Economic growth and economic development 678

Introduction to Modern Economic Growth
where, as usual, ρ > 0. The supply side is represented by an aggregate production
function combining the outputs of two intermediate sectors:
i ε
h
ε−1
ε−1 ε−1
,
(15.3)
Y (t) = γYL (t) ε + (1 − γ)YH (t) ε

where YL (t) and YH (t) denote the outputs of two intermediate goods. As the indices
indicate, the first is L-intensive, while the second is H-intensive. The parameter
ε ∈ [0, ∞) is the elasticity of substitution between these two intermediate goods,

while γ is again a distribution parameter determining the importance of the two

intermediate goods for aggregate output. The resource constraint of the economy
at time t takes the form
C (t) + X (t) + Z (t) ≤ Y (t) ,

(15.4)

where, as before, X (t) denotes total spending on machines and Z (t) is aggregate
R&D spending.
The two intermediate goods are produced competitively with the following production functions:
(15.5)

1
YL (t) =
1−β

and
(15.6)

1
YH (t) =
1−β

ÃZ

NL (t)

xL (ν, t)1−β dν

!

Lβ

xH (ν, t)1−β dν

!

Hβ,

0

ÃZ

NH (t)

0

where xL (ν, t) and xH (ν, t) denote the quantities of the different types of machines
(used in the production of one or the other intermediate good) and β ∈ (0, 1).1

These machines are again assumed to depreciate after use. The parallel between
these production functions and the aggregate production function of the economy
in the baseline expanding product varieties model of Chapter 13 is obvious. There
are two important differences, however. First, these are now production functions
for intermediate goods rather than the final good. Second, the two production
functions (15.5) and (15.6) use different types of machines. The range of machines
complementing labor, L, is [0, NL (t)], while the range of machines complementing
factor H is [0, NH (t)].
1Note

that the range of machines used in the two sectors are different (there are two disjoint
sets of machines); we use the index ν to denote either set of machines for notational simplicity.

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