The Cobb-Douglas Production Function is a particular form of the Production Function.

It takes the following form:

Q(L,K) = A L^β K^α

• L:labor
• K:capital
• Q:output
• A>0
• 0<α<1
• 0<β<1

1. The marginal product is positive and decreasing.
2. Output elasticity is constant, equal to α for L or β for K.
3. Return to scale are α+β

Plot of a Cobb-Douglas production function:

This behavior is usually seen in a lot of real world examples. To find the marginal product of a production factor, we derivate the total output with respect that factor. For example, to find the marginal product of capital:

∂Q / ∂K =

= α * (A L^β ) K^(α-1)

Given that α is positive and lower than 1, the marginal product is positive and decreasing.

Graphically:

Output elasticity is defined as the percent change in output, when there is a percent change in one production factor.

In the case of the Cobb-Douglas production function, output elasticity is constant. Output elasticity of labor is β and output elasticity of capital is α.

By definition, output elasticity is:

(∂Q/Q) / (∂L/L) =

= (∂Q/∂L) / (Q/L) That is the marginal product of labor divided the medium product of labor.

= [ Aβ L^β-1 ) K^α ] / [ A L^β K^α / L ]

Given that 1/L is L^-1 , AL^β K^α /L is AL^β-1 K^α . It follows that output elasticity is:

= AβL^β-1 K^α / AL^β-1 K^α

The only difference between the numerator and the denominator is β; then:

Output elasticity = AβL^β-1 K^α / AL^β-1 K^α = β

Returns to scale measure the proportional change in output, given a proportional change in the quantity of every factor of production.

If we increase every factor in a given constant c, the new output level will be:

Q^' = A(cL)^β (cK)^α

= Ac^β L^β c^α K^α

= c^β c^α AL^β K^α

= c^β+α Q

This means that if we increase every production factor by c, the output level will increase in c^β+α .

If β+α = 1, the output will increase in c. In this case, the Cobb-Douglas production function has constant return to scale.

If β+α < 1, the proportional increase in output will be lower than the proportional increase in production factors. In this case, the Cobb-Douglas production function has decreasing returns to scale.

If β+α > 1, the proportional increase in output will be higher than the proportional increase in production factors. In this case, the production function has increasing returns to scale.

Cobb-Douglas Output Elasticity