Question

A firm has the following demand function 𝑷 = 𝟐𝟎𝟎 − 𝟐𝑸 and the average cost of 𝑨𝑪 = 𝟏𝟎𝟎/𝑸 + 𝟑𝑸 − 𝟐𝟎.

a. Find the profit function.

b. Estimate the marginal cost function.

c. Obtain the production that maximizes the profit.

d. Evaluate the average cost and the marginal cost at the maximizing production level.

Answer 1

a. R(x) = p(q)$\cdot$ q = 200⋅ q - $2 \cdot q^2$

b. TC = $(𝟏𝟎𝟎/𝑸 + 𝟑𝑸 − 𝟐𝟎 ) \cdot Q = 100 + 3Q^2 - 20Q$

$MC = dQ(100 + 3Q^2 - 20Q) = 6Q - 20$

c. In absolute compatetive market P = MR, so MR = MC is the maximizes the profit.

$6Q - 20 = 200 - 2Q$

$8Q = 220$

Q = 27,5

d. AC = $100/27.5 + 3\cdot 27.5 − 20 = 66.13$

MC = $6 \cdot 27.5 - 20 = 145$