Distance functions and optimal taxation

Governments use taxes to pay for some of their expenditures. Setting aside the benefits of the expenditures, the taxes economic agents have to pay reduce their well being. One objective of the optimal taxation literature is to find tax systems that minimize the loss of well being, given the government's revenue requirement. Thus one way to frame the optimization problem is to have the government choose tax rates (or prices) to maximize individual utility given the revenue requirement. Given the prevalence of price-times-quantity expressions in budget constraints, taking derivatives with respect to tax rates or prices yields rules expressed in terms of quantities. One of Terence Gorman's many insights was that if the objective is to find rules about prices, reframe the problem so that the government chooses quantities; derivatives of p times q with respect to q yield rules about p or tax rates. Below I show that some optimal tax problems are simplifi ed by assuming the government chooses quantities to maximize revenue subject to a fixed level of individual utility. The distance function, which is de nied as the number by which one must scale the arguments of the utility function to yield a particular level of utility, plays a central role.