DWL with a Cobb-Douglas

Consider a consumer with an income given by I=10, who spends the income on two consumption goods, X and Y, and whose utility is given by: U(X,Y)=X*Y. The prices of each unit of X and Y are given respectively by Px=Py=1.

a) Solve for the optimal bundle under the benchmark conditions (with no taxes in place). (Hint: you should specify an optimal tangency condition, MRS=Px/Py; and solve a system of two equations, where the additional equation is given by the budget constraint).

b) Suppose that the government levies a pre-unit tax on Y given by Ty=1, so the after-tax price of Y is now given by Py=2). Solve for the new optimal bundle and calculate the tax revenues raised by the government (AT). Calculate the utility level derived by the consumer (U1).

c) Suppose that the government replaces the per-unit tax levied on Y by a lump-sum tax. Denote the lump-tax by T. Solve for the optimal bundle under the lump-sum regime (as a function of T). Calculate the tax revenues and the utility derived by the consumer (as a function of T).

d) What should be the value of T that would yield the same level of utility as derived under the per-unit tax regime? (T=PT).

e) What is the excess burden of the per-unit tax regime? (EB=PT-AT)

f) Provide an illustrative figure (you should draw three budget curves associated with the benchmark-, the per-unit tax- and the lump-sum regimes. Stick to the notation used in the notes:E, E* and E^).

Remark: Intermediate Public Economics, Jean Hindricds and Gareth Myles, MIT Press, 2006 (Main Textbook)