Production Functions

A production function is a relationship between inputs (factors of production) and outputs (products). It illustrates how many workers and machines it might take to produce, for example, 1 bushel of wheat, 2 bushels of wheat, or 1,000 bushels of wheat.

Short Run versus Long Run

The short run is a time period during which a business cannot vary one or more factors of production. In other words, at least one input is fixed.

The long run is a time period during which the firm has the flexibility to change all inputs. It can buy more or bigger machines, hire more workers, and expand the building.

The length of the short and long runs varies for each company. For example, if a steel plant cannot vary the size of its factory and the number of its machines for a period of three years, then the short run for this company is a period shorter than three years, and the long run is a period longer than 3 years. On the other hand, if an Internet-based company can vary all of its factors of production, including the space in which it operates, the number of computers, and the number of workers, within three weeks, then the short run for this Internet-based company is a period shorter than three weeks, and the long run is a period longer than three weeks.

Fixed and Variable Inputs

Fixed inputs remain constant in the short run, even as production decreases or increases. Examples of resources that are typically fixed in the short run include land, heavy machinery, buildings, and workers on long-term contracts.

Variable inputs can be varied in the short run. They increase or decrease as production increases or decreases. Inputs that are typically variable in the short run include hourly and part-time labor, office supplies, energy, and raw materials.

Total, Average, and Marginal Production

The table below contains a hypothetical car manufacturer’s production schedule during a short-run period of time. We assume that the firm has fixed and variable inputs. For example, it has a fixed amount of land (and buildings) and a fixed number of machines with which to work. The only variable input is the number of workers.

Marginal means “additional.” Marginal production of labor is how much one additional worker adds to the total production. If a fourth worker joins the company and total production increases from 15 to 19, then the marginal production of this fourth worker is 4. Average production of labor is the production per worker. It is the total production divided by the number of workers. If total production is 15 at 3 workers, then average product of labor is 15 divided 3, or 5.

Number of Workers
Amount of Land in Acres
Number of Machines
Total Production of Cars Average Production of Labor
Marginal Production of Labor
0 2 5 0
1 2 5 3 3 3
2 2 5 7 3.5 4
3 2 5 15 5 8
4 2 5 19 4.75 4
5 2 5 22 4.4 3
6 2 5 23 3.83 1

The Law of Diminishing Marginal Production

The law of diminishing marginal production states that when a firm uses a variable input, such as labor, the additional productivity of workers who are hired at a later stage is less than the additional productivity of workers who were hired first. In the table above, the firm can hire from 0 to 6 workers. The additional production of the first worker is 3 products. The additional production of the second worker is 4 products, and the additional production of the third worker is 8 products. Thus, the marginal production for the first three workers rises. This occurs because these first three workers are able to specialize in the work that is best suited to them.

The specialization opportunities diminish for workers 4, 5, and 6. Because of the fixed inputs, there are not enough machines and offices to comfortably accommodate these additional employees. Subsequently, these workers are not as productive as the first three workers. Additional output of the fourth worker is only 4 products. The fifth worker adds only 3 products, and the sixth employee only contributes one additional product. The decrease in marginal production after a certain number of workers is known as the Law of Diminishing Marginal Production.

The law of diminishing marginal production only occurs in the short run. In the long run, it is possible to add machines and increase the size of an office or factory. Therefore, the short-run reasons for diminishing marginal production are non-existent in the long run.

In the long run, average and marginal production can decrease, but this occurs for different reasons. A company can grow too large and bureaucratic and lose efficiency. When this happens, we experience “decreasing returns to scale” and “diseconomies of scale.” These concepts are further explained in the next unit (Unit 5).

Total, Marginal, and Average Production Graphs
Below are graphs of the total production, marginal production, and average production of labor. The graphs are based on the data from the table above.

Another Example of a Firm’s Production Function

In the table above, the number of workers changed by one with each row. Below we look at a production function in which the number of workers changes by 5 with each row. This changes how we compute marginal production.

Number of Workers
Amount of Land in Acres
Number of Machines
Total Production of Cars Average Production of Labor
Marginal Production of Labor
0 20 15 0
5 20 15 10 2 2
10 20 15 30 3 4
15 20 15 70 4.67 8
20 20 15 100 5 6

To obtain the values for marginal production in the table above, we have to divide the change in total production by the change in the number of workers (5 in the table above).

In the first table on this page, the number of workers changes by only 1 for each row. In this case, the marginal production is the change in total production divided by 1, which is simply the change in total production. In the second table, the number of workers changes by 5 with each additional row. Therefore, in the second table, the marginal production is the change in total production divided by 5.

Example
Problem: What is the marginal production at a level of 15 workers?

Solution: The total production at 15 workers is 70 cars. The total production in the previous row with 10 workers is 30 cars. Marginal production, therefore, is 40 (70 minus 30) divided by 5 (which equals 8).

Video Explanation
For a video explanation of a production function and the calculation of average and marginal production, please watch: