2 marksMicroeconomicsCosts of productionDeriving cost functions and cost curvesHardabout 3 min

GATE 2023 Question 59: Costs of production

Consider the following short-run cost function:

$C(q) = 10q^3-80q^2+300q+50$

At the minimum average variable cost (AVC), the value of marginal cost (MC) is ______ (in integer).

Answer and worked solution

Answer: 140

Variable cost is the cost excluding the fixed cost (the constant term, 50): $VC(q) = 10q^3-80q^2+300q$ .

$AVC(q) = \frac{VC(q)}{q} = 10q^2-80q+300$

Minimizing AVC: $\dfrac{d(AVC)}{dq} = 20q-80=0 \Rightarrow q=4$ .

(Second-order check: $\dfrac{d^2(AVC)}{dq^2}=20>0$ , confirming a minimum.)

Marginal cost: $MC(q) = \dfrac{dC}{dq} = 30q^2-160q+300$ .

Evaluate MC at $q=4$ (a well-known property: MC always crosses AVC exactly at AVC's minimum point, which also serves as a useful cross-check here):

$MC(4) = 30(16) - 160(4) + 300 = 480 - 640 + 300 = 140$

Therefore the marginal cost at minimum AVC is 140.

The trap: 20

A common error is computing only $q=4$ (the quantity that minimizes AVC) and mistakenly reporting that quantity itself, or the value of AVC at that quantity, instead of correctly evaluating the separate marginal cost function MC(q) at $q=4$ .

Plain-text version

Consider the following short-run cost function: C(q) = 10q³-80q²+300q+50 At the minimum average variable cost (AVC), the value of marginal cost (MC) is (in integer).

Also written as: Consider the following short-run cost function: C(q) = 10q^3-80q^2+300q+50 At the minimum average variable cost (AVC), the value of marginal cost (MC) is (in integer).

Costs of production has come up 83 times in 7 exams since 2006.

What this question tests

This question is on Deriving cost functions and cost curves, part of Costs of production in Microeconomics.

Working out cost curves from a cost function, a production function or a marginal cost schedule.

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