What Industry Should We Privatize?: Mixed Oligopoly and Externality

Transcription

1 What Industry Should We Privatize?: Mixed Oligopoly and Externality Susumu Cato May 11, 2006 Abstract The purpose of this paper is to investigate a model of mixed market under external diseconomies. In our model, firm s production generates emissions, which make environmental damage. Since private firms only consider their own profit and ignore such an environmental damage, then their emission level is high. On the other hand, the state-owned public firm considers environmental damage, and maximizes social welfare. We show that, under sufficient high external diseconomy, the mixed oligopoly is better than the pure oligopoly for social welfare, even if the market is competitive. Moreover, in mixed oligopoly, an increase of external diseconomy might improve welfare. JEL classification H42, L13, L33, Q28 Keywords substitution environmental damage, mixed oligopoly, privatization, production I thank to Hiroaki Ino, Akifumi Isihara, Katsuhito Iwai, Tomohiro Kawamori, Toshihiro Matsumura and Noriyuki Yanagawa for their helpful conversation and comments. Needless to say, all errors are of course mine. mailto:ksusumu@hotmail.co.jp, Graduate school of Economics, University of Tokyo 1

2 1 Introduction Recently, there exist many literatures on mixed market, including the state-owned public firm and private firms. 1 In fact, in many countries, we can observe the mixed market, in which public enterprises compete against private firms. In most papers on the mixed market, it is assumed that firm s production activity do not produce the external damage. However, since the classical paper Meade (1952), the relation beteween externality and market structure has been regarded as the important issue. 2 In this paper we examine the effect of external diseconomy in the context of mixed oligopoly, and argue what industry we should privatize. In the typical studies of mixed markets, it is assumes that the public firm maximizes social welfare and private firms maximize their own profit. By privatization policy, the public firm s goal changes into his own profit. The well-known result of the standard model is privatization of the public firm might improve welfare. Especially, in the market that is sufficient competitive (the number of firm in the market is sufficient large), privatization is better. In this means, the standard for privatization is the number of firms in the market. The key term in the study of the mixed oligopoly is the production substitution. 3 In the mixed oligopoly, by the strategic interaction, the public firm supplies excessively, and marginal cost of the public firm is higher than private firms. 4 Then, privatization makes the production substitution from the public firm to private firms, and makes two effects to welfare, positive effect and negative effect. First, since the public firm s output decrease by privatization, then there exists cost reduction effect which increase social welfare. Second, since total output in the mixed oligopoly is smaller than in the pure oligopoly, then consumer surplus decrease by privatization. If the former force dominate the latter, then the privatization of the public firm is better in view of the social welfare. Otherwise, the mixed oligopoly is better. Most works in this field suppose that there is no external diseconomy in the market, but actually firm s production activity often make the environmental damage. And most industry where there exist state-owned public firms, such as energy industry and a water industry, often make high environmental damage. Should we privatize such industry? In our framework, there are n private firms and 1 state-owned public firm, 1 See De fraja and Delbono (1989) and Matsumura (1998). See also Vickers and Yarrow (1991), Dewenter,K.L and Malatesta (2001) and Megginson,W.L and Netter,J.M (2001). 2 See Baumol and Oates (1988) and Buchanan (1969). See also Barnett (1980), Lee (1975) and Misolek (1980). 3 Lahiri and Ono (1988) gives the discussion of welfare-enhancing product substitution effect. 4 In the model of mixed oligopoly, we usually assume convexity of cost function. Then, the cost of the public firm become high. 2

3 competing in Cournot fashion, and they pollute the environment and produce homogeneous goods. Therefore, the technologies of firm depend on not only the quantity of output but also the level of pollution. And the social welfare comprises the consumer surplus, the profit and the environmental damage. Since the public firm s goal is the welfare, he considers the environmental damage, while public firms maximize their own profit and ignore the environmental damage. First, we show that an increase in the degree of external damage causes the production substitution from the public firm to private firms. As a result, we find that when the emission makes sufficient high environmental damage, social welfare make worse by privatization even if there exist a number of private firms. We try to highlight a reason why the mixed oligopoly is better than the pure oligopoly with sufficient high externality in the context of the production substitution from the public firm to private firms. Our result suggests that the industry with high externality should not be privatized, and while privatization of the industry with low externality may make welfare improving if the market is sufficient competitive. The standard for privatization policy are not only the number of firm and technology but also the degree of external damage. Moreover, it is worth to emphasize the possibility that, in the mixed oligopoly, the existence of the external diseconomy is better for social welfare. Then, the larger external diseconomy might improve the social welfare when the state-owned public firm exists. This result is counter to our intuition. In fact, in the pure oligopoly situation with externality, larger externality causes more extensive damage to the social welfare. The paper organized as follows. In section2 we formulate the mixed oligopoly model with externality. In section3 we show our comparative static result and main result. Section4 concludes the paper. 2 The Model We consider an industry in which N +1 firms produce a homogeneous goods. In the market there are 1 state-owned public firm and N are private firms which compete in quantities. The zero-th one is public firm and his object is to maximize social welfare. The remaining n firms are private, and seek to maximize their own profit. Denote the inverse demand function by P (Q) where P is the price, Q is the total output. P (Q) is twice continuously differentiable with P (Q) < 0. We further assume that P (Q) + qp (Q) < 0. This is the standard assumption and guarantees the stability of equilibrium: each firm s best-reply is downward sloping. 5 5 Precisely, this condition implies only downward sloping of private firm s best-reply. Downward sloping of public firm s best-reply is guaranteed by P (Q) < 0. 3

4 In our model, ith firm s technology depend on not only output q i, but also the amount of emission e i. Therefore, ith firm decides his output level q i R + and the amount of emission e i R +. We assume that all firm have same technology, and the typical cost function is C(e i, q i ) where C e < 0 for [0, ē] and C e > 0 for [ē, ), C q > 0, C qq > 0 and C eq = C qe < 0 for [0, ē]. 6 By our assumption, each firm have no incentives to choose e i > ē. In this means, ē is the maximum emission level, then e i [0, ē] for all i {0,..., n}. 7 We measure the pecuniary environmental damage generated by the production activity by D(E; γ) where γ R +, the total emissions generated E = N i=0 e i, D E (E; γ) > 0, D γ (E; γ) > 0 and D Eγ = D γe > 0. The parameter γ represents the degree of the environmental damage. Particularly, if D(E; γ) = γd(e) where d (E) > 0, we interpret γ as social weight of environmental damage. Therefore, a large value of γ means strong external diseconomy in the industry and we assume that D(E; 0) = 0. This means that if γ = 0, there exists no externality in the market. We further assume that C ee + D EE (E; γ) > 0. Social welfare consists of consumer surplus, firms profit and the pecuniary environmental damage; W = Q 0 P (q)dq pq + where Π i is firm i s profit. N Π i D(E; γ) = i=0 Q 0 P (q)dq N C(e i, q i ) D(E; γ) (1) We investigate two regimes, (i) the mixed oligopoly and (ii) the pure oligopoly. 8 In the mixed oligopoly, there exists one public firm. On the other hand, in the pure oligopoly, the public firm is privatized, and maximizes his-own profit. Then, in this case, (N+1) private firms with homogeneous technology compete in quantities. Let denote W M and W P welfares in the mixed oligopoly and the pure private oligopoly, respectively. Note that if γ = 0, since all firms including the public firm choose the highest technology, i.e, e i = ē, i {0,..., n}, then our model corresponds to typical mixed oligopoly model such as De Fraja and Delbono (1989). In this means, our model is the generalization of the standard mixed oligopoly model. We suppose that W P > W M if γ = 0, and privatization is better. This assumption is not essential. By many paper of the mixed oligopoly, we know that if market is sufficient competitive, this inequality is holds. 9 i=0 6 If the public firm s cost function different from the private firm s, we can obtain same result of this paper. For simplicity, we assume identical technology between public firm and private firm. 7 We can also interpret ē as the environmental standard given by the government or a treaty. 8 There exist other regimes. In De Fraja and Delbono (1989), Stackelberg situation where the public firm behaves as Stackelberg leader is considered. partial privatization. 9 See De Fraja and Delbono (1989) and Matsumura (1998). Matsumura (1998) considers the possibility of the 4

5 3 The Result Clearly, since private firms ignore environmental damage, then each of them chooses the maximum emission level, i.e, e i = ē, i {1,..., n}. Then, the reduced maximization problem of private firm is as follows; max q i P (Q)q i C(ē, q i ) Therefore, there are N + 2 first order conditions; W = 0 and W = 0 (public firm) q 0 e 0 Π i q i = 0 for i = 1,..., N (private firms) Clearly, the second order condition of private firm is satisfied. Moreover, we assume that the second order condition of public firm is also satisfied. 10 We interest in the equilibrium in which outcome of private firm is symmetric, i.e., q 1 = q 2 =,..., = q n. Let q s denote the equilibrium output of public firm and q p denote the equilibrium output of each private firm. As we mentioned before, the equilibrium emission level of private firm e p is ē. Moreover, by first order conditions, the equilibrium emission level of public, e s, the equilibrium output of private firm and public firm, q s and q p, satisfy the following equations 11 ; P (Nq p + q s )q p + P (Nq p + q s ) C q (ē, q p ) = 0 (2) P (Nq p + q s ) C q (e s, q s ) = 0 (3) C e (e s, q s ) D E (E; γ) = 0 (4) First, we present the following comparative static results. Proposition 1. (i) qs < 0, (ii) qp > 0, (iii) Q < 0 and (iv) es < 0 Proof. See Appendix. Q.E.D. C eq 10 The hessian of public firm s object is H = P C qq. The hessian H is C eq (C ee + D EE ) negative definite if and only if P C qq < 0 and {P C qq }{ C ee D EE } C eq 2 > 0. By assumptions, the former is satisfied. And we suppose the latter is satisfied. 11 We suppose the interior solution of e s 5

6 This proposition shows that there exists the production substitution from the public firm to private firms. That is, an increase in the degree of external damage will increase the equilibrium output of private firm, and decrease the public firm s output. Moreover, the equilibrium emission level of public firm and the total output in equilibrium decrease by an increase in the degree of external damage. The intuition of this result is as follows. First, since the public firm consider environmental damage, when the degree of externality is higher, he chooses lower emission level. Since choosing lower emission level means that the public firm s technology relatively deceases, the output of the public firm decrease. Finally, reply function of private firms is downward sloping and they increase the output. What industry we should privatize is the question. Next, we consider this important question, and will obtain the answer by a simple proposition. To begin with, by simple application of the envelope theorem for the maximization problem of the public firm, we can obtain the following. W M = {P (Q ) C q (ē, q p )}N qp D γ(nē + e s ; γ) (5) W P = D γ((n + 1)ē; γ) (6) The following inequality is the sufficient condition of single crossing property. Figure1 shows the typical situation when this inequality holds. W M > W P D γ ((N + 1)ē; γ) D γ (Nē + e s ; γ) {C q (ē, q p ) P (Q )}N qp D γ ((N + 1)ē; γ) D γ (Nē + e s ; γ) P (Q )q p N qp > 0 Since e s [0, ē), by assumption, D γ ((N + 1)ē; γ) D γ (Nē + e s ; γ) > 0. By Figure 1: Welfare Comparison 6

7 proposition 1, P (Q )q p N qp W M > 0. Therefore, Now we present our main results as follows. > W P holds. Proposition 2. Suppose that W p > W M for γ = 0. There exists γ such that W P < W M for γ > γ, and W P > W M for γ < γ. This proposition contains simple and important policy implication. Suppose that a market is competitive and privatization is better under no externality (γ = 0). However, if externality is sufficient strong (γ > γ ), the government plan to privatize public firm reduces total surplus. This suggest that when the government decides to privatize state-owned public firm or not to privatize, he must check the degree of external damage, γ. The industry in which there exists the sufficient strong external diseconomy, should not privatize. This is our answer for the question in title. The intuition why under the sufficient large externality the mixed oligopoly is better than the pure private oligopoly, is as follows. We should emphasize that there exist two effect, direct effect and indirect effect. The direct effect is emission reduction by public firm, then environmental damage in mixed oligopoly is smaller than in pure oligopoly. The indirect effect is the production substitution from the public firm to private firm. The direct effect, reducing emission by the public firm, decrease the public firm s output and increase private firm s output, then the product substitution occurs. Since the public firm is welfare-maximizer, then a slight decreasing in q 0 does not harm to the welfare, i.e, W M / q 0 = 0. On the other hand, the private firm maximize his own profit, so W M / q i = P (Q ) C q (ē, q p ) > 0 for all i {1,..., n} Then a silght increasing in the production of private firm improve the welfare. Therefore, by the production substitution from the public firm to private firm improves the social welfare. 12 Note that the first term on the right hand side of equation(5) is positive while the second term is negative. The farmer is the effect of production substitution, and the later is one of environmental damage. In this sense, the direct effect is relatively better for the social welfare, while the production substitution absolutely improve the welfare. To sum up, in the mixed oligopoly, an increase in the degree of externality has two oppositing effects on welfare. It is commonly believed that an increase of external diseconomy decrease welfare. Conversely, it is said that if the positive effect of production substitution is sufficient large, in mixed oligopoly, an increase of external diseconomy might make welfare improving rather than no externality. Figure2 show this case. In general, 12 This argument is similar to the explanation of non-optimality of the full nationalization in Matsumura(1998). The effect caused by the increase in the degree of the external diseconomy is the same as one caused by slight privatization from the full nationalization. 7

8 if the following inequality holds, then above situation occurs. W M = P (Q )q pr N qp D γ (Nē + e s ; γ) > 0 (7) γ= γ γ= γ Figure 2: Welfare Comparison Example We assume the following. e s P (Q) = a Q the cost function of state-owned firm; C i (e i, q i ) = K + (1/2)α q2 i e i the cost function of private firm; C i (e i, q i ) = K + (1/2)β q2 i e i the damage function; D(E; γ) = γe We have outputs in equilblium. We have inputs of environment factor. q s β + (N + 1)ē = a (2αγ) 1/2, β + ē (8) q pr = ē β + ē (2αγ)1/2 (9) ( α ) 1/2a e s β + (N + 1)ē = α, (10) 2γ β + ē We can easiliy check Proposition1, that is, qs e pr = ē (11) < 0, qp > 0, Q < 0, and < 0. Using these equations, we obtain the following as the equliblium total output and price. Q = a (2αγ) 1/2, (12) P = (2αγ) 1/2 (13) 8

9 We can easiliy check Proposition1, that is, qs W M qp < 0, Q > 0, es < 0, and < 0. [ ē ] 2αN = (Nē + e s ) (14) ē + β W P = (N + 1)ē (15) 4 Concluding Remarks In this paper, we investigate the industry with externality in which public firm compete against private firms and consider the standard for privatization. We find that the industry in which a degree of external damage is strong should not privatize, even if the market is sufficient competitive. So, we can adopt the degree of externality as the standard for privatization. It is worth to note that, with external diseconomy, the mixed oligopoly has two advantages over the pure oligopoly in the view of the social welfare. First, since the public firm considers the environmental damage, then the damage is smaller in the mixed oligopoly. Second, with the higher externality, since the public firm cuts down the his excess production, then the production substitution from the public firm to private firms is caused. Since the price is strictly larger than the marginal cost of the private firm, then private firms in the mixed market make too little production in the view of social welfare. so this production substitution improves the welfare. Moreover, if the positive effect of production substitution is sufficient large, there exists the possibility that the increasing in external diseconomy improve the total surplus. In our model, the tax for emission is not considered. However, actually, levying Pigovian tax rule is a natural policy for the government, and is the standard issue in environmental policy in market structure. For example, Buchanan (1969), Lee (1975), Barnett(1980) and Misolek (1980) treat the Pigovian tax rule under monopoly. With the tax for emission, our results may not be robust. There issue remain for future research. 9

10 Appendix Proof of Proposition1 Proof. (i)(ii)(iv) From equations(2)(3)(4), by using the implicit function theorem, we can obtain the following. J P q + P 0 q pr / NP P C qq C eq q s / = 0 C eq D EE C ee e s / where J = NP q + (N + 1)P C qq. Therefore, by Cramer s fomula, 0 0 D Eγ (16) q p = C eqd Eγ (P q + P ) (17) q s = D EγC eq {NP q + (N + 1)P C qq } (18) e s = NP D Eγ (P q + P ) (P C qq )D Eγ {NP q + (N + 1)P C qq } (19) = C qqd Eγ {NP q + (N + 1)P C qq } P D Eγ {P C qq } (20) where = {C 2 eq + (C ee + D EE )(P C qq )}{NP q + (N + 1)P C qq } NP (C ee + D Eγ )(P q +P ). By second order condition, C 2 eq +(C ee +D EE )(P C qq ) < 0. And by assumption, NP q + (N + 1)P C qq < 0 and P (C ee + D Eγ )(P q + P ) < 0. Therefore, > 0. This implies qp qs > 0 and < 0. Moreover, Our assumption implies that C qq D Eγ {NP q + (N + 1)P C qq } < 0 and P D Eγ {P C qq } > 0. Then, es < 0. (iii) Q = N qp + qs. By using equation(17)(21), we obtain the following. Q = D EγC eq {P C qq } < 0 (21) Q.E.D. 10

11 References [1] Baumol,W.J and Oates,W.E (1988) The Theory of Environmental Policy, 2nd Ed. Cambridge University Press [2] Barnett,A.H. (1980) The Pigouvian Tax Rule under Monopoly American Economic Review, 70(5), pp [3] Buchanan,J,M (1969) External Diseconomies, Corrective Taxes, and Market Structure American Economic Review, 59, [4] Dewenter,K.L and Malatesta (2001) State-Owned and Private Owned firms: An Empirical Analysis of Profitability, Leverage, and Labor Intensity American Economic Review 91(1), pp [5] De Fraja,G. and Delbono,F. (1989) Alternative strategies of public enterprise in oligopoly, Oxford Economic Papers, 41(2), pp [6] De Fraja,G. and Delbono,F. (1990) Game Theorotic Models of Mixed Oligopoly Journal of Economic Survry,4(1) pp.1-17 [7] Lee, D.R. (1975) Efficiency of Pollution and Market Structure Journal of Environmental Economics and Management, 2, pp [8] Lahiri,S and Ono,Y. (1988) Helping minor firms reduce welfare Economic Journal, vol.98(393), pp [9] Matsumura,T (1998) Partial privatization in mixed oligopoly, Journal of Public Economics, 70(3), pp [10] Meade,J.E (1952) External Economies and Diseconomies in a Competitive Situation Economic Journal, 62, pp [11] Megginson,W.L and Netter,J.M (2001) From State to Market: A Survey of Empirical Studies on Privatization Journal of Economic Literature pp [12] Misolek, W.S. (1980) Effluent of Taxation in Monopoly Markets Journal of Environmental Economics and Management, 7, pp [13] Vickers,J and Yarrow,G (1991) Economic Perspective on Privatization Journal of Economic Perspectives 5(2) pp