## Aggregate Production Function Background

Before we jump into the definition of the aggregate production function, let's first explain what it involves. Economists need to increase the four main factors of production to increase productivity. These factors of production are labor, land, capital, and entrepreneurship. So, how do economists measure total factor productivity? They measure it by dividing output by an index of all the factors of production or inputs.

Let's say economists want to measure the productivity of one input, say, labor. They do this by measuring the output per unit of labor. However, for output to increase faster than inputs, there should be a factor that increases the productivity of the inputs. For instance, in the case of labor, technological advancements will improve the output per unit of labor since workers will have better machines or improved processes at their disposal.

So, what are technological advancements? Technological advancements refer to changes in production processes or the invention of new products that serve as inputs. The invention of electricity is an example of a change in production processes, whereas the invention of computers is an example of a product invention.

**Technological advancements** refer to changes in production processes or the invention of new products that serve as inputs.

The production function typically considers physical capital (machinery or equipment) and labor.

We are economists, and this means we need to quantify how these contribute to productivity. This is where the aggregate production function comes in.

## What is Aggregate Production Function?

The aggregate production function is a hypothetical equation that shows how factors of production contribute to total output. The aggregate production function relates total output to the number of inputs.

**The aggregate production function** is a hypothetical equation that shows how factors of production contribute to total output.

Technology advances over time, and this increases the productivity of factors like physical capital and labor. Let's consider the example below.

Consider an economy where coffee processing plants are used to process the coffee manually. However, these plants now have automated coffee processing machines due to technological advancements, which led to the invention of the machines. The addition of machines to the coffee processing plants means that capital has increased due to advancements in technology. As a result, employees can now be more productive as the machines improve their productivity.

So, let's say the productivity of these inputs has increased due to technological advancements; what impact have they had on total output? Economists use the aggregate production function to answer this question.

## Aggregate Production Function Equation

The equation for the aggregate production function is derived by equating total output to the inputs and technology. The aggregate production function can be written as:

\(Q=AF(K, L, R)\)

Where:

\(Q\) represents total output,

\(K\) represents the contribution of capital,

\(L\) represents the contribution of labor, and

\(R\) represents the contribution of natural resources.

\(A\) represents advancements in technology, whereas

\(F\) represents the production function.

Note that other factors of production can be inserted into the equation. It just depends on which factor(s) you are focusing on.

## Aggregate Production Function Macroeconomics

So, how is the aggregate production function applied in macroeconomics? To understand this, we must look at the neoclassical economic growth model, which **assumes an economy with a single homogenous good produced using only labor and capital as inputs**. In this model, we equate the total output per worker to the capital per worker. This is written as:

\(Q=F(K/L)\)

Where Q represents output per worker, and K/L represents capital per worker.

You must be wondering how capital can increase and contribute to productivity. Economists refer to this as **capital deepening**. It refers to an increase in capital that is faster than the increase in labor.

**Capital deepening** refers to an increase in capital at a faster rate than the increase in labor.

All other things remaining equal, capital deepening will increase output per worker. Let's look at a few instances where capital deepening can increase output per worker.

Consider a coffee farm that wants to harvest 200 bags of coffee. The farm owner could use 20 unskilled people to manually harvest the coffee or employ 1 skilled operator and provide an automated harvester.

In the above example, capital deepening occurs when the farm owner keeps 1 skilled operator and provides an automated harvester.

A bank operates in a busy city but has only one branch with 5 human tellers. This results in long bank queues, so the bank installs 50 Automated Teller Machines (ATMs) for customers to use for both deposits and withdrawals.

The example described above is capital deepening as the bank increases the number of machines (physical capital) without increasing the quantity of labor (human tellers).

## Aggregate Production Function Graph

Here, we will illustrate the aggregate production function using the aggregate production function graph. The aggregate production function graph shows the relationship between the output per worker and capital per worker. It is plotted with output per worker on the vertical axis and capital per worker on the horizontal axis. The aggregate production function graph is presented in Figure 1.

The **aggregate production function graph** is the graphical illustration of the relationship between output per worker and capital per worker.

Fig. 1 - Aggregate Production Function Graph

As shown in Figure 1, the output per worker (Q/L) increases as capital per worker (K/L) increases. This is because, with all other things held constant, an increase in capital will result in an increase in output. We also see that the curve bends and gets flatter as capital per worker increases. This is because of the diminishing returns to capital. The diminishing returns to capital are such that all other things being equal, each additional unit of capital per worker results in a smaller increase in productivity. Therefore, as capital per labor increases, output per worker also increases, but at a decreasing rate.

The **diminishing returns to capital** are such that all other things being equal, each increase in capital per worker results in a smaller increase in output per worker.

We have shown what happens when capital per labor increases; however, technology also tends to advance over time. So, what happens to the aggregate production function graph when technology increases? It shifts upward! Have a look at Figure 2 to visualize the effect of technology.

Fig. 2 - Effect of Technology on Aggregate Production Function Graph

The aggregate production function shifts upward over time (from APF_{0} to APF_{1}) due to improvements in technology. This means that technological improvements and capital deepening result in greater output per worker over time, holding all other factors equal.

- The aggregate production function shifts upward over time due to improvements in technology.

## Aggregate Production Function Example

We can now look at a simple example of the aggregate production function.

In an economy, the physical capital per worker and the output per worker increase as follows:

Capital Per Worker ($000) | Output Per Worker ($000) |

0 | 0 |

50 | 100 |

100 | 180 |

150 | 250 |

Table 1. Aggregate Production Function Example.

Assuming that all other factors have remained equal, draw and explain the aggregate production function graph.

First, let's plot the graph with the output per worker on the vertical axis and capital per worker on the horizontal axis in Figure 3.

Fig. 3 - Aggregate Production Function Graph Example

As Figure 3 shows, even though capital per worker is increasing in equal units, output per worker is increasing at a decreasing rate. This shows the diminishing returns to capital, which suggests that each unit of increase in capital per worker will result in a smaller increase in output per worker.

The example also illustrates capital deepening as capital increases faster than labor.

Read our article on Growth Rates to learn more.

## Aggregate Production Function - Key takeaways

- The aggregate production function is a hypothetical equation that shows how factors of production contribute to productivity or the real gross domestic product (GDP).
- The aggregate production function can be written as:
Q=AF(K, L, R)

Where: Q represents total output, K represents the contribution of capital, L represents the contribution of labor, and R represents the contribution of natural resources. A represents advancements in technology, whereas F represents the production function.

Capital deepening refers to an increase in capital at a faster rate than the increase in labor.

The diminishing returns to capital are such that all other things being equal, each additional unit of capital per worker results in a smaller increase in productivity.

The aggregate production function shifts upward over time due to improvements in technology.

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##### Frequently Asked Questions about Aggregate Production Function

What are the components of the aggregate production function?

The components of the aggregate production function are inputs and total output.

What is the aggregate production function?

The aggregate production function is a hypothetical equation that shows how factors of production contribute to total output.

What is aggregate production function with GDP?

The aggregate production function is a hypothetical equation that shows how factors of production contribute to the real gross domestic product (GDP).

How do you calculate aggregate production function?

The equation for the aggregate production function is derived by equating total output to the inputs and technology. The aggregate production function can be written as:

Q=AF(K, L, R)

Where: Q represents total output, K represents the contribution of capital, L represents the contribution of labor, and R represents the contribution of natural resources. A represents advancements in technology, whereas F represents the production function.

Why is aggregate production function important?

Economists use the aggregate production function to determine the contribution of the factors of production to total output.

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