Cobb-Douglas Output Elasticity

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====== Cobb-Douglas Output Elasticity ======

The [[en:cobb-douglas-production-function|Cobb-Douglas]] [[en:elasticity-of-production|Output Elasticity]] is constant and equal to α or β.

If the Cobb-Douglas production function is **Q(L,K) = A L<sup>β</sup>K<sup>α</sup>**, 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:
<code>
(∂Q/Q) / (∂L/L) 
</code>
This is equal to:
<code>
(∂Q/∂L) / (Q/L)
</code>
Now we have the marginal product divided by the average product. Applying the Cobb-Douglas production function:

= [ Aβ L<sup>(β-1)</sup> K<sup>α</sup> ] / [ A L<sup>β</sup> K<sup>α</sup> / L ]
= [ Aβ L<sup>(β-1)</sup> K<sup>α</sup> ] / [ A L<sup>(β-1)</sup> K<sup>α</sup> ]
= 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]]

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