Production Function

Production Function Definition

All firms operate in a way that they produce goods and services for their customers. They operate by transforming quantities of inputs into quantities of outputs. Inputs represent factors of production such as labor, physical capital, land, etc. And outputs are the goods and services the firm creates for its customers. A production function shows the relationship between these inputs and outputs.

Production Function Example

Let's say there is a farming company that plants apples. For simplicity, let's assume that the firm's factors of production are labor, land, and physical capital. The farm has around 1000 apple trees already implanted. It is harvesting time, and the firm wants to employ labor to harvest the apples. Here in this example, the farm's inputs are the land, the machinery, and the labor. And the output is the number of apples it produces. Suppose we are to draw a production function for this company. In that case, we can show the relationship between the inputs (the quantity of labor, the size of the land, and the machinery used) and the quantity of output (the quantity of apple that was produced).

Production Function Graph

Before we consider the production function graph, let's consider some data from a made-up company in Table 1 below as an example.

Table 1. Production function table

As mentioned before, the production function shows the relationship between the quantities of inputs and outputs of a company. Let’s draw a graph for our example in a very simple way. On the y-axis, we have the quantity of apples in tons, and on the x-axis, we have the number of workers. Notice here, for now, we are not considering the other factors of production (land and machinery) because we consider them as fixed inputs.

Figure 1 shows the production function graph. Let’s say that every unit of labor increases output by 0.5 tons. Each unit of labor represents one worker. So the firm’s output increases by an increment of 0.5 tons of apple for every worker it hires. The straight line in figure 1 represents the total production curve. The total production curve shows how variable inputs affect the quantity of output. This example is a linear curve because every extra worker increased the output by exactly 0.5 tons. This shape of the production function is called a linear production function.

However, in reality, many constraints make it difficult for extra workers to produce the same amount as already existing workers. For example, if an office fits only two workers and you hire four workers to work there, then the extra two workers will not be able to produce the same amount of additional output as the two original workers, as there aren’t enough spaces for them to work. In our example, the first worker will be able to harvest the most because he has access to all the available resources without being limited by other co-workers. This is called diminishing marginal returns to labor.

Before explaining the meaning of diminishing marginal returns to an input, let’s understand the meaning of the marginal product of an input.

If we increase the number of workers by one, our total product increases by 0.5 tons, so the marginal product of labor (MPL) equals 0.5. In a linear production function, the marginal product is constant. However, in reality, when the number of workers increases by one, the number of apples produced will increase by less than 0.5 tons. This is due to the diminishing marginal returns to labor.

So, in reality, in our example, when the firm hires one more worker, the total output will not increase by 0.5 tons but by less than that. In other words, the marginal product of labor decreases when the firm uses more labor and other inputs are fixed. Additionally, the average product produced drops as the number of workers increases. Think about it, as an additional worker adds less to the overall production, the average product produced will also drop. So let’s modify our graph so that it represents a more realistic situation.

Figure 2 shows the production function when diminishing marginal returns from labor are taken into account. Notice that the total apples produced do not increase as much as before when the number of workers increases. This represents a more realistic production function that you can find in the real world, as all firms exhibit diminishing marginal returns in the short run with other fixed inputs.

Now, let's consider fixed inputs in our analysis. We mentioned earlier that fixed inputs are inputs that take a longer time to be changed. But what happens to the total product curve when a fixed input changes.

Figure 3 shows how the total product curve and the marginal product of labor change when a piece of new machinery is bought. As we can see, the total production curve shifts upward from TP₁ to TP₂ because each worker's marginal product increases. By changing the fixed inputs, the firm was able to increase the marginal product of each worker and the total quantity of apples produced.

Production in the Short Run versus the Long Run

It takes time for firms to change the quantity of output. To change output, firms need to change the quantity of input. However, there are some constraints to that. Firms have time constraints. Some variables, such as labor, can be changed relatively quicker than other variables. It takes longer to expand the land and buy new pieces of machinery than to hire a new worker. This time constraint creates a limitation in production capacity in the short run. However, in the long run, the firm can vary its input to produce any quantity of output.

The short-run is the period of time where at least one of the factors of production cannot be changed. In our example, labor can be changed easily relative to other inputs. By increasing or decreasing labor, the firm can change the output quantity of apples. Land and machinery cannot be changed in the short run.

The long-run is the period of time where all inputs are variable. The firm can expand its land and implant more trees not in the short run but in the long run. In our example, if it takes the firm three years to buy more acres of land and implant new trees, then the short-run production period is less than three years for this company.

Production Function formula

The most common form of the Production function formula is as follows

\(q=Af(K,L) \)

From the equation, q represents the total output, and A represents technology. \(f(K,L)\)represents the function of inputs. K for capital and L for labor.

Technology in the production function means a technological process that enables firms to increase production without changing the quantities of inputs. In the apple production example, technology could represent the creation of genetically modified apple seeds that could produce two times more apples.

You might also see a simplified production function that does not include technology:

\(q=f(K,L) \)

Marginal Product Formula

The marginal product of an input is the change in output divided by the change in the input:

\(MP=\frac{\hbox{change in output}}{\hbox{change in input}}\)

The marginal product of labor (MPL):

\(MPL=\frac{\hbox{change in output}}{\hbox{change in the number of workers}}\)

If the output from adding an extra worker change from 3 to 5, then the marginal product of labor is equal to \(5-3=2\)

Short-run vs. long-run production functions

There are two main types of production functions: the short-run production function and the long-run production function.

The short-run production function is the type of production function where at least one of the inputs cannot be changed. Usually, you have the number of workers changing in the short-run while the capital remains fixed.

The long-run production function is the type of production function where all input can change. As it is concerned with the long run, both labor and capital can change.