_{1}

^{*}

This paper explores the capacity choice for a public firm that is a welfare-maximizer and for a private firm that is a pure-profit-maximizer in the context of a price-setting mixed duopoly with a simple mechanism of network effects where the surplus that a firm’s client gets increases with the number of other clients of that firm. In this paper, we show that the public firm chooses over-capacity irrespective of the strength of network effects and the demand parameter, and that the difference between the output level and capacity level of the private firm strictly depends on the values of both the strength of network effects and the demand parameter. More precisely, the private firm chooses over-capacity when the strength of network effects is high relative to the demand parameter, while it chooses under-capacity otherwise.

This paper investigates the capacity choice issue for a public firm that is a welfare-maximizer and for a private firm which is a pure-profit-maximizer in the context of a price-setting mixed duopoly with network effects where the surplus that a firm’s client gets increases along with the number of other clients of that firm^{1}. The purpose of this paper is particularly to show that the difference between the output level and the capacity level of the private firm strictly depends on both the strength of network effects and the demand parameter^{2}.

Nishimori and Ogawa [^{3}. Ogawa [^{4}. Subsequently, from the viewpoints of both a quantitysetting and a price-setting mixed duopoly with differentiated goods, Tomaru et al. [^{5}. Most recently, Nakamura and Saito [^{6}.

In the context of a price-setting mixed duopoly, this paper checks the robustness of the result on the differences between the output and capacity levels of both the public firm and the private firm against introducing the strength of the network effects such that the necessity to build market share is stronger. For tractability, we concentrate on the simple mechanism of network effects as studied in Katz and Shapiro [^{7}. The public firm has a strong incentive to increase mixed duopolistic market competition to increase of consumer surplus since its objective function is social welfare. Then, in the model of this paper, by taking into consideration the fact that the price level of the private firm is negatively associated with the capacity level of the public firm for any strength of network effects and demand parameter, the public firm makes the private firm behave aggressively in the market by increasing its capacity level. As a consequence, the public firm chooses overcapacity for arbitrary values of both the strength of network effects and the demand parameter.

In contrast, in this paper, we show that the difference between the output level and capacity level of the private firm strictly depends on both the strength of network effects and the demand parameter. In particular when the strength of network effects is high relative to the demand parameter, we obtain the result that the private firm chooses over-capacity, which is strikingly different from the result obtained in Bárcena-Ruiz and Garzón [

The remainder of this paper is organized as follows. In Section 2, we formulate a price-setting mixed duopolistic model with capacity choice of both the public firm and the private firm and with network effects. In Section 3, we consider the difference between the output and capacity levels of both firms. Section 4 concludes with several remarks. Each firm’s equilibrium price level is relegated to the Appendix^{8}.

We formulate a price-setting competition model in a mixed duopoly with the capacity choice of both the public firm and the private firm and with an additional term that reflects network effects in the fashion of Katz and Shapiro [^{9}.

We assume that firm 0 is a public firm that is a welfare-maximizer whereas firm 1 is a private firm that is a pure-profit-maximizer. Similar to Hoernig [

where and are demand parameters. indicates the strength of network effects, and is consumers’ expectation on firm i’s equilibrium market share. As explained in Hoernig [

where m denotes the income of the representative consumer and represents some symmetric function of expectations. In this paper, as in Hoernig [

^{10}.

We suppose that both firms adopt identical technologies represented by cost function, where is the capacity level of firm. Following Vives [^{11}. This cost function implies that if each firm’s output level equals its capacity level, , then the long-run average cost is minimized. The profit of firm i 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 0 and 1,. Finally, we suppose that social welfare in this paper is equal to the sum of consumer surplus and producer surplus.

We investigate the game with the following two stages: In the first stage, firms 0 and 1 simultaneously set their capacity levels. In the second stage, after both the firms observe each other’s capacity level, they engage in a price-setting competition. In the fashion of Hoernig [

We solve the game by backward induction from the second stage to obtain the “rational expectations” subgame perfect Nash equilibrium. In the second stage, firm 0 maximizes social welfare W with respect to, whereas firm 1 maximizes its pure profit with respect to. The best-response functions of both the firms in the second stage are given as follows:

(1)

From Equations (1) and (2), we find that for any strength of network effects, is increasing in, and thus the price levels of both firms 0 and 1 are strategic complements.

Furthermore, we obtain the rational expectations Nash equilibrium of the price-setting stage by substituting the two conditions and into the best-response functions of both firms 0 and 1. Then, we obtain

In the first stage, both firms 0 and 1 know that their capacity choices affect their price levels in the second stage. Given Equations (3) and (4), firms 0 and 1 simultaneously and independently set their capacity levels with respect to social welfare and own relative profit, respectively. Thus, by solving the first-order conditions of firms 0 and 1 in the first stage, we have

yielding

Note that superscript ^{*} is used to represent the subgame perfect equilibrium market outcomes with consumers’ rational expectations. Thus, the output levels of both the firms in the equilibrium are given as follows^{12}:

Hence, from easy calculations, we obtain the following results on the difference between the output and capacity levels of both firms 0 and 1:

Thus, we recognize that the public firm chooses overcapacity irrespective of the strength of network effects, n, and the demand parameter, b. In contrast, we find that the difference between the private firm’s output level and capacity level strictly depends on both n and b. By summing the above two facts, we obtain the next proposition on the differences between the output levels and capacity levels of both firms 0 and 1.

Proposition 1. Public firm 0 chooses over-capacity, , for any value of the demand parameter and any strength of network effects. In contrast, the difference between private firm 1’s output and capacity levels strictly depends on both n and b. More precisely, firm 1 chooses over-capacity if n is high relative to b, whereas it selects under-capacity, , otherwise. Moreover, the area in the -plane where in firm 1 selects under-capacity becomes wider as b increases.

Proposition 1 gives the result that irrespective of both n and b, public firm 0 chooses over-capacity, which corresponds to the case of substitutable goods presented in Bárcena-Ruiz and Garzón [

In contrast, in Proposition 1, it is stated that the difference between the output level and capacity level of firm 1 strictly depends on both the strength of network effects n and the demand parameter b. In ^{13} Thus, when the strength of network effects n is low relative to the demand parameter b, the capacity level of private firm 1 becomes comparatively low owing to a downward spiral induced by the relatively strong strategic substitutability between them, implying the result that firm 1 chooses under-capacity. In reverse, when the strength of network effects n is high relative to the demand parameter b, firm 1 is less likely to set a lower capacity level compared to the case wherein n is low relative to b. Consequently, we obtain the surprising result that private firm 1 can choose over-capacity when the strength of network effects n is high relative to the demand parameter b.

Finally, Proposition 1 provides that given the strength of network effects n, as the demand parameter b increases, firm 1 strategically tends to choose under-capacity. The intuition behind this result is given as follows. In general, firm 1 attempts to raise the price level of firm 1 by setting a relatively low capacity level, since it has a strong incentive to decrease the market competition. From Equation (3), on the basis of this incentive of firm 1, it chooses a lower capacity level when b is sufficiently high. Therefore, as b increases, it is likely that the area in the -plane wherein the difference between firm 1’s output and capacity levels is positive, becomes wider.

This paper explored capacity choice for a public firm that is a welfare-maximizer and a private firm that is a pureprofit-maximizer in the context of a price-setting mixed duopoly with network effects. With regard to network effects, we consider the situation wherein consumers’ expectation on each firm’s equilibrium market share is directly reflected in the demand for its product. More precisely, in the model of this paper, the degree of consumers' expectation on each firm’s equilibrium market share that is reflected in the demand for its product implies the strength of network effects. Then, we derived the subgame perfect Nash equilibrium with consumers’ rational expectations such that each firm’s output level is equal to its capacity level, which was introduced in Katz and Shapiro [

Although Barcena-Ruiz and Garzon [

Finally, similar to Nakamura and Saito [