Bounding the relative profitability of price discrimination

We derive bounds on the ratio of a monopolist's profit from third-degree price discrimination to that from uniform pricing. If the monopolist serves N independent markets, demand is continuous, and the cost function is superadditive, then the profit ratio is bounded by N. A linear-demand example is provided coming arbitrarily close to this bound. We provide examples showing the profit ratio can be unboundedly large when marginal cost is decreasing, demand is discontinuous, or fixed cost is positive. If the monopolist has access to certain demand-rationing strategies under uniform pricing, we can bound the profit ratio even for discontinuous demand functions and multiproduct cost functions.