A Note on Topological Properties of Outcomes in a Monopoly Market

A monopolist with a nonnegative constant marginal cost faces an arbitrary nondecreasing and upper-semicontinuous demand function on R+ that takes a value in {0,1} outside of a fixed compact interval. This note derives topological properties of outcomes induced by this monopolist’s optimal pricing problem. Specifically, the monopolist’s optimal profit is continuous in both the marginal cost and the demand (under the weak-* topology); the induced output is lower (upper)-semicontinuous in both the marginal cost and the demand when the monopolist always charges the highest (lowest) optimal price; the optimal price correspondence is upper-hemicontinuous in both the marginal cost and the demand, which in turn implies that the consumer surplus is upper (lower)-semicontinuous in both the marginal cost and the demand when the monopolist always charges the lowest (highest) optimal price. These results further imply similar topological properties of outcomes in settings that feature either second-degree price discrimination or third-degree price discrimination