Essays on Monotone Comparative Statics for Constrained Optimization Problems with Applications

Abstract

My dissertation consists of three chapters. Chapters 1 and 2 both address extensions to existing monotone comparative statics results for constrained optimization problems using lattice programming techniques. Chapter 3 applies monotone comparative statics results to the question how environmental regulation affects investment in innovation in imperfectly competitive markets. Generally we can distinguish between two types of comparative statics problems. The first type of problem considers the change of the optimal solution to a maximization problem as the objective function changes, the other type considers the change due to a change in the constraint set. Lattice-based comparative statics theorems have been developed for both types of problems in the literature. The strengths of these lattice-theoretic comparative statics results are that they don't depend on the usual smoothness, interiority and concavity assumptions as required by the classical approach based on the Implicit Function Theorem, as well as convexity of the constraint set. Moreover, these comparative statics results also apply in the case of non-unique solutions.Quah (2007) expanded existing results by Milgrom and Shannon (1994) by making them applicable to some non-lattice constraint sets. In the first chapter, I extend existing comparative statics theorems to parametrized objective functions and non-lattice constraint sets. This generalization makes it possible to analyze a variety of economic optimization problems that fall into this class of problems which cannot be addressed using existing lattice-based techniques. I provide examples from consumer theory, producer theory and environmental economics that show the result's broad scope of applications. The second chapter studies monotone comparative statics with respect to price changes in the consumer's utility maximization problem. Most attempts to derive this property rely on aspects of the demand curve, and it has been hard to derive this property using assumptions on the primitive utility function. Using new results on the comparative statics of demand in Quah (2007), I provide simple and easy conditions on utility functions that yield the gross substitutes property. Quah (2007) provides conditions on utility functions that yield normal demand. I add an assumption on elasticity of marginal rate of substitution, which combined with Quah's assumptions yields gross substitutes. I apply this assumption to the family of constant elasticity of substitution preferences. My approach is grounded in the standard comparative statics decomposition of a change in demand due to a change in price into a substitution effect and an income effect. Quah's assumptions are helpful to sign the income effect. Combined with the elasticity assumption, we can sign the overall effect. As a by-product, I also present conditions which yield the gross complements property. Chapter 3 is an application of monotone comparative statics results to the question how environmental regulation affects incentives for R&D investment. For decades, there has been debate among economists whether environmental regulation hurts firms by restricting their choices or provides them with a comparative advantage through investment in innovation in more efficient technology. This chapter studies the effect of environmental regulation on firms' investment in R&D in imperfectly competitive output markets using a monotone comparative statics approach. I provide necessary and sufficient conditions on the profit function that guarantee nondecreasing R&D investment as regulation is tightened and find that a form of weak complementarity between environmental R&D investment and the policy variable plays a crucial role. Moreover, I provide properties of such profit functions through assumptions on demand, Cournot output and cost functions.