Consider again that a rm operates in the long run, using the following productio
ID: 1164514 • Letter: C
Question
Consider again that a rm operates in the long run, using the following production function: Q = 2 L K , where Q is units of output, L is units of labor, and K is units of capital. Suppose that the firm produces total 24 units of output. The price of labor, w, is $10, and the price of capital, r, is $50. Answer the following questions.
(a) Find six integer combinations of L and K producing 24 units of Q.
(b) Calculate the total cost of each combination of L and K obtained in part (a), and find the optimal combination of L and K that minimizes the total cost.
Explanation / Answer
a. Given the production function, the six integers combinations of L and K are.
1. L=1, K=12; Q=24
2. L=2, K=6; Q=24
3. L=3, K=4; Q=24
4. L=4, K=3; Q=24
5. L=6, K=2; Q=24
6. L=12, K=1; Q=24
b. The formula for total cost is,
Total Cost = w.L + r.k.............................(1)
w is the price of labour and r is the price of capital.
L and K are quantities of labour and capital, respectively.
Putting the values of w, r, L and K in equation 1, the total cost for each unit is calculated.
The following table shows the Total cost for each combination of L and K.
From the above table, the minimum cost is $160 with the combination, L=6 and K=1. Hence, the optimal combination of L and K that minimizes the cost are L=6 and K=2
L K Total Cost 1 12 610 2 6 320 3 4 230 4 3 190 6 2 160 12 1 170