29
3.2
A Benchmark Model of Growth and Structural Transformation
We use the model of the previous section as the starting point for our analysis of structural
transformation in the context of the growth model.
3.2.1
Set up of the Benchmark Model
As in the previous section, we assume an infinitely lived stand–in household that has prefer
ences characterized by (1) and is endowed with one unit of time and an initial capital stock.
Di
ff
erent than in the previous section, we now assume that
C
t
is a composite of agricultural
consumption (
c
at
), manufacturing consumption (
c
mt
) and service consumption (
c
st
):
C
t
=
ω
1
ε
a
(
c
at

¯
c
a
)
ε

1
ε
+
ω
1
ε
m
(
c
mt
)
ε

1
ε
+
ω
1
ε
s
(
c
st
+
¯
c
s
)
ε

1
ε
ε
ε

1
(12)
where ¯
c
i
, ω
i
≥
0 and
ε >
0. The functional form (12) is a parsimonious choice that allows us
to capture two features on the demand side that are potentially important for understanding the
reallocation of activity across these three sectors: how the demand of the household reacts to
changes in income and in relative prices. In particular, the presence of the two terms ¯
c
a
and
¯
c
s
allows for the period utility function to be non–homothetic and therefore the possibility that
changes in income will lead to changes in expenditure shares even if relative prices are constant.
The parameter
ε
influences the elasticity of substitution between the three goods, and hence the
response of nominal expenditure shares to changes in relative prices. Note, however, that in the
above specification the elasticity of substitution is not equal to
ε
because it also depends on the
non–homotheticity terms.
We generalize the previous model to allow for four Cobb–Douglas production functions,
one for each of the three consumption goods and one for the investment good. Formally, the
30
production functions are given by:
20
c
it
=
k
θ
it
(
A
it
n
it
)
1

θ
,
i
∈ {
a
,
m
,
s
}
(13)
X
t
=
k
θ
xt
(
A
xt
n
xt
)
1

θ
(14)
There is a tradition in the literature of working with only three production functions, with the
assumption that all investment is produced by the manufacturing sector. Under this assumption,
the output of the manufacturing sector can be used as either consumption or investment whereas
the output of the other two sectors can only be used as consumption. We have not adopted this
specification for two reasons. First, despite the apparent reasonableness of the claim that in
vestment is to first approximation produced exclusively by the manufacturing sector, it turns out
that this is not supported by the data. Moreover, such an assumption is becoming increasingly
at odds with the data over time, due at least in part to the fact that software is both a sizeable
and increasing component of investment, and most software innovation takes place in the ser
vice sector. In fact, total investment has exceeded the size of the entire manufacturing sector
in the US since 2000. The second reason for considering a separate investment sector derives
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 Summer '19
 The American, Trigraph, gross domestic product, Value added, National accounts, uT uT uT