Question

A plant's production function is f(L,K)=LK. For this production function, MPK​=L and MPL​= K. The price of, labor, w, is $15 and of capital, r, is $30 per unit.

a) In the short run, the plant's capital is fixed at Kˉ=50. Find the amount of labor it must employ to produce Q=28,800 units of output.
b) How much money is the firm losing by not having the ability to choose its level of capital optimally? ..

Answer 1

In the short run, the plant must employ 576 units of labor to produce 28,800 units of output. The firm is losing $924 by not having the ability to choose its level of capital optimally.

Step-by-step explanation:

In the short run, with fixed capital (K), the plant's production function becomes Q = f(L). To find the amount of labor required to produce 28,800 units (Q), we can use the production function and rearrange the equation as L = Q/K. Substituting the given values, we get L = 28,800/50 = 576 units of labor.

To calculate the money the firm is losing by not having the ability to choose its level of capital optimally, we need to find the difference in cost between the fixed capital and the optimal capital. The optimal capital can be calculated by dividing the marginal product of capital (MPK) by the capital price (r), which gives us K_optimal = MPK/r = L/30 = 576/30 = 19.2 units of capital. The cost difference is the difference in the cost of fixed capital and optimal capital, which is (50 - 19.2) * r = 30.8 * 30 = $924.