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.