Price equilibrium and willingness to pay in a vertically differentiated mixed duopoly

In the framework of a vertically differentiated mixed duopoly, with uncovered market and zero costs, we study the existence of a price equilibrium when a welfare-maximizing public firm producing low quality goods competes against a profit-maximizing private firm producing high quality goods. We show that a price equilibrium exists if the quality spectrum is wide enough vis à vis a measure of the convexity of the distribution of the consumers’ willingness to pay, and that such equilibrium is unique if this sufficient condition is tightened. Logconcavity of the income distribution is inconsistent with the existence of equilibrium.