To find the consumer surplus, we need to calculate the area between the demand curve and the market price of $5.
The demand function is:
p = -0.01x² - 0.3x + 15
Step 1: Find the quantity demanded when the price is $5.
Substitute p = 5 into the demand equation:
5 = -0.01x² - 0.3x + 15
Rearrange:
0.01x² + 0.3x - 10 = 0
Multiply everything by 100:
x² + 30x - 1000 = 0
Factor:
(x + 50)(x - 20) = 0
x = -50 or x = 20
Since quantity cannot be negative, x = 20.
Remember that x is measured in thousands, so this represents 20,000 cartridges per week.
Step 2: Set up the consumer surplus integral.
Consumer surplus is the area under the demand curve but above the market price.
CS = Integral from 0 to 20 of [(-0.01x² - 0.3x + 15) - 5] dx
Simplify:
CS = Integral from 0 to 20 of (-0.01x² - 0.3x + 10) dx
Step 3: Integrate.
CS = [-0.01x³/3 - 0.15x² + 10x] evaluated from 0 to 20
Substitute x = 20:
CS = [-0.01(20³)/3 - 0.15(20²) + 10(20)]
CS = -26.6667 - 60 + 200
CS = 113.3333
Step 4: Convert to actual dollars.
Since x represents thousands of cartridges, we must multiply the result by 1,000.
CS = 113.3333 × 1,000
CS = $113,333.33
Final Answer: $113,333.33
This means consumers receive approximately $113,333.33 in total weekly economic benefit beyond what they actually pay for the cartridges.
The most important detail is remembering to multiply by 1,000 because the quantity in the original demand function is measured in thousands of units.
