Assume that the demand for travel over a bridge takes the form Y =1,000,000 - 50,000P, where Y is the number of trips over the bridge and P is the bridge toll (in dollars).
a) Draw the demand curve for travel over a bridge.
b) Calculate the consumer surplus if the bridge toll is $0, $1, and $20.
c) Assume that the cost of the bridge is $1,800,000. Calculate the toll at which the bridge owner breaks even. What is the consumer surplus at the breakeven toll?
d) Assume that the cost of the bridge is $8 million. Explain why the bridge should be built even though there is no toll that will cover the cost.
Consumer surplus is a measure of the welfare that people gain from consuming goods and services.
Consumer surplus is the difference between the total amount that consumers are willing and able to pay for a good or service (indicated by the demand curve) and the total amount that they actually do pay (i.e. the market price). [See first weblink below]
If a person pays less than they are willing or able to pay for goods or services (in this case, travel across a toll bridge) then they can use the money they would have paid for the goods/services for something else.
The market price is the actual amount paid, P, in this case possibly $0, $1 or $20 dollars. The price consumers are willing to pay however, is defined by the the demand curve, here given by
Y = 1,000,000 - 50,000P (1)
where Y is the number of journeys made across the bridge.
a) see graph attached of demand curve (1)
b) Given the price is fixed at P, the triangular area depicted in the graph, which equals the consumer surplus is (using that the area of a triangle is given by 1/2 x base x height)
C = 1/2 x (1,000,000 - 50,000P) x (1,000,000/50,000 - P)
= 1/2 x (1,000,000 - 50,000P) x (20 - P)
where 1,000,000 - 50,000P is the base of the triangle in question and (20 - P) is the height.
So if P = $0, $1, $20 respectively, these prices correspond to a consumer surplus of C = $10,000,000,
C = 19/2 x (1,000,000 - 50,000) = 19/2 x 950,000 = $9,025,000
C = $0 respectively.
c) If the cost of the bridge is $1,800,000, the toll P at which the bridge owner breaks even is given by
P = 1,800,000/Y
where Y is the number of journeys made across the bridge by consumers.
The consumer surplus at this toll is given by
C = 1/2 x (1,000,000 - 1,800,000/Y) x (20 - 1,800,000/Y)
d) The demand for the bridge for prices beyond P = $20 is zero, from the demand curve. If the cost of the bridge is $8 million = $8,000,000 this implies that the toll should be set at
P = 8,000,000/Y
to break even, where Y is the number of journeys made. But the maximum number of journeys made, from the demand curve, is 1,000,000 (with a toll of P = $0, a free toll). This would imply that the number of journeys made Y would need to be infinite for the bridge owner to break even. This of course, in practice, is an impossibility. However, as Y becomes very large, P can approach zero for the bridge owner to break even. If the bridge is very popular, as they often are, the bridge owner could break even or more even with quite a low toll. The demand could increase compared to the estimated demand curve, which in advance can only be a guess.