economics
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ELSEVIER
Economics Letters 53 (1996) 189-195
letters
Mixed oligopoly, privatization and subsidization Mark D. White* Department of Economics, University of Cincinnati, ML 0371, Cincinnati, OH 45221-0371, USA Received 29 April 1996; accepted 23 September 1996 Abstract
This paper demonstrates three effects of domestic production subsidies in a mixed oligopoly industry regarding privatization and efficiency. First, if subsidies are used before and after privatization, welfare is unchanged by privatization. Second, if subsidies are used only before privatization, then privatization always lowers welfare, regardless of the number of private firms in the industry. Third, the subsidy contributes to overall efficiency in a mixed oligopoly due to cost distribution effects. Keywords: Mixed oligopoly; Privatization; Subsidy JEL classification: D43; t.~,,,, ,,,~u, L32
1. Introduction
Privatization is often studied using mixed oligopoly models, models that incorporate public-owned welfare-maximizing firms in a standard oligopoly framework (see DeFraja and Delbono (1990) for a review). The standard result in the literature is that privatization will lower welfare if there are relatively few private firms in the market and will raise welfare if there are relatively many private firms, assuming Cournot competition and no regulation apart from the operation of the public firm. The purpose of this paper is to investigate the role that production subsidies play in a mixed oligopoly and how they may influence the privatization decision, which have not yet been done. In this paper, we present three main results. First, we find that, when optimal subsidies are used before and after privatization, privatization does not change welfare; in fact, the optimal subsidy, outputs, price, profits and consumer surplus are all the same under mixed or private oligopoly when both are subsidized. Second, if the mixed oligopoly is subsidized but the private oligopoly is not, privatization always lowers welfare; this contrasts with the standard * Correspondence address: Department of Economics, University of Cincinnati, ML {)371, Cincinnati, OH 45221-0371, USA. Tel.: (513) 556-2280; fax: (513) 556-2669; e-mail:
[email protected]. 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved Pii S0165-1765(96)00916-0
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M.D. White / Economics Letters 53 (1996) 189-195
result in the literature, which states that, in the absence of subsidies, privatization raises welfare if there are enough private firms in the market. Third, the previous two results are explained by the cost distribution effect of a production subsidy in mixed oligopoly, which ceteris paribus increases welfare through minimizing total industry costs, which enhances the standard positive welfare effect of a subsidy in an imperfectly competitive industry (Tirole 1988, pp. 68-69). In this paper, we examine four regimes: mixed and private oligopoly, each with and without subsidies. In the regimes without subsidies, a simple Cournot quantity-setting game is constructed and solved. In the regimes with subsidies, a two-period game is constructed, wherein the government sets the subsidy in the first period, and in the second period all firms observe the subsidy and simultaneously set their production levels. In all four regimes we assume linear demand and quadratic cost, with fixed cost equal to zero with no loss of generality (since issues of entry are not examined). In Section 2 we solve the four models and comment on each. In Section 3 we compare the three regime~, and we conclude in Section 4.
2. Results of the four models We assume there are m private firms and one public firm operating in a homogeneous goods market with inverse demand given by p = a - Q, where Q is total output. The firms have identical technologies, represented by the quadratic cost function C(q) = F + ½kq z. We assume F = 0 with no loss of generality, since entry decisions are not considered. Each private firm i chooses its output q: to maximize its own profits, given by
m 7r~ - q~ a - ~
)
I
q~ - q .
- ~kq;
(1)
+ sq~ ,
~1
where i-- 1. . . . , m, qo represents the output of the public firm, and s is the subsidy. The public firm's profits are given by ~'o =
qo
ta
m
- ~ . q~ - q o
)
I "~
- ~kq~ + sqo ,
is|
(2)
and social welfare is given by W - CS
+ ~ro +
~. Ir~ i= l
s
q~ + q o \i= l
,
(3)
where C S - - ~ Q " represents consumer surplus. The subsidies are included in the welfare expression as both a component of profits and a state expenditure; hence, tile direct effect of the subsidy on welfare is zero. In this sense, the subsidy paid to the public firm does not directly affect its output decision; the subsidy only influences the public firm through its effect on the private firms' output. In the mixed oligopoly models (Sections 2.1 and 2.2), the public firm chooses qo to
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maximize welfare (3), and in the private oligopoly model (Sections 2.3 and 2.4), it is privatized and therefore maximizes its own profit (2). All results are presented as functions of the subsidy, and the superscript indicates the nature of the oligopoly (M for mixed; P for private).
2.1. Unsubsidized mixed oligopoly As a benchmark, we present the Cournot-Nash equilibrium values of outputs, the price, profits, consumer surplus and welfare, obtained by maximizing (1) and (3) simultaneously, when there is no subsidy (s = 0):
a(k + 1)
ak q~M(O) = k 2 + (m + 2 ) k + 1 '
q"M(O) = k' ~-(m + 2)k + I
aIk(m + 1) + 1]
(5)
QM(O) = k 2 + (m + 2)k + 1 '
a2k(k + 1) 2
a2k2(k + 2) zr'M(O) = 2[k 2 + (m + 2)k + 112,
7r°M(O) = 2[k 2 + (m + 2)k + 1] 2 '
a2[k(m + 1) + 1] 2 c s M ( o ) - 21k 2 + (m + 2)k + 1] 2,
wM(o)=a2[(k
+ 1) 3 +mk(k
(4)
2 + (m + 4)k + 2)1
2[k 2 + (m + 2)k + 1] 2
(6)
(7) (8)
Note that, contrary to common belief, the public firm makes a strictly positive profit (assuming low fixed costs), regardless of the parameter values. Also note that the output of the public firm is unambiguously higher than that of a private firm, implying that the public firm has higher marginal and total costs than a private firm.
2.2. Subsidized mixed oligopoly We now consider the mixed oligopoly regime when the government considers setting a production subsidy. We solve the model by backward induction, solving for the second-stage equilibrium expressions first. Maximizing (1) and (3) simultaneously, we arrive at the second-stage Cournot-Nash equilibrium outputs in terms of the subsidy s:
ak+s(k+l) , qoM(S)=k2a(k+l)-sm q~M(s) = k 2 + (m + 2)k + 1 + (m + 2)k + 1
(9)
As expected, a private firm's output is positively related to the subsidy. However, the public firm's output is negatively related to the subsidy, which is due to the potential of cost redistribution. Even though the subsidy has no direct effect on the public firm's output choice, as mentioned above, it will reduce the public firm's output indirectly through its reaction to the increased private output. As shown above, q0(0) > q~(0); if there is no subsidy, the output,
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M.D. WRite / Economics Letters 53 (1996) 189-I95
marginal and total costs of the public firm are higher than that of a private firm. Administering a positive subsidy has the effect of redistributing output from the higher-marginal-cost public firm to the lower-marginal-cost private firms, thereby lowering total industry costs, which ceteris paribus increases welfare. It is easily seen that total industry output, and therefore consumer surplus, increase with the subsidy, as does each private firm's profit° The general effects of the subsidy on the public firm's profit and welfare are ambiguous, but the derivatives of both with respect to s are positive at s = 0, indicating that the public firm's profit and welfare will both initially rise if a small subsidy is applied. We now solve the first stage of the game. Taking into account the social welfare function (3) and all firms' reactions to the subsidy, the government determines the welfare-maximizing subsidy: a s*
m+k+ 1"
(10)
Note that (10) is strictly positive, regardless of the parameters; a welfare-maximizing government will always grant a positive subsidy, both to equalize production among the firms and to boost overall output. Also, the optimal subsidy is negatively related to the number of private firms m and the cost parameter k. The more private firms there are, the more equally production will be spread among firms in the absence of a subsidy, and so less subsidy is needed to achieve complete equalization; also, more private firms implies more competition, which also reduces the need for a subsidy to boost production. The higher the cost parameter k, the smaller are the potential gains from cost distribution, which reduces the positive welfare effect of a subsidy; also, a higher value of k implies a lower efficient level of production, reducing the need for the production-expanding subsidy. The firms observe this subsidy and produce accordingly, resulting in the following subgame perfect Nash equilibrium (SPNE) outcomes: a
q~U(s*) - q°M(s*) -- m + k + 1 ' a(m + 1)
Q"(s*) = m + k + l
(12)
'
a2(k + 2) 2(m + k + I)2 ,
CSM(s*)- a2(m +
(ll)
1) 2
2(m + k + 1)2 '
(13) (14)
aZ(m + 1)
WM(S*)--2(m + k + 1) "
(15)
By comparing (6)-(8) with (13)-(15), it can be shown that consumer surplus, the private firms' profits, and welfare are all higher with the optimal subsidy. The public firm's profit is higher when subsidized, but only for relatively high values of m; even when this is the case, the positive effects on private profit and consumer surplus dominate, and welfare increases
M.D. White I Economics Letters 53 (1996) 189-195
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when subsidies are used. Note that, when subsidized, all firms (public and private) produce equally, which minimizes total costs. This is due to 1he cost distribution effect of ~.he subsidy; the government sets the subsidy to the level which will equalize public and private firms' production and minimize total industry costs given ltotal output (which also increases due to the standard subsidy effect). The preceding results are summarized in Proposition 1.
Proposition I. When optimal subsidies are used in a mixed oligopoly with one public firm and m private firms, output is equalized between public and private firms, efficiently allocating costs, increasing output, and increasing welfare. 2.3. Unsubsidized private oligopoly As a benchmark, we solve for the Cournot-Nash equilibrium outcomes of the unsubsidized private oligopoly model, wherein all firms maximize their own profits, (1) or (2), and s = 0. The results are as follows: a
qiP(O) = q°P(O) - m + k + 2 ' a(m + 1) Q v ( O ) - m + k +2 ' a2(k + 2) ~'iP(0) = ~'°P(0) = 2(m + k + 2) 2 '
(16) (17) (18)
a2(m + 1) 2 CSP(O) - 2(m + k + 2) 2.'
(19)
Wr(O ) _ a2(m + 1)(m + k + 3) 2(m + k + 2) 2
(20)
2.4. Subsidized private oligopoly In this section, we follow the methodology of Section 2.2; we first solve for the firms' output in terms of the subsidy s, and then the government chooses s to maximize welfare. The effects
of the subsidy on the private oligopoly results are summarized in Proposition 2. Proposition 2. Used optimally, production subsidies can raise output and welfare in a private oligopoly. As stated in the introduction, this result is standard; assuming that subsidies involve no deadweight loss, an optimal subsidy will raise welfare, with some potentially undesirable distributional effects. Recognizing such political considerations, privatization of a subsidized mixed oligopoly to both subsidized and unsubsidized private oligopoly are considered below. It happens that the optimal subsidy, outputs, price, profits, consumel surplus and welfare in this case are identical with those in Section 2.2, subsidized mixed oligopoly. Hence,
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expressions (11)-(15) also represent the relevant expressions for the subsidized private oligopoly. This result will be elaborated upon in the next section (Proposition 4).
3. Comparisons First, we compare the two unsubsidized cases (Sections 2.1 and 2.3); this is parallel to the standard privatization analysis in the mixed oligopoly literature. The difference between welfare in the unsubsidized mixed and private oligopoly regimes is
a2[(k + 1)3 - km(m + l)! l " ' WM(O) - Wv(O) = 2(m + k + 2)2[k 2 + (m + 2)k + 1
(21)
This indicates that mixed oligopoly dominates private oligopoly for relatively low values of m, and vice versa for relatively large m. This verifies the standard finding (DeFraja and Delbono 1989) that, in the absence of subsidies, privatization of a single public firm will maximize welfare in the presence of a relatively large number of private firms. Next, we compare the subsidized mixed oligopoly outcomes with those of unsubsidized private oligopoly. This may be relevant if there is political opposition toward subsidizing a private oligopoly because of distributional effects, but subsidization is acceptable if there is already government involvement in the industry in the form of a public firm. Proposition 3 states the result of this comparison.
Proposition 3. When optimal subsidies are used before but not after privatization of a public firm in a mixed oligopoly, welfare is lower after privatization, regardless of the number of private firms already in the market. Note that the only difference between (13)-(14) and (18)-(19), the expressions for profits and consumer surplus in the two regimes, are the denominators, which are larger in the private oligopoly case. Thus profits and consumer surplus, and therefore welfare, are higher in the subsidized mixed oligopoly, regardless of the value of m. ~'he use of optimally chosen subsidies makes the mixed oligopoly regime unambiguously "better" than the unsubsidized private oligopoly, compo.red with the ambiguous superiority of the unsubsidized mixed oligopoly regime. Finally, we compare the two subsidized regimes, which will indicate the welfare effects of privatization when subsidies are used before and after the action. This result is presented in Proposition 4 and was hinted at previously.
Proposition 4. When optimal subsidies are used before and after privatization of a public firm #~ a mixed oligopoly, the optimal subsidy and welfare are unchanged. This result is particularly surprising, because in previous studies there has always been a change in welfare due to privatization, whether it be positive or negative. This result indicates that, when subsidies are set optimally, there may be no welfare-related reason to favor one
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system over the other; other considerations, such as political climate and voter sentiment, would become more prominent in the privatization decision.
4. Conclusions In this paper, we introduce the cost distribution effect of a production subsidy and find that it has important implications regarding the welfare effects of privatization. When optimal subsidies are used before and after privatization, welfare is unchanged by privatization. When optimal subsidies are used only before privatization, welfare is always lower after privatization, regardless of the number of private firms already in the market. Both results contrast with the findings of the existing mixed oligopoly literature, which has never before considered the use of subsidies in any capacity. These results indicate a new direclion in the study of mixed oligopoly, by introducing a new policy tool, the production subsidy, to be used in regulating such an industry. As an extension to international mixed oligopoly, Pal and White (1996) consider the effects of subsidies used as a strategic trade policy instrument, where the subsidy has an import-deterring effect in addition to the cost distribution effect detailed here. In this case, production is not equalized between public and private firms, because the trade effect of increased subsidization must also be taken into account.
Acknowledgements I thank Debashis Pal, Wolfgang Mayer and Marina White for helpful comments; any remaining errors are my own.
References DeFraja, G. and F. Delbono, 1989, Alternative strategies of a public enterprise in oligopoly, Oxford Economic Papers 41. DeFraja, G. and F. Delbono, 1990, Game-theoretic models of mixed oligopoly, Journal of Economic Surveys 4. Pal. D. and M. White, 1996, Mixed oligopoly, privatization and strategic trade policy, Mimeo, University of Cincinnati. Tirole, J., 1988, The theory of industrial organization (MIT Press, Cambrdige, MA).