Price competition in a mixed duopoly with discontinuous marginal cost function

In this paper, we study price competition in a mixed duopoly with discontinuous marginal cost function. We obtain that the set of pure strategy Nash equilibrium is a range of prices. We show that when the market size is sufficiently big, the set of Nash equilibrium in the mixed duopoly is the same as the Bertrand competition with two firms. Hence, social welfare is also similar in this case. However, the upper bound of the set of pure strategy Nash equilibrium can be lower in the case of mixed duopoly than Bertrand competition with two firms, when the market size is not sufficiently big. Therefore, the nature of ownership is relevant in the case of smaller markets and not in the case of big markets. When the cost function of one of the firms is only discontinuous as specified in section 2 and the marginal cost of the other firm is constant, we get unique pure strategy Nash equilibrium in certain cases.