Question

Suppose you are in charge of a toll bridge that costs essentially nothing to operate. The demand for bridge crossingsQis given byP= 15 - (1/2)Q.

a. Draw the demand curve for bridge crossings.

b. How many people would cross the bridge if there were no toll?

c. What is the loss of consumer surplus associated with a bridge toll of \(5?

d. The toll-bridge operator is considering an increase in the toll to \)7. At this higher price, how many people would cross the bridge? Would the toll-bridge revenue increase or decrease? What does your answer tell you about the elasticity of demand?

e. Find the lost consumer surplus associated with the increase in the price of the toll from \(5 to \)7.

Short answer

a. The demand curve for bridge crossings is given below:

b. 30 people will use cross the bridge at no toll.

c. The loss of consumer surplus is $125.

d. At a toll of $7, 16people will cross the bridge.

The toll-bridge revenue will increase.

The demand is elastic as the revenue increases with an increase in price.

e. The loss in the consumer surplus is $36.

Step-by-step solution

Explanation for part (a)

The graph below depicts the toll or prices on the x-axis and bridge crossing or demand on the x-axis.

Since the demand equation is given as; P = 15 - (1/2)Q. Based on this equation, the two intercepts are 30 on the x-axis (toll price is zero) and 15 on the y-axis (Q=0).

Explanation for part (b)

If toll or price is zero, then:

p = 0

0 = 15 - 1/2Q

Q = 15 x 2

= 30

Therefore, 30 people will cross the bridge at 0 tolls.

Explanation for  part (c)

Consider the diagram below:

In the graph above, at P=0, the consumer surplus will become the area of the triangle formed by the demand curve:

1/2 x 15 x 30 = 225

Given the demand equation, if the toll is set at $5, then demand would be:

p = 5

5 = 15 - 1/2Q

Q = 15 - 5 - 1/2

= 10 - 1/2

= 20

Thus; when the toll will be P=5, then consumer surplus will be:

1/2 x 5 x 20 = 100

Therefore, the loss in consumer surplus will be 225-100=125.

Explanation for part (d)

The number of people who will cross the bridge when the toll is increased to $7:

p = 7

7 = 15 - 1/2Q

Q = 15 - 7 - 1/2

= 8 - 1/2

= 16

You can see that the number of people who will cross at $7 toll will be 16. Considering the area of the triangle which would be, the revenue of this toll would be:

1/2 x 7 x 16 = 112

The initial toll was 100. The new toll is 112;it can be concluded that the toll has increased.

Since the revenue has increased even after the increase in the toll, the demand for crossing bridges is inelastic towards price or toll.

Explanation for part (e)

The loss in the consumer surplus is depicted through 2 arrows in the graph; when these areas pointed by two arrows are added, the lost consumer surplus will be (16)(7 – 5) + (1/2)(20 – 16)(7 – 5) = $36.