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)
"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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