The demand equation for a company is P = 200 -3x
And the cost function is C(x) = 75 + 80x - x2
a) Determine the value of x and the corresponding price that maximize the profit
b) If the government imposes a tax on the company of R4 per unit quantity produced, determine the new price that maximizes the profit.
a)
The firm makes profit at the point where MR=MC
"profit=TR-TC"
"TR=P\\times Q"
"P = 200 -3x"
"\\therefore TR=(200-3x)x"
"=200x-3x^2"
"MR=200-6x"
"C(x) = 75 + 80x - x^2"
"MC=80-2x"
"MR=MC"
"200-6x=80-2x\\\\200-80=-2x+6x\\\\120=4x\\\\x=30"
b)
After tax, cost increases by 4x
"C(x)=75+80x\u2212x^2+4x\\\\C(x)=75+84x\u2212x^2\\\\MC=84-2x\\\\MC=MRv\\\\84-2x=200-6x\\\\200-84=4x\\\\116=4x\\\\x=29"
"P = 200 -3(29)=113"
Quantity=29 and price=113
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