en:cobb-douglas-output-elasticity

The Cobb-Douglas Output Elasticity is constant and equal to α or β.

If the Cobb-Douglas production function is **Q(L,K) = A L ^{β}K^{α}**, the output elasticity with respect to labor (L) is β and the output elasticity with respect to capital (K) is α.

To prove this, we must take into account the definition of output elasticity: it is the porcentual change in output in respond to a porcentual change in levels of either labor or capital.

If we want to calculate the output elasticity with respect to labor, we must use the following equation:

(∂Q/Q) / (∂L/L)

This is equal to:

(∂Q/∂L) / (Q/L)

Now we have the marginal product divided by the average product. Applying the Cobb-Douglas production function:

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

The same applies to the Cobb Douglas output elasticity with respect to capital.

It is noteworthy that both output elasticities are constant and greater than 0 and smaller than 1.

See Also: Elasticity of Production

en/cobb-douglas-output-elasticity.txt · Last modified: 2017/08/31 14:36 by federico

## Discussion