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Production Function and Total Product of Labor

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A firm's production function is given by:
Q= Le^-0.2L

Find the value of L which maximizes the TOTAL PRODUCT of labor. In obtaining this result ensure that you:

(a) Derive the first-order condition for the firm's product maximization.
(b) Verify that, at the critical point, the second-order sufficient condition holds.
(c) Check that average product equals marginal product at the critical point.

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A neat and step-by-step solution to determine the production function and total product of labor.

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Q = L e^-0.2L

(a) dQ/dL = L (-0.2 e^-0.2L) + e^-0.2L

= e^-0.2L (-0.2L + 1)

For Q to be maximum, dQ/dL = 0

Therefore e^-0.2L (-0.2L + 1) = 0

This gives -0.2L + 1 = 0

-0.2L = -1

L = 5

The critical point is L = 5.

(b) d^2Q/dL^2 = e^-0.2L (-0.2) + (-0.2L ...

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