Welfare impact of policy in incomplete markets: Theory and computation

Chapter 1 extends the theory of demand to incomplete markets. It starts with smoothness of demand, Slutsky decompositions, and properties of Slutsky matrices. It defines Slutsky perturbations as perturbations of Slutsky matrices that arise from some symmetric perturbation of the Hessian of utility. Finally, it identifies Slutsky perturbations as the solutions to a linear system of equations with budget variables as coefficients.Chapters 2 and 3 examine the welfare impact of taxation and of financial innovation in incomplete markets. Taking tax policy or financial innovation policy as primitives, it studies the generic existence of Pareto improving policy parameters, their computation, and the size of Pareto improvement. Generic existence obtains if the price adjustment implied by the introduction of tax rates is sufficiently sensitive to the risk aversion of the economy, and if both incompleteness and policy parameters outnumber household heterogeneity. Several known and new tax policies pass this sensitivity test, so does a new financial innovation policy, all therefore supporting Pareto improvements. It is chapter 1's identification of Slutsky perturbations that verifies they pass this test.Chapter 4 illustrates Pareto improving taxation on current income and asset purchases. The Pareto improvement following taxes is small. This is bounded above by the improvement following the removal of all future uncertainty, also small.Chapter 5 synthesizes research on the transfer paradox. It reinterprets Samuelson's equivalence of the paradox with instability, as identifying the threshold, the minimum level of trade beyond which the transfer paradox appears. Although the equivalence is false in general, and later research focused on qualifying or debunking it, this reinterpretation generalizes while quantifying the later research.Chapter 6 documents two Mathematica programs for chapter 4's example, where utility is von Neumann-Morgenstern. In the simpler one the state index is a quadratic transformation of Cobb-Douglas in the more elaborate one, it is a HARA transformation of CES. To find Pareto improvements from the envelope theorem, the derivative of demand is needed. The former has a closed formula for demand, and computes its derivative symbolically with Mathematica the latter has not, and computes its derivative instead with chapter 1's Slutsky decompositions.