Final answer:
a. When K = 2 and L = 3, 9 units of output are produced. b. The cost-minimizing input mix for producing 9 units of output is 0 units of capital and 3 units of labor.
Step-by-step explanation:
a. In order to determine the amount of output produced, we substitute K = 2 and L = 3 into the production function. Q = F(K,L) = min {9K,3L}. So, Q = min {9(2), 3(3)} = min {18, 9} = 9. Therefore, 9 units of output are produced when K = 2 and L = 3.
b. The cost-minimizing input mix for producing 9 units of output can be found by comparing the marginal product of capital (MPK) and the marginal product of labor (MPL) with their respective prices. Let's calculate MPK and MPL for the given production function:
MPK = ∂Q/∂K = 9 if 9K ≤ 3L, and MPK = 0 if 9K > 3L
MPL = ∂Q/∂L = 3 if 3L < 9K, and MPL = 0 if 3L ≥ 9K
If the wage rate is $70 per hour and the rental rate on capital is $45 per hour, we can compare the ratios of prices to marginal products to determine the cost-minimizing input mix. In this case, we need to find a combination of K and L such that (MPK/PK) = (MPL/PL). Let's calculate:
(MPK/PK) = 9/45 = 0.2
(MPL/PL) = 3/70 ≈ 0.043
Since (MPK/PK) > (MPL/PL), we should use more labor and less capital to minimize costs. Therefore, the cost-minimizing input mix for producing 9 units of output is:
Capital: 0 units
Labor: 3 units