Question

Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant’s long-run production function is Q = 4K0.75L0.25, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. For this production function, the MPL = K0.75/L0.75 and the MPK = 3L0.25/K0.25. The weekly cost function for the paper plant is C = 10K + 2L, where C is the total weekly cost. What ratio of capital to labor minimizes Miguel and Jake’s total costs? How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week? How much will hiring these inputs cost them?

Answer 1

To minimize total costs, Miguel and Jake need to find the ratio of capital to labor where the marginal product of capital equals the marginal product of labor. They can then determine the amount of capital and labor needed to produce 1,000 reams of paper each week by solving the production function equation. The cost of hiring these inputs can be calculated using the cost function.

Step-by-step explanation:

To minimize Miguel and Jake’s total costs, they need to find the point at which the ratio of marginal product of capital (MPK) to marginal product of labor (MPL) is equal to the ratio of capital cost (MK) to labor cost (ML). In this case, the ratio is MPK/MPL = 3L^0.25/K^0.25 = MK/ML = 10K/2L. Simplifying this equation, we get 3L^0.25/K^0.25 = 5K/L. To find the minimum cost, we need to solve for the ratio of L to K:

L^0.25/K^0.25 = 5/3

L^0.25 = 5/3*K^0.25

L = (5/3)^4 * K

To produce 1,000 reams of paper each week, Miguel and Jake need to rent and hire enough capital and labor to satisfy the production function Q = 4K^0.75L^0.25 = 1,000. Plugging this value into the production function, we get:

(4K^0.75)(L^0.25) = 1,000

Substituting the value we obtained for L in terms of K, we can solve for the amount of capital required:

(4K^0.75)((5/3)^4 * K)^0.25 = 1,000

Solving this equation will give us the value of K, and we can then calculate the amount of labor required using the equation L = (5/3)^4 * K. Finally, we can calculate the cost of hiring these inputs using the cost function C = 10K + 2L.