Introduction to Modern Economic Growth Example 2.1 (The Cobb-Douglas Production Function) Let us consider the most common example of production function used in macroeconomics, the CobbDouglas production function–and already add the caveat that even though the Cobb-Douglas production function is convenient and widely used, it is a very special production function and many interesting phenomena are ruled out by this production function as we will discuss later in this book The Cobb-Douglas production function can be written as Y (t) = F [K (t) , L (t)] = AK (t)α L (t)1−α , < α < (2.15) It can easily be verified that this production function satisfies Assumptions and 2, including the constant returns to scale feature imposed in Assumption Dividing both sides by L (t), we have the representation of the production function in per capita terms as in (2.13): y (t) = Ak (t)α , with y (t) as output per worker and k (t) capital-labor ratio as defined in (2.12) The representation of factor prices as in (2.14) can also be verified From the per capita production function representation, in particular equation (2.14), the rental price of capital can be expressed as R (t) = ∂Ak (t)α , ∂k (t) = αAk (t)−(1−α) Alternatively, in terms of the original production function (2.15), the rental price of capital in (2.6) is given by R (t) = αAK (t)α−1 L (t)1−α = αAk (t)−(1−α) , which is equal to the previous expression and thus verifies the form of the marginal product given in equation (2.14) Similarly, from (2.14), w (t) = Ak (t)α − αAk (t)−(1−α) × k (t) = (1 − α) AK (t)α L (t)−α , 52