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Privatization neutrality theorem revisited
Economics Letters 118 (2013) 324–326
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Privatization neutrality theorem revisited Toshihiro Matsumura a , Yasunori Okumura b,∗ a
Institute of Social Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-3300, Japan
b
Faculty of Economics, Hannan University, 5-4-33, Amami Higashi, Matsubara-shi, Osaka, 580-8502, Japan
article
info
Article history: Received 31 October 2012 Received in revised form 20 November 2012 Accepted 24 November 2012 Available online 29 November 2012
abstract Fjell and Heywood (2004) show that privatization is not necessarily welfare neutral in mixed oligopolies under a production subsidy if firms move sequentially. We find that the neutrality holds for any time structure if instead an output floor is introduced. © 2012 Elsevier B.V. All rights reserved.
JEL classification: H42 L13 Keywords: Minimum quantity regulation Mixed oligopolies Stackelberg Cournot Irrelevance results
1. Introduction Despite the wave of privatization across the globe, public firms still exist and compete against private firms in a wide range of industries. Such markets, called mixed oligopolies, are still important in many developed, developing, and former communist countries in transition.1 Furthermore, in the recent financial crisis, many private enterprises facing financial problems have been either fully or partially nationalized. Studies on mixed oligopolies and the re-privatization of public enterprises have also become increasingly common. Many works point out that privatization can either improve or reduce welfare depending on the competition structure, but they assume that the government does not adopt direct regulations or tax-subsidy policies. White (1996) shows that the implications of privatization as described in this literature on mixed oligopoly change drastically if such policies are explicitly considered. He investigates a Cournot model and shows that a simple nondiscriminatory unit production subsidy yields the first–best outcome in both mixed and private oligopolies, and thus privatization does not matter under the optimal subsidy policy (the privatization neutrality theorem).
∗
Corresponding author. Tel.: +81 72 332 1224; fax: +81 72 336 2633. E-mail addresses: [email protected], [email protected] (Y. Okumura). 1 See Gil-Moltó et al. (2011) and the works cited by them. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.11.028
Many studies following White (1996) suggest that this result is robust and holds true in various economic circumstances. Hashimzade et al. (2007) consider product differentiation, Tomaru (2006) adopts a partial privatization approach formulated by Matsumura (1998) and Kato and Tomaru (2007) consider nonprofit-maximizing private firms. The theorem holds in all these cases. Poyago-Theotoky (2001) and Tomaru and Saito (2010) consider Stackelberg duopolies where the public firm is the leader and the follower, respectively, and show that privatization does not affect welfare in either case. However, all of these models assume that firms move simultaneously when a public firm is privatized. Fjell and Heywood (2004) investigate a two-period duopoly model and show that the privatization neutrality theorem crucially depends on this assumption. Unless a discriminative policy is allowed (i.e., if a uniform subsidy rate is adopted), the privatization neutrality theorem does not hold if two firms move sequentially after privatization. As Fjell and Heywood (2004) point out, it may be unnatural to assume that privatization changes the timing of firms’ actions. We revisit this problem by considering a different policy, output floor regulation.2 We show that whether the timing of the output
2 We can show that our result holds in the Bertrand and endogenous contract models with a differentiated product market discussed in Matsumura and Ogawa (2012) if we consider a price ceiling regulation (Bertrand) or a combination of price ceiling and output floor regulations (endogenous price–quantity contract).
T. Matsumura, Y. Okumura / Economics Letters 118 (2013) 324–326
choice follows Cournot, Stackelberg, or a more complicated time structure, the privatization neutrality theorem holds if we use output floor regulations.3 2. The model Let N = {1, 2, . . . , n} be the set of firms. These firms produce a homogeneous good, and firm i chooses its output xi . Let x = (x1 , x2 , . . . , xn ), X = x1 +· · ·+ xn and X−i = X − xi . The cost functions of the firms are the same and given by C (·), which satisfies C ′ > 0 and C ′′ > 0. Let P (X ) be the inverse demand function. The profit of firm i is given by πi (xi ) = P (X )xi − C (xi ). We assume that the marginal revenue is decreasing; i.e., P ′ (X ) + P ′′ (X )xi ≤ 0 for all X ≥ xi ≥ 0. These assumptions are sufficient for the concavity of πi (xi ). Social welfare W is defined as the sum of consumer surplus and the profits of the firms and is given as W (x) =
X
P (z )dz − P (X )X +
n
0
πi .
i =1
Each firm maximizes the following objective function with respect to its output xi ,
(1 − αi ) W (x) + αi πi (xi ), where αi ∈ [0, 1]. If αi = 1, then firm i is a private firm that maximizes its profit. If αi = 0, then firm i is a public firm that maximizes social welfare. If αi ∈ (0, 1), then firm i is a partially privatized firm that maximizes a weighted average of social welfare and its profit. That is, αi represents the degree of privatization of firm i (Matsumura, 1998). In the literature on mixed oligopolies, it is assumed that α1 < 1 and α2 = · · · = αn = 1 before the privatization of firm 1 and that α1 = · · · = αn = 1 after the privatization of firm 1. In this setting, we will generalize to the cases where two or more firms are public (Matsumura and Shimizu, 2010) or some private firms care about social welfare before and after privatization because of concerns for social responsibility (Ghosh and Mitra, 2010). Let α = (α1 , . . . , αn ). Let bi = bi (X−i ) be the best reply of firm i given the outputs of the other firms. This is derived from the following first-order condition, P (X ) + αi P ′ (X )xi − C ′ (xi ) = 0. The second-order condition is satisfied under the above assumptions. We have b′i ≤ 0 for all i. Next, we derive the first-best outcome at which W (x) is maximized. The first-best output of each firm xfb satisfies P (nxfb ) − C ′ (xfb ) = 0. We have the following result regarding this first–best output and the best reply functions: Lemma 1. If X−i ≥ (n − 1) xfb , then bi (X−i ) ≤ xfb for all i. Proof. First, we show bi (n − 1) xfb ≤ xfb . By the first-order condition of firm i, bi (n − 1) xfb satisfies
+ αi P ′ (n − 1) xfb + bi (n − 1) xfb bi (n − 1) xfb − C ′ bi (n − 1) xfb = 0.
P (n − 1) xfb + bi (n − 1) xfb
Therefore, bi (n − 1) xfb Lemma 1.
We consider the timing of the game. Let T = {T1 , T2 , . . . , Tm } be a partition of N. The firms in Ti are the ith movers. That is, the firms in Ti simultaneously choose their outputs given the outputs i−1 of the firms in k=1 Tk . If T1 = N, then all firms simultaneously choose their outputs; that is, they engage in Cournot competition. If T1 = {1} and T2 = {2, . . . , n}, then firm 1 is the Stackelberg leader and the other firms are the followers. Moreover, if T1 = {1}, T2 = {2, . . . , n}, α1 = 0 and αi = 1 for all i = 2, . . . , n, then firm 1 is a public firm and a leader and the other firms are private firms and followers. 3. Privatization neutrality theorem under output floor regulation We consider an output floor regulation. Let x be the minimum output level; that is, no firm can choose an output below x. Let x∗ (x) be the equilibrium output vector for the given mini mum output level. The best reply of firm i under x is bˆ i X−i , x = max{bi (X−i ) , x}. We derive the optimal output floor that maximizes social welfare. Note that if x∗i (x) = xfb for all i, then x must be the optimal level. Therefore, we have the following result. Theorem 1. For any α and T , x = xfb is the optimal output floor and yields the first-best outcome. Proof. Let T = {T1 , T2 , . . . , Tm } and let j be a firm in Tj . Suppose x = xfb . We derive the equilibrium using backward induction. First, we consider the mth movers’ outputs. Because X−m ≥ (n − 1)xfb and Lemma 1 holds, bˆ m X−m , x = xfb for all m ∈ Tm . Next, we examine the m − 1th movers’ outputs. Because the firms in Tm will choose xfb , X−(m−1) ≥ (n−1)xfb . Therefore, bˆ m−1 X−(m−1) , x = xfb for all (m−1) ∈ T(m−1) . Similarly, we can show that all firms choose xfb and thus x∗i (xfb ) = xfb for all i. This theorem implies that under the optimal output floor regulation, privatization does not affect welfare regardless of the time structure T and the degree of privatization α . Our result also implies that privatization does not matter regardless of whether it changes the time structure. Thus, the privatization neutrality theorem holds in much more general settings under an output floor than under a production subsidy. Acknowledgments We are indebted to an anonymous referee for his/her precious and constructive comments and suggestions. Needless to say, we are responsible for any remaining errors. We gratefully acknowledge the financial support of Grant-in-Aid from the Japanese Ministry of Education, Culture, Sports, Science and Technology and the Japan Society for the Promotion of Science. References
Because αi ∈ [0, 1], P (n − 1) xfb + bi (n − 1) xfb
325
− C ′ bi (n − 1) xfb ≥ 0.
≤ xfb . Because b′i ≤ 0, we have
3 For examples and properties of output floors, see De Fraja and Iossa (1998).
De Fraja, G., Iossa, E., 1998. Price caps and output floors: a comparison of simple regulatory rules. The Economic Journal 108, 1404–1421. Fjell, K., Heywood, J.S., 2004. Mixed oligopoly, subsidization and the order of firm’s moves: the relevance of privatization. Economics Letters 83, 411–416. Ghosh, A., Mitra, M., 2010. On that old rivalry: Bertrand versus Cournot. Working Paper, University of New South Wales. Gil-Moltó, M.J., Poyago-Theotoky, J., Zikos, V., 2011. R&D subsidies, spillovers and privatization in mixed markets. Southern Economic Journal 78, 233–255. Hashimzade, N., Khodavaisi, H., Myles, G., 2007. An irrelevance result with differentiated goods. Economics Bulletin 8, 1–7. Kato, K., Tomaru, Y., 2007. Mixed oligopoly, privatization, subsidization and the order of firms’ moves: several types of objectives. Economics Letters 96, 287–292. Matsumura, T., 1998. Partial privatization in mixed duopoly. Journal of Public Economics 70, 473–483.
326
T. Matsumura, Y. Okumura / Economics Letters 118 (2013) 324–326
Matsumura, T., Ogawa, A., 2012. Price versus quantity in a mixed duopoly. Economics Letters 116, 174–177. Matsumura, T., Shimizu, D., 2010. Privatization waves. The Manchester School 78, 609–625. Poyago-Theotoky, J., 2001. Mixed oligopoly, subsidization, and the order of firm’s moves: an irrelevance result. Economics Bulletin 12, 1–5.
Tomaru, Y., 2006. Mixed oligopoly, partial privatization and subsidization. Economics Bulletin 12, 1–6. Tomaru, Y., Saito, M., 2010. Mixed duopoly, privatization and subsidization in an endogenous timing framework. The Manchester School 78, 41–59. White, M.D., 1996. Mixed oligopoly, privatization and subsidization. Economics Letters 53, 189–195.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Privatization neutrality theorem revisited Toshihiro Matsumura a , Yasunori Okumura b,∗ a
Institute of Social Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-3300, Japan
b
Faculty of Economics, Hannan University, 5-4-33, Amami Higashi, Matsubara-shi, Osaka, 580-8502, Japan
article
info
Article history: Received 31 October 2012 Received in revised form 20 November 2012 Accepted 24 November 2012 Available online 29 November 2012
abstract Fjell and Heywood (2004) show that privatization is not necessarily welfare neutral in mixed oligopolies under a production subsidy if firms move sequentially. We find that the neutrality holds for any time structure if instead an output floor is introduced. © 2012 Elsevier B.V. All rights reserved.
JEL classification: H42 L13 Keywords: Minimum quantity regulation Mixed oligopolies Stackelberg Cournot Irrelevance results
1. Introduction Despite the wave of privatization across the globe, public firms still exist and compete against private firms in a wide range of industries. Such markets, called mixed oligopolies, are still important in many developed, developing, and former communist countries in transition.1 Furthermore, in the recent financial crisis, many private enterprises facing financial problems have been either fully or partially nationalized. Studies on mixed oligopolies and the re-privatization of public enterprises have also become increasingly common. Many works point out that privatization can either improve or reduce welfare depending on the competition structure, but they assume that the government does not adopt direct regulations or tax-subsidy policies. White (1996) shows that the implications of privatization as described in this literature on mixed oligopoly change drastically if such policies are explicitly considered. He investigates a Cournot model and shows that a simple nondiscriminatory unit production subsidy yields the first–best outcome in both mixed and private oligopolies, and thus privatization does not matter under the optimal subsidy policy (the privatization neutrality theorem).
∗
Corresponding author. Tel.: +81 72 332 1224; fax: +81 72 336 2633. E-mail addresses: [email protected], [email protected] (Y. Okumura). 1 See Gil-Moltó et al. (2011) and the works cited by them. 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.11.028
Many studies following White (1996) suggest that this result is robust and holds true in various economic circumstances. Hashimzade et al. (2007) consider product differentiation, Tomaru (2006) adopts a partial privatization approach formulated by Matsumura (1998) and Kato and Tomaru (2007) consider nonprofit-maximizing private firms. The theorem holds in all these cases. Poyago-Theotoky (2001) and Tomaru and Saito (2010) consider Stackelberg duopolies where the public firm is the leader and the follower, respectively, and show that privatization does not affect welfare in either case. However, all of these models assume that firms move simultaneously when a public firm is privatized. Fjell and Heywood (2004) investigate a two-period duopoly model and show that the privatization neutrality theorem crucially depends on this assumption. Unless a discriminative policy is allowed (i.e., if a uniform subsidy rate is adopted), the privatization neutrality theorem does not hold if two firms move sequentially after privatization. As Fjell and Heywood (2004) point out, it may be unnatural to assume that privatization changes the timing of firms’ actions. We revisit this problem by considering a different policy, output floor regulation.2 We show that whether the timing of the output
2 We can show that our result holds in the Bertrand and endogenous contract models with a differentiated product market discussed in Matsumura and Ogawa (2012) if we consider a price ceiling regulation (Bertrand) or a combination of price ceiling and output floor regulations (endogenous price–quantity contract).
T. Matsumura, Y. Okumura / Economics Letters 118 (2013) 324–326
choice follows Cournot, Stackelberg, or a more complicated time structure, the privatization neutrality theorem holds if we use output floor regulations.3 2. The model Let N = {1, 2, . . . , n} be the set of firms. These firms produce a homogeneous good, and firm i chooses its output xi . Let x = (x1 , x2 , . . . , xn ), X = x1 +· · ·+ xn and X−i = X − xi . The cost functions of the firms are the same and given by C (·), which satisfies C ′ > 0 and C ′′ > 0. Let P (X ) be the inverse demand function. The profit of firm i is given by πi (xi ) = P (X )xi − C (xi ). We assume that the marginal revenue is decreasing; i.e., P ′ (X ) + P ′′ (X )xi ≤ 0 for all X ≥ xi ≥ 0. These assumptions are sufficient for the concavity of πi (xi ). Social welfare W is defined as the sum of consumer surplus and the profits of the firms and is given as W (x) =
X
P (z )dz − P (X )X +
n
0
πi .
i =1
Each firm maximizes the following objective function with respect to its output xi ,
(1 − αi ) W (x) + αi πi (xi ), where αi ∈ [0, 1]. If αi = 1, then firm i is a private firm that maximizes its profit. If αi = 0, then firm i is a public firm that maximizes social welfare. If αi ∈ (0, 1), then firm i is a partially privatized firm that maximizes a weighted average of social welfare and its profit. That is, αi represents the degree of privatization of firm i (Matsumura, 1998). In the literature on mixed oligopolies, it is assumed that α1 < 1 and α2 = · · · = αn = 1 before the privatization of firm 1 and that α1 = · · · = αn = 1 after the privatization of firm 1. In this setting, we will generalize to the cases where two or more firms are public (Matsumura and Shimizu, 2010) or some private firms care about social welfare before and after privatization because of concerns for social responsibility (Ghosh and Mitra, 2010). Let α = (α1 , . . . , αn ). Let bi = bi (X−i ) be the best reply of firm i given the outputs of the other firms. This is derived from the following first-order condition, P (X ) + αi P ′ (X )xi − C ′ (xi ) = 0. The second-order condition is satisfied under the above assumptions. We have b′i ≤ 0 for all i. Next, we derive the first-best outcome at which W (x) is maximized. The first-best output of each firm xfb satisfies P (nxfb ) − C ′ (xfb ) = 0. We have the following result regarding this first–best output and the best reply functions: Lemma 1. If X−i ≥ (n − 1) xfb , then bi (X−i ) ≤ xfb for all i. Proof. First, we show bi (n − 1) xfb ≤ xfb . By the first-order condition of firm i, bi (n − 1) xfb satisfies
+ αi P ′ (n − 1) xfb + bi (n − 1) xfb bi (n − 1) xfb − C ′ bi (n − 1) xfb = 0.
P (n − 1) xfb + bi (n − 1) xfb
Therefore, bi (n − 1) xfb Lemma 1.
We consider the timing of the game. Let T = {T1 , T2 , . . . , Tm } be a partition of N. The firms in Ti are the ith movers. That is, the firms in Ti simultaneously choose their outputs given the outputs i−1 of the firms in k=1 Tk . If T1 = N, then all firms simultaneously choose their outputs; that is, they engage in Cournot competition. If T1 = {1} and T2 = {2, . . . , n}, then firm 1 is the Stackelberg leader and the other firms are the followers. Moreover, if T1 = {1}, T2 = {2, . . . , n}, α1 = 0 and αi = 1 for all i = 2, . . . , n, then firm 1 is a public firm and a leader and the other firms are private firms and followers. 3. Privatization neutrality theorem under output floor regulation We consider an output floor regulation. Let x be the minimum output level; that is, no firm can choose an output below x. Let x∗ (x) be the equilibrium output vector for the given mini mum output level. The best reply of firm i under x is bˆ i X−i , x = max{bi (X−i ) , x}. We derive the optimal output floor that maximizes social welfare. Note that if x∗i (x) = xfb for all i, then x must be the optimal level. Therefore, we have the following result. Theorem 1. For any α and T , x = xfb is the optimal output floor and yields the first-best outcome. Proof. Let T = {T1 , T2 , . . . , Tm } and let j be a firm in Tj . Suppose x = xfb . We derive the equilibrium using backward induction. First, we consider the mth movers’ outputs. Because X−m ≥ (n − 1)xfb and Lemma 1 holds, bˆ m X−m , x = xfb for all m ∈ Tm . Next, we examine the m − 1th movers’ outputs. Because the firms in Tm will choose xfb , X−(m−1) ≥ (n−1)xfb . Therefore, bˆ m−1 X−(m−1) , x = xfb for all (m−1) ∈ T(m−1) . Similarly, we can show that all firms choose xfb and thus x∗i (xfb ) = xfb for all i. This theorem implies that under the optimal output floor regulation, privatization does not affect welfare regardless of the time structure T and the degree of privatization α . Our result also implies that privatization does not matter regardless of whether it changes the time structure. Thus, the privatization neutrality theorem holds in much more general settings under an output floor than under a production subsidy. Acknowledgments We are indebted to an anonymous referee for his/her precious and constructive comments and suggestions. Needless to say, we are responsible for any remaining errors. We gratefully acknowledge the financial support of Grant-in-Aid from the Japanese Ministry of Education, Culture, Sports, Science and Technology and the Japan Society for the Promotion of Science. References
Because αi ∈ [0, 1], P (n − 1) xfb + bi (n − 1) xfb
325
− C ′ bi (n − 1) xfb ≥ 0.
≤ xfb . Because b′i ≤ 0, we have
3 For examples and properties of output floors, see De Fraja and Iossa (1998).
De Fraja, G., Iossa, E., 1998. Price caps and output floors: a comparison of simple regulatory rules. The Economic Journal 108, 1404–1421. Fjell, K., Heywood, J.S., 2004. Mixed oligopoly, subsidization and the order of firm’s moves: the relevance of privatization. Economics Letters 83, 411–416. Ghosh, A., Mitra, M., 2010. On that old rivalry: Bertrand versus Cournot. Working Paper, University of New South Wales. Gil-Moltó, M.J., Poyago-Theotoky, J., Zikos, V., 2011. R&D subsidies, spillovers and privatization in mixed markets. Southern Economic Journal 78, 233–255. Hashimzade, N., Khodavaisi, H., Myles, G., 2007. An irrelevance result with differentiated goods. Economics Bulletin 8, 1–7. Kato, K., Tomaru, Y., 2007. Mixed oligopoly, privatization, subsidization and the order of firms’ moves: several types of objectives. Economics Letters 96, 287–292. Matsumura, T., 1998. Partial privatization in mixed duopoly. Journal of Public Economics 70, 473–483.
326
T. Matsumura, Y. Okumura / Economics Letters 118 (2013) 324–326
Matsumura, T., Ogawa, A., 2012. Price versus quantity in a mixed duopoly. Economics Letters 116, 174–177. Matsumura, T., Shimizu, D., 2010. Privatization waves. The Manchester School 78, 609–625. Poyago-Theotoky, J., 2001. Mixed oligopoly, subsidization, and the order of firm’s moves: an irrelevance result. Economics Bulletin 12, 1–5.
Tomaru, Y., 2006. Mixed oligopoly, partial privatization and subsidization. Economics Bulletin 12, 1–6. Tomaru, Y., Saito, M., 2010. Mixed duopoly, privatization and subsidization in an endogenous timing framework. The Manchester School 78, 41–59. White, M.D., 1996. Mixed oligopoly, privatization and subsidization. Economics Letters 53, 189–195.