Pollution and Pigouvian tax.

Suppose that, in the basic one-period model, there is no government spending and no taxes. Production by the representative firm produces pollution (χ) in proportion to the amount of output produced (Y ). Assume χ = a·Y with a > 0. Given any consumption bundle (a consumption-leisure pair), the consumer is worse off the more pollution there is. Assume the utility function is U = U(C −χ,l).

(a) In a diagram, show the competitive equilibrium and the Pareto optimum. Show that the competitive equilibrium is not Pareto optimal, and explain why. Is more or less output produced in the competitive equilibrium than at the Pareto optimum?

(b) Now, suppose that the government imposes a proportional tax t on the output of the firm, and rebates the proceeds of the tax in a lump-sum fashion, as a transfer TR to the representative consumer. Show that the tax can be set in such a way that the competitive equilibrium is Pareto optimal.