Question

Rivertown has two bridges and no tolls.   The hourly demand curves for vehicular bridge crossings are Bridge A, P = 125 – Q and Bridge B,   P = 60 – 0.5Q where P is the price for crossing the bridge and Q is the number of vehicles.

a. Find the total crossings and the total consumer surplus at the current price.

b. The Rivertown administration wishes to reduce vehicular traffic on both bridges to a total of 200 vehicles per hour and charge the same toll on both bridges.    Find the toll for both bridges and the traffic on each bridge that accomplishes that goal.

c. Find the change in consumer surplus from moving to no tolls to the the toll in part b.

d. The price elasticity of demand for each bridge at the toll price in part.

Answer 1

The hourly demand curves for vehicular bridge crossings are Bridge A, P = 125 – Q and Bridge B, P = 60 – 0.5Q where P is the price for crossing the bridge and Q is the number of vehicles.

a. Find the total crossings and the total consumer surplus at the current price.

With no toll, there is no price for crossing. Hence the number of crossings on bridge A is 125 – Q = 0 or Q = 125 crossings. On bridge B it is 60 = 0.5Q or Q = 120 crossings. This implies that there are a total of 245 crossings. Consumer surplus under bridge A is CS1 = 0.5*125*125 = 7812.5. Consumer surplus under bridge B is CS2 = 0.5*60*120 = 3600. Hence total consumer surplus = 11412.50.  

b. The Rivertown administration wishes to reduce vehicular traffic on both bridges to a total of 200 vehicles per hour and charge the same toll on both bridges. Find the toll for both bridges and the traffic on each bridge that accomplishes that goal.

Under this case, QA + QB = 200 and P1 = P2. Hence we have

125 – QA = 60 - 0.5QB

65 = QA – 0.5QB

Use QA = 200 – QB

65 = 200 – 1.5QB

QB* = 90 and QA = 110.

Hence The toll for the bridges is P = 125 – 110 = $15 and the traffic on bridge A = 110 crossings, on bridge B = 90 crossings.

c. Find the change in consumer surplus from moving to no tolls to the the toll in part b.

Now the consumer surpluses are

CS1 = 0.5*(125 – 15)*110 = $6050 and CS2 = 0.5*(60 – 15)*90 = 2025. Total consumer surplus is now $8075 which is lower than the consumer surplus with no toll at $11412.50

d. The price elasticity of demand for each bridge at the toll price

ed (bridge A) = -1 * (15/110) = -0.136. ed (bridge B) = (-1/0.5)*15/90 = -0.333