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) [1] = (∂Q/∂L) / (Q/L) [2] The first part of [2] (the dividend) is the marginal product of labor. The second part of [2] (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 L^β K^α . Applying this to the formula [2] (∂Q/∂L) / (Q/L) [2] = [ Aβ L^(β-1) K^α ] / [ A L^β K^α / L ] [3] = [ Aβ L^(β-1) K^α ] / [ A L^(β-1) K^α ] [4] = β [5] 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%.