Question

Determine the best level of output for a perfectly competitive firm that sells its product at P = $4 and faces TC = 0.04Q3– 0.9Q2 + 10Q + 5. Will the firm produce this level of output? Why?                                                                               2 marks      

Answer 1

Total revenue= $P*Q$ =$4*Q$ =$4Q$

$Profit= TR-TC$

$Profit=4Q-(0.04Q^3-0.9Q^2+10Q+5)$

$\frac{dP} {dQ}$ =$–0.12Q^{2} +1.8Q –6$

$\frac{dP} {dQ} =0$

$–0.12Q^{2} +1.8Q–6$$=0$ divide across by 0.12

$\frac{-0.12Q^2} {0.12} + \frac{1.8} {0.12}-\frac{6} {0.12} =\frac{0} {0.12}$

$–Q^{2} +15Q-50=0$

$(-Q+10)(Q-5)=0$

$(Q=5) and (Q=10)$

For Q=5; Profit = $–0.04(5)^{3} +0.9(5)^{2} –6(5)–5$ = –17.5

For Q=10; Profit = $–0.04(10)^{3} +0.9(10)^{2} –6(10)–5$ = –15

No. The firm would not produce this level of output since there are losses when selling the output at $4 dollars per unit.