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This paper explores the equilibrium market outcomes in the contexts of both quantity-setting and price-setting private duopolies with the consistent conjectures of two private firms, wherein they maximize the weighted sum of their own profits and their respective opponent firm’s profit. Similar to the private duopoly without network effects wherein the two private firms maximize their genuine relative profits, in the private duopoly with network effects such that both firms maximize the weighted sum of their own profits and their respective opponent firm’s profit, we show that the equilibrium outcomes in the quantity-setting competition with the consistent conjectures of both firms are equivalent to those in the price-setting competition with the consistent conjectures of both firms.

This paper tackles the problem of whether or not the consistent conjectures of two relative profit maximizing private firms yield the same equilibrium outcomes between a quantity-setting competition and a price-setting competition in the context of a private duopolistic market with differentiated and substitutable goods and with network effects. Conjectural variations in oligopolistic markets have been investigated for a long time. For example, Bresnahan [^{1}. More recently, in private duopoly with the linear demand function and constant marginal cost functions composed of two symmetric private firms, Tanaka [^{2}.

As indicated in Matsumura et al. [

In this paper, we consider the equilibrium market outcomes between the quantity-setting competition and the price-setting competition in a private duopoly with network effects by adopting the maximization of the weighted sum of their own profit and the profit of their respective opponent firm including the case of their genuine relative profit maximization (the “extended” relative profit). The network effects that we consider in this paper were introduced in Katz and Shapiro [

Except for the question of whether or not there exists the presence of network effects à la Katz and Shapiro [^{3}. In this paper, we show that even if we take into account both the network effects and the possibility of the weighted sum of each firm’s profit and its opponent firm’s profit, the equilibrium market outcomes in the quantity-setting competition are equivalent to those in the price-setting competition. Thus, the equivalence of Cournot and Bertrand equilibria in the private duopoly with differentiated and substitutable goods still holds against the introduction of network effects à la Katz and Shapiro [

The remainder of this paper is organized as follows: in Section 2, we formulate the basis model employed in this paper. In Section 3, we derive the equilibrium outcomes in both the quantity-setting competition and price-setting competition with differentiated and substitutable goods in the private duopoly with network effects à la Katz and Shapiro [

We formulate a private duopolistic model with differentiated and substitutable goods and consistent conjectures composed of two extended relative profit-maximizing private firms with an additional term that reflects the network effects introduced in Katz and Shapiro [

where and are demand parameters^{4}. indicates the strength of network effects, and is consumers’ expectations of firm’s equilibrium market share. The ordinary demand function for the good of firm is obtained from the inverse demand function given in Equation (1) as follows:

As explained in Hoernig [

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

^{5}.

We consider a private duopolistic market composed of two extended relative profit maximizing private firms (firms 0 and 1). We use and to represent firm’s output and price levels, respectively,. We adopt the constant marginal cost function, where is a common marginal cost between firms 0 and 1, similar to Hoernig [^{ 6}. The marginal cost of production of both firms 0 and 1 is commonly assumed to be. The profit function of firm is given by

where is given in Equation (1) and is given in Equation (2). Consumer surplus is expressed as the representative consumer’s utility 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 is defined as the sum of consumer surplus and producer surplus. We consider the “rational expectations” subgame perfect Nash equilibrium by imposing the rational expectations condition that and à la Katz and Shapiro [

In this section, we derive the equilibrium market outcomes with firms 0 and 1 in the contexts of both the quantity-setting competition and price-setting competition with their consistent conjectures in the private duopoly with differentiated and substitutabled goods wherein they maximize the extended relative profit.

In this subsection, we consider the situation wherein the strategic variables of firms 0 and 1 are their output levels. The objective functions of firms 0 and 1 are given as follows:

where ^{7}.

Firm 0 decides its output level in order to maximize assuming that the reaction of the output level of firm 1 to the output level of firm 0 is given as follows:

On the other hand, firm 1 decides its output level in order to maximize assuming that the reaction of the output level of firm 0 to the output level of firm 1 is given as follows:

The first-order conditions of firms 0 and 1 in the quantity-setting market competition are given, and their real reaction functions of firms are obtained as follows^{8}:^{}

From the real reaction function of the output level of firm to the output level of firm, we obtain the following result:

The conditions of the consistency of the conjectural variations of firms 0 and 1 are, respectively,

yielding

From the symmetry of firms 0 and 1, we notice that. The above values of firms 0 and 1 are the equilibrium consistent conjectures in the quantity-setting competition under the assumption that and. Thus, by substituting the rational expectations assumption that and, the equilibrium output levels and price levels of firms 0 and 1 under the assumption that and are obtained as follows:

and

In this subsection, we consider the situation wherein the strategic variables of firms 0 and 1 are their price levels. The objective functions of firms 0 and 1 are given as follows:

Firm 0 decides its price level in order to maximize assuming that the reaction of the price level of firm 1 to the price level of firm 0 is given as follows:

On the other hand, firm 1 decides its price level in order to maximize assuming that the reaction of the price level of firm 0 to the price level of firm 1 is given as follows:

The first-order conditions of firms 0 and 1 in the price-setting competition are given, and the real reaction functions of firms are obtained as follows^{9}:

From the real reaction of the price level of firm to the price level of firm, we obtain the following result:

The conditions of the consistency of the conjectural variations of firms 0 and 1 are, respectively,

yielding

The above values of firms 0 and 1 are the equilibrium consistent conjectures in the price-setting competition under the assumption that and . Note that each firm’s consistent conjectural variation in the price-setting competition is different from that in the quantity-setting competition^{10}. Thus, by substituting the rational expectations assumption that and, the equilibrium price level and output level under the assumption that and are obtained as follows:

and

Thus, we have the result that and ,. Summing up the rational expectations equilibrium market outcomes with consistent conjectures including the output and price levels of firms 0 and 1 between the quantity-setting competition and pricesetting competition, we obtain the following proposition:

Proposition 1 In the private duopoly with consistent conjectural variations composed of the two extended relative profit maximizing private firms, the rational expectations equilibrium outcomes including their output and price levels, profit, consumer surplus, and social welfare in the quantity-setting competition are equivalent to those in the price-setting competition.

Note that the statement of Proposition 1 is relevant to the private duopoly composed of extended relative profitmaximizing private firms that is without network effects à la Katz and Shapiro [

In this paper, we considered the equilibrium market outcomes in a private duopoly with differentiated and substitutable goods and with an additional term that reflects network effects in the fashion of Katz and Shapiro [

Finally, we identify several topics to be addressed in our future research. In a symmetric private duopoly with differentiated and substitutable goods wherein two private firms maximize their genuine relative profits, Tanaka [