Q. 4 For a firm with a Total Cost function TC = C(Q), recall that the definitions of Marginal Cost (MC) and Average !3! Total Cost (ATC) are given as follows: Marginal Cost (MC) = C'(Q) Average Total Cost (ATC) = C(Q)/Q Prove that the marginal cost is equal to the average total cost (MC = ATC) when the average total cost is at its minimum value. (Hint: By using the definition of average cost function, find the quantity that minimizes it by applying unconstraint optimization. Show that the FOC of the minimization problem implies that MC=ATC has to be satisfied at the minimum level of average cost.)
Q. 4 For a firm with a Total Cost function TC = C(Q), recall that the definitions of Marginal Cost (MC) and Average !3! Total Cost (ATC) are given as follows: Marginal Cost (MC) = C'(Q) Average Total Cost (ATC) = C(Q)/Q Prove that the marginal cost is equal to the average total cost (MC = ATC) when the average total cost is at its minimum value. (Hint: By using the definition of average cost function, find the quantity that minimizes it by applying unconstraint optimization. Show that the FOC of the minimization problem implies that MC=ATC has to be satisfied at the minimum level of average cost.)
Essentials of Economics (MindTap Course List)
8th Edition
ISBN:9781337091992
Author:N. Gregory Mankiw
Publisher:N. Gregory Mankiw
Chapter12: The Cost Of Production
Section: Chapter Questions
Problem 3CQQ
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Q. 4
For a firm with a Total Cost function TC = C(Q), recall that the definitions of Marginal Cost (MC) and Average !3! Total Cost (ATC) are given as follows:
Marginal Cost (MC) = C'(Q)
Average Total Cost (ATC) = C(Q)/Q
Prove that the marginal cost is equal to the average total cost (MC = ATC) when the average total cost is at its minimum value.
(Hint: By using the definition of average cost function, find the quantity that minimizes it by applying unconstraint optimization. Show that the FOC of the minimization problem implies that MC=ATC has to be satisfied at the minimum level of average cost.)
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