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Price versus quantity in a mixed duopoly with foreign penetration
Research in Economics 68 (2014) 338–353
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Price versus quantity in a mixed duopoly with foreign penetration$ Junichi Haraguchi a,n, Toshihiro Matsumura b a b
Graduate School of Economics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Institute of Social Science, The University of Tokyo, Japan
a r t i c l e in f o
abstract
Article history: Received 5 March 2014 Accepted 27 September 2014 Available online 13 October 2014
We characterize the endogenous competition structure (in prices or quantities) in a differentiated duopoly between a public firm that maximizes domestic welfare and a private firm that can be owned by domestic or foreign investors. The market for which they compete can be domestic or integrated: in the first case Bertrand competition emerges endogenously and in the second case Cournot competition can emerge if the fraction of domestic consumers in the integrated market is low enough. We also determine the optimal degree of foreign penetration showing the optimality of a partial foreign ownership. Finally, we extend the model to increasing marginal cost confirming the robustness of the results. & 2014 University of Venice. Published by Elsevier Ltd. All rights reserved.
Keywords: Cournot Bertrand Mixed markets International competition Trade
1. Introduction The comparison between price and quantity competition has been extensively discussed in the literature. In oligopolies between private firms it is well known that price competition is tougher, resulting in a lower level of profits, compared to quantity competition.1 Ghosh and Mitra (2010) have revisited this classic result in a mixed oligopoly, namely when a welfare-maximizing public firm competes with a profit-maximizing private firm.2 They have shown that, in contrast to the private duopoly, quantity competition is tougher than price competition, resulting in a smaller profit for the private firm.3 A related literature, started by Singh and Vives (1984), has endogenized the structure of competition (in prices or in quantities). As they pointed out, firms often choose whether to adopt a price contract or a quantity contract. In a private duopoly where both firms maximize profits, assuming linear demand and product differentiation, Singh and Vives (1984) have shown that the choice of a quantity contract is the dominant strategy for each firm when goods are substitutes, and the choice of a price contract is the dominant strategy when goods are complements. Cheng (1985), Tanaka (2001a,b) and Tasnádi (2006) have extended their analysis to asymmetric oligopolies, more general demand and cost conditions, and
☆ We are indebted to two anonymous referees and the editor for their valuable and constructive suggestions. Needless to say, we are responsible for any remaining errors. Financial support of the Grant-in-Aid from Murata Science Foundation and Japanese Ministry of Education, Science and Culture is greatly appreciated. n Corresponding author. Tel.: þ81 5841 4932; fax: þ81 5841 4905. E-mail address: [email protected] (J. Haraguchi). 1 See Shubik and Levitan (1980) and Vives (1985). 2 The pioneering work on mixed oligopolies was by Merrill and Schneider (1966). This and many other studies in the field assume that a public firm maximizes welfare (consumer surplus plus firms' profits) while private firms maximize profits. 3 See also Nakamura (2013) under network externality.
http://dx.doi.org/10.1016/j.rie.2014.09.001 1090-9443/& 2014 University of Venice. Published by Elsevier Ltd. All rights reserved.
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vertical product differentiation, confirming the robustness of the results. Recently, Matsumura and Ogawa (2012) have obtained a contrasting example in case of a mixed duopoly: when one of the two firms is public, the choice of a price contract is the dominant strategy for both private and public firms, regardless of whether goods are substitutes or complements. All the mentioned papers, including Ghosh and Mitra (2010) and Matsumura and Ogawa (2012), assume that private firms are domestically owned and firms compete in the domestic market. In other words, they ignore any aspect of international competition, while we know that the nationality of private firms is often crucial in shaping mixed oligopolies (see the literature starting with Corneo and Jeanne, 1994, and Fjell and Pal, 1996)4 and the nationality of the consumers for which private and public firms are competing is equally crucial (see the trade literature staring with Brander and Spencer, 1985, even if the existence of public firms is usually neglected in that literature). In this work we present a general model of the endogenous form of competition where the ownership of the private firm can be domestic, foreign or mixed, and consumers can be entirely or partially domestic. Therefore the model covers the standard case of domestic firms competing in the domestic market (as already in Ghosh and Mitra, 2010, and Matsumura and Ogawa, 2012) and cases that are typically analyzed in the international trade literature, as when domestic and foreign firms compete in the home market, when they compete in a fully integrated market, and when they compete in a third market (without domestic consumers, as in Brander and Spencer, 1985). We find that foreign penetration can change radically the comparison between Bertrand and Cournot price and production strategies, but the endogenous competition structure depends on the kind of market for which the firms compete: in the traditional case of a domestic market Bertrand competition emerges always as the endogenous competition structure and independently from the ownership share of foreign owners in the private firm, but in the case of an integrated market where firms serve both domestic and foreign consumers Cournot competition can emerge as long as the fraction of domestic consumers is small enough. We also characterize the optimal degree of foreign penetration showing that it is optimal to have a minority share for foreign owners: in particular, when the firms produce almost homogeneous goods it is optimal to maintain fully domestic the private firm, while when they are independent monopolists, it becomes optimal to sell half of the stocks to foreign investors. Extending the model to increasing marginal costs we confirm the robustness of our results. The paper is organized as follows. Section 2 describes the model and compare Cournot and Bertrand equilibria. Section 3 characterizes the endogenous competition structure and the optimal level of foreign penetration. Section 4 extends the model in different directions. Section 5 concludes. 2. The model We adopt a standard duopoly model with differentiated goods and linear demand (Dixit, 1979).5 The quasi-linear utility function of the representative consumer is β U q0 ; q1 ; y ¼ α q0 þ q1 q20 þ 2δq0 q1 þ q21 þ y; 2
ð1Þ
where q0 is the consumption of good 0 produced by the public firm, q1 is the consumption of good 1 produced by the private firms and y is the consumption of an outside good that is competitively provided (with a unitary price). Parameters α and βare positive constants and δ A ð0; 1Þ6 represents the degree of product differentiation: a smaller δ indicates a larger degree of product differentiation. The inverse demand functions for goods i¼0,1with i aj are pi ¼ α βqi βδqj ;
ð2Þ
where pi is the price of firm i. The marginal cost of production is constant for both firms. Let us denote with ci the marginal cost of firm i, assuming α 4ci . Firm 0 is a state-owned public firm whose payoff is the domestic social surplus (welfare). This is given by βðq20 þ2δq0 q1 þq21 Þ p0 q0 p1 q1 ; ð3Þ SW ¼ p0 c0 q0 þ 1 θ p1 c1 q1 þ α q0 þ q1 2
4 Pal and White (1998) and Bárcena-Ruiz and Garzón (2005) discussed trade policies. Mukherjee and Suetrong (2009) discussed the relationship between foreign direct investment and privatization policies. Matsumura et al. (2009) used a monopolistic competition model by Anderson et al. (1997) to show that under foreign ownership privatization is more likely to improve welfare in the long run, whereas the opposite result is derived in the short run. Matsushima and Matsumura (2006), Ogawa and Sanjo (2007), and Heywood and Ye (2009) have shown that foreign ownership affects firms' locations. Han and Ogawa (2008) and Lin and Matsumura (2012) adopted the partial privatization approach formulated by Matsumura (1998) showing that foreign penetration affects the welfare implications of privatization policies; and Wang and Chen (2010) and Cato and Matsumura (2012) demonstrated the same property for free entry markets. Wang and Lee (2013) introduced foreign firms into the framework of Ino and Matsumura (2010) showing that foreign ownership matters in Stackelberg models. Matsumura and Tomaru (2012) revisited the privatization neutrality theorem presented by White (1996) showing that his result does not hold under foreign ownership of the private firms. 5 This demand function is popular in the literature on mixed oligopolies. See Bárcena-Ruiz (2007), Fujiwara (2007), Ishida and Matsushima (2009), Matsumura and Shimizu (2010), and Nakamura (2013). 6 If δ 4 ð o Þ 0, the products are substitutes (complements). Although we restrict our attention to the case of substitute products only, we can show that our main propositions hold when δ A ð 1; 0Þ as well.
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where θ A ½0; 1 is the ownership share of foreign investors in firm 1, which can be potentially affected by policymakers acting on capital liberalization. Firm 1 is a private firm and its payoff is its own profit:
π 1 ¼ ðp1 c1 Þq1 :
ð4Þ
Notice that we are now assuming that the representative consumer is a domestic one, therefore we are focusing on competition for the domestic market: in a later section we will move to the analysis of an integrated market where domestic consumers are only some of the consumers and the public firm maximizes the sum of their surplus and total domestic profits. We now move to the characterization of the Cournot and Bertrand equilibria. We assume that the solutions in all the following games are interior, that is, equilibrium prices and quantities of both firms are strictly positive for any θ. This happens if α is large, jc0 c1 j is not too large, and c0 is not too small. Let us define ai α ci , and let us adopt the superscript ‘ij’ to denote the equilibrium outcome when firm 0 chooses i A fp; qg and firm 1 chooses j A fp; qg. 2.1. Cournot model (q q game) First, we discuss the Cournot model (q q game) in which both firms choose quantities. The first-order conditions for firms 0 and 1 are, respectively, ∂SW ¼ a0 βq0 βδq1 þ θβδq1 ¼ 0; ∂q0 ∂π 1 ¼ a1 2βq1 βδq0 ¼ 0: ∂q1 The second-order conditions are satisfied. From the first-order conditions, we obtain the following reaction functions for firms 0 and 1, respectively: a0 βδq1 þ θβδq1 Rqq ; 0 q1 ¼ Rqq 1 q0
β
a1 βδq0 : ¼ 2β
These functions lead to the following expressions for the equilibrium quantity of each firm, respectively: qqq 0 ¼
2a0 δa1 þ θδa1
;
βð2 δ2 Þ þ θβδ2 a1 δa0 : qqq 1 ¼ βð2 δ2 Þ þ θβδ2
Note that the assumption of the interior solutions implies that 2a0 δa1 4 0 and a1 δa0 4 0. Substituting these equilibrium quantities into the demand and payoff functions, we have the following resulting domestic welfare and firm 1's profit, respectively: SW qq ¼
π qq 1 ¼
H1 2βð2 δ ð1 θÞÞ2 2
ða1 δa0 Þ2
βð2 δ2 ð1 θÞÞ2
;
:
ð5Þ
ð6Þ
where the constant H1 and other constants are reported in the Appendix. 2.2. Bertrand model (p p game) We now characterize the Bertrand model (p p game) in which both firms choose prices. The direct demand functions can be derived as q0 ¼
α αδ p0 þ δp1 ; βð1 δ2 Þ
ð7Þ
q1 ¼
α αδ p1 þ δp0 : βð1 δ2 Þ
ð8Þ
The first-order conditions for firms 0 and 1 are respectively ∂SW c0 p0 δc1 þ δp1 θδðp1 c1 Þ ¼ ¼ 0; ∂p0 βð1 δ2 Þ
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∂π 1 c1 2p1 þ α þ δp0 αδ ¼ ¼ 0: ∂p1 βð1 δ2 Þ The second-order conditions are satisfied, and we can solve for the following reaction functions for firms 0 and 1: Rpp 0 p1 ¼ c0 þ δ p1 c1 θδ p1 c1 ; c1 þ α þ p0 δ αδ : Rpp 1 p0 ¼ 2 These functions lead to the following expressions for the equilibrium prices of each firm:
αδ αδ2 þ 2c0 c1 δ θðαδ αδ2 δc1 Þ ; 2 2 2δ þδ θ α αδ þc1 þc0 δ c1 δ2 þ θδ2 c1 : ppp 1 ¼ 2 2 2δ þδ θ ppp 0 ¼
Substituting these equilibrium prices into the payoff functions, we have the following resulting domestic welfare and firm 1's profit, respectively: SW pp ¼
π pp 1 ¼
H2 2βð1 δ Þðδ θ δ þ2Þ2 2
2
2
ða1 δa0 Þ2
βð1 δ2 Þðδ2 θ δ2 þ 2Þ2
;
ð9Þ
:
ð10Þ
2.3. Cournot–Bertrand comparison Before presenting our main results, we present some supplementary results. Lemmas 1 and 2 show how the quantity ranking between Bertrand and Cournot depends on θ: qq pp qq Lemma 1. When θ ¼ 0, (i) qpp 0 o q0 and (ii) q1 4q1 .
Proof. By direct computation we have qq qpp 0 q0 ¼
δðδð1 δÞθc1 ða1 δa0 ÞÞ ; βð1 δ2 Þð2 δ2 ð1 θÞÞ
ð11Þ
qq qpp 1 q1 ¼
δðδða1 δa0 Þ ð1 δÞθc1 Þ : βð1 δ2 Þð2 δ2 ð1 θÞÞ
ð12Þ
Substituting θ ¼ 0 into (11)–(12), we obtain δða1 δa0 Þ o 0; βð1 δ2 Þð2 δ2 Þ δ2 ða1 δa0 Þ qq 4 0: qpp 1 q1 ¼ βð1 δ2 Þð2 δ2 Þ qq qpp 0 q0 ¼
Note that a1 δa0 4 0 under the interior solution assumption made in this paper.
□
As Singh and Vives (1984) pointed out, demand is more elastic under Bertrand competition compared to Cournot competition. Thus, the private firm tends to choose a lower price, resulting in a larger output under Bertrand competition. The public firm, however, has an incentive to increase the rival's output, and obtains this by choosing a higher price and a smaller output under Bertrand competition. Because products are substitutes, firm 1's low price provides and additional reason to reduce the resulting output of firm 0: qq pp qq pp qq pp qq Lemma 2. When θ ¼ 1, (i) both qpp 0 o q0 and q0 4 q0 are possible, and (ii) both q1 4q1 and q1 o q1 are possible.
Proof. Substituting θ ¼ 1 into (11) and (12), we have qq qpp 0 q0 ¼
δðδð1 δÞc1 ða1 δa0 ÞÞ ; 2 2βð1 δ Þ
ð13Þ
qq qpp 1 q1 ¼
δðδða1 δa0 Þ ð1 δÞc1 Þ : 2 2βð1 δ Þ
ð14Þ
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qq (i) Substituting δ ¼ 1=2; α ¼ 50; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (13), we have qpp 0 q0 ¼ 13=2 o 0. Substituting δ ¼ 3=7;
qq α ¼ 10; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (13), we have qpp 0 q0 ¼ 57=560 40. (ii) Substituting δ ¼ 1=2; α ¼ 50; β ¼ 1; c0 ¼ 9, and pp qq c1 ¼ 8 into (14), we have q1 q1 ¼ 9=4 4 0. Substituting δ ¼ 1=4; α ¼ 10; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (14), we have
qq qpp 1 q1 ¼ 89=120 o 0.
□
When the share of foreign ownership in the private firm is higher, the public firm becomes more aggressive in the attempt to reduce the private firm's profit outflow. As mentioned above, demand is more elastic under Bertrand competition; thus, it is relatively easier for the public firm to reduce its rival's profit. This results in the public firm behaving more aggressively. When this effect is sufficiently strong, the output-level ranking in firm 0 is reversed. Since goods are substitutes, the output-level ranking in firm 1 can also be reversed. We now present our main results. First, we compare firm 1's profit in the two games. As Lemmas 1 and 2 state, the quantity ranking between Bertrand and Cournot depends on θ. However, Proposition 1 shows that, surprisingly, the profit ranking does not depend on θ: Proposition 1. (i) The profits of the private firm are always decreasing in the ownership share of foreign investors, (ii) Bertrand competition leads always to larger profits for the private firm relative to Cournot competition and (iii) the difference in profits is always decreasing in the ownership share of foreign investors. Proof. From (6) and (10), (i) is obvious. From (6) and (10), we have qq π pp 1 π1 ¼
δ2 ða1 δa0 Þ2 4 0: βð1 δ2 Þð2 δ2 ð1 θÞÞ2
whose sign is positive for any
δ and which is obviously decreasing in θ A ½0; 1. This proves (ii) and (iii).
□
An increase in the ownership share of foreign investors strengthens competition in both Bertrand and Cournot duopolies. The outflow to foreign investors is proportional to θπ 1 . Therefore, an increase in θ increases the incentive for firm 0 to reduce the outflow to foreign investors, becoming more aggressive. We label this effect as the “outflow-restricting effect”. pp qq qq This is stronger when π1 is larger. Thus, an increase in θ reduces π1 more significantly than π1 as long as π pp 1 4 π1 . When θ ¼ 0 (and thus, when there is no outflow-restricting effect), Ghosh and Mitra (2010) have already shown that Bertrand competition leads to larger profits for the private firm relative to Cournot competition. As Singh and Vives (1984) have pointed out, demand is more elastic with respect to its own price and to the rival's price under Bertrand competition. Suppose that substituting ðp0 ; p1 Þ ¼ ðp00 ; p01 Þ into the demand function yields ðq0 ; q1 Þ ¼ ðq00 ; q01 Þ. From the demand functions in the p-p and q-q games, we have that firm 1's demand elasticity under Bertrand competition when p0 ¼ p00 , 1 . Because firm 1's ð∂q1 =∂p1 Þðp01 =q01 Þ, is larger than that under Cournot competition when q0 ¼ q00 , ð∂p1 =∂q1 Þðq01 =p01 Þ demand is more elastic with respect to its own price under Bertrand competition at the point ðp0 ; p1 Þ ¼ ðp00 ; p01 Þ than under Cournot competition at the point ðq0 ; q1 Þ ¼ ðq00 ; q01 Þ, the private firm becomes more aggressive under Bertrand competition. This results in a larger output of firm 1 in equilibrium under Bertrand competition. The public firm prefers such a larger output. Similarly, firm 1's demand is more elastic with respect to firm 0's price under Bertrand competition. Since the public firm is concerned with welfare, it prefers a larger output of the private firm. Thus, the public firm becomes less aggressive under Bertrand competition, which leads to a smaller output for firm 0 under Bertrand competition. Both effects increase the private firm's profit. Thus, without outflow-restricting effect, Bertrand yields a larger profit for the private firm. We call this effect “demand-elasticity effect”. Because the demand functions given by (2) and (8) do not depend on θ, this effect does
not depend on θ.
π
qq When θ ¼ 0, π pp 1 4 π 1 . As mentioned above, as
π
qq 1 .
A decrease in π
qq θ increases, outflow-restricting effect reduces π pp 1 π 1 , as long as
π reduces the difference in outflow-restricting effect between Bertrand and Cournot models. θ increases, qq π would be close to zero. On the other hand, demand-elasticity effect remains unchanged and it converges to zero if π pp 1 1 as θ increases. Thus, difference in outflow-restricting effect between Bertrand and Cournot models is never large enough to pp 1 4
pp 1
qq 1
On one hand, the difference in outflow-restricting effect between Bertrand and Cournot models is smaller as
dominate demand-elasticity effect and never reverses the profit ranking, even when the private firm is entirely foreign owned. We finally compare welfare in the two models: Proposition 2. (i) Bertrand competition leads always to larger domestic welfare relative to Cournot competition and (ii) the difference is always decreasing in the ownership share of foreign investors. Proof. From (5) and (9), we have SW pp SW qq ¼
δ2 ða1 δa0 Þ2 4 0; 2 2 2βð1 δ Þð2 δ ð1 θÞÞ2
whose sign is positive for any
δ and which is obviously decreasing in θ A ½0; 1.
□
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Ghosh and Mitra (2010) have already shown that SW pp 4 SW qq when θ ¼ 0: here we extend the result to any ownership share in spite of counteracting forces. Indeed, Cournot competition yields more severe competition compared to Bertrand competition, which restricts the profit outflow toward foreign investors more effectively. This becomes more important when θ increases. However, a larger θ accelerates competition as well and reduces overall profits. Because of this, the outflow-restricting effect mentioned above is never strong enough to reverse the welfare ranking, even when the private firm is entirely foreign owned. The comparison between equilibria and the general result for which Bertrand yields softer competition will be at the source of our analysis of the endogenous competition structure, but may also be important in the analysis of the stability of collusion, at least when aimed at maximizing welfare (with side payments to the private firm).7 While this is clearly beyond the scope of this work, a remark is in order. If we consider a grim trigger strategy (Friedman, 1971; Deneckere, 1983), our result implies that one can always find a more severe punishment for the private firm's deviation under Cournot competition rather than under Bertrand competition, which can lead to more stable collusion under Cournot competition.8 This is against standard results emerging in duopolies with private firms.
3. Endogenous competition structure In this section, we endogenize the competition structure (either price or quantity). We follow the standard model formulated by Singh and Vives (1984). The game runs as follows. In the first stage, each firm chooses whether to adopt a price contract or a quantity contract. In the second stage, after observing the rival's choice in the first stage, each firm simultaneously chooses its own strategy, according to the decision in the first stage. There are four possible subgames: both firms choose quantity contracts (q q game), both firms choose price contracts (p p game), only firm 0 chooses the quantity contract (q p game), or only firm 0 chooses the price contract (p q game). Again, we assume that the solutions in these four games are interior, that is, equilibrium prices and quantities of both firms are strictly positive. We have already solved the p p and q q games; we now solve the remaining two games. 3.1. p q game Consider the situation in which firm 0 chooses the price contract and firm 1 chooses the quantity contract. The relevant direct and inverse demand functions are given by q0 ¼
α βδq1 p0 β
ð15Þ
p1 ¼ ðβδ βÞq1 þ δp0 αδ þ α 2
ð16Þ
The first-order conditions are respectively ∂SW c0 p0 ¼ θδq1 ¼ 0; ∂p0 β ∂π 1 2 ¼ a1 þ δp0 2β ð1 δ Þq1 αδ ¼ 0: ∂q1 7 Profit-maximizing collusion is impossible even if we consider transfers from the private firm to the public one. Welfare maximizing collusion requires both firms to set following prices:
ðδc1 þ αδ αδÞθ þ ð δc1 þ ð2 δ Þc0 αδ þ αδÞθ þ ðδ 1Þc0 2
p0 ¼
2
2
2
2
δ2 θ2 þ ð2 2δ2 Þθ þ δ2 1 2 2 2 δ c1 θ þ ðð1 2δ Þc1 þ δc0 αδ þ αÞθ þ ðδ2 1Þc1 p1 ¼ δ2 θ2 þ ð2 2δ2 Þθ þ δ2 1 or the corresponding quantities. This leads to the following profits for the private firm:
π1 ¼
ða1 δa0 Þ2 ðθ 1Þθ βðδ2 θ2 2δ2 θ þ 2θ þ δ2 1Þ2
The one-shot deviation strategies and the associated profits can be easily derived under both price and quantity competition. 8 Consider the infinitely repeated game of our one-shot game. Only if the private firm chooses the collusive welfare-maximizing strategy and there was no deviation in the past periods, it obtains a transfer from the public firm. Once the private firm deviates from collusion (the public one has no incentives to do it), the Bertrand or Cournot equilibrium holds forever. Since the punishment is more severe under Cournot competition, it is natural to guess that collusion is more stable in this case. However, the deviation from collusion is more profitable under Cournot competition, which goes in the opposite direction. Assume θ ¼ 0, α ¼ 10, c0 ¼ 8, c1 ¼ 6, β ¼ 1, and consider different values of the degree of substitutability δ. Then, it can be shown that when goods are highly substitutes (δ 4 0:17) collusion is more stable under Bertrand, but when they are poorly substitutes (δ o 0:17) it becomes more stable under Cournot.
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and lead to the following reaction functions for firms 0 and 1: Rpq 0 q1 ¼ c0 βθδq1 ; a1 þ δp0 αδ : Rpq 1 p0 ¼ 2 2βð1 δ Þ These functions lead to the following expressions for the respective equilibrium choice of each firm: 2ð1 δ Þc0 θδða1 δαÞ 2
ppq 0 ¼ qpq 1 ¼
2ð1 δ Þ þ θδ 2
2
a1 δa0 2β ð1 δ Þ þ βθδ 2
2
;
:
Substituting these into the payoff functions, we have the resulting domestic welfare and firm 1's profit, respectively. SW pq ¼
π pq 1 ¼
H3 2βðδ θ þ 2ð1 δ ÞÞ2 2
2
ð1 δ Þða1 δa0 Þ2
;
ð17Þ
2
βðδ2 θ þ2ð1 δ2 ÞÞ2
:
ð18Þ
3.2. q p game Consider the situation in which firm 0 chooses the quantity contract and firm 1 chooses the price contract. The relevant demand functions are p0 ¼ ðβδ β Þq0 þ δp1 αδ þ α 2
q1 ¼
ð19Þ
α βδq0 p1 β
ð20Þ
The first-order conditions are respectively ∂SW 2 ¼ a0 δa1 βð1 δ Þq0 þ θδ p1 c1 ¼ 0; ∂q0 ∂π 1 c1 2p1 þ α βδq0 ¼ ¼ 0: ∂p1 β From this system we obtain the following reaction functions: a0 δa1 þ θδðp1 c1 Þ ; Rqp 0 p1 ¼ βð1 δ2 Þ α þ c1 βδq0 : Rqp 1 q0 ¼ 2 which lead to the following expressions for the respective equilibrium choice of each firm: qqp 0 ¼
2ða0 δa1 Þ þ θδa1 2β ð1 δ Þ þ βθδ 2
2
;
c1 þ α þ c0 δ δα 2c1 δ þ θδ c1 2
pqp 1 ¼
2ð1 δ Þ þ θδ 2
2
2
:
Substituting these into the payoff functions, we have the resulting domestic welfare and profit of firm 1 SW qp ¼
π qp 1 ¼
H4 2βðδ θ þ 2ð1 δ ÞÞ2 2
2
ða1 δa0 Þ2
βðδ θ þ2ð1 δ2 ÞÞ2 2
:
;
ð21Þ
ð22Þ
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Table 1 First stage game. 0\1
Quantity
Price
Quantity
ðSW ; π qq 1 Þ ðSW pq ; π pq 1 Þ
ðSW qp ; π qp 1 Þ
Price
qq
ðSW pp ; π pp 1 Þ
3.3. Equilibrium We now discuss the choice at the first stage. Table 1 summarizes the game. The best responses of each firm can be easily derived as follows: pq qp qq Lemma 3. (i) SW pq 4SW qq , (ii) SW pp 4SW qp , (iii) π pp 1 4 π 1 , and (iv) π 1 4 π 1 .
Proof. (i) From (5) and (17), we have SW pq SW qq ¼
δ2 ða1 δa0 Þ2 H5 : 2 2 2 2 2βðδ θ þ 2 δ Þ2 ðδ θ þ2ð1 δ ÞÞ2
H5 is positive under the assumption of interior solution. (ii) From (9) and (21), we have SW pp SW qp ¼
δ2 ða1 δa0 Þ2 H6 : 2 2 2 2β ð1 δ Þðδ θ 2δ þ2Þ2 ðδ θ δ þ 2Þ2 2
2
H6 is positive under the assumption of interior solution. (iii) From (10) and (18), we have pq π pp 1 π1 ¼
δ2 ða1 δa0 Þ2 ðð2 δ2 Þδ4 θ2 þ 2ð1 δ2 Þð3 δ2 Þθ þ ð4 δ2 Þð1 δ2 Þ2 Þ : βð1 δ2 Þðδ2 θ 2δ2 þ2Þ2 ðδ2 θ þ 2 δ2 Þ2
which is positive under the assumption of interior solution. (iv) From (6) and (22), we have qq π qp 1 π1 ¼
δ2 ða1 δa0 Þ2 ð2δ2 θ þ 4 3δ2 Þ : βð2 δ2 ð2 θÞÞ2 ð2 δ2 ð1 θÞÞ2
which is positive under the assumption of interior solution.
□
We now present our second main result: Proposition 3. Bertrand competition is the endogenous competition structure for any ownership share of foreign investors. Proof. Lemma 3(i) and Lemma 3(ii) imply that choosing p is the dominant strategy for firm 0. Lemma 3(iii) and Lemma 3(iv) imply that choosing p is the dominant strategy for firm 1. □ To explain the intuition behind Proposition 3 for any degree of foreign ownership, we first show that firm 0 chooses the price contract in the first stage if firm 1 chooses the price contract. From (8) and (20), for any ðp00 ; q00 Þ, we have ∂q 1 ∂q1 1 ¼ ¼ : 1 4 2 ∂p1 p0 ¼ p0 ∂p1 q0 ¼ q0 β β ð1 δ Þ 0 0 In other words, an increase in p1 at the second stage reduces more significantly firm 1's quantity when firm 0 chooses the price contract than the quantity contract. Thus, firm 1 tends to adopt a lower price in the second stage when firm 0 chooses the price contract in the first stage. Therefore, by choosing a price contract in the first stage, the public firm induces the private one to be more aggressive in the competition stage. The lower price of the private firm improves welfare therefore the public firm prefers the price contract. Next, we show that firm 1 chooses the price contract in the first stage if firm 0 chooses the price contract. Suppose that firm 1 chooses the price contract: in the second stage, firm 0 chooses p0 against p1. Because firm 0 prefers a larger quantity of firm 1, an increase in p0 increases q1 and improves welfare (at the cost of deviating from firm 0's optimal price). Suppose now that firm 1 chooses the quantity contract: in the second stage, firm 0 chooses p0 against q1. Since firm 0 prefers a lower price of firm 1, a decrease in p0 increases p1 and improves welfare (at the cost deviating from firm 0's optimal price). Therefore, firm 0 naturally sets a higher price when firm 1 chooses the price contract rather than a quantity contract. Accordingly, by choosing the price contract in the first stage, firm 1 induces firm 0's to be more accommodating in the competition stage. Thus, choosing the price contract is firm 1's best reply. We now consider the two extreme cases, those of the domestic firm ðθ ¼ 0Þ and the foreign firm ðθ ¼ 1Þ. We plot four points of equilibrium prices in four fixed contract games in the two cases in Fig. 1.9 Point ijk (i; j ¼ fp; qg; k ¼ fD; Fg) denotes 9
A similar figure is developed by Cheng (1985).
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Fig. 1. Equilibrium prices (θ ¼0 or
θ ¼1).
Fig. 2. Equilibrium relative prices (θ ¼ 0 or
θ ¼ 1).
the equilibrium pair of prices. i (res. j) denotes the choice of public (res. private) firm and the superscript k ¼ D (res. k¼F) denotes that θ ¼ 0 (res. θ ¼ 1). From Fig. 1, we can see that foreign penetration makes the public firm more aggressive, and it reduces the private firm's price through the strategic interaction. Although the price levels are lower in case of foreign penetration, we can see a similar relationship among the four points, whether θ ¼ 0 or θ ¼ 1. Therefore, similar qualitative results hold regardless of θ. We also plot in Fig. 2 the equilibrium relative price p0 =p1 for these cases. We can see that price competition yields a higher relative price p0 =p1 and foreign penetration yields a lower relative price, which in this case can be interpreted as the terms of trade. Both quantity competition and foreign penetration accelerate competition, and reduce the public firm's price more significantly than the private firm's price, thereby yielding a lower relative price. To complete the picture on endogenous competition structures in a duopoly with differentiated goods, it is useful to mention what happens in the endogenous timing model formulated by Hamilton and Slutsky (1990), where each firm can decide when to commit to a quantity or price strategy in one of two subsequent periods. Pal (1998) and Matsumura (2003) have analyzed endogenous timing in a mixed duopoly with quantity competition. The first (second) work shows that in the case of a domestic (foreign) private firm, the public firm becomes endogenously the follower (Stackelberg leader).10 Bárcena-Ruiz (2007) has investigated endogenous timing in a mixed duopoly with price competition. He shows that when the private firm is domestic, a simultaneous-move outcome (Bertrand competition) appears endogenously in equilibrium, and the same result holds when the private firm is foreign.11 It would be interesting to compare equilibrium welfare and profits under the two forms of competition and, eventually, determine the equilibrium of anex ante stage in which the firms choose between price or quantity contracts. 3.4. Welfare and the optimal degree of foreign penetration Finally, we discuss the relationship between θ and welfare to derive our third main result. World welfare is the sum of consumer surplus and profits of both firms, which are crucially affected by the share of the private firm that is owned by 10 Notice that in the domestic private firm case, the equilibrium with the public firm as follower yields a second best outcome. Now, suppose that the private firm becomes foreign-owned and the public firm remains the follower. Then, after observing the private firm's action, the public firm becomes more aggressive in order to reduce the private firm's price, and expecting this aggressive behavior of the public firm, the private firm becomes less aggressive, which reduces welfare. For this reason the public firm prefers to become a leader. Strictly speaking, multiple equilibria exist in a mixed duopoly when the private firm is domestic. However, Matsumura and Ogawa (2010) have shown that private leadership is risk dominant. 11 In the price competition model, whether the private firm is domestic or foreign, the private (public) firm sets a higher (lower) price when it is the leader than when both firms move simultaneously. Thus, both firms do not want to be the follower, and thus, the simultaneous-move outcome appears in equilibrium.
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foreign investors. Since the degree of capital liberalization can affect such a share, it is interesting to determine its optimal level from a global perspective. Proposition 4. Welfare is maximized by the following ownership share of foreign investors: 2 1δ 1 θn ¼ A 0; 2 2 2δ which is increasing in product differentiation. Proof. World welfare is given by βðq20 þ2δq0 q1 þq21 Þ p0 q0 p1 q1 : V ¼ p0 c0 q0 þ p1 c1 q1 þ α q0 þ q1 2 Substituting equilibrium prices under Bertrand competition, we have V pp ¼
H7 2β ð1 δ Þðδ θ δ þ 2Þ2 2
2
2
:
Differentiating Vpp with respect to θ, we have ∂V pp δ ða1 δa0 Þ2 ð1 δ ð2 δ ÞθÞ p : ∂θ βð1 δ2 Þð2 δ2 ð1 θÞÞ3 2
2
2
This is negative (positive) if θ 4ð o Þð1 δ Þ=ð2 δ Þ. Of course ∂θ =∂δ ¼ 2δ=ð2 δ Þ2 o0 with θ ¼ 1=2 for δ ¼ 0 and θ -0 for δ-1. □ 2
2
n
2
n
n
World welfare is nonmonotonic with respect to the degree of foreign penetration θ. Suppose that θ is close to zero. A slight increase in θ makes the public firm more aggressive, and through strategic interaction, it reduces the private firm's price: this improves world welfare. When θ is large, a further increase in θ makes the public firm too aggressive for world welfare because the public firm is not concerned about the private firm's profit and the difference between domestic and world welfare is large. In other words, the public firm's price is too low for world welfare. Thus, further aggressive pricing by the public firm caused by a larger θ reduces world welfare. The optimal ownership share of foreign investors is decreasing in δ, and in particular when firms produce almost homogeneous goods (δ-1) it is optimal to maintain fully domestic private firm, while when they produce fully independent goods (δ ¼ 0), that is they are independent monopolists, it becomes optimal to sell half of the stocks to foreign investors. This suggests that more differentiation between the goods provided by the firms leads to the optimality of a larger foreign penetration. As mentioned above, an increase in θ makes the public firm more aggressive and it is harmful for world welfare when θ is large because it reduces the resulting q1. When product a more differentiated, a lower p1 reduces q1 less significantly and thus, the welfare loss by aggressive pricing of firm 1 is weaker. Therefore, the range of θ for which an increase in θ improves world welfare is wider, and thus, the optimal degree of capital liberalization is higher. What is important to remark is that the optimal degree of foreign penetration can be optimal also from the perspective of the domestic country. Suppose that at the beginning all the stocks in firm 1 are owned by atomistic domestic investors (θ ¼ 0 initially). Suppose that some of them are sold in a competitive and efficient financial market: then, the price of the n stocks must match the fraction of the expected profits obtained by the private firm. In such a case, θ is also the degree of foreign penetration that maximizes domestic welfare ex ante. The problem, however, is that domestic owners may not sell their stocks if they can internalize the negative impact of an increase of θ on the profits of the private firm: this is the case, for instance, if domestic investors are not negligible.
4. Extensions This section extends the model in two main directions. The first is the consideration of competition for an integrated market with both domestic and foreign consumers, which can lead to a different endogenous competition structure with Cournot competition. The second is the consideration of a different cost function for the two firms, which however confirms the baseline results. 4.1. Competition in an integrated market Until now, we have assumed that all consumers are domestic. However, it is possible that the market is comprised of both domestic and foreign consumers and the public and private firms compete for all this integrated market. To investigate this case, we simply assume that the fraction of domestic consumers is k A ½0; 1 and take this into account in the welfare evaluation. This changes the perspective of the public firm and can fundamentally undermine the endogenous
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competition structure determined above. Indeed, we will show that in an integrated market, Cournot competition can finally emerge.12 If k¼1 we are back to the previous model of a domestic market, for intermediate values we are considering an integrated market where k is the relative size of domestic consumers, and if k¼0, we are in the case of competition for a third market (Brander and Spencer, 1985): all these cases are extensively discussed in the trade literature. Social welfare is now given by βðq20 þ 2δq0 q1 þ q21 Þ SW ¼ p0 c0 q0 þ 1 θ p1 c1 q1 þk α q0 þq1 p0 q0 p1 q1 ; 2 Under Cournot competition, the equilibrium quantity of the public firm can be derived as qqq 0 ¼
2a0 δð2 θ kÞa1
;
βðδ2 θ þ ð2 kÞð2 δ2 ÞÞ
and that of the private firm is qqq 1 ¼
ð2 kÞa1 δa0 : βðδ2 θ þ ð2 kÞð2 δ2 ÞÞ
Substituting these equilibrium quantities into the demand and payoff functions, we have the following domestic welfare and firm 1's profit: SW qq ¼
π qq 1 ¼
H8 ; 2 2 2βðδ θ þ ð2 δ Þð2 kÞÞ ðδa0 ð2 kÞa1 Þ2
βðδ2 θ þð2 δ2 Þð2 kÞÞ
ð23Þ
:
ð24Þ
Under Bertrand competition, the equilibrium price of the public firm is ppp 0 ¼
αð1 δÞð2ð1 þ δÞ kð2 þ δÞ δθÞ þ δðθ kÞc1 þ 2c0 ; δ2 θ þð2 δ2 Þð2 kÞ
and that of the private firm is ppp 1 ¼
αð1 δÞð2 þ δ kð1 þ δÞÞ þ ð2 k δ2 ð1 θÞÞc1 þ δc0 : δ2 θ þ ð2 δ2 Þð2 kÞ
Substituting these equilibrium prices into the payoff functions, we have SW pp ¼
π pp 1 ¼
H9 2β ð1 δ Þðδ θ þ ð2 δ Þð2 kÞÞ2 2
2
2
ðð1 þð1 δ Þð1 kÞÞa1 δa0 Þ2
;
ð25Þ
2
βð1 δ Þðδ θ þð2 δ2 Þð2 kÞÞ2 2
2
:
ð26Þ
From (23) and (25) we have SW pp SW qq ¼
δ2 ð2 kÞðða1 δa0 Þ2 ð1 δ2 Þð1 kÞ2 a21 Þ : 2 2 2β ð1 δÞ2 ðδ ðθ þkÞ þ 2ð2 k δ ÞÞ2
This is positive if and only if ða1 δa0 Þ2 ð1 δ Þð1 kÞ2 a21 is positive. Solving we obtain: 2
1 δa0 =a1 1 δa0 =a1 n k 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi o k o1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 2 1δ 1δ n
Since k 4 1 always holds, we ignore the upper bound k . In case of symmetric firms or a small enough cost difference k 4 0 n therefore SW pp SW qq is positive if and only if k 4k . From (23) and (26) we have qq π pp 1 π1 ¼
δ2 ðða1 δa0 Þ2 ð1 δ2 Þð1 kÞ2 a21 Þ : βð1 δ2 Þðδ2 ðθ þ kÞ þ 2ð2 k δ2 ÞÞ2
which requires the same condition for being positive. Remarkably, both firms agree on the relative preference for the form of competition independently from the degree of foreign penetration. Cournot competition is preferred when the fraction of domestic consumers is small, and therefore the public firm is not much concerned about consumers surplus (this is exactly the case of trade models of competition for a third market). This is less likely to happen when there is more differentiation 12
We are grateful to a referee for suggesting this extension.
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between goods: indeed for δ ¼ 0 we have k ¼ 0 and the Bertrand outcome is always preferred by both firms, but an increase pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n in δ tends to increase the cut-off, as can be easily verified in case of cost symmetry, where k ¼ 1 ð1 δa0 =a1 Þ= 1 δ . Analogous calculations (available from the authors) allow one to compare the mixed cases and determine the endogenous competition structure as follows: n
Proposition 5. (i) Cournot competition leads to larger profits for the private firm and larger welfare relative to Bertrand competition if and only if 1 δa0 =a1 n k ok ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1δ (ii) If θ ¼ 0 there exists a cut-off k A ð0; 1Þ such that both firms choose the quantity contract in equilibrium if k rk . (iii) If θ ¼ 1 nnn nnn there exists a cut-off k A ð0; 1Þ such that both firms choose the quantity contract in equilibrium if k rk . nn
nn
If the share of domestic consumers is small, Cournot payoffs dominate Bertrand payoffs for both firms, and Cournot competition can emerge endogenously in equilibrium. This result is in sharp contrast to that presented in the previous sections, which states that Bertrand competition emerges in equilibrium regardless of the nationality of the private firm. The intuition is simple: in the case of an integrated market where firms compete for both domestic and foreign consumers, the public firm has lower incentives to reduce prices in order to improve consumer surplus. In other words, the public firm becomes more profit-oriented, and has an incentive to adopt a quantity contract that mitigates competition.13 This is why Cournot competition emerges in equilibrium when k is small enough. 4.2. Increasing marginal cost Until now, we have adopted a model with constant marginal costs, a setting that is quite popular in the literature.14 However, it is important to verify our baseline results in case of increasing marginal costs.15 Incidentally, there is another advantage in considering the case of increasing marginal costs. When the marginal cost is constant, the profit of the public firm that competes with the foreign private firm is negative unless θ ¼ 0, however, this may not be very realistic. When the marginal cost is increasing, instead, the profit of the public firm can be positive as well. For simplicity, let us return to the case of a domestic market and assume that the cost function is symmetric and quadratic, that is C i ðqi Þ ¼ γ q2i =2, where γ is a positive constant. The proofs are presented in the Appendix. First, we compare welfare and private firm's profit levels between Cournot and Bertrand competition when the private firm is domestic. The result on welfare is the same as in the case of constant marginal costs, but this is not the case for profits of the private firm. qq pp qq Lemma 4. When θ ¼ 0, (i) SW pp 4 SW qq and (ii) both π pp 1 o π 1 and π 1 4 π 1 are possible.
Next, we consider the case of foreign private firm. The results are the same as in the case of constant marginal costs. qq Lemma 5. When θ ¼ 1, (i) SW pp 4 SW qq and (ii) π pp 1 4 π1 .
Accordingly, the result that Bertrand competition yields greater welfare appears to be fairly robust, but the result for which Bertrand competition yields larger profit for the private firm is not. Furthermore, from Lemmas 4 and 5, we can see that foreign ownership in the private firm strengthens the profit-enhancing effect of Bertrand competition. We now discuss the endogenous choice of variables (price or quantity), starting with the following result:. qq pp pq Lemma 6. Both when θ ¼ 0 and θ ¼ 1, (i) SW pq 4 SW qq (ii) SW pp 4 SW qp , (iii) π qp 1 4 π 1 and (iv) π 1 4 π 1 .
From Lemma 6, we find that choosing the price contract is a dominant strategy for both firms, whether the private firm is domestic or foreign. Thus, we confirm the same result obtained under constant marginal costs: Proposition 6. Under quadratic costs, Bertrand competition is the endogenous competition structure for any ownership share of foreign investors. 5. Concluding remarks In this study, we revisited the classic discussion of the comparison between price and quantity competition, but in a mixed duopoly under international competition. Ghosh and Mitra (2010) considered the domestic private firm and showed that in a mixed duopoly, price competition yields a larger profit for the private firm. We find that regardless of the share of foreign ownership in the private firm, price competition yields higher domestic welfare and a larger profit for the private firm. We also endogenize the choice of price or quantity contract. Matsumura and Ogawa (2012) considered the domestic private firm and showed that choosing a price contract is the dominant strategy for both firms. We find that both firms 13 14 15
When k is small, firm 1's profit is larger under Cournot competition than under Bertrand competition. See Proposition 5-i. See among others, Mujumdar and Pal (1998), Pal (1998), Matsumura (2003), Bárcena-Ruiz (2007), and Matsumura and Ogawa (2010, 2012). See De Fraja and Delbono (1989), Fjell and Pal (1996) and Matsumura and Shimizu (2010).
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choose price contracts in the unique equilibrium regardless of the ownership share of foreign investors. This indicates the importance of price competition in mixed oligopolies. However, this is not true and quantity competition appears in equilibrium when we consider an integrated market rather than a domestic market. This result suggests the importance of investigating an integrated market in mixed oligopolies. In this study, we endogenized the competition structure by endogenizing the choice of a price or quantity contract. In this context, endogenizing the number of firms is also important. In the literature on mixed oligopolies, many works have already investigated free entry markets, but most works assumed that the form of contract (whether price or quantity) is given exogenously.16 Endogenizing both these factors remains an area for future research. Appendix A The following positive constants appear in the text: H 1 δ ð2a0 δ a1 Þa1 θ þ 2ð2δð1 δ Þa0 a1 þ δ a20 ð1 δ Þa21 Þθ þð4 δ Þa20 þ ð3 δ Þa21 2
2
2
2
2
2
2
2δð3 δ Þa0 a1 2
H 2 δ ð1 δ Þð2δa0 a1 Þa1 θ þ 2δð1 δ Þð2ð1 δ Þa0 a1 þ ð1 δ Þa21 þ δa20 Þθ þ ðδ 3δ þ 3Þa21 2
2
2
2
2
2
4
2
þ ð2δ 5δ þ 4Þa20 2δðδ 3δ þ 3Þa0 a1 4
2
4
2
H 3 δ ð2a0 δ a1 Þa1 θ þ 2ð2δð1 2δ Þa0 a1 ð1 2δ Þa21 þ δ a20 Þθ 2
2
2
2
2
þ ð1 δ Þðð4 δ Þa20 þ3a21 6δa0 a1 Þ 2
δ
2
δ θ
δ
δ
δ
θ
δ
δ
2 2 2 2 2 2 2 þ 2ð2ð1 2 Þa0 a1 2ð1 2 Þa21 þ a20 Þ þ ð4 5 Þa20 þ ð3 4 Þa21 H 4 ð2a0 a1 a21 Þ 2 2 ð3 4 Þa0 a1 4 2 2 2 2 H5 þ2 þ ð1 Þð4 Þ 4 2 2 2 2 2 H6 þ 4 ð1 Þ þ ð1 Þð4 3 Þ 2 2 2 2 5 4 H 7 ð1 Þð2 a0 a1 Þa1 þ 2 ð2 Þða21 þ a20 2 a0 a1 Þ 2a0 a1 þ ð2a20 þ a21 Þ 3 2 þ 6a0 a1 ð5a20 þ3a21 Þ 6a0 a1 þ 4a20 þ 3a21 2 2 3 2 3 2 2 H 8 ð2a0 ð2 kÞa1 Þa1 2ða1 k ð a0 a1 a21 2 a0 a1 þ 4a21 Þk þ 3 a0 a1 ða20 þ 2a21 Þ
δ δ δ θ δ θ δ δ δ θ δ δ θ δ δ δ δ δ θ δ δ δ θ δ δ δ δ δ δ θ δ δ δ δ δ 2 2 3 2 3 2 2 2 2 2 4δa0 a1 þ 4a1 Þθ þ ð1 δ Þa1 k 2ðδa0 þ ð1 2δ Þa1 Þk ð2a0 a1 δ þ ð4a1 a0 Þδ 12δa0 a1 3 2 þ 4a21 þ 4a20 Þk þ 2ð2a0 a1 δ δ a20 8δa0 a1 þ 4a20 þ 4a21 Þ 2 2 2 2 2 2 2 H 9 a1 δ ð1 δ Þð2δa0 ð2 kÞa1 Þθ 2ð1 δ Þða21 k a1 ða1 ð4 δ Þ a0 δð1 δ ÞÞk 3 2 5 3 2 2 4 þ ð3a0 a1 δ δ ð2a1 þ a0 Þ þ 4a1 ða1 δa0 ÞÞθ ð2 kÞð2a0 a1 δ ð2a0 þ a1 Þδ 8δ a0 a1 2 þ δ ð5a20 þ 4a21 Þ þ8δa0 a1 4ða20 þ a21 ÞÞ 2 3 2 4 2 2 4 2 2 2 H 10 ðγ 3 ð2 3δ Þ þ 2γ 2 βðδ 7δ þ 5Þ þ γβ ð2 δ Þðδ þ 8ð1 δ ÞÞþ 2β ð1 δ Þð2 δ Þ2 Þ 2 2 2 2 2 H 11 ðγ 2 ð2 δ Þ þ 2γβð3 2δ Þ þ β ð1 δ Þð4 3δ ÞÞ
A.1. Proof of Lemma 4 Under Cournot competition, the equilibrium quantity of the public firm is qqq 0 ¼
αγ þ αβð2 ð1 θÞδÞ ; γ 2 þ 3βγ ð1 θÞβ2 δ2 þ 2β2
and that of the private firm is qqq 1 ¼
αγ þ ð1 δÞαβ : γ 2 þ 3βγ ð1 θÞβ2 δ2 þ 2β2
Substituting these equilibrium quantities into the demand and payoff functions, we have the following domestic welfare (up to a positive constant) and firm 1's profit: SW qq p
1 2ðγ 2 þ 3β þð2 ð1 θÞδ Þβ Þ2 2
2
;
ð27Þ
16 For discussions of free entry markets in mixed oligopolies, see Matsumura and Kanda (2005), Brandão and Castro (2007), and works mentioned in Introduction. For recent developments in this field, see Cato and Matsumura (2013), Ghosh et al. (2014), and Ghosh and Sen (2012).
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π qq 1 ¼
351
α2 γ þ 2 β γ þ ð1 δÞβ 2 : 2 2 2ðγ 2 þ 3β þ ð2 ð1 θÞδ Þβ Þ2
ð28Þ
Under Bertrand competition, the equilibrium price of public firm is ppp 0 ¼
αðγ 2 þ γβð1 δÞð2 þ δÞ þð1 θÞβ2 δð1 δÞ2 ð1 þ δÞÞ ; γ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 ð1 θÞδ2 Þ
and that of private firm is ppp 1 ¼
αðγ 2 þ γβð1 δÞð2 þ δÞ þ β2 ð1 δÞ2 ð1 þ δÞÞ : γ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 ð1 θÞδ2 Þ
Substituting these equilibrium prices into the payoff functions, we have the following resulting domestic welfare (up to a positive constant) and firm 1's profit: SW pp p
π pp 1 ¼
1 2ðγ
2þ
γβð3 δ Þ þ β ð1 δ2 Þðθδ2 þ ð2 δ2 ÞÞÞ2 2
2
;
ð29Þ
α2 ððγ þ βð1 δÞÞ2 Þðγ þ2βð1 δ2 ÞÞ ; 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þðθδ þ ð2 δ ÞÞÞ2
ð30Þ
From (27) and (29), for θ ¼ 0 we have (up to a positive constant) SW pp SW qq p
2ðγ
2 þ3
α2 β2 δ2 ðγ þ βð1 δÞÞ2 4 0: γβ β ð2 δ2 ÞÞ2 ðγ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 δ2 ÞÞ2 2
From (28) and (30), for θ ¼ 0 we have (up to a positive constant) qq π pp 1 π1 ¼
α2 β2 δ2 ðγ βδ þ βÞ2 H10 ; 2 2 2 2 2 2ðγ 2 þ 3γβ β ð2 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2 2
qq Substituting α ¼ 1; β ¼ 1; γ ¼ 1; δ ¼ 24=25, we have π pp 1 π 1 0:0003 o 0. Substituting α ¼ 1; β ¼ 1; γ ¼ 1; δ ¼ 1=2, we have pp qq π 1 π 1 0:0076 4 0. □
A.2. Proof of Lemma 5 From (27) and (29), for θ ¼ 1 we have SW pp SW qq ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 3 þ γ 2 βð5 δ2 Þ þ 4γβ2 ð2 δ2 Þ þ 4β3 ð1 δ2 ÞÞ 4 0: 2 2 2 2 2ðγ 2 þ 3βγ þ 2β Þ2 ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2
From (28) and (30), for θ ¼ 1 we have qq π pp 1 π1 ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 3 ð2 δ2 Þ þ 2γ 2 βð5 3δ2 Þ þ 4γβ2 ð4 3δ2 Þ þ 8β3 ð1 δ2 ÞÞ 4 0: 2 2 2 2 2ðγ 2 þ3βγ þ 2β Þ2 ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2
□
A.3. Proof of Lemma 6 In p q game, the equilibrium price of the public firm is ppq 0 ¼
αðγ 2 þ γβð2 δ δ2 Þ β2 δθð1 δÞÞ ; γ 2 þ γβð3 δ2 Þ þ β2 ðδ2 θ 2δ2 þ 2Þ
and the equilibrium quantity of the private firm is qpq 1 ¼
αðγ þ βð1 δÞÞ : γ 2 þ γβð3 δ2 Þ þ β2 ðδ2 θ 2δ2 þ 2Þ
Substituting these into the payoff functions, we have the following expressions for domestic welfare (up to a positive constant) and firm 1's profit SW pq p
1 2ðγ
2þ
γβð3 δ Þ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞÞ2 2
;
ð31Þ
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π pq 1 ¼
α2 ðγ þ βð1 δÞÞ2 ðγ þ 2βð1 δ2 ÞÞ : γβð3 δ2 Þ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞÞ2 2ðγ
ð32Þ
2þ
In q p game, the equilibrium quantity of the public firm is qqp 0 ¼
γ
αðγ þ βð2ð1 δÞδθÞÞ ; γβ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞ
2 þ3
and the equilibrium price of the private firm is pqp 1 ¼
αðγ 2 þ γβð2 δÞ þ β2 ð1 δÞÞ : γ 2 þ 3γβ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞ
Substituting these into the payoff functions, we have the resulting domestic welfare (up to a positive constant) and profit of firm 1 SW qp p
π qp 1 ¼
1 2ðγ
2 þ3
;
ð33Þ
γβ þ β ð2ð1 δ2 Þδ2 θÞÞ 2
ðα2 ðγ þ 2βÞðγ þ βð1 δÞÞÞ2
: 2 2 2 2ðγ 2 þ3γβ þ β ð2ð1 δ Þδ θÞÞ
ð34Þ
For θ ¼ 0 we have SW pq SW qq ¼
α2 β2 δ2 ðγ þ βÞðγ þ βð1 δÞÞ2 ðγ 2 ð2 δ2 Þ þ2γβð3 2δ2 Þ þ β2 ð1 δ2 Þð4 δ2 ÞÞ 4 0; 2 2 2 2 2 2ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2 ðγ 2 γβð3 δ Þ þ2β ð1 δ ÞÞ2
SW pp SW qp ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ ð1 þ δ2 Þ þ βð1 δ2 ÞÞH11 40; 2 2 2 2 2 2 2ðγ 2 þ3γβ þ 2β ð1 δ ÞÞ2 ðγ 2 þ γβ ð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2
π qp π qq ¼
α2 β2 δ2 ðγ þ2βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ6γβ þ β2 ð4 3δ2 ÞÞ 40 2 2 2 2 2ðγ 2 þ 3γβ þ 2β ð1 δ ÞÞ2 ðγ 2 þ3γβ þ β ð2 δ ÞÞ2
π pp π pq ¼
α2 β2 δ2 ð1 δ2 Þðγ þ βð1 δÞÞ2 ðγ þ 2βð1 δ2 ÞÞð2γ 2 þ2γβð3 δ2 Þ þ β2 ð1 δ2 Þð4 δ2 ÞÞ 40: 2 2 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2
and
For θ ¼ 1 we have SW pq SW qq ¼
α2 β2 δ2 ðγ þ βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ γβð6 δ2 Þ þ β2 ð4 δ2 ÞÞ 40; 2 2 2 2 2ðγ 2 þ 3γβ þ2β Þ2 ðγ 2 þ γβð3 δ Þ þ β ð2 δ ÞÞ2
SW pp SW qp ¼
α2 β2 δ2 ðγ þ βð1 δ2 ÞÞðγ þ βð1 δÞÞ2 ð2γ 2 þ γβð6 δ2 Þ þ β2 ð4 3δ2 ÞÞ 4 0; 2 2 2 2 2 2ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ2β ð1 δ ÞÞ2
π qp π qq ¼
α2 β2 δ2 ðγ þ2βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ6γβ þ β2 ð4 δ2 ÞÞ 40 2 2 2 2ðγ 2 þ3γβ þ 2β Þ2 ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2
π pp π pq ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 2βð1 δ2 ÞÞð2γ 2 þ 2γβð3 δ2 Þ þ β2 ð4 3δ2 ÞÞ 4 0: 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ2β ð1 δÞð1 þ δÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð2 δ ÞÞ2
and □
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Contents lists available at ScienceDirect
Research in Economics journal homepage: www.elsevier.com/locate/rie
Price versus quantity in a mixed duopoly with foreign penetration$ Junichi Haraguchi a,n, Toshihiro Matsumura b a b
Graduate School of Economics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Institute of Social Science, The University of Tokyo, Japan
a r t i c l e in f o
abstract
Article history: Received 5 March 2014 Accepted 27 September 2014 Available online 13 October 2014
We characterize the endogenous competition structure (in prices or quantities) in a differentiated duopoly between a public firm that maximizes domestic welfare and a private firm that can be owned by domestic or foreign investors. The market for which they compete can be domestic or integrated: in the first case Bertrand competition emerges endogenously and in the second case Cournot competition can emerge if the fraction of domestic consumers in the integrated market is low enough. We also determine the optimal degree of foreign penetration showing the optimality of a partial foreign ownership. Finally, we extend the model to increasing marginal cost confirming the robustness of the results. & 2014 University of Venice. Published by Elsevier Ltd. All rights reserved.
Keywords: Cournot Bertrand Mixed markets International competition Trade
1. Introduction The comparison between price and quantity competition has been extensively discussed in the literature. In oligopolies between private firms it is well known that price competition is tougher, resulting in a lower level of profits, compared to quantity competition.1 Ghosh and Mitra (2010) have revisited this classic result in a mixed oligopoly, namely when a welfare-maximizing public firm competes with a profit-maximizing private firm.2 They have shown that, in contrast to the private duopoly, quantity competition is tougher than price competition, resulting in a smaller profit for the private firm.3 A related literature, started by Singh and Vives (1984), has endogenized the structure of competition (in prices or in quantities). As they pointed out, firms often choose whether to adopt a price contract or a quantity contract. In a private duopoly where both firms maximize profits, assuming linear demand and product differentiation, Singh and Vives (1984) have shown that the choice of a quantity contract is the dominant strategy for each firm when goods are substitutes, and the choice of a price contract is the dominant strategy when goods are complements. Cheng (1985), Tanaka (2001a,b) and Tasnádi (2006) have extended their analysis to asymmetric oligopolies, more general demand and cost conditions, and
☆ We are indebted to two anonymous referees and the editor for their valuable and constructive suggestions. Needless to say, we are responsible for any remaining errors. Financial support of the Grant-in-Aid from Murata Science Foundation and Japanese Ministry of Education, Science and Culture is greatly appreciated. n Corresponding author. Tel.: þ81 5841 4932; fax: þ81 5841 4905. E-mail address: [email protected] (J. Haraguchi). 1 See Shubik and Levitan (1980) and Vives (1985). 2 The pioneering work on mixed oligopolies was by Merrill and Schneider (1966). This and many other studies in the field assume that a public firm maximizes welfare (consumer surplus plus firms' profits) while private firms maximize profits. 3 See also Nakamura (2013) under network externality.
http://dx.doi.org/10.1016/j.rie.2014.09.001 1090-9443/& 2014 University of Venice. Published by Elsevier Ltd. All rights reserved.
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339
vertical product differentiation, confirming the robustness of the results. Recently, Matsumura and Ogawa (2012) have obtained a contrasting example in case of a mixed duopoly: when one of the two firms is public, the choice of a price contract is the dominant strategy for both private and public firms, regardless of whether goods are substitutes or complements. All the mentioned papers, including Ghosh and Mitra (2010) and Matsumura and Ogawa (2012), assume that private firms are domestically owned and firms compete in the domestic market. In other words, they ignore any aspect of international competition, while we know that the nationality of private firms is often crucial in shaping mixed oligopolies (see the literature starting with Corneo and Jeanne, 1994, and Fjell and Pal, 1996)4 and the nationality of the consumers for which private and public firms are competing is equally crucial (see the trade literature staring with Brander and Spencer, 1985, even if the existence of public firms is usually neglected in that literature). In this work we present a general model of the endogenous form of competition where the ownership of the private firm can be domestic, foreign or mixed, and consumers can be entirely or partially domestic. Therefore the model covers the standard case of domestic firms competing in the domestic market (as already in Ghosh and Mitra, 2010, and Matsumura and Ogawa, 2012) and cases that are typically analyzed in the international trade literature, as when domestic and foreign firms compete in the home market, when they compete in a fully integrated market, and when they compete in a third market (without domestic consumers, as in Brander and Spencer, 1985). We find that foreign penetration can change radically the comparison between Bertrand and Cournot price and production strategies, but the endogenous competition structure depends on the kind of market for which the firms compete: in the traditional case of a domestic market Bertrand competition emerges always as the endogenous competition structure and independently from the ownership share of foreign owners in the private firm, but in the case of an integrated market where firms serve both domestic and foreign consumers Cournot competition can emerge as long as the fraction of domestic consumers is small enough. We also characterize the optimal degree of foreign penetration showing that it is optimal to have a minority share for foreign owners: in particular, when the firms produce almost homogeneous goods it is optimal to maintain fully domestic the private firm, while when they are independent monopolists, it becomes optimal to sell half of the stocks to foreign investors. Extending the model to increasing marginal costs we confirm the robustness of our results. The paper is organized as follows. Section 2 describes the model and compare Cournot and Bertrand equilibria. Section 3 characterizes the endogenous competition structure and the optimal level of foreign penetration. Section 4 extends the model in different directions. Section 5 concludes. 2. The model We adopt a standard duopoly model with differentiated goods and linear demand (Dixit, 1979).5 The quasi-linear utility function of the representative consumer is β U q0 ; q1 ; y ¼ α q0 þ q1 q20 þ 2δq0 q1 þ q21 þ y; 2
ð1Þ
where q0 is the consumption of good 0 produced by the public firm, q1 is the consumption of good 1 produced by the private firms and y is the consumption of an outside good that is competitively provided (with a unitary price). Parameters α and βare positive constants and δ A ð0; 1Þ6 represents the degree of product differentiation: a smaller δ indicates a larger degree of product differentiation. The inverse demand functions for goods i¼0,1with i aj are pi ¼ α βqi βδqj ;
ð2Þ
where pi is the price of firm i. The marginal cost of production is constant for both firms. Let us denote with ci the marginal cost of firm i, assuming α 4ci . Firm 0 is a state-owned public firm whose payoff is the domestic social surplus (welfare). This is given by βðq20 þ2δq0 q1 þq21 Þ p0 q0 p1 q1 ; ð3Þ SW ¼ p0 c0 q0 þ 1 θ p1 c1 q1 þ α q0 þ q1 2
4 Pal and White (1998) and Bárcena-Ruiz and Garzón (2005) discussed trade policies. Mukherjee and Suetrong (2009) discussed the relationship between foreign direct investment and privatization policies. Matsumura et al. (2009) used a monopolistic competition model by Anderson et al. (1997) to show that under foreign ownership privatization is more likely to improve welfare in the long run, whereas the opposite result is derived in the short run. Matsushima and Matsumura (2006), Ogawa and Sanjo (2007), and Heywood and Ye (2009) have shown that foreign ownership affects firms' locations. Han and Ogawa (2008) and Lin and Matsumura (2012) adopted the partial privatization approach formulated by Matsumura (1998) showing that foreign penetration affects the welfare implications of privatization policies; and Wang and Chen (2010) and Cato and Matsumura (2012) demonstrated the same property for free entry markets. Wang and Lee (2013) introduced foreign firms into the framework of Ino and Matsumura (2010) showing that foreign ownership matters in Stackelberg models. Matsumura and Tomaru (2012) revisited the privatization neutrality theorem presented by White (1996) showing that his result does not hold under foreign ownership of the private firms. 5 This demand function is popular in the literature on mixed oligopolies. See Bárcena-Ruiz (2007), Fujiwara (2007), Ishida and Matsushima (2009), Matsumura and Shimizu (2010), and Nakamura (2013). 6 If δ 4 ð o Þ 0, the products are substitutes (complements). Although we restrict our attention to the case of substitute products only, we can show that our main propositions hold when δ A ð 1; 0Þ as well.
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where θ A ½0; 1 is the ownership share of foreign investors in firm 1, which can be potentially affected by policymakers acting on capital liberalization. Firm 1 is a private firm and its payoff is its own profit:
π 1 ¼ ðp1 c1 Þq1 :
ð4Þ
Notice that we are now assuming that the representative consumer is a domestic one, therefore we are focusing on competition for the domestic market: in a later section we will move to the analysis of an integrated market where domestic consumers are only some of the consumers and the public firm maximizes the sum of their surplus and total domestic profits. We now move to the characterization of the Cournot and Bertrand equilibria. We assume that the solutions in all the following games are interior, that is, equilibrium prices and quantities of both firms are strictly positive for any θ. This happens if α is large, jc0 c1 j is not too large, and c0 is not too small. Let us define ai α ci , and let us adopt the superscript ‘ij’ to denote the equilibrium outcome when firm 0 chooses i A fp; qg and firm 1 chooses j A fp; qg. 2.1. Cournot model (q q game) First, we discuss the Cournot model (q q game) in which both firms choose quantities. The first-order conditions for firms 0 and 1 are, respectively, ∂SW ¼ a0 βq0 βδq1 þ θβδq1 ¼ 0; ∂q0 ∂π 1 ¼ a1 2βq1 βδq0 ¼ 0: ∂q1 The second-order conditions are satisfied. From the first-order conditions, we obtain the following reaction functions for firms 0 and 1, respectively: a0 βδq1 þ θβδq1 Rqq ; 0 q1 ¼ Rqq 1 q0
β
a1 βδq0 : ¼ 2β
These functions lead to the following expressions for the equilibrium quantity of each firm, respectively: qqq 0 ¼
2a0 δa1 þ θδa1
;
βð2 δ2 Þ þ θβδ2 a1 δa0 : qqq 1 ¼ βð2 δ2 Þ þ θβδ2
Note that the assumption of the interior solutions implies that 2a0 δa1 4 0 and a1 δa0 4 0. Substituting these equilibrium quantities into the demand and payoff functions, we have the following resulting domestic welfare and firm 1's profit, respectively: SW qq ¼
π qq 1 ¼
H1 2βð2 δ ð1 θÞÞ2 2
ða1 δa0 Þ2
βð2 δ2 ð1 θÞÞ2
;
:
ð5Þ
ð6Þ
where the constant H1 and other constants are reported in the Appendix. 2.2. Bertrand model (p p game) We now characterize the Bertrand model (p p game) in which both firms choose prices. The direct demand functions can be derived as q0 ¼
α αδ p0 þ δp1 ; βð1 δ2 Þ
ð7Þ
q1 ¼
α αδ p1 þ δp0 : βð1 δ2 Þ
ð8Þ
The first-order conditions for firms 0 and 1 are respectively ∂SW c0 p0 δc1 þ δp1 θδðp1 c1 Þ ¼ ¼ 0; ∂p0 βð1 δ2 Þ
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∂π 1 c1 2p1 þ α þ δp0 αδ ¼ ¼ 0: ∂p1 βð1 δ2 Þ The second-order conditions are satisfied, and we can solve for the following reaction functions for firms 0 and 1: Rpp 0 p1 ¼ c0 þ δ p1 c1 θδ p1 c1 ; c1 þ α þ p0 δ αδ : Rpp 1 p0 ¼ 2 These functions lead to the following expressions for the equilibrium prices of each firm:
αδ αδ2 þ 2c0 c1 δ θðαδ αδ2 δc1 Þ ; 2 2 2δ þδ θ α αδ þc1 þc0 δ c1 δ2 þ θδ2 c1 : ppp 1 ¼ 2 2 2δ þδ θ ppp 0 ¼
Substituting these equilibrium prices into the payoff functions, we have the following resulting domestic welfare and firm 1's profit, respectively: SW pp ¼
π pp 1 ¼
H2 2βð1 δ Þðδ θ δ þ2Þ2 2
2
2
ða1 δa0 Þ2
βð1 δ2 Þðδ2 θ δ2 þ 2Þ2
;
ð9Þ
:
ð10Þ
2.3. Cournot–Bertrand comparison Before presenting our main results, we present some supplementary results. Lemmas 1 and 2 show how the quantity ranking between Bertrand and Cournot depends on θ: qq pp qq Lemma 1. When θ ¼ 0, (i) qpp 0 o q0 and (ii) q1 4q1 .
Proof. By direct computation we have qq qpp 0 q0 ¼
δðδð1 δÞθc1 ða1 δa0 ÞÞ ; βð1 δ2 Þð2 δ2 ð1 θÞÞ
ð11Þ
qq qpp 1 q1 ¼
δðδða1 δa0 Þ ð1 δÞθc1 Þ : βð1 δ2 Þð2 δ2 ð1 θÞÞ
ð12Þ
Substituting θ ¼ 0 into (11)–(12), we obtain δða1 δa0 Þ o 0; βð1 δ2 Þð2 δ2 Þ δ2 ða1 δa0 Þ qq 4 0: qpp 1 q1 ¼ βð1 δ2 Þð2 δ2 Þ qq qpp 0 q0 ¼
Note that a1 δa0 4 0 under the interior solution assumption made in this paper.
□
As Singh and Vives (1984) pointed out, demand is more elastic under Bertrand competition compared to Cournot competition. Thus, the private firm tends to choose a lower price, resulting in a larger output under Bertrand competition. The public firm, however, has an incentive to increase the rival's output, and obtains this by choosing a higher price and a smaller output under Bertrand competition. Because products are substitutes, firm 1's low price provides and additional reason to reduce the resulting output of firm 0: qq pp qq pp qq pp qq Lemma 2. When θ ¼ 1, (i) both qpp 0 o q0 and q0 4 q0 are possible, and (ii) both q1 4q1 and q1 o q1 are possible.
Proof. Substituting θ ¼ 1 into (11) and (12), we have qq qpp 0 q0 ¼
δðδð1 δÞc1 ða1 δa0 ÞÞ ; 2 2βð1 δ Þ
ð13Þ
qq qpp 1 q1 ¼
δðδða1 δa0 Þ ð1 δÞc1 Þ : 2 2βð1 δ Þ
ð14Þ
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qq (i) Substituting δ ¼ 1=2; α ¼ 50; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (13), we have qpp 0 q0 ¼ 13=2 o 0. Substituting δ ¼ 3=7;
qq α ¼ 10; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (13), we have qpp 0 q0 ¼ 57=560 40. (ii) Substituting δ ¼ 1=2; α ¼ 50; β ¼ 1; c0 ¼ 9, and pp qq c1 ¼ 8 into (14), we have q1 q1 ¼ 9=4 4 0. Substituting δ ¼ 1=4; α ¼ 10; β ¼ 1; c0 ¼ 9, and c1 ¼ 8 into (14), we have
qq qpp 1 q1 ¼ 89=120 o 0.
□
When the share of foreign ownership in the private firm is higher, the public firm becomes more aggressive in the attempt to reduce the private firm's profit outflow. As mentioned above, demand is more elastic under Bertrand competition; thus, it is relatively easier for the public firm to reduce its rival's profit. This results in the public firm behaving more aggressively. When this effect is sufficiently strong, the output-level ranking in firm 0 is reversed. Since goods are substitutes, the output-level ranking in firm 1 can also be reversed. We now present our main results. First, we compare firm 1's profit in the two games. As Lemmas 1 and 2 state, the quantity ranking between Bertrand and Cournot depends on θ. However, Proposition 1 shows that, surprisingly, the profit ranking does not depend on θ: Proposition 1. (i) The profits of the private firm are always decreasing in the ownership share of foreign investors, (ii) Bertrand competition leads always to larger profits for the private firm relative to Cournot competition and (iii) the difference in profits is always decreasing in the ownership share of foreign investors. Proof. From (6) and (10), (i) is obvious. From (6) and (10), we have qq π pp 1 π1 ¼
δ2 ða1 δa0 Þ2 4 0: βð1 δ2 Þð2 δ2 ð1 θÞÞ2
whose sign is positive for any
δ and which is obviously decreasing in θ A ½0; 1. This proves (ii) and (iii).
□
An increase in the ownership share of foreign investors strengthens competition in both Bertrand and Cournot duopolies. The outflow to foreign investors is proportional to θπ 1 . Therefore, an increase in θ increases the incentive for firm 0 to reduce the outflow to foreign investors, becoming more aggressive. We label this effect as the “outflow-restricting effect”. pp qq qq This is stronger when π1 is larger. Thus, an increase in θ reduces π1 more significantly than π1 as long as π pp 1 4 π1 . When θ ¼ 0 (and thus, when there is no outflow-restricting effect), Ghosh and Mitra (2010) have already shown that Bertrand competition leads to larger profits for the private firm relative to Cournot competition. As Singh and Vives (1984) have pointed out, demand is more elastic with respect to its own price and to the rival's price under Bertrand competition. Suppose that substituting ðp0 ; p1 Þ ¼ ðp00 ; p01 Þ into the demand function yields ðq0 ; q1 Þ ¼ ðq00 ; q01 Þ. From the demand functions in the p-p and q-q games, we have that firm 1's demand elasticity under Bertrand competition when p0 ¼ p00 , 1 . Because firm 1's ð∂q1 =∂p1 Þðp01 =q01 Þ, is larger than that under Cournot competition when q0 ¼ q00 , ð∂p1 =∂q1 Þðq01 =p01 Þ demand is more elastic with respect to its own price under Bertrand competition at the point ðp0 ; p1 Þ ¼ ðp00 ; p01 Þ than under Cournot competition at the point ðq0 ; q1 Þ ¼ ðq00 ; q01 Þ, the private firm becomes more aggressive under Bertrand competition. This results in a larger output of firm 1 in equilibrium under Bertrand competition. The public firm prefers such a larger output. Similarly, firm 1's demand is more elastic with respect to firm 0's price under Bertrand competition. Since the public firm is concerned with welfare, it prefers a larger output of the private firm. Thus, the public firm becomes less aggressive under Bertrand competition, which leads to a smaller output for firm 0 under Bertrand competition. Both effects increase the private firm's profit. Thus, without outflow-restricting effect, Bertrand yields a larger profit for the private firm. We call this effect “demand-elasticity effect”. Because the demand functions given by (2) and (8) do not depend on θ, this effect does
not depend on θ.
π
qq When θ ¼ 0, π pp 1 4 π 1 . As mentioned above, as
π
qq 1 .
A decrease in π
qq θ increases, outflow-restricting effect reduces π pp 1 π 1 , as long as
π reduces the difference in outflow-restricting effect between Bertrand and Cournot models. θ increases, qq π would be close to zero. On the other hand, demand-elasticity effect remains unchanged and it converges to zero if π pp 1 1 as θ increases. Thus, difference in outflow-restricting effect between Bertrand and Cournot models is never large enough to pp 1 4
pp 1
qq 1
On one hand, the difference in outflow-restricting effect between Bertrand and Cournot models is smaller as
dominate demand-elasticity effect and never reverses the profit ranking, even when the private firm is entirely foreign owned. We finally compare welfare in the two models: Proposition 2. (i) Bertrand competition leads always to larger domestic welfare relative to Cournot competition and (ii) the difference is always decreasing in the ownership share of foreign investors. Proof. From (5) and (9), we have SW pp SW qq ¼
δ2 ða1 δa0 Þ2 4 0; 2 2 2βð1 δ Þð2 δ ð1 θÞÞ2
whose sign is positive for any
δ and which is obviously decreasing in θ A ½0; 1.
□
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Ghosh and Mitra (2010) have already shown that SW pp 4 SW qq when θ ¼ 0: here we extend the result to any ownership share in spite of counteracting forces. Indeed, Cournot competition yields more severe competition compared to Bertrand competition, which restricts the profit outflow toward foreign investors more effectively. This becomes more important when θ increases. However, a larger θ accelerates competition as well and reduces overall profits. Because of this, the outflow-restricting effect mentioned above is never strong enough to reverse the welfare ranking, even when the private firm is entirely foreign owned. The comparison between equilibria and the general result for which Bertrand yields softer competition will be at the source of our analysis of the endogenous competition structure, but may also be important in the analysis of the stability of collusion, at least when aimed at maximizing welfare (with side payments to the private firm).7 While this is clearly beyond the scope of this work, a remark is in order. If we consider a grim trigger strategy (Friedman, 1971; Deneckere, 1983), our result implies that one can always find a more severe punishment for the private firm's deviation under Cournot competition rather than under Bertrand competition, which can lead to more stable collusion under Cournot competition.8 This is against standard results emerging in duopolies with private firms.
3. Endogenous competition structure In this section, we endogenize the competition structure (either price or quantity). We follow the standard model formulated by Singh and Vives (1984). The game runs as follows. In the first stage, each firm chooses whether to adopt a price contract or a quantity contract. In the second stage, after observing the rival's choice in the first stage, each firm simultaneously chooses its own strategy, according to the decision in the first stage. There are four possible subgames: both firms choose quantity contracts (q q game), both firms choose price contracts (p p game), only firm 0 chooses the quantity contract (q p game), or only firm 0 chooses the price contract (p q game). Again, we assume that the solutions in these four games are interior, that is, equilibrium prices and quantities of both firms are strictly positive. We have already solved the p p and q q games; we now solve the remaining two games. 3.1. p q game Consider the situation in which firm 0 chooses the price contract and firm 1 chooses the quantity contract. The relevant direct and inverse demand functions are given by q0 ¼
α βδq1 p0 β
ð15Þ
p1 ¼ ðβδ βÞq1 þ δp0 αδ þ α 2
ð16Þ
The first-order conditions are respectively ∂SW c0 p0 ¼ θδq1 ¼ 0; ∂p0 β ∂π 1 2 ¼ a1 þ δp0 2β ð1 δ Þq1 αδ ¼ 0: ∂q1 7 Profit-maximizing collusion is impossible even if we consider transfers from the private firm to the public one. Welfare maximizing collusion requires both firms to set following prices:
ðδc1 þ αδ αδÞθ þ ð δc1 þ ð2 δ Þc0 αδ þ αδÞθ þ ðδ 1Þc0 2
p0 ¼
2
2
2
2
δ2 θ2 þ ð2 2δ2 Þθ þ δ2 1 2 2 2 δ c1 θ þ ðð1 2δ Þc1 þ δc0 αδ þ αÞθ þ ðδ2 1Þc1 p1 ¼ δ2 θ2 þ ð2 2δ2 Þθ þ δ2 1 or the corresponding quantities. This leads to the following profits for the private firm:
π1 ¼
ða1 δa0 Þ2 ðθ 1Þθ βðδ2 θ2 2δ2 θ þ 2θ þ δ2 1Þ2
The one-shot deviation strategies and the associated profits can be easily derived under both price and quantity competition. 8 Consider the infinitely repeated game of our one-shot game. Only if the private firm chooses the collusive welfare-maximizing strategy and there was no deviation in the past periods, it obtains a transfer from the public firm. Once the private firm deviates from collusion (the public one has no incentives to do it), the Bertrand or Cournot equilibrium holds forever. Since the punishment is more severe under Cournot competition, it is natural to guess that collusion is more stable in this case. However, the deviation from collusion is more profitable under Cournot competition, which goes in the opposite direction. Assume θ ¼ 0, α ¼ 10, c0 ¼ 8, c1 ¼ 6, β ¼ 1, and consider different values of the degree of substitutability δ. Then, it can be shown that when goods are highly substitutes (δ 4 0:17) collusion is more stable under Bertrand, but when they are poorly substitutes (δ o 0:17) it becomes more stable under Cournot.
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and lead to the following reaction functions for firms 0 and 1: Rpq 0 q1 ¼ c0 βθδq1 ; a1 þ δp0 αδ : Rpq 1 p0 ¼ 2 2βð1 δ Þ These functions lead to the following expressions for the respective equilibrium choice of each firm: 2ð1 δ Þc0 θδða1 δαÞ 2
ppq 0 ¼ qpq 1 ¼
2ð1 δ Þ þ θδ 2
2
a1 δa0 2β ð1 δ Þ þ βθδ 2
2
;
:
Substituting these into the payoff functions, we have the resulting domestic welfare and firm 1's profit, respectively. SW pq ¼
π pq 1 ¼
H3 2βðδ θ þ 2ð1 δ ÞÞ2 2
2
ð1 δ Þða1 δa0 Þ2
;
ð17Þ
2
βðδ2 θ þ2ð1 δ2 ÞÞ2
:
ð18Þ
3.2. q p game Consider the situation in which firm 0 chooses the quantity contract and firm 1 chooses the price contract. The relevant demand functions are p0 ¼ ðβδ β Þq0 þ δp1 αδ þ α 2
q1 ¼
ð19Þ
α βδq0 p1 β
ð20Þ
The first-order conditions are respectively ∂SW 2 ¼ a0 δa1 βð1 δ Þq0 þ θδ p1 c1 ¼ 0; ∂q0 ∂π 1 c1 2p1 þ α βδq0 ¼ ¼ 0: ∂p1 β From this system we obtain the following reaction functions: a0 δa1 þ θδðp1 c1 Þ ; Rqp 0 p1 ¼ βð1 δ2 Þ α þ c1 βδq0 : Rqp 1 q0 ¼ 2 which lead to the following expressions for the respective equilibrium choice of each firm: qqp 0 ¼
2ða0 δa1 Þ þ θδa1 2β ð1 δ Þ þ βθδ 2
2
;
c1 þ α þ c0 δ δα 2c1 δ þ θδ c1 2
pqp 1 ¼
2ð1 δ Þ þ θδ 2
2
2
:
Substituting these into the payoff functions, we have the resulting domestic welfare and profit of firm 1 SW qp ¼
π qp 1 ¼
H4 2βðδ θ þ 2ð1 δ ÞÞ2 2
2
ða1 δa0 Þ2
βðδ θ þ2ð1 δ2 ÞÞ2 2
:
;
ð21Þ
ð22Þ
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Table 1 First stage game. 0\1
Quantity
Price
Quantity
ðSW ; π qq 1 Þ ðSW pq ; π pq 1 Þ
ðSW qp ; π qp 1 Þ
Price
ðSW pp ; π pp 1 Þ
3.3. Equilibrium We now discuss the choice at the first stage. Table 1 summarizes the game. The best responses of each firm can be easily derived as follows: pq qp qq Lemma 3. (i) SW pq 4SW qq , (ii) SW pp 4SW qp , (iii) π pp 1 4 π 1 , and (iv) π 1 4 π 1 .
Proof. (i) From (5) and (17), we have SW pq SW qq ¼
δ2 ða1 δa0 Þ2 H5 : 2 2 2 2 2βðδ θ þ 2 δ Þ2 ðδ θ þ2ð1 δ ÞÞ2
H5 is positive under the assumption of interior solution. (ii) From (9) and (21), we have SW pp SW qp ¼
δ2 ða1 δa0 Þ2 H6 : 2 2 2 2β ð1 δ Þðδ θ 2δ þ2Þ2 ðδ θ δ þ 2Þ2 2
2
H6 is positive under the assumption of interior solution. (iii) From (10) and (18), we have pq π pp 1 π1 ¼
δ2 ða1 δa0 Þ2 ðð2 δ2 Þδ4 θ2 þ 2ð1 δ2 Þð3 δ2 Þθ þ ð4 δ2 Þð1 δ2 Þ2 Þ : βð1 δ2 Þðδ2 θ 2δ2 þ2Þ2 ðδ2 θ þ 2 δ2 Þ2
which is positive under the assumption of interior solution. (iv) From (6) and (22), we have qq π qp 1 π1 ¼
δ2 ða1 δa0 Þ2 ð2δ2 θ þ 4 3δ2 Þ : βð2 δ2 ð2 θÞÞ2 ð2 δ2 ð1 θÞÞ2
which is positive under the assumption of interior solution.
□
We now present our second main result: Proposition 3. Bertrand competition is the endogenous competition structure for any ownership share of foreign investors. Proof. Lemma 3(i) and Lemma 3(ii) imply that choosing p is the dominant strategy for firm 0. Lemma 3(iii) and Lemma 3(iv) imply that choosing p is the dominant strategy for firm 1. □ To explain the intuition behind Proposition 3 for any degree of foreign ownership, we first show that firm 0 chooses the price contract in the first stage if firm 1 chooses the price contract. From (8) and (20), for any ðp00 ; q00 Þ, we have ∂q 1 ∂q1 1 ¼ ¼ : 1 4 2 ∂p1 p0 ¼ p0 ∂p1 q0 ¼ q0 β β ð1 δ Þ 0 0 In other words, an increase in p1 at the second stage reduces more significantly firm 1's quantity when firm 0 chooses the price contract than the quantity contract. Thus, firm 1 tends to adopt a lower price in the second stage when firm 0 chooses the price contract in the first stage. Therefore, by choosing a price contract in the first stage, the public firm induces the private one to be more aggressive in the competition stage. The lower price of the private firm improves welfare therefore the public firm prefers the price contract. Next, we show that firm 1 chooses the price contract in the first stage if firm 0 chooses the price contract. Suppose that firm 1 chooses the price contract: in the second stage, firm 0 chooses p0 against p1. Because firm 0 prefers a larger quantity of firm 1, an increase in p0 increases q1 and improves welfare (at the cost of deviating from firm 0's optimal price). Suppose now that firm 1 chooses the quantity contract: in the second stage, firm 0 chooses p0 against q1. Since firm 0 prefers a lower price of firm 1, a decrease in p0 increases p1 and improves welfare (at the cost deviating from firm 0's optimal price). Therefore, firm 0 naturally sets a higher price when firm 1 chooses the price contract rather than a quantity contract. Accordingly, by choosing the price contract in the first stage, firm 1 induces firm 0's to be more accommodating in the competition stage. Thus, choosing the price contract is firm 1's best reply. We now consider the two extreme cases, those of the domestic firm ðθ ¼ 0Þ and the foreign firm ðθ ¼ 1Þ. We plot four points of equilibrium prices in four fixed contract games in the two cases in Fig. 1.9 Point ijk (i; j ¼ fp; qg; k ¼ fD; Fg) denotes 9
A similar figure is developed by Cheng (1985).
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Fig. 1. Equilibrium prices (θ ¼0 or
θ ¼1).
Fig. 2. Equilibrium relative prices (θ ¼ 0 or
θ ¼ 1).
the equilibrium pair of prices. i (res. j) denotes the choice of public (res. private) firm and the superscript k ¼ D (res. k¼F) denotes that θ ¼ 0 (res. θ ¼ 1). From Fig. 1, we can see that foreign penetration makes the public firm more aggressive, and it reduces the private firm's price through the strategic interaction. Although the price levels are lower in case of foreign penetration, we can see a similar relationship among the four points, whether θ ¼ 0 or θ ¼ 1. Therefore, similar qualitative results hold regardless of θ. We also plot in Fig. 2 the equilibrium relative price p0 =p1 for these cases. We can see that price competition yields a higher relative price p0 =p1 and foreign penetration yields a lower relative price, which in this case can be interpreted as the terms of trade. Both quantity competition and foreign penetration accelerate competition, and reduce the public firm's price more significantly than the private firm's price, thereby yielding a lower relative price. To complete the picture on endogenous competition structures in a duopoly with differentiated goods, it is useful to mention what happens in the endogenous timing model formulated by Hamilton and Slutsky (1990), where each firm can decide when to commit to a quantity or price strategy in one of two subsequent periods. Pal (1998) and Matsumura (2003) have analyzed endogenous timing in a mixed duopoly with quantity competition. The first (second) work shows that in the case of a domestic (foreign) private firm, the public firm becomes endogenously the follower (Stackelberg leader).10 Bárcena-Ruiz (2007) has investigated endogenous timing in a mixed duopoly with price competition. He shows that when the private firm is domestic, a simultaneous-move outcome (Bertrand competition) appears endogenously in equilibrium, and the same result holds when the private firm is foreign.11 It would be interesting to compare equilibrium welfare and profits under the two forms of competition and, eventually, determine the equilibrium of anex ante stage in which the firms choose between price or quantity contracts. 3.4. Welfare and the optimal degree of foreign penetration Finally, we discuss the relationship between θ and welfare to derive our third main result. World welfare is the sum of consumer surplus and profits of both firms, which are crucially affected by the share of the private firm that is owned by 10 Notice that in the domestic private firm case, the equilibrium with the public firm as follower yields a second best outcome. Now, suppose that the private firm becomes foreign-owned and the public firm remains the follower. Then, after observing the private firm's action, the public firm becomes more aggressive in order to reduce the private firm's price, and expecting this aggressive behavior of the public firm, the private firm becomes less aggressive, which reduces welfare. For this reason the public firm prefers to become a leader. Strictly speaking, multiple equilibria exist in a mixed duopoly when the private firm is domestic. However, Matsumura and Ogawa (2010) have shown that private leadership is risk dominant. 11 In the price competition model, whether the private firm is domestic or foreign, the private (public) firm sets a higher (lower) price when it is the leader than when both firms move simultaneously. Thus, both firms do not want to be the follower, and thus, the simultaneous-move outcome appears in equilibrium.
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foreign investors. Since the degree of capital liberalization can affect such a share, it is interesting to determine its optimal level from a global perspective. Proposition 4. Welfare is maximized by the following ownership share of foreign investors: 2 1δ 1 θn ¼ A 0; 2 2 2δ which is increasing in product differentiation. Proof. World welfare is given by βðq20 þ2δq0 q1 þq21 Þ p0 q0 p1 q1 : V ¼ p0 c0 q0 þ p1 c1 q1 þ α q0 þ q1 2 Substituting equilibrium prices under Bertrand competition, we have V pp ¼
H7 2β ð1 δ Þðδ θ δ þ 2Þ2 2
2
2
:
Differentiating Vpp with respect to θ, we have ∂V pp δ ða1 δa0 Þ2 ð1 δ ð2 δ ÞθÞ p : ∂θ βð1 δ2 Þð2 δ2 ð1 θÞÞ3 2
2
2
This is negative (positive) if θ 4ð o Þð1 δ Þ=ð2 δ Þ. Of course ∂θ =∂δ ¼ 2δ=ð2 δ Þ2 o0 with θ ¼ 1=2 for δ ¼ 0 and θ -0 for δ-1. □ 2
2
n
2
n
n
World welfare is nonmonotonic with respect to the degree of foreign penetration θ. Suppose that θ is close to zero. A slight increase in θ makes the public firm more aggressive, and through strategic interaction, it reduces the private firm's price: this improves world welfare. When θ is large, a further increase in θ makes the public firm too aggressive for world welfare because the public firm is not concerned about the private firm's profit and the difference between domestic and world welfare is large. In other words, the public firm's price is too low for world welfare. Thus, further aggressive pricing by the public firm caused by a larger θ reduces world welfare. The optimal ownership share of foreign investors is decreasing in δ, and in particular when firms produce almost homogeneous goods (δ-1) it is optimal to maintain fully domestic private firm, while when they produce fully independent goods (δ ¼ 0), that is they are independent monopolists, it becomes optimal to sell half of the stocks to foreign investors. This suggests that more differentiation between the goods provided by the firms leads to the optimality of a larger foreign penetration. As mentioned above, an increase in θ makes the public firm more aggressive and it is harmful for world welfare when θ is large because it reduces the resulting q1. When product a more differentiated, a lower p1 reduces q1 less significantly and thus, the welfare loss by aggressive pricing of firm 1 is weaker. Therefore, the range of θ for which an increase in θ improves world welfare is wider, and thus, the optimal degree of capital liberalization is higher. What is important to remark is that the optimal degree of foreign penetration can be optimal also from the perspective of the domestic country. Suppose that at the beginning all the stocks in firm 1 are owned by atomistic domestic investors (θ ¼ 0 initially). Suppose that some of them are sold in a competitive and efficient financial market: then, the price of the n stocks must match the fraction of the expected profits obtained by the private firm. In such a case, θ is also the degree of foreign penetration that maximizes domestic welfare ex ante. The problem, however, is that domestic owners may not sell their stocks if they can internalize the negative impact of an increase of θ on the profits of the private firm: this is the case, for instance, if domestic investors are not negligible.
4. Extensions This section extends the model in two main directions. The first is the consideration of competition for an integrated market with both domestic and foreign consumers, which can lead to a different endogenous competition structure with Cournot competition. The second is the consideration of a different cost function for the two firms, which however confirms the baseline results. 4.1. Competition in an integrated market Until now, we have assumed that all consumers are domestic. However, it is possible that the market is comprised of both domestic and foreign consumers and the public and private firms compete for all this integrated market. To investigate this case, we simply assume that the fraction of domestic consumers is k A ½0; 1 and take this into account in the welfare evaluation. This changes the perspective of the public firm and can fundamentally undermine the endogenous
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competition structure determined above. Indeed, we will show that in an integrated market, Cournot competition can finally emerge.12 If k¼1 we are back to the previous model of a domestic market, for intermediate values we are considering an integrated market where k is the relative size of domestic consumers, and if k¼0, we are in the case of competition for a third market (Brander and Spencer, 1985): all these cases are extensively discussed in the trade literature. Social welfare is now given by βðq20 þ 2δq0 q1 þ q21 Þ SW ¼ p0 c0 q0 þ 1 θ p1 c1 q1 þk α q0 þq1 p0 q0 p1 q1 ; 2 Under Cournot competition, the equilibrium quantity of the public firm can be derived as qqq 0 ¼
2a0 δð2 θ kÞa1
;
βðδ2 θ þ ð2 kÞð2 δ2 ÞÞ
and that of the private firm is qqq 1 ¼
ð2 kÞa1 δa0 : βðδ2 θ þ ð2 kÞð2 δ2 ÞÞ
Substituting these equilibrium quantities into the demand and payoff functions, we have the following domestic welfare and firm 1's profit: SW qq ¼
π qq 1 ¼
H8 ; 2 2 2βðδ θ þ ð2 δ Þð2 kÞÞ ðδa0 ð2 kÞa1 Þ2
βðδ2 θ þð2 δ2 Þð2 kÞÞ
ð23Þ
:
ð24Þ
Under Bertrand competition, the equilibrium price of the public firm is ppp 0 ¼
αð1 δÞð2ð1 þ δÞ kð2 þ δÞ δθÞ þ δðθ kÞc1 þ 2c0 ; δ2 θ þð2 δ2 Þð2 kÞ
and that of the private firm is ppp 1 ¼
αð1 δÞð2 þ δ kð1 þ δÞÞ þ ð2 k δ2 ð1 θÞÞc1 þ δc0 : δ2 θ þ ð2 δ2 Þð2 kÞ
Substituting these equilibrium prices into the payoff functions, we have SW pp ¼
π pp 1 ¼
H9 2β ð1 δ Þðδ θ þ ð2 δ Þð2 kÞÞ2 2
2
2
ðð1 þð1 δ Þð1 kÞÞa1 δa0 Þ2
;
ð25Þ
2
βð1 δ Þðδ θ þð2 δ2 Þð2 kÞÞ2 2
2
:
ð26Þ
From (23) and (25) we have SW pp SW qq ¼
δ2 ð2 kÞðða1 δa0 Þ2 ð1 δ2 Þð1 kÞ2 a21 Þ : 2 2 2β ð1 δÞ2 ðδ ðθ þkÞ þ 2ð2 k δ ÞÞ2
This is positive if and only if ða1 δa0 Þ2 ð1 δ Þð1 kÞ2 a21 is positive. Solving we obtain: 2
1 δa0 =a1 1 δa0 =a1 n k 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi o k o1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 2 1δ 1δ n
Since k 4 1 always holds, we ignore the upper bound k . In case of symmetric firms or a small enough cost difference k 4 0 n therefore SW pp SW qq is positive if and only if k 4k . From (23) and (26) we have qq π pp 1 π1 ¼
δ2 ðða1 δa0 Þ2 ð1 δ2 Þð1 kÞ2 a21 Þ : βð1 δ2 Þðδ2 ðθ þ kÞ þ 2ð2 k δ2 ÞÞ2
which requires the same condition for being positive. Remarkably, both firms agree on the relative preference for the form of competition independently from the degree of foreign penetration. Cournot competition is preferred when the fraction of domestic consumers is small, and therefore the public firm is not much concerned about consumers surplus (this is exactly the case of trade models of competition for a third market). This is less likely to happen when there is more differentiation 12
We are grateful to a referee for suggesting this extension.
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between goods: indeed for δ ¼ 0 we have k ¼ 0 and the Bertrand outcome is always preferred by both firms, but an increase pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n in δ tends to increase the cut-off, as can be easily verified in case of cost symmetry, where k ¼ 1 ð1 δa0 =a1 Þ= 1 δ . Analogous calculations (available from the authors) allow one to compare the mixed cases and determine the endogenous competition structure as follows: n
Proposition 5. (i) Cournot competition leads to larger profits for the private firm and larger welfare relative to Bertrand competition if and only if 1 δa0 =a1 n k ok ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1δ (ii) If θ ¼ 0 there exists a cut-off k A ð0; 1Þ such that both firms choose the quantity contract in equilibrium if k rk . (iii) If θ ¼ 1 nnn nnn there exists a cut-off k A ð0; 1Þ such that both firms choose the quantity contract in equilibrium if k rk . nn
nn
If the share of domestic consumers is small, Cournot payoffs dominate Bertrand payoffs for both firms, and Cournot competition can emerge endogenously in equilibrium. This result is in sharp contrast to that presented in the previous sections, which states that Bertrand competition emerges in equilibrium regardless of the nationality of the private firm. The intuition is simple: in the case of an integrated market where firms compete for both domestic and foreign consumers, the public firm has lower incentives to reduce prices in order to improve consumer surplus. In other words, the public firm becomes more profit-oriented, and has an incentive to adopt a quantity contract that mitigates competition.13 This is why Cournot competition emerges in equilibrium when k is small enough. 4.2. Increasing marginal cost Until now, we have adopted a model with constant marginal costs, a setting that is quite popular in the literature.14 However, it is important to verify our baseline results in case of increasing marginal costs.15 Incidentally, there is another advantage in considering the case of increasing marginal costs. When the marginal cost is constant, the profit of the public firm that competes with the foreign private firm is negative unless θ ¼ 0, however, this may not be very realistic. When the marginal cost is increasing, instead, the profit of the public firm can be positive as well. For simplicity, let us return to the case of a domestic market and assume that the cost function is symmetric and quadratic, that is C i ðqi Þ ¼ γ q2i =2, where γ is a positive constant. The proofs are presented in the Appendix. First, we compare welfare and private firm's profit levels between Cournot and Bertrand competition when the private firm is domestic. The result on welfare is the same as in the case of constant marginal costs, but this is not the case for profits of the private firm. qq pp qq Lemma 4. When θ ¼ 0, (i) SW pp 4 SW qq and (ii) both π pp 1 o π 1 and π 1 4 π 1 are possible.
Next, we consider the case of foreign private firm. The results are the same as in the case of constant marginal costs. qq Lemma 5. When θ ¼ 1, (i) SW pp 4 SW qq and (ii) π pp 1 4 π1 .
Accordingly, the result that Bertrand competition yields greater welfare appears to be fairly robust, but the result for which Bertrand competition yields larger profit for the private firm is not. Furthermore, from Lemmas 4 and 5, we can see that foreign ownership in the private firm strengthens the profit-enhancing effect of Bertrand competition. We now discuss the endogenous choice of variables (price or quantity), starting with the following result:. qq pp pq Lemma 6. Both when θ ¼ 0 and θ ¼ 1, (i) SW pq 4 SW qq (ii) SW pp 4 SW qp , (iii) π qp 1 4 π 1 and (iv) π 1 4 π 1 .
From Lemma 6, we find that choosing the price contract is a dominant strategy for both firms, whether the private firm is domestic or foreign. Thus, we confirm the same result obtained under constant marginal costs: Proposition 6. Under quadratic costs, Bertrand competition is the endogenous competition structure for any ownership share of foreign investors. 5. Concluding remarks In this study, we revisited the classic discussion of the comparison between price and quantity competition, but in a mixed duopoly under international competition. Ghosh and Mitra (2010) considered the domestic private firm and showed that in a mixed duopoly, price competition yields a larger profit for the private firm. We find that regardless of the share of foreign ownership in the private firm, price competition yields higher domestic welfare and a larger profit for the private firm. We also endogenize the choice of price or quantity contract. Matsumura and Ogawa (2012) considered the domestic private firm and showed that choosing a price contract is the dominant strategy for both firms. We find that both firms 13 14 15
When k is small, firm 1's profit is larger under Cournot competition than under Bertrand competition. See Proposition 5-i. See among others, Mujumdar and Pal (1998), Pal (1998), Matsumura (2003), Bárcena-Ruiz (2007), and Matsumura and Ogawa (2010, 2012). See De Fraja and Delbono (1989), Fjell and Pal (1996) and Matsumura and Shimizu (2010).
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choose price contracts in the unique equilibrium regardless of the ownership share of foreign investors. This indicates the importance of price competition in mixed oligopolies. However, this is not true and quantity competition appears in equilibrium when we consider an integrated market rather than a domestic market. This result suggests the importance of investigating an integrated market in mixed oligopolies. In this study, we endogenized the competition structure by endogenizing the choice of a price or quantity contract. In this context, endogenizing the number of firms is also important. In the literature on mixed oligopolies, many works have already investigated free entry markets, but most works assumed that the form of contract (whether price or quantity) is given exogenously.16 Endogenizing both these factors remains an area for future research. Appendix A The following positive constants appear in the text: H 1 δ ð2a0 δ a1 Þa1 θ þ 2ð2δð1 δ Þa0 a1 þ δ a20 ð1 δ Þa21 Þθ þð4 δ Þa20 þ ð3 δ Þa21 2
2
2
2
2
2
2
2δð3 δ Þa0 a1 2
H 2 δ ð1 δ Þð2δa0 a1 Þa1 θ þ 2δð1 δ Þð2ð1 δ Þa0 a1 þ ð1 δ Þa21 þ δa20 Þθ þ ðδ 3δ þ 3Þa21 2
2
2
2
2
2
4
2
þ ð2δ 5δ þ 4Þa20 2δðδ 3δ þ 3Þa0 a1 4
2
4
2
H 3 δ ð2a0 δ a1 Þa1 θ þ 2ð2δð1 2δ Þa0 a1 ð1 2δ Þa21 þ δ a20 Þθ 2
2
2
2
2
þ ð1 δ Þðð4 δ Þa20 þ3a21 6δa0 a1 Þ 2
δ
2
δ θ
δ
δ
δ
θ
δ
δ
2 2 2 2 2 2 2 þ 2ð2ð1 2 Þa0 a1 2ð1 2 Þa21 þ a20 Þ þ ð4 5 Þa20 þ ð3 4 Þa21 H 4 ð2a0 a1 a21 Þ 2 2 ð3 4 Þa0 a1 4 2 2 2 2 H5 þ2 þ ð1 Þð4 Þ 4 2 2 2 2 2 H6 þ 4 ð1 Þ þ ð1 Þð4 3 Þ 2 2 2 2 5 4 H 7 ð1 Þð2 a0 a1 Þa1 þ 2 ð2 Þða21 þ a20 2 a0 a1 Þ 2a0 a1 þ ð2a20 þ a21 Þ 3 2 þ 6a0 a1 ð5a20 þ3a21 Þ 6a0 a1 þ 4a20 þ 3a21 2 2 3 2 3 2 2 H 8 ð2a0 ð2 kÞa1 Þa1 2ða1 k ð a0 a1 a21 2 a0 a1 þ 4a21 Þk þ 3 a0 a1 ða20 þ 2a21 Þ
δ δ δ θ δ θ δ δ δ θ δ δ θ δ δ δ δ δ θ δ δ δ θ δ δ δ δ δ δ θ δ δ δ δ δ 2 2 3 2 3 2 2 2 2 2 4δa0 a1 þ 4a1 Þθ þ ð1 δ Þa1 k 2ðδa0 þ ð1 2δ Þa1 Þk ð2a0 a1 δ þ ð4a1 a0 Þδ 12δa0 a1 3 2 þ 4a21 þ 4a20 Þk þ 2ð2a0 a1 δ δ a20 8δa0 a1 þ 4a20 þ 4a21 Þ 2 2 2 2 2 2 2 H 9 a1 δ ð1 δ Þð2δa0 ð2 kÞa1 Þθ 2ð1 δ Þða21 k a1 ða1 ð4 δ Þ a0 δð1 δ ÞÞk 3 2 5 3 2 2 4 þ ð3a0 a1 δ δ ð2a1 þ a0 Þ þ 4a1 ða1 δa0 ÞÞθ ð2 kÞð2a0 a1 δ ð2a0 þ a1 Þδ 8δ a0 a1 2 þ δ ð5a20 þ 4a21 Þ þ8δa0 a1 4ða20 þ a21 ÞÞ 2 3 2 4 2 2 4 2 2 2 H 10 ðγ 3 ð2 3δ Þ þ 2γ 2 βðδ 7δ þ 5Þ þ γβ ð2 δ Þðδ þ 8ð1 δ ÞÞþ 2β ð1 δ Þð2 δ Þ2 Þ 2 2 2 2 2 H 11 ðγ 2 ð2 δ Þ þ 2γβð3 2δ Þ þ β ð1 δ Þð4 3δ ÞÞ
A.1. Proof of Lemma 4 Under Cournot competition, the equilibrium quantity of the public firm is qqq 0 ¼
αγ þ αβð2 ð1 θÞδÞ ; γ 2 þ 3βγ ð1 θÞβ2 δ2 þ 2β2
and that of the private firm is qqq 1 ¼
αγ þ ð1 δÞαβ : γ 2 þ 3βγ ð1 θÞβ2 δ2 þ 2β2
Substituting these equilibrium quantities into the demand and payoff functions, we have the following domestic welfare (up to a positive constant) and firm 1's profit: SW qq p
1 2ðγ 2 þ 3β þð2 ð1 θÞδ Þβ Þ2 2
2
;
ð27Þ
16 For discussions of free entry markets in mixed oligopolies, see Matsumura and Kanda (2005), Brandão and Castro (2007), and works mentioned in Introduction. For recent developments in this field, see Cato and Matsumura (2013), Ghosh et al. (2014), and Ghosh and Sen (2012).
J. Haraguchi, T. Matsumura / Research in Economics 68 (2014) 338–353
π qq 1 ¼
351
α2 γ þ 2 β γ þ ð1 δÞβ 2 : 2 2 2ðγ 2 þ 3β þ ð2 ð1 θÞδ Þβ Þ2
ð28Þ
Under Bertrand competition, the equilibrium price of public firm is ppp 0 ¼
αðγ 2 þ γβð1 δÞð2 þ δÞ þð1 θÞβ2 δð1 δÞ2 ð1 þ δÞÞ ; γ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 ð1 θÞδ2 Þ
and that of private firm is ppp 1 ¼
αðγ 2 þ γβð1 δÞð2 þ δÞ þ β2 ð1 δÞ2 ð1 þ δÞÞ : γ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 ð1 θÞδ2 Þ
Substituting these equilibrium prices into the payoff functions, we have the following resulting domestic welfare (up to a positive constant) and firm 1's profit: SW pp p
π pp 1 ¼
1 2ðγ
2þ
γβð3 δ Þ þ β ð1 δ2 Þðθδ2 þ ð2 δ2 ÞÞÞ2 2
2
;
ð29Þ
α2 ððγ þ βð1 δÞÞ2 Þðγ þ2βð1 δ2 ÞÞ ; 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þðθδ þ ð2 δ ÞÞÞ2
ð30Þ
From (27) and (29), for θ ¼ 0 we have (up to a positive constant) SW pp SW qq p
2ðγ
2 þ3
α2 β2 δ2 ðγ þ βð1 δÞÞ2 4 0: γβ β ð2 δ2 ÞÞ2 ðγ 2 þ γβð3 δ2 Þ þ β2 ð1 δ2 Þð2 δ2 ÞÞ2 2
From (28) and (30), for θ ¼ 0 we have (up to a positive constant) qq π pp 1 π1 ¼
α2 β2 δ2 ðγ βδ þ βÞ2 H10 ; 2 2 2 2 2 2ðγ 2 þ 3γβ β ð2 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2 2
qq Substituting α ¼ 1; β ¼ 1; γ ¼ 1; δ ¼ 24=25, we have π pp 1 π 1 0:0003 o 0. Substituting α ¼ 1; β ¼ 1; γ ¼ 1; δ ¼ 1=2, we have pp qq π 1 π 1 0:0076 4 0. □
A.2. Proof of Lemma 5 From (27) and (29), for θ ¼ 1 we have SW pp SW qq ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 3 þ γ 2 βð5 δ2 Þ þ 4γβ2 ð2 δ2 Þ þ 4β3 ð1 δ2 ÞÞ 4 0: 2 2 2 2 2ðγ 2 þ 3βγ þ 2β Þ2 ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2
From (28) and (30), for θ ¼ 1 we have qq π pp 1 π1 ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 3 ð2 δ2 Þ þ 2γ 2 βð5 3δ2 Þ þ 4γβ2 ð4 3δ2 Þ þ 8β3 ð1 δ2 ÞÞ 4 0: 2 2 2 2 2ðγ 2 þ3βγ þ 2β Þ2 ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2
□
A.3. Proof of Lemma 6 In p q game, the equilibrium price of the public firm is ppq 0 ¼
αðγ 2 þ γβð2 δ δ2 Þ β2 δθð1 δÞÞ ; γ 2 þ γβð3 δ2 Þ þ β2 ðδ2 θ 2δ2 þ 2Þ
and the equilibrium quantity of the private firm is qpq 1 ¼
αðγ þ βð1 δÞÞ : γ 2 þ γβð3 δ2 Þ þ β2 ðδ2 θ 2δ2 þ 2Þ
Substituting these into the payoff functions, we have the following expressions for domestic welfare (up to a positive constant) and firm 1's profit SW pq p
1 2ðγ
2þ
γβð3 δ Þ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞÞ2 2
;
ð31Þ
352
J. Haraguchi, T. Matsumura / Research in Economics 68 (2014) 338–353
π pq 1 ¼
α2 ðγ þ βð1 δÞÞ2 ðγ þ 2βð1 δ2 ÞÞ : γβð3 δ2 Þ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞÞ2 2ðγ
ð32Þ
2þ
In q p game, the equilibrium quantity of the public firm is qqp 0 ¼
γ
αðγ þ βð2ð1 δÞδθÞÞ ; γβ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞ
2 þ3
and the equilibrium price of the private firm is pqp 1 ¼
αðγ 2 þ γβð2 δÞ þ β2 ð1 δÞÞ : γ 2 þ 3γβ þ β2 ðδ2 θ þ 2ð1 δ2 ÞÞ
Substituting these into the payoff functions, we have the resulting domestic welfare (up to a positive constant) and profit of firm 1 SW qp p
π qp 1 ¼
1 2ðγ
2 þ3
;
ð33Þ
γβ þ β ð2ð1 δ2 Þδ2 θÞÞ 2
ðα2 ðγ þ 2βÞðγ þ βð1 δÞÞÞ2
: 2 2 2 2ðγ 2 þ3γβ þ β ð2ð1 δ Þδ θÞÞ
ð34Þ
For θ ¼ 0 we have SW pq SW qq ¼
α2 β2 δ2 ðγ þ βÞðγ þ βð1 δÞÞ2 ðγ 2 ð2 δ2 Þ þ2γβð3 2δ2 Þ þ β2 ð1 δ2 Þð4 δ2 ÞÞ 4 0; 2 2 2 2 2 2ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2 ðγ 2 γβð3 δ Þ þ2β ð1 δ ÞÞ2
SW pp SW qp ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ ð1 þ δ2 Þ þ βð1 δ2 ÞÞH11 40; 2 2 2 2 2 2 2ðγ 2 þ3γβ þ 2β ð1 δ ÞÞ2 ðγ 2 þ γβ ð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2
π qp π qq ¼
α2 β2 δ2 ðγ þ2βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ6γβ þ β2 ð4 3δ2 ÞÞ 40 2 2 2 2 2ðγ 2 þ 3γβ þ 2β ð1 δ ÞÞ2 ðγ 2 þ3γβ þ β ð2 δ ÞÞ2
π pp π pq ¼
α2 β2 δ2 ð1 δ2 Þðγ þ βð1 δÞÞ2 ðγ þ 2βð1 δ2 ÞÞð2γ 2 þ2γβð3 δ2 Þ þ β2 ð1 δ2 Þð4 δ2 ÞÞ 40: 2 2 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ 2β ð1 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð1 δ Þð2 δ ÞÞ2
and
For θ ¼ 1 we have SW pq SW qq ¼
α2 β2 δ2 ðγ þ βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ γβð6 δ2 Þ þ β2 ð4 δ2 ÞÞ 40; 2 2 2 2 2ðγ 2 þ 3γβ þ2β Þ2 ðγ 2 þ γβð3 δ Þ þ β ð2 δ ÞÞ2
SW pp SW qp ¼
α2 β2 δ2 ðγ þ βð1 δ2 ÞÞðγ þ βð1 δÞÞ2 ð2γ 2 þ γβð6 δ2 Þ þ β2 ð4 3δ2 ÞÞ 4 0; 2 2 2 2 2 2ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2 ðγ 2 þ γβð3 δ Þ þ2β ð1 δ ÞÞ2
π qp π qq ¼
α2 β2 δ2 ðγ þ2βÞðγ þ βð1 δÞÞ2 ð2γ 2 þ6γβ þ β2 ð4 δ2 ÞÞ 40 2 2 2 2ðγ 2 þ3γβ þ 2β Þ2 ðγ 2 þ 3γβ þ β ð2 δ ÞÞ2
π pp π pq ¼
α2 β2 δ2 ðγ þ βð1 δÞÞ2 ðγ 2βð1 δ2 ÞÞð2γ 2 þ 2γβð3 δ2 Þ þ β2 ð4 3δ2 ÞÞ 4 0: 2 2 2 2 2 2ðγ 2 þ γβð3 δ Þ þ2β ð1 δÞð1 þ δÞÞ2 ðγ 2 þ γβð3 δ Þ þ β ð2 δ ÞÞ2
and □
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