# Cobb Douglas production function

The Cobb Douglas production function is a neoclassical approach to estimating a country's production function. In this way, thus being able to project its expected economic growth.

To represent the relationships between the output obtained, it uses the variations in the inputs capital (K) and labor (L), to which technology was later added, also called total factor productivity (TFP). It is a production function frequently used in economics.

The origin of the Cobb Douglas function is found in the empirical observation of the distribution of total national income in the United States between capital and labor. According to what the data showed, the distribution remained relatively constant over time. Specifically, work took 70% and capital 30%. In this way, the Cobb Douglas function represents a relationship, where the proportions of labor and capital, with respect to the total product, are constant.

## Cobb Douglas production function formula

Where:

- Y = Production
- A = Technological progress (exogenous), also called Total Factor Productivity (TFP)
- K = Capital stock
- L = Number of employees
- α and β = parameters that represent the weight of the factors (K and L) in income. The parameters vary between 0 and 1.

## Properties of the Cobb Douglas production function

The Cobb Douglas function has certain special features that facilitate the explanation of theories such as utility and production. Below we describe three of its most relevant characteristics.

- Constant returns to scale that depend on the sum of α and β: Returns to scale measure the variation in production before a proportional change in all factors.
- α + β = 1: There will be constant returns to scale.

- α + β> 1: There will be increasing returns to scale.

- α + β <1: There will be decreasing returns to scale.

- Positive and decreasing marginal productivity: This property reflects the law of diminishing returns of the factors. Therefore, it indicates that, as one of the factors of production increases, while the rest remains constant, its productivity declines.
- Constant production elasticity: The production elasticity measures the percentage change in production, given a change in the inputs used. In the case of the Cobb Douglas function, it is constant and equal to α for capital and β for labor. Thus, for example, if β equals 0.2, and labor increases by 10%, production will increase by 2%.

## Simplification of the Cobb Douglas function

To estimate future economic growth, it is more useful to reformulate the Cobb Douglas function, for this, applying natural logarithms.

In this sense, assuming that α + β = 1 (constant returns to scale), and a few more small assumptions, we can establish the economic growth rate as a function of the changes in the factors of production:

% ΔY ≅ (% ΔA) + α (% ΔK) + (1-α) (% ΔL)

Where:

- % ΔY = Expected GDP variation rate
- % ΔTFP = Total Factor Productivity Growth (TFP)
- % ΔK = Capital Stock Growth
- % ΔL = Growth in the number of employees
- α = Elasticity of capital over production

This formula is widely used in the stock market to estimate economic growth. Empirical studies suggest that it would be reasonable to assume that employment growth (L) has a linear effect on employment growth.

## Cobb Douglas function example

We are going to calculate economic growth assuming that TFP, capital (K) and employment (L) grow by 1.5%, 0.2% and 1.7% respectively, if the elasticity of capital (α) is equal to 0.35:

% ΔY = 1.5% + 0.35 (0.2%) + (1-0.35) = 2.675%

## Human capital in the Cobb Douglas function

Human capital is considered a very important factor of production. So much so that, in the studies by Uzawa and Lucas, it was introduced as the main variable of the Cobb-Douglas production function. In this way, substituting the labor factor (L), by the human capital factor (H), and maintaining the technology (A) and the financial capital (k):