A monopoly has a total cost function
C = 1000 + 108x + 12x^2
for its product, which has demand function
p = 324 − 3x − 2x^2.
Find the consumer's surplus at the point where the monopoly has maximum profit. (Round your answer to the nearest cent.)
To find the consumer's surplus at the point where the monopoly has maximum profit, we need to follow these steps:
Step 1: Determine the monopoly's profit-maximizing level of output.
Step 2: Find the equilibrium price corresponding to this level of output.
Step 3: Calculate the consumer's surplus using the equilibrium price.
Let's go step by step:
Step 1: Determine the monopoly's profit-maximizing level of output.
The monopoly maximizes its profit by setting its marginal cost equal to the marginal revenue. To find the profit-maximizing output level (x), we need to find the point where the marginal cost (MC) is equal to the marginal revenue (MR).
MC(x) = dC/dx = 108 + 24x
MR(x) = p(x) + x * dp(x)/dx = 324 − 3x − 2x^2 + x * (-6x - 2)
Setting MC(x) = MR(x):
108 + 24x = 324 − 3x − 2x^2 − 6x^2 − 2x
Simplifying the equation:
8x^2 + 31x - 216 = 0
Now, we solve this quadratic equation to find the value(s) of x.
Step 2: Find the equilibrium price corresponding to this level of output.
To find the equilibrium price, substitute the value of x (found in step 1) into the demand function:
p = 324 − 3x − 2x^2
Step 3: Calculate the consumer's surplus using the equilibrium price.
Consumer's surplus represents the difference between what consumers are willing to pay for a product and what they actually pay. It is the area under the demand curve.
To calculate the consumer's surplus, we integrate the demand function from 0 to the equilibrium quantity (x) and subtract it from the corresponding price:
Consumer's surplus = ∫[0,x] (p - (324 − 3x − 2x^2)) dx
Once we have the equilibrium price and the consumer's surplus, we can calculate the answer to the question.