Question

Suppose a perfectly competitive firm with total cost function given as:
TC= 400+20Q-2Q2+Q3.
A. Find profit maximizing level of output and the maximum profit if the average revenue equals $180.
B. Calculate the shutdown level of output and price

Answer 1

Given the values:

$TC=400+20Q−2Q^2+Q^3\\AR = MR = \$180$

a:

$TC=400+20Q−2Q^2+Q^3\\MC =\frac{ dTC}{dQ}=20−4Q+3Q^2\\Now,\space MC = MR\\20−4Q+3Q^2=180\\0=180 − (20−4Q+3Q^2)\\0=180−20+4Q−3Q^2\\0=160+4Q−3Q^2$

Now solve for the Q,  thus, Q=8 and $Q=\frac{−20}{ 3}$

So consider the positive value Q=8.

Thus, profit maximizing level of output is = 8

$Profit = TR − TC\\Profit = (MR×Q)−(400+20Q−2Q^2+Q^3)\\Profit = (180×8)−(400+20×8−2×8^2+8^3)\\Profit = 1440−(400+160−128+512)\\Profit = 1440−(944)\\Profit = \$496$

b:

Shutdown point,

$AVC = MC\\TC = FC+VC\\Thus, \space VC = 20Q−2Q^2+Q^3\\AVC=\frac{VC}{Q}\\AVC=\frac{20Q−2Q^2+Q^3}{Q}\\AVC=20−2Q+Q^2\\At\space shutdown\space point,\\ MC=AVC\\20−4Q+3Q^2=20−2Q+Q^2\\20−4Q+3Q^2−(20−2Q+Q^2)=0\\20−4Q+3Q^2−20+2Q−Q^2=0\\−2Q+2Q^2=0\\2Q(−1+Q)=0\\−1+Q=0\\Q=1$

shutdown level of quantity = 1

$P=MC\\P=20−4Q+3Q^2\\P=20−4(1)+3(1)^2\\P=20−4+3\\P=\$19$

shutdown price =$19