Question 8.3.2: Suppose that the total cost of producing Q > 0 units of a......
Suppose that the total cost of producing Q > 0 units of a commodity is C(Q) =aQ² + bQ + c, where a, b, and c are positive constants.
(a) Find the value of Q that minimizes the average cost defined by A(Q) = C(Q)/Q in the special case when C(Q) = 2Q² + 10Q + 32.
(b) Show that in the general case, the average cost function has a minimum at Q^{∗} =\sqrt{c/a}.
In the same coordinate system, draw the graphs of the average cost, the marginal cost,and the straight line P = aQ + b.
(a) We find that here A(Q) = 2Q + 10 + 32/Q, so A^{\prime}(Q) = 2 − 32/Q^{2} and A^{\prime\prime}(Q) = 64/Q^{3}. Since A^{\prime\prime}(Q) > 0 for all Q > 0, the function A is convex, and since A^{\prime}(Q) = 0 for Q = 4, this is a minimum point.
(b) We find that here A(Q) = aQ + b + c/Q, A^{\prime}(Q) = a − c/Q^{2} and A^{\prime\prime}(Q) = 2c/Q^{3}. Since A^{\prime\prime}(Q) > 0 for all Q > 0, the function A is convex, and since A^{\prime}(Q) = 0 for Q^{∗} =\sqrt{c/a}, this is a minimum point. The graphs are drawn in Fig. 8.3.3. Note that at the minimum point Q^{∗}, marginal cost is equal to average cost. This is no coincidence, because it is true in general that A^{\prime}(Q) = 0 if and only if C^{\prime}(Q) = A(Q).^{5}
^{5} See Example 6.7.6. The minimum average cost is A(Q^{\ast})=a\sqrt{c/a}+b+c/\sqrt{c/a}=\sqrt{a c}+b+{\sqrt{a c}}=2{\sqrt{a c}}+b.
