7. Suppose that a market is described by the following supply and demand equations: Q S =2P Q D = 300-P a. Solve for the equilibrium price and equilibrium quantity. Setting quantity supplied equal to quantity demanded gives 2P = 300 – P. Adding P to both sides of the equation gives 3P = 300. Dividing both sides by 3 gives P = 100. Plugging P = 100 back into either equation for quantity demanded or supplied gives Q = 200. b. Suppose that a tax of T is placed on buyers, so the new demand equation is: Q D =300- (P+T) Solve for the new equilibrium. What happens to the price received by sellers, the price paid by buyers, and the quantity sold? Now P is the price received by sellers and P +T is the price paid by buyers. Equating quantity demanded to quantity supplied gives 2P = 300 − (P+T). Adding P to both sides of the equation gives 3P = 300 – T. Dividing both sides by 3 gives P = 100 –T/3. This is the price received by sellers. The buyers pay a price equal to the price received by sellers plus the tax (P +T = 100 + 2T/3). The quantity sold is now Q = 2P = 200 – 2T/3. c. Tax revenue is T x Q. Use your answer to part (b) to solve for tax revenue as a function of T. Graph this relationship for T between 0 and 300. Because tax revenue is equal to T x Q and Q = 200 – 2T/3, tax revenue equals 200T − 2T 2 /3. The figure below shows a graph of this relationship. Tax revenue is zero at T = 0 and at T = 300 d. The deadweight loss of a tax is the area of the triangle between the supply and demand curves. Recalling that the area of a triangle is ½ BH, solve for deadweight loss as a function of T. Graph this relationship for T between 0 and 300. (Hint: Looking sideways, the base of the deadweight loss triangle is T, and the height is the difference between the quantity sold with the tax and the quantity sold without the tax.)
As the figure above shows, the area of the triangle (laid on its side) that represents the deadweight loss is 1/2 × base × height, where the base is the change in the price, which is the size of the tax (T) and the height is the amount of the decline in quantity (2T/3). So the deadweight loss equals 1/2 × T × 2T/3 = T 2 /3. This rises exponentially from 0 (when T = 0) to 30,000 when T = 300, as shown in the figure below. e. The government now levies a tax on this good of $200 per unit. Is this a good policy? Why or why not? Can you propose a better policy? A tax of $200 per unit is a bad idea, because it is in a region in which tax revenue is declining. The government could reduce the tax to $150 per unit, get more tax revenue ($15,000 when the tax is $150 versus $13,333 when the tax is $200), and reduce the deadweight loss (7,500 when the tax is $150 compared to 13,333 when the tax is $200).
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