Question

If demand P=100-Q, TC=800+20Q+Q2 and MC=20+2Q what is the profit maximizing price and the rate of output

Answer 1

profit=total revenue(TR)-total cost(TC)

Given,

$TC = 800+20Q+Q^2\\ P = 100 - Q\\ TR = Price (P) \times Quantity (Q)\\ TR = (100 - Q)\times Q\\ TR = 100Q - Q^2\\ So\space here,\\ Profit = 100Q - Q^2 - (800+20Q+Q^2)\\ = 100Q - Q^2 - 800 - 20Q - Q^2\\ = 80Q -2Q^2 - 800$

Now, for maximizing profit we will derivate profit with respect to Q and will put it equal to zero

$\frac{∂\space Profit }{ ∂Q} = 80 - 4Q\\ Putting\\ \frac{∂Profit }{ ∂Q} = 0\\ 80 - 4Q = 0\\ 80 = 4Q\\ Q = 20$

So, rate of output is 20

Putting value of Q in P

$P = 100 - 20\\ P = 80$

So, profit maximizing price is 80 and rate of output is 20.