en:cobb-douglas-production-function

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

- The marginal product is positive and decreasing.
- Output elasticity is constant, equal to α for L or β for K.
- 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.

en/cobb-douglas-production-function.txt · Last modified: 2019/04/06 08:27 by federico

## Discussion

If it is possible can someone post a real applicable example of how to calculate the elasticity of output with respect to labor and capital on Cobb Douglas function

Example function:

`[1] Q=10 L^0.4 K^0.6`

Output elasticity (OE) with respect to K:

`[2] (∂Q/∂K) / (Q/K)`

`[3] (∂Q/∂K) = 10 L^0.4 0.6 K^-0.4`

`[4] (Q/K) = 10 L^0.4 K^-0.4`

Then, the output elasticity is: [3]/[4] (notice that almost all components of [3] and [4] all the same, so they cancel each other.

`[5] Output Elasticity = 0.6`

Complete case example should be better.