Marginal Products and the Least-Cost Rule

Every firm must decide how to produce its output. Should electricity be produced with oil or coal? Should cars be assembled in the United States or Mexico? Should classes be taught by faculty or graduate  students? We now complete the link between
production and cost by using the marginal product concept to illustrate how firms select the least-cost combinations of inputs.

In our analysis, we will rely on the fundamental assumption that firms minimize their costs of production.

This cost-minimization assumption actual makes good sense not only for perfectly competitive firms but for monopolists or even nonprofit organizations like. colleges or ° hospitals. It chimp states that the firm should strive to produce its output at the lowest possible cost and there vb have maximum amount of revenue left over for pr or for other objectives.

A simple example will illustrate h might decide between different input functions. Say a firm’s engineers ha -e calculated the desired output level of 9 uni diced with two possible options, energy (E) costs $2 per unit.



The shaped marginal cost curve in (b) arises from the shaped of the marginal product curve in (a). With fixed land and nriable labor, the marginal product of labor in (a) to the left of B, peaks at B, and then falls at D as returns to labor set in.
marginal cost curve derives from production data region to the left of Bin (b)-such as at point A arginal product means that marginal cost is falling; peak marginal product occurs at minimum marginal the right of B (e.g., at D). the marginal cost of pro output increases as the marginal product of labor tall, increasing and then diminishing marginal to the variable factor gives a Ll-shaped marginal ‘e.

per hour. Under option I, the input mix is E = 10 and L = 2. Option 2 has E = 4 and L = 5. Which is the preferred option? At the market prices for inputs, total production costs for option 1 are ($2 X 10) + ($5 X 2) = $30, while total costs for option 2 are ($2 X 4) + ($5 X 5) = $33. Therefore, option .1 would be the preferred least-cost combination of inputs.

More generally, there are usually many possible input combinations, not just two. But we don’t have to calculate the cost of every different combination of inputs in order to find the one which costs the least. Here’s a simple way to find the least-cost combination:

Start by calculating the marginal product of each input, as we did in Chapter 6. Then divide the marginal product of each input by its factor price. This gives you the matginal product per dollar of input. The cost-minimizing combination of inputs comes when the marginal product per dollar of in- put is equal for all inputs. That is, the marginal contribution to output of each dollar’s worth of labor, of land, of oil, and so forth, must be just the same.

Following this reasoning; a firm will minimize its total cost of production when the marginal product per dollar of input is  qualized for each factor of production. This is called the least-cost rule. Least-cost rule: To produce a given level of outplll at least cost, a firm should buy inputs until it has equalized the marginal product per dollar spent on each input. This implies that arginal product of L Price of L mar!4inal product of A price 01″ A . This rule for firms is exactly analogous to what consumers do when they maximize utilities, as we saw in Chapter 5. In analyzing consumer choice, we saw that to maximize utility, consumers should
buy goods so that the marginal utility per dollar spent on each consumer good is equalized for all commodities.

One way of understanding the least-cost rule is the following: Break each factor into packages worth $1 each. (In our earlier energy-labor example, $1 of would be one-fifth of an hour, while $1 of energv would be Ih unit.) Then the least-cost rule
states that the marginal product of each dollar-unit of input must be equalized. If the marginal products per $1 of inputs were not equal, you could reduce the low-MP-per-dollar input and increase the -MP-per-do))ar input and produce the same output at lower cost. A corollary of the least-cost r:ule is the substitution rule. Substitution rule: If the price of one factor falls while all other factor prices remain the same, firms will profit by substituting the now-cheaper factor for the other factors until the marginal products per dollar are equal for all inputs. Let’s take the case of labor (L). A fall in the price of labor will raise the ratio
ratio for other inputs. Raising the employment of L lowers MPL by the law of diminishing returns and therefore lowers Lower price and MP of labor then bring the marginal product per dollar for labor back into equality with that ratio for other factors.

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