Third-degree price discrimination versus uniform pricing

We compare the profit of optimal third-degree price discrimination policy against a uniform pricing policy. A offers same to all segments market. Our main result establishes that for broad class problems with concave functions (in space) and common support, is guaranteed achieve one half monopoly profits. This bound holds any number prices seller might use under discrimination. establish these conditions are tight weakening either support or concavity can lead arbitrarily poor comparisons even regular monotone hazard rate distributions.