Part 1A firm has two plants that produce identical output. The cost functions areUpper C 1 equals 399 q minus 12 q squared plus 0.5 q cubedC1=399q−12q2+0.5q3 and Upper C 2 equals 399 q minus 16 q squared plus 1.0 q cubedC2=399q−16q2+1.0q3.Part 2First, note that if AC 1AC1equals=399399minus−1212qplus+0.50.5q squaredq2, then StartFraction dAC 1 Over dq EndFraction equals negative 12 plus 2 left parenthesis 0.5 right parenthesis qdAC1dq=−12+2(0.5)q. Similarly, if AC 2AC2equals=399399minus−1616qplus+1.01.0q squaredq2, then StartFraction dAC 2 Over dq EndFraction equals negative 16 plus 2 left parenthesis 1.0 right parenthesis qdAC2dq=−16+2(1.0)q.Part 3At what output level does the average cost curve of each plant reach its minimum?Part 4The first plant reaches minimum average cost at [input]1212 units of output. (Enter a numeric response using an integer.)Part 5The second plant reaches minimum average cost at [input]enter your response here units of output. (Enter a numeric response using an integer.)

Question

Part 1A firm has two plants that produce identical output. The cost functions areUpper C 1 equals 399 q minus 12 q squared plus 0.5 q cubedC1=399q−12q2+0.5q3 and Upper C 2 equals 399 q minus 16 q squared plus 1.0 q cubedC2=399q−16q2+1.0q3.Part 2​First, note that if AC 1AC1equals=399399minus−1212qplus+0.50.5q squaredq2​, then StartFraction dAC 1 Over dq EndFraction equals negative 12 plus 2 left parenthesis 0.5 right parenthesis qdAC1dq=−12+2(0.5)q. ​Similarly, if AC 2AC2equals=399399minus−1616qplus+1.01.0q squaredq2​, then StartFraction dAC 2 Over dq EndFraction equals negative 16 plus 2 left parenthesis 1.0 right parenthesis qdAC2dq=−16+2(1.0)q.Part 3At what output level does the average cost curve of each plant reach its​ minimum?Part 4The first plant reaches minimum average cost at [input]1212 units of output. ​(Enter a numeric response using an​ integer.)Part 5The second plant reaches minimum average cost at [input]enter your response here units of output. ​(Enter a numeric response using an​ integer.)

BlackTom AI · Accepted Answer

To determine where the average cost curve for each plant is minimized, we first express average cost (AC) as AC = C(q)/q for each plant.
For Plant 1: C1(q) = 399q − 12q^2 + 0.5q^3. Dividing by q gives AC1 = 399 − 12q + 0.5q^2. The minimization occurs where the derivative d(AC1)/dq = −12 + 2(0.5)q simplifies to −12 + q. Setting this equal to zero yields q = 12. Therefore, the first plant’s minimum average cost occurs at q = 12 units of output.
For Plant 2: C2(q) = 399q − 16q^2 + 1.0q^3. Dividing by q gives AC2 = 399 − 16q + 1.0q^2. Differentiate to get d(AC2)/dq = −16 + 2(1.0)q, which simplifies to −16 + 2q. Setting this equal to zero gives q = 8. Thus, the second plant’s minimum average cost occurs at q = 8 units of output.

In terms of common pitfalls, it’s important to minimize AC by differentiating AC, not merely inspecting C(q) or MC, and to ensure the derivative calculations reflect the division by q when forming AC. If one confused AC with total cost directly or used the marginal cost condition MC = AC without the AC expression, the result would be incorrect. Here, the algebra cleanly leads to q = 12 for Plant 1 and q = 8 for Plant 2, which matches the provided answers.

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