Question

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.

Answer 1

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

$C(x)=75+80x−x^2+4x\\C(x)=75+84x−x^2\\MC=84-2x\\MC=MRv\\84-2x=200-6x\\200-84=4x\\116=4x\\x=29$

$P = 200 -3(29)=113$

Quantity=29 and price=113