Marginal Productivity Theory

We cannot say exactly how many electricians or computer programmers will be demanded within a given time period, but we can present some general rules about how the numbers of people demanded in these occupations might rise or fall. In this analysis, we reencounter many principles. All apply equally to the demand for resource inputs, including labor.

The Profit-Maximizing Level of Employment

The extra output that each additional unit of labor adds to total output is called the marginal product of labor:

MP_L = ∆TP,

∆L

where MP _L is the marginal product of labor;

∆TP is the change in total product;

∆L is adding one unit of labor.

As we saw adding units of the variable input to other fixed inputs will eventually decrease the marginal product of the variable input in the short run.

Marginal product of labor: The change in total output that results from employing one more unit of labor.

Table 2

A Marginal Revenue Product Schedule for Labor

(1) Units of Labor (worker-hours)	(2) Marginal Product of Labor MPL (bushels of corn)	(3) Marginal Revenue MR=P (dollars per bushel)	(4) Marginal Revenue Product MRP=MRx MPL=∆TR/∆L	(5) Marginal Factor Cost (dollars per hour) MFC = Wage
		$2	$28

The marginal revenue product of labor n column (4) falls as more units of labor are employed because the marginal product of labor (column 2) falls. The firm will hire additional units of labor up to the point where the extra revenue these units generate just equals the extra cost of paying for them. At this point, 6 units of labor, MRP = MFC.

When the extra com produced by each additional unit of labor is sold, the resulting increase in total revenue is called marginal revenue product (MRP):

MRP_L = ∆TR/∆L,

where MRP_L is the marginal revenue product of labor;

∆TR is the change to total revenue;

∆L is the change in the amount of labor hired.

Another method of expressing MRP_L, is

MRP_L = MR x MP_L,

where MR is the marginal revenue (the increase in total revenue resulting

from selling one more unit of output);

MP_L is the marginal product of labor.

Marginal revenue product: The change in total revenue that results from employing one more unit of a variable input.

The extra cost of hiring one more unit of labor is called the marginal factor cost of labor:

MFC_L = ∆TC/∆L,

where MFC_L, is the marginal factor cost;

∆TC is the change in total cost.

Marginal factor cost: The change in total cost that results from employing one more unit of a variable input.

The profit-maximizing firm will hire labor to the point where MRP_L = MFC_L.

The numbers used in Table 2 are presented in graph form in Figure 2. The MP_L, curve is drawn in part a by plotting the points from columns (1) and (2). The demand curve for the farmer's com is shown in part b, where P = $2. The MRP_L is found by multiplying marginal revenue by marginal product; the result is the marginal revenue product curve in part c.

The prevailing wage is $8, stated as MFC_L. The profit-maximizing quantity of labor is the point where the MFC_L curve crosses the MRP_L curve. At this point, the marginal revenue product equals the marginal factor cost of labor.

Figure 2. The Firm's Marginal Revenue Product Curve

(a) The farmer's marginal product of labor (MP_L) curve is plotted from columns (1) and (2) in Table 2.

(b) The demand curve for the farmer's product is determined by the product price (P = MR in this perfectly competitive market).

(c) The marginal revenue product (MRP_L) curve is found by multiplying MP_L by MR. The profit-maximizing firm hires labor up to the point where marginal revenue product of labor equals marginal factor cost of labor (MFC_L). With a wage of $8 and a product price of $2, the firm will hire up to 6 units of labor.

Profit-Maximizing Rule for All Inputs

This rule applies to all inputs, such as tractors, seed, land, tractor drivers, and water. The profit-maximizing quantity of all inputs is the amount that makes the MRP of each input equal to the MFC of each input:

MRP ₁ = MRP ₂ = MRP ₃ = … = MRP _N = 1,

MFC₁ MFC₂ MFC₃ MFC_N

where the subscripts 1, 2,and 3,... indicate the MRP and MRC of the first, second, third, and so on, inputs;

Nindicates the total number of inputs used.

Each ratio of MRP to MFC is equal to 1 because the MRP of each is equal to the MFC of each. Under this condition, all inputs are employed in the profit-maximizing amounts.

 

The Demand for Labor

 

Marginal productivity theory gives insight into the purchasing behavior of firms. In the short run, when labor is the only variable input, the firm is willing to purchase or hire additional units up to the point at which MRP = MFC. The firm's marginal revenue product curve (Figure 2c) thus traces the relation between the price of labor (the wage rate) and the amount of labor a firm is willing to purchase. In other words, the marginal revenue product curve is also the firm's short-run demand curve for labor. Figure 3 illustrates this. At wage W_o the firm chooses L_o units of labor. A higher wage such as W₁results in less labor hired. For each wage, the firm adjusts the level of employment to maintain the equation MRP_L = MFC_L.

Figure 3. The Firm's Short-Run Demand for Labor

The firm hires labor up to the point where the wage is equal to the marginal revenue product of labor (MRP_L). The marginal revenue product curve shows the various quantities of labor the firm is willing to hire at all the different wage rates when labor is the only variable input.

 

A single firm's demand for labor in the short run is equivalent to its marginal revenue product of labor. To obtain the overall market demand for labor in the short run, we must make some additional calculations.

Each firm will, of course, demand labor up to the point where MRP_L = MFC_L. Thus, at a particular wage, the market demand for labor may be shown as the sum of all the firms' MRP_L curves. Figure 4 shows such a curve, MRP_LO. This curve represents simply the sum of all the firms' short-run demand curves for a particular type of labor. The curve labeled MRP_LO, however, is not the short-run market demand curve for this type of labor, but it does establish one point on the short-run market demand curve, D.

To establish the whole demand curve, we must examine what happens when the wage rate changes. Imagine the overall demand for corn huskers. Many farmers employ huskers at wage W₀, the total number of corn huskers employed at W₀ is L_0. Recall that the MRP_L curve is obtained by multiplying the price of corn by the marginal product of labor. If the wage rate rises to W₁, then each firm's costs of production will rise. The increased costs of production will decrease the supply of com and increase the equilibrium price. To obtain the new MRP_L curve (MRP_L1), we multiply the marginal product of labor by the new, higher price of corn, and we sum this quantity over all the firms. The new wage, W₁, and level of employment, L₁, establish a second point on the short run market demand curve. The short run market demand curve, D, differs from the sum of the MRP_L curves because it allows for increases in the price of the product that result from wage increases.

Figure 4. The Sort-Run Market Demand for Labor

MRP_LO represents the sum of all the firms' marginal revenue product curves for a particular occupation. When the wage for this occupation is W₀, the firms hire L_o units of labor. If the wage rises to W₁, the firms initially hire the lower L_o1 units of labor. However the increase in wages causes an increase in the price of the product. The higher product price shifts the sum of the firms MRP_LO to MRP_L1. At W₁, when labor is the only variable input the firms hire L₁ units of labor. D represents the short-run market demand curve for labor because it allows for changes in the wage rate.