Question

Suppose a firm's production function is Q = 4K⁰·⁵ L⁰·⁵. Its level of capital is fixed at 1 unit, the price of labor is PL = 32 per unit, and the price of capital is PK = 400 per unit. Given this information, its short run production function and average product of labor function may be determined. Which of the following is correct?

1) Q = 2L⁰·⁵
2) Q = 4L⁰·⁵
3) APL = 2L⁻⁰·⁵
4) APL = 4L⁻⁰·⁵

Answer 1

The correct option 0.5 Q=4L option B.

Step-by-step explanation:

Given the production function

�

=

4

�

0.5

�

0.5

Q=4K

0.5

L

0.5

and the fixed level of capital (

�

=

1

K=1), we can substitute

�

=

1

K=1 into the production function to simplify it:

�

=

4

(

1

)

0.5

�

0.5

=

4

�

0.5

Q=4(1)

0.5

L

0.5

=4L

0.5

This implies that the short-run production function is

�

=

4

�

0.5

Q=4L

0.5

, and option 2,

�

=

4

�

0.5

Q=4L

0.5

, is correct.

To determine the average product of labor (APL), we use the formula

�

�

�

=

�

�

APL=

L

Q

​

. Substituting the short-run production function into this formula:

�

�

�

=

4

�

0.5

�

=

4

�

−

0.5

APL=

L

4L

0.5

​

=4L

−0.5

Therefore, the correct expression for the average product of labor is

�

�

�

=

4

�

−

0.5

APL=4L

−0.5

, and option 4,

�

�

�

=

4

�

−

0.5

APL=4L

−0.5

, is correct.

Understanding these functions is crucial for firms to optimize their production processes and make informed decisions about resource allocation. In this scenario, the short-run production function

�

=

4

�

0.5

Q=4L

0.5

highlights the relationship between labor input and output, while the average product of labor function

�

�

�

=

4

�

−

0.5

APL=4L

−0.5

provides insights into labor productivity as a function of the variable input

�

L.