Answer to Question #265629 in Microeconomics for Ade

Question #265629

Given a demand function of the monopolist as Q=50-0.5P and cost function of the nature C=50-4Q.


1. Calculate profit maximising output


2. Calculate the price at which profit is maximised


3. Compute the total profit

1
Expert's answer
2021-11-14T17:39:55-0500

(1)

"Q=50-0.5P"

"C=50-4Q"

Add 0.5P to both sides of the demand equation:

"Q+0.5P=50-0.5P+0.5P"

"Q+0.5P=50"

Subtract Q from both sides of the above equation:

"Q+0.5P-Q=50-Q"

"0.5P=50-Q"

Dividing both sides of the equation by 0.5:

"\\frac{0.5P}{0.5}=\\frac{50-Q}{0.5}"

"P=100-2Q"

To determine Total Revenue (TC), multiply both sides of the equation by Q:

"TR=PQ=(100-2Q)\\times Q"

"TR=100Q-2Q^2"


"\\pi=TR-TC"

"=(100Q-2Q^2)-(50-4Q)"

"=-50+104Q-2Q^2"

Differentiating the above equation w.r.t. Q:

"\\frac{d\\pi}{dQ}=104-4Q"

"104-4Q=0"

"\\implies Q=26."


(2)

"Q=50-0.5P"

"26=50-0.5P"

"P=48"


(3)

Total Profit:

"\\pi=-50+104Q-2Q^2"

"\\pi=-50+104(26)-2(26)^2"

"\\pi=1,302"


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