Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36 + 3/Q, calculate the level of output Q which a) maximizes total revenue b) maximizes profits

Answer 1

Revenue = Quantity*Price
R = Q*P = 24Q - .5Q^2
Revenue is a maximum when
dR/dQ = 0
24 -Q = 0
Q = 24 units sold @ a price of P = 12

Profit (P) is
P = R - Q*(AC)= Q*P - Q*(AC)
= 24Q -0.5Q^2 -Q*(Q^2 -8Q +36 +3/Q)
= 24Q -0.5Q^2 -Q^3 +8Q^2 +36Q +3
= -Q^3 +7.5 Q^2 +60Q +3
Set dP/dQ = 0 to solve for the maximum-profit production level, Q.
-3Q^2 +15Q +60 = 0
The positive root is
Q = (-1/6)[-15 -sqrt(289+720)]
Q = 7.8