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This paper studies capacity choice in a quantity-setting mixed duopoly with differentiated goods, when the objective function of the private firm is its relative profit. In this paper, we show that the differences between the output levels and capacity levels between both the public firm and the private firm strictly depend on both the degrees of product differentiation and of importance of the private firm’s relative performance. More precisely, we find that the public firm chooses over-capacity when both the degrees of importance of the private firm’s relative performance and of product differentiation are sufficiently high whereas it chooses under-capacity otherwise, and further the private firm chooses under-capacity when the degree of importance of its relative performance is high as compared the degree of product differentiation whereas it chooses over-capacity otherwise.

This paper reconsiders the capacity choices of a public firm that is a welfare-maximizer and a private firm that is a relative-profit-maximizer in the context of a mixed duopoly. In particular, we focus on the effect of the degree of importance of the private firm’s relative performance on the differences of both the public firm’s and the private firm’s output and capacity levels.

Many researchers have been considering capacity choice in a private oligopoly composed of only private firms since the works of Dixit [^{1} In contrast, in the context of mixed oligopoly, researches on the capacity choices of both the public firm and the private firm are relatively recent. Nishimori and Ogawa [^{2} Tomaru et al. [^{3}

In this paper, in a quantity-setting market with differentiated goods, we assume that the objective function of the private firm is its relative profit, that is, , where is its profit and is the public firm’s profit, and.^{4} In particularly, we investigate the effect of both and the degree of product differentiation on the differences between the output and capacity levels of both firms. The one of reasons why we consider the relative profit as the objective of the private firm is that evaluations of managerial performances are frequently based on not the absolute performance of managers on but their relative performance.^{5} In addition, the theoretical rationale such that the value of is positive is composed of the following two facts: 1) The advantage of outperforming managers in the management job market; 2) The evolutionary stability and the spiteful behaviors of managers that is closely related to the objective functions based on their relative performance, which are indicated in the experimental works.^{6} Furthermore, in the real world mixed oligopolistic market in Japan, the idea of yardstick competition has been introduced as yardstick regulations in the industries including natural gas and rail in order for the authority to regulate the managerial efficiencies of all the public and private firms.^{7} Of course, under the yardstick competition, the private firm must pay attention to the profits of all the rival firms including the public firm. In such industries, it is likely that the objective function of the private firm is its relative profit. Additionally, in particular in developing countries, it is considered that the private firm envies the public firm which is overly protected by the corresponding authority or the government. Such a situation corresponds to the case such that the private firm has some sort of positive in its relative profit.^{8}

^{7}In Laffont and Tirole [

^{8}See also Matsumura and Matsushima [

The purpose of this paper is to ascertain the effect of the degree of importance of the private firm’s relative performance on the difference between the output and capacity levels of both the firms. In this paper, we first show that the public firm chooses over-capacity when both the degrees of importance of the private firm’s relative performance and of product differentiation are sufficiently high, whereas it chooses under-capacity otherwise. The intuition behind the result on the difference between the output level and capacity level of the public firm obtained except for the case wherein both the degrees are sufficiently high is similar to that presented in Ogawa [

In addition, in this paper, it is also shown that the difference between the output and capacity levels of the private firm strictly depends on both the degree of importance of the private firm’s relative performance and the degree of product differentiation. More precisely, when the degree of importance of the private firm’s relative performance is high relative to the degree of product differentiation, the private firm aggressively behaves in the market, implying under-capacity by the private firm. Otherwise, that is, when the degree of importance of the private firm’s relative performance is low relative to the degree of product differentiation, the private firm strategically chooses over-capacity in order to increase its payoff by enlarging its market share because its capacity level is negatively associated with the output level of the public firm. The main contribution of this paper is that even if the relation between the products of the public firm and the private firm is restricted to substitutability, the differences between the output levels and capacity levels of both the public firm and the private firm can change according to the degree of importance of the private firm’s relative performance. Therefore, whether both the public firm and the private firm select undercapacity or over-capacity depends on not only the degree of product differentiation, which is shown in the existing literature in this field, but also on the degree of importance of the private firm’s relative performance.

The remainder of this paper is organized as follows. In Section 2, we formulate a quantity-setting mixed duopolistic model with differentiated goods with the capacity choices of both the public firm and the private firm that will be investigated throughout this paper. In Section 3, we consider the difference between the output and capacity levels of both firms using the model built in Section 2. Section 4 concludes with several remarks. The market outcomes including each firm’s equilibrium profit, the equilibrium consumer surplus, and the equilibrium social welfare are relegated to the Appendix.

We formulate a quantity-setting mixed duopolistic model with choices of not only the output levels but also the capacity levels for both the public firm (firm 0) and the private firm (firm 1). The basic structure of the model follows a standard product differentiation model as in Singh and Vives [^{9} Note that and are demand parameters with denoting the degree of product differentiation.^{10} This inverse demand system can be described from a quasi-linear representative consumer utility function of the form

where represents the numeraire goods.

Let be firm’s profit and let be firm’s relative profit. Then, in particular, the payoff of firm 1 is given by whereas the payoff of firm 0 is given by, where is the total social surplus (the sum of consumer surplus and the profits of firms 0 and 1).^{11} Moreover, we assume that.^{12} The parameter indicates the degree of importance of the relative performance of the private firm, firm 1.

Both firms adopt identical technologies represented by the cost function, where is the capacity level of firm. Following Vives [

. This cost function implies that if each firm’s output level equals its capacity level, , the long-run average cost is minimized. The profit of firm is given by

. Consumer surplus as the representative consumer utility is represented as follows:

whereas producer surplus is given by the sum of the profits of both firms:.

We investigate the game with the following orders of each firm’s moves: in the first stage, firms 0 and 1 simultaneously set their capacity levels. In the second stage, after both firms observe their capacity levels, they engage in quantity competition with each other.

We solve the game by backward induction from the second stage to obtain a subgame perfect Nash equilibrium. In the second stage, firm 0 maximizes social welfare with respect to, whereas firm 1 maximizes its payoff with respect to. The first-order conditions of firms 0 and 1 are, respectively, given as

yielding

In the first stage, both the firms know that their decisions regarding capacity affect their decisions regarding output in the second stage. Given Equations (3) and (4), firms 0 and 1 set simultaneously and independently their capacity levels with respect to social welfare and firm 1’s relative profit, respectively. Thus, by solving the firstorder conditions of firms 0 and 1, we have

yielding

Note that superscript ^{*} is used to represent the equilibrium outcomes of firms 0 and 1. Thus, the output levels of both firms in the equilibrium are given as follows:

Hence, from easy calculations, we obtain the following results on the difference between the output and capacity levels of both firms:

From the above calculations, the differences between the output levels and capacity levels between both firms 0 and 1 strictly depend on the degree of product differentiation and the degree of importance of firm 1’s relative performance. By summing these facts, we obtain the next proposition on the difference between the output and capacity levels of firms 0 and 1.

Proposition 1. Public firm 0 chooses under-capacity, in the almost area of in the -plane on the degree of product differentiation and the degree of importance of firm 1’s relative performance whereas it chooses over-capacity, only when both and are sufficiently high. The difference between private firm 1’s output level and capacity level strictly also depends on both the degrees of product differentiation and of the importance of its relative performance. More precisely, private firm 1 selects under-capacity, , if is high relative to b, whereas it selects over-capacity, , otherwise. Moreover, the area in the -plane where private firm 1 selects over-capacity widens as the value of the degree of product differentiation b increases.

The intuition behind the result in Proposition 1 that public firm 0 chooses under-capacity, , except for the situation wherein both the degree of product differentiation b and the degree of firm 1’s relative performance are sufficiently high is similar to that presented in Ogawa [

However, only when both the degree of product differentiation and the degree of importance of firm 1’s relative performance are sufficiently high, the difference of the output level and capacity level of firm 0 becomes reverse, and it chooses over-capacity, which is described in

level of firm 1, since the output level of firm 1 is almost independent of the capacity level of firm 0. Consequently, when both the degree of product differentiation and the degree of importance of firm 1’s relative performance are sufficiently high, public firm 0 chooses over-capacity, which is different from that in Ogawa [

At the same time, Proposition 1 also gives that the private firm surprisingly chooses under-capacity when the degree of importance of its relative performance is high relative to the degree of product differentiation, which is described in ^{13} We consider the two effects of the degree of importance of the private firm’s relative performance and of the degree of product differentiation on its output level and its capacity level. First, as the degree of importance of the private firm’s relative performance increases, it aggressively behaves in the market.^{14}

Thus, as the degree of importance of the private firm’s relative performance increases, its output level tends to rise. Second, from Equation (3), the private firm enlarges its market share through a decrease in the output level of the public firm by increasing its capacity level. Depending on which of the above two effects is relatively strong, the difference between the private firm’s output level and capacity level is determined. When the degree of importance of the private firm’s relative performance is high relative to the degree of product differentiation, the effect of on the difference between its output level and its capacity level is consequently strong. Thus, the private firm strategically selects under-capacity. Otherwise, the effect of on the difference between the private firm’s output level and capacity level is comparatively strong. Thus, the private firm strategically chooses over-capacity in order to enlarge its market share through a decrease in the output level of the public firm by increasing its capacity level.

Finally, Proposition 1 provides the result that the area in the -plane wherein the private firm, firm 1, chooses over-capacity widens as the value of the degree of product differentiation increases. This is so because the effect of the degree of product differentiation on the difference between the private firm’s output level and capacity level is comparatively strong relative to the effect of the degree of importance of the private firm’s relative performance.

This paper investigated the difference between the output and capacity levels of both the public firm and the private firm in a quantity-setting mixed duopoly with differentiated goods, in particular when the objective function of the private firm is its relative profit. In the existing literature on the difference between the output and capacity levels of both the public firm and the private firm including Nishimori and Ogawa [

Finally, we mention an issue to be tackled in the future. In this paper, we considered the differences between the output and capacity levels of both the public firm and the private firm in quantity-setting mixed duopolistic market. However, recent research on mixed oligopoly has been investigating the endogenous choice of the strategic variables of both the public firm and the private firm, and among them, Matsumura and Ogawa [^{15} Future research must deal with the above problems.