Asked by Taylor Eakin on Jun 18, 2024
Verified
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
Demand Function
A mathematical representation showing the quantity of a good or service demanded at different prices.
Consumer's Surplus
The difference between the total amount that consumers are willing to pay for a good or service and the total amount that they actually pay.
Mead
An alcoholic beverage created by fermenting honey with water, sometimes with various fruits, spices, grains, or hops added.
- Acquire an understanding of how demand functions operate and the role of price in affecting demand.
- Learn about the interaction between consumer surplus and variations in pricing.
Verified Answer
FV
Fanny VelazquezJun 21, 2024
Final Answer :
D
Explanation :
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.
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.
Learning Objectives
- Acquire an understanding of how demand functions operate and the role of price in affecting demand.
- Learn about the interaction between consumer surplus and variations in pricing.