Minimum Wage Laws And How Firms Find The Optimal Mix Of Capital And Labor

Firms use both capital and labor. They need to use the optimal mix to maximize profits. If they do so, the following equation will be true

MPL/W = MPK/R

MPL is the marginal product of labor. It means how many more units get produced if you add 1 worker. MPK is something similar for capital. W is the wage rate or cost of 1 unit of labor. R is the cost of 1 unit of capital. This is also sometimes called the least-cost rule.

When those two fractions are equal, a firm minimizes the cost of producing a given level of output. It means that spending $1 more on labor increases output just as much as spending $1 more on capital. If this were not true, the firm would spend more on labor (if the labor side of the relation were greater).

But if you add workers, their MP falls because of the law of diminishing returns. So the labor side of the relation falls. The firm also spends less on capital. The capital side of the relation rises. So even when they start out unequal, they eventually end up that way.

Now what if wages go up? This will happen if the government imposes a minimum wage. This will make the labor side of the relation lower. Then the firm has an incentive to substitute capital for labor, resulting in job losses.

Here is an example:

MPL = 12
W = 2
MPK = 12
R = 2

MPL/W = 12/2 = 6 = MPK/R = 6 = 12/2

Now the government says the wage has to be 3. Then MPL/W = 12/3 = 4

That means the last dollar spent on labor increases the firm's output by 4 but the last dollar spent on capital increases output by 6. So capital is giving the firm a better deal now. So they will want to use more capital. That will require them spending less on labor. That results in job losses.