12.11 The Ramsey formula for optimal taxation The development of optimal tax policy has been a major topic in public finance for centuries.17 Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualized the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss.18 Specifically, suppose there are n goods (xi with prices pi) to be taxed with a sequence of ad valorem taxes (see Problem 12.10) whose rates are given by ti (i ¼ 1, n). Therefore, total tax revenue is given by T ¼ Pn i¼1 tipixi. Ramsey’s problem is for a fixed T to choose tax rates that will minimize total deadweight loss DW ¼ Pn i¼1 DWðtiÞ.

a. Use the Lagrange multiplier method to show that the solution to Ramsey’s problem requires ti ¼ l(1/eS & 1/eD), where l is the Lagrange multiplier for the tax constraint.

b. Interpret the Ramsey result intuitively.

c. Describe some shortcomings of the Ramsey approach to optimal taxation.