Download THE NATURAL-RESOURCE SEE-SAW: Resource extraction and

THE NATURAL-RESOURCE SEE-SAW:
Resource extraction and consumption with directed technological change
Rob Hart
Department of Economics, SLU,
Box 7013, 750 07 Uppsala, Sweden.
Tel +46 18 67 17 34, fax +46 18 67 35 02
[email protected]
Acknowledgements. Thanks to anonymous referees for helpful comments on earlier work, to Sjak
Smulders, Yves Surry, Clas Eriksson, Richard Woodward, Christian Bogmans, and Francesco Ricci for
advice and encouragement, and to Daniel Wikström for programming help.
THE NATURAL-RESOURCE SEE-SAW:
Resource extraction and consumption with directed technological change
ABSTRACT.
Non-renewable resource prices tend to be constant in the long run, whereas con-
sumption rates rise. We explain this, and predict a transition to a balanced growth path along
which consumption is constant and prices rise, tracking wages. On the balanced path resource
abundance determines the consumption rate, and ease of substitution in production determines
the factor share. These conclusions are based on an innovative and carefully constructed general
equilibrium model (but note that the discount rate is exogenous) including: a detailed model
of the resource-extraction sector; endogenous directed technological change in the final-good
sector; arbitrage between the sectors.
1. Introduction
It is almost eighty years since Hotelling (1931) published his famous paper on resource pricing by a mine owner, where he derives a series of seminal results based on the insight that a
resource owner should treat resources in the ground as an asset, and that a delay in selling an
asset must be compensated for by a higher price (given a positive interest rate). It follows un5
der fairly general conditions that the price path of extraction rights to a single resource deposit
should rise at the interest rate.
Hotelling’s result is only a first step towards explaining and predicting resource prices and
consumption rates; two further steps remain. Firstly, it remains to model extraction costs as a
function of extraction technology, the price of extraction inputs, and the quality of the resource
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deposit, the latter being a function of cumulative extraction. Secondly, it remains to model shifts
in the demand function for the resource as a function of expectations about future prices; high
prices must lead to investment such that the demand curve shifts downwards. That is, a model
of endogenous directed technological change in a general equilibrium context is required.1
Only a handful of models in the literature attempt to make the above two steps, and no model
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successfully explains stylized facts about resource prices and consumption rates — long-run resource prices tend to be constant whereas consumption rates rise — while simultaneously making credible predictions about future prices. The lack of such a model is troubling given not only
This version April 20, 2009.
1
‘General equilibrium’ is used in the sense that the model accounts for the allocation decisions of all the agents
in the economy.
1
2
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the importance of many such resources — not least fossil fuels — to the economy, but also the
links between fossil-fuel use and climate change; it is difficult to evaluate alternative policies
20
to control emissions without a model of dynamic prices and quantities in the absence of such
policies.
We build a model which makes the two steps above, and hence is generally applicable to
the prediction of long-run paths of price and quantity of non-renewable resources. We carefully justify our assumptions by reference to a series of properties of the economy, including
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the resource sector, and we compare the model predictions to stylized facts about resource use.
We demonstrate that the explanatory power of the model is much greater than that of existing
models, and indicate ways in which it could be improved further. In developing the model of resource prices, we also make a methodological contribution concerning the modelling of directed
technological change generally, in that we develop an alternative approach to that exemplified
30
by Acemoglu (2007); the model may be interpreted as giving micro-economic foundations to
the ideas of Hicks (1932) and Kennedy (1964) concerning induced bias of innovation, and the
conclusions concerning links between factor prices and the direction of technological change
are straightforward and highly intuitive.
The literature on non-renewable resources is huge,2 but only a few papers are really relevant
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to this one, due to a combination of the existing models’ lack of explanatory power, and their
omission of crucial processes such as directed technological change. Consider for instance the
literature on the optimal depletion of non-renewable resources and sustainable growth, including
seminal papers such as Dasgupta and Heal (1974), Solow (1974), Stiglitz (1974), and Hartwick
(1977), and more recent extensions which include endogenous growth mechanisms such as Gri-
40
maud and Rougé (2003), Schou (2002), and Groth and Schou (2007). All of the papers cited
above are immediately ruled out as candidates to explain the stylized facts about resource consumption and prices since extraction costs are zero in the models, and hence the resource price
simply follows the Hotelling path, i.e. it rises at the rate of interest. Tsur and Zemel (2005) is a
related paper where extraction costs are instead an increasing (but unchanging) function of the
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extraction rate until physical resource exhaustion; hence the prediction of rising resource price
applies a fortiori compared to the models above, given the observation of rising consumption
rates.
Turning now to the more relevant papers in the literature, we describe three categories: firstly,
those where the supply-side model is of particular interest; secondly, where the demand-side
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model of of interest; and finally simulation models.
2
For a review see Krautkraemer (1998).
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3
There are many theoretical papers in the first category, with rich models of resource supply;
important insights can be found in such papers, but without a dynamic link between supply and
demand it is not possible to make credible predictions about overall resource prices and consumption rates. A good example is Heal (1976), who assumes infinite stocks, and that marginal
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extraction costs rise in cumulative extraction, up to some point at which some effectively infinite, homogeneous, reserve of the resource is tapped. He shows that price is above marginal cost
because (p.372) ‘present extraction, by raising the total extracted, pushes up future extraction
costs’. Thus as cumulative extraction approaches the level at which the homogeneous reserve
is tapped, the difference between price and marginal extraction cost approaches zero. See also
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Cairns and Quyen (1998) (and references therein), who develop a sophisticated model of exploration and exploitation of heterogeneous reserves by competing producers, and show that
the price path for individual deposits is quite different from the path for the finished resource.
An important paper in this category is that of Tahvonen and Salo (2001), who build a general
equilibrium model focused on energy. There are renewable and non-renewable energy sources,
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and extraction costs of non-renewables rise in total extraction. They assume a finite stock, and
in the long run there is zero non-renewable resource use, prior to which non-renewable resource
use first rises and then falls. However, there is no directed technological change in the resourcedemanding (production) sector, and hence no possibility for energy-saving technology to be
developed as a response to rising prices; as a result, a necessary condition for long-run growth is
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exponentially increasing energy supply. Furthermore, technology in the resource-extraction sector is tied to the rate of extraction, since they apply the learning-by-doing type of technological
change (cf. Arrow 1962, Romer 1986).
Consider now papers in the second category, i.e. papers in which the modelling of the resourcedemanding sector is of interest. Firstly, a number of authors — see for instance Dasgupta et al.
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(1983) — model short-run adoption games played by a resource supplier and a consumer; these
are potentially relevant, but these authors make no attempt to explain long-run price and consumption trends. Secondly, some papers on climate policy, such as Popp (2006), include models
of endogenous technological change and resource prices. However, these papers are essentially
empirical in nature and the modelling of (for instance) endogenous technological change is
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highly stylized. Thus Popp (2006) fixes the social returns to research investment at four times
the private returns, whereas energy R&D is assumed to crowd out other R&D at a rate of 50
percent.3 Thirdly, Smulders and de Nooij (2003) and André and Smulders (2004) both model
3
Note also Pindyck (1978), who uses a model with lagged changes in demand; demand is a decreasing function
of current prices, and an increasing function of demand in the previous period. This can be seen as an ad hoc model
of directed technological change on the demand side, but it plays only a peripheral role in the paper.
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directed technological change in energy demand in the context of a general equilibrium analysis with growth, using the approach of Acemoglu (2002). Smulders and de Nooij (2003) treat
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energy supply as exogenous, whereas André and Smulders (2004) also include a model of resource supply in which there is a finite quantity of a homogeneous resource, no inputs are used
in resource extraction, but there are ‘iceberg’ costs which decline over time.4 In this context they
show how a decline in iceberg costs, or falling taxes, could if it were sufficiently steep cancel
out the baseline (Hotelling-rule) rise in the resource price and lead to temporarily falling prices.
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The paper is valuable in that it shows how directed technological change in final-good production may be combined with a Hotelling-type model of resource supply, and how technological
change in extraction may lead to temporary price declines in this context. However, the extraction model does not meet our criteria for a general model stated above, that extraction costs
should be a function of extraction technology, the price of extraction inputs, and the quality of
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the resource deposit, the latter being a function of cumulative extraction.
There are two papers in the third category, in which stylized facts about resource prices and
consumption are linked to changes in demand and supply. Wils (2001) uses simulation models
to generate various paths of prices and consumption for resources, roughly corresponding to
observed trends; unfortunately the modelling is ad hoc and there are no analytical results, so
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the insights from the model are limited. Cynthia Lin and Wagner (2007) set up supply and
demand functions for resources with parameters for exogenous rates of technological progress
and stock effects in the supply of minerals, and test whether their assumptions are compatible
with constant resource prices for 14 minerals. They show that the assumptions are compatible,
but in no way do they explain why the rates of progress, and stock effects, are such that constant
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prices result, nor does their approach allow us to make predictions about future prices and
consumption rates.
In our model firms may work in one of two sectors, a production sector and a resourceextraction sector; there is an arbitrage condition. In the production sector, there is directed technological change such that high expected resource prices lead to high investment in resource-
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saving technology; knowledge spillovers between firms are modelled explicitly. In the extraction sector resource holders employ firms to extract resources; holders choose how many firms
to employ and how much to invest in extraction technology, given a link between cumulative
extraction and extraction costs. There are thus four state variables. We show that the economy
approaches a balanced growth path (b.g.p.) along which resource prices are constant relative
4
Iceberg costs are the loss of a fixed proportion of the good, originally in transit, here in the extraction process.
See Samuelson (1954).
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5
to wages, and resource use is constant. The abundance of the resource determines the level of
resource use, while the ease of substituting for the resource in production determines resource
expenditures. Furthermore — given reasonable assumptions about initial conditions — we show
that resource use passes through a frontier phase on the way to the b.g.p., during which resource
use grows at close to the overall growth rate whereas the real resource price is approximately
120
constant.
In Section 2 we set out and justify essential properties of a satisfactory dynamic model of
resource demand and supply. The aim of the remainder of the paper is to set up, solve, and
analyse a model with these properties. In Section 3 we present and solve models of resource
demand (in a final-good sector) and resource supply (from an extraction sector) which together
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have all of these essential properties. In Section 4 we put these two halves together into an
overall model and solve it, finding a closed-form solution for the b.g.p. In Section 5 we set up a
primitive economy and investigate the transition path to the b.g.p. both analytically and through
simulation. In Section 6 we discuss the intuition behind the results, their robustness, and links
to existing literature. Section 7 concludes.
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2. Resource economics: demand, supply, price, and quantity
In the introduction we claimed that a satisfactory model must include the evolution of both
supply of and demand for a resource given the presence of endogenous directed technological change and a link between cumulative extraction and extraction costs, and argued that no
existing literature meets these criteria. Here we substantiate the former claim by stating and
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justifying essential properties of resource demand and supply.
2.1. Essential properties of final-good production and resource demand. We assume that
resources are demanded solely as an input to final-good production. Given this assumption, a
model of the production side of the economy should have the following three properties.
Property D1. If the supply of physical inputs (including the resource) is constant then the pro-
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duction sector approaches a balanced growth path driven by endogenous technical change.
Normally, non-renewable resources are not included in endogenous growth models; instead
we have labour and capital inputs. A key property of these models is that if physical inputs
are held constant then production increases proportionally to the increase in the overall level of
technology, or ‘general knowledge’, and the prices of the physical inputs rise at the growth rate;
145
see for instance Romer (1990) and Aghion and Howitt (1992). We make the same assumption,
although we differentiate a non-renewable resource as a separate physical input. Furthermore,
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THE NATURAL-RESOURCE SEE-SAW
we follow the endogenous growth tradition in assuming that knowledge is a non-rival good
developed through deliberate investment, but that private returns to a given investment attenuate
as the knowledge spills over into the public domain.
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Property D2. Long-run resource demand depends on directed technology investment decisions
made by production firms; a rise in expected future resource prices leads to a rise in resourcesaving technological change — where technology is to be interpreted broadly and may be embodied in capital or labour — and hence a fall in demand.
As explained above, it is common in the literature on resource pricing to ignore technological
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change on the demand side. Effectively, demand is treated either as constant or exogenous,
evolving according to a pre-determined path unaffected by prices. To see how unreasonable
this is, perform the following thought experiment. Imagine a world where hydrocarbon fuels —
whether sourced from fossil-fuel deposits or otherwise — are extremely scarce, but electricity
is plentiful and cheap. Global supply corresponds to just a few barrels of oil per year, despite
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the price being thousands of dollars per barrel. Nevertheless, the capital stock (and embodied
technologies) have evolved just as in our own economy, and hence the short-run demand curve in
this economy is also identical to our own. Millions of cars, aeroplanes, power stations, etc. stand
permanently idle for want of fuel, despite the availability of cheap electricity.
In his groundbreaking work on directed technological change, Acemoglu (2002, 2007) places
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great emphasis on the possibility that, due to directed technological change, the long-run aggregate demand curve for an input factor (such as a non-renewable resource) may be upwardsloping. However, he also explains that this result is only possible in economies where technology choices are not made by technology users (i.e. producers); instead, technology choices must
be controlled by intermediate monopolists. Given the above thought experiment, and empirical
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evidence concerning the effect of resource prices on technology — see for example Newell et al.
(1999) — we follow Hart (2008) in assuming that producers also control technology investment.
Property D3. Short-run resource demand is inelastic.
This property is straightforward; observed short-run elasticities of demand for the naturalresource inputs typically take values between 0 and −1 (see for instance Espey, 1998).
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2.2. Essential properties of resource extraction and supply.
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7
Property S1. Extraction costs vary depending both on the physical nature of the resource deposit, and the quality of extraction technology.5
This property is straightforward; given inhomogeneous deposits, we then expect extraction
costs to rise in cumulative extraction. Recall that many models in the literature fail to account
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for links between cumulative extraction and extraction costs, and very few account for changes
in extraction technology.
Property S2. Marginal extraction costs are rising in the rate of extraction if capital is held
constant, but constant if extraction capital can be replicated.
For given extraction capital — a given mine with given equipment — we expect marginal ex-
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traction costs to rise steeply in the rate of extraction. On the other hand, if we allow for a
scaling-up of extraction effort (opening new mines) then — if there are constant returns to scale
— marginal extraction costs will be independent of the rate of extraction, depending instead on
resource remoteness and extraction technology (Property S1).
Property S3. Resource reserves are heterogeneous and extraction costs increase without bound
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in cumulative extraction, as the most easily accessible resources are extracted first. There is
however no theoretical limit to total extraction; total reserves are infinite.
The basic idea of Property S3 is well established, following (among others) Heal (1976) and
Goeller and Weinberg (1976). Concerning exhaustion, when resources are chemical elements
— iron, copper, zinc, etc. — the argument is trivial if we limit the timescale to, say, one thou-
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sand years, given the vast reserves in the earth’s crust, and the fact that such elements are not
destroyed when they are used in the economy, merely dispersed. Turning to fossil fuels, exhaustion can clearly be envisaged. However, considering the full range of fuels available — crude
oil, coal, natural gas, oil shales, etc. — the picture of large but inhomogeneous stocks where
extraction costs (including costs of discovery, extraction and refinement) rise in cumulative ex-
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traction fits well. Concerning energy supplies in the very long run, see Section 6.2.
Property S4. Returns to investment in extraction technology decline the higher the level of
extraction technology relative to production technology.
Property S4 links the level of extraction technology to production technology. The link follows because technologies used in different industries are related to each other, and progress in
5Note that we use extraction costs as a shorthand for all costs involved in turning a resource in the ground into
an input in the production process, ready to use. In other words, it includes costs of exploration, extraction and
refinement.
8
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THE NATURAL-RESOURCE SEE-SAW
one industry spills over to others. If one industry — small relative to the sum of all other industries — invests heavily then it will advance technologically relative to the other industries, and
the more it advances the less it will benefit from spillovers from these industries; conversely, if
an industry falls behind due to lack of investment then the potential for spillovers increases.
To see the link more clearly, perform another thought experiment. An extraction industry
210
is, from 1950 onwards, technologically isolated from the global economy. The industry suffers
technologically as it (i) fails to take advantage of advances in computing, measurement equipment, transport equipment, etc. which are made in the economy as a whole, and (ii) does not
have the resources to make these advances for itself. The industry emerges from its isolation in
2008, and a period of very high returns to investment in extraction technology follows.
3. A model of two halves
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In this section we first mention some key properties of the overall framework (Section 3.1),
before setting up separate models of a final-good sector in which the number of production
firms and supply of the resource are exogenous (Section 3.2) and an extraction sector in which
the number of extracting firms and demand for the resource are exogenous (Section 3.3). Recall
220
that in Section 4 we put the two together to build a model in which all variables are endogenously
determined.
3.1. Overall framework. In the overall model there are two sets of infinitely-lived agents: a
mass L of identical workers, each owning a single machine and supplying their labour inelastically; and a unit mass of identical competitive resource holders, each endowed with extraction
225
rights to separate, identical, resource deposits. A machine-owning worker is also denoted a firm.
All agents maximize the discounted sum of their lifetime consumption, U . The marginal utility
of consumption is assumed constant such that the (constant) time preference factor ρ is also
equal to the discount factor per period; ρ ∈ (0, 1).6 Thus we have
∞
(1)
U = ∑ ρ t xt ,
t=0
where xt is consumption in period t.
6Note that this can be taken at face value, as in Aghion and Howitt (1992). Alternatively, if the economy is
always close to the b.g.p. then the discount factor will be constant even when the underlying utility function is of the
C.E.S. type.
THE NATURAL-RESOURCE SEE-SAW
230
9
Resource holders do not work; instead, they each employ lte machine-owning workers (firms)
as the sole extraction input. The total mass of firms employed in extraction is thus given by
(2)
Lte
=
Z 1
0
e
li,t
di,
where i indexes resource holders. (Note that we suppress this index except where doing so would
cause confusion.) The remaining firms, mass Lty , work in final-good production, for which they
buy non-renewable resource inputs qt . The total quantity of resource consumed is
(3)
235
Qt =
Z Lty
0
q j,t d j
where j indexes production firms.
Firms and resource holders are always price takers, and their mass is fixed by construction in
the model; they cannot merge in order to gain market power or benefit from scale economies.
(Note that the size of machines — and hence the number of firms — could be made endogenous
in an expanded model with a continuum of final goods and a love of variety. In that case scale
240
economies would be offset in equilibrium by the loss of variety due to mergers.) Furthermore,
irrespective of the choice of sector, workers and machines are used in fixed proportions; each
machine requires one worker to operate it (Leontief). Since firms are price takers, if a firm
operates, it will operate at full capacity (one machine, one worker). We can thus combine
workers and machines, a combination we refer to as labour; furthermore, the cost of hiring a
245
unit of labour is referred to as the wage.7
3.2. Model of final-good production and resource demand. The modelling of the final-good
sector is a simplified version of Hart (2008), borrowing from Solow et al. (1966). In a given period t, the sector contains a mass of identical firms Lty , treated as exogenous in this section. Each
firm produces a single final good price pty (normalized to unity), quantity yt . For concreteness
250
we measure production y in units of utils per period, and measure money in dollars (not to be
confused with any real currency).
In addition to labour, final-good production requires resource input qt , which is supplied
exogenously at price pt . Again, the production function is Leontief, such that in a given period,
the flow of output is proportional to the flow of the non-renewable resource input, up to the point
255
at which the unit of labour operates at full capacity. A firm may boost its productivity over time
through investment, in two respects; raising production technology a or raising resource-saving
7Since the firm invests in the quality of its worker and machine, which are then used in production, then they may
also be thought of as the firm’s capital. The physical quantity of a firm’s capital is then constant, whereas its quality
(and hence value) may rise through investment.
10
THE NATURAL-RESOURCE SEE-SAW
technology b. A single firm’s production function is
(4)
yt = at min{1, qt (bt /at )}.
To understand the function, assume labour runs at full capacity. Then a rise in a leads to proportional increases in both final-good production and resource demand, whereas an increase in
b leaves final-good production unchanged but reduces resource demand. Since firms are price
takers it follows that if firms produce at all (when profits are positive) then:8
(5)
qt = at /bt ;
(6)
yt = at .
Now define the factor shares of labour and resources as σtl and σtq respectively. Then
(7)
σtl = 1 − pt /bt ;
(8)
σtq = pt /bt .
The flow of consumption x (utils per period) to the owner of the firm is profits divided by
final-good price (unity), minus investment (utils per period),
(9)
260
xt = at (1 − pt /bt ) − (αt + βt ),
where α and β are investments of final goods in boosting technologies a and b respectively.
Investment in period t boosts firm technology in period t + 1, building on general knowledge for
the respective technologies, At and Bt :
(10)
at+1 = At ((αt /At )/α0 )φ ;
(11)
bt+1 = Bt ((βt /At )/β0 )φ .
Here φ is a parameter (less than unity) determining the elasticity of next-period knowledge to investment, and α0 and β0 are parameters. General knowledge is the integral of firms’ knowledge,
265
hence
(12)
At =
Z Ly
t
0
ai,t di.
(The analogous equation applies to Bt .) We also call At total production, as it is the quantity of
final-good production in equilibrium (from equation 6).
8Note that, given positive profits, a firm’s short-run resource demand is a function of its technology levels but not
the resource price. In an alternative version of the model, based on the model in Hart (2008), short-run demand for
the resource is of constant elasticity, less than unity. This complicates the model somewhat, without changing any of
the key results.
THE NATURAL-RESOURCE SEE-SAW
11
Note that equations 10 and 11 imply that the benefits of firms’ investments αt and βt are
completely private in period t + 1, and completely public thereafter. This is of course a simpli270
fication; in Hart (2008) there is carryover of technology within the firm from one period to the
next, and the results are qualitatively similar. Furthermore, note that there is a scale effect in
the model; if the number of firms increases, the growth rate will increase for given investment
rates. There are well-known methodologies to extend growth models such as the one presented
here in order to ‘sterilize’ such scale effects, and potentially increase the power of the model
275
in other respects. See Jones (1999) for an introductory discussion. Finally, note that there is
no strategic interaction between firms; there is, for instance, no trading of production rights for
patented technologies.
Each final-good firm faces a constrained optimization problem over infinite time. However,
since firm knowledge is private for one period only and the marginal utility of consumption
280
is constant (equation 1), we can divide the problem up into a series of independent problems
in which the firm maximizes net returns to final-good production — denoted wt — period by
period. Net returns are equal to the profits from final-good production in period t minus the
present value of the optimal investments made in period t − 1:
(13)
wt = at − pt qt − (αt−1 + βt−1 )/ρ .
Furthermore, we already have the optimal choice of q in each period, equation 5. So for period-t
285
production each firm chooses {αt , βt } to maximize wt+1 subject to the constraints on knowledge
growth (equations 10 and 11) and the choice of q (equation 5):9
αt φ
At αt β0 φ αt + βt
(14)
wt+1 = At
− pt+1
−
.
At α0
Bt βt α0
ρ
Take first-order conditions in α and a, and for β and b, to derive
αt φ
qt+1
(15)
αt = ρφ At
1 − pt+1
.
At α0
at+1
αt φ
qt+1
(16)
βt = ρφ At
pt+1
.
At α0
at+1
Furthermore, second-order conditions show that the production function is concave given the
restriction that production yt+1 must be positive. Hence for given prices and quantities, any
290
solution for α and β which satisfies the first order conditions is a unique global optimum, and
hence there is a symmetric competitive equilibrium, and we have
(17)
At = Lty at ;
Bt = Lty bt ;
Qt = Lty qt .
9Note that w is also the cost of hiring firms for resource extraction in the full model of Section 4.
t
12
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Finally, denote the growth factor for final-good production per period as θt , and use equation 10
to write
θt = at+1 /at = Lty (αt /(At α0 ))φ .
(18)
Then we have the following proposition, where a tilde denotes a constant value for a variable.
295
Proposition F1. If there are Ley workers in the final-good sector, and the resource price grows
by a constant factor θ p per period, then there is a b.g.p. in the final-good sector along which
resource consumption per firm q grows by a constant factor θ q per period, and production
technology a grows by θe per period, such that θ p θ q = θe.
Proof. See Appendix A.
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The intuition behind Proposition F1 is straightforward. Firstly, note that θ p θ q = θe implies
that pq/a is constant, hence the factor shares of labour and resources are constant along the
b.g.p. Secondly, the growth equations show that if constant shares of production are invested in
technologies a and b then they will grow at constant rates. Finally, we know from equation 5
305
that θ b θ q = θe, hence we have balanced growth when b/a is such that θ b = θ p .
For further intuition, consider the following two special cases.
Case 1. When the resource price pt tracks production at such that pt /at is constant, then on the
b.g.p. qt is also constant.
Case 2. When the resource price pt is constant, then on the b.g.p. resource use q grows at θe.
In Case 1 the prices of both inputs grow at the same rate, while the quantity of labour is
310
fixed. Given the symmetry in the technology growth functions we expect Hicks-neutral balanced
growth where the quantity of the resource inputs, and hence factor shares, are also constant. In
Case 2 the price of the resource falls at a constant rate relative to the price of labour, hence the
ratio b/a falls over time. On the b.g.p. b is constant and hence (from equation 5) resource use q
grows at the overall growth rate; factor shares are again constant.
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3.3. Model of resource extraction. In this section we model resource extraction. There is a
unit mass of resource holders, each with identical endowments, and each employing lte firms
as the sole extraction input. In period t a single holder supplies quantity nt , and total supply
Qt =
R1
0
nt,i di, where i indexes the resource holders.10 Holders must pay firms the net returns
10Note that Q is thus doing double service; we assume that total extraction is equal to total consumption in each
t
period. This is harmless since there is perfect information in the model; there is no reason to extract resources which
cannot be sold.
THE NATURAL-RESOURCE SEE-SAW
13
they could otherwise earn in final-good production, wt . In the full model wt is given by equation
320
13, but here we treat it as exogenous. The price holders receive for finished resources, pt , is also
exogenous, as is the overall level of production technology At .
The resource holder actually agrees to hire labour in period t − 1, and then invests γt−1 in the
quality of the labour, i.e. the level of extraction technology ct . This quality builds on general
knowledge in both extraction and production in period t − 1, as follows:
(19)
325
ζ
1−ζ
ct = Ct−1 At−1 ((γt−1 /At−1 )/γ0 )φ ,
where ζ ∈ [0, 1) and γ0 > 0. Consider the corner values for ζ . When ζ = 0 then extraction
technology is effectively reinvented each period based on the latest production technology in the
final-good sector; as ζ approaches unity then extraction technology grows almost independently
of production technology, and spillovers from the final-good sector approach zero. General
knowledge Ct grows in a way analogous to production knowledge, equation 12:
(20)
330
Ct =
Z Le
t
0
ci,t di.
These equations are consistent with Property S4; extraction technology improves through investment and spillovers from the production sector, and these spillovers decline in importance
the higher the level of extraction technology compared to production technology.
Effective extractive effort for a given resource holder is ct lte . However, the rate of extraction
depends on a further variable, the physical difficulty of getting the resource out of the ground.
335
This variable, denoted remoteness R, is an increasing function of cumulative extraction, since we
assume that the most accessible (least remote) resources are extracted first. Denoting cumulative
extraction at the end of period t as Nt (thus Nt = ∑ts=0 ns , where nt is extraction in period t), we
define
(21)
Rt = R0 exp[Nt /N],
where N > 0 is the abundance of the resource and R0 is initial remoteness.
340
Equation 21 implies that total resource stocks are infinite, but that the remoteness of marginal resources increases exponentially for a constant rate of extraction. Furthermore, the more
abundant the resource (the greater is N), the lower the rate at which remoteness increases for
a given rate of extraction. Consider iron ore (high abundance) and platinum (low abundance).
Extraction of 1000 tons of iron ore has a negligible effect on the remoteness of the marginal
345
remaining stocks, whereas extraction of the 1000 most accessible tons of platinum has an appreciable effect. Note that Nt and N are both measured in tons.11
11The assumption of a perfect exponential increase in remoteness is an approximation, the attraction of which is
that it allows for a b.g.p. in the overall economy. In Section 6 we discuss the consequences when the assumption does
14
THE NATURAL-RESOURCE SEE-SAW
Now define nt , Zt , and zt as total extraction over period t, total extraction costs for period t,
and the marginal cost of an increase in nt respectively. Furthermore, define st,τ as the flow rate
of extraction after fraction τ of period t, hence nt =
350
R1
0 st,τ dτ .
Extraction technology and labour
are (by construction) constant within a given period, but remoteness increases continuously. We
define
st,τ =
(22)
ct e
l ,
Rt,τ t
that is the flow of resource extraction increases linearly in labour and the level of extraction
technology, but falls in the remoteness of the marginal resource. Recall that resource holders
must pay wt per unit of labour. Total extraction costs are thus given by
Zt = wt lte .
(23)
355
They can also be expressed as a function of nt :12
(24)
Zt = Nwt (Rt−1 /ct )(exp(nt /N) − 1) = Nwt (Rt /ct )(1 − exp(−nt /N)).
Differentiate w.r.t. nt to yield
zt = wt (Rt /ct ).
(25)
Equations 24 and 25 are consistent with Properties S1 and S2; total extraction costs rise in
cumulative extraction and fall in the level of extraction technology, whereas marginal extraction
costs are independent of the rate of extraction. The latter result follows given that firstly we
360
abstract from short-run adjustment costs of capital,13 and secondly the resource holders are
price takers, and can therefore adjust their employment of labour without affecting its price.
Finally, note that ceteris paribus (i.e. when R and c are both constant) extraction costs rise at the
wage rate. Furthermore, equations 21 and 24 are consistent with Property S3; extraction costs
not hold; here we present two physical models of resource distribution which could give rise to the assumed trend.
Firstly, assume that the representative resource owner owns an infinite vertical column of the resource in pure form,
so dt = Nt /N, where dt is the depth of the most accessible remaining resource. Furthermore, remoteness Rt = edt ;
for given technology, extraction costs increase exponentially with depth. Equation 21 then follows. Alternatively,
assume that the representative owner owns an infinite one-dimensional strip of the resource on the surface. At
the start of the strip, point l = 0, the resource is in pure form (concentration k = 1), but moving along the strip
concentration declines such that k = (l/l + 1)−1/2 (where l = N/2). If in addition costs of extraction and refinement
rise exponentially in k−1 such that Rt = exp[(1 − kt )/kt ], then equation 21 is again returned.
12Set up an equation for R
t,τ ,
use it to set up a differential equation in st,τ , and solve it.
13If we were to allow for such adjustment costs then some dynamic effects due to lags in changes in extraction
capital would in general emerge, but the b.g.p. would be unaffected.
THE NATURAL-RESOURCE SEE-SAW
15
rise in cumulative extraction, and indeed they approach infinity in the absence of technological
365
progress. However, there is no physical resource exhaustion.
Recall that resource holders, like workers, maximize discounted consumption; equation 1.
The problem facing a resource holder in period t − 1 is then to choose how many workers lte
to employ, and how much γt−1 to invest in workers’ machinery. Returns are gross revenue pt nt
minus extraction costs Zt minus the present value of investment costs lte γt−1 /ρ , plus the value
370
of changes in knowledge stocks (equation 19) and resource stocks (Nt = Nt−1 + nt ). Note that
Zt is a function of state variables and nt (equation 24), and lte = Zt /wt (equation 23). Hence we
can formulate a resource holder’s problem as the choice of {ns , γs }∞
s=0 , and write the problem as
a Lagrangian:14
∞
(26)
L
=
∑ ρt
t=0
pt nt − N(Rt−1 /ct )[exp(nt /N) − 1][wt + γt−1 /ρ ]
ζ
1−ζ
c
−ρνt+1
[ct+1 −Ct At
((γt /At )/γ0 )φ ] + λt [Nt − (Nt−1 + nt )] ,
where ν c is the shadow price of the extraction knowledge held by the le agents employed by
375
the resource holder, and λt is the shadow cost of extraction. Take first-order conditions in firm
investment γt and technology ct , and substitute for c from equation 19, to yield
γt =
(27)
ρφ
wt+1 ;
1−φ
investment chosen by the resource holder in a given unit of labour depends only on the price
of that unit, and not on the holder’s extraction rate. This is what we should expect given that
the resource holders are price takers; if they want to extract more in a given period they employ
380
more units of labour, but invest the same amount in the technology levels of each unit. The
condition in extraction nt yields
(28)
pt (1 − φ ) = zt + ρ
Zt+1
Zt+2
+ ρ2
+ ··· .
N
N
The firm sets quantity such that the price (exogenous to the firm) is equal to marginal extraction
cost plus the discounted sum of effects on future extraction costs, all divided by (1 − φ ). The
factor (1 − φ ) reflects the fact that increased extraction also entails increased investment costs.
385
We thus have two equations in two unknowns for a single resource holder, given: (i) the
evolution of variables exogenous to that holder, i.e. A, B, C, w, py , and p; (ii) the equations
linking a resource holder’s extraction costs to the extraction rate, extraction labour, and levels of
the state variables (i.e. 23–25); and (iii) the equations governing the evolution of state variables
14For an introduction to the use of the Lagrangian method to solve dynamic problems in economics see Chow
(1997).
16
THE NATURAL-RESOURCE SEE-SAW
c and R (i.e. 19 and 21). Assuming a symmetric equilibrium we have
(29)
390
Ct = Lte ct ;
Lte = lte ;
Qt = nt .
Finally, we analyse the properties of the sector, presenting two propositions which apply
under alternative assumptions about the wage w, the resource price p, and the level of production
technology a.
Proposition E1. If the w, p, and a all grow by a constant factor θe per period, then there is a
b.g.p. in the extraction sector along which the rate of extraction is constant.
395
Proof. See Appendix B.
The intuition here is that increases in technology a compensate for increases in w, and the net
effect on real extraction costs is zero. If the competitive resource price p is to rise at the growth
rate this implies that remoteness R must rise at this rate, which (given the nature of the stocks)
implies a constant rate of extraction.
400
Proposition E2. If w and a grow by a constant factor θe per period, but (i) p is constant and
(ii) resource abundance N is infinite and hence Rt is constant, then there is a growth path in the
extraction sector where the rate of extraction grows by a factor θe per period.
Proof. See Appendix B.
Since R is constant, technological change allows exponentially increasing extraction using
405
the same (constant) labour force. Unit costs are constant, not falling, since the wage increases
at the same rate.
4. The overall model
In this section we put together the final-good and extraction sectors to build a model which
accounts for the allocation decisions of all the agents in the economy; in this sense it is a general
410
equilibrium model, although it is not general equilibrium in the fullest sense since the marginal
utility of consumption is assumed constant, implying that the interest rate is also constant. We
solve the model for a balanced growth path, and go on to investigate the properties of the b.g.p.,
and the transition path to the b.g.p. from a hypothesized primitive economy.
Recall that in the final-good and extraction sectors we had Lty and Lte firms respectively. In
415
the overall model we have a fixed total number of firms, i.e.
(30)
L = Lty + Lte ,
THE NATURAL-RESOURCE SEE-SAW
17
and Lty ≥ 0, Lte ≥ 0. Furthermore, we have an arbitrage condition between the sectors such that
resource holders must pay the firms which they hire wt per period, where wt is given by equation
13, i.e. it is the net returns that labour can earn when employed in the final-good sector in the
same period.
420
4.1. Balanced growth. Given that we have a competitive equilibrium in the overall economy
then have the following proposition.
Proposition O1. There is a b.g.p. in the overall economy, in which: (i) the growth factor per
period θe and the allocation of labour between sectors are constant; and (ii) the resource price
is constant relative to the price of labour, and resource consumption is constant.
425
Proof. See Appendix C.
Intuitively, Proposition O1 follows from Propositions F1 and E1, with the addition of an
arbitrage condition determining the quantities of labour in production and extraction respectively. There are two key pairs of equations for the b.g.p.: (i) growth–arbitrage; and (ii) resource
supply–demand.
0.01
0.4
Growth rate, θe
0.2
θe = 1/ρ
Demand, Eq. 33
0.008
Price g
p/a, $/ton
0.6
1
0.8
0.6
0.004
Growth, Eq. 31
0.4
0.002
0.2
0
Supply, Eq. 34
0.006
0
0.5
e
Ly = L
Arbitrage, Eq. 32
1
Production labour Ley
0
0
20
40
60
80
Quantity ne, tons/period
F IGURE 1. The steady state: (a) Growth–arbitrage; (b) resource supply/demand.
Growth–arbitrage. The growth and arbitrage equations on the b.g.p. are as follows:
φ /(1−φ )
ρφ
e
(31)
θ=
Ley
α0 + β0
(32)
430
Lee (θe − 1)(1 − ρ θe)(1 − φ )
β0
=
,
y
e
e
e
α
−
φ
(
α0 + β0 )
L
0
(1 − ρ )θ ln θ
as illustrated in Figure 1(a) for the baseline parameterization (Appendix D). The growth equation shows how growth increases linearly in the size of the final-good sector (recall the discussion on the scale effect in Section 3.2). The arbitrage equation shows how labour moves
18
THE NATURAL-RESOURCE SEE-SAW
into production from extraction the higher is the growth rate; this is because a high growth rate
lowers the effective discount rate, hence resource holders — who are most forward-looking —
435
reduce their extraction rate. When θe = 1/ρ then extraction stops altogether.
Supply–Demand. The demand and supply equations for the resource in the steady state are as
follows (where a tilde now also denotes a constant value for the ratio of two variables):
(33)
(34)
/a =
pf
β0 Ley
α0 + β0 ne
ne = N ln θe,
as illustrated in Figure 1(b) for the baseline parameterization (Appendix D). Note also that the
wage rate tracks overall growth:
(35)
g
/a =
w
α0
− φ.
α0 + β0
The demand equation follows from Proposition F1: if the resource price is constant relative
to the prices of other (fixed) inputs, then the economy approaches a growth path along which
440
technological change is Hicks-neutral and resource use is constant. The supply equation follows
from Proposition E1: if the resource price is constant relative to the prices of other (fixed) inputs,
resource extraction approaches a rate ne such that the remoteness of the most accessible resources
remaining in the ground rises at the same rate as the overall rate of technological progress in the
445
economy, θe. Put the two together to understand why the economy converges on the steady state:
if the extraction rate n > ne then remoteness will rise at a rate greater than θe, hence the resource
price will rise relative to other inputs, reducing demand and hence extraction. And vice versa.
4.2. Comparative statics and stability of the b.g.p. Consider now comparative statics, starting with the growth–arbitrage equations (Figure 1(a)). The key point here — given that our focus
is on the role of the resource in economic development — is that the higher is Le , the lower is
450
the growth rate.15 The main factor determining Le (given L) is β0 /α0 , which measures the substitutability of resource inputs by other inputs (labour) in the long run. How easy (relatively) is
it to develop technologies which economize on resource use? The easier it is (low β0 /α0 ) the
lower is Le and hence the higher the growth rate. In the base parameterization this ratio is small
and hence Le /L, and the factor share of resource inputs, is small. This means in turn that Ly /L
455
is close to unity and effects of changes in the resource sector on overall growth are small.
Consider now resource supply and demand (Figure 1(b)). The supply curve is vertical, and
hence the b.g.p. level of resource consumption does not depend on demand for the resource.
Rather, it depends on the nature of resource deposits and the overall growth rate of technology;
15Other effects, such as that of the discount factor on growth, are as expected.
THE NATURAL-RESOURCE SEE-SAW
19
these determine the extraction rate such that increasing resource remoteness is balanced by
460
improvement in extraction technology. A rise in either resource abundance or the overall growth
rate shifts the long-run supply curve to the right, such that quantity increases and price falls
(Figure 1(b)). Given the vertical long-run supply curve, shifts in long-run demand affect the
price while leaving quantity unaffected. Long-run demand is decreasing in the substitutability
of the resource β0 , i.e. the baseline cost of developing resource-saving technology.
465
From equations 5 and 8, the identities 17 and 29, and the demand equation 33 we have
(36)
fq = β0 /(α0 + β0 ).
σ
Thus the factor share of resources in final-good production depends only on the relative substitutability of the resource, and not at all on resource abundance N. To understand this, consider
the units. Recall that abundance N is measured in unit of mass (tons). It is then intuitive that the
price, in $/ton, should be declining in abundance (tons), whereas resource expenditure, price ×
470
quantity and hence measured in $/period, should not be a function of abundance, but rather the
economic variables such as substitutability of the resource. In terms of the model, an increase
in N leads to an increase in b.g.p. resource consumption and a compensating reduction in the
b.g.p. price (long-run demand is downward-sloping).
We now turn to the properties of the b.g.p. Is it optimal, unique and stable? The reduced-
475
form state variables b/a, c/a, and R/a are constant given balanced growth; that is, we have
a steady state, and we can in principle perform a linear approximation around the steady-state
solution to investigate its properties. We then need expressions for αt , βt , nt+1 and γt , close
to the steady state, in terms of (b/a)t , (R/a)t and (c/a)t . Unfortunately the expressions are
cumbersome, and we therefore restrict ourselves to numerical analysis. Figure 2 shows results
480
for one parameterization. Note that c/a is not shown; given the choice of ζ it moves rapidly to
its b.g.p. value.16 For all the parameterizations tested we have a stable steady state.
5. Transition path
The b.g.p. predicted by the model — in which the resource price tracks the wage while consumption is constant — does not tally well with the observation that prices tend to be constant
485
while consumption increases rapidly. On the other hand, steady growth in consumption rates
cannot go on indefinitely on a finite planet, hence there must at some stage be a transition to
some other longer-run trend. This raises the question of what the model predicts about the
transition path towards the b.g.p.
16See Appendix D for details concerning the simulations.
20
THE NATURAL-RESOURCE SEE-SAW
g
Remoteness, (R/a)t /R/a
1.6
?
1.4
1.2
6
-
1
?
0.8
6
b/a constant
0.6
0.6
0.7
0.8
0.9
R/a constant
1
1.1
1.2
1.3
1.4
/a
Resource-saving tech., (b/a)t /bf
F IGURE 2. Phase diagram for base parameterization. The dotted lines show
simulated transition paths for different starting values, successive dots showing values of state variables for successive periods. The continuous lines are
derived by estimating the turning points of the dotted lines.
Trends in prices and consumption rates along any transition path will depend on the starting
490
point of that path; hence it is important to determine the starting point in a non-arbitrary way.
We begin by doing this, defining a pre-resource primitive economy. Given that there is no
closed-form solution to the model away from the b.g.p., we then pursue two alternatives: firstly
we investigate closed-form solutions to a simplified model; secondly we perform numerical
simulations with the full model. We show that the economy passes through a frontier phase on
495
the way to the b.g.p., during which the resource price is approximately constant while resource
consumption grows at approximately the overall growth rate.
5.1. The primitive economy. In the primitive economy an alternative (primitive) resource is
used by final-good producers, price Pt , where units are normalized such that the primitive resource is a perfect substitute for the main resource. Furthermore, by construction Pt /at is con500
/a, the long-run b.g.p. price (equations 31–35). The
stant — denoted P/a — and greater than pf
final-good sector is on a b.g.p. as defined by Proposition F1, Case 1, given P/a, hence the ratio
of extraction to production knowledge, c/a, is constant (equation A27 in the appendix). We
assume that this knowledge can be transferred to the new resource with the loss of a fixed proportion: ct /at = ψ · c/a, where ψ ∈ (0, 1). The reason that the main resource is not used is that
505
extraction costs are above Pt , due to the primitive extraction technology.
In the primitive economy R is constant at R0 , since there is no extraction. Hence we know,
from equation 25, that marginal extraction costs are given by zt = R0 (wt /ct ). The wage wt
rises at the overall growth rate of at , as does the hypothetical level of ct assuming a decision
to produce followed by optimal investment γt−1 . Thus marginal extraction costs are constant in
THE NATURAL-RESOURCE SEE-SAW
510
21
real terms, whereas the price of the primitive resource tracks the wage. Thus within finite time
extraction costs fall below the price of the primitive resource, and extraction begins.17
/a and ψ close to unity then extraction will start at close to its b.g.p. level;
If P/a is close to pf
there is a smooth transition from one resource to the other. However, we are more interested in
the case where the normalized b.g.p. price of the new resource is much lower than that of the
515
/a ≪ P/a) but the primitiveness of technology hinders its extraction
primitive resource (i.e. pf
initially. In that case there is a frontier phase.
5.2. The simplified model. In order to investigate the transition path simply and analytically
we simplify the model in two respects. Firstly, recall that we are interested in the case where
520
/a ≪ P/a, and hence the initial extraction rate n is low compared to the b.g.p. rate. In the
pf
simplified model we assume a limiting case where initial extraction is so low that it has no effect
on Rt , the remoteness of marginal resource stocks. Hence we have Rt = R0 , and furthermore
the shadow price of the stock, λt , is zero (equation 26). Secondly, we assume that the resource
sector is a small part of the overall economy, such that values of α , w, and θ can be calculated
independent of changes in resource prices and quantities. Given this simplification we can solve
525
the demand-side model explicitly for α , β , and w.18
The solution is straightforward. From equations 13–16 we have wt /at = 1− φ . Use equations
19, 25, 27, 28, and A27 to show that extraction begins when
" #1/(1−ζ )
L ζ γ0 φ
R0
(37)
Pt ≥
,
ψ
α0
Lee
whereas on the b.g.p. of the simplified economy
" #1/(1−ζ )
L ζ γ0 φ
.
(38)
pt = R0
α0
Lee
Then from Proposition F1 we know that resource consumption grows at the overall growth rate
φ /(1−φ )
nt+1 at+1
ρφ
(39)
=
=L
.
nt
at
α0
530
Thus the simplified model predicts that the economy will approach a b.g.p. along which the
resource price is constant whereas resource consumption grows at the overall growth rate. The
17Note that the actual start of extraction will also depend on λ , the shadow price of resource stocks; for an explicit
solution see the simplified model below.
18For the vast majority of resources — perhaps all with the exception of energy — this is a reasonable assumption.
In the case of energy, very large fluctuations in prices do affect overall growth and wages. However, the fluctuations
we observe are of a short-run nature and far greater than anything predicted by the model, and the effects are largely
on the business cycle rather than affecting the overall growth trend. Even here, the effects of long-run, more gradual
changes in prices on overall growth rates are likely to be small.
22
THE NATURAL-RESOURCE SEE-SAW
interpretation is as follows. As long as extraction n is low compared to abundance N, then
extraction has an insignificant effect on remoteness. There are then two effects: firstly, the
quality of extraction technology grows at the growth rate; secondly, the wage grows at the same
535
rate. These effects cancel each other out, and the resource price is constant. However, a constant
resource price implies that the price of resource inputs falls relative to the price of other inputs
(the wage), hence demand increases. Returning to the extraction sector, we can think of the
increase in extraction as being achieved through a geographical expansion of extraction efforts
— opening more mines, wells, etc. — to exploit reserves which are not significantly different to
540
reserves already under exploitation.
Along the b.g.p. of the simplified economy the factor share of resources is constant, thus if it
starts very low (in accordance with the simplifying assumptions) it will remain low. However,
the extraction rate grows exponentially, and hence the first simplifying assumption — that extraction is so low that its effect on remoteness R can be ignored — will hold less and less well
545
over time in the full model. That is why the b.g.p. of the simplified economy can never be a
b.g.p. in the full model; given a constant growth rate of resource extraction, gradually the effect
on remoteness Rt of the marginal remaining resources becomes significant; that is, lower quality
reserves must be exploited. This leads to rises in the resource price and slows the growth rate in
resource consumption. In the long run — as shown by Proposition O1 — we have a b.g.p. along
550
which the real resource price tracks the price of labour wt , and resource consumption is constant.
/a. If
The frontier phase is thus transitory, and its length is a function of the ratio P/a/ pf
initial price is very high (and hence initial consumption is very low) due to the difficulty of
extraction, then it may take hundreds of years for increases in the productivity of labour to
generate sufficient demand for the resource such that physical remoteness R rises at something
555
approaching the overall growth rate.
5.3. Numerical simulation. In Figure 3 we see simulation results — for the baseline parameter
/a, so the (normalvalues and ψ = 1 (Appendix D) — starting in period T = 0, with P/a = 100 pf
ized) price of the primitive resource is 100 times the long-run b.g.p. resource price. Consumption
is initially slow to rise as there is a lag in the fall in b/a; once this lag has worked through we
560
see a period of 100–200 years where the real price is approximately constant and consumption
rises at a rate close to the overall growth rate. Once price begins to rise, tracking wages, there
is another lag in the slowing in growth of resource consumption, now caused by a lag in the rise
of resource efficiency on the demand side, b/a.
Normalized price, quantity
THE NATURAL-RESOURCE SEE-SAW
wt /w0
1000
@
R
@
23
nt /n0
?
100
@
I
@
pt /p0
0
0
10
20
30
40
Period (10 yrs)
F IGURE 3. Transition to the steady state, starting at the start of the frontier
phase (period 0). The vertical axis gives the variables normalized by their starting values; note the logarithmic scale.
6. Discussion
565
In this section we relate the results to stylized facts about existing trends, and go on to discuss
the robustness of the predictions.
6.1. Explaining stylized facts about resource prices and consumption. We present four stylized facts about resource use, justify them, and then discuss them in relation to the model results
and alternative explanations in the literature.
570
SF 1. The prices and quantities of different resources vary widely.
SF 2. Real prices fluctuate significantly in the short and medium term.
SF 3. The long-run movements in real prices are very slow.
SF 4. The long-run trend in resource consumption is rising.
SF 1 is uncontroversial; consider for example diamonds, fossil fuels, and sand. Concerning
575
SF 2 and 3, there is much empirical research on trends in resource prices, triggered by Barnett
and Morse (1963) who examined resource prices in the U.S. from 1870 to 1957, and concluded
that the real price of natural resources did not generally increase over the period examined.
The majority of subsequent authors conclude that natural resource prices are non-stationary and
have unit roots; Withagen (1998) has a useful introduction to this question. However, in a recent
580
paper Lee et al. (2006) argue on the contrary that natural resource prices are stationary around
deterministic trends with structural breaks. In either case, the trends are weak compared to the
trend in the overall growth in wages, while the breaks are large. SF 4, increasing consumption,
is straightforward on the face of it, but it is scarcely dealt with in the mainstream resource literature, where the focus is almost entirely on prices. However, Johnson et al. (1980) present
24
585
THE NATURAL-RESOURCE SEE-SAW
data showing large increases in output of a range of U.S. extractive industries over the period
1870–1966.19 Furthermore, the US Geological Survey present data showing that global production/consumption rates of a range of minerals rose steeply during the 20th century (Kelly and
Matos, 2005).
Turning to the results, the model does encompass the wide variations in the prices and con-
590
sumption levels of different resources (SF 1). Recall that on a b.g.p.: (i) the level of resource
consumption depends only on the nature of resource deposits (resource abundance N); (ii) the
factor share of the resource sector, σ q , is not a function of resource abundance N but depends
on the substitutability of the resource; and (iii) the resource price (compared to wages) is a decreasing function of resource abundance and the substitutability of the resource. In the light of
595
points (i)–(iii), consider the following heuristic examples.
Diamon is of very low abundance and, although it has many potential uses it is easily substitutable in most.20 Hence: (i) the factor share is low; (ii) total expenditure on Diamon is
low; (iii) the price level depends on the balance between two factors — abundance and
substitutability — working in opposite directions, and if abundance is sufficiently low,
600
the price per ton will be high.
Fofuel is abundant, but hard to substitute for across a wide range of economic activities. Hence:
(i) consumption is high; (ii) factor share is high; (iii) abundance and substitutability
work in opposite directions, and if Fofuel is sufficiently abundant, the price per ton will
be low.
605
Granit is both abundant and easily substitutable. Hence consumption is high, while both factor
share and price are low.
Focusing on SF 2 (price fluctuations), there is perfect information in the model and hence
price fluctuations caused by unexpected events are excluded. An obvious reason for short-run
price spikes is that both supply and demand are inelastic in the short run, hence if demand is
610
unexpectedly high this pushes up the price significantly. In principle it would be straightforward
to extend the model to include such effects.21
Focus now on SF 3 and 4, flat real prices and increasing resource consumption. The model
results are congruent with both of these if we assume that the economy is close to a frontier
growth path as discussed in Section 5; recall that we expect the economy to pass through a
19Output increases by a factor of approximately 5.
20For instance, it makes ideal hard-core for road-building, but crushed rock is equally good.
21Such an extension could be to add a stochastic factor in resource demand, plus short-run adaptation costs on
the supply side.
THE NATURAL-RESOURCE SEE-SAW
615
25
primitive–frontier–b.g.p. progression where during the frontier phase real prices are approximately constant and consumption rises at the overall growth rate. Furthermore, in Figure 3 we
see that the frontier phase may be very long (hundreds of years).
Is the specific prediction that resource consumption should grow at the overall growth rate
when real prices are constant backed up by data? We do not pretend to answer this question,
620
however we do take a couple of examples which suggest that it may be. In Figure 4 we show data
on price and consumption rates from Kelly and Matos (2005) for resources — copper and iron
ore — whose prices trends are nearly flat, compared to data on real global GDP from Maddison
(2007).
(a) Copper
(b) Iron
31.6
31.6
Normalized price, quantity
GDPt
qt
10
3.16
GDPt
10
@
R
@
6
qt
3.16
@
R
@
0
−3.16
0
(p/py )t 6
1920
1940
@
R
@
1960
1980
Period (10 yrs)
2000
−3.16
(p/py )t 6
1920
1940
1960
1980
2000
Period (10 yrs)
F IGURE 4. Consumption rates and real price, compared to global GDP, for
(a) copper and (b) iron. The data is presented in the same logarithmic form
as Figure 3. Resource data averaged over 10-year periods; GDP data sporadic
prior to 1950.
6.2. Robust prediction of future trends. There are many ways of testing the robustness of a
625
model’s predictions, but none (except perhaps hindsight) is foolproof. Here we take a closer look
at some aspects of the modelling in comparison to the literature. We begin, for historical reasons,
by linking the model to the Hotelling rule. We then evaluate the model — and its predictions —
in comparison to the two most relevant papers in the literature, André and Smulders (2004) and
Tahvonen and Salo (2001). Finally we study two aspects of the model, firstly the modelling of
630
technological change in the final-good sector, and secondly the modelling of resource supply.
To link Hotelling to our model, note that: (i) in the model the price of extraction rights to a
given small deposit would, if such rights were traded, rise at the interest rate, but every small
deposit has a different price, depending on its remoteness, and hence there is no direct link
between this price rise and the price of the finished resource; and (ii) if our model is altered such
26
635
THE NATURAL-RESOURCE SEE-SAW
that resource deposits are finite and extraction costs zero up to the point of exhaustion, then the
simple Hotelling price path is returned. Furthermore, note that the frontier phase in our model is
actually extremely simple; prices fail to rise because stocks are large and hence the user cost of
extraction is small. This begs the question of why resource prices, across a range of resources,
seem to be marked up well above the marginal extraction costs of many producers. Ellis and
640
Halvorsen (2002), in an econometric study of a large actor’s behaviour in international nickel
market, show that the user cost accounts for only a very small proportion of this markup, with
the majority attributable to the exercise of market power. This evidence, although anecdotal
in the sense that it comes from a study of just one industry, supports our model of the frontier
economy. (Although we abstract from market power, adding this to the model would not change
645
the fundamentals.)
Consider now André and Smulders (2004) and Tahvonen and Salo (2001). The former model
directed technological change on the demand side, but have a stylized extraction sector with
iceberg costs but no need for inputs. The latter have no directed technological change on the
demand side, but a fairly rich extraction model. However, in both cases the observation of
650
constant or declining resource prices is explained by one rate of change matching or outstripping
another. In the former, a decline in iceberg costs, or falling taxes, can balance the baseline
(Hotelling-rule) rise in the resource price; in the latter, the benefits of learning-by-doing in
extraction can balance the effects of cumulative extraction. Both focus on the energy sector, but
if we were to extend the models to the interpretation of resource prices across many different
655
types of resource, the observation of approximately constant long-run prices would appear to
be a coincidence rather than something which should be expected based on the structure of the
economy.
Furthermore, both models make predictions for the long run which differ drastically from
ours. Recall that on the demand side in Tahvonen and Salo (2001) there is no directed techno-
660
logical change; energy is one of two inputs in a Cobb–Douglas production function, and hence
exponentially increasing energy consumption is required to maintain long-run growth. This is
tantamount to predicting that an inability to maintain growth in energy supplies will prevent
long-run growth; there is no corresponding result in our model, where increasing prices in the
long run lead to increasing resource efficiency in production. On the other hand, the supply side
665
of André and Smulders (2004) approaches the simplest Hotelling case in the long run, with a
finite stock and zero extraction costs; thus the extraction rate approaches zero while the price
rises at the interest rate, implying that the ratio of the resource price to the prices of other inputs (e.g. labour) approaches infinity. By contrast our model predicts, in the baseline case, that
THE NATURAL-RESOURCE SEE-SAW
27
some non-zero flow of resources to the production sector can always be maintained, and that the
670
above ratio will approach some constant level. The difference between the long-run predictions
of our model and those of André and Smulders (2004) hangs on assumptions about the nature
of the stock; in André and Smulders it is finite and homogeneous, whereas here it is infinite but
inhomogeneous. What applies for a given resource is an empirical question, but for no resource
will the actual stocks exactly match model assumptions.
675
Apart from the nature of the stocks, further empirical information or assumptions — concerning for instance market structure, and the availability of substitutes — would be required to parameterize the model and make predictions for a specific resource. For instance, for fossil fuels
our model fails to reflect long-run energy markets in three respects: equation 21 — the equation for Rt as a function of cumulative extraction — does not fit the nature of fossil-fuel stocks;
680
there is no allowance for substitute energy sources, which could be both renewable (e.g. wind)
or non-renewable (e.g. uranium);22 and resource holders have no market power. Each of these
deficiencies could easily be corrected in theory within the model framework, although generating credible empirical estimates of parameters would of course be challenging. The key to the
outcome in the very long run is the potential of the substitute energy sources: the worst case for
685
long-run growth is if energy flows must come from renewable sources with bounded potential,
in which case prices track wages and energy consumption is constant (e.g. a constant flow from
renewable sources); this case is similar to the b.g.p. in the base case. The more optimistic case
is that boundless sources of energy are found, hence energy consumption can rise in the long
run and hence prices will continue to fall relative to wages.
690
Consider now the modelling of directed technological change in the final-good sector, where
our model differs significantly from the dominant approach in the literature, that of Acemoglu
(2002, 2007). The key difference is that in our model those who use the technology (final-good
producers) also make the decisions about what investments to make; there is no intermediate
monopolist who imposes technologies on final-good producers. Furthermore, we fix the size
695
of final-good firms by construction. This means that there is no ‘market size effect’ in the
model, and there is no possibility that if an input becomes (exogenously) more abundant, its
price rises due to technological change which makes producers much more dependent on that
input. The market size effect adds interest and power to Acemoglu’s model; however, the model
we present has the advantage of simplicity, both analytically and in terms of the results. In fact,
700
the model gives microeconomic foundations to ideas about directed technological change (or
induced bias in innovation) presented by Kennedy (1964), who argues (p. 545) that ‘there is
22For other resources the inclusion of recycling could be important.
28
THE NATURAL-RESOURCE SEE-SAW
a fundamental technological bias in innovation possibilities’, and that depending on this bias
‘there will be determinate long-run equilibrium values of the [shares of labour and capital],
and that any departure from these equilibrium values will induce a bias in innovation tending
705
to restore them’. In our model of final-good production, the bias is determined by the ration
α0 /β0 , and the constant-share result can be seen in equation 33.
Staying with resource demand, an interesting feature of the data in Figure 4 is that resource
consumption appears to be negatively correlated with the price, with a time lag. This is exactly
what we would expect from the model of the demand side. See Hart (2008) for a related model
710
of technology investment on the demand side which is parameterized for data — showing similar
lagged reaction to price changes — on petroleum prices and consumption in the U.S. It would
be interesting to parameterize and test the demand model based on fitting consumption patterns
based on observed price trends across a wide range of resources.
Consider now resource supply, where it would be interesting to study empirically links be-
715
tween cumulative resource extraction, decreasing accessibility, and extraction technology, and
link these observations to the model. Several recent papers are relevant. Cuddington and Moss
(2001), for instance, find that technological change largely counteracts the effects of decreasing
accessibility of U.S. natural gas, although the impact on finding costs for crude oil reserves has
been more modest. Managi et al. (2004), in one of a series of papers, examine depletion in
720
offshore oil and gas more broadly. Using micro-level data over a 50-year period, they show that
productivity increases have offset depletion effects in the Gulf of Mexico over that period. Note
that, from the perspective of our model, constant real prices should be seen as technological
change outpacing — not merely offsetting — decreasing accessibility. Given constant technology, but rising wages, extraction costs will track the wage; given in addition rising remoteness
725
extraction costs will rise faster than wages. Hence these papers seem to add weight to our argument that we are in a transition phase where resource remoteness is rising more slowly than the
pace of technological change, hence allowing extraction costs to fall relative to wages.
Returning to Figure 4, an apparent difference between the data and the model predictions
for the frontier phase is that real resource prices actually fall slightly over the period. This
730
could be because there are actually increasing returns in the scale of resource extraction, so unit
extraction costs decrease (for given physical remoteness) as the scale of extraction (measured
in tons/year) increases. Alternatively, given spillovers within the extraction sector (i.e. ζ > 0)
then, if extraction knowledge starts at a low level, it may take a long time to ‘catch up’ with
production knowledge; by definition, during such a catching up phase c/a would be rising, and
735
hence marginal extraction costs would fall as long as R were constant.
THE NATURAL-RESOURCE SEE-SAW
29
7. Conclusions
The basic model predicts that long-run resource prices will grow at the overall growth rate,
and hence follow wage growth, whereas resource consumption will be constant. This prediction
is by contrast to current trends, where prices are constant and consumption grows in line with
740
overall growth. In future new technologies will increase the value of output from given physical
quantities of resource inputs in the same way that today’s new technologies increase the value
of output from given physical inputs of labour and capital goods. This occurs through a transformation of capital goods, increasing their quality, but not through long-run increases in the
physical quantity of capital goods.
745
These conclusions are based on a number of assumptions about smooth and predictable
processes in the economy: there is no uncertainty; technological progress occurs according to
fixed functions of investment and existing technology; and resource remoteness grows smoothly
given steady extraction. In practice none of these assumptions is likely to hold. However, there
is no obvious bias in these assumptions which would lead the model to predict systematically
750
incorrect long-run trends for resource prices. The Hotelling rule, based on supply-side considerations only, suggests that ceteris paribus resource prices should rise at the interest rate. Here
we show, based on analysis of both supply and demand for resources, that it is more reasonable
to suppose that resource prices should rise, in the long run, at the same rate as the prices of other
factor inputs.
Appendix A. Proof of Proposition F1
755
The existence of a b.g.p. follows more-or-less directly from the growth equations 10 and 11,
the first-order conditions 15 and 16, and the equations (17) linking firm-level and aggregate
variables. Recall (equation 8) that σtq is the factor share of resources. On the b.g.p. we have
fq =
σ
(A1)
Then θe is the solution to
(A2)
760
(θe)1/φ −
(θ p /θe)1/φ
.
α0 /β0 + (θ p /θe)1/φ
ρφ ey (1−φ )/φ e β0 p 1/φ
(L )
θ + (θ ) = 0.
α0
α0
Assume a unique solution, and define
(A3)
αt∗ = (αt /At )/α0 ;
βt∗ = (βt /At )/β0 ;
Then the control variables are
(A4)
f∗ = (θe/Ley )1/φ ,
α
γt∗ = (γt /At )/γ0 .
30
THE NATURAL-RESOURCE SEE-SAW
and
βf∗ = (θ p /Ley )1/φ .
(A5)
Now to verify that there is a unique solution. If the constant term in A2, β0 /α0 (θ p )1/φ , is zero
then there is a unique solution implying β = 0 and zero factor share of resources. Now allow the
765
constant term to rise. There are always two solutions, the upper of which is never allowed, since
it implies a negative factor share of resources. The lower solution is allowed up to the point at
which θe becomes zero, at which point the factor share of labour is zero.
Appendix B. Proof of Propositions E1 and E2
The proofs are straightforward, and we therefore only give an outline. First, define the vari-
770
g
g
/a as the b.g.p. values of the fractions p/a, w/a, c/a, and R/a respec/a, w
/a, cf
/a, and R
ables pf
tively.
Now, for Proposition E1, assume
(A6)
(A7)
and
(A8)
775
/a
(1 − φ )(1 − ρ θe)(θe − 1) pf
·
,
Lee = N
e
g
w/a
(1 − ρ )θ
/a =
cf
Lee
Ley
!ζ /(1−ζ )
g
/a =
R
g
(α0 + β0 ) w/a
γ0
1−φ
!φ /(1−ζ )
,
/a
(1 − φ )(1 − ρ θe) pf
/a.
cf
g
(1 − ρ )
/a
w
Then from the first-order conditions it is straightforward to derive optimal values for γ ∗ , nt , and
lte for this initial state, and to show that these are consistent with the constant values of c/a, R/a,
and Le .
Now for E2, where p and R are constants. Then assume
(A9)
780
/a =
cf
g
/a R
w
,
(1 − φ ) p
/a. Then from the first-order conditions — and
while Le satisfies equation A7 for this value of cf
using the fact that N is infinite and hence Zt = nt zt — it is straightforward to derive optimal
values for γ ∗ , and lte for this initial state, and to show that these are consistent with the constant
values of c/a and Le . Furthermore, the resulting equation for the extraction rate is
(A10)
nt = at Lee
g
/a
w
;
(1 − φ )p
hence extraction increases at the overall growth rate.
THE NATURAL-RESOURCE SEE-SAW
31
Appendix C. Solving for the b.g.p.
Assuming a symmetric equilibrium we have the following equations linking aggregate and
firm-level variables. We have equations 17,
At = Lty at
Bt = Lty bt
Qt = Lty qt ,
and equations 29
Ct = Lte ct
Lte = lte
Qt = nt .
Given these equations, and the definitions of α ∗ , β ∗ , and γ ∗ in A3, we can set the model up with
just three state variables, as follows:
(A11)
(b/a)t+1 = (βt∗ /αt∗ )φ (b/a)t ;
(A12)
(c/a)t+1 = (Lte /Lty )ζ (γt∗ /αt∗ )φ (c/a)t ;
(A13)
(R/a)t+1 = exp[nt+1 /N]/(Lty (αt∗ )φ )(R/a)t .
ζ
Furthermore, we can write equations for the remaining variables in terms of state and control
variables.
nt /Lty = 1/(b/a)t ;
(A14)
(A15)
∗
∗
∗
(w/a)t = 1 − (p/a)t n(t)/Lty − (α0 αt−1
+ β0 βt−1
)/(ρ (αt−1
)φ );
(A16)
Lty + Lte = L;
(A17)
(z/a)t = (w/a)t (R/a)t /(c/a)t ;
(A18)
(Z/a)t = (z/a)t N(1 − exp(−nt /N));
Lte = N(Rt−1 /ct )(exp(nt /N) − 1).
(A19)
Finally, rewrite the first order conditions:
ρφ ∗ φ
y (αt ) 1 − (p/a)t+1 nt+1 /Lt+1
;
α0
ρφ ∗ φ
y
βt∗ =
(αt ) (p/a)t+1 nt+1 /Lt+1
;
β0
αt∗ =
(A20)
(A21)
(A22)
(A23)
785
ρφ ∗ φ (w/a)t+1
(α )
;
γ0 t
1−φ
(p/a)t (1 − φ ) − (z/a)t = ρ Lty (αt∗ )φ (Z/a)t+1 /N + (p/a)t+1 (1 − φ ) − (z/a)t+1 .
γt∗ =
It is straightforward to demonstrate the existence of a steady state for this system, i.e. a set of
values for the state variables in period t such that both state and control variables are constant
for all periods T where T ≥ t.
First, a lemma.
32
790
THE NATURAL-RESOURCE SEE-SAW
Lemma AC1. If equations 31–34 have a solution for the control variable ne and the aggregate
ee and Ley , then it is unique.
/a, and L
variables θe, pf
Proof. Take equation 32 and show that Ley is an increasing function of θe over the allowed range,
θe ∈ (0, 1/ρ ), and that when θe = 0 then Ley = 0 and ∂ θe/∂ Ley = 0. Then the result follows from
equation 31.
Given Lemma AC1, proof of the existence of a steady state is simply a matter of verifying
795
that the following additional equations — which define steady-states values of the remaining
control variables α ∗ , β ∗ , and γ ∗ , and the state variables b/a, c/a and R/a — satisfy the firstorder conditions while simultaneously leading to an unchanged state in the following period.
1/(1−φ )
ρφ
f
∗
∗
f
(A24)
α =β =
.
α0 + β0
(A25)
γe∗ =
(A26)
800
(A27)
(A28)
/a =
cf
Lee
Ley
ρφ
α0 + β0
1/(1−φ )
1 − φ (1 + β0 /α0 ) α0
.
1−φ
γ0
ey /e
/a = L
bf
n.
!ζ /(1−ζ ) 1 − φ (1 + β0 /α0 ) α0
1−φ
γ0
φ /(1−ζ )
.
/a
(1 − φ )(1 − ρ θe) pf
g
/a =
/a.
R
cf
g
(1 − ρ )
/a
w
Appendix D. Numerical solution
We set up the full model in GAMS, using the equations in Appendix C. Note that the GAMS
model is not an optimization, it is fully determined, since we insert the first-order conditions
805
into the model. Having set up the model we then run simulations for various parameter choices.
Baseline parameters are chosen as follows. The time period is set to 10 years; that is, technologies are renewed every ten years. We set the discount rate per year to 4 percent, therefore
ρ = 0.675564169, and the b.g.p. growth rate to around 2 percent, which more-or-less fixes α0 ,
the key determinant of long-run growth given that the resource share is small. The ratio α0 /β0 is
810
then set to make the resource factor around 1 percent, while γ0 is close to unity to ensure that total investments in extraction technology are in line with the size of the sector: α0 = 0.01126709,
β0 = 0.0001, γ0 = 0.9. Parameter N = 200. The elasticity of knowledge to investment φ = 0.1;
this value may seem low, but this is necessary given the simplifying assumption of 100 percent
depreciation of private knowledge each period; with a higher value, higher investments would be
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33
needed to maintain constant levels of technology. Finally, ζ = 0.1, hence extraction knowledge
c is tied closely to production knowledge a.
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