Impact of Mergers on Consumer Surplus in Imperfectly Competitive Industries: Role of Forwa, Lecture notes of Statistics

Download Impact of Mergers on Consumer Surplus in Imperfectly Competitive Industries: Role of Forwa and more Lecture notes Statistics in PDF only on Docsity! Forward Contracts, Market Structure, and the Welfare Effects of Mergers∗ Nathan H. Miller† Georgetown University Joseph U. Podwol‡ U.S. Department of Justice June 28, 2019 Abstract We examine how forward contracts affect economic outcomes under generalized market structures. In the model, forward contracts discipline the exercise of market power by making profit less sensitive to changes in output. This impact is greatest in markets with intermediate levels of concentration. Mergers reduce the use of forward contracts in equi- librium and, in markets that are sufficiently concentrated, this amplifies the adverse effects on consumer surplus. Additional analyses of merger profitability and collusion are provided. Throughout, we illustrate and extend the theoretical results using Monte Carlo simulations. We discuss the practical relevance for antitrust enforcement. Keywords: forward contracts; hedging; mergers; antitrust policy JEL classification: L13; L41; L44 ∗This paper subsumes an earlier working paper titled “Forward Contracting and the Wel- fare Effects of Mergers” (2013). We thank Jeff Lien and Jeremy Verlinda for valuable com- ments and Jan Bouckaert, Patrick Greenlee and Louis Kaplow for helpful discussions. The views expressed herein are entirely those of the authors and should not be purported to reflect those of the U.S. Department of Justice. †Georgetown University, McDonough School of Business, 37th and O Streets NW, Wash- ington, DC 20057. Email: nhm27@georgetown.edu. ‡U.S. Department of Justice, Antitrust Division, Economic Analysis Group, 450 5th St. NW, Washington, DC 20530. Part of this research was conducted while I was the Victor Kramer fellow at Harvard Law School, Cambridge, MA. Email: joseph.podwol@usdoj.gov. 1 Introduction A long-standing result in the theoretical literature is that forward markets can increase output and lower prices in imperfectly competitive industries (Allaz and Vila (1993)). Underlying the result is that forward sales discipline the ex- ercise of market power in the spot market by making profit less sensitive to the changes in output. Little attention has been played, however, to the role of com- petition in determining the magnitude of these effect, as existing articles focus on symmetric duopoly. This limits the usefulness of the literature for merger review in industries such as wholesale electricity, where forward commitments are a prominent feature.1 In the present study, we examine the effects of for- ward markets under generalized market structures, and obtain results that are of practical relevance to antitrust practitioners. Our model features an oligopolistic industry in which firms sell a homoge- neous product and compete through their choices of quantities. Competition happens first in one or more contract markets, and later in a spot market. Fol- lowing Perry and Porter (1985), firms have heterogeneous marginal cost sched- ules that reflect their respective capacities. The model can incorporate any arbitrary number of firms and any combination of capacities, and thereby facil- itates an analysis of market structure. Thus, we bring together two established theoretical literatures: one on strategic forward contracts (e.g., Allaz and Vila (1993)), and the other on the effects of horizontal mergers with homogeneous products (e.g., Perry and Porter (1985); Farrell and Shapiro (1990)). We establish that the presence of forward markets weakly increases aggregate output in equilibrium, relative to a Cournot benchmark, regardless of market structure. Forward markets allow firms to make strategic commitments, and the ensuing competition for Stackleberg leadership increases output relative to a Cournot baseline. This effect is largest for intermediate levels of market con- centration, and converges to zero as market structure approaches the limit cases of monopoly and perfect competition. The non-monotonicity arises because in- creasing the number of firms intensifies the competition for Stackleberg leader- ship and thereby pushes the industry toward a perfectly competitive equilibrium faster than would be the case under Cournot competition. However, there are 1Recent wholesale electricity mergers investigated by the U.S. Department of Justice in- clude Exelon/PSEG in 2005, FirstEnergy/Allegheny Energy and Mirant/RRI in 2010, and Exelon/Constellation in 2012. The academic literature has emphasized the importance of forward commitments in these settings (Morris and Oska (2008), Wolak and McRae (2009)). Anderson and Sundaresan (1984) and Newberry (1984) discuss other imperfectly competitive industries characterized by forward markets, such as tin, aluminum, copper, coffee, and cocoa. 1 with infinitely many contracting rounds. Liski and Montero (2006) consider the role of forward contracting in sustaining collusive outcomes. Breitmoser (2012) allows firms to pre-order inputs. Breitmoser (2013) shows that if firms have upward-sloping marginal costs then the competitive effects of forward markets are diminished. Ritz (2014) shows that the Allaz and Vila model of forward con- tracts is strategically equivalent to a model of managerial delegation in which managers maximize an endogenously-determined mix of profit and revenue. Em- pirical evidence on the importance of forward contracting is presented in Wolak (2000), Bushnell (2007), Bushnell, Mansur and Saravia (2008), Hortacsu and Puller (2008) and Brown and Eckert (2017). Among the aforementioned studies, the closest to our research are Bushnell (2007), Breitmoser (2013), and Brown and Eckert (2017). Bushnell examines the welfare impact of a forward market for a symmetric N -firm oligopoly, in- creasing marginal costs, and a single round of forward contracting. The obtained results suggest that the impact of forward contracting is maximized for inter- mediate levels of competition, a result we generalize substantially.2 Breitmoser (2013) examines a symmetric duopoly model with increasing marginal costs and arbitrarily-many rounds of forward contracting. Again we provide the general- ization to asymmetric oligopoly. Finally, Brown and Eckert (2017) simulate the effects of an electricity merger in Canada and determine that forward markets amplify post-merger price increases. We prove this is a specific rather than general result, as forward markets can mitigate or amplify merger effects. The paper proceeds as follows. Section 2 describes the model of multistage quantity competition and solves for equilibrium strategies using backward in- duction. Section 3 analyzes the welfare impact of forward contracting, showing that the welfare impact of a forward market is non-monotonic in concentration. Section 4 formally models the welfare impacts of mergers highlighting how the results differ from the baseline model of Cournot competition. Section 5 pro- vides an extension to collusion and Section 6 concludes with a discussion of the applicability of our results. 2The empirical focus of Bushnell (2007) is on deregulated electricity markets. The model is calibrated to match market data and used to assess the impact of forward markets on equilibrium prices and output. 4 2 Model 2.1 Overview We consider a modified Cournot model that features T contracting stages. The model is a variant of Allaz and Vila (1993) but we allow for an arbitrary num- ber of producers with heterogeneous production technologies as in McAfee and Williams (1992). In each of T periods prior to production, firms can contract at a set price to buy or sell output to be delivered at time t = 0. Denote each of these contracting stages as T, . . . , t, . . . , 1 such that stage t occurs t periods be- fore production. Following the conclusion of each contracting stage, contracted quantities are observed by all market participants and are taken into account in the subgame that follows. At t = 0, production takes place, contracts are set- tled, and producers compete via Cournot to sell any residual output in the spot market. The solution concept is Subgame Perfect Nash Equilibrium (“SPE”). Formally, let f ti denote the quantity contracted by producer i ∈ {1, . . . , N} in stage t, and let qti = ∑T τ=t+1 f τ i denote the producer’s forward position at the beginning of period t. Forward contracts in stage t are agreed upon taking as given the forward price, P t, and the vector of forward positions, qt = {qt1, ..., qtN}, and with knowledge of the corresponding subgame equilibrium that follows. At t = 0, each producer sells qsi in the spot market taking into account the vector of forward positions q0 = { q01 , ..., q 0 N } and given other producers’ output. This determines the producer’s output, qi, as the sum of its contracted and spot sales. Producers are “short” in the spot market if q0i > 0. Total output is the sum of all firms’ output and is denoted Q = ∑ i qi. Buyers are passive entities and are represented by the linear inverse demand schedule P (Q) = a− bQ, for a, b > 0. Each producer i is characterized by its capital stock, ki, a proxy for its productive capacity. Total costs are Ci (qi) = cqi + q2i /2ki, so that marginal costs, C ′i (qi) = c+ qi/ki, are increasing in output but decreasing in the capital stock. As a result, firms with greater capital stocks will have higher market shares owing to this cost advantage. We assume a > c ≥ 0 to ensure that gains to trade exist. 2.2 Spot market subgame Solutions are obtained via backward induction: first considering the output deci- sions of producers in the spot market, given any vector of contracted quantities, 5 and then considering the contract market. The spot price is determined by total output, Q ( q0 ) , which is itself a function of the vector of forward positions, q0. Producer i chooses its output, qi (the sum of forward and spot market quanti- ties), taking as given q0 as well as the vector of other producers’ output, q−i, to maximize the profit function, πsi ( qi; q 0,q−i ) = P ( Q ( q0,q−i )) ( qi ( q0,q−i ) − q0i ) − Ci ( qi ( q0,q−i )) . Suppressing dependence on q0 and q−i, the first-order condition implies that P (Q) + ( qi − q0i ) P ′ (Q) = C ′i (qi) . (1) If the producer holds a short position (i.e. q0i > 0), then the inclusion of q0i in equation (1) says that, relative to Cournot, revenue is less sensitive to output because selling an additional unit has no effect on the price received from forward sales. This amounts to an outward shift in the firm’s marginal revenue function, holding fixed the output of other producers.3 If competing producers increase their output relative to Cournot due to their own forward positions, this will cause i’s marginal revenue function to shift back somewhat. We derive closed-form expressions for equilibrium price and quantities by making use of the following terms: βi = bki bki + 1 , B = ∑ i βi, B−i = ∑ j 6=i βj , F 0 = ∑ i βiq 0 i , F 0 −i = ∑ j 6=i βjq 0 j , Proposition 1 In the spot market subgame with vector of forward positions, q0, there exists a unique Nash equilibrium in which price, total output and individual firms’ output are given by: P ( q0 ) = c+ a− c 1 +B − bF 0 1 +B Q ( q0 ) = ( a− c b ) B 1 +B + F 0 1 +B qi ( q0 ) = ( a− c b ) βi 1 +B + βi 1 +B [ (1 +B−i) q 0 i − F 0 −i ] 3Anderson and Sundaresan (1984) use this very argument to show that given a short forward position, a monopolist will necessarily increase output relative to Cournot. They rely on risk aversion to explain why a monopolist would hold a short position in the first place. 6 behavior in the subgame beginning in stage τ is, Rτ+1 i = − Mτ −i 1 +Mτ −i . We can use Lemma 1 to show how the firm’s problem is impacted by the presence of a forward market. It is evident that the marginal revenue curve facing firm i in the contract market as expressed in equation (2) is flatter in own output than it would be under Cournot. Since 1 + Rτi < 1, a marginal increase in firm i’s contracted quantity does not reduce the price by as much as it would under Cournot because other firms respond by reducing their own output. Holding all other firms’ output fixed at their Cournot levels and assuming no forward position in period τ (i.e., qτi = 0), the inclusion of 1+Rτi in equation (2) pivots firm i’s marginal revenue curve up from the vertical axis, which suggests firm i will increase output relative to Cournot. As we saw in the spot market subgame, incorporating a short position shifts the firm’s marginal revenue curve outward, thereby reinforcing this effect. However, if the same incentives facing firm i lead other firms to increase their output relative to Cournot, firm i’s marginal revenue curve shifts down because quantities are strategic substitutes. This shift curbs firm i’s incentive to increase output relative to Cournot and may even decrease it if other firms increase their output by a large enough amount. We can now derive the equilibrium of the full game. Let Mτ = ∑ i µ τ i for any τ ≥ 1 and for completeness of notation, let Rti = 0 for all i when t = 0. Proposition 2 There exists a unique SPE of the game beginning in period T such that in each period, a producer anticipates producing qi and sells a strictly positive fraction of its uncommitted anticipated output which rationalizes qi as an equilibrium. The equilibrium is characterized by a vector of outputs, {qi}i, a sequence of forward sales, {f ti }i,t, total output, Q, and price, P , satisfying: 9 qi = ( a− c b ) µTi 1 +MT fτi = Rτ−1i −Rτi 1 +Rτ−1i ( qi − T∑ t=τ+1 f ti ) Q = ( a− c b ) MT 1 +MT P = c+ a− c 1 +MT Absent a contract market (i.e., Rti = 0 ∀ i, t), µTi and MT reduce to βi and B, respectively, so that the price and quantities in Proposition 2 collapse to their values in the Cournot game of McAfee and Williams (1992). We can assess the impact of a forward market more broadly by analyzing changes in equilibrium outcomes as T increases from zero as in Cournot to positive values. We have that, Corollary 1 For any T ∈ {0, 1, . . .}, price is (weakly) lower and total output is (weakly) higher in the SPE of the game with T + 1 contracting rounds than with T . Each inequality is strict outside of the monopoly case. An individual producer’s output can nevertheless be lower in the game with T + 1 contracting rounds relative to T if its capital stock is sufficiently small relative to that of its competitors. Allaz and Vila (1993) provide a special case of this result for a symmetric two-firm oligopoly. When firms are symmetric, our model shows that all firms increase their output as T increases, as they do in Allaz and Vila (1993). Corol- lary 1 shows that this may no longer be the case when firms are asymmetric. This result suggests that the introduction of a forward market may increase concentration as measured by output, even as it improves welfare. The impact of the forward market on output can be substantial. Consider the special case with a single contracting stage (T = 1) and constant marginal cost, which we model as the limiting case as capital stocks become infinite. In this case, βi = 1 so that R1 i = −N−1N , µ1 i = N , and M1 = N2. The presence of a forward market increases output by 140 percent when N = 2 and by nearly 600 percent when N = 6. These increases would be somewhat smaller if 10 marginal costs were instead increasing (ki <∞) and larger with multiple rounds of contracting (T > 1). 3 Market Structure and Welfare We now examine the role of market structure in evaluating the impact of a for- ward market on welfare. Whereas Allaz and Vila (1993) showed that welfare can span duopoly-Cournot to perfect competition levels as the number of contract- ing rounds increases, our focus is on how the welfare impact of a forward market is influenced by market structure. As such, we treat T as fixed, determined by the particulars of the industry.7 3.1 Market structure and hedge rates The welfare impact of a forward market is related to the fraction of each firm’s output that is contracted in the forward market, i.e. its “hedge rate.” The following result aids the understanding of this relationship. Lemma 2 Given equilibrium strategies within the SPE of the (T + 1)-stage game, the hedge rate can be expressed as hi ≡ q0i qi = |RTi | = MT−1 −i 1+MT−1 −i . The result is fairly general in that the first equality in Lemma 2, hi = |RTi |, does not rely on the shape of the demand or cost functions. It follows from the fact that a firm, when deciding how much to supply on the contract mar- ket, takes into account that a marginal increase in supply will be met by a decrease in its competitors’ sales in subsequent periods. Thus, while a marginal increase in contracted supply on its own causes the price to decline, the corre- sponding decrease in competitors’ outputs partially offsets this. The optimum equates marginal revenue across each of T + 1 stages much in the way that a third-degree price discriminating monopolist equates marginal revenue across customer segments. A firm’s hedge rate depends at a first order on the amount of capital stock controlled by its competitors as well as the distribution of capital stock among them.8 Competitors with larger capital stocks produce more irrespective of 7Bushnell (2007) discusses the institutional details of forward sales within wholesale elec- tricity. 8In the game with T = 1 contracting stages, a firm’s hedge rate, hi = B−i/(1 + B−i), depends only on the capital stocks of its competitors. But when T > 1, the hedge rate depends on µT−i, each of which depends on firm i’s capital stock through its influence on every other firm’s hedge rate. The effect of βi on h−i is of a second-order magnitude, however. 11 In the remainder of this section, we use numerical techniques to illustrate the result. We first compare the welfare statistics obtained with T = 1 rounds of forward contracting to those obtained in Cournot equilibrium (T = 0). To do so, we create data on 9,500 “industries,” evenly split between N = 1, 2, . . . , 20. For each industry, we calibrate the structural parameters of the model (a, b, c, k) such that Cournot equilibrium exactly matches randomly-allocated market shares, an average margin, and normalizations on price and total output.11 We then obtain the welfare statistics that arise in Cournot equilibrium and with a single round of forward contracting. Consumer surplus and total surplus can be expressed as functions of total output and the average price-cost margin:12 CS = b 2 Q2 TS = Q 2 [ a− c+ ∑ i si ( P − C ′ i )] (5) Figure 1 summarizes the results of the numerical exercise. In each panel, the vertical axis provides the ratio of surplus with forward contracting to sur- plus with Cournot. The horizontal axes shows the Herfindahl-Hirschman index (“HHI”). The HHI is the sum of squared market shares, which attains a maxi- mum of unity in the monopoly extreme and asymptotically approaches zero as the market approaches perfect competition. The HHI is an appealing statistic due to its well-known theoretical connection to welfare in the baseline Cournot model;13 it also features prominently in the Merger Guidelines of the U.S. De- partment of Justice and Federal Trade Commission. In the graphs, each dot represents a single industry, and the lines provide nonparametric fits of the data. As shown, consumer surplus and total surplus are greater with forward con- tracting than with Cournot (because all dots exceed unity). Further, consistent with Corollary 2, the impact of a forward market is greatest at intermediate levels of competition.14 The gain in consumer surplus is maximized at an HHI 11We normalize P = Q = 100 and use an average margin of 0.40. 12Derivations are in the Appendix. 13Notice that when all hi = 0, LI = HHI/ε. 14As there is not a one-to-one correspondence between HHI and consumer or total surplus, we view these results as illustrative. The advantage to using HHI to measure concentration is that it offers a complete ordering of any two capital allocations and hence allows us to plot the results. In the following section, we analyze a more theoretically-robust measure of concentration that, while it does not offer a complete concentration-ordering of allocations, it confirms the interpretation of Figure 1 that the ratio of consumer surplus with forward contracting to consumer surplus with Cournot is increasing (decreasing) in concentration at 14 1 1.05 1.1 1.15 1.2 1.25 1.3 Pe rfo rm an ce R el at iv e to C ou rn ot 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 HHI Consumer Surplus .4 .5 .6 .7 .8 .9 1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 HHI Producer Surplus 1 1.01 1.02 1.03 1.04 1.05 1.06 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 HHI Total Surplus Figure 1: Welfare Statistics with Heterogeneous Capital Stocks around 0.30, which corresponds roughly to a symmetric three firm oligopoly. The gain in total surplus is maximized at an HHI around 0.40, between the symmetric triopoly and duopoly levels. The figure also shows that forward markets diminish producer surplus, particularly in unconcentrated markets. It is also possible to compare the welfare statistics that arise with forward contracts to those obtained with perfect competition. This is especially tractable in the special case of symmetric firms and constant marginal costs. Constant marginal costs are obtained as the limiting case as capital stocks become infinite so that βi → 1. The expressions in (5) can be presented as functions of the common hedge rate: CS(h(T )) = (a− c)2 2 ( N N + 1− h(T ) )2 TS(h(T )) = (a− c)2 2 ( N N + 1− h(T ) − 1 2 ( N N + 1− h(T ) )2 ) low (high) levels of concentration. 15 The analogous expressions with perfect competition are CS(1) = TS(1) = 1 2 (a− c)2. Thus, the levels of consumer surplus and of total surplus with for- ward contracts, relative to perfect competition, are free of the demand and cost parameters and depend only on the number of firms and the hedge rate. This holds for any given hedge rate, including the SPE rates h(T ). Figure 2 plots the ratios CS(h(T ))/CS(1) andW (h(T ))/W (1) for T = 0, . . . , 3. Again, T = 0 corresponds to Cournot competition and h(0) = 0. The horizontal axis in each panel is the number of firms (N = 1, ..., 10) which, under symmetry, is a sufficient statistic for concentration. As shown, consumer surplus and total surplus increase with N under Cournot equilibrium; in the limit as N → ∞ these welfare statistics approach the perfectly competitive level. Incorporating each round of contracting adds curvature to the relationship between surplus and the number of firms, such that surplus approaches the perfectly competi- tive level faster as N grows large. The “gap” between surplus with Cournot and surplus with forward contracts is largest for intermediate N , again consistent with Corollary 2. Lastly, the figure is highly suggestive that forward markets amplify the impacts of market structure changes (e.g., mergers) on welfare in concentrated markets, but diminish impacts otherwise. We provide a more so- phisticated analytical treatment of market structure changes in the next section. 4 Mergers In this section, we analyze the welfare impacts of consolidation, which we treat as the transfer of capital stock from small to large firms. Mergers are inherently consolidating regardless of whether the larger or smaller firm is the acquirer because the merged firm’s capital stock will be larger than either of the merging firms’. Our interest extends beyond mergers to partial acquisitions as many real- world applications involve the sale of individual plants. Even when evaluating full mergers, antitrust authorities must often consider whether and to what extent a partial divestiture might offset the anticompetitive harm. 4.1 Effects on consumer surplus We begin by analyzing the effect of consolidation on consumer surplus. To the extent that antitrust agencies review mergers under a consumer surplus 16 Keeping in mind that µi = βi 1+βiRi (Lemma 1), we have that, dµi =  ( µi βi )2 dβi − µ2 i · dRi if i = 1, 2 −µ2 i · dRi if i 6= 1, 2 Collecting the dβi terms and the dRi terms, respectively, the change in consumer surplus is, dCS = SE +HE, where, SE ≡ a− c (1 +M) 2 [( µ1 β1 )2 dβ1 + ( µ2 β2 )2 dβ2 ] HE ≡ − a− c (1 +M) 2 ∑ i µ2 i · dRi This decomposition allows us to state the following proposition. Proposition 3 All consolidating transfers reduce consumer surplus in the pres- ence of a forward market. The loss of consumer surplus due to a consolidating transfer is mitigated if each firm’s hedge rate remains fixed at its pre-transfer value. That consolidation leads to lower output should not be surprising as the result holds within the baseline model of Cournot competition. What it inter- esting is that the reduction is output is magnified when firms adjust their hedge rates in response to consolidation as they do in the SPE of the two-stage game. This follows from the fact that SE,HE < 0. The structural effect is negative for the standard reasons: The capital transfer leads the inside firms to reduce output, while outside firms react by expanding their output. The total expan- sion across all outside firms only partially offsets the output reduction by the inside firms, leading to a net decrease in industry output.17 The hedging effect follows from Lemma 4 and the subsequent discussion. It is natural to ask whether the effect of a consolidating transfer is more pronounced within the two-stage game relative to the baseline Cournot game. The answer, as suggested by Corollary 2 and Figure 2, depends on the level of concentration in the industry prior to the transfer. Unlike the symmetric case shown in Figure 2, when firms are asymmetric, the number of firms is not a sufficient statistic for concentration. To make progress, we focus on the limit- ing cases of monopoly and perfect competition. Beginning with any arbitrary 17See Farrell and Shapiro (1990) for this result in Cournot oligopoly. 19 allocation of capital k, we say that an alternative allocation, k′, is more concen- trated than k if k′ can be obtained from k via a series of capital transfers from small to large firms.18 According to our definition, the monopoly case, wherein all capital is allocated to firm 1, is more concentrated than any alternative al- location. At the opposite extreme, perfect competition requires a large number of firms with positive capital endowments such that from the standpoint of any one firm, the amount of capital held by all other firms is quite large.19 Suppose that for some arbitrary allocation of capital k, every firm is replicated l-times so that the resulting allocation k′ has l-times as many firms and l-times as much capital stock as k. In that case, we say that k′ is an l-replication of k. Beginning with any arbitrary allocation of capital k, we say that an alternative allocation, k′, is less concentrated than k if k′ can be obtained from k by: (i) a series of capital transfers from large to small firms; followed by (ii) an l-replication of the post-transfer allocation for some positive integer l. Proposition 4 There exists a capital allocation k such that the reduction in consumer surplus due to a consolidating transfer is greater within the SPE of the two-stage model than in Cournot. There exists a capital allocation k′ that is less concentrated than k such that the reduction in consumer surplus due to a consolidating transfer is greater in Cournot than in the SPE of the two-stage model. Proposition 4 says that the welfare effects of consolidating transfers (from firm 2 to firm 1, k1 > k2) within the two-stage model are greater than Cournot in industries that are sufficiently concentrated and smaller than Cournot in indus- tries that are unconcentrated. In the proof we consider consolidating transfers first in unconcentrated markets and then in concentrated markets. In the former case, we formalize the intuition that (i) the high rate of hedging in the two-stage model leads the structural effect to be smaller than the loss of consumer sur- plus under Cournot, and (ii) consolidating transfers do not affect hedging much, leading to a small hedging effect. In the latter case, we formalize that (i) the low rate of equilibrium hedging in the two-stage model leads the structural effect to approach the consumer loss under Cournot, and thus that (ii) the hedging effect, which exacerbates loss in the two-stage model, is determinative. 18Waehrer and Perry (2003) dub this definition of concentration as the “transfer principle.” 19To see this, note that from Proposition 2, price converges to c as MT →∞, which requires: 1) the capital stock to grow arbitrarily large; and 2) that capital stock be allocated across an arbitrarily large set of firms. 20 We revisit the Monte Carlo experiments to illustrate and extend the analyses beyond first-order effects to full mergers.20 We create data on 9,000 industries evenly split between N = 2, 3, . . . , 20, and calibrate the structural parameters of the model to match randomly-allocated market shares, an average margin of 0.40, and normalizations on price and total output. We then simulate eco- nomic outcomes using the obtained structural parameters under the alternative assumption of Cournot competition (T = 0). Finally, to simulate mergers, we combine the capital stocks of the first and second firm of each industry and recompute equilibria both with the two-stage model and with Cournot. Figure 3 summarizes the results. The vertical axis is the loss of consumer surplus in the two-stage model divided by the loss under Cournot; this is greater than unity if forward markets amplify loss. The horizontal axis is the post- merger HHI. Each dot represents a single industry, and the line provides a nonparametric fit of the data. As shown, the relative consumer surplus loss with forward contracts increases in the post-merger HHI, consistent with Proposition 5. The threshold level above which forward contracts tend to amplify consumer surplus loss is around a post-merger HHI of 0.40, roughly between symmetric triopoly and duopoly levels. 4.2 Profitability It is notoriously difficult to analyze the effect of mergers on firm profitability in models such as ours, even in the absence of forward markets (Perry and Porter (1985); Farrell and Shapiro (1990)). Thus, we begin this section with a simple numerical analysis. Revisiting the Monte Carlo exercise described above, we plot the change in the inside firms’ profits against the post-merger HHI. Figure 4 shows the results for the two-stage model (Panel A) and Cournot (Panel B). The striking result is that all mergers within the two-stage model are profitable whereas in Cournot, many are not. We provide the following conjecture: Conjecture 1 All mergers are profitable in the T + 1-stage model. This result may help offer a more complete response to the “merger para- dox.” Salant, Switzer and Reynolds (1983) examined the incentive to merge 20Because Proposition 4 is a statement about first-order effects, it is theoretically ambiguous whether it extends to large transactions including full mergers. For example, it may be the case that allocation k is sufficiently unconcentrated that an incremental transfer would reduce consumer surplus more under Cournot but a larger transfer would reduce consumer surplus more in the contracting model. 21 a small capital transfer leads the inside firms to reduce output, so this change of variables is without loss. Further, fixing the magnitude of the inside firms’ output reduction allows us to focus on the our object of interest, outsiders’ expansion, which we denote, dQO. The change in insiders’ profits is, dπI = g + [ P + ( 1 +RTI ) QIP ′ − C ′I ] dQI + [ dQO/dQI −RTI ] QIP ′dQI (6) The first summand, g (≥ 0) is the cost savings incurred by the inside firms upon rationalizing output across their combined capital assets.23 The second summand, [ P + ( 1 +RTI ) QIP ′ − C ′I ] dQI is the marginal increase in insiders’ profit due to their output reduction.24 The third summand and the focus of our inquiry, [ dQO/dQI −RTI ] QIP ′dQI , is the change in insiders’ profit due to outsiders’ expansion. Since P ′ < 0 and dQI < 0, the change in insiders’ profit due to outsiders’ expansion takes the sign of dQO/dQI −RTI , which we will see, is negative. The term, dQO/dQI , denotes the aggregate expansion in all outsiders’ output to a change in insiders’ output. In Cournot, this is derived from the typical reaction functions where it is assumed that each outsider takes the insiders’ output as given. This interpretation is less straightforward in the presence of a forward market since a firm’s output reflects decisions made in each of T + 1 periods, all but the first of which is influenced by choices made by rivals in prior periods. In this way, a portion of the reaction of outsiders will be internalized by the insiders through the intertemporal effects of forward sales, which is reflected in RTI . Recall from Lemma 4 that outsiders’ hedge rates decline with consolidation. As a result, a merger incrementally increases the insiders’ ability to act as a Stackelberg leader with respect to rivals’ sales in subsequent periods which allows insiders to mitigate the impact of rivals’ expansion on its profit. This intuition is consistent with existing results in a related setting. Daugh- ety (1990) models an industry with symmetric firms and constant marginal costs 23The cost savings is strictly positive when the merging firms are asymmetric pre-merger as marginal unit of output from the smaller firm is produced at a higher cost than the marginal unit from the larger firm. This component is absent from Salant, Switzer and Reynolds (1983) given their focus on the symmetric case and is not instrumental in our results. 24To see this, let C′I denote the inside firms’ marginal cost function evaluated at the pre- merger output and RTI the inside firms’ period-T conjecture. Because insiders reduce their output in equilibrium, it must be the case that at the pre-merger equilibrium output, its marginal cost exceeds its marginal revenue. From the inside firms’ period-T first-order condi- tion, this is equivalent to [ P + ( 1 +RTI ) QIP ′ − C′I ] < 0. Since the pre-merger output puts the insiders on the downward sloping portion of πI with respect to QI , a small decrease in QI increases profit by the slope of πI with respect to QI , [ P + ( 1 +RTI ) QIP ′ − C′I ] , multiplied by the output decrease, dQI . 24 (as in Salant, Switzer and Reynolds (1983)), but where an arbitrary number of firms behave as Stackelberg leaders. In his model, a merger among two Stack- elberg followers that causes the combined firm to become a leader is always profitable. Our model extends this result to incremental changes in leadership behavior.25 To formalize this intution for the current setting, consider the solution for firm j’s problem in period T :26 P (Q) + qj ( 1 +RTj ) P ′ (Q) = C ′j (qj) (7) Differentiating both sides of (7) with respect toQ−j = ∑ k 6=j qk and using P ′ = b and C ′ = c+ qj/kj , we obtain firm j’s reaction function, rj ≡ dqj dQ−j = − µTj 1 + µTj (8) From dqj = rjdQ−j , we have that, dqj (1 + rj) = rj (dqj + dQ−j) = rjdQ, or equivalently, dqj = − ( rj 1 + rj ) dQ = −µTj dQ (9) Summing (9) over all j ∈ O, we have that, dQO = −MT −IdQ. Since dQ = (dQO + dQI) we can rearrange terms so that, dQO dQI = − MT −I 1 +MT −I (10) Using Lemma 2, expression (10) says that −dQO/dQI is equivalent to the inside firms’ hedge rate in the game with T + 1 rounds of contracting, which we denote h (T+1) I . Whereas, Lemma 2 says that −RTI is the inside firms’ hedge rate in a game with T rounds of contracting, h (T ) I , we then have that, dQO/dQI −RTI = − ( h (T+1) I − h(T ) I ) , (11) which is negative. We want to show that expression (11) is larger than the equivalent expression under Cournot, −BI/ (1 +BI). Since −BI/ (1 +BI) ≡ h(1)I from Lemma 2, this 25There are other differences between the two models. In Daughety (1990), mergers can lead to higher output and hence surplus due to the addition of a Stackelberg leader, whereas mergers always lead to lower output in the current setting. 26This expression is the period-T analogue to expression (1). 25 is equivalent to showing that, h (T+1) I − h(T ) I < h (1) I (12) It is sufficient to show that h (τ) I is concave in τ . We know that expression (12) is true for very large T since limT→∞ ( h (T+1) I − h(T ) I ) = 0.27 Establishing this result for intermediate values of T is complicated due to asymmetry. To make progress, we focus on the symmetric case. Proposition 5 When firms are symmetric, h (T+1) I −h(T ) I < hTI for any T ≥ 1, so that all else equal, expansion from outside firms has a smaller effect on insid- ers’ profits in the presence of a forward market. Further, ( h (T+1) I − h(T ) I ) → 0 as T → ∞ so that the impact of outsiders’ expansion becomes negligible when the insiders are able to fully internalize outsiders’ output decisions. We view Proposition 5 as illustrating the mechanism underlying Figure 4. To that end, Proposition Proposition 5 suggests that mergers should be even more profitable with more rounds of forward contracts. 5 Collusion We now investigate collusion in the presence of forward markets. We place the model into a standard repeated-game setting with an infinite number of trading periods indexed t = 0, 1, 2, . . . . In each period, firms simultaneously sell output in a spot market and contract for output up to T periods ahead. The discount factor is δ. Following Liski and Montero (2006), which examines the case of duopoly, we impose constant marginal costs and hence symmetry in order to improve the tractability of the incentive compatibility constraints. We advance the literature primarily by considering an arbitrary number of firms, N . We focus on a particular set of strategies under which firms collectively pro- duce the monopoly output, Qm = (a − c)/2, in each period. Let f t,t+τi denote the quantity contracted by firm i during period t for delivery τ = 1, 2, . . . , T periods later. Along the collusive path, firms trade in the forward market ac- cording to f t,t+1 i = xQm/N and f t,t+τi = 0 for all τ > 1 and trade in the spot market according to qsi = (1 − x)Qm/N . We consider x ∈ [−1, 1] so that firms can be long (x < 0) or short (x > 0) in the spot market. If any firm devi- ates from this collusive path, then competition in all subsequent periods reverts 27This follows from the fact that: 1) hτi is monotonically increase in τ ; and 2) hτi ≤ 1. 26 .3 .4 .5 .6 .7 .8 C rit ic al D is co un t R at e 2 3 4 5 6 7 8 9 10 Number of Firms Cournot Optimal Strategies (x=x*) Figure 5: Effect of Forward Markets on Critical Discount Rates more likely that, in the presence of a forward market, firms will switch from competition to collusion in response to an increase in concentration.28 The relationship shown in Figure 5 derives from the “hedging effect” identi- fied in Section 4, whereby consolidation leads firms to reduce forward sales under the strategies described by Proposition 2, thereby providing an additional boost to profits. Under Cournot, the critical discount rate decreases in concentration because greater concentration causes the incremental gain from deviation rel- ative to cooperation to decline at a greater rate than the incremental gain of deviation relative to the punishment. Under contracting, the incremental gain from deviation relative to punishment declines at an even lower rate due to the hedging effect, leading to an even larger decline in the critical discount rate under contracting. That in the presence of a forward market, the critical discount rate is in- creasing faster in N , is robust to the forward position dictated by the collusive strategy. Suppose that rather than x∗, firms sold a fraction h∗ = (N − 1)/N of the collusive output in the forward market, where h∗ is the hedge rate in the stationary equilibrium (see equation 3). Table 1 shows that this has little im- pact on the critical discount rate. It is evident that δ(h∗) lies between δ(x∗) and the critical discount rate under Cournot, so that our conclusion is unchanged. 28These results are robust to T > 1 in all of the numerical specifications we have explored: a large T discourages deviation by making punishment harsher, but encourages deviation by providing a longer period of Stackleberg leadership. The net effect appears to be small. 29 6 Conclusions We analyze mergers in the presence of a forward market. Our core finding is that forward markets exacerbate the loss of consumer surplus caused by mergers if the market is sufficiently concentrated, but mitigate loss otherwise. The result obtains from the combination of two considerations: (i) forward contracts disciple the exercise of market power, and (ii) mergers lessen the incentive to sign forward contracts. The first effect dominates if there are many firms but second effect dominates if there are few. The forward contracts we examine can be characterized as a constraint on market power that arises due to endogenous firm behaviors. A series of consol- idating events in a market with an endogenous constraint may initially appear benign, but then produce a surprisingly sudden shift toward supra-competitive prices. In practice, it may be difficult to identify the precise “tipping point” at which further consolidation would lead firms to eliminate or substantially lessen the endogenous constraint on market power. Thus, especially to the ex- tent a merger produces insubstantial verifiable efficiencies, aggressive merger enforcement may be warranted. At the broadest level, appropriate merger re- view should proceed cautiously in interpreting endogenous firm behaviors as a mitigating consideration to an otherwise anticompetitive merger. While our general results are relevant for policy makers in the merger review process, an appropriate level of caution should be exercised in interpreting the specific thresholds and relationships we develop. The model of capital stocks which we have employed uses a simple characterization of firm’s cost functions, and, in practice, mergers may change the shape of firms’ marginal cost functions. We have also assumed the strategic variable to be quantity. In wholesale electricity markets, spot prices are determined based on price-quantity sched- ules submitted by firms. In the supply-function equilibrium model of Klemperer and Meyer (1989), supply functions can be strategic substitutes or complements. Mahenc and Salanie (2004) study strategic complements in the context of dif- ferentiated Bertrand spot market competition and find that forward contracting increases spot market prices. However, we are aware of no studies that analyze the effect of mergers within this context. Finally, we have assumed that all agents have perfect foresight so that the only motive for firms to sell in the contract market is to influence spot market competition. As we do not believe this to be the case in practice, our assumption of perfect foresight was made for the sake of tractability. Allaz (1992) and 30 Hughes and Kao (1997) show that when foresight is imperfect and firms are risk averse, equilibrium hedge rates are higher than in the perfect-foresight case. How hedge rates change in response to a merger in this setting has not been explored to our knowledge. However, it is conceivable that our basic findings would still obtain. Consolidation, by increasing market power, increases the value to the merged firm of withholding output. To the extent that forward contracting even for the sake of hedging risk comes at the expense of exercising market power, mergers may well limit the incentive for firms to forward contract. We leave this issue and the other issues posed in this section to future research. 31 Perry, Martin K. and Robert H. Porter, “Oligopoly and the Incentive for Horizontal Merger,” American Economic Review, 1985, 75 (1), 219–227. Ritz, Robert A., “On Welfare Losses Due to Imperfect Competition,” Journal of Industrial Economics, 2014, 62 (1), 167–190. Salant, Stephen W., Sheldon Switzer, and Robert J. Reynolds, “Losses from Horizontal Merger: The Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium,” Quarterly Journal of Economics, 1983, 98, 185–200. Shapiro, Carl, “Theories of Oligopoly Behavior,” in R. Schmalensee and R. Willig, eds., Handbook of Industrial Organization vol 1, New York: El- sevier Science Publishers, 1989, chapter 6, pp. 330–414. Sweeting, Andrew and Xuezhen Tao, “Dynamic Oligopoly Pricing with Asymmetric Information: Implications for Mergers,” 2016. working paper. Waehrer, Keith and Martin K. Perry, “The Effects of Mergers in Open Auction Markets,” RAND Journal of Economics, 2003, 34 (2), 287–304. Wolak, Frank A., “An Empirical Analysis of the Impact of Hedge Contracts on Bidding Behavior in a Competitive Electricity Market,” International Economic Journal, 2000, 14 (2), 1–39. and Shaun D. McRae, “Merger Analysis in Restructured Electricity Supply Industries: The Proposed PSEG and Exelon Merger (2006),” in John E. Kwoka Jr and Lawrence J. White, eds., The Antitrust Revolution, New York: Oxford University Press, 2009, pp. 30–66. 34 Appendices A Proofs A.1 Proof of Proposition 1 Fixing the price at a candidate equilibrium value, P , and using the definition of βi given in the text, we can express equation (1) as, qi = ( ki bki + 1 ) (P − c) + ( bki bki + 1 ) q0i = βi b (P − c) + βiq 0 i Using the definitions of B and F 0 from the text, we can express total output as, Q = ∑ i qi = B b (P − c) + F 0 Substituting the identity Q = (a− P ) /b into the left-hand side of the above expression yields a− P b = B b (P − c) + F 0 It is straightforward to solve the above for the equilibrium value of P , which we then plug into the above expressions for qi and Q to obtain their equilibrium values. A.2 Proof of Lemma 1 Consider t = 1. From the expression of qi in Proposition 1, we have that, ∂qi ∂f1i = βi (1 +B−i) 1 +B . (A.1) From the same expression of qi, we also have that, ∂qj ∂f1i = − βiβj 1 +B . so that ∑ j 6=i ∂qj ∂f1i = −βiB−i 1 +B . (A.2) Using (A.1) and (A.2), we have that, R1 i ≡ ∑ j 6=i ∂qj ∂f1i / ∂qi ∂f1i = − B−i 1 +B−i 35 Now consider any t = τ > 1. Fixing price at some candidate equilibrium, P , and using the definition of µτi from the statement of the lemma, we can express equation (2) as, qi = µτi ( P − c b ) + µτi (1 +Rτi ) qτi (A.3) Define the following terms: F τ = ∑ i µτi (1 +Rτi ) qτi , F τ −i = ∑ j 6=i µτj ( 1 +Rτj ) qτj We can then express total output as, Q = ∑ i qi = Mτ ( P − c b ) + F τ (A.4) Substituting Q = (a− P ) /b into the above yields, a− P b = Mτ ( P − c b ) + F τ (A.5) It is straightforward to solve the above expression for the equilibrium value of P , which we then plug into (A.3) to obtain, qi (qτ ) = ( a− c b ) µτi 1 +Mτ + µτi 1 +Mτ [( 1 +Mτ −i ) (1 +Rτi ) qτi − F τ−i ] (A.6) Differentiating qi (qτ ) with respect to the firm’s own forward position yields, ∂qi (qτ ) ∂fτi = µτi (1 +Rτi ) 1 +Mτ ( 1 +Mτ −i ) (A.7) Differentiating with respect to another firm’s position yields, ∂qj (qτ ) ∂fτi = µτi (1 +Rτi ) 1 +Mτ µτ so that, ∑ j 6=i ∂qj (qτ ) ∂fτi = µτi (1 +Rτi ) 1 +Mτ Mτ −i (A.8) Using (A.7) and (A.8), we have that, Rτ+1 i ≡ ∑ j 6=i ∂qj ∂fτ+1 i / ∂qi ∂fτ+1 i = − Mτ −i 1 +Mτ −i 36 After manipulating terms, this is equivalent to, βi > 1 R1 i ( 1 +B−i 1 +M1 −i − 1 ) The right-hand side of the above expression is bounded above zero in all but the monopoly case. Therefore, when there are at least three firms, the right- hand side remains bounded above zero even as βi → 0. It follows that for βi sufficiently close to zero, the condition fails. This suggests the existence of a critical level of capital that conditional on the configuration of rivals’ capital stocks, a firm whose capital stock is less than the critical level decreases its output with more rounds of forward contracting. A.5 Proof of Lemma 2 It was established in the proof of Proposition 2 that a producer’s marginal revenue is equal across each period. Equating its period-T marginal revenue with its period-0 marginal revenue, we have, qi ( 1 +RTi ) = qi − q0i Rearranging terms, we have that, q0i qi = ∣∣RTi ∣∣ A.6 Proof of Lemma 3 The solution to the producer’s problem in period T is characterized by a mod- ified version of equation (2) in which τ = T and qTi = 0 for all i. Rearranging terms, we have, P − C ′i P = −qi P P ′ (Q) ( 1 +RTi ) = −Q P P ′ (Q) si (1− hi) = si (1− hi) ε where the second line uses the result of Lemma 2 that hi = RTi and uses the substitution, qi = siQ. The third line uses the definition of demand elasticity, ε. Pre-multiplying by si then summing over all i obtains the result. A.7 Derivation of consumer and total surplus Consumer surplus is social surplus net of expenditures, so that, 39 CS = ∫ Q 0 (a− bx− P ) dx = (a− P )Q− b 2 Q2 = b 2 Q2. Total surplus is social surplus net of costs, so that, TS = ∫ Q 0 (a− bx) dx− ∑ i Ci = aQ− b 2 Q2 − ∑ i Ci (A.12) By construction, Ci = cqi + q2i /2ki, which implies that marginal cost is of the form, C ′ i = c+ qi/ki. It follows that, ∑ i Ci = ∑ i qi [ c+ 1 2 ( C ′ i − c )] = 1 2 [ (c+ P )Q− ∑ i qi ( P − C ′ i )] Substituting qi = Qsi and P = c+ bQ/M (from Proposition 2), we have, ∑ i Ci = Q 2 [ 2c+ b M Q− ∑ i si ( P − C ′ i )] (A.13) Combining (A.12) and (A.13), we have, TS = Q 2 [ 2 (a− c)− b (1 +M) M Q+ ∑ i si ( P − C ′ i )] (A.14) Finally, from Proposition 2, b(1+M)Q/M = a−c. Substituting this into (A.14) yields the desired expression. A.8 Proof of Lemma 4 Recall from Lemma 2 that firm i’s hedge rate is, hi = −Ri. We have that, dRi = − dB−i (1 +B−i) 2 , (A.15) where, dB−i =  dβ2 if i = 1 dβ1 if i = 2 dβ1 + dβ2 if i > 2 (A.16) Consider first the outside firms. The change in an outside firm’s hedge rate due to a consolidating transfer takes the sign of dB−i = dβ1 + dβ2. Let δ ≡ k1 − k2. Using, dβ1 = b ( β1 bk1 )2 dk (A.17) 40 and, dβ2 = −b ( β2 bk2 )2 dk = − ( β2 bk2 )2( β1 bk1 )−2 dβ1 (A.18) we have that, dβ1 + dβ2 = [ 1− ( β2 bk2 )2( β1 bk1 )−2] dβ1 = [( β1 bk1 )2 − ( β2 bk2 )2 ]( β1 bk1 )−2 dβ1 = [ β1 bk1 − β2 bk2 ]( β1 bk1 + β2 bk2 )( β1 bk1 )−2 dβ1 = − ( β2δ bk1k2 )( β1 bk1 + β2 bk2 )( β1 bk1 )−2 dβ1 ≤ 0 (A.19) The inequality in (A.19) implies that sign (dhi) = sign (dBi), which is negative. Therefore, all outisde firms reduce their hedge rate post-consolidation. Next, consider firm 1, the (weakly) larger of the two inside firms. We have that, dh1 = ( 1 1 +B−1 )2 dβ2 = − ( 1 1 +B−1 )2( β2 bk2 )2( β1 bk1 )−2 dβ1 ≤ 0 For firm 2, the (weakly) smaller of the two inside firms, we have, dh2 = ( 1 1 +B−2 )2 dβ1 ≥ 0 Evidently, firm 1’s hedge rate decreases while firm 2’s increases as a result of the capital transfer. It remains to show that the absolute change in firm 1’s hedge rate exceeds the change in firm 2’s hedge rate. Let B−m ≡ ∑ j 6=1,2 βj . We have that, 41 A.10 Proof of Proposition 4 In Cournot, the change in consumer surplus due to a consolidating transfer is, dCS0 = a− c (1 +B) 2 (dβ1 + dβ2) Recall that in the two-stage model, the reduction in consumer surplus from a consolidating transfer is the sum of the structural and hedging effects. Formally, the change in consumer surplus is dCS = SE +HE, where SE ≡ a− c (1 +M) 2 [( µ1 β1 )2 dβ1 + ( µ2 β2 )2 dβ2 ] HE ≡ − a− c (1 +M) 2 ∑ i µ2 i · dRi For the remainder of the proof, and without loss of generality, we consider a consolidating transfer from firm 2 to firm 1, which implies k1 > k2. Lemma 7 dCS0 ≤ SE ≤ 0. Both inequalities are strict outside the monopoly case. Proof. Since Ri < 0 for any firm i, it follows from equation (A.21) that µi > βi for all i, which implies that M ≡ ∑ i µi > B ≡ ∑ i βi. Therefore, if the bracketed term within the definition of SE is in the interval, (dβ1 + dβ2, 0), we have the desired result. For this to be true, it is sufficient to show that in all but the monopoly case, ( µ1 β1 )2 > ( µ2 β2 )2 . Using equation (A.21), this expression reduces to (1 +B) (B − β1 − β2) + β1β2 > 0 (A.23) which is true by construction since βi > 0 for all i and B ≡ ∑ i βi ≥ β1 + β2. In the monopoly case, µi = βi for all i, so that dCS0 = SE. Lemma 7 is notable for two reasons. The first is its implication that if the hedging effect is sufficiently small then the reduction in consumer surplus is larger under Cournot. We establish (next) that this applies to industries that are are sufficiently unconcentrated. The second is that it establishes that in markets that are nearly monopolized, the Cournot effect and the structural effect are nearly equal, so that the hedging effect is determinative. The following Lemma shows that as the industry structure approaches per- fect competition, the hedging effect vanishes. We model perfect competition as the limiting case of a reduction in concentration due to an l-replication of any capital allocation as l →∞. For the sake of notation, let κ denote the fraction of industry capital held by firm 1, the acquiring firm. Further let κ C−→ 0 denote κ going to 0 due to a reduction in concentration. Lemma 8 lim κ C−→0 HE = 0. 44 Proof. Consider first what happens to equilibrium hedge rates as the industry approaches perfect competition i.e., κ C−→ 0. From Lemma 2, the equilibrium hedge rate in the two-stage (T = 1) model is, hi = B−i/ (1 +B−i). κ C−→ 0 corresponds to a situation where the number of firms is increasing without limit so that B−i → ∞ while βi remains fixed for any i. It follows that for any firm i, lim κ C−→0 hi = 1. Consider now what happens to each term in HE as given in equation (A.22). Since hi → 1 as κ C−→ 0, it follows that µi = βi/ (1− hiβi)→ βi (1− βi) for every firm i. Hence, µi is finite for every i. It follows that since B−i → ∞ for every firm i, µi/ (1 +B−i) → 0 as κ C−→ 0. Finally, since dβ1 and dβ2 are unaffected by a change in the capital held by other firms ((A.17) and (A.18) confirm this), it follows that HE → 0 as κ C−→ 0. From Lemma 7 and Lemma 8, we have that, lim κ C−→0 [ dCS0 − (SE +HE) ] < 0 (A.24) so that in highly unconcentrated industries, the reduction in consumer surplus from a consolidating transfer is larger under Cournot. This proves the second claim of the proposition. We now show that the inequality is flipped in highly concentrated industries. Consider the limiting case as all capital is consolidated in firm 1 i.e., as κ C−→ 1. Lemma 9 lim κ C−→1 SE = lim κ C−→1 dCS0 < 0. Proof. In the limit as κ C−→ 1, βj → 0 for all j 6= 1 so that B → β1. From (A.21), we have that, lim κ C−→1 ( µ1 β1 ) = 1 (1 +B) (1−B) +B2 = 1 and lim κ C−→1 ( µ2 β2 ) = 1 +B 1 +B = 1. It follows that, lim κ C−→1 SE = lim κ C−→1 dCS0 = lim κ C−→1 (dβ1 + dβ2) = ( 1 (bk1 + 1) 2 − 1 ) b · dk < 0 Meanwhile, since lim κ C−→1 HE = −bB · dk < 0, it follows that there exist highly concentrated industry structures such that dCS0 > SE + HE. This proves the first claim of the proof. 45 A.11 Proof of Proposition 5 In the symmetric case, the hedge hedge rate with T rounds of contracting is given by expression (4). Rearranging terms, we have that, h(T+1) − h(T ) = (N − 1)β2 ( h(T ) − h(T−1) )[ 1 + (N − 1)β − βh(T ) ] [ 1 + (N − 1)β − βh(T−1) ] < (N − 1)β2 [1 + (N − 2)β] 2 ( h(T ) − h(T−1) ) Since (N−1)β2 [1+(N−2)β]2 < 1, we have that h(T+1) − h(T ) < h(T ) − h(T−1) for any arbitrary T , which establishes the first result. Further, the Banach fixed- point theorem says that the sequence {h(t)} converges to a fixed point, which establishes the second result. A.12 Proof of Proposition 6 Let V d (δ, x) denote the present value of the most profitable deviation. It follows that the collusive strategy constitutes a SPE if V c (δ, x) ≥ V d (δ, x). In what follows, we derive the profit terms in expression (14). τ = 0 : Prior to the opening of the spot market in period t (the period in which deviation takes place), each firm has a forward position of xQm/N from contracts signed in period t−1 under the collusive strategy. Firm i’s spot-market deviation solves, πd,0 = max ( a− xQm − N − 1 N (1− x)Qm − q − c ) q Because the monopoly output is Qm = (a− c)/2, the deviation output is, qd,0 = (a− c)2 4N (N + 1− x) It follows that, πd,0 = (a− c)2 4N (N + 1− x) 2 4N = πm (N + 1− x) 2 4N (A.25) which denotes the profit from production in period t. τ = 1 : Under the collusive strategy, the quantity traded by all firms j 6= i in period t for production to be delivered in period t+ 1 is f t,t+1 = xQm/N . In determining the optimal deviation in the market for one-period forward quantity, firm i takes into account that rival firms will detect deviations in period t and will begin the punishment phase in the period t+ 1 spot market. Using Proposition 46