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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

  • engineering -

    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

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