On the Robustness of Private Leadership in Mixed Duopoly

Transcription

1 On the Robustness of Private Leadership in Mixed Duopoly Toshihiro Matsumura Institute of Social Science, University of Tokyo and Akira Ogawa College of Liberal Arts, International Christian University April 16, 2008 Abstract We investigate a mixed duopoly where a state-owned public enterprise competes against a profit-maximizing private enterprise. We analyze whether private leadership or public leadership is robust in observable delay game. We find that private leadership is always risk dominant. We also investigate how ownership structure in a public firm affects the equilibrium distribution of roles. We find that the roles are as follows: (1) Cournot, when the degree of privatization is low, (2) private leadership, when it is middle, (3) both private leadership and public leadership, when it is high. The result implies that private leadership is again more robust. JEL classification numbers: H42, H44, L13 Key words: observable delay, risk dominance, Stackelberg, partial privatization Corresponding author: Akira Ogawa, College of Liberal Arts, International Christian University, , Osawa, Mitaka-shi, Tokyo , Japan. Phone: (81) Fax: (81) ogawaa@icu.ac.jp 1

2 1 Introduction We have observed a worldwide wave of privatization of state-owned public enterprises. Nevertheless, public firms and semipublic firms (partial ownership by public sectors) still exist and many of them compete with private firms in private goods markets. Competition between public and private firms existed, or still exists, in a range of industries including the airline, rail, telecommunications, natural gas, electricity, steel, and overnight delivery industries, as well as services including banking, home loans, health care, life insurance, hospitals, broadcasting, and education. Mixed oligopoly is still important in many developed, developing, and former communist transitional countries. 1 Many studies on mixed oligopoly analyzed the Cournot model and some other studies analyzed the Stackelberg models. In both the Cournot and Stackelberg models, the role of each firm is given exogenously. However, it is often more reasonable to assume that firms choose not only what actions to take, but also when to take them. Endogenous timing is important because an alternative ordering of moves often gives rise to different results. In mixed duopoly, in particular, whether the public firm becomes the leader or the follower is important. Under the private leadership, the public firm plays a complementary role as a potential competitor against private firm and the private firm mainly supplies the products in equilibrium. Under public leadership, the public firm cannot play such a role and the private leadership is better than the public leadership from the normative viewpoint. 2 Pal (1998) investigates this problem by using observable delay game formulated by Hamilton 1 This interest in mixed oligopolies is due to their importance to the economies of many countries, except for the United States. Although they are less significant in the United States, there are some examples of mixed oligopolies such as the packaging and overnight-delivery industries. The idea of mixed oligopoly dates at least to Merrill and Schneider (1966). Recently, the literature on mixed oligopoly has become richer and more diverse. For example, Ishibashi and Matsumura (2006) and Nishimori and Ogawa (2002) investigate R&D competition between public and private sectors. Mujumdar and Pal (1998) consider tax effects. Bárcena-Ruiz and Garzón (2005a,b) investigate policy interaction between market integration and privatization policy. Corneo and Jeanne (1994), Fjell and Pal (1996), Pal and White (1998), and Fjell and Heywood (2002) investigate international competition. Bárcena-Ruiz and Garzón (2006) analyze environmental policies. Cremer et al. (1991) analyzes endogenous product differentiation. 2 See Pal (1998) and Matsumura (2003a). 2

3 and Slutsky (1990). 3 He finds that the public firm can be either the follower or the leader in duopoly (multiple equilibria). 4 The indeterminacy of the equilibrium role in mixed duopoly can be an obstacle of the subsequent researches because it does not provide the model we should adopt in mixed duopoly. In this paper, we tackle this problem. In this paper we take a close look at whether private or public leadership is robust by using observable delay game. First, we discuss which leadership is more robust by using the criterion of risk dominance in Harsanyi and Selten (1988). We find that private leadership is always risk dominant. This result contains important implications. As Pal (1998) shows, the unique equilibrium outcome is private leadership in oligopoly with more than three private firms, whereas the duopoly yields multiple equilibria, private leadership and public leadership. Our result solves this curious discontinuity in mixed oligopoly, and suggest that we should focus on private leadership only. Second, we discuss partial privatization in the public firm. We introduce partial ownership into Pal (1998). Partial ownership is observed all over the world, and has become increasingly popular in this field. 5 We again find that public leadership is less robust than private leadership because the parameter values for the existence of public leadership is much smaller than for the existence 3 For the extension of this game, see Normann (2002). For the experimental studies on this game, see Fonseca et al. (2006). 4 See also Jacques (2004), who shows the multiplicity of equilibria in duopoly even when the number of periods is more than two. For the robustness of multiplicity of equilibria under more general demand and cost conditions, see Tomaru and Kiyono (2005). For the subsequent works on observable delay game in mixed oligopoly, see Matsumura (2003b), Lu (2006), and Bárcena-Ruiz (2007). Matsumura (2003a) investigates this problem by using two-production period model formulated by Saloner (1987) and show that public leadership des not arise in equilibrium. However, there are continuum configuration of equilibria in the model, so it might be more natural to use observable delay for discussing endogenous roles in mixed duopoly. For the discussion of two production period model in private duopoly, see also Pal (1991,1996). 5 Partial privatization has been intensively discussed in the literature. For a general discussion, see Matsumura (1998). See Chang (2005), Chao and Yu (2006), and Fujiwara (2006) in the context of international trade, Kato (2006) and Ohori (2006) for environmental problems, White (1996), Mujumdar and Pal (1998), Fjell and Heywood (2004), Tomaru (2006), Kato and Tomaru (2007) for tax and subsidy policies, Matsumura and Kanda (2005), Fujiwara (2007) and Heywood and Guangliang (2008) in free entry markets, Lu and Poddar (2007) for product differentiation, Lee and Hwang (2003) for the agency problem, Lee (2006) in the context of regulation policy, Beladi and Chao (2006) in the context of urban unemployment, Bárcena-Ruiz and Garzón (2003) for merger problems, and Futagami and Yanagihara (2008) in the context of macroecomomics. 3

4 of private leadership. We think that we should focus on the private leadership equilibrium in the context of mixed duopoly. 2 The Model Firms produce perfectly substitutable commodities for which the market demand function is given by p = a Q (price as a function of quantity). Firm 0 is a semipublic (partially privatized) firm and it is assumed to maximize U 0 = θw +(1 θ)π 6 0, while firm 1 is a private firm and it maximizes π 1. π i (i =0, 1) is firm i s profit and W is the total social surplus (profits of firms plus consumer surplus). θ [0, 1] indicates the degree of privatization. We normalize the private firm s marginal cost to 0 (following Pal (1998). Let c denote firm 0 s marginal cost. We assume that a>4c so as to ensure the interior solution in the Cournot game and in the Stackelberg game with public leadership. 7 We consider the observable delay game of Hamilton and Slutsky (1990), where firms first choose the timing of their production. There are two possible time periods for quantity choice and each firm chooses its quantity in only one of the two periods. In the first stage, firms simultaneously choose whether they produce in period 1 or in period 2. In the second stage, each firm i selects its quantity q i knowing when the other firm chooses the quantity. If both firms choose the same period, they face Cournot competition. If only firm i chooses period 1, the firms face Stackelberg competition with firm i s leadership. 3 Fixed Timing In this section, we consider three subgames, Cournot, Stackelberg with public leadership, and Stackelberg with private leadership, where timing is given exogenously. 6 This is a standard formulation of the payoff function of semipublic firm. See the works mentioned in footnote 3. 7 In the Stackelberg game with private leadership, sufficiently large demand does not guarantee the interior solution. 4

5 3.1 Cournot All firms produce simultaneously. The first order conditions of semipublic and private firms are respectively: a c 2q 0 q 1 + θq 0 = 0, (1) a q 0 2q 1 = 0. (2) From them, we have the following reaction functions and then the equilibrium output: R 0 (q 1 ) = a c q 1, 2 θ (3) R 1 (q 0 ) = a q 0, 2 (4) q0 S = a 2c 3 2θ, (5) q1 S = a(1 θ)+c, 3 2θ (6) where superscript S denotes the Cournot game (simultaneous-move game). The resulting payoffs are: π S 1 = [a(1 θ)+c]2 (3 2θ) 2, U S 0 = 8ac +8acθ +2a2 +8c 2 +4a 2 θ 5c 2 θ 2acθ 2 8a 2 θ 2 +3a 2 θ 3 2(3 2θ) 2. (7) 3.2 Stackelberg with public leadership Firm 0 produces and then firm 1 produces. Firm 0 chooses q 0 so as to maximize U 0 (q 0,R 1 (q 0 )), where: The first order condition is: U 0 (q 0,R 1 (q 0 )) = ( 4+3θ)q2 0 +(4a 2aθ 8c)q 0 +3a 2 θ. (8) 8 ( 4+3θ)q 0 +(2a aθ 4c) =0. (9) 5

6 From (9) them we have: q Pub 0 = 2a aθ 4c, (10) 4 3θ where superscript Pub denotes the equilibrium outcome in the Stackelberg game with public leadership. Substituting it into R 1 (q 0 ) yields q Pub 1 = a(1 θ)+2c. (11) 4 3θ The resulting payoffs are: π Pub 1 = U Pub [a(1 θ)+2c]2 (4 3θ) 2, 0 = 4ac +2acθ + a2 +4c 2 +2a 2 θ 2a 2 θ 2. (12) 2(4 3θ) 3.3 Stackelberg with private leadership Firm 1 produces and then firm 0 produces. Firm 1 chooses q 1 so as to maximize π 1 (R 0 (q 1 ),q 1 ), where: π 1 (R 0 (q 1 ),q 1 )= (a aθ + c q 1 + θq 1 )q 1. (13) 2 θ If θ <(a 3c)/(a 2c), the game has the interior solution. Otherwise, it has the corner solution where firm 0 produces nothing (and firm 1 produces a c). We obtain the following equilibrium outputs: { } a(1 θ)+c q1 Pri = min,a c, (14) 2(1 θ) { } a(1 θ) 3c +2cθ q0 Pri = max, 0, (15) 2(2 θ)(1 θ) where superscript Pri denotes the equilibrium outcome in the Stackelberg game with private leadership. When θ<(a 3c)/(a 2c), the equilibrium payoffs are: π Pri 1 = U Pri [a(1 θ)+c]2 4(1 θ)(2 θ), 0 = 6ac +8acθ + a2 +9c 2 +5a 2 θ 5c 2 θ 2acθ 2 9a 2 θ 2 +3a 2 θ 3. (16) 8(1 θ)(2 θ) 6

7 When θ (a 3c)/(a 2c), the equilibrium payoffs are: π Pri 1 = c(a c), U Pri 0 = 4 Equilibrium in the mixed duopoly θ(a + c)(a c). (17) 2 We now restrict our attention to the case where θ = 1, which is discussed by Pal (1998). He showed that both private leadership and public leadership are equilibrium outcomes in the observable delay game. We now investigate which is robust from the viewpoint of risk dominance. Proposition 1 Private leadership is always risk dominant. Proof See Appendix A.1. As is well known risk dominance is closely related to evolutional stability. Proposition 1 implies that the private leadership is more stable than the public leadership. 5 Equilibrium under partial privatization In this section we investigate how the degree of privatization affect the equilibrium pattern. First, we compare the payoffs among three fixed timing game to derive the equilibrium with endogenous timing. Lemma (i) π1 Pri >π1 S ; (ii) U Pub 0 >U0 S; (iii) πs 1 > (=)πpub 1 if and only if θ<(=)y; (iv) U0 S > (= )U0 Pri if and only if θ<(=)z; and (v) z<y, where y (2a c c 2 +4ac)/(2a) and z is the smallest solution of the equation 17c +7a 18aθ +17cθ +15aθ 2 4aθ 3 4cθ 2 =0. Proof See Appendix A.2. We present our main results summarized in Figure 1. Proposition 2 (i) For any (θ, c), either Cournot or private leadership is an equilibrium outcome; (ii) if public leadership is an equilibrium, then private leadership is also an equilibrium; however the inverse is not true. Proof See Appendix A.3. 7

8 [Figure 1 will be around here.] Proposition 2(ii) indicates that private leadership appears in equilibrium for a wider range of the parameter values than public leadership. This indicates that private leadership is more robust and so more plausible than public leadership. We believe that we should focus on private leadership rather than public leadership in mixed duopoly. Finally, we make some remarks on the robustness of our results. We show that the private leadership is risk dominant if θ = 1 at the previous section. We can also show the statement even if θ [y, 1]. In other words, the private leadership is always risk dominant if multiple equilibrium arise. However, such extension needs a long and winding proof and therefore we only present a rough sketch of proof at Appendix A.4, instead of a formal Proposition and a detailed proof. In this paper, we assume that c does not depends on θ, i.e., privatization does not affect the relative inefficiency of firm 0. Instead, if we assume that c depends on θ and c (θ) > 0, Proposition 2 holds true. In this paper we consider a duopoly model. If we extend it to an n-private firm model, Proposition 2 still holds true. In this case, private leadership means that all private firms simultaneously produce and then firm 0 produces, and public leadership means that firm 0 produces and then all private firms simultaneously produce. 6 Concluding Remarks In this paper we revisited endogenous timing in mixed duopoly discussed by Pal (1998). We show that private leadership is more robust than public leadership for two reasons. First, we show that private leadership is robust from the viewpoint of risk dominance (and so evolutionary stable). Second, private leadership is more robust under partial privatization. Private Leadership means that the private firm obtains a large market share and the public firm plays the role of potential competitor. This distribution of roles among two firms is efficient for social welfare. Our results indicate that the desirable distribution of roles (private become the leader and the public becomes the follower) are more likely realized in equilibrium than the other inefficient one. 8

9 Appendix A.1 Proof of Proposition 1 Substituting θ = 1 into (7), (12), and (17), we obtain π1 S = c2, U0 S = a2 2ac +3c 2, 2 1 =4c 2, U0 S = a2 2ac +4c 2, 2 π1 S = ac c2, U0 S = a2 c 2. (18) 2 π Pub Under the equilibrium in mixed strategy, the choice probabilities Pr i, the probability that firm i chooses period 1, must satisfy 8 Arranging them, we obtain Since Pr 0 a=4c =2/5 and Pr 0 a Pr 1 U S 0 +(1 Pr 1 )U Pub 0 = Pr 1 U Pri 0 +(1 Pr 1 )U S 0, Pr 0 π S 1 +(1 Pr 0)π Pri 1 = Pr 0 π Pub 1 +(1 Pr 0 )π S 1. (19) Pr 0 = Pr 1 = π1 S πpri 1 2π1 S πpri 1 π1 Pub U0 S U 0 Pub 2U0 S U 0 Pub U0 Pri = a 2c a + c, = c 2a 3c. (20) =3c/(a+c) 2 > 0, Pr 0 > 2/5. On the other hand, Pr 1 a=4c =1/5 and Pr 1 a = 2c/(2a 3c) 2 < 0 implies Pr 1 < 1/5. Hence, Pr 0 >Pr 1, i.e., Pr 0 (1 Pr 1 ) > (1 Pr 0 )Pr 1. Q.E.D. A.2 Proof of Lemma Consider Lemma(i) and (ii). By definition, π Pri 1 π S 1 and U Pub 0 U S 0 hold (the Stackelberg leader can choose its Cournot output; thus its payoff is never smaller than the Cournot counterpart). The strict inequalities are derived from q Pri 1 q S 1 and qpub 0 q S 0. π1 S a(4 3θ)(1 θ)+c(10 7θ) πpub 1 = (3 2θ) 2 (4 3θ) 2 (a 2c 2aθ + cθ + aθ 2 ). (21) 8 Under the equilibrium in mixed strategy, each firm cannot increase the value of objective function. In other words, the expected value when the firm chooses period 1 must be equal to the one when the firm chooses period 2. 9

10 Since θ [0, 1], the sign of (21) is equal to the sign of a 2c 2aθ + cθ + aθ 2. For θ [0, 1], a 2c 2aθ + cθ + aθ 2 0 if and only if θ (2a) 1 (2a c c 2 +4ac) and equality holds when θ = y. This implies Lemma(iii). U S 0 U Pri 0 = [a(1 θ)+c] 8(3 2θ) 2 (2 θ)(1 θ) f(θ) if θ<a 3c a 2c, (22) (a 2c)(2 θ) 2(2θ 3) 2 (a 2c 2aθ 2cθ + aθ 2 +2cθ 2 ) otherwise, (23) where f(θ) 17c +7a 18aθ +17cθ +15aθ 2 4aθ 3 4cθ 2. (23) is negative under the assumption a>4c. The first term in (22) is positive. Thus, the sign of (22) is equal to the sign of f(θ). Since f(0) > 0,f((a 3c)/(a 2c)) < 0, and f (θ) > 0 for θ [0, 1], f(θ) = 0 has one solution, z, and f(θ) 0 if and only if θ z (the equality holds when θ = z. This implies Lemma(iv). From Lemma(iv) we have that U0 S U 0 Pri < 0 if and only if θ>z.substituting θ = y( (2a c c 2 +4ac)/(2a)) into f(θ), which is given in the proof of Lemma(iv), we have f = c(a + c + c 2 +4ac)/a < 0 (and so U0 S U 0 Pri < 0 when θ = y.) This implies Lemma(v). Q.E.D. A.3 Proof of Proposition 2 (a): Both firms choose period 2 (then the equilibrium outcome is Cournot) if and only if π S 1 πpri 1 and U S 0 U Pub 0. These inequalities are never satisfied (Lemma(i) and (ii)). (b): Both firms choose period 1 (then the equilibrium outcome is Cournot) if and only if π S 1 πpub 1 and U S 0 U Pri 0. These inequalities are satisfied if and only if θ z (Lemma(iii)-(v)). (c): Only firm 1 chooses period 1 (then the equilibrium outcome is Stackelberg with private leadership) if and only if π1 Pri π1 S θ z (Lemma(i) and (iv)). and U Pri 0 U0 S. These inequalities are satisfied if and only if (d): Only firm 0 chooses period 1 (then the equilibrium outcome is Stackelberg with public leadership) if and only if π1 Pub π1 S θ y (Lemma(ii) and (iii)). and U Pub 0 U0 S. These inequalities are satisfied if and only if (b) and (c) imply statement (i). (c), (d), and Lemma(v) imply statement (ii). Q.E.D. 10

11 A.4 Rough sketch of proof of extension of Proposition 1 (under partial privatization) As we mentioned at Appendix A.1, Pr 0 >Pr 1 implies Pr 0 (1 Pr 1 ) > (1 Pr 0 )Pr 1. The following sketch describes steps to show Pr 0 >Pr 1. Step 1. Comparing (12) with (16) and (17), U Pri 0 >U Pub 0. By definition, U Pub 0 >U S 0. Therefore, Pr 1 < 1/2. Step 2. Comparing (12) with (16), π Pri 1 >π Pub 1 if θ <(a 3c)/(a 2c). Comparing (12) with (17), π Pri 1 <π Pub 1 if and only if the following conditions are satisfied; a (4c, v 1c (θ)c), (24) θ > max{(12 2 3)/11, (a 3c)/(a 2c)}, (25) where v 1c (θ) 9θ2 20θ θ 4 192θ θ 2 224θ +64 2(1 θ) 2. Therefore, π1 Pri <π1 Pub (24) and (25). By definition, π1 Pri >π1 S. It implies that Pr 0 > 1/2, i.e. Pr 0 > Pr 1 unless conditions (24) and (25) are satisfied. We note that ( v 1c )/( θ) > 0 if condition (25) are satisfied, and that v 1c (1) = 5. Step 3. Suppose conditions (24) and (25) are satisfied. Pr 0 Pr 1 = X(v,θ)(2θ 3) 2, where v a/c, (26) A 0c (v,θ) A 1c (v,θ) and A ic are denominators of Pr i. Step 4. By the following substeps, we can show A 0c < 0. i. ( 2 A 0c )/( v θ) =0 v = η 1 (θ), ( 3 A 0c )/( v 2 θ) =2(θ 1)(28θ 2 68θ + 41). ii. ( η 1 )/( θ) =η 2 (θ)/[2(θ 1) 2 (28θ 2 68θ + 41) 2 ]. iii. ( 2 η 2 )/( 2 θ) > 0. Since ( η 2 )/( θ) θ=1 < 0, ( η 2 )/( θ) < 0. Because η 2 (1) > 0, ( η 1 )/( θ) > 0. 11

12 iv. Since η 1 (3/4) > 5, η 1 (θ) >v 1c (θ), θ. Because ( 3 A 0c )/( v 2 θ) < (=)0 if θ<(=)1, ( 2 A 0c )/( v θ) > 0. Since ( A 0c )/( θ) v=4 > 0, ( A 0c )/( θ) > 0. Because A 0c (v,1) < 0, A 0c < 0 v, θ. Step 5. By the following substeps, we can show A 1c < 0. i. ( A 1c )/( v) =0 v = ν 1 (θ). ( 2 A 1c )/( v 2 )=2(θ 1) 2 (2θ 2 8θ + 7). ii. Since ( ν 1 )/( θ) > 0 and ν 1 (1/2) > 5, ν 1 (θ) >v 1c (θ), θ. Hence, ( A 1c )/( v) < 0. iii. A 1c (4) = (θ 1)ν 2 (θ) 5. Since ( 2 ν 2 )/( θ 2 ) < 0, ν 2 (1/2) > 0, and ν 2 (1) > 0, ν 2 (θ) > 0 θ. Hence, A 1c < 0 v, θ. Step 6. By the following substeps, we can show X>0. i. ( 3 X)/( v 3 )=0 v = ρ 1 (θ), and ( 4 X)/( v 4 ) > 0. Since ( ρ 1 )/( θ) > 0 and ρ 1 (7/10) > 5, ( 3 X)/( v 3 ) < 0. ii. ( 2 X)/( v 2 ) v=5 = ρ 2 (θ), We define ( k ρ 2 )/( v k ) γ k (θ). We note that γ 3 (θ) is a cubic function of θ. iii. Since γ 3 (3/5) < 0, γ 3 (7/10) > 0, γ 3 (1) > 0, γ 3 (6/5) < 0, and γ 3 (3/2) > 0, γ 3 (θ) > 0. 9 iv. Since γ 2 (7/10) < 0 and γ 2 (1) > 0, γ 1 (θ) has one local minimum point in (7/10, 1). v. Since γ 1 (7/10) > 0 and γ 2 (1) < 0, ρ 2 (θ) has one local maximum point in (7/10, 1). Because ρ 2 (7/10) > 0 and ρ 2 (1) > 0, ρ 2 (θ) > 0. Hence, ( 2 X)/( v 2 ) > 0. vi. ( X)/( v) v=4 =2θ{(1 θ)[(1 θ)ρ 3 (θ) 39] + 4} and ρ 3 (θ) is a cubic function of θ. Since ( ρ 3 )/( θ) = 0 has one solution in [0, 1], ρ 3 (7/10) > 0, and ρ 3 (1) > 0, ρ 3 (θ) > 0 and therefore ( X)/( v) > 0. vii. X(4,θ)=[θ( 14 4) 2 + 4(1 θ 2 )+( )θ 2 ]ρ 4 (θ) and ρ 4 is a cubic function of θ. Since ( ρ 4 )/( θ) < 0in[7/10, 1] and ρ 4 (1) > 0, ρ 4 (θ) > 0. It implies that X(4,θ) > 0 and therefore X(v,θ) > 0. 9 Here (and hereafter) we use the intermediate value theorem. We note that (25) ensures θ>7/10. 12

13 Step 7. Steps 3-6 implies Pr 0 >Pr 1 if conditions (24) and (25) are satisfied. Step 2 shows it unless these conditions are satisfied. Therefore, Pr 0 >Pr 1 if θ [y, 1]. 13

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