Question

Suppose a perfect in competitive firmà ƒ ƒ ¢ € ™s short
run cost function is given by:
TC= 1/3 Q3 + 3 Q2 + 10Q +40
if the market price of the commodity Is Birr 26 per unit.
A. Determine the profit maximizing Level of output
b, find AFC AC, AVC and MC of firm
at optimum level a output.
c, find maximum profit of the firm.

Answer 1

A

The firm should produce at MR=MC to maximize its profit.

$TC=\frac{1}{3} Q^3+3Q^2+10Q+40$

$MC=Q^2+6Q+10$

$MR=MC\\TR=PQ=26Q\\MR=26\\26=Q^2+6Q+10\\Q^2+6Q-16=0\\Q^2+8Q-2Q-16=0\\Q(Q+8)-2(Q+8)=0\\(Q-2)(Q+8)=0\\Q=2 \space or\space Q=-8$

The profit maximizing level is 2 units

b.

$AFC=\frac{FC}{Q}$

$=\frac{40}{2}=20$

$AC=\frac{TC}{Q}=\frac{\frac{1}{3}(2)^3+3(2)^2+10(2)+40}{2}=37.3333333$

$AVC=\frac{VC}{Q}=\frac{\frac{1}{3}(2)^3+3(2)^2+10(2)}{2}=17.333333$

$MC=\frac{\delta TC}{\delta Q}=Q^2+6Q+10$

c.

$profit =TR-TC\\TR=P\times Q=26\times 2=52\\TC=\frac{1}{3} (2)^3+3(2)^2+10(2)+40=74.66666\\profit =52-74.66666=-22.66666$