The Cobb Douglas Production Function: Definition, Formula & Example

The Cobb Douglas Production Function: Definition, Formula & Example
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  • 0:00 Factors of Production
  • 1:36 Cobb-Douglas…
  • 3:27 Marginal Profit
  • 4:33 Returns to Scale
  • 5:36 Lesson Summary
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Lesson Transcript
Instructor: Ronald Price

Ron has held a variety of positions in higher ed and business, including 25+ years as an instructor and 20+ years as a corporate senior manager, and consultant.

The Cobb-Douglas production function represents the relationship between two or more inputs - typically physical capital and labor - and the number of outputs that can be produced. It's a commonly used function in macroeconomics and forecast production.

Factors of Production

You can't make something from nothing. You need supplies, equipment, resources, and some know-how, too. How much you have of these things can affect your production. In economics, a production function is a way of calculating what comes out of production to what has gone into it. The formula attempts to calculate the maximum amount of output you can get from a certain number of inputs.

In macroeconomics, the factors of production are:

  • Physical capital (K), or tangible assets that are created for use in the production process. This includes such things as buildings, machines, computers, and other equipment.
  • Labor (L), or input of skilled and unskilled activities of human workers.
  • Land (P), which includes natural resources, raw materials, and energy sources, such as oil, gas, and coal.
  • Entrepreneurship (H), which is the quality of the business intelligence that is applied to the production function.

The production function is expressed in the formula: Q = f(K, L, P, H), where the quantity produced is a function of the combined input amounts of each factor. Of course, not all businesses require the same factors of production or number of inputs. Another form of the production function reduces the inputs to just labor and physical capital. The formula for this form is: Q = f(L, K), in which labor and capital are the two factors of production with the greatest impact on the quantity of output.

Cobb-Douglas Production Function

In 1928, Charles Cobb and Paul Douglas presented the view that production output is the result of the amount of labor and physical capital invested. This analysis produced a calculation that is still in use today, largely because of its accuracy.

The Cobb-Douglas production function reflects the relationships between its inputs - namely physical capital and labor - and the amount of output produced. It's a means for calculating the impact of changes in the inputs, the relevant efficiencies, and the yields of a production activity. Here's the basic form of the Cobb-Douglas production function:


The Cobb-Douglas production function
basic formula


In this formula, Q is the quantity produced from the inputs L and K. L is the amount of labor expended, which is typically expressed in hours. K represents the amount of physical capital input, such as the number of hours for a particular machine, operation, or perhaps factory. A, which appears as a lower case b in some versions of this formula, represents the total factor productivity (TFP) that measures the change in output that isn't the result of the inputs. Typically, this change in TFP is the result of an improvement in efficiency or technology. The Greek characters alpha and beta reflect the output elasticity of the inputs. Output elasticity is the change in the output that results from a change in either labor or physical capital.

For example, if the output elasticity for physical capital (K) is 0.60 and K is increased by 20 percent, then output increases by 3 percent (0.6/0.2). The same is true for the output elasticity of labor: an increase of 10 percent in L with an output elasticity of 0.40 increases the output by 4 percent (0.4/0.1).

Marginal Product

Another concept associated with the Cobb-Douglas production function is marginal product, which is the change in the output that results from one additional unit of a single production factor with all other factors held constant. Or, as the economists say, ceteris parabis, which means 'all other things equal.' Marginal product is measured in physical units, which is why it is also called marginal physical product.

For example, consider a company called WeeBee Toys. When there are no workers in the factory, there is no output even though physical capital is present. When a single worker shows up, three units are produced per labor hour. When two workers come in, output increases to five units per hour.

The addition of the labor of the second worker results in two more units per hour, or a marginal product of two. Because the marginal product is directly related to the increase in labor, this is also called the marginal product of labor. Had the increase in output been a result of new technology or physical capital, the change would be marginal product of capital.

Returns to Scale

In a production function, the amount of output can change as a result of changes in the input amounts. Returns to scale measures the change in output that results from a proportional change to the inputs.

A constant returns to scale (CRS) is when the change in output is proportional to the changes in the inputs. If output increases from a proportional change to the inputs, we have an increasing returns to scale (IRS). A reduction in the output from a proportional change is a decreasing returns to scale (DRS).

The factors that most affect the outcome of the returns to scale calculation are the output elasticity of the inputs. The alpha (a) and beta (b) factors in the Cobb-Douglas production function can be used to predict the result of the returns to scale:

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