If marginal cost equals average cost, then what must be TRUE?

Question

BlackTom AI · Accepted Answer

When marginal cost (MC) equals average cost (AC), the condition characterizes an extremum of the average cost curve and, in standard cost shapes, that extremum is the minimum.

1) Relevant knowledge points and concepts
- Average cost AC(Q) = C(Q)/Q, where C(Q) is total cost and Q is output.
- Marginal cost MC(Q) = dC/dQ.
- The relationship between the slope of the average and the difference between marginal and average: the sign of dAC/dQ is determined by MC − AC.

2) Complete reasoning process step by step
Start with AC(Q) = C(Q)/Q.
Differentiate AC with respect to Q using the quotient rule:
d(AC)/dQ = d/dQ [C(Q)/Q] = [Q·C'(Q) − C(Q)] / Q^2.

3) Intermediate calculations
Replace C'(Q) with MC(Q) and note AC(Q) = C(Q)/Q:
d(AC)/dQ = [Q·MC − Q·AC] / Q^2 = (MC − AC) / Q.

4) Why the correct answer is correct
Set MC = AC. Then d(AC)/dQ = (MC − AC)/Q = 0, so the average cost has zero slope at that output — an extremum. In the usual U-shaped cost context (fixed costs spread at low Q, rising marginal costs at high Q), the average cost falls while MC < AC and rises while MC > AC. Therefore the point where MC = AC is the turning point from falling to rising AC, i.e., the minimum of AC. Thus "Average cost is at its minimum point" is correct.

5) Why the other options are incorrect
- a. "Average cost must be falling." Incorrect: if MC = AC the slope of AC is zero, not negative; AC is not falling at that point.
- c. "Marginal cost at its minimum point." Incorrect: MC = AC does not imply d(MC)/dQ = 0. The minimum of MC is where its derivative is zero, a different condition.
- d. "Average cost must be rising." Incorrect: as with (a), equality implies zero slope, not positive slope.

6) Conclusion
Because d(AC)/dQ = (MC − AC)/Q and MC = AC implies d(AC)/dQ = 0 and, given typical cost behavior, that point is the minimum of AC, the correct answer is: b. Average cost is at its minimum point. This matches the provided answer.

If marginal cost equals average cost, then what must be TRUE?单项选择题

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