Optimal Use of an Input

Extra workers are assigned to less productive tasks. These workers generate additional output but at a diminishing rate.

The law of diminishing returns means that the firm faces a basic trade-off in determining its level of production. By using more of a variable input, the firm obtains a direct benefit—increased output—in return for incurring an additional input cost. What level of the input maximizes profits? As before, we look at the firm’s marginal profit, but this time we look at marginal profit per unit of input. We increase the input until the marginal profit per unit of input is zero.

In analyzing this input decision, a definition is helpful. Marginal revenue product is the formal name for the marginal revenue associated with increased use of an input. An input’s marginal revenue product is the extra revenue that results from a unit increase in the input. To illustrate, suppose the auto parts supplier is considering increasing labor from 20 to 30 workers. According to Table 5.2, the resulting marginal product is 4.5 parts per worker. Suppose further that the supplier’s marginal revenue per part is constant. It can sell as many parts as it wants at a going market price of $40 per part. Therefore, labor’s marginal revenue product (MRPL) is ($40)(4.5) $180 per worker. Similarly, the MRPL for a move from 30 to 40 workers is ($40)(5.0) $200 per worker. More generally, labor’s marginal revenue product can be expressed as

[5.2]

where MR denotes marginal revenue per unit of output.3

Now consider the marginal cost of using additional labor. The marginal cost of an input is simply the amount an additional unit of the input adds to the firm’s total cost.4 If the firm can hire as many additional workers as it wishes at a constant wage (say, $160 per day), then the marginal cost of labor is MCL $160. (In some cases, however, the firm may have to bid up the price of labor to obtain additional workers.)

Now note that the additional profit from adding one more worker is the revenue generated by adding the worker less the worker’s marginal cost.

MRPL (MR)(MPL),

M L MRPL MCL.

3In calculus terms, MRPL dR/dL (dR/dQ)(dQ/dL) (MR)(MPL). 4It is important to distinguish between the marginal cost of an input and the marginal cost of an additional unit of output. Taking labor as an example, MCL is defined as C/ L, the cost of hiring an extra worker. In contrast, the added cost of producing an extra unit of output is MC C/ Q.

The firm should continue to increase its labor force as long as the amount of additional profit from doing so is positive, that is, as long as the additional revenue (MRPL) is greater than the additional cost (MCL). Due to diminishing marginal returns, labor’s marginal revenue product eventually will fall. When MRPL exactly matches MCL (that is, when M L 0), increasing the labor force any further will be unprofitable, which leads to the following principle:

To maximize profit, the firm should increase the amount of a variable input up to the point at which the input’s marginal revenue product equals its marginal cost, that is, until:

MRPL MCL. [5.3]

After this point, the marginal cost of labor will exceed the marginal revenue product of labor and profits will decline.

EXAMPLE 1 The human resources manager of the auto parts firm with a 10,000-square-foot plant estimates that the marginal cost of hiring an extra worker is PL $160 per day. Earlier we noted that a move from 20 to 30 workers implies an MRPL of $180 per worker (per day). Since this exceeds the daily cost per worker, $160, this move is profitable. So, too, is a move from 30 to 40 workers (MRPL $200). But an increase from 40 to 50 workers is unprofitable. The resulting MRPL is ($40)(3.3) $132, which falls well short of the marginal labor cost. After this, MRPL continues to decline due to diminishing returns. Thus, the optimal size of the firm’s labor force is 40 workers.

What would be the firm’s optimal labor force if it had in place a 30,000square-foot plant? From Table 5.1, we see that a move from 50 to 60 workers results in an MRPL of $212, a move from 60 to 70 workers an MRPL of $168, and a move from 70 to 80 workers an MRPL of $128. Given a labor price of $160 per day, the firm profits by increasing its labor force up to a total of 70 workers (since MRPL MCL in this range). But an increase beyond this level reduces profitability (MRPL MCL). The firm would best utilize the 30,000-square-foot plant by using 70 workers and producing 520 parts per day.

Let MR $40 and MCL $160 per day. Using the relevant information from Table 5.1,

determine the firm’s optimal number of workers for a 20,000-square-foot plant. Repeat the calculation for a 40,000-square-foot plant.

EXAMPLE 2 Suppose that a firm’s production function is described by

where Q measures units of output and L is the number of labor hours. Suppose that output sells for $2 per unit and the cost of labor is MCL $16 per hour.

Q 60L L2 ,

CHECK STATION 2