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# 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 LβKα, 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) Kα ] / [ A Lβ Kα / L ] = [ Aβ L(β-1) Kα ] / [ A L(β-1) Kα ] = 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. 