Travelers driving through Gotham City can use a freeway or the Cross Town Tollway to get through the city. The tollway charges $1.00 per car during the morning rush hour (6-9 AM) and the afternoon rush hour (4-7 PM), and the toll is $0

Question

40 per car at all other times. The weekly demand for using the tollway during rush hour is Q1 = 800 - 200P1 where quantity demanded is measured in thousands of cars, and the weekly demand for the non-rush hour period is Q2 = 2000 - 1000P2. Gotham City's marginal cost of operating the tollway is MC = 0.02 + 0.001Q per car. a. What are the marginal revenue curves associated with the two demand curves? b. Has the city set the profit maximizing tolls for the Cross Town Tollway? If not, do the current tolls generate too much or too little traffic on the tollway?

Answer 1

a.
The price-dependent expressions of the rush hour demand curve is P1 = 4 - 0.005Q1, and the expression for the non-rush hour demand is P2 = 2 - 0.001Q2. The associated marginal revenue curves are MR1 = 4 - 0.01Q1 and MR2 = 2 - 0.002Q2.

b.
The profit maximizing level of rush hour traffic is found by solving MC = MR1, which provides 4 - 0.01Q1 = 0.02 + 0.001Q1 so that Q1 = 3.98/0.011 = 361.8 thousand cars. Accordingly, the profit maximizing price (toll) at this level of rush hour traffic is P1 = 4 - 0.005(361.8 ) = $2.19. The profit maximizing level of non-rush hour traffic is found by solving MC = MR2, which provides 2 - 0.002Q2 = 0.02 + 0.001Q2 so that Q2 = 1.98/0.003 = 660.0 thousand cars. Accordingly, the profit maximizing price (toll) at this level of non-rush hour traffic is P2 = 2 - 0.001(660 ) = $1.34. Thus, the current tolls are below the profit maximizing levels, and the tollway attracts more traffic than the optimal level during rush hour and non-rush hour periods.