(a) Suppose that the demand for packs of cigarettes is given by D(P) = 180 40P where P is the price of a pack of cigarettes. The supply of cigarettes is given by S(P) = 20 + 10P. Solve for the equilibrium price and quantity of packs of cigarettes. (b) Now suppose that the government adds a tax of $t per pack of cigarettes to be paid by consumers. What is the new equilibrium quantity of packs of cigarettes? What is the after-tax price paid by consumers? What is the price received by sellers? Does the proportion of the tax burden payed by consumers depend on the size of the tax? (Your answers may have t in them.) (c) Find an expression for tax revenue as a function of t. Graph tax revenue as a function of t. If the government wanted to maximize tax revenues, what tax would they choose? (d) Find an expression for deadweight loss as a function of t. Graph deadweight loss as a function of t. (e) Suppose the government cares about getting more tax revenue but also cares about keeping the deadweight loss from the tax relatively small. Consider the following objective function for the government: max t f(, T R, DW L) (2) where T R is tax revenue, DW L is deadweight loss and is a number between zero and one that captures how much weight the government places on generating tax revenue relative to creating deadweight loss. If equals one, the government cares only about tax revenue and the objective function should be reduced to simply being maxt T R. If equals zero, the government cares only about minimizing deadweight loss and the objective function should be reduced to simply being mint DW L (which is the same as maxt DW L). Come up with a function f(, T R, DW L) that satisfies these properties. Specifically, your function should do the following: f(, T R, DW L) should increase as T R increases. f(, T R, DW L) should decrease as DW L increases. f(, T R, DW L) should reduce to just a function of T R when equals one. f(, T R, DW L) should reduce to just a function of DW L when equals zero. (f) Use your function from the previous part to find the optimal tax t as a function of , the weight the government places on tax revenue relative to deadweight loss.