In economics, a production function is an equation that describes the relationship between input and output, or what goes into making a certain product, and a Cobb-Douglas production function is a specific standard equation that is applied to describe how much output two or more inputs into a production process make, with capital and labor being the typical inputs described.

Developed by economist Paul Douglas and mathematician Charles Cobb, Cobb-Douglas production functions are commonly used in both macroeconomics and microeconomics models because they have a number of convenient and realistic properties.

The equation for the Cobb-Douglas production formula, wherein K represents capital, L represents labor input and a, b, and c represent non-negative constants, is as follows:

f(K,L) = bK^{a}L^{c}

If a+c=1 this production function has constant returns to scale, and it would thus be considered linearly homogeneous. As this is a standard case, one often writes (1-a) in place of c. It's also important to note that technically a Cobb-Douglas production function could have more than two inputs, and the functional form, in this case, is analogous to what is shown above.

### The Elements of Cobb-Douglas: Capital and Labor

When Douglas and Cobb were conducting research on mathematics and economies from 1927 to 1947, they observed sparse statistical data sets from that time period and came to a conclusion about economies in developed countries around the world: there was a direct correlation between capital and labor and the real value of all goods produced within a timeframe.

It's important to understand how capital and labor are defined in these terms, as the assumption by Douglas and Cobb make sense in the context of economic theory and rhetoric. Here, capital indicates the real value of all machinery, parts, equipment, facilities, and buildings while labor accounts for the total number of hours worked within a timeframe by employees.

Basically, this theory then posits that the value of the machinery and the number of person-hours worked directly relate to the gross output of production. Although this concept is reasonably sound on the surface, there were a number of criticisms Cobb-Douglas production functions received when first published in 1947.

### The Importance of Cobb-Douglas Production Functions

Fortunately, most early criticism of the Cobb-Douglas functions was based on their methodology of research into the matter-essentially economists argued that the pair did not have enough statistical evidence to observe at the time as it related to true production business capital, labor hours worked, or complete total production outputs at the time.

With the introduction of this unifying theory on national economies, Cobb and Douglas shifted the global discourse at it related to micro- and macroeconomic perspective. Furthermore, the theory stood true after 20 years of research when the 1947 United States Census data came out and the Cobb-Douglas model was applied to its data.

Since then, a number of other similar aggregate and economy-wide theories, functions, and formulas have been developed to ease the process of statistical correlation; the Cobb-Douglas production functions are still used in analyses of economies of modern, developed, and stable nations around the world.