# 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) ``

= (∂Q/∂L) / (Q/L) ``

The first part of  (the dividend) is the marginal product of labor. The second part of  (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 

(∂Q/∂L) / (Q/L) ``

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

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

= β ``

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%.