A Production Function That Uses Only Labor

We summarize the technological possibilities of a firm using a production functionA description of how much output a firm can produce as it varies its inputs., which is a description of how much output a firm can produce as it varies its inputs. Even though a typical firm’s production function contains many different inputs, we can understand most of the key features of the production function using an example where labor is the only factor of production. Although there are few goods or services that literally require no inputs other than labor, there are many services that are highly labor intensive, such as babysitting, housecleaning, and personal training at a gym.

To be concrete, think about housecleaning and suppose it has the following production function:

output = productivity × hours of labor input,

where we think of productivity as just some number. If output measures clean houses, and if it takes 5 hours of labor to produce one clean house, then productivity is 0.2, and the production function is

output = 0.2 × hours of labor input.

The production function tells us the level of output of a firm for given levels of labor input. Labor input is the total hours of labor time used by a firm. At this point, we are not distinguishing between hours worked per person and the number of people working, so a firm with 8 employees each working 20 hours per week has the same weekly labor input as a firm with 4 employees each working 40 hours per week. Table 9.1 "Production Function for Housecleaning" lists the amount of output that a housecleaning firm can obtain from various levels of input. We call this a linear production function because its graph is a straight line, as shown in Figure 9.6 "A Linear Production Function".

The marginal product of laborThe amount of extra output produced from one extra hour of labor input. is the amount of extra output produced from one extra hour of labor input and is defined as

$$\text{marginal product of labor}=\frac{\text{change in output}}{\text{change in labor input}}\text{.}$$

When the production function is linear, the marginal product of labor is constant. It is equal to the number we labeled productivity in our original production function.

In most cases, the marginal product of labor is not constant. To understand why, imagine you are managing a Starbucks outlet. You already have the machines to produce espresso, and you have lots of coffee beans on hand. You also have 500 square feet of space for making coffee and charging customers. But you still need labor. If you have no barista to operate the espresso machine, then you will have no output. If you hire one worker, you will be able to serve coffee to people. Adding the first worker will increase output considerably. However, that person must not only make the coffee but also clear the tables and handle the cash register. Adding a second worker to help with the register and clear tables will increase output even more. Now suppose you keep increasing the number of workers in the 500 square feet of space. After the third or fourth worker, they will start to bump into each other, and the barista will start to be very annoyed and unproductive. In other words, because one of your inputs—the amount of available space—is fixed, each additional worker contributes less and less to output. We call this the diminishing marginal product of laborThe idea that each additional hour of labor input contributes a smaller and smaller amount to output..

Table 9.2 "Production Function for Coffee with a Diminishing Marginal Product of Labor" is an example of a production function with a diminishing marginal product of labor. In creating this table, the labor input is changed while holding all other inputs (the size of the café, the number of espresso machines, etc.) fixed.

The marginal product of labor is shown in the third column. For the first few entries, you can calculate it directly from the table because you can easily determine how much output changes from one row to the next. For example, the marginal product of the third hour of labor is 17.3 – 14.1 = 3.2. Finding the marginal product of, say, the 40th unit of labor from the table is trickier because the table doesn’t tell us how much we can produce with 39 hours of labor. Looking back at the formula for the marginal product of labor, however, we can calculate it:

$$\text{marginal product of labor}=\frac{\text{change in output}}{\text{change in labor input}}=\frac{63.2-59.2}{40-35}=\frac{4}{5}=\mathrm{0.8.}$$

We illustrate this production function in Figure 9.7 "A Production Function with a Diminishing Marginal Product of Labor". Notice that while the slope of the production function is always positive, the slope decreases as the labor input increases.

Marginal Cost

The cost function in Figure 9.8 "The Cost Function" is linear. Because the production function has a constant marginal product of labor, the cost function displays constant marginal cost. What about the case in which the production function has a diminishing marginal product? Then additional labor provides less and less output. Turning this around, it follows that producing each additional unit of output requires more and more labor. We show this in Figure 9.9 "The Cost Function with a Decreasing Marginal Productivity of Labor". In this figure, the marginal cost is increasing, so the cost function gets steeper as we produce more output.

We show the marginal cost curve in Figure 9.10 "The Marginal Cost Function". In this example, marginal cost is a straight line, but this need not be the case in general.

Marginal cost depends on the following:

• The cost of inputs into the production process
• The productivity of the inputs into the production process

If the costs of inputs increase, then the marginal cost is higher is well. If the productivity of the inputs into the production function is higher, then the marginal cost is lower. In fact, marginal cost can be written as

$$\text{marginal cost}=\frac{\text{wage}}{\text{marginal product of labor}}\text{.}$$

We can see from this equation that when the marginal product of labor decreases, the marginal cost of production increases. We see also that an increase in the cost of inputs—in this case an increase in wages—leads to an increase in the marginal cost of production.

The Choice of Inputs

There is one issue that we are ignoring here. Firms typically have many different ways in which they can produce the same quantity of output. A firm might have a choice between two production processes: (1) a process that is simple and cheap to operate but wasteful of raw materials and (2) a process with recycling that uses fewer materials but is more complicated and costly to run. As another example, if a construction company needs to dig a ditch, it could employ 20 people and equip each with a shovel, or it could hire a single individual and a backhoe. Economists say that the first process is labor intensive because it requires a lot of labor relative to capital; they call the second process capital intensive because it requires a relatively large amount of capital.

In medium-sized or large firms, there is usually a specific functional area, called operations, that decides how to produce output. Operational decisions are governed in large part by technical or engineering considerations: what are the ways in which it is physically possible to transform inputs into the desired output? Operational decisions also have an economic component. Given that there may be many different ways to get the same final amount of output, which is the most cost effective? In economics, not surprisingly, we focus on the second of these questions and leave the first to engineers and other technical experts.

The basic principle is intuitive: operations managers tend to choose methods of production that economize on relatively expensive inputs. For example, much garment manufacture takes place in countries like China or Vietnam, where wages are low (that is, labor is relatively cheap). As a result, the production processes tend to be highly labor intensive, using a lot of workers relative to the amount of machinery. By contrast, garment manufacture in richer countries (where labor is much more expensive) tends to use methods of production that require fewer people and more machines.

We simply presume that the operations function of a firm is doing a good job and has succeeded in finding the cheapest way of producing the firm’s output, taking into account both the technical aspects of production and the costs of different inputs. When we talk about the cost function of a firm, therefore, we are assuming that it gives us the lowest cost for producing each given level of output.

Changes in the Real Wage

Changes in the cost of labor are one reason firms create or destroy jobs. Decreases in the real wage lead firms to produce more output and hire more workers, thus creating jobs. Increases in the real wage cause firms to produce less output and lay off workers. Going deeper, we can ask why the real wage might change. The answer comes from looking back at Figure 9.4 "Labor Market Equilibrium". The real wage changes, causing a change in the quantity of labor demanded, if the labor supply curve shifts.

Population growth is one source of a shift in labor supply. As the number of workers in the economy grows, then total labor supply will shift. At a given wage, there will be more workers and hence the labor supply curve will shift to the right. Other things equal, this causes a decrease in the real wage.

Changes in labor market participation also shift the labor supply curve. A leading example of this is the increased participation of women in the labor market. In the United States, the fraction of women in the labor force rose from about 20 percent in 1950 to about 70 percent in 2000. Participation might also change because of workers’ expectations about the future state of the labor market. If you are worried you won’t have a job next year, you might want to work this year.

Changes in Productivity

Changes in productivity—more precisely, in the marginal product of labor—work exactly like changes in the real wage. Remember that marginal cost depends on both real wages and productivity. If productivity increases, perhaps because a firm has upgraded its capital equipment, then marginal cost decreases. Firms will produce more output and hire more labor. The opposite is true if productivity decreases: in this case, firms will produce less and destroy jobs.

Over long periods of time, productivity in an economy increases. This increase in productivity is driven largely by technological advances: firms get better at producing goods and services and so are able to produce them more cheaply. As workers’ productivity increases, firms demand more labor at any given wage.

Changes in Demand

When a firm’s product becomes more desirable in the market, the demand curve that it faces shifts outward. This shift in demand typically leads to an outward shift in marginal revenue, inducing a firm to produce more output and demand more labor. We show this in Figure 9.14 "The Effect of a Change in Demand on a Firm’s Choice of Output and Employment": an outward shift of a firm’s demand curve typically leads to an outward shift in labor demand.