- Email: [email protected]
Competition and collusion with fixed output
Economics Letters 120 (2013) 259–261
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Competition and collusion with fixed output✩ Hans Zenger ∗ Charles River Associates, Avenue Louise 81, 1050 Brussels, Belgium
highlights • When output is fixed, firms compete by allocating supplies between markets. • Collusion then leads firms to shift output from high-margin markets to low-margin markets. • This reduces welfare although prices in some markets decrease.
article
info
Article history: Received 5 March 2013 Received in revised form 7 April 2013 Accepted 19 April 2013 Available online 3 May 2013
abstract In many industries, output is fixed by exogenous constraints, so firms compete by allocating a given stock of supplies between different markets. This paper shows that collusion in such industries leads firms to shift output from high-margin markets to low-margin markets. As a result, welfare is generally reduced although prices decrease in some markets and increase in others. © 2013 Elsevier B.V. All rights reserved.
JEL classification: L41 L13 Keywords: Collusion Fixed output Price discrimination
1. Introduction In many industries, output is capped by an exogenous constraint such as a fixed stock of input supplies. For instance, in the airline industry the number of flights is often determined by binding slot constraints at international airports. Similarly, in natural gas markets the capacity available to local suppliers is frequently dictated by long-term supply agreements with upstream producers. Finally, in agricultural markets, total volumes are often determined by government-imposed quotas. In analyzing such markets, the economic literature has mainly focused on firms’ strategic incentives to hoard or withhold capacity.1 Yet, in many real world markets firms lack the ability or incentive to hold capacity idle. For instance, landing slots at
✩ The views expressed in this article represent those of the author only and do not necessarily reflect the opinions of other CRA staff or of CRA’s clients. ∗ Tel.: +32 2 6271400. E-mail addresses: [email protected], [email protected]. 1 E.g., see Esö et al. (2010) and Stahl (1988) for static models and Benoît and
Krishna (1987) and Davidson and Deneckere (1990) for dynamic models. 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.04.036
slot-constrained airports are granted on the condition that airlines actually use them.2 Similarly, gas companies often lack the incentive to withhold contracted capacities from the market, because take-or-pay provisions in their supply agreements require payment for contracted volumes even if they are not actually used. Finally, agricultural quotas are typically so tight that even monopolists use their assigned quota in full. Accordingly, real world antitrust investigations have often dealt with agreements between firms that jointly allocate a given stock of output across markets to maximize profits. For instance, in E.ON/GDF two gas companies were found to conspire not to enter each others’ respective home markets, although their capacities were fixed through long-term supply contracts with an upstream producer.3 Similarly, in Bananas a cartel of major international fruit exporters was uncovered that operated under the shadow of binding import quotas (which, however, left firms room to assign
2 See Gale and O’Brien (2013) for an analysis of such use-or-lose provisions. 3 See Case COMP/39.401 E.ON/GDF (European Commission decision of 8 July 2009).
260
H. Zenger / Economics Letters 120 (2013) 259–261
output to different local markets and periods of sale).4 In such cases, firms do not collude on how much to sell, but where and when. These antitrust investigations have been of some significance. For instance, the agreement in E.ON/GDF led to a fine of more than one billion euro for the involved companies—the largest fine ever imposed by the European Commission in an antitrust proceeding. Such judgments also tend to be controversial, since collusion with fixed output leads to price increases in some markets and price reductions in others. The impact on welfare is therefore prima facie unclear. This paper thus tries to shed some light on the economic effects of competition and collusion in industries with fixed output. It is shown that when firms compete by strategically allocating supplies between different markets, collusion leads firms to collectively shift output from high-margin markets to lowmargin markets. As a result, social welfare is generally reduced, although prices decrease in some markets and increase in others. 2. The model Consider two markets A and B with downward-sloping inverse demands pA (qA ) and pB (qB ), where qj denotes output in market j = A, B. There are n firms i = 1, . . . , n, each of which controls a fixed stock of output q¯ i that can be allocated to either of the two markets at marginal cost c ≥ 0. It is assumed that firms lack either 5 the ability or the incentive to withhold capacity from the market. Total industry output is therefore fixed and given by i q¯ i := q¯ . Let qij denote the output firm i assigns to market j, so i qij = qj and ¯ i . Each firm’s strategy is then uniquely determined by the j qij = q output qiA it allocates to market A, as qiB = q¯ i − qiA . Since demands are downward-sloping, any shift of output from one market to the other must increase prices in the first market and decrease them in the second. To determine its competitive strategy, each firm i maximizes its profit max [pA (qA ) − c] qiA + [pB (¯q − qA ) − c] (¯qi − qiA ) . An interior solution to this maximization problem is characterized by the n first-order conditions6 pB (¯q − qA ) − pA (qA ) pA (qA ) + pB (¯q − qA ) ′
′
+
p′B (¯q − qA ) pA (qA ) + p′B (¯q − qA ) ′
q¯ i
for i = 1, . . . , n.
(1)
Adding these up yields qA =
pB (¯q − qA ) − pA (qA )
p′B (¯q − qA ) n+ ′ q¯ . pA (qA ) + pB (¯q − qA ) pA (qA ) + p′B (¯q − qA ) ′
′
qiA q¯ i
=
pB − pA 1 p′A + p′B q¯ i
+
p′B p′A + p′B
for i = 1, . . . , n.
Since we must have p′A + p′B < 0 in equilibrium, this expression is decreasing in q¯ i if and only if pA > pB . Smaller firms are therefore more aggressive than large firms in putting pressure on margins in higher-priced markets. Third, note from (1) that in markets where prices are identical in equilibrium(pA = pB ), every firm allocates the same proportion of output p′B / p′A + p′B to market A. An important special case in which such price convergence arises is when demand functions are identical up to a transformation of size; i.e., pB (qB ) = pA (γ qB ) for some γ > 0. In that case (1) becomes qiA =
pA (γ (¯q − qA )) − pA (qA ) p′A (qA ) + γ p′A (γ (¯q − qA ))
+
γ p′A (γ (¯q − qA )) q¯ i for i = 1, . . . , n. pA (qA ) + γ p′A (γ (¯q − qA )) ′
(2)
Note that (2) is an implicit function qA (¯q), which simultaneously determines qB (¯q) = q¯ − qA (¯q) (and hence the overall distribution of output across markets). By applying the implicit function theorem to (2) and making use of the second-order conditions of firms’ maximization problems, it is straightforward to show that both qA and qB satisfy dqi (¯q) /dq¯ ≥ 0, as one would expect. The equilibrium allocation qA (¯q) has a number of interesting properties. First, note that (2) is independent of the distribution
4 See Case COMP/39.188 Bananas (European Commission decision of 15 October 2008). 5 The latter occurs when total industry capacity is sufficiently small or when opportunity costs of holding capacity idle are sufficiently large. 6 The existence of an interior solution requires that the initial distribution of capacity among firms is not too asymmetric. For simplicity, this is assumed throughout.
(3)
Consider the candidate solution qA =
γ q¯ . 1+γ
(4)
Substituting (4) into (3) and rearranging then yields qiA =
γ 1+γ
q¯ i .
(5)
Adding (5) up for all i gives (4) again, so the strategies described in (5) indeed form an equilibrium. Moreover, (4) implies that prices in the two markets must be identical, since pB (¯q − qA ) = pA (γ (¯q − qA )) = pA
qiA
qiA =
of individual stocks q¯ i . Hence, in markets with fixed output, traditional concentration-based antitrust measures (such as market shares or HHIs) are no meaningful indicators of market power. Second, it is easy to verify that smaller firms allocate a larger proportion of their output to high-margin markets than large firms. Specifically, from (1) we have
γ q¯ = pA (qA ) . 1+γ
In summary, when demand functions are identical up to a transformation of size, each firm allocates the same proportion of its output to the two markets and there is price convergence— irrespective of the number of firms in the industry. We can therefore conclude that neither mergers nor collusion affect equilibrium price levels in such markets. We are now ready to explore the more general competitive implications of the market equilibrium. Without loss of generality, consider the case where pA ≥ pB . Since p′A + p′B < 0 in equilibrium, the first term on the right-hand-side of (2) is increasing in n. Thus, when the number of firms in the industry increases for some given level of capacity q¯ , firms have an incentive to allocate a larger proportion of their endowment to the higher-margin market A. In other words, fragmentation among suppliers reduces price divergence across markets. Indeed, when n → ∞ we must have pA = pB from (2), so prices fully with each firm allocating the same converge, proportion p′B / p′A + p′B of its output to market A. These results provide important insights into the economic effects of collusion in markets with fixed output. When firms collude, their collective incentives are equivalent to the incentives of a monopoly provider. With a sufficiently large discount factor, the collusive market equilibrium can therefore be determined by letting n = 1 in (2). This gives the following result. Proposition 1. When firms collude in markets with fixed output, they allocate a smaller proportion of their output to the higherpriced market than absent collusion. Hence, collusion leads to price divergence across markets.
H. Zenger / Economics Letters 120 (2013) 259–261
It is well-known that third degree price discrimination by a monopolist cannot increase welfare unless output increases (Varian, 1985). Given that output is fixed in our model, one may therefore conjecture that social welfare is decreasing in the degree of price divergence. Indeed, it is easy to show that this is the case. Total welfare in our model is given by
∞
[pA (x) − c] dx +
W = qA
∞
[pB (x) − c] dx. q¯ −qA
Taking the derivative with respect to qA then yields dW dqA
= pA (qA ) − pB (¯q − qA ) .
This term is positive if and only if pA > pB , so increasing social welfare indeed calls for shifting output to the higher-priced market until prices have converged. We can thus state the following. Proposition 2. Since collusion with fixed output leads to price divergence across markets, it generically reduces welfare. As an immediate corollary of Proposition 2, note that consumer surplus must also be reduced by collusion, since total welfare decreases although profits increase. 3. Policy implications The results in the previous section can be immediately applied to the antitrust assessment of collusion in industries with fixed output. Indeed, a number of relevant implications flow from the model for such cases. First, defendants in investigations such as the Bananas case discussed in the Introduction typically argue that collusion should not be expected to cause harm to consumers, since prices in some markets increase, whereas prices in other markets fall. Yet, the foregoing analysis has shown that collusion should nonetheless be expected to cause social harm, because it generically leads to inefficient price divergence across markets. Of course, when markets are fairly similar, the induced price divergence may turn out to be small. In particular, when markets differ only on account of their size, collusion causes no price divergence in the first place. Second, firms that have engaged in market sharing agreements (as in E.ON/GDF ) often argue that they would have lacked an incentive to enter the other market even absent the collusive agreement. Indeed, when output is fixed, winning a sale in the foreign market implies losing a sale in the home market. It is therefore often claimed that there is no incentive to enter new markets unless foreign margins appreciably exceed domestic
261
margins. The foregoing analysis has shown, however, that firms’ incentives to enter new markets are strong even in the absence of appreciable margin differentials. In fact, significant mutual entry occurs in the model even when pre-entry margins in the two markets are identical, so none of the firms is attracted by higher margins abroad. For instance, consider the special case where demand functions in the two markets are identical and there are two firms with equal endowments. In this case equilibrium prices in the two markets are the same, irrespective of whether or not firms collude. This notwithstanding, (5) implies that each firm will shift half of its output to the respective foreign market when it follows its unilateral incentives, thereby inducing mutually offsetting trade-flows. By continuity, it follows that firms have an incentive to shift output to foreign markets even if foreign margins are lower than domestic margins (at least as long as the difference in margins is not too large). To understand this surprising result, consider the special case where each firm initially sells the monopoly quantity in its home market. In that case, moving small volumes of output to the respective other market generates only a second-order loss in the home market (this follows from the envelope theorem). Yet, there is a first-order gain in the foreign market, which is equal to the margin in that market. Firms are therefore not primarily attracted to neighboring markets by higher margins, but by business stealing incentives. Finally, the previous point also suggests that, in the presence of entry costs, market sharing agreements in industries with fixed output may involve a trade-off between allocative efficiency (which tends to call for mutual entry to drive down price differentials) and productive efficiency (which tends to call for lack of entry to reduce entry costs). For instance, when price differences across markets are small and entry costs are significant, market sharing agreements are likely to increase social welfare in industries with fixed output. References Benoît, J.-P., Krishna, V., 1987. Dynamic duopoly: prices and quantities. Review of Economic Studies 54, 23–35. Davidson, C., Deneckere, R., 1990. Excess capacity and collusion. International Economic Review 31, 521–541. Esö, P., Nocke, V., White, L., 2010. Competition for scarce resources. RAND Journal of Economics 41, 524–548. Gale, I., O’Brien, D., 2013. The welfare effects of use-or-lose provisions in markets with dominant firms. American Economic Journal: Microeconomics 5, 175–193. Stahl, D.L., 1988. Bertrand competition for inputs and Walrasian outcomes. American Economic Review 78, 189–201. Varian, H.R., 1985. Price discrimination and social welfare. American Economic Review 75, 870–875.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Competition and collusion with fixed output✩ Hans Zenger ∗ Charles River Associates, Avenue Louise 81, 1050 Brussels, Belgium
highlights • When output is fixed, firms compete by allocating supplies between markets. • Collusion then leads firms to shift output from high-margin markets to low-margin markets. • This reduces welfare although prices in some markets decrease.
article
info
Article history: Received 5 March 2013 Received in revised form 7 April 2013 Accepted 19 April 2013 Available online 3 May 2013
abstract In many industries, output is fixed by exogenous constraints, so firms compete by allocating a given stock of supplies between different markets. This paper shows that collusion in such industries leads firms to shift output from high-margin markets to low-margin markets. As a result, welfare is generally reduced although prices decrease in some markets and increase in others. © 2013 Elsevier B.V. All rights reserved.
JEL classification: L41 L13 Keywords: Collusion Fixed output Price discrimination
1. Introduction In many industries, output is capped by an exogenous constraint such as a fixed stock of input supplies. For instance, in the airline industry the number of flights is often determined by binding slot constraints at international airports. Similarly, in natural gas markets the capacity available to local suppliers is frequently dictated by long-term supply agreements with upstream producers. Finally, in agricultural markets, total volumes are often determined by government-imposed quotas. In analyzing such markets, the economic literature has mainly focused on firms’ strategic incentives to hoard or withhold capacity.1 Yet, in many real world markets firms lack the ability or incentive to hold capacity idle. For instance, landing slots at
✩ The views expressed in this article represent those of the author only and do not necessarily reflect the opinions of other CRA staff or of CRA’s clients. ∗ Tel.: +32 2 6271400. E-mail addresses: [email protected], [email protected]. 1 E.g., see Esö et al. (2010) and Stahl (1988) for static models and Benoît and
Krishna (1987) and Davidson and Deneckere (1990) for dynamic models. 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.04.036
slot-constrained airports are granted on the condition that airlines actually use them.2 Similarly, gas companies often lack the incentive to withhold contracted capacities from the market, because take-or-pay provisions in their supply agreements require payment for contracted volumes even if they are not actually used. Finally, agricultural quotas are typically so tight that even monopolists use their assigned quota in full. Accordingly, real world antitrust investigations have often dealt with agreements between firms that jointly allocate a given stock of output across markets to maximize profits. For instance, in E.ON/GDF two gas companies were found to conspire not to enter each others’ respective home markets, although their capacities were fixed through long-term supply contracts with an upstream producer.3 Similarly, in Bananas a cartel of major international fruit exporters was uncovered that operated under the shadow of binding import quotas (which, however, left firms room to assign
2 See Gale and O’Brien (2013) for an analysis of such use-or-lose provisions. 3 See Case COMP/39.401 E.ON/GDF (European Commission decision of 8 July 2009).
260
H. Zenger / Economics Letters 120 (2013) 259–261
output to different local markets and periods of sale).4 In such cases, firms do not collude on how much to sell, but where and when. These antitrust investigations have been of some significance. For instance, the agreement in E.ON/GDF led to a fine of more than one billion euro for the involved companies—the largest fine ever imposed by the European Commission in an antitrust proceeding. Such judgments also tend to be controversial, since collusion with fixed output leads to price increases in some markets and price reductions in others. The impact on welfare is therefore prima facie unclear. This paper thus tries to shed some light on the economic effects of competition and collusion in industries with fixed output. It is shown that when firms compete by strategically allocating supplies between different markets, collusion leads firms to collectively shift output from high-margin markets to lowmargin markets. As a result, social welfare is generally reduced, although prices decrease in some markets and increase in others. 2. The model Consider two markets A and B with downward-sloping inverse demands pA (qA ) and pB (qB ), where qj denotes output in market j = A, B. There are n firms i = 1, . . . , n, each of which controls a fixed stock of output q¯ i that can be allocated to either of the two markets at marginal cost c ≥ 0. It is assumed that firms lack either 5 the ability or the incentive to withhold capacity from the market. Total industry output is therefore fixed and given by i q¯ i := q¯ . Let qij denote the output firm i assigns to market j, so i qij = qj and ¯ i . Each firm’s strategy is then uniquely determined by the j qij = q output qiA it allocates to market A, as qiB = q¯ i − qiA . Since demands are downward-sloping, any shift of output from one market to the other must increase prices in the first market and decrease them in the second. To determine its competitive strategy, each firm i maximizes its profit max [pA (qA ) − c] qiA + [pB (¯q − qA ) − c] (¯qi − qiA ) . An interior solution to this maximization problem is characterized by the n first-order conditions6 pB (¯q − qA ) − pA (qA ) pA (qA ) + pB (¯q − qA ) ′
′
+
p′B (¯q − qA ) pA (qA ) + p′B (¯q − qA ) ′
q¯ i
for i = 1, . . . , n.
(1)
Adding these up yields qA =
pB (¯q − qA ) − pA (qA )
p′B (¯q − qA ) n+ ′ q¯ . pA (qA ) + pB (¯q − qA ) pA (qA ) + p′B (¯q − qA ) ′
′
qiA q¯ i
=
pB − pA 1 p′A + p′B q¯ i
+
p′B p′A + p′B
for i = 1, . . . , n.
Since we must have p′A + p′B < 0 in equilibrium, this expression is decreasing in q¯ i if and only if pA > pB . Smaller firms are therefore more aggressive than large firms in putting pressure on margins in higher-priced markets. Third, note from (1) that in markets where prices are identical in equilibrium(pA = pB ), every firm allocates the same proportion of output p′B / p′A + p′B to market A. An important special case in which such price convergence arises is when demand functions are identical up to a transformation of size; i.e., pB (qB ) = pA (γ qB ) for some γ > 0. In that case (1) becomes qiA =
pA (γ (¯q − qA )) − pA (qA ) p′A (qA ) + γ p′A (γ (¯q − qA ))
+
γ p′A (γ (¯q − qA )) q¯ i for i = 1, . . . , n. pA (qA ) + γ p′A (γ (¯q − qA )) ′
(2)
Note that (2) is an implicit function qA (¯q), which simultaneously determines qB (¯q) = q¯ − qA (¯q) (and hence the overall distribution of output across markets). By applying the implicit function theorem to (2) and making use of the second-order conditions of firms’ maximization problems, it is straightforward to show that both qA and qB satisfy dqi (¯q) /dq¯ ≥ 0, as one would expect. The equilibrium allocation qA (¯q) has a number of interesting properties. First, note that (2) is independent of the distribution
4 See Case COMP/39.188 Bananas (European Commission decision of 15 October 2008). 5 The latter occurs when total industry capacity is sufficiently small or when opportunity costs of holding capacity idle are sufficiently large. 6 The existence of an interior solution requires that the initial distribution of capacity among firms is not too asymmetric. For simplicity, this is assumed throughout.
(3)
Consider the candidate solution qA =
γ q¯ . 1+γ
(4)
Substituting (4) into (3) and rearranging then yields qiA =
γ 1+γ
q¯ i .
(5)
Adding (5) up for all i gives (4) again, so the strategies described in (5) indeed form an equilibrium. Moreover, (4) implies that prices in the two markets must be identical, since pB (¯q − qA ) = pA (γ (¯q − qA )) = pA
qiA
qiA =
of individual stocks q¯ i . Hence, in markets with fixed output, traditional concentration-based antitrust measures (such as market shares or HHIs) are no meaningful indicators of market power. Second, it is easy to verify that smaller firms allocate a larger proportion of their output to high-margin markets than large firms. Specifically, from (1) we have
γ q¯ = pA (qA ) . 1+γ
In summary, when demand functions are identical up to a transformation of size, each firm allocates the same proportion of its output to the two markets and there is price convergence— irrespective of the number of firms in the industry. We can therefore conclude that neither mergers nor collusion affect equilibrium price levels in such markets. We are now ready to explore the more general competitive implications of the market equilibrium. Without loss of generality, consider the case where pA ≥ pB . Since p′A + p′B < 0 in equilibrium, the first term on the right-hand-side of (2) is increasing in n. Thus, when the number of firms in the industry increases for some given level of capacity q¯ , firms have an incentive to allocate a larger proportion of their endowment to the higher-margin market A. In other words, fragmentation among suppliers reduces price divergence across markets. Indeed, when n → ∞ we must have pA = pB from (2), so prices fully with each firm allocating the same converge, proportion p′B / p′A + p′B of its output to market A. These results provide important insights into the economic effects of collusion in markets with fixed output. When firms collude, their collective incentives are equivalent to the incentives of a monopoly provider. With a sufficiently large discount factor, the collusive market equilibrium can therefore be determined by letting n = 1 in (2). This gives the following result. Proposition 1. When firms collude in markets with fixed output, they allocate a smaller proportion of their output to the higherpriced market than absent collusion. Hence, collusion leads to price divergence across markets.
H. Zenger / Economics Letters 120 (2013) 259–261
It is well-known that third degree price discrimination by a monopolist cannot increase welfare unless output increases (Varian, 1985). Given that output is fixed in our model, one may therefore conjecture that social welfare is decreasing in the degree of price divergence. Indeed, it is easy to show that this is the case. Total welfare in our model is given by
∞
[pA (x) − c] dx +
W = qA
∞
[pB (x) − c] dx. q¯ −qA
Taking the derivative with respect to qA then yields dW dqA
= pA (qA ) − pB (¯q − qA ) .
This term is positive if and only if pA > pB , so increasing social welfare indeed calls for shifting output to the higher-priced market until prices have converged. We can thus state the following. Proposition 2. Since collusion with fixed output leads to price divergence across markets, it generically reduces welfare. As an immediate corollary of Proposition 2, note that consumer surplus must also be reduced by collusion, since total welfare decreases although profits increase. 3. Policy implications The results in the previous section can be immediately applied to the antitrust assessment of collusion in industries with fixed output. Indeed, a number of relevant implications flow from the model for such cases. First, defendants in investigations such as the Bananas case discussed in the Introduction typically argue that collusion should not be expected to cause harm to consumers, since prices in some markets increase, whereas prices in other markets fall. Yet, the foregoing analysis has shown that collusion should nonetheless be expected to cause social harm, because it generically leads to inefficient price divergence across markets. Of course, when markets are fairly similar, the induced price divergence may turn out to be small. In particular, when markets differ only on account of their size, collusion causes no price divergence in the first place. Second, firms that have engaged in market sharing agreements (as in E.ON/GDF ) often argue that they would have lacked an incentive to enter the other market even absent the collusive agreement. Indeed, when output is fixed, winning a sale in the foreign market implies losing a sale in the home market. It is therefore often claimed that there is no incentive to enter new markets unless foreign margins appreciably exceed domestic
261
margins. The foregoing analysis has shown, however, that firms’ incentives to enter new markets are strong even in the absence of appreciable margin differentials. In fact, significant mutual entry occurs in the model even when pre-entry margins in the two markets are identical, so none of the firms is attracted by higher margins abroad. For instance, consider the special case where demand functions in the two markets are identical and there are two firms with equal endowments. In this case equilibrium prices in the two markets are the same, irrespective of whether or not firms collude. This notwithstanding, (5) implies that each firm will shift half of its output to the respective foreign market when it follows its unilateral incentives, thereby inducing mutually offsetting trade-flows. By continuity, it follows that firms have an incentive to shift output to foreign markets even if foreign margins are lower than domestic margins (at least as long as the difference in margins is not too large). To understand this surprising result, consider the special case where each firm initially sells the monopoly quantity in its home market. In that case, moving small volumes of output to the respective other market generates only a second-order loss in the home market (this follows from the envelope theorem). Yet, there is a first-order gain in the foreign market, which is equal to the margin in that market. Firms are therefore not primarily attracted to neighboring markets by higher margins, but by business stealing incentives. Finally, the previous point also suggests that, in the presence of entry costs, market sharing agreements in industries with fixed output may involve a trade-off between allocative efficiency (which tends to call for mutual entry to drive down price differentials) and productive efficiency (which tends to call for lack of entry to reduce entry costs). For instance, when price differences across markets are small and entry costs are significant, market sharing agreements are likely to increase social welfare in industries with fixed output. References Benoît, J.-P., Krishna, V., 1987. Dynamic duopoly: prices and quantities. Review of Economic Studies 54, 23–35. Davidson, C., Deneckere, R., 1990. Excess capacity and collusion. International Economic Review 31, 521–541. Esö, P., Nocke, V., White, L., 2010. Competition for scarce resources. RAND Journal of Economics 41, 524–548. Gale, I., O’Brien, D., 2013. The welfare effects of use-or-lose provisions in markets with dominant firms. American Economic Journal: Microeconomics 5, 175–193. Stahl, D.L., 1988. Bertrand competition for inputs and Walrasian outcomes. American Economic Review 78, 189–201. Varian, H.R., 1985. Price discrimination and social welfare. American Economic Review 75, 870–875.