Question

Numerical problem on consumer surplus: 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. Calculate the consumer surplus if the bridge toll is $0, $1, and $20.

b. 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?

c. 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

Answer 1

a)

Y = 1,000,000 - 50,000P

Y = Represents the number of trips over the bridge

 P = Represents the bridge toll

Inverse demand is calculated as;

$-50,000 P = Y - 1,000,000$

$P =20 - \frac{Y}{50,000}$

To find the consumer surplus

$Consumer surplus = ½ \times base \times height$

Base = Y ( number of trips per price)

Height = ( willingness price - Actual price)

We need to get the willingness price when Y =0

$= P = 20 - \frac{Y}{50,000}$

$P = 20 - \frac{0}{50,000}$

$P = 20$

 

Consumer surplus when the toll is $0

$Consumer surplus = ½ \times base \times height$

$Consumer surplus = ½ \times 1,000,000 \times (20-0)$

$Consumer surplus =10,000,000$

Consumer surplus when the toll is $1

$Consumer surplus = ½ \times base \times height$

$Consumer surplus = ½ \times 950,000 \times (20-1)$

$Consumer surplus =9,025,000$

Consumer surplus when the toll is $20

$Consumer surplus = ½ \times base \times height$

$Consumer surplus = ½ \times 0 \times (20-20)$

$Consumer surplus =0$

 b)

At Break even point

$= TR - TC = 0$

$TR = P\times Y$

$TR = 20 - \frac{Y}{50,000}Y$

$TR = 20Y - \frac{Y^2} {50,000}$

$0 = 20Y - \frac{Y^2} { 50,000} - 1,800,000$

$1,800,000 =20Y - \frac{Y^2} {50,000}$

$90,000,000,000 = 1,000,000Y - Y2$

$-Y^2 +1000,000Y - 90,000,000,000$

Use the quadratic formula

$X = -b +- √ \frac {b^2 -4ac} {2a}$

$X = - 1,000,000 +- √\frac { 1,000,0002 -4\times -1\times 90,000,000,000} {2\times -1}$

$X = 100,000 units\space or \space900,000 \space units$

$Y = 900,000 \space or\space 100,000$

We find the price by replacing Y

$P = 20 - \frac{Y}{50,000}$

$P = 20 - \frac {900,000}{50,000}$

$p=2$

Or

$P = 20 - \frac {Y}{50,000} \space P = 20 - \frac {100,000}{50,000}$

$P = 18$

Then we find the consumer surplus

Consumer surplus when the toll is $2

$Consumer surplus = ½ \times base \times height$

$Consumer surplus = ½ \times 900,000 \times (20-2)$

$Consumer surplus =8,100,000$

Consumer surplus when the toll is $18

$Consumer surplus = ½ \times base \times height$

$Consumer surplus = ½ \times 100,000 \times (20-18)$

$Consumer surplus =100,000$

C )

Given that

$TC = \$ \space8,000,000$

$Y = 1,000,000 - 50,000P$

 When toll is P = 0

$Y = 1,000,000$

Because when the toll is zero (0) there will be a Consumer surplus of 10,000,000 which is a benefit to the society.