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This paper analyzes a model in which both the owner of a social welfare-maximizing public firm and the owner of an absolute profit-maximizing private firm can hire biased managers for strategic reasons in a mixed duopoly in the contexts of both a price competition and a quantity competition. In this paper, in a mixed duopoly, we show that in the contexts of both a price competition and a quantity competition, the owners of both firms employ more aggressive managers. In particular, in the result obtained in the price competition, the attitude of the manager employed by the owner of the private firm reverses to that obtained in the case of classical strategic delegation works.

As shown in Kaplan et al. [^{1}. Following the empirical works of Kaplan et al. [

In the context of mixed oligopoly, Barros [^{2}. Departing from the classical approach introduced in Fershtman and Judd [

In this paper, in the mixed duopoly, we show that in both the price competition and quantity competition, the owners of both the public firm and private firm employ more aggressive managers than in the case of absolute profit-maximizers. In particular, the result obtained in the price competition is strikingly different from that obtained in Nakamura and Inoue [^{3}.

The remainder of this paper is organized as follows. In Section 2, we build the basic model employed in this paper. In Section 3, we attempt to derive the Nash equilibrium market outcomes in the context of the price-set- ting competition. In Section 4, we attempt to derive the Nash equilibrium market outcomes in the context of the quantity-setting competition. Section 5 concludes with several remarks.

We consider a mixed duopolistic market composed of one social welfare-maximizing public firm (firm 0) and one absolute profit-maximizing private firm, (firm 1). The basic structure of the model follows a standard pro- duct differentiation model as in Dixit [

where ^{4}. These inverse and ordinary demand functions result from the following representative consumer’s utility:

Following Englmaier and Reisinger [

Note that ^{5}. The marginal costs of production of firms 0 and 1 are commonly assumed to be^{6}. The consumer surplus is represented by using the representative consumer utility in Equation (1) as follows:

We investigate the game with the following orders: In the first stage, firm^{7}. In the second stage, the hired managers of firms 0 and 1 compete in the market by maximizing the objective functions of firms 0 and 1 with respect to

Similar to Englmaier and Reisinger [^{8}. In this paper, to simplify the exposition, we assume that the expectation of firm

^{4} indicates that the relation between the goods produced by firms 0 and 1 is substitutable. We omit the trivial case wherein

^{5}Firm

^{6}We assume that

^{7}Although one wonders whether or not firm

^{8}As indicated in Englmaier and Reisinger [

In this section, we consider the price-setting competition by using backward induction in order to derive the subgame perfect Nash equilibrium (SPNE), and thus we start to conduct the analysis of the second stage. In the second stage, the manager of firm

Given the values of both

From easy calculations we obtain the following results:

Thus, the optimal price level of firm ^{9}

Next, we consider the analysis of the first stage, the determination of

where the values of

^{9}As indicated in Englmaier and Reisinger [

^{10}The second-order conditions are satisfied, and hence there is a unique equilibrium in this model.

The owners of firms 0 and 1 maximize social welfare and their absolute profit with respect to ^{10}

yielding

Similar to Englmaier and Reisinger [

Summing the above result, we obtain the following result:

Proposition 1. In the game with a price-setting mixed duopoly, the owners of both public firm 0 and private firm 1 hire aggressive managers,

The intuition behind the results described in Proposition 1 is as follows: the owner of public firm 0 takes the consumer surplus into account, and thus s/he makes her/his manager behave aggressively in the market by decreasing the value of

Thus, since the strategic relation between

Thus, the result obtained in Propostioin 1 is similar to that obtained in the standard private duopoly explored in Englmaier and Reisinger [

In this section, we consider the quantity-setting competition by conducting a similar analysis to that of the price- setting competition. From the same processes presented in Section 3, we obtain the following Nash equilibrium quantities in the second stage:

Furthermore, from easy calculations, we obtain the following result:

Thus, the owner of firm i employs an aggressive manager as the value of k_{i} increases, as shown in Englmaier and Reisinger [

Furthermore, from easy calculations, we obtain the following result:

Furthermore, we have

Summing up all the above results, we have the following proposition.

Proposition 2. Similar to the price-setting competition, in the game with a quantity-setting mixed duopoly wherein firm 0 maximizes social welfare and firm 1 maximizes its absolute profit, respectively, the owners of both firms hire aggressive managers,

The intuition behind the results described in Proposition 2 is given as follows: we find that in the first stage, the strategic relation between

Similar to the price competition, in the quantity competition, the owner of public firm 0 employs a manager who becomes more aggressively in the market since that firm takes consumer surplus into account. On the other hand, if the owner of private firm 1 employs a manager with

The results obtained in Proposition 2 are similar to those obtained in the works on strategic managerial delegation in a quantity-setting mixed duopoly, which includes Nakamura and Inoue [

This paper explored the situation wherein it was possible for each firm’s owner to hire a biased manager in a mixed duopoly composed of one social welfare-maximizing public firm and one absolute profit-maximizing private firm in the contexts of both a price competition and a quantity competition.

In this paper, we showed that in the contexts of both a price competition and a quantity competition in a mixed duopoly, the owners of both the public firm and the private firm hire aggressive managers. In particular, in a price-setting mixed duopolistic market, the owner of the private firm can employ an aggressive manager, which is strikingly different from the existing works on strategic managerial delegation in mixed oligopoly, which includes Nakamura and Inoue [^{11}.

Finally, we mention the open problems to be tackled in our future research. First, taking into account that firms’ owners evaluate their managers not on the basis of their absolute profit, but on the basis of their relative profit, which is the weighted sum of their own absolute profit and the absolute profits of the opponent firms in the real world economy, we should address the situation wherein the owners of competing firms can hire biased managers with respect to the market size they face, and in which the objective functions of their managers are their relative profits. Second, in recent works on strategic managerial delegation, the results obtained in classical strategic delegation studies Fershtman and Judd [

We would like to thank three anonymous referees for their helpful comments and suggestions. We are grateful for the financial support of KAKENHI (25870113). All remaining errors are our own.