# Consider the following short-run production function: q = 5L2 – (1/3) L3. a. At what level of L do diminishing marginal returns begin? Show your derivation.b. At what level of L do diminishing...

Consider the following short-run production function: q = 5L2 – (1/3) L3.

a. At what level of L do diminishing marginal returns begin? Show your derivation.

b. At what level of L do diminishing returns begin? Show your derivation.

c. At what level of L does the marginal product of labor equal the average product of labor? Show your derivation.

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### 1 Answer

The short-run production function is given by q = 5L^2 – (1/3)*L^3 where L is the labor and q is the output.

The derivative of q with respect to L gives the marginal return, or the change in output with change in labor employed.

q' = 10*L - L^2

When the derivative of q' is negative a stage of diminishing marginal returns is reached.

q'' = 10 - 2L

10 - 2L < 0

10 < 2L

L > 5

At values of L greater than 5, the marginal returns start to diminish.

The returns start to diminish when q'<0 or

10*L - L^2 < 0

L^2 > 10*L

L > 10

The average product of labor is q/L

APL = (5L^2 – (1/3)*L^3)/L

APL = 5L - L^2/3

The average product of labor is equal to the marginal product of labor when

5*L - L^2/3 = 10*L - L^2

(2/3)*L^2 = 5*L

L = (5*3)/2 = 7.5

At L = 7.5, the average product of labor is equal to marginal product of labor.

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