Question

9. A firm produces output according to a production function Q = F(K,L) min [2K,4L]. a. How much output is produced when K-2 and L = 3? b. If the wage rate is $30 per hour and the rental rate on capital is $10 per hour, what is the cost-minimizing input mix for producing 4 units of output? How does your answer to part b change if the wage rate decreases to $10 per hour but the rental rate on capital remains at $10 per hour? c.

Answer 1

Solution:

We are given production function: Q = F(K, L) = min{2K, 4L}

a) With K = 2 and L = 3, output produced is

Q = min{2*2, 4*3} = min{4, 12} = 4 units

b) Given wage rate, w = $30 per hour; rental rate, r = $10 per hour

We have to find the cost-minimizing input mix to produce 4 units of output.

Total cost = w*L + r*K = 30L + 10K

Cost minimizing input mix in case of such Leontief production function occurs at the kink. Given the function, we know kink occurs at 2K = 4L

Cost minimizing condition gives 2K* = 4L*

Also, we want Q = 4 units. Then, with 4 = 2K* = 4L*

Solving these equations give:

2K = 4, giving K* = 2

4L = 4, giving L* = 1

Thus, cost minimizing input mix in order to produce 4 units of output is (L*, K*) = (1, 2)

(Minimum) Total cost in this case = 30*1 + 10*2 = $50

c) Now, the wage rate, w = $10 per hour; r = $10 per hour

Notice that the production function is still unchanged, and thus the kink point is still same.

Overall, we have the same cost-minimizing condition as in part (b). For Leontief production function, change in wage or rental rate have no effect on the optimal input mix, only the total cost changes.

Now, Total cost, TC = 10*1 + 10*2 = $30