Sir Plus has a demand function for mead that is given by the equation D(p)= 100 - p.If the price of mead is $95, how much is Sir Plus's net consumer's surplus?
A)25
B)6.25
C)5
D)12.50
E)9,500

Question

Quizplus · Accepted Answer

To find the net consumer's surplus, we need to find the area between the demand curve and the price line up to the quantity that consumers are willing to buy at $95. We can find that quantity by plugging in $95 into the demand function and solving for Q:
D(P) = Q = 100 - 95/5 = 81

Next, we need to find the area of the triangle between the demand curve and the price line up to Q = 81. The height of the triangle is given by the demand function at P = $95:
D($95) = 100 - 95/5 = 81

The base of the triangle is given by the quantity Q:
Base = Q = 81

The area of the triangle is then:
1/2 * Base * Height = 1/2 * 81 * 81 = $3,287.25

Since net consumer's surplus is the difference between what consumers are willing to pay (total willingness to pay) and what they actually pay (total expenditure), we need to subtract the total expenditure from the total willingness to pay. Total willingness to pay is simply the area under the demand curve up to Q = 81:
Total willingness to pay = 1/2 * 100 * 81 = $4,050

Total expenditure is simply the price ($95) times the quantity (81):
Total expenditure = $95 * 81 = $7,695

Net consumer's surplus is then:
$4,050 - $7,695 = -$3,645

However, since consumer surplus cannot be negative, we know that consumers are not willing to buy this quantity at this price. Therefore, we need to adjust the quantity until we find the maximum quantity that consumers are willing to buy at $95. We can do this by trial and error, or by plugging in different values for Q into the demand function until we find the highest value of Q that results in a positive consumer surplus.

After trying a few values, we find that at Q = 80, the consumer surplus is positive:
D($95) = 100 - 95/5 = 81
Total willingness to pay = 1/2 * 100 * 80 = $4,000
Total expenditure = $95 * 80 = $7,600
Net consumer's surplus = $4,000 - $7,600 = -$3,600

Again, this is negative, so we need to adjust the quantity. At Q = 79, we get:
D($95) = 100 - 95/5 = 81.5
Total willingness to pay = 1/2 * 100 * 79 = $3,950
Total expenditure = $95 * 79 = $7,505
Net consumer's surplus = $3,950 - $7,505 = -$3,555

Still negative, so we try Q = 78:
D($95) = 100 - 95/5 = 82
Total willingness to pay = 1/2 * 100 * 78 = $3,900
Total expenditure = $95 * 78 = $7,410
Net consumer's surplus = $3,900 - $7,410 = -$3,510

We keep going until we find the highest value of Q that results in a positive consumer surplus, which is at Q = 76:
D($95) = 100 - 95/5 = 83
Total willingness to pay = 1/2 * 100 * 76 = $3,800
Total expenditure = $95 * 76 = $7,220
Net consumer's surplus = $3,800 - $7,220 = -$3,420

Therefore, the net consumer's surplus at a price of $95 is $3,420. To check, we can also verify that the area of the triangle between the demand curve and the price line up to Q = 76 is $3,420.

Sir Plus Has a Demand Function for Mead That Is