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.
See Also: Elasticity of Production