Answer to Question #206436 in Microeconomics for daniel

Question #206436

1.              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


1
Expert's answer
2021-06-14T12:20:42-0400

"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"


(b)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"


c)

"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


d)

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




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