FMA - Cost Minimizing Combination of Inputs Lesson

Cost Minimizing Combination of Inputs Lesson

Recall from a previous module, consumers seek to maximize benefits given a budget constraint. The consumer was brought into the utility maximizing level of consumption by consuming at a rate that would make the MU/$ spent for Good A equal to the MU/$ spent for Good B. Firms behave similarly in determining the optimal amounts of labor and capital to employ. In reality, firms have a budget, as well, because they do not have unlimited funds with which to produce. Therefore, firms want to divide their limited funds between labor and capital (and land, for that matter) in such a way as to get the most "bang for every dollar spent." Minimizing costs is one step firms can take to maximize profits.

To ensure the firm is using the least cost combination of inputs, it should seek to employ resources so that:

$LaTeX: \frac{MP_L}{P_L}=\frac{MP_K}{P_K}$ $\frac{MP_L}{P_L}=\frac{MP_K}{P_K}$

The MP_L represents the marginal product of labor and P_L represents the price of labor. The MP_K represents the marginal product of capital and P_K represents the price of capital. Suppose for a given firm the wage paid to the last unit of labor is $10 and that unit of labor has a marginal product of 50 units. Further, suppose the rent paid for the last unit of capital is $50 and that unit of capital has a marginal product of 150 units. Then:

50/$10 ≠ 150/$50

This inequality tells us that for the last unit of labor, each dollar spent by the firm on that unit of labor is allowing the firm to produce 5 additional units of output. Each dollar spent by the firm on the last unit of capital is allowing the firm to produce 3 additional units of output. What should the firm do? It should employ more units of the resource that is rendering a greater amount of output per dollar and less of the resource that is rendering a smaller amount of output per dollar. As the number of laborers is increased by 1 unit, the marginal productivity will decrease (due to diminishing returns to labor). This causes the value of the left side of the equality to decrease. As less capital is employed (view this as moving back up the production schedule), the marginal product of the last unit of capital will increase. The right side of the equality will get larger. By changing the combination of resources in such a way, the firm will bring about a cost minimizing combination of inputs, where it is purchasing units of labor and capital in such a way that the last dollar spent on each is yielding the same marginal product.

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