The estimated demand equation is Q = 112,000 – 500P + 5(20,000) = 212,000 – 500P.
The inverse demand function is P = 424 – 0.002Q
The marginal revenue is MR = 424 – 0.004Q
The average variable cost is AVC = 200 – 0.012Q + 0.000002Q2
The marginal cost is SMC = 200 – 0.024Q + 0.000006Q2
a. How many carpets should the firm produce in order to maximize profit?
To maximize profit, marginal revenue is set to equal marginal cost.
MR = SMC = 424 – 0.004Q = 200 – 0.024Q + 0.000006Q2 = 224 + 0.020Q – 0.000006Q2
Solving for Q, it is equal to 4666.67 and –8000. Since there can be no negative output, the firm should produce 4667 carpets.
b. What is the profit-maximizing price of carpets?
The profit maximizing price P = 424 – 0.002Q, when Q = 4667, the price is $414.33.
c. What is the maximum amount of profit that the firm can earn selling carpets?
The maximum amount of profit the firm can earn selling carpets is:
(P * Q) – [(AVC * Q) + TFC] = (414.33 * 4667) – [(187.56 * 4667) + 100,000)] = $958,341.48
d. Answer parts a through c if consumer’s income per capita is expected to be $30,000 instead.
The estimated demand equation is Q = 112,000 – 500P + 5(30,000) = 262,000 – 500P.
The inverse demand function is P = 524 – 0.002Q
The marginal revenue is MR = 524 – 0.004Q
The average variable cost is AVC = 200 – 0.012Q + 0.000002Q2
The marginal cost is SMC = 200 – 0.024Q + 0.000006Q2
To maximize profit, marginal revenue is set to equal marginal cost.
MR = SMC = 524 – 0.004Q = 200 – 0.024Q + 0.000006Q2 = 1476 – 0.020Q + 0.000006Q2
Solving for Q, it is equal to 5833.33 and –9166.67. Since there can be no negative output, the firm should produce 5834 carpets.
The profit maximizing price P = 524 – 0.002Q, when Q = 5834, the price is $512.33.
The maximum amount of profit the firm can earn selling carpets is:
(P * Q) – [(AVC * Q) + TFC] = (512.33 * 5834) – [(198 * 5834) + 100,000)] = $1,733,801.22.