Aggregate Production Function

The Aggregate Production Function is the function that shows a technical relationship between aggregate inputs and aggregate outputs. It is a mathematical model that economists use to illustrate the change in productivity because of the changes in factors of production. It helps an economy to produce its potential level of output.

There are numerous uses for estimating this production function, including calculating the total factor productivity (TFP) and the potential output gap in the economy. So, economists, policymakers, and businesses use this concept as it can explain production creation and economic growth in a relatively easy method.

• The aggregate production function is one of the many tools to determine productivity and economic growth. So, economists use it to measure the quality of inputs and the resultant efficiency of the outcome.
• Cobb and Douglas envisioned the concept. It formed the input for formulating the total factor productivity and output gap and the Solow growth model.
• Capital and labor are two major components that affect an economy's production function. 
• Other factors include land, entrepreneurship, the number of natural resources, human capital, social infrastructure, and technology.

The Aggregate Production Function is also known as the Cobb-Douglas production function (CDPF) as Economists Cobb, and Douglas first estimated and introduced it in the late 1920s. From then on, it proved its significance. For example, Robert Solow used Cobb-Douglas functions to pioneer his Solow growth model and its subsequent growth formula. The Solow-Swan model, which is known as the exogenous growth model, is an economic model that calculates long-term economic growth. Technical advancement primarily fuels factors such as capital accumulation, labor or population expansion, and an increase in productivity. This production function considers these factors to explain long-run economic growth. 

In an economy, production (output) increases with an increase in input. It may be in the form of raw material supplied, equipment, tools, machinery, and other factors, along with the employees' knowledge of how to do things. The amount of each of the items mentioned above significantly impacts production. For example, productivity suffers when machinery is present but labor is insufficient. It is the same when there is enough labor but with machinery shortage. It would take much time and reduce efficiency if firms carried out all manufacturing processes mechanically. Therefore, the production function calculates what goes inside as inputs and the efficiency derived through the output produced.

The upgraded version of this production function is widely used in the transcendental logarithmic (translog) or constant elasticity of substitution (CES) because it is the fundamental base.

Aggregate output depends on the following factors of production that majorly influence the economy:

• Physical Capital: It accounts for the assets that firms or governments create and employ in the process of production. These include assets such as buildings, plants, machines, equipment, computers, and other production facilities.
• Human Capital: It accounts for the skill and education of the economic workforce.
• Labor: It is skilled and unskilled labor input. It includes entrepreneurship, which constitutes business intelligence that firms and institutions apply to the production function. 
• Land: It also includes land, its natural resources, and the raw materials available.
• Knowledge: It accounts for the technical or scientific expertise that helps the production process.
• Infrastructure: It includes social, legal, business, and cultural setup in an economy.
• Factors other than physical and human capital and labor are termed technology.

1. Economic growth increases as functions of aggregate production increase due to technological, human capital, knowledge, and social infrastructure changes.
2. Marginal output is positive due to an increase in the marginal input, i.e., natural resources, land, and labor.
3. Total factor productivity shows output increasing due to technological changes. Therefore, changes not attributed to physical or human capital determine growth due to technology.
4. Economists assume that there are no diminishing returns for human capital and technology as they do not have a prominent unit of measure. At the same time, diminishing returns are assumed for labor and capital as their units are hours of work and hours of capital usage, respectively.
5. Labor and capital show a diminishing return when only one-factor increases and the others are held constant (ceteris paribus). It is because it causes a lower increase in productivity over time. For example, a single firm can increase its output in the short term if it increases its number of workers, considering other factors remaining the same. However, in the long term, with an increase in labor, the firm's capital per worker will fall as workers will not have enough stock of equipment to work. So, the marginal output will decrease.

The aggregate production function graph below demonstrates a pictorial representation of the concept:

It depicts a relationship between output and capital where other factors are constant. Output increases with an increase in capital input. However, the changes in production are subject to diminishing marginal returns.

The above graph showcases technical change and the aggregate production function. It is simple, with two components: capital stock and output. When there are additional inputs through technological advancements or an increase in labor supply, they can significantly impact the overall productivity, efficiency, and economic growth. Technological advances can push the economy forward and make processes more efficient by making things quick and easy. Similarly, when more people are involved in the production, there are enough manufactured goods in the economy. It increases profit and boosts economic activity. Therefore, it is safe to conclude that technical change and the aggregate production function are connected.

Here is a formula to calculate the production function:

Y = A*F (K, L)

It can also be written as follows:

Y = AK^0.25 L^0.75

Y denotes the real GDP, i.e., aggregate output in an economy.

A represents the technological factor. It is a measure of the economy's overall productivity. 

K is the total quantity of non-human capital input into the economy. It is measured in dollars or physical units.

L is the number of employees in the economy (human capital).

Here is an aggregate production function example:

Solve for real GDP, real wage, and real rental cost of capital. Show that the share of income to capital and share of income to labor are 0.25 and 0.75, respectively.

Suppose K or the total quantity of capital input = $160000, 

L or the number of employees in the economy =10,000, and 

A, which measures the economy's overall productivity = 10.

Using the formula Y = A*F (K, L) or Y = AK^0.25 L^0.75

Y= 10 (160,000) ^0.25 (10,000)^0.75

After simplification 

Y= 10*20*1000 = $2,00,000 (Real Gross Domestic Product)

The marginal product of labor of Y = AK^0.25 L^0.75

MP_L= ∂Y/∂L

     0.75A K^0.25 L^-0.25

      0.75(10) (160,000)^0.25 (10,000)^-0.25 (substituting the above given values)

      7.5 (20) (0.10) (simplifying the above equation)

      = $15

One additional employee will increase the real Gross Domestic Product by $15.

Real wage = wage/price level

So here, the marginal product of labor = real wage = $15

Now,

The marginal product of capital of Y= AK^0.25 L^0.75

MP_K= ∂Y/∂K

     0.25A K^-0.75 L^0.75

      0.25(10) (160,000)^-0.75 (10,000)^0.75 (substituting the above given values)

      2.5 (0.000125) (1,000) (simplifying the above equation)

     = $0.3125 per dollar of capital.

One additional dollar increase in the capital stock will increase the real Gross Domestic Product by $0.3125

Here, the marginal product of capital = real rental cost of capital = $0.3125

Final Step,

Total Income = Y = $200,000

Total capital income = rental rate of capital × capital stock = 0.3125 × 160,000 = $50,000

Share of income to capital = $50,000/$200,000 = 0.25

Total labor income = real wage × labor = $15 × 10,000 = $150,000

Share of income to labour = $150,000/$200,000 = 0.75

This has been a guide to Aggregate Production Function and its definition. Here we discuss aggregate production function graph with its function, formula and examples. You can learn more from the following articles - 

• Aggregate Supply
• Aggregate Demand (AD)
• IS-LM Model