study of Cartel Stability: The Joint Exec- from Forward Contracting in Wholesale Electricity utive Committee, 1880-1886.” Bell Journal of Eco- Markets.” Journal of Applied Econometrics, 22, 2007, nomics, 14, 1983, 301–14. 1179–1209. Rey, Patrick, and Jean Tirole. “A Primer on Foreclosure.” Wolfram, Catherine D. “Measuring Duopoly Power in the Handbook of Industrial Organization, Vol. III. Ams- British Electricity Spot Market.” American Economic terdam: North-Holland, 2008. Review, 89(4), 1999, 805–26. Copyright of Economic Inquiry is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.