Question

The Interboro Rapid Transit Authority (IRTA) has estimated the following peak and off-peak demand equations for its commuter rail service:

QP=50-2P

QOP=20-4POP

Where Q is measured in thousands of passenger miles. IRTA’s marginal cost of providing commuter rail service is constant at $3 per passenger mile. IRTA’s capacity is 20 thousand passenger miles, which it reaches during peak, rush hours. What fare should IRTA change during peak and off-peak hours?

Answer 1

Profit is maximized when in each segment, Marginal revenue (MR) equals Marginal cost (MC).

In Peak segment,

QP = 50 - 2P

2P = 50 - QP

P = 25 - 0.5QP

Total revenue (TR) = P x QP = 25QP - 0.5QP²

MR = dTR/dQP = 25 - QP

Equating with MC,

25 - QP = 3

QP = 22

P = 25 - (0.5 x 22) = 25 - 11 = $14

For Off-peak segment,

QOP = 20 - 4POP

4POP - 20 - QOP

POP = 5 - 0.25QOP

TR = POP x QOP = 5QOP - 0.25QOP²

MR = dTR/dQOP = 5 - 0.5QOP

Equating with MC,

5 - 0.5QOP = 3

0.5QOP = 2

QOP = 4

POP = 5 - (0.25 x 4) = 5 - 1 = $4