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CMA Foundation · Fundamentals of Business Economics and Management · Theory of Production

A firm's short-run production function is TP = 30L + 9L^2 - L^3, where L is the number of workers. Using calculus, at how many workers does the marginal product reach its maximum?

Marginal product is maximum at 3 workers. Differentiating TP gives MP = 30 + 18L - 3L squared. Setting the derivative of MP, 18 - 6L, equal to zero gives L = 3, and the negative second derivative confirms a maximum.

  1. AL = 2
  2. BL = 3Correct
  3. CL = 5
  4. DL = 9

Explanation

MP = dTP/dL = 30 + 18L - 3L^2. Setting dMP/dL = 18 - 6L = 0 gives L = 3, and the second derivative -6 is negative, confirming a maximum. L = 5 solves MP = 0 for the wrong function, namely 30 + 18L - 3L^2 = 0 gives L about 7.4, so it is not a root of any step. L = 9 would be setting TP's quadratic coefficient equal to the answer incorrectly.

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