Refer to the following:
The price of capital (r) is $20.

-What combination of K and L should the firm choose to produce 14,000 units of output at the lowest cost?
A) 180K, 120L
B) 180K, 0L
C) 60K, 120L
D) 90K, 60L
E) none of the above

Question

Refer to the following:
The price of capital (r) is $20.
 
-What combination of K and L should the firm choose to produce 14,000 units of output at the lowest cost?
A) 180K, 120L 
B) 180K, 0L 
C) 60K, 120L 
D) 90K, 60L 
E) none of the above

Quizplus · Accepted Answer

We need to use the formula for the cost-minimizing combination of capital and labor, which is:

MPK/PL = r/w

Where:
- MPK = marginal product of capital
- MPL = marginal product of labor
- r = price of capital
- w = price of labor

We are given that r = $20, so we just need to find the values of MPK, MPL, and w.

To do this, we can use the production function given in the previous question:

Q = 10K^0.5 L^0.5

Taking the partial derivative of this function with respect to K and L, we get:

MPK = 5Q/K
MPL = 5Q/L

We also know that the firm needs to produce 14,000 units of output, so:

Q = 14,000

Substituting these values into the MPK/PL = r/w formula, we get:

5(14,000)/60L = 20/w

Simplifying this equation, we get:

L = 90K/7

Substituting this back into the production function, we get:

14,000 = 10K^0.5 (90K/7)^0.5

Solving for K, we get:

K = 655.2

Substituting this back into the equation for L, we get:

L = 841.4

So the cost-minimizing combination of capital and labor is:

K = 655.2
L = 841.4

Rounding to the nearest whole number, we get:

K = 655
L = 841

Therefore, the answer is D.

Refer to the Following