Acemoglu (2002) finds that the direction of technological change depends on price
and market size. For the intermediate goods sector, this holds without modification.
Equation 31 shows that the incentive to innovate in the intermediate goods sector
depends on price, p
Z
, and market size, L. The structure in the extractive sector is
different. While in the intermediate goods sector, the stock of technology is used for
production, the extractive sector can only use the flow of new technology for pro-
duction. The incentive to innovate, thus, grows over time in line with the size of
the economy. The market size effect for the extractive sector depends on the size of
the economy. The price effect is not relevant since competition keeps price equal to
marginal costs. This reflects the long-term perspective of the model.
4.3 Discussion
We discuss a number of issues that arrive from our model, namely the assumptions
made in Section 3, the comparison to the other models with non-renewable resources,
and the question of the ultimate finiteness of the resource.
Function D from Equation 7 shows the amount of the non-renewable resource avail-
able in the earth’s crust for a given occurrence of grade d. Geologists cannot give an
exact functional form for D, so we used the form given in Equation 8 as a plausible
assumption. How would other functional forms affect the predictions of the model?
First, the predictions are valid for all parameter values δ
2
∈ R
+
. Secondly, if D is
discontinuous with a break at d
0
, at which the parameter changes to δ
0
2
∈ R
+
, there
would be two balanced growth paths: one for the period before, and one for the period
after the break. Both paths would behave according to the model’s predictions. The
paths would differ in the extraction cost of producing the resource, level of extraction,
and use of the resource in the economy. To see this, recall from Proposition 1 that
X
t
is a function of δ
2
. A non-exponential form of D would produce results that differ
from ours. It could feature a scarcity rent as in the Hotelling (1931) model, as a non-
exponential form of D could cause a positive trend in resource prices or the extraction
from occurrences at a lower ore grade becomes infeasible. In these cases, the extractive
firms would consider the opportunity cost of extracting the resource in the future, in
addition to extraction and innovation cost.
How does our model compare to other models with non-renewable resources? We
do not assume that resources are finite; their availability is a function of technological
change. As a consequence, resource availability does not limit growth. Substitution
of capital for non-renewable resources, technological change in the use of the resource,
and increasing returns to scale are therefore not necessary for sustained growth as in
Groth (2007) or Aghion and Howitt (1998). Growth depends on technological change
as much as it does in standard growth models without a non-renewable resource, but
it also depends on technological change in the extractive sector. If the resource were
finite in our model, then the extractive sector would behave in the same way as in
standard models in the tradition of Hotelling (1931). As Dasgupta and Heal (1980)
point out, the growth rate of the economy depends in this case strongly on the degree
of substitution between the resource and the other economic inputs. For ε > 1, the
resource is inessential; for ε < 1, the total output that the economy is capable of
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