The development of optimal tax policy has been a major topic in public finance for centuries. Probably the most famous result in the theory of optimal taxation is due to the English economist Frank...

The development of optimal tax policy has been a major topic in public finance for centuries. Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualised the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss. Specifically, suppose there are n goods (x_i
with prices p_i) to be taxed with a sequence of ad valorem taxes whose rates are given by t_i
(i= 1, n). Therefore, total tax revenue is given by  Ramsey’s problem is for a fixed T to choose tax rates that will minimise total deadweight loss

a. Use the Lagrange multiplier method to show that the solution to Ramsey’s problem requires t_i= λ(1/e_S−1/e_D), where λ 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.