# Cobb Douglas Output Elasticity

To calculate the output elasticity of a Cobb-Douglas production function, we must derive the total output with respect to the level of a production input. For example labor or capital.

The output elasticity with respect to labor is:

(∂Q/Q) / (∂L/L) `[1]`

= (∂Q/∂L) / (Q/L) `[2]`

The first part of [2] (the dividend) is the marginal product of labor. The second part of [2] (the divisor) is the average product of labor.

In the case of the Cobb Douglas production function, the output elasticity can be measured quite easily:

A general Cobb Douglas production function is: Q(L,K) = A Lβ Kα . Applying this to the formula [2]

(∂Q/∂L) / (Q/L) `[2]`

= [ Aβ L(β-1) Kα ] / [ A Lβ Kα / L ] `[3]`

= [ Aβ L(β-1) Kα ] / [ A L(β-1) Kα ] `[4]`

= β `[5]`

Output elasticity with respect to labor is constant and equal to β. If β is 0.4 and labor increases in 10%, output will increase 4%.

The same conclusion applies to the output elasticity with respect to capital: The output elasticity with respect to capital is constant and equal to α. If α is 0.6 and capital increases in 10%, output will increase 6%.