Question

Suppose perfect competitive firm short run cost function

total cost=1/3q3+3q2+10Q+40

. if the market price of the commodity is birr 26 per unit

A, determine the profit maximizing level of out put

B find average fixed cost ,average cost ,average variable cost and marginal cost of firm at optimum level of out put

C find maximum profit of the firm

Answer 1

$Given\\ Total\space Cost = \frac{1}{3}q^3+3q^2+10Q+40\\ Market \space Price = 26$

a.

Profit Maximizing output is archived where MR = MC

Now Calculate MC

$MC = \frac{∂TC}{∂q}\\MC = q^2 + 6q + 10$

Now put the value in the condition where MR = MC

$q^2 + 6q + 10 = 26\\q^2 + 6q − 16 = 0\\(q−2)(q+8) = 0\\q = 2 \space and\space −8$

quantity should be always positive therefore q = 2

hence when q = 2, profit should be maximized.

b.

$Total\space Cost = \frac{1}{3}q^3+3q^2+10Q+40\\$

Calculation of Average Fixed Cost

$AFC = \frac{TFC}{q}\\AFC = \frac{40}{q}\\where\space q = 2\\AFC =\frac{40}{2} = 20$

Calculation of Average Cost

$AC = \frac{TC}{q}\\AC = \frac{\frac{1}{3}q^3+3q^2+10q+40}{q}\\AC = \frac{2.67 + 12 +20+40}{2}\\AC = 37.335$

Calculation of Average Cost

$AVC = \frac{TVC}{q}\\AVC = \frac{\frac{1}{3}q^3+3q^2+10q}{q}\\AVC =\frac{ 2.67 + 12 +20}{2}\\AVC = 17.335$

Calculation of MC

$MC = q^2 + 6q + 10\\MC = 22 + 6\times 2 + 10\\MC = 4 + 12 + 10\\MC = 26$

c.

$Profit = TR - TC\\ Profit = 26\times 2 - \frac{1}{3}\times 2^3+3\times 2^2+10\times2+40\\ Profit = 112 - 74.67\\ Profit = 37.33$