In 2007, Americans smoked 19.2 billion packs of cigarettes. They paid an average retail price of $4.50 per pack.
a. Given that the elasticity of supply is 0.5 and the elasticity of demand is -0.4, derive linear demand and supply curves for cigarettes.
b. Cigarettes are subject to a federal tax, which was about 40 cents per pack in 2007.What does this tax do to the market-clearing price and quantity?
c. How much of the federal tax will consumers pay? What part will producers pay?
a. Let the demand curve be of the general linear form Q = a bP and the supply curve be Q = c dP, where a, b, c, and d are positive constants that we have to find from the information given above. To begin, recall the formula for the price elasticity of demand Elasticity of demand = (P/Q d )( ?Q d / ?P) We know the values of the elasticity, P, and Q, so we can solve for the slope, which is b in the above formula for the demand curve. -4 = (4.50/19.2)(-b) b= 0.4 (19.2/4.50) = 1.71 To find the constant a, substitute for Q, P, and b in the demand curve formula: 19.2 = a 1.71(4.50). Solving yields a = 26.9. The equation for demand is therefore Q = 26.9 1.71P. To find the supply curve, recall the formula for the elasticity of supply and follow the same method as above: Elasticity ofsupply = (P/Q s )( ?Q S / ?P) 0.5 =(4.50/19.2)(d) d = 0.5(19.2/4.50) = 2.13 To find the constant c, substitute for Q, P, and d in the supply formula, which yields 19.2 = c 2.13(4.50). Therefore, c = 9.62, and the equation for the supply curve is Q = 9.62 2.13P. b) The tax drives a wedge between supply and demand. At the new equilibrium, the price buyers pay, P b , will be 40 cents higher than the price sellers receive, P s…
. Also, the quantity buyers demand at P b must equal the quantity supplied at price P s . These two conditions are: P b P s = 0.40 and 26.9 1.71P b = 9.62 2.13P s . Solving these simultaneously,Ps = $4.32 and P b = $4.72. The new quantity will be Q = 26.9 1.71(4.72) = 18.8 billion packs. So the price consumers pay will increase from $4.50 to $4.72 and consumption will fall from 19.2 to 18.8 billion packs per year. c) Consumers pay $4.72 4.50 = $0.22 and producers pay the remaining $4.50 4.32 = $0.18 per pack. We could also find these amounts using the pass-through formula. The fraction of the tax paid by consumers is E S /(E S E D ) = 0.5/[0.5 (0.4)] = 0.5/0.9 =.556. Therefore, consumers will pay 55.6% of the 40-cent tax, which is 22 cents, and suppliers will pay the remaining 18 cents