Combining the endogenous choice of the timing of setting incentive parameters and the contents of strategic contracts in a managerial mixed duopoly

This study considers a game in which both the timing of setting the incentive parameters and the contents of the strategic contracts are determined endogenously. We investigate this game in a managerial mixed duopoly comprising a public firm with a welfare-maximizing owner and a private firm with a profit-maximizing owner. We suppose that the managers of both firms are employed based on their sales delegation contracts, which are equal to the weighted sum of their profits and sales revenues with respect to their FJSV incentive parameters. We show that two equilibrium market structures are possible in this game: (1) the market structure in which the owner of the public firm is the follower with a quantity contract and the owner of the private firm is the leader with a price contract and (2) the market structure in which the owners of both firms use price contracts and set their FJSV incentive parameters simultaneously in the late period. Furthermore, we show that the highest degree of social welfare can be achieved with the first equilibrium market structure. Therefore, the game considered in this study differs from the game in which both the timing of setting the incentive parameters and the contents of the strategic contracts are determined endogenously in a managerial private duopoly composed of two firms with profit-maximizing owners. Specifically, in the game considered in this study, it is unnecessary for the corresponding authority, such as the government, to regulate the free determination of both the timing of setting the incentive parameters and the contents of the strategic contracts based on each firm's individual incentive. In addition, since the cost functions and their quantities are represented under the technologies of both firms, we confirm the results concerning the equilibrium market structures and welfare implications obtained in the case where both firms have constant marginal cost functions with respect to their quantities.