====== Cobb-Douglas Production Function ====== The Cobb-Douglas Production Function is a particular form of the [[:en:production-function|]]. It takes the following form: Q(L,K) = A L^β K^α * L:labor * K:capital * Q:output * A>0 * 0<α<1 * 0<β<1 ======= Properties of Cobb Douglas Production Function ======== - The [[:en:marginal-product|marginal product]] is positive and decreasing. - [[:en:cobb-douglas-output-elasticity|Output elasticity]] is constant, equal to α for L or β for K. - [[:en:return-to-scale|Return to scale]] are α+β Plot of a Cobb-Douglas production function: {{:en:cobb-douglas-ew.jpg?nolink&|cobb-douglas-ew.jpg}}{{ :en:cobb-douglas-1.svg.png?nolink&}} ===== The marginal product is positive and decreasing ===== This behavior is usually seen in a lot of real world examples. To find the marginal product of a production factor, we derivate the total output with respect that factor. For example, to find the marginal product of capital: ∂Q / ∂K = = α * (A L^β ) K^(α-1) Given that α is positive and lower than 1, the marginal product is positive and decreasing. Graphically: {{:es:cobb-douglas-marginal.png?nolink&}} ===== Output elasticity is constant ===== Output elasticity is defined as the percent change in output, when there is a percent change in one production factor. In the case of the Cobb-Douglas production function, output elasticity is constant. Output elasticity of labor is β and output elasticity of capital is α. ==== Proof ==== By definition, output elasticity is: (∂Q/Q) / (∂L/L) = = (∂Q/∂L) / (Q/L) That is the marginal product of labor divided the medium product of labor. = [ Aβ L^β-1 ) K^α ] / [ A L^β K^α / L ] Given that 1/L is L^-1 , AL^β K^α /L is AL^β-1 K^α . It follows that output elasticity is: = AβL^β-1 K^α / AL^β-1 K^α The only difference between the numerator and the denominator is β; then: Output elasticity = AβL^β-1 K^α / AL^β-1 K^α = β ===== Return to scale are α+β ===== Returns to scale measure the proportional change in output, given a proportional change in the quantity of every factor of production. ==== Proof ==== If we increase every factor in a given constant c, the new output level will be: Q^' = A(cL)^β (cK)^α = Ac^β L^β c^α K^α = c^β c^α AL^β K^α = c^β+α Q This means that if we increase every production factor by c, the output level will increase in c^β+α . If β+α = 1, the output will increase in c. In this case, the Cobb-Douglas production function has constant return to scale. If β+α < 1, the proportional increase in output will be lower than the proportional increase in production factors. In this case, the Cobb-Douglas production function has decreasing returns to scale. If β+α > 1, the proportional increase in output will be higher than the proportional increase in production factors. In this case, the production function has increasing returns to scale. [[:en:cobb-douglas-output-elasticity|]]