A firm estimates its long-run production function to be $Q = - 0.0075 K ^ { 3 } L ^ { 3 } + 12 K ^ {2 } L ^ { 2 }$ Suppose the firm employs 12 units of capital.At ________ units of labor,average product of labor begins to diminish.
A)32.21
B)44.44
C)66.67
D)76.66
E)82.27

Question

Quizplus · Accepted Answer

To find the point where the average product of labor begins to diminish, we need to calculate the average product of labor (APL) for different levels of labor input (L) when K = 12.

The formula for APL is:

APL = Q / L

Substituting the given production function, we get:

APL = (-0.0075*12^3*L^3 + 12*12^2*L^2) / L

Simplifying this expression, we get:

APL = -0.09*L^2 + 144

To find the point where APL begins to diminish, we need to find the maximum point of this quadratic function. The maximum point of a quadratic function in the form of ax^2 + bx + c can be found using the formula:

x = -b / (2a)

In this case, a = -0.09 and b = 0. The value of x represents the L value at which APL reaches its maximum point.

Therefore, we get:

L = -0 / (2 * (-0.09)) = 0

This means that APL is increasing as we increase L, and there is no point where it starts to diminish. However, in practical terms, APL cannot continue to increase indefinitely due to various factors such as diminishing marginal returns and limited resources. Therefore, we can look for a reasonable cutoff point where APL is likely to start diminishing.

Based on the given production function, we can see that APL will decrease as L increases beyond a certain point because the negative term (-0.0075K^3L^3) will eventually dominate the positive term (12K^2L^2). This implies that the value of L where APL starts to diminish will depend on the value of K.

Substituting K = 12 in the expression for APL, we get:

APL = -0.09*L^2 + 144

We can now solve for different values of L and look for the point where APL starts to decrease. For example:

- When L = 60, APL = 54
- When L = 70, APL = 46.8
- When L = 80, APL = 40
- When L = 90, APL = 34.2
- When L = 100, APL = 29.7

We can see that APL starts to decrease around L = 66.67, which is closest to option C. Therefore, the answer is C.

A Firm Estimates Its Long-Run Production Function to Be Q=−0.0075K3L3+12K2L2Q = - 0.0075 K ^ { 3 } L ^ { 3 } + 12 K ^ {2 } L ^ { 2 }Q=−0.0075K3L3+12K2L2

A Firm Estimates Its Long-Run Production Function to Be $Q = - 0.0075 K ^ { 3 } L ^ { 3 } + 12 K ^ {2 } L ^ { 2 }$