Strategic Regional Competition among Local Government Firms

Transcription

1 Strategic Regional Competition among Local Government Firms Kazuharu Kiyono Takao Ohkawa Makoto Okamura Yang Yu December, 2005 Abstract China, like other transition economies, holds various firms differing in their ownership structures. One striking feature, compared with other market economies, is that many local governments hold their own firms competing among one another. Along with the progress in the so-called socialist market economy, local governments have greater discretion over their own economy and social life. They also have an option to set up the own manufacturing industry when there are other regionss enjoying rents. But what if the country allows each local government to freely nurture the own industry? Our theoretical analysis shows that there tends to be excessive entry as in private free entry-exit Cournot equilibrium. 1 Introduction The world had experienced great changes in the political economic systems towards the end of the twentieth century. There is no more Berlin wall, and the Soviet Union has broken apart. The Soviet-type socialist doctrine has now been replaced with the market-based principle. The former Soviet Union and the East Europe had attempted to make drastic reforms in the economic system as well as the political and social ones. Very Preliminary. Please do not quote without the authors permission. School of Political Science & Economics, Waseda University Faculty of Economics, Ritsumeikan University Economics Department, Hiroshima University Graduate School of Economics, Waseda University 1

2 Number Output Employment Net value of Profits and taxes (%) (%) (%) fixed assets (%) (%) Large Medium Small tab:soe Table 1: State-Owned Industrial Enterprises by Sie, 1993 (Souce:?) Year Urban Area Rural Area (% in total) (% in total) ,514 However, China, another leader of socialism, has decided to pursue its own program towards the future under the banner of the so-called socialist market economy. These are the state firms, distinguished from the collective firms which are owned and controlled by lower levels of government, i.e. districts/counties and below. Collectives are owned by the workers rather than by all the people. Of the state-run enterprises and collectives, some (such as steel manufacturers) are large and centralized, and others are smaller and are owned by a town, county, or other administrative unit. 2 Model Consider a country consisting of N symmetric regions, each of which has the own local government. Each region consumes two goods, one of which is called the numeraire and privately produced subject to constant returns to scale in perfect competition and the other of which is produced by several regions subject to the local government s management. Let S N where N := {1,..., N} denotes the set of all the regions. We also let C := N \ S, i.e., the set of the regions not producing the good in question. We call the regions in S the producing regions and those in C the consuming regions. The availabe technology is given by the total cost function C(x) where x denotes the output. We assume Assumption 1 The total cost function C(x) satisfies A?? as:cost -1: lim x +0 C(x) > 0 2

3 A?? as:cost -2: C x (x) > 0 and C xx (x) 0 for all x > 0. as:cost Assume that the utility function of the representative household in each region is given in the same quasi-liner form, and focus our attention on the production and consumption of the good other than the numeraire. Then region i s consumer surplus is given by S i = u(y i ) p i y i, where y i denotes region i s consumption of the good in question (i.e., the good other than the numeraire) and p i its domestic price. Let us assume Assumption 2 The utility or gross consumer-surplus function u(y) satisfies A?? as:u -1: u(0) = 0 A?? as:u -2: lim y +0 u y(y) = + A?? as:u -3: u y (y) > 0 and u yy (y) > 0 for all y > 0. as:u Assumption 3 A?? as:econ -1: The good is freely traded wihtout any transport costs among the regions, so that there holds one market price over the country. A?? -2: as:econ Every household can freely purchase the good so as to maximize the own utility. as:econ-c A?? as:econ -3: Every local government firm maximizes the region s welfare. as:econ By Assumption?? as:econ -?? as:econ-c, the household s behavior and consumer surplus are expressed as functions of the market price only. Let d(p i ) := arg max {y i } {u(y i) p i y i } denote the demand function of region i and S(p i ) := u (d(p)) pd(p) 3

4 its consumer surplus. The total demand function is then given by D(p) := nd(p) where p denotes the market price. We express its inverse by p = P (X) where X denotes the total output over the country. By using this inverse market demand function, we may describe each region s demand as a function of the total output as below. δ(x) := d (P (X)) = X n. The profit of region i s local government firm, which we often call firm i, is expressed by π i (x i, X i ) := P (x i + X i )x i C(x i ) for i S, where X i := k i x k denotes the aggregate output other than firm i. Then the welfare of each region is expressed as below. w i S (P (x i + X i )) + π i (x i, X i ) for i S, (x i, X i ) := S (P (x i + X i )) for i C, where use was made of x i = 0 for all i C. For simplicity of exposition, we assume Assumption 4 The welfare of each producing region is strictly concave in the as:concave own output. Given the above market structure, we focus our attention on the following two-stage game. Stage 1: Each local government simultaneously decides whether to foster the own indiginous industry. Stage 2: After observing the industry setup decision by all the local governments, each local government firm simultaneously chooses how much to produce. We call the resulting equilibrium number of producing regions or active local government firms the equilibrium number of local government firms, which we denote by n e u. This number of active local government firms is a result of regional competition. However, as is the problem in China and other market economies, each local government may privatize the decision to set up the indigenous industries. When all the local governments privatize the industries, then the market becomes private oligopoly under free entry and exit. This equiliribum is what? and? 4

5 explored. They showed that the resulting free entry-exit equilibrium number of active firms, denoted by n e r in the succeeding discussion, exceeds what they call the second-best number of firms that maximizes the aggregate social welfare. Our interest lies in the following two questions. Question 1: Does this equilibrium number of local government firms maximizes the country s aggregate welfare? Question 2: How does each local government s privatization affect the equilibrium number of active firms and the individual welfare of each region? We explore the sub-game perfect Nash equilibrium. For this purpose we first focus our attention on the second stage equilibrium. 3 Entry-Consgtrained Regional Competition For comparing the competition between local government firms and private firms, we may formulate the payoff function of each individual firm as follows. v i (x i, X i, λ i ) := (1 λ i )S (P (x i + X i )) + π(x i, X i ), (1) where λ i [0, 1] is a constant. The firm is a local government firm maximizing the regional welfare for λ i = 0, while it is a private firm maximizing only the profit for λ i = 1. We call λ i the privatization parameter for producing region i. 3.1 Quasi-Reaction Function Given the number of active firms n and the privatization parameter profile λ := (λ i ) i S, each active firm i maximizes its payoff. The associated first-order condition is expressed by 1 0 = vi (x i, X i, λ i ) x i = P X (X) (x i (1 λ i )δ(x)) + (P (X) C x (x i )). (2) eq:vi-foc In the succeeding analysis, we use the so-called quasi-reaction function defining each firm s best-response output as a function of the total output, instead of the standard reaction function This tool is often so useful when 1 The second-order condition is given by 0 > P XX (X) (x i (1 λ i )δ(x)) + P X (X) 2 1 λ «i C xx (x i ) n 5

6 we discuss the oligopoly with more than two active firms. For this purpose, we may regard (?? ) as eq:vi-foc a function of x i, X and λ i, which we express by ϕ(x, X, λ) := P X (X) (x (1 λ)δ(x)) + (P (X) C x (x)). Since it is straightforward to see ϕ x (x, X, λ) = P X (X) C xx (x) < 0, we may apply the implicit function theorem and obtain firm i s quasi-reaction function, which we represent by γ(x, λ i ). This quasi-reaction function satisfies the following properties. Lemma 1 The individual quasi-reaction function γ(x, λ) satisfies the following properties. Property 1: γ i,u 2 X (X) < 1. Property 2: γ X (X, λ) < 0 if and only if each firm s output is a strategic substitute to the aggregate output other than its own. 3 Property 3: γ λ (X, λ) < 0. 4 In view of Property 1 above we call each firm s output a strategic substitute to the total output when there holds γ X (X,, λ) < 0 in the succeeding discussion. 2 The result holds because of the following relations. so that there holds ϕ x (x, X, λ) = P X (X) C xx (x) < 0, ϕ X (x, X, λ) = P XX (X) (x (1 λ)δ(x)) + P X (X) 3 We first note that there holds ϕ x(x, X, λ) ϕ X (x, X, λ) = 2 v i (x, X, λ) x 2 i γ X (X, λ) = ϕ X (γ(x, λ), X, λ) ϕ x (γ(x, λ), X, λ) ϕ X (γ(x, λ), X, λ), where use was made of γ x (x, X, λ) < 0. But here holds 1 1 λ «, n > 0. ϕ X (x, X, λ) = P XX (x (1 λ)δ(x)) + (1 δ X (X)) P X (X) = 2 v i (x i, X i, λ) X i x i. This establishes the result in the text. 4 In view of γ x (x, X, λ) < 0, it suffices to confirm ϕ λ (x, X, λ) = δ(x)p X (X) > 0. 6

7 Property 2 holds, for the private firm s loss in the terms of trade caused by sales expansion outweighs the local public firm s counterpart. This is because the former is proportional to the firm s sales, but the latter is proportional to the region s export, the whole sales minus the local consumption nd-Stage Equilibrium Given the properties of the individual quasi-reaction function, the second-stage equilibrium is governed by X e = i S γ(x e, λ i ), (3) eq:2nd-eq where X e denotes the equilibrium total output. Andthe output of each firm is governed by the quasi-reaction function γ(x, λ i ). This is shown by Figure??. fig:non-coop-eq For the later comparative statics, we need the stability condition for the present entry-constrained second-stage equilibrium. Specifically we assume à la Cournot that the outputs are adjusted in disequilibrium subject to { } Ẋ = α γ(x, λ i ) X, i S where α is a strictly positive constant representing the adjustment speed. The equilibrium is stable under the above adjustment process when there holds Assumption 5 For any number of active firms n, the privatization parameter profile λ and all X > 0, there holds γ X (X, λ i ) < 1, i S for all X > 0. as:stab Note that this assumption always holds when each firm s output is a strategic substitute to the total output. For example, this condition holds when we assume a linear demand function. Now (?? eq:2nd-eq ) implies that the equilibrium total output depends on the number of active firms n as well as the privatization parameter profile λ. We express this relation by X e (n, λ). 7

8 X γ i (X) i S E 45 X e X Figure 1: Non-cooperative equilibrium among local government firms fig:non-coop-eq 8

9 3.3 Short-run Effect of Privatization Let us first explore the features of the second-stage entry-constrained equilibrium. Although the equilibrium total output generally hinges on the privatizatin parameter profile over the producing regions, one may obtain the following stronger result when the marginal cost of production is constant. Proposition 1 Suppose that the marginal cost is constant. Then given the number of active firms, the equilibrium total otuput is determined by the sum of the privatization parameters over the producing regions, i.e., Λ := i S λ i. Sum (?? eq:vi-foc ) over the producing regions at the equilibrium and obtain 0 = (X e (n Λ)δ(X e )) P X (X e ) + n (P (X e ) c), (4) where c denotes the constant marginal cost. Since the right-hand side is a function of X, Λ and n, it is straightforward to see that Proposition 1 holds. The effect of privatization by local governments is also straightforward to derive. When producing region i privatizes its industry, then its privatization parameter λ i changes from zero to unity. However since γ λ (X, λ) < 0, such privatization should always decrease the equilibrium total output in view of Assumption??. as:stab Proposition 2 When any local government privatizes the industry, then the total output declines and the market price rises. prop:privatization Thus privatization by any region should lower the welfare of every consuming region. What are the effects on the producing regions? Privatization may increase or decrease the individual firm s output. When there is a region enforcing privatization, the firms in the other regions increase their output and the profits if and only if it is a strategic substitute to the total output. The output of the privatized firm decreases but it is generally ambiguous whether its profit decreases or increases. Let us explore these effects more in detail. For this purpose, let us represent the equilibrium profit of firm i S with π i,e (n, λ) := P (X e (n, λ)) γ (X e (n, λ), λ i ) C (γ (X e (n, λ), λ i )). 9

10 The effect of privatization by some other region j i is given by π i,e (n, λ) λ j = x i P x (X e ) Xe λ j + (P (X e ) C x (x i )) γ X X e λ j (where x i = γ(x e, λ i )) = {x (x (1 λ i )δ(x e X e ( )) γ X } P X (?? ) ) eq:vi-foc λ j ) (x (1 λ i )δ(x e ))γ X x ( P X (X) < 0 and Xe > 0 λ j < (x (1 λ i )δ(x e )) x ( γ X (X, λ i ) < 1 and x i δ(x e ) > 0 for all i S) = (1 λ i )δ(x e ) 0. Thus as stated above, privatization by some region increases the profit of the firms in the other regions. What is the effect on the profit of the firm which is privatized? the result is generally ambiguous. However when Assumption?? as:stab holds, then privatization lowers the profit of the privatized firm as expressed by π i,e (n, λ) λ i = x i P x (X e ) Xe + (P (X e X ) C x (x i )) (γ e ) X + γ λ λ i λ i (where x i = γ(x e, λ i )) = {x i (x i (1 λ i )δ(x e X e )) γ X } P X (x i (1 λ i )δ(x e ))P X γ λ λ ( i (?? ) ) eq:vi-foc = {x i (x i (1 λ i )δ(x e )) γ X } P X γ i λ 1 k S γk X (x i (1 λ i )δ(x e ))P X γ λ {(x i (1 λ i )δ(x e ))γ X x i } (x i (1 λ i )δ(x e )) ) ( P X (X) < 0 and Xe > 0 λ j = x i (x i (1 λ i )δ(x e )) 1 where use was made of Assumption??. as:stab Thus we have established k S,k i γ k X < 0, ( 1 k S γ k X Proposition 3 Privatization by a region increases the profit of the other firms ) 10

11 but decreases that of the privatized firm. prop:priv-profit What is the effect of an increase in the number of active firms? As in the proof of the effect of privatization, one can prove the following results in a straightforward fashion by virtue of Assumption??. as:stab Proposition 4 An increase in the number of active firms gives rise to the followign effects. Effect 1: The total output increases. Effect 2: The consuming regions are all better off. Effect 3: The profit of every incumbent firm decreases. Note that it is generally ambiguous whether the producing countries are worse off. We may illustrate the relationship between the number of active firms n (or the number of producing regions) and the welfare of each region as shown by Figure?? fig:free-entry. The curve named w N shows the welfare of the consuming region and w S that of the producing region. 4 Race for Indigenous Local Industries Let us now discuss the first stage of the entry decision. For the time being, we assume that all the firms are set up as local public firms, and consider the incentive of each regional government to nurture the own local public industry. 4.1 Race for Local Public Industries If the active local public firms earn strictly positive profits, then the region with an indigenous industry enjoys the higher welfare than those regions without indigenous industries. Then each region without its own industry has an incentive to set up its own. This incentive vanishes only when each country is indifferent between becoming a producing region and staying a consuming one, which is shown by point E u in Figure?? fig:free-entry. Clearly the local public firm s profit should be zero at this free-entry equilibrium with the equilibrium number of local public firms n e u. It is also straightforward to see that this number of local public firms constitues an equilibrium for the 1st stage of entry decision. Proposition 5 Suppose that all the firms are set as local public firms. Then when the regions freely choose their setup, the resulting free-entry equilibrium entails zero profit for the local public firms. 11

12 w N u E u w S n e u n Figure 2: Regional Incentive for Industry Set-Up fig:free-entry 12

13 The race for industry setup ceases at the number of active firms n e u where the profit becomes zero as in the free entry-exit equilibrium of private oligopoly. The problem is whether such regional incentive for industry set-up leads to efficient resource allocation from the viewpoint of the country as a whole. For the succeeding analysis, let us confine our attention onto the symmetric case of λ = λ/i for all i S. Then the aggregate welfare of the country is given in general by w T (n, λ) := nw S (n, λ) + (N n)w N (n, λ). (5) An increase in the number of active firms gives rise to the change in the aggregate welfare expressed by 5 w T n (n, λ) = { w S (n, λ) w N (n, λ) } + nw S n(n, λ) + (N n)w N n (n, λ). (6) Then, the social marginal value of industry setup is measured by w T n (n, λ) = n ( P T (X e (n, λ)) C x (x e (n, λ)) ) x e n(n, λ) + π e (n, λ) This equation is just the same as? once derived for the private free-entry quasi-cournot oligopoly market. Thus there holds the following proposition. Proposition 6 When each local public firm s output is a strategic substitute to the others, then the free-entry equilibrium number of local public firms exceeds the second-best number of local public firms maximizing the aggregate welfare. 4.2 Privatization and Free-Entry Equilibrium We have thus shown that even when the firms are local-government owned, there is a tendency of socially excess entry to the market as in private oligopoly. Then the problem is whether privatization enhances the aggregate welfare. For this discussion, we have to explore the effect of privatization on each region, a producing region and a consuming one. In view of Proposition??, pri- prop:privatization vatization dereases the aggregate output and lowers the welfare of the consuming region. Thus the welfare curve of the consuming region in Figure?? should fig:free-entry shift downward from w C u to w C r by privatization in every producing region. The privatization effect on the producing region is ambiguous at a first glance, for privatization decreases the consumer surplus but increases the profit of the firms. However one can establish 5 Hereafter by λ we mean λ with λ i = λ for all i S. 13

14 Proposition 7 Privatization by every region enhances the welfare of the producing region. In Figure?? fig:privatization, the welfare curve of the producing region is thus shown to have shifted upward from wu S to wr S as a result of privatization. The resulting equilibrium is shown by point E r with the equilibrium number of firms n e r, which is greater than n e u. w C u E u E r w C r w S r w S u n e u n e r n Figure 3: Privatization and Free-Entry Equilibrium fig:privatization As shown in Figure?? fig:privatization, it is straightforward to estblish Proposition 8 The free-entry equilibrium number of private firms is greater than the free-entry equilibrium number of local public firms. Now what is the resulting effect of privatization on each region s welfare? When we focus our attention on the consuming region s welfare at the equilibrium, it is higher as the total output increases. Thus it suffices to see whether privatization increases the total output at the free entry equilibrium. Our intuition suggests that the private oligopoly equilibrium entails the smaller total 14

15 output than the local-public oligopoly one. After some tedious calculation, one can confirm this result. Proposition 9 The total output is greater and the individual region s welfare is higher at the local-public firm free-entry equilibrium than at the private firm free-entry equilibrium. 5 Concluding Remarks In market economies, one of the authors proved that there tends to be excessive entry into a Cournot oligopoly. The present paper generalizes this result into a competition among local government firms. The result implies that the central government should restrict and coordinate the industry setups by local governments. We can extend the result to the international setting in which each country deliberately chooses to set up the own indigenous industries in perfect competition using taxes and subsidies. Our result again implies excessive industry setups by too many countries. Thus from the global point of view, again, it is necessary to restrict such industry protection policies. We can extend the analysis in the present paper in several directions. First, one may References Cao, Y., Y Qian and B. R. Weingast (1999), From Federalism, Chinese style to privatization, Chinese style, Economics of Transition, 7, pp Mankiw, N. G., and M. D. Whinston (1986), Free Entry and Social Inefficinecy, Rand Journal of Economics, 17, pp Ohkawa, T., and M. Okamura (2003), The Uniqueness of the Welfare- Maximizing Number of Firms under Cournot Oligopoly, Bulletin of Economic Research, 55, pp Okuno-Fujiwara, M., and K. Suzumura (1993), Symmetric Cournot Oligopoly and Economic Welfare: A Synthesis, Economic Theory, 3, pp Suzumura, K., and K. Kiyono (1987), Entry Barriers and Economic Welfare, Review of Economic Studies, 54, pp Qian, Y., and B. R. Weingast (1997), Federalism as a Commtiment to Preserving Markete Incentives, Journal of Economic Perspectives, 11, pp

16 To prove the proposition, we must first derive the relationship between the equilibrium number of firms and the privatization parameters. As in the text, we assume that the privatization parameters are the same for all the regions, i.e., λ i = λ for all i S. Then the equilibrium number of firms given the privatization parameter λ, denoted by n e (λ), should satisfy π e (n e (λ), λ) = 0. Proposition AA states that the equilibrium individual profit is strictly decreasing in the number of firms (i.e., π e n(n, λ) < 0) and increasing in the privatization parameter (i.e., πλ e (n, λ) > 0), we may apply the implicit function theorem and prove in a straightforward fashion n e λ (λ) > 0. For exploring the effect of further privatization on the total output, we have to use the following relations. Effect of more firms on the profit πn(n, e λ) = (p C x ) xe n + xp X e X n x e = ((1 λ)δ x) P X which further implies ( (1 λ)x = X N n n + xp X ) x e P X X e n n + X n P X πn(n, e ( λ) 1 λ = XP X N 1 ) x e n n + 1 X e n n X e n, ( (?? )) eq:vi-foc Since each firm s output is a strategic substitute to the others, there holds x e n Xe < 0, while we have n = xe +n xe n > 0 by virtue of the stability condition. Thus we may rewrite the above equation as below. πn(n, e ( λ) 1 λ = XP X N 1 ) x e n n + 1 X e n n ( 1 λ = N 1 ) X e γ X n n + 1 X e n n 16

17 π e n(n, λ) XP X X e n ( 1 λ = N 1 n ) γ X + 1 n = 1 n 2 N (N n(1 λ)) nγ X + 1 n 1 n 2 N (N n(1 λ)) + 1 n = 1 λ nn 1 n n = 1 λ nn + n 1 n 2 > 0, which establishes the desired result. We note also the following relation. π e n(n, λ) XP X X e n ( 1 λ = N 1 n Effect of further privatization on the profit which establishes πλ e (n, λ) x P e X λ X = 1 λ N 1 n + 1 ) γ X + 1 n = 1 λ N + n 1 > 0, n π e λ(n, λ) > 0. Thus we have also established n e λ(λ) = 1 λ N ( 1 λ + n 1 n N 1 n) γx + 1 n γλ x Effect of more firms on the total output output is measured by Now the effect on the total dx e dλ = Xe λ + Xe n ne λ(λ) x = n e λ + nγ λ 1 nγ X 1 nγ X = 17