Growth Models with Exogenous Saving Rates
27
1. Constant returns to scale.
The function
F
(·
) exhibits constant returns to scale. That
is, if we multiply capital and labor by the same positive constant,
λ, we get
λ the amount
of output:
F
(λK, λL, T ) =
λ ·
F(K, L, T ) for all
λ > 0
(1.4)
This property is also known as
homogeneity of degree one in K and L. It is important to
note that the definition of scale includes only the two rival inputs, capital and labor. In other
words, we did not define constant returns to scale as
F
(λK, λL, λT ) =
λ ·
F(K, L, T ).
To get some intuition on why our assumption makes economic sense, we can use the
following
replication argument. Imagine that plant 1 produces
Y units of output using the
production function
F and combining
K and
L units of capital and labor, respectively, and
using formula
T . It makes sense to assume that if we create an identical plant somewhere
else (that is, if we
replicate the plant), we should be able to produce the same amount of
output. In order to replicate the plant, however, we need a new set of machines and workers,
but we can use the same formula in both plants. The reason is that, while capital and labor are
rival goods, the formula is a nonrival good and can be used in both plants at the same time.
Hence, because technology is a nonrival input, our definition of returns to scale makes sense.
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