In economics and econometrics, the Cobb-Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by those inputs. The function used to model production is defined by:
P(L, K) = bL^αK^(1-α)
where P is the total production (the monetary value of all goods produced in a year), L is the amount of labor (the total number of person-hours worked in a year), and K is the amount of capital invested (the monetary worth of all machinery, equipment, and buildings). Its domain is {(L, K) | L ≥ 0, K ≥ 0} because L and K represent labor and capital and are therefore never negative.
1. Show that the Cobb-Douglas production function can be written as:
P(L, K) = bL^αK^(1-α) ⇒ ln P/K = ln b + α ln L/K
2. Show that the Cobb-Douglas production function P = bL^αK^(1-α), where β = 1 - α satisfies the equation:
∂P/∂L + K∂P/∂K = (α + β)P