Answer to Question #206064 in Macroeconomics for Nablis

Question

For a monopolist firm the demand and the total cost functions are given as Q = 20-

0.5P and TC= 4Q2-8Q+15, respectively. (8 marks)

Find

a) the optimum quantity and the optimum price level

b) the profit/loss on these levels

c) at what price should the monopolist shut down?

d) Show the economic profit (loss) of the firm in a graphic representation

Answer 1

$Q=20-0.5P\\TC=4Q^2-8Q+15$

a)

$Q=20-0.5P\\0.5P=20-Q\\\implies P=\frac{20-Q}{0.5}$

$\implies 40-2Q\\TR=P\times Q\\TR=(40-2Q)Q$

$MR=\frac{dTR}{dQ}\\MR=40-4Q\\and\\TC=4Q^2-8Q\\MC=\frac{dTC}{dQ}=8Q-8$

Equilibrium

$MR=MC\\\implies40-4Q=8Q-8\\\implies40+8=8Q+4Q\\\implies12Q=48\\\implies Q=\frac{48}{12}=4$

Optimum quantity

$Q=4\\P=40-2(Q)\\=40-2(4)\\=40-8\\P=32$

now

$TR=P\times Q=32\times 4=128\\TC=4Q^2-8Q+15\\=4(4^2)-8(4)+15\\TC=47\\profit=TR-TC=128-47=81$

b)

$TC=4Q^2-8Q+15\\VC=4Q^2-8Q\\AVC=\frac{VC}{Q}=\frac{4Q^2-8Q}{Q}\\AVC=4Q-8$

for P<AVC

$\implies40-2Q<4Q+2Q\\\implies40+8<4Q+2Q\\\implies48<6Q\\\implies \frac{48}{6}<Q\\\implies8<Q$

for Q>8 we have P<AVC

let Q=8

$P=40-2Q\\P=40-2(8)\\P=40-16\\\implies P=24\\P<24$

firm shut down

c)

$ATC=\frac{TC}{Q}=\frac{4Q^2-8Q+15}{Q}\\=4Q-8+\frac{15}{Q}\\ATC=4(4)-8\frac{15}{4}=11.75$