Title: The Relative-Profit-Maximization Objective of Private Firms and Endogenous Timing in a Mixed Oligopoly

Transcription

1 Working Paper Series No (Econ) China Economics and Management Academy China Institute for Advanced Study Central University of Finance and Economics Title: The Relative-Profit-Maximization Objective of Private Firms and Endogenous Timing in a Mixed Oligopoly by Yuanzhu Lu

2 The Relative-Profit-Maximization Objective of Private Firms and Endogenous Timing in a Mixed Oligopoly Yuanzhu Lu China Economics and Management Academy, Central University of Finance and Economics Abstract This paper investigates whether the relative-profit-maximization objective of private firms affects endogenous timing in a mixed oligopoly in the linear demand case. Assuming firms have constant marginal costs and symmetric private firms are more efficient than the public firm, I find that such an objective does not affect endogenous timing at all compared with the absolute-profit-maximization case. When the equilibrium involves the public firm acting as a leader, social welfare increases compared with the level in the absolute-profit-maximization case. When the equilibrium involves the public firm acting as a follower, social welfare remains unchanged. Keywords: Mixed oligopoly; Endogenous timing; Stackelberg; Relative profits JEL Classification: L3, D43, H4 I am grateful to two anonymous referees, Subhashini Muthukrishnan, and participants at the fourth APEA (Asia-Pacific Economic Association) conference for their helpful comments on earlier versions of this paper. All errors remain mine. Yuanzhu Lu: China Economics and Management Academy, Central University of Finance and Economics, No 39 South College Road, CHINA, 0008; Tel: (860)688397, Fax: (860)688376, yuanzhulu@cufe.edu.cn.

3 . Introduction Mixed oligopolies are common in many countries. The oil industry, heavy manufacturing, telecommunications and tourism industry are very good examples of mixed oligopolies. There have been a lot of studies on mixed oligopolies including partial privatization, capacity choice, endogenous timing and so on. One strand of literature on mixed oligopoly (especially mixed duopoly) has focused on endogenous timing, since alternate order of moves often produces significantly different results and thus leads to different welfare levels. These papers adopt Hamilton and Slutsky (990) s extended game with observable delay. For example, Pal (998) analyzed endogenous order of moves in quantity choice in a mixed oligopoly consisting of a single public firm and n domestic private firms. Matsumura (003) considered endogenous roles of firms in a mixed duopoly market where a state-owned public firm and a foreign private firm compete. Lu (006) investigated endogenous timing in a mixed oligopoly with one public firm, n domestic private firms and m foreign private firms. All these papers find that in equilibrium a public firm never moves simultaneously with domestic private firms. Most of the papers on mixed oligopoly make standard assumptions about a firm s objective. They assume that a private firm is an absolute-profit maximizer while a public firm is a social welfare maximizer. However, according to managerial theory of the firm, it has long been proposed that private firms in reality do not act to maximize absolute See De Fraja and Delbono (990) and Nett (993) for general reviews of the mixed oligopoly models. For recent literature on mixed oligopoly (duopoly), see Sun, Zhang and Li (005), Lu and Poddar (006), Cantos-Sánchez and Moner-Colonques (006), Matsushima and Matsumura (006), Kato and Tomaru (007), etc. Jacques (004) and Lu (007) slightly correct Proposition 4. of Pal (998).

4 payoffs. 3 In this paper I assume that the public firm s objective is to maximize social welfare which is defined as the sum of consumer surplus and the firm s absolute profits, while a private firm s objective is to maximize relative profit which is defined as the difference between the firm s absolute profit and the average absolute profit of all the firms. Why do I consider relative profit as a private firm s objective? Agent theory suggests that there are benefits associated with evaluating agents on the basis of their relative performances when the agent s performances are affected by a common shock term. The use of relative performance evaluation has been empirically supported by Gibbons and Murphy (990) who test the presence of relative performance evaluation for CEOs using data on 608 chief executive officers (CEOs) from 049 corporations from 974 to 986. Their results strongly support the hypothesis that relative performance evaluation is used in compensation and retention decisions affecting CEOs. In recent years, maximizing relative profit instead of absolute profit has aroused the interest of economists from different fields. From an evolutionary perspective, Schaffer (989) demonstrates with a Darwinian model of economic natural selection, that if firms have market power, profit-maximizers are not necessarily the best survivors. According to Schaffer (989), a unilateral deviation from Cournot equilibrium decreases the profit of the deviator, but decreases the other firm s profit even more. In other words, on the condition of being better than other competitors, firms that deviate from Cournot equilibrium achieve higher payoffs than the payoffs they receive under Cournot equilibrium. In Vega-Redondo (997), it is further argued that, under a general 3 Kaneda and Mastui (003) provided analytical review of the literature.

5 equilibrium framework, if firms maximize relative profit, a Walrasian equilibrium can be induced. On the other hand, Lundgren (996) shows that by making managerial compensation depend on relative profits rather than absolute profits, the incentives for oligopoly collusion can be eliminated. Kockesen, et al. (000) have shown that under some conditions a firm which strives to maximize relative profit will outperform a firm which maximizes absolute profit. Bolton and Ockenfels (000) and Fehr and Schmidt (999, 006) conducted an analysis considering an individual utility function that brings about a feeling of compassion towards an individual with a relatively lower material payoff and simultaneously brings about envy of other individuals with a higher material payoff. Morgan, et al. (003) studied auctions where bidders have independent private values but attach a disutility to the surplus of rivals. The purpose of this study is to find answers to the following questions: Does the relative-profit-maximization objective of private firms affect endogenous timing in a mixed oligopoly? Does social welfare increase compared to the absolute-profit-maximization case? Using a constant marginal cost function, I find that the objective does not affect endogenous timing at all. The equilibrium configuration is exactly the same as that characterized in Pal (998), Jacques (00) and Lu (007). When the equilibrium involves the public firm acting as a leader, social welfare increases compared with the level in the absolute-profit-maximization case. When the equilibrium involves the public firm acting as a follower, social welfare remains unchanged. The organization of the paper is as follows. In Section, I describe the model. 3

6 Section 3 presents three fixed timing games. The SPNEs in the observable delay game are presented in Section 4. Section 5 closes the paper.. The Model A mixed oligopoly market is considered with one public firm, called firm 0, and N private firms, all producing a single homogeneous good. The market price is determined by the inverse demand function p= a Q, where p is market price, Q q q N = 0 + j= j is total output and i q denotes the output of firm i ( = 0,,..., N ). Assume that a is sufficiently large. All private firms marginal costs are constant and identical, normalized to 0. The public firm is assumed to be less efficient than private firms and has a constant marginal cost c> 0. 4 Fixed costs are assumed to be zero for all firms. Each private firm i ( = 0,,..., N ) is a relative-profit-maximizer, and firm 0 is a public firm maximizing social welfare which is defined as the sum of consumer surplus and all firms profits. So a private firm i s objective function is π = π = ( a Q) q (( a Q) Q cq0), () + N+ N r i i π j i N j= 0 and the public firm s objective function is W = p( x) dx cq = aq Q cq Q 0 0. () 0 I consider the observable delay game of Hamilton and Slutsky (990) in the context 4 If the public firm is more or equally efficient than the private firms, it would produce a quantity such that the market price equals its marginal cost, resulting in a public monopoly. 4

7 of a quantity setting mixed oligopoly where the firms choose the timing for choosing their quantities. There are M periods for quantity choice and each firm cannot produce in more than one period. Specifically, I consider a two-stage game. In stage one, firms announce in which period they will choose their quantities and are committed to this choice. In stage two, after the announcement, firms choose their quantities knowing when the other firms will make their quantity choice and the market then clears after each firm has produced its output. My objective is to solve for the subgame perfect Nash equilibria (SPNE) of this extended quantity setting mixed oligopoly game. 3. Fixed Timing Games I investigate one Cournot and two Stackelberg models of fixed timing. The two Stackelberg models are called public follower model and public leader model. The equilibria of the observable delay game are determined in the next section. Without loss of generality, I assume that there are only two periods for quantity choice in this section. 3. Cournot Model In this model, all firms produce output simultaneously. The first-order condition for a private firm i s relative-profit-maximization problem is qi+ a Q ( a Q) = 0, i=,..., N, N+ (3) which implies that the reaction function of firm i is a N Ri q j = qi = q j. j i N j i (4) And the first-order condition for a public firm s social welfare maximization problem is 5

8 which implies that the reaction function of firm 0 is a Q= c, (5) N N R0 q j = q0 = a c q j. j= j= (6) The above equation says that the public firm will produce output until the market price is equal to its marginal cost. So in both the Cournot model and the public follower model, the market price is equal to c ; thus the public firm earns zero profit and a private firm r i s relative profit is then π π ( Nπ ) / ( N ) π / ( N ) cq / ( N ) = + = + = +, taking into i i i i i account that the symmetric private firms produce the same quantity level in the symmetric equilibrium. Solving for the equilibrium output of the Cournot model yields ( ) ( ) a N + c C C a+ N c q0 =, qi =, i=,..., N, N+ N+ (7) where the superscript C denotes the equilibrium outcome of the Cournot game. Each firm s payoff is then r ( π i ) C + ( ) ( N+ ) a N c = c, i=,..., N, (8) ( ) a N + c C W = ( a c ) c. N+ (9) 3. Public Follower Model In this model, all private firms produce simultaneously in period and the public firm produces in period. As pointed out before, the public firm will produce until 6

9 p = c in period. In period, private firms produce output taking into account the public firm s reaction function (6). Clearly, private firms will produce as much as possible subject to q0 0. Thus, F F a c q0 = 0, qi =, i=,..., N, (0) N r ( π i ) F ( c) ( ), i,..., N +, c a = = N N () F W = ( a c ), () where the superscript F denotes the equilibrium outcome of the public follower game. So social welfare in the public follower model is higher than in the Cournot model: W F C > W. This is because the market price is the same in both models and the less efficient public firm produces nothing in the public follower model while it produces a positive amount in the Cournot model. Therefore, the public firm and all private firms producing simultaneously in period cannot be sustained as an equilibrium. 3.3 Public Leader Model In this model, the public firm produces in period and all private firms produce simultaneously in period. In period, each private firm i s reaction function is given by (4). It is straightforward to obtain each private firm i s quantity in period as a function of q 0 : q i = ( ) 0 Na N q N +. (3) In period, the public firm produces output taking into account each private firm i s reaction function (3). Its objective function can then be written as 7

10 W = ( ( ) ) ( ( ) ) a N a N q0 N a N q 0 cq0. N It is straightforward to obtain ( N ) ( N )( N + ) L a N + L a 0 i q = c, q = + c, i=,..., N, N+ N+ N+ N+ r ( π i ) L c = a+ ( N+ ) ( ) ( N )( N + ) N+ c, (4) (5) + L N ac W = a + c, N+ N+ (6) where the superscript L denotes the equilibrium outcome of the public leader game. Clearly, the social welfare in this model is never lower than in the Cournot model C since the public firm can always choose its quantity equal to q 0. Since it chooses a L C different amount q0 q0 when N, L W must be higher than C W : W L C > W. L C However, when N =, q0 = q0 = a / c and thus W L N, C = W. Therefore, when the public firm and all private firms producing simultaneously in period cannot be sustained as an equilibrium. It is also clear that W F L > W since in the public follower model, the price is equal to the public firm s marginal cost and the less efficient public firm produces nothing, which maximizes social welfare. 4. Equilibria in the Observable Delay Game I now determine the equilibrium in the observable delay game. Proposition 8

11 summarizes the results of welfare comparisons among the three models of fixed timing. Proposition : When N, F L C W > W > W ; when N =, F L C W > W = W. This result tells us that the public firm acting as a follower of all private firms is socially optimal. To maximize social welfare, the public firm should make credible commitment on the role of a follower. As pointed out before, () the public firm and all private firms producing simultaneously in period cannot be sustained as an equilibrium, and () when N, the public firm and all private firms producing simultaneously in period cannot be sustained as an equilibrium. To determine whether the public firm and the private firm ( N = ) producing simultaneously in period can be sustained as an equilibrium, I will check whether the private firm has an incentive to be a leader. Note that the private firm s relative profit as a leader is never lower than the level as a simultaneous-mover. Moreover, since the private firm chooses a different amount q F i q ( a c a / ), the C i private firm s relative profit as a leader must be strictly higher. Thus, Proposition : The public firm and all private firms producing simultaneously cannot be sustained as an equilibrium. This proposition holds true regardless of the number of possible periods for quantity choice ( M ). Next I will consider the public follower and public leader candidate equilibria. I first conisder the case when M =. Consider the public follower candidate equilibrium. Clearly, the public firm has no incentive to deviate. Does a private firm, say firm, have an incentive to deviate to produce in period? Suppose it deviated. Then in period, the public firm s reaction function (6) and firm s reaction function (4) imply q ( ) N a c N 0 N+ j= = q and j 9

12 ( ( ) ) q = a+ N c. Since the public firm will produce until p= c, each private firm N+ i=,..., N chooses to produce as large as possible such that q 0 = 0. Firm s relative profit is then ( ) r c a c N cq N+ N+ π = = c, which is lower than ( ) ( ) c a c N N + since a is sufficiently large. Therefore, firm has no incentive to deviate. Similarly, it can also be shown that a subgroup of private firms has no incentive to deviate. Suppose firms,,, m ( m N ) deviate to produce in period. Then in period i ( ( ) ) q = a+ N c, i=,..., m. The market price is still c, and q 0 = 0, thus each N+ private firm i s ( i=,..., m ) relative profit is = ( i=,..., m ). So the public r N π i N+ c follower candidate equilibrium is an equilibrium indeed. Consider the public leader candidate equilibrium. Clearly, the public firm has no incentive to deviate. Does a private firm, say firm, have an incentive to produce in period? Suppose it deviated. Then in period, each private firm i=,..., N will produce ( )( ) q Na N q0+ q i N N + =. In period, the social welfare and firm s relative profit can be written as ( N N) a+ ( N+ )( q0+ q) ( N N) a+ ( N+ )( q0+ q) W = a cq 0, N N+ N N+ π = a ( N N) a+ ( N+ )( q0+ q) r N N+ q ( N N) a+ ( N+ )( q0+ q) ( N N) a+ ( N+ )( q0+ q) a cq N+ N N+ N N+ It is straightforward to obtain 0. 0

13 q 0 a N q a N r a c N c N N N N N, c N N N N N, N N N N N N N 3 c. Comparing firm s relative profit when deviating and the relative profit when producing in period yields a ( N ) ( N ) = ( )( + )( + ) c N N N N N N c a+ ( N+ ) c ( N ) ( N )( N + ) N N 3N + 4N 5N + N, ( N+ ) which is negative when N and positive when N >. So when N, a private firm has no incentive to deviate from the public leader candidate equilibrium. It can also be shown that the two private firms (when N = ) have no incentive to deviate as a group. 3 c Therefore the candidate equilibrium is an equilibrium indeed. Proposition 3: Suppose there are only two possible periods for quantity choice ( M = ). If N >, there is a unique SPNE: public follower. At this equilibrium, the market price equals the marginal cost of the public firm and the public firm produces nothing. If N, then there is a second SPNE: public leader. When there are more than two possible periods for quantity choice ( M > ), the public firm has an incentive to be a follower. So any public leader equilibrium must involve all private firms producing in the last period. However, if there are at least two c

14 private firms, a private firm has an incentive to produce before the other private firms. 5 So the public leader equilibrium arises if and only if N = ; moreover, the public leader equilibrium must involve the public firm producing in the first period since otherwise the private firm has an incentive to produce as a leader. As for the public follower equilibrium, again since each private firm has an incentive to produce before the other private firms, such an equilibrium must involve all private firms producing in the first period if N and the public firm producing in a subsequent period while if N =, it involves the private firm producing in any period except the last one and the public firm producing in a subsequent period. Proposition 4: Suppose there are more than two possible periods for quantity choice ( M > ). If N, there is a unique SPNE, in which all private firms produce in the fist period and the public firm produces in a subsequent period. If N =, then there are two types of equilibria: public follower equilibrium in which the private firm produces in any period except the last one and the public firm produces in a subsequent period, and public leader equilibrium in which the public firm produces in the first period and the private firm produces in the last period. 4. Comparison between the Relative-Profit-Maximization Case and the Absolute-Profit-Maximization Case Now I compare the endogenous order of moves between the relative-profit-maximization case and the absolute-profit-maximization case. Pal (998), 5 According to our analysis above, this is clear when N. When N, it is straightforward to show that a private firm has an incentive to be a leader of the other private firm (still a follower of the public firm).

15 Jacques (004) and Lu (007) investigate the absolute-profit-maximization case. It is clear that the endogenous order of moves in these two cases is exactly the same. This is the main result of this paper. Proposition 5: In the linear demand and constant marginal cost case, endogenous timing in a mixed oligopoly is the same regardless of the relative-profit-maximization or absolute-profit-maximization objective of private firms. The social welfare level when the order of moves is endogenized can also be compared between the two cases. It is clear that in the public follower equilibrium, the social welfare level is exactly the same since in each case the total amount of the quantity is a c and the public firm produces nothing. In the public leader equilibrium, when + private firms are relative-profit maximizers, ( N ) W = a + c ; when private L N ac + N+ firms are absolute-profit maximizers, it is straightforward to obtain ( ) L W a N c ac = + +. Since a is sufficiently large, the social welfare level is higher in the relative-profit-maximization case. To understand why, note that when the public firm acts as a leader, the first-order condition for a private firm i s absolute-profit maximization problem is qi+ a Q= which implies that the reaction function of firm i is 0, a Ri q j = qi = q j. j i j i Comparing this function with (4) tells us that a private firm will produce more output in the relative-payoff-maximization case. Taking this into account, a public firm as a leader produces less. Moreover, the total quantity is higher than in the 3

16 absolute-payoff-maximization case. Proposition 6: In the public follower equilibrium, private firms relative-profit-maximization objective has no effect on social welfare; however, in the public leader equilibrium, social welfare increases when the private firms are relative-profit maximizers. 5. Concluding Remarks In this paper I investigate the effect of private firms relative-payoff-maximization objective on endogenous timing in a mixed oligopoly in the linear demand and constant marginal cost case. I find that endogenous timing in a mixed oligopoly is the same regardless of the relative-profit-maximization or absolute-profit-maximization objective of private firms. I also find that in the public follower equilibrium, private firms relative-profit-maximization objective has no effect on social welfare; however, in the public leader equilibrium, social welfare increases when the private firms are relative-profit maximizers. This suggests that, it is socially desirable that private firms focus on relative profits instead of absolute profits. I believe that the main results of this paper will continue to hold for more general demand and cost functions. First, note that generally (except in very special cases) a private firm has an incentive to produce more output in the relative-profit-maximization case. When a general (inverse) concave and downward sloping demand function p( Q ) and cost function C0( q 0) for the public and convex cost function C( q ) for the private 4

17 firm are used, 6,7 a private firm i s relative profit is N N r π i = π i π j = p( Q) qi C( qi) p( Q) Q C0( q0) C( q j), N+ j= 0 N+ j= and the first order condition can be written as p Q qi p Q p Q q j C qi N ( ) + ( ) ( ) = ( ) It is then clear that once q 0, 8 a relative-profit-maximizing private firm i j i j > produces more than an absolute-profit-maximizing private firm. Second, given this fact, it can be expected that generally all firms producing simultaneously cannot be sustained as an equilibrium outcome. Suppose they produce in the first period. The public firm has an incentive to deviate to be a follower so that private firms produce more. Suppose they produce in a subsequent period. Either the public firm or a private firm has an incentive j i. to deviate to be a leader. Third, it can also be expected that when N is sufficiently large, the public leader configuration cannot be sustained as an equilibrium outcome since a private firm has an incentive to produce before other private firms. So private firms relative-payoff-maximization objective has not much, if any, effect on endogenous order of moves in a mixed oligopoly. Generally, social welfare increases in the relative-profit-maximization case compared with the absolute-profit-maximization case since private firms have incentives 6 The inverse demand function p( Q) satisfies p < 0, p 0and cost function C( q) satisfies C 0. 7 For the sake of simplicity, the symmetry of the private firms is again imposed. The public firm is again assumed to be less efficient to avoid a public monopoly, i.e., C ( q) C ( q) 0 >. j i q j > 8 As long as the public firm is not a follower or there are at least two private firms, 0. 5

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