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Modern Economy, 2011, 2, 45-48 doi:10.4236/me.2011.21007 Published Online February 2011 (http://www.SciRP.org/journal/me) Copyright © 2011 SciRes. ME Partial Privatization in Price - Setting Mixed Duopolies with Complementary Goods Kazuhiro Ohnishi Institute for Basic Economic Science, Osaka, Japan E-mail: ohnishi@e.people.or.jp Received October 7, 2010; revised October 22 , 20 10; accepted October 29, 2010 Abstract We consider a domestic (resp. international) mixed duopoly model in which a domestic public firm and a domestic (resp. foreign) private firm produce complementary goods. First, the domestic government chooses the level of privatization to maximize domestic social welfare. Second, observing the level of privatization, the firms simultaneously and independently choose prices. We present the equilibrium outcomes of the two mixed duopoly models and shows that our result is in marked contrast to that of the price-setting mixed du- opoly model with substitute goods. Keywords: Partial Privatization, Price - Setting Model, Domestic Mixed Duopoly, International Mixed Duopoly, Complementary Goods 1. Introduction The theoretical analysis of partial privatization of state- owned public firms has received significant attention in recent years and has been extensively studied by many economists, such as [1-17]. However, these studies analyze partial privatization in mixed duopoly competition in which public and private firms produce substitutable goods. To the best of my knowledge, the analysis of partial privatization in Ber- trand mixed markets with public and private firms pro- ducing complementary goods has been ignored. Therefore, we study partial privatization in price-set- ting mixed duopoly competition with complementary goods. We extend the analysis of Ohnishi [17], which investigates a price-setting mixed duopoly model in- volving a domestic public firm and a domestic private firm to reassess the welfare effect of partial privatization. Ohnishi demonstrates that partial privatization is not a reasonable choice for the government that wishes to maximize social welfare. We consider both domestic and international mixed duopoly models with complementary goods.1 We con- sider the following situation. In the first stage, the do- mestic government chooses the degree of privatization to maximize domestic social welfare. In the second stage, observing the degree of privatization, the firms simulta- neously and independently choose prices. The main purpose of this paper is to present the equi- librium outcomes of the two mixed duopoly models and to show that this result is in marked contrast to that of the price - setting mixed duopoly model with substitute goods. The remainder of the paper is organized as follows. In Section 2, we examine a domestic mixed duopoly model. Section 3 examines an international mixed duopoly model. Section 4 concludes the paper. 2. Domestic Mixed Duopoly In this section, we consider a domestic mixed model with two firms (firm P and firm D) and the government. These firms produce complementary goods. There is no possi- bility of entry or exit. On the consumption side, there is a continuum of consumers of the same type whose utility function is linear. Subscripts P and D denote firm P and firm D, respectively. Following Bárcena-Ruiz [18], we assume that the representative consumer maximizes PDPP DD ,Uq qpqpq , where i q is the amount of 1As is well known, international mixed oligopolies are common in developed and developing countries as well as in former communist countries. Public firms compete against foreign private firms in many industries, such as banking, life insurance, automobiles, airlines, steel, shipbuilding and tobacco. For example, in the tobacco industries o f France, Italy, Russia, Spain, Austria, Turkey, China, Japan, etc, we can find real-world examples in which public firms compete or competed against foreign private f i r ms such as Philip Morris and R. J. Reynolds. K. OHNISHI Copyright © 2011 SciRes. ME 46 good i and i p is its price P,Di. The function PD ,Uq q is quadratic, strictly concave and symmetric in P q and D q: 22 PDP DPPD D 1 ,2 2 Uq qaqqqbqqq (1) where a is a constant and 1, 0b is a measure of the degree of complementarity among products. The demand function is gi v e n by 2 1 (,P,D; ) 1 ij i abpbp qijij b (2) For simplicity, we assume that 1a and 0.5b . Each firm’s profit is (P,D), iiii pcq i (3) where i c is the marginal cost of firm i. Since the result of this paper is not affected by i c, we normalize it to zero. Firm D aims to maximize its profit. Furthermore, domestic social welfare, which is the sum of consumer surplus and profits, is given by PD WCS . (4) The objective function of firm P is given by P1,VW (5) where 0,1 is the level of privatization. That is, if 0 firm P is purely public, whereas if 1 it is purely private. The game is constructed by the following two-stage decision-making. In the first stage, the government chooses the level of privatization, , to maximize do- mestic social welfare. Observing , the firms non-co- operatively choose prices in the second stage. Through- out this paper, the subgame perfect Nash equilibrium of the price-setting game is examined. We now examine the welfare effect of partial privati- zation in the domestic mixed duopoly game. We obtain the reaction functions in prices of the two firms: D P 332 , 23 p R (6) P D3 4 p R (7) From (6) and (7), the equilibrium can be derived as follows: PD PD 12 52 , 72 72 47 4452 , 37 237 2 pp qq Comparative static results yield P0dp d , D0dp d , P0dq d , and D0dq d . Thus, the privatization decreases each firm’s output and firm D’s price, and increases firm P’s price. Furthermore, the profits, consumer surplus, and social welfare can be expressed as follows: P2 41 274, 37 2 (8) 2 D2 45 2, 37 2 (9) 2 2 829 192, 37 2 CS (10) In the first stage, the government chooses the level of privatization. Substituting (8), (9) and (10) into (4), the objective function of the government is obtained as 2 815 8. 72 W (11) Social welfare W is illustrated as a function of . When 0 , 22249 2.449W, and when 1 , 26 252.24W . In addition, when 0.1 , W 2 1322892.457. This can be stated in the following proposition. Proposition 1. In the domestic mixed market with complementary goods, neither full nationalization nor full privatization is a reasonable choice for the govern- ment that wishes to maximize domestic social welfare; that is, the optimal solution is partial privatization. 3. International Mixed Duopoly In this section, we consider an international mixed du- opoly model in which a state-owned public firm (firm P) competes against a foreign private firm (firm F). Sub- scripts P and F denote firm P and firm F, respectively. In addition, an asterisk denotes the international mixed du- opoly model. Firm F aims to maximize its own profit. Furthermore, domestic social welfare is given by *** P.WCS (12) The objective function of firm P is given by ** * P1,VW (13) where 0, 1 is the level of privatization. That is, if 0 firm P is purely public, whereas if 1 it is purely private. The timing of this model is identical to the domestic mixed duopoly model. We examine the welfare effect of partial privatization in the international mixed duopoly model. We obtain the K. OHNISHI Copyright © 2011 SciRes. ME 47 reaction functions in pr i ces of the two firms: * F * P 32 , 23 p R (14) * *P F3. 4 p R (15) From (14) and (15), the equilibrium can be derived as follows: ** PF ** PF 3235 2 , , 22 722 7 213745 2 , . 22 722 7 pp qq Comparative static results yield * P0dp d , * F0dp d , * P0dq d , and * F0dq d . Thus, the privatization decreases each firm’s output and firm F’s price, and increases firm P’s output. Furthermore, the profits and consumer surplus can be expressed as follows: * P2 6213 7, 22 7 (16) 2 * F2 12 52, 22 7 (17) 2 * 2 2 36726340. 22 7 CS (18) Substituting (16) and (18) into (12), the objective function of the government is obtained as 2 * 2 18 55343. 22 7 W (19) Social welfare * W is illustrated as a function of . When 0 , *211 2422.045W, and when 1 , *12325 1.92W. In addition, when 0.2 , *2526 106092.050W. This is stated in the fol- lowing proposition. Proposition 2. In the international mixed market with complementary goods, neither full nationalization nor full privatization is a reasonable choice for the govern- ment that wishes to maximize domestic social welfare; that is, the optimal solution is partial privatization. 4. Conclusions We have studied partial privatization in price-setting mixed duopolies with complementary goods. First, the government chooses the level of privatization to maxi- mize social welfare. Second, observing the level of pri- vatization, the firms simultaneously and independently choose prices. We have then presented the equilibrium outcomes of the two mixed duopoly markets. We have demonstrated that in each market, neither full nationali- zation nor full privatization is a reasonable choice for the government that wishes to maximize domestic social welfare; that is, the optimal solution is partial privatiza- tion. 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