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IAI Actuarial Core Principles · Business Economics · Production, costs, revenue and profit in price and output decisions

A Hyderabad insurer's long-run total cost is LTC = 500Q - 8Q^2 + Q^3/3 in suitable units. At which output level does the long-run average cost reach its minimum, indicating the end of economies of scale in this model?

Minimum long-run average cost occurs at Q = 12. Dividing total cost by Q gives LAC = 500 - 8Q + Q²/3; setting its derivative -8 + 2Q/3 to zero gives Q = 12, and the positive second derivative confirms it is a minimum.

  1. AQ = 8
  2. BQ = 12Correct
  3. CQ = 16
  4. DQ = 24
  5. Q = 30

Explanation

LAC = 500 - 8Q + Q^2/3. Differentiate: -8 + 2Q/3 = 0, giving Q = 12. The second derivative 2/3 is positive, confirming a minimum. Q = 24 results from forgetting to divide by 2/3 properly, and Q = 8 comes from setting the Q coefficient wrongly.

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