This dissertation combines my work on two topics: optimal regulation of screening monopolists (Price Distribution Regulation), and linear optimization models in economic theory (Sharpening Winkler's Extreme Point Theorem: Economic Applications, and Concavification Bounds and Mechanism Simplicity).

The first paper, Price Distribution Regulation, studies the optimal regulation of a second-degree price discriminating (Mussa-Rosen) monopolist, under the assumption that the regulator has access to complete data about transacted prices, but none about product quality, reflecting difficulties in defining and enforcing quality standards. I characterize what such a regulator can achieve: despite the quality friction, she is quite powerful, and can induce the monopolist to use any monotone mechanism which leaves the lowest consumer type with 0 surplus -- notably, including the efficient mechanism. I additionally show that, since gains from regulation accrue primarily to high-type consumers, a regulator who puts more weight on low-type consumers will temper regulation. Finally, I show that, under the appropriate regularity condition, the monopolist has no incentive to "cheat" regulation by using a randomized mechanism.

The second paper, Sharpening Winkler's Extreme Point Theorem: Economic Applications, refines an important mathematical result due to Winkler (1988). Winkler gave a necessary condition for a point to be an extreme point of a (linearly) constrained set: namely, that it be a convex combination of at most m+1 extreme points of the unconstrained set, where m is the number of constraints. I give additional "complementary slackness"-style conditions which, when combined with Winkler's condition, characterize the extreme points of the constrained set. Applied to economic theory, my characterization enables an analyst to produce a set of potential optimal policies, each of which is optimal for some model primitives.

In the third paper, Concavification Bounds and Mechanism Simplicity, I again work in the linear functional optimization setting where extreme point techniques, and particularly Winkler's theorem, are useful. In this setting, Winkler's theorem gives an upper bound on the complexity (defined as the number of service classes) in an optimal mechanism; I ask the question: How much could a designer give up, in the worst case (over model primitives), by using a less complex mechanism? I compute this upper bound precisely in the capacity-constrained selling model studied by Bulow and Roberts; the key insight there is that the worst-case buyer valuation distribution is supported on exactly two prices.