Consider two consumers with utility functions

Both consumers have income M and the (before-tax) price of both goods is 1.

a. Calculate the market equilibrium.

b. Calculate the social optimum for a utilitarian social welfare function.

c. Show that the optimum can be sustained by a tax placed on good 1 (so the after-tax price becomes 1 + t) with the revenue returned equally to the consumers in a lump-sum manner.

d. Assume now that preferences are given by

Calculate the taxes necessary to decentralize the optimum.

e. For preferences of part d and income M = 20, contrast the outcome when taxes can and cannot be differentiated between consumers.