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 costless quality choice, 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. Log-concavity of the income distribution is inconsistent with the existence of equilibrium.