Question

May you please walk me through this problem step by step? I got the answers, but I would like to know why. I am having a hard time comprehending where the numbers are coming from.

A firm produces output according to a production function: Q = F(K,L) = min {5K,5L}.

a. How much output is produced when K = 2 and L = 3?

b. If the wage rate is $60 per hour and the rental rate on capital is $35 per hour, what is the cost-minimizing input mix for producing 10 units of output?

Capital:

Labor:

NOTE: What is being added, subtracted, multiplied and/or divided to get Capital and Labor?

c. How does your answer to part b change if the wage rate decreases to $35 per hour but the rental rate on capital remains at $35 per hour?

It does not change.

Capital decreases and labor increases.

Capital and labor increase.

Capital increases and labor decreases.

Answer 1

(a)

Q = min [5K, 5L]

Put K = 2 and L =3

=> Q = min[5*2 , 5*3]

=> Q = min [ 10, 15]

Q = 10.

10 units of output will be produced when K =2 and L =3.

Note: Production function is min production function. So, the minimum value inside the bracket will be the solution.

------------------------------------------------------------------------------------

(b) Q = min[5K, 5L]

The above production function is min production function, it implies inputs K and L are complements to each other and will be used in a fixed proportion to produce each unit of output (Q).

In that case at optimal point; Q = 5K = 5L

=> Q = 5K

Put Q =10

=> 10 = 5K

=>K = (10 / 5)

=> K = 2

and

Q = 5L

Put Q = 10

=> 10 = 5L

=> L = (10 / 5)

=> L = 2

Thus, cost minimizing combination of K and L is 2 units of K and 2 units of L in order to produce 10 units of output.

-----------------------------

(c) The answer in part (b) would not change if there is any change in input price (i.e. if wage rate decrease)

Because K and L will be used in fixed proportion to produce each unit of output regardless of input prices.

Answer: It does not change