Economic growth and economic development 66
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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 , 0 < α < 1.
(2.15)
It can easily be verified that this production function satisfies Assumptions 1 and 2,
including the constant returns to scale feature imposed in Assumption 1. 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 ,
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