There are two goods in this economy, Cigarettes and "all other goods," sold at a per unit price of p_c and p_o= $1. Consider a representative consumer with Cobb-Douglas preferences u(x_c, x_o) = x_c^1/3x_o^2/3 where x_c is the quantity of cigarettes and x_o is the quantity of "all other goods" (assume both amounts can be any positive real number). This consumer has a $9 income. Suppose congress enacts a $0.75 quantity tax on cigarettes (a quantity tax is a per unit tax). Show the original and new budget constraint for a representative consumer.

(1) Compute the pre- and post-tax optimal bundle. Then decompose the change in cigarette consumption into income and substitution effects.

(2) What is the maximum amount the smoker would have been willing to pay the government not to impose the quantity tax? What is the name of this concept?

(3) What amount of income would have to be given to the smoker after the tax to restore his or her utility to the pre-tax level? What is the name of this concept?

(4) What is amount of taxes the government collects from the consumer? Show that the government could levy a lump sum tax on the consumer without changing the price of cigarettes, collect the same amount of revenue, and leave the consumer better off.