endogenous extraction in the solow model – hotelling’s rule
DESCRIPTION
Bachelor Thesis in EconomicsTRANSCRIPT
E n d o g e n o u s E x t r a c t i o n
i n t h e S o l o w M o d e l
– H o t e l l i n g ’ s R u l e
B ac h e l o r t h e s i s i n e c o n o m i c s
F r e d e r i k Dy e r b e r g G r e i s e n
L a r s H v e l p l u n d
S u p e rv i s o r : H a n s J ø r g e n Wh i t ta - J a c o b s e n
A p r i l 2 0 1 2
D e pa rt m e n t o f E c o n o m i c s
U n i v e r s i t y o f C o p e n h a g e n
c© Lars Hvelplund & Frederik Dyerberg Greisen, 2012.
This thesis is written in 12pt and 1.5 linespread.
Layout and typography by the authors by the use of LATEX, the memoir -class.
Figures are made by hand, and data graphs are made in MS Excel.
Printed by Arbejdsmarkedsstyrelsen.
Contents
Abstract iii
1 Introduction 11.1 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Scope and restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Data and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Empirics 32.1 Characteristics of a nonrenewable resource . . . . . . . . . . . . . . . . . . 32.2 Nonrenewable resource prices . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Nonrenewable resource extraction rates . . . . . . . . . . . . . . . . . . . . 6
3 The Solow model 93.1 The Solow model with a nonrenewable resource . . . . . . . . . . . . . . . 103.2 The law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Hotelling’s rule 144.1 The rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Cost modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 The model 165.1 The Solow model with Hotelling’s rule . . . . . . . . . . . . . . . . . . . . 165.2 The law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Analysis 236.1 Characteristics and long run growth . . . . . . . . . . . . . . . . . . . . . . 236.2 Initial and feasible values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7 Conclusion 30
8 Perspective 31
Bibliography 32
A Appendix 33A.1 SAS test output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2 Capital-output ratio statistics . . . . . . . . . . . . . . . . . . . . . . . . . 34A.3 Analysis - law of motion for the extraction rate . . . . . . . . . . . . . . . 35A.4 Calibration of the endogenous model . . . . . . . . . . . . . . . . . . . . . 39
List of Figures2.1 Growth rate of crude oil prices 1862-2010 . . . . . . . . . . . . . . . . . . . . . 42.2 Level of crude oil prices 1861-2010 . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Crude oil extraction rates 1980-2010 . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Law of motion and steady state for the capital-output ratio . . . . . . . . . . . 12
5.1 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1 Phase diagram with feasible band . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.1 OxMetrics output for test of the growth rate in crude oil prices 1862 – 2010 . 33A.2 OxMetrics output for test of the growth rate in crude oil prices 1900 – 2010 . 33A.3 SAS output for test of crude oil prices 1861 – 2010 . . . . . . . . . . . . . . . 33A.4 SAS output for test of crude oil prices 1900 – 2010 . . . . . . . . . . . . . . . 34A.5 OxMetrics output for test of the extraction rate 1980 – 2008 . . . . . . . . . . 34A.6 Capital-output ratio for Denmark 1966 – 2010 . . . . . . . . . . . . . . . . . . 34A.7 Law of motion for the extraction rate . . . . . . . . . . . . . . . . . . . . . . . 35A.8 Estimations of the integration of q - 10.000 and 100.000 repetitions . . . . . . 38A.9 Estimation of the evolution of R - 500 periods . . . . . . . . . . . . . . . . . . 38A.10 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
List of Tables2.1 Growth rate in the crude oil price - test . . . . . . . . . . . . . . . . . . . . . 52.2 Level of the crude oil price- test . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Extraction rate - test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.1 Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
ABSTRACT iii
Endogenous Extraction
in the Solow Model– Hotelling’s Rule
Abstract
The importance of oil and other nonrenewable resources have been increasing rapidly
over the recent years. The dependence of a resource that is nonreproducible and
exhaustible will at some point result in restrictions on the production. It is therefore
interesting to model an economy characterized by such a restriction.
In this thesis we analyze an endogenization of the extraction behavior of a nonre-
newable resource in a Solow model. The model is endogenized by an implementation
of Hotelling’s rule - an equilibrium condition for the asset markets. The rule denotes
that the growth rate of the asset price must be equal to the real interest rate.
We start by describing the empirical characteristics of crude oil, as our example
of a nonrenewable resource. Especially we examine the evolution of the real price
and the extraction rate. Hereafter we present the basis of our analysis - the Solow
model with a constant extraction rate. We analyze the law of motion, steady state
and growth path of the model, and after conferring with the empirics, we conclude
that the model does not provide a good fit.
Based on this finding we implement Hotelling’s rule in the extraction behavior.
This leads to a setup with a higher degree of general equilibrium, since the model
now includes optimizing behavior in the extraction sector. The model’s fit with the
empirics is not improved drastically, but the economic explanations of the dynamics
is enhanced towards a more realistic description.
We compare the long run growth results of the exogenous and the endogenous
model and observe the somewhat unforeseen result, that the exogenous model pro-
duce a higher growth path of output per worker. Finally, we analyze a shock to the
resource reserves - an analysis that is highly relevant given the recurring oil findings.
We find our analysis to provide an exact economic explanation of the interac-
tion between the extraction and production sector caused by the essentiality of a
nonrenewable resource.
1 Introduction
1.1 Observation
During the last century economies of the world have become increasingly dependent on
nonrenewable resources. A major part of the global consumption is based on such re-
sources and they are essential in production. Though there are many nonrenewable re-
sources to consider when carrying out an analysis, we have chosen to focus on crude oil,
since it is one of the most important and debated resources in modern times.
The focus of this thesis is the long run, and we consequently want to create a model
that can explain the long run effects of being dependent on nonrenewable resources in
production.
1.2 Motivation
In this bachelor thesis we will shed light on the implications of endogenizing the constant
extraction rate in a Solow model with a nonrenewable resource. The Solow model is in
some aspects a general equilibrium model, since the firms are optimizing their behavior
and since it constitutes a macroeconomic equilibrium. The implementation of Hotelling’s
rule will endogenize the extraction behavior. Since the transition will incorporate opti-
mizing behavior in the extraction sector, the model will move towards a higher degree of
general equilibrium, compared to the model with an exogenous extraction sector.
The thesis will be testing the relation between theory and empirics regarding the price,
extraction behavior and growth in an economy with a nonrenewable resource.
1.3 Problem statement
What are the implications of an endogenization of the extraction rate of a nonrenewable
resource in a Solow model?
What are the results of an exogenous extraction rate in the Solow model?
What qualitative and quantitative aspects will change if applying Hotelling’s rule?
What are the prospects of long run growth in output per worker in the models?
Can an exogenous setup be interpreted as a special case of the endogenous setup?
What are the effects of an unexpected increase in the resource reserves?
2 1. INTRODUCTION
1.4 Scope and restrictions
We have chosen the Solow model with a constant extraction rate as the basis for our
analysis. A similar analysis could have been carried out in a more complex model such
as the Ramsey model. We have chosen not to do so, since a more complex framework
would require a substantially larger setting. Furthermore the Solow model is sufficient to
incorporate the endogenization of the extraction behavior and will not lead to important
losses with respect to our target of analysis. We have chosen to formulate the model
in continuous time since it is simpler to model the endogenization in such terms. For
convenience we analyze the models in terms of the capital-output ratio.
1.5 Thesis structure
We have outlined our thesis through the endogenization of the nonrenewable resource sec-
tor in the Solow model. We start by providing a thorough background for the empirics of
nonrenewable resources and the theory of the Solow model. We then introduce Hotelling’s
rule to the model, and in the end we analyze the implications of the modification, dis-
cuss the results, and compare the different models. In the conclusion we will deal with
the problem statement through a summary of our main results. In the end we give our
perspective of the consequences of our results.
1.6 Data and methodology
Data on world crude oil prices and extraction rates are taken from BP - Statistical Review
of World Energy June 2011. Data on the Danish capital-output ratio is taken from
Statistics Denmark. We consider these data sources to be highly valid. Analysis of the
data has been carried out in SAS 9.2, OxMetrics 6.10 and Maple 15.01. The thesis is
written in a macroeconomic theoretical perspective, and for this reason we have chosen
our econometric analysis to be rather simple and mainly informative. One could possibly
achieve some very interesting conclusions if going further in depth with the empirical
analysis. Nonetheless, due to the structural limits of this thesis, we have chosen to mainly
focus on the theoretical analysis.
2 Empirics
2.1 Characteristics of a nonrenewable resource
Before analyzing the empirics of a nonrenewable resource, we define the characteristics of
the market of such an asset. A nonrenewable resource can be regarded as any other asset
from the viewpoint of the owner. Holding an asset is profitable when the asset yields a
positive return. In general there are two options in which an asset can yield a positive
return.
Option 1: Sell the asset, invest the proceeds and obtain the investment return.
Option 2: Obtain yield from asset value appreciation. The second option can be decom-
posed in the following way, c.f. [1]:
The first component is given by the marginal product of holding the asset.
The second component is the physical depreciation.
The third component is created by increases in the market value of the asset i.e.
capital gains.
When regarding the return of a nonrenewable resource according to these three compo-
nents, the origins of the asset return can be determined.
A nonrenewable resource, will not contribute with a marginal product when held in
situ. That is, in the ground. Furthermore the asset is non-reproducible, and the existing
stock will therefore not be able to increase over time, apart from new findings. The first
component will always be zero when considering a nonrenewable resource in situ.
Regarding the second component, a nonrenewable resource will not physically deteri-
orate over time as long as it is kept in situ. Thus a nonrenewable resource will not be
exposed to depreciation as regular physical assets and the second component will therefore
also be equal to zero.
The third component is the only basis for an appreciation of the asset. Capital gains
are thus the only way holding a nonrenewable resource can lead to a positive return.
2.2 Nonrenewable resource prices
We have given a description of the market pricing of a nonrenewable resource, and now
turn to the empirics to create a basis for validating the theoretical results presented later
in the thesis. In figure 2.1 the annual growth rate of real crude oil prices in 2010-USD
4 2. EMPIRICS
from 1861 – 2010 is depicted. As seen from the figure the growth rate has roughly been
centered around a mean of zero. Furthermore the price changes have been subject to high
volatility at varying degrees.
Figure 2.1: Growth rate of crude oil prices 1862-2010
To analyze whether the growth rate of real crude oil prices (ut) is significantly different
from zero, we perform a Wald-test. On the basis of the observed graph, we see that
the time series has a stationary unconditional mean, but a non-constant variance. Using
heteroskedasticity consistent standard errors, the data fulfills the OLS requirements and
the Wald test is thus consistent. We construct the null hypothesis that the growth rate
i.e. the intercept of the OLS-regression is equal to zero (H0 : β0 = 0). We test against a
two-sided alternative hypothesis (HA : β0 6= 0).
ut − ut−1
ut= β0 + εt (2.1)
The results of the test are given in table 2.1, and imply that the test is borderline between
rejecting and not rejecting the null hypothesis. Hence we cannot give a definite conclusion.
In the first part of the considered period, the uncertainty on the aggregate crude oil
reserves was large. This can explain the high volatility in crude oil prices from 1862 –
1900. We thus want to test if the growth rate has been zero, when considering the period
2.2. NONRENEWABLE RESOURCE PRICES 5
1900 – 2010. For this period our test result show, that we cannot reject the null hypothesis
of a growth rate equal to zero.
Table 2.1: Growth rate in the crude oil price - test
Parameter Estimate s.e.* t-value p-value
β0 : 1862− 2010 0.0540 0.0278 1.9500 0.0533
β0 : 1900− 2010 0.0379 0.0280 1.3400 0.1821
* Using heteroskedasticity consistent standard errors.
Note: Both tests were carried out in OxMetrics. For output see appendix A.1, figure A.1 and A.2
A consequence of calculating growth rates or first differences is a loss of information. The
high volatility of the time series gives a somewhat blurry picture of the overall level of
the crude oil price. To obtain a more detailed picture of the historic level of the crude oil
price, the development of real crude oil prices in 2010-USD is plotted in figure 2.2.
Figure 2.2: Level of crude oil prices 1861-2010
The figure shows that the real price of crude oil is highly volatile - just as we saw in figure
2.1 with the growth rate of the oil price. Using the level data, we test for a constant growth
rate in the crude oil price, and consequently we estimate a model with exponential growth.
ut = γ0 · exp(γ1t) · εt (2.2)
6 2. EMPIRICS
ln(ut) = ln(γ0) + γ1t+ ln(εt) (2.3)
We construct the null hypothesis that the oil price is constant (H0 : γ1 = 0). Again we
test against a two-sided alternative hypothesis (HA : γ1 6= 0).
The test results are given in table 2.2. Since the t-value is low and thus the p-value is
high, we cannot reject the null hypothesis of constant real crude oil prices for the period
1861 – 2010.
The high volatility in the period 1861 – 1900 makes the test uncertain, and from
the beginning of the 20th century it might look like, that the real crude oil prices have
been growing exponentially. We therefore test if the growth rate has been constant in
the period 1900 – 2010. As seen from table 2.2, we can reject the null hypothesis of a
constant oil price. The test suggests that the price have been exponentially growing at a
relatively low positive growth rate of approximately 1 percent per year. The fitted values
are shown in figure 2.2.
Table 2.2: Level of the crude oil price- test
Parameter Estimate s.e.* t-value p-value
γ1 : 1861− 2010 0.0012 0.0013 0.9200 0.3574
γ1 : 1900− 2010 0.0095 0.0014 6.8400 <.0001
* Using heteroskedasticity consistent standard errors.
Note: Both tests were carried out in SAS. For output see appendix A.1, figure A.3 and A.4.
We now have two weak conclusions regarding the evolution of the crude oil price. The
first is that we cannot reject that the growth rate of oil prices have been zero, while the
second suggests that oil prices have been growing exponentially at a rate of around 1
percent. It is consequently difficult to give a definite conclusion on the evolution of prices.
2.3 Nonrenewable resource extraction rates
The extraction rate of a nonrenewable resource is another important characteristic, that
provides information on how fast the resource is depleted. The extraction rate is thus a
fundamental part in the analysis of the long run growth of an economy, of which nonre-
newable resources are essential parts of production. Figure 2.3 shows the extraction rate
of crude oil from 1980 – 2010.
As seen from the figure the extraction rate has been located at a level of around 2 –
2.5 percent during the last 20 years, with a declining trend. The extraction rate has been
2.3. NONRENEWABLE RESOURCE EXTRACTION RATES 7
calculated as the ratio between total production and total proved reserves in the given
year (q = ER).
Figure 2.3: Crude oil extraction rates 1980-2010
It should be noted that the actual extraction rate is lower than in figure 2.3, since the
proved reserves most likely are lower than the actual reserves. This will imply that the
actual extraction rate is lower, e.g. 0.5 percent as suggested by Nordhaus, see [6, pp.
15–16].
To test whether the extraction rate has been declining exponentially, we test the
regression in (2.5). The null hypothesis is, that the extraction rate has been constant
(H0 : α1 = 0). We test against a one-sided alternative (HA : α1 < 0).
qt = α0 · exp(α1t) · εt (2.4)
ln(qt) = ln(α0) + α1t+ ln(εt) (2.5)
The test results are shown in table 2.3, and based on these we reject the null hypothesis.
The results suggest that the extraction rate has been exponentially declining at a rate of
0.008 percentage points1 per year. Based on this, we conclude that the extraction rate
1Since the test is log-level and the regressant is a rate.
8 2. EMPIRICS
does not seem to be constant, but more likely declining.
Table 2.3: Extraction rate - test
Parameter Estimate s.e.* t-value p-value
α1 -0.0082 0.0019 -4.42 0.0001
* Using heteroskedasticity consistent standard errors.
Note: Testing in OxMetrics. For full output see appendix A.1, figure A.5.
The estimated regression for the extraction does not give a good graphical fit with the
data. The curvature of the exponential function is not high enough, which results in an
almost linear function. One could have achieved a better fit using a power regression or
a logarithmic regression. We have chosen not to do so, since our regressor is the trend
variable, year. Taking logarithms of time regressors makes the estimates unintuitive and
hard to interpret.
Given these initial facts regarding the crude oil price and the extraction rate, we
continue to the theoretical modeling of nonrenewable resources in an economy. First, we
will introduce the basis of our analysis – the Solow model.
3 The Solow model
In order to describe the long run effects of natural resources, we use the growth model
introduced by Solow in 1956, see [8]. The Solow model is a neoclassical growth model
formulated to explain the evolution of capital and output in a closed economy in the long
run. As we want to model the world economy the Solow model thus fits the purpose.
The Solow model is used in order to specify a setup where the prices of input factors
are determined according to a production function. This is the first step towards obtaining
a model that can describe the price of a nonrenewable resource.
There are three commodities in the model; labor, capital, and output, and two types
of agents; consumers and firms. Since the model describes the economy in the long run it
is assumed that the markets clear continuously i.e. supply equals demand. This implies
that the production is determined by the supply of goods, and therefore by the input
factors of production; capital, labor, and technology. Production is given by the total
output of the firms, and the evolution of output is thus given by the growth of the input
factors.
The production function of the model has diminishing returns to the input factors –
labor and capital, and it is assumed to have constant returns to scale due to the replication
argument. The Cobb-Douglas production function fulfills the specifications stated above,
and is thus used in the model. The Cobb-Douglas production function implies that the
income shares of the input factors are constant and equal to their output elasticities, which
is consistent with the empirics, see [10, pp. 47 - 48]. The technology input is assumed to
be labor augmenting, but this assumption does not affect the analysis. Technology and
population are assumed to grow at constant rates.
Total income is divided by consumers between consumption and savings. The savings
rate is assumed to be an exogenous constant, which is somewhat in accordance with
the empirics, see [10, pp. 64 - 65]. In the market for capital, total savings will equal
total investment since the model describes a closed economy. Capital is accumulated as
investment at a given point in time minus the depreciation of the existing capital stock.
The markets for goods and input factors are characterized by identical firms, perfect
information, and homogenous goods, implying perfect competition. Factor prices are
given by their marginal products, due to perfect competition in the input factor markets.
For now, this is the only mechanism determining factor prices. The price of output is
numéraire and is thus normalized to one.
10 3. THE SOLOW MODEL
3.1 The Solow model with a nonrenewable resource
In order to analyze the dependence of the nonrenewable resource, we define the resource
input to be essential to production. That is, if there is no resource input, there will be
no production. The nonrenewable resource is defined by an initial stock that is depleted
when the resource is used as input in production. The flow of resource used in production
is thus equal to the change in the stock of the resource, at a given point in time. The
extraction sector is modeled exogenously, which in this case implies that the extraction
rate is a constant. As in general, the resource is utilized to the point, where the marginal
product equals the factor price. The model now has four input factors.
The variables in the model are total output Y , the capital stock K, the technological
level A, the labor force L, the energy derived from resource utilization E, the resource
stock R, the real rental rate r, the real wage rate ω, and the resource price u. The
parameters are the income share of capital α, the income share of labor β, the income
share of resource ε , the savings rate s, the capital depreciation rate δ, the labor force
growth rate n, the growth rate of technology g, and the constant extraction rate sE.
Dotted variables are time derivatives. The model can be summarized as follows.
Y = Kα(AL)βEε, α, β, ε > 0, α + β + ε = 1 (3.1)
r = αKα−1 (AL)β Eε (3.2)
ω = βKαAβLβ−1Eε (3.3)
u = εKα (AL)β Eε−1 (3.4)
R = −E, R0 given (3.5)
E = sER (3.6)
K = sY − δK, K0 given (3.7)
L
L= n, L0 given (3.8)
A
A= g, A0 given (3.9)
3.2 The law of motion
It is convenient to analyze the model in terms of the capital-output ratio, z ≡ KY. First
we calculate the law of motion for the capital-output ratio.
z ≡ K
Y=
K
Kα(AL)βEε= K1−α(AL)−βE−ε (3.10)
3.3. STEADY STATE 11
Taking logarithms of z and differentiating with respect to time to find the growth rate of
the capital-output ratio.
z
z= (1− α)
K
K− β
(A
A+L
L
)− εE
E(3.11)
= (1− α)K
K− β (g + n)− εE
E(3.12)
In order to find the law of motion for z, we derive the growth rate of the energy input, E.
Taking logarithms and differentiating the equation for the extraction rate (3.6), we find
the growth rate of E.
E
E=
˙sEsE
+R
R=R
R, since sE is constant (3.13)
E
E=−ER
= −sE, due to (3.5) (3.14)
We insert the growth rate of energy input into (3.12) to find the law of motion for the
capital-output ratio.
z
z= (1− α)
K
K− β (g + n) + εsE (3.15)
= (1− α)
(sY − δK
K
)− β (g + n) + εsE
= (β + ε)(sz− δ)− β (g + n) + εsE
= (β + ε)s
z− β(n+ g + δ) + ε(sE − δ)
z = (β + ε)s+ (ε(sE − δ)− β(n+ g + δ))z (3.16)
The law of motion determines the growth path of the capital-output ratio. For t→∞ the
capital-output ratio converges towards the steady state z∗, since z is negatively dependent
on z and thus equilibrium correcting, given that sE < δ. This is illustrated in figure 3.1.
3.3 Steady state
To find the steady state, we set the absolute change in the capital-output ratio z equal
to zero (z = 0).
z∗ =(β + ε)s
β(n+ g + δ)− ε(sE − δ)(3.17)
Since the capital-output ratio is constant in steady state, capital and output must be
growing at the same rate. The growth rate of output can be determined by taking
12 3. THE SOLOW MODEL
Figure 3.1: Law of motion and steady state for the capital-output ratio
logarithms and differentiating the production function (3.1) with respect to time.
Y
Y= α
K
K+ β
(A
A+L
L
)+ ε
E
E(3.18)
The growth rate of E from equation (3.14) is inserted.
gY = αgK + β(g + n)− εsE (3.19)
In steady state gY = gK since z = 0⇒ zz
= KK− Y
Y= 0.
gY =β
β + ε(g + n)− ε
β + εsE (3.20)
We now find the growth rate of output per worker in steady state. Using that y ≡ YL⇒
yy
= YY− L
L= gY − gL.
gy =β
β + ε(g + n)− ε
β + εsE − n (3.21)
gy =β
β + εg − ε
β + ε(sE + n) (3.22)
We see from (3.22) that the growth rate of output per worker is negatively influenced by
the essentiality of a nonrenewable resource. Having determined the long run growth path
of the economy, we now want to use the model to find the steady state for the input factor
prices. The factor price of an input is given by the marginal product, and from (3.2) and
(3.3) we get.
r = αKα−1(AL)βEε = αY
K=α
z(3.23)
ω = βKαAβLβ−1Eε = βY
L= βy (3.24)
3.3. STEADY STATE 13
The real rental rate is constant in steady state, and consequently the real interest rate,
ρ = r−δ, is constant as well. In steady state the real wage rate grows at the rate of output
per worker. These results are in accordance with the classical theory and are known as
the balanced growth path.
The price of the nonrenewable resource is also determined by the marginal product.
To find the growth rate of the resource price in steady state, we use that the income share
of the resource is constant and equal to its output elasticity.
uE
Y=εKα (AL)β Eε−1E
Y= ε
Y
Y= ε⇔ u =
εY
E(3.25)
Taking logarithms and differentiating with respect to time gives.(u
u
)∗=ε
ε+
(Y
Y
)∗− E
E(3.26)
=β
β + ε(g + n)− ε
β + εsE + sE
=β
β + ε(g + n+ sE) (3.27)
From (3.27) we see that the growth rate of the resource price is constant in steady state,
implying an exponential growth of the resource price. The extraction rate, sE = ER, is
constant and exogenous per definition. Thus the absolute amount extracted and used in
production will decline, and the reserves R will asymptotically be fully exhausted in the
long run. The growing resource price can be explained by the increasing capital and labor
inputs and the declining energy input.
The empirical tests of section »2.1« show that in the long run, the evolution of the
nonrenewable resource price is uncertain. In this model an exogenous extraction rate
leads to exponential growth in the price of the nonrenewable resource. This feature might
be in accordance with the empirics.
The choice of an exogenous extraction rate partially determines the evolution of the
resource price, which is an unsatisfactory aspect of the model. One could however argue
that the size of the extraction rate is low compared to the growth rates of technology and
the labor force, hence limiting the effect on the growth rate of the resource price. We still
find it interesting to analyze the implications of an endogenous extraction rate, since it is
a more realistic assumption for the extraction behavior.
4 Hotelling’s rule
In order to modify the model, we need a theory for the extraction of the nonrenewable
resource. Instead of an exogenous extraction rate we want to use a microeconomic founded
theory. The equilibrium theory Hotelling’s rule formulated by Hotelling [2], satisfies our
purpose.
4.1 The rule
Hotelling’s rule is based on a no-arbitrage condition in the asset markets. The rule states
that agents in the asset markets holding a nonrenewable resource in situ, has two options.
Keeping the resource in situ with the intent of selling at higher price, or extracting the
resource and investing the proceeds in an interest bearing asset.
Looking at the first option, a nonrenewable resource will only generate a positive
return if the asset appreciates i.e. the price u(t) increases c.f. section »2.1«. In order
to sell the asset the resource must be extracted from the reserves. The marginal cost of
extraction so will be denoted by the function c(t). The net price from selling the asset is
given by the marginal profit.
π(t) = u(t)− c(t) (4.1)
The second option will yield the real interest rate denoted by ρ. In equilibrium there can
be no arbitrage between the expected return on the two alternatives1. The rate of change
in the net resource price must therefore be equal to the real interest rate.
π(t)
π(t)= ρ = r − δ (4.2)
Thus in order for the asset markets to be in equilibrium the net price of the nonrenewable
resource must grow at the rate of interest. This is the main result of Hotelling’s rule.
Hotelling’s rule is a microeconomic approach of finding the social optimal depletion of
a nonrenewable resource [7]. Originally the rule was formulated in terms of utility. This
meant that the prices were denominated in units of utility, and that ρ was the utility
discount rate. This version is difficult to apply to reality, and the profit maximizing
version is used instead.
Further Hotelling’s rule is a stable equilibrium condition. That is, if the economy is
located outside equilibrium the economic mechanisms of the rule will correct towards the1It is assumed that agents are risk neutral - a realistic assumption on a global scale.
4.2. COST MODELING 15
equilibrium and the economy will converge. E.g. if the economy is at a point where the
real interest rate is lower than the growth rate of the resource price, there will be an
incentive to keep the resource in situ. Thus the supply of the resource will fall, the price
of the resource will increase in the current period and the growth rate of the price will
fall. Hence the economy will converge towards equilibrium.
4.2 Cost modeling
To simplify the analysis we assume that the marginal cost of extracting the nonrenewable
resource is constant, c(t) = c. This yields the following evolution of the resource price.
u(t)
u(t)= ρ
(1− c
u(t)
)(4.3)
As u(t) rises the growth rate of the nonrenewable resource price will converge towards the
real interest rate. Solving the differential equation leads to the specific increasing price
path of the resource.
u(t) =c
ρ+ C · eρ·t (4.4)
C is an arbitrary constant. To simplify the analysis further we assume that the marginal
cost of extracting the resource is equal to zero, c = 0. This assumption is not entirely true,
but it is a rather close approximation of the nonrenewable resource production. The fixed
costs of setting up production are high, but marginal costs are relatively small compared
to the price of the resource, and thus c is approximately equal to zero. This leads to.
u(t)
u(t)= ρ = r − δ (4.5)
For Hotelling’s rule to be consistent with a constant real interest rate, the crude oil price
would have to rise exponentially. The prediction of Hotelling’s rule might thus be in
accordance with the empirical data cf. the tests of section »2.2«, given that r − δ is
constant.
However Hotelling’s rule itself is not satisfactory to determine the evolution of resource
prices, since it does not take the production sector of the economy into account. In line
with the conclusion in section »3.3«, we want the optimal model to take the effects of
both the Solow model and Hotelling’s rule into account. The model will determine the
evolution of r − δ and hence uu. This will be the objective of the following chapter.
5 The model
We create a full microeconomic founded model for the extraction behavior of the nonre-
newable resource. The model will take the results of both the Solow model and Hotelling’s
rule into account. Some aspects of the model are inspired by Stiglitz [9].
5.1 The Solow model with Hotelling’s rule
Hotelling’s rule provides a microeconomic foundation for the extraction sector. The rule
itself describes the profit maximizing behavior of the extraction sector – for a full mathe-
matical proof, see [3]. The optimizing behavior in the extraction sector will endogenize the
extraction rate of the nonrenewable resource. This will improve our model by achieving
a higher degree of general equilibrium.
The model is similar to the exogenous setup in many aspects. We now denote the
endogenous extraction rate by q, instead of the exogenous rate sE. Furthermore the model
includes a relation for Hotelling’s rule (5.5), and an efficiency condition for extraction
of the resource (5.7). We consider R0 to be the proved reserves. The model can be
summarized as follows.
Y = Kα(AL)βEε, α, β, ε > 0, α + β + ε = 1 (5.1)
r = αKα−1 (AL)β Eε (5.2)
ω = βKαAβLβ−1Eε (5.3)
u = εKα (AL)β Eε−1 (5.4)u
u= r − δ (5.5)
q =E
R(5.6)∫ ∞
0
E(t)dt ≤ R0 (5.7)
R = −E, R0 given (5.8)
K = sY − δK, K0 given (5.9)
L
L= n, L0 given (5.10)
A
A= g, A0 given (5.11)
5.2. THE LAW OF MOTION 17
5.2 The law of motion
As in the exogenous model, we analyze the model in terms of the capital-output ratio z.
To find the law of motion for z we take logarithms and differentiate with respect to time.
z ≡ K
Y=
K
Kα(AL)βEε= K1−α(AL)−βE−ε (5.12)
z
z= (1− α)
K
K− β(
A
A+L
L)− εE
E(5.13)
= (1− α)(s
z− δ)− εE
E− β(g + n), since
K
K=s
z− δ
= (β + ε)s
z− εE
E− β(g + n+ δ)− εδ (5.14)
In order to find the growth rate of the energy input EE, we start by finding the growth
rate of u with regards to the marginal product of the resource (5.4).
u
u= α
K
K+ β(g + n) + (ε− 1)
E
E(5.15)
In the exogenous case, the resource price was indirectly depending on capital through the
marginal product in production. In contrast the endogenous model requires the extraction
sector to be directly dependent on the interest rate. We thus use Hotelling’s rule (5.5) to
connect the capital markets and the extraction sector. We start by inserting the growth
rate of the resource price into Hotelling’s rule, (5.5).
αK
K+ β(g + n) + (ε− 1)
E
E= r − δ (5.16)
Rewriting the marginal product of capital from equation (5.2).
r = αKα−1 (AL)β Eε =α
z(5.17)
Inserting (5.17) into (5.16), and isolating the growth rate of E.
α(s
z− δ) + β(g + n) + (ε− 1)
E
E=α
z− δ
αs− 1
z− (α + β)
E
E= (α− 1)δ − β(g + n)
E
E=
1
α + β
(αs− 1
z+ β(g + n+ δ) + εδ
)(5.18)
We further see from equations (5.5) and (5.17), that the growth rate of the resource price
is negatively dependent on the capital-output ratio z.
u
u=α
z− δ (5.19)
18 5. THE MODEL
We find the law of motion for the capital-output ratio, with z being the sole endogenous
variable. For doing so we insert the growth rate of E from (5.18) into the law of motion
for z, (5.14). The law of motion for z will thus take the direct interaction between the
extraction sector and the capital markets into account.
z
z=(β + ε)
s
z− ε
α + β
(αs− 1
z+ β(g + n+ δ) + εδ
)− β(g + n+ δ)− εδ (5.20)
=
((β + ε)s− αε(s− 1)
α + β
)1
z−(
1 +ε
α + β
)(β(g + n+ δ) + εδ)
=(α + β)(β + ε)s− αε(s− 1)
α + β· 1
z− β(g + n+ δ) + εδ
α + β
=(1− α)(1− ε)s− αε(s− 1)
α + β· 1
z− β(g + n+ δ) + εδ
α + β
=(1− α− ε)s− αε)
α + β· 1
z− β(g + n+ δ) + εδ
α + β
=βs+ αε
α + β· 1
z− β(g + n+ δ) + εδ
α + β
z =βs+ αε
α + β− β(g + n+ δ) + εδ
α + βz (5.21)
We proceed to derive the dynamics of the extraction rate in the endogenous model. The
extraction rate is defined as (5.6). Taking logarithms, differentiating with respect to time,
and using (5.8) to find the growth of the extraction rate.
q
q=E
E− R
R=E
E+E
R=E
E+ q (5.22)
To find the law of motion for q we insert the growth rate of E as a function of z (5.18).
q
q=
1
α + β
(αs− 1
z+ β(g + n+ δ) + εδ
)+ q (5.23)
5.3 Steady state
The law of motion for z solely depends on z, and the dynamics of z are the same as in
figure 3.1. This is different from the law of motion for q, which depends on the evolution
of both z and q. The evolution of q is thereby described by the two differential equations.
For convenience the law of motion for both z and q are restated.
z =βs+ αε
α + β− β(g + n+ δ) + εδ
α + βz
q
q=
1
α + β
(αs− 1
z+ β(g + n+ δ) + εδ
)+ q
5.3. STEADY STATE 19
The steady state value for the capital-output ratio is derived by setting z = 0.
z∗ =βs+ αε
β(g + n+ δ) + εδ(5.24)
Having found the steady state for z, we are now able to calculate the steady state for
the other endogenous variables. We start by finding the steady state value for the energy
growth rate, by inserting (5.24) into (5.18).(E
E
)∗=
1
α + β
(α(s− 1)(β(g + n+ δ) + εδ)
βs+ αε+ β(g + n+ δ) + εδ
)(5.25)
=1
α + β
(α(s− 1) + βs+ αε
βs+ αε
)(β(g + n+ δ) + εδ)
=1
α + β
(s(α + β)− α(α + β)
βs+ αε
)(β(g + n+ δ) + εδ)
=s− αβs+ αε
(β(g + n+ δ) + εδ) (5.26)
In order for the model to fulfil (5.7) we must have that(EE
)∗< 0. This implies that
α > s, which is plausible regarding empirical data, see [10, pp. 79 - 80]. This will be
assumed throughout this thesis. By inserting (5.24) into the modified Hotelling’s rule
(5.19), we find the steady state of the growth rate in resource prices.(u
u
)∗=α(β(g + n+ δ) + εδ)
βs+ αε− δ (5.27)
Having determined the steady state for z and EE, we find the steady state for q. We set
q = 0 and insert the steady state of the extraction growth rate(EE
)∗.
q
q=E
E+ q = 0 (5.28)
q∗ = −
(E
E
)∗=
α− sβs+ αε
(β(g + n+ δ) + εδ) (5.29)
In order for a meaningful solution to exist, we need q∗ to be non-negative. This is fulfilled
since α > s.
Having determined the dynamic equations and steady state values of q, z and uu, we
now turn to the remaining input factor prices. The real rental rate r is given by the
marginal product of capital, (5.17). Inserting the steady state value of the capital-output
ratio (5.24).
r∗ =α
z∗= α
β(g + n+ δ) + εδ
βs+ αε(5.30)
20 5. THE MODEL
Subtracting the capital depreciation rate gives the constant steady state real interest rate.
ρ∗ = r∗ − δ = αβ(g + n+ δ) + εδ
βs+ αε− δ (5.31)
The real wage rate is likewise determined by the marginal product according to equation
(5.3). This implies that the real wage rate will grow at the rate of output per worker in
steady state.
The growth rate of the labor force is exogenously given, so to determine the growth
rate of output per worker, we have to determine the growth rate of output. Taking
logarithms and differentiating the production function (5.1) with respect to time yields.
gY = αgK + β(g + n) + εE
E(5.32)
Inserting(EE
)∗and setting the growth rates of capital and output to be equal, since the
capital-output ratio is constant in steady state.
gY =β
β + ε(g + n)− ε
β + ε· α− sβs+ αε
(β(g + n+ δ) + εδ) (5.33)
Subtracting the growth rate of labor gives the growth rate of output per worker.
gy =βg − εnβ + ε
− ε
β + ε· α− sβs+ αε
(β(g + n+ δ) + εδ) (5.34)
Thus we arrive at the general result – the economy follows a balanced growth path in
steady state.
5.4 Phase diagram
Since z and q are determined by a system of two differential equations, the solution to
the system can be illustrated in a phase diagram. In the phase diagram we have z along
the horizontal axis and q along the vertical axis. We draw the steady paths of z and q,
that is the z = 0 and q = 0 loci.
z = 0 : z =βs+ αε
β(g + n+ δ) + εδ(5.35)
q = 0 : q =α1− sα + β
· 1
z− β(g + n+ δ) + εδ
α + β(5.36)
The z = 0 locus is a vertical line at z∗, and the q = 0 locus is a hyperbola with a positive
intersection with the z-axis. The loci define the possible outcomes where z and q are
separately steady.
5.4. PHASE DIAGRAM 21
Figure 5.1: Phase diagram
The motion of z and q outside the loci are as follows. When z is larger than z∗, z will be
negative and thus z will fall until it reaches z∗ – and vice versa when z is lower than z∗.
The capital-output ratio is hence equilibrium correcting and stable. This is not the case
for q. Being on the q = 0 locus, a larger value of q will cause q to be positive and thus q
will rise. A lower value of q will make q negative and cause q to decline. The extraction
rate q itself is hence unstable and will not correct towards equilibrium.
Since the z-differential equation is stable, and the q-differential equation is unstable,
the system is saddle point stable. The motions of the phase diagram determines the
unique saddle path of the economy.1 Once the economy is located on the saddle path, it
will converge towards the equilibrium. The equilibrium denoted E, is thus a stationary
point of the system. Once the economy reaches the equilibrium E, the system will never
revert unless it is hit by a shock.
But can we be certain that the economy will ever be on the saddle path? Let us start
by analyzing the paths diverging from equilibrium. In the system of the two differential
equations the diverging paths are feasible, but economically neither rational nor realistic.
Taking the initial value of z to be given at a point in time, a value of q larger than the
1A mathematical proof would require a complete solution to the system of differential equations. Notethat the saddle path has been drawn as a linear function. We can however not be sure that the saddlepath is linear, since we have not derived an analytical solution to the system of differential equations.Due to the motions of the phase diagram, we can though be sure that the saddle path is declining. Thelinearity is assumed for convenience, and will not alter the analysis.
22 5. THE MODEL
corresponding point on the saddle path, will lead to q > 0, and thereby an exponentially
growing extraction rate. This will lead to q diverging towards infinity, and the total sum
of extractions would become larger than the finite resource stock. This is not feasible
according to (5.7), and furthermore q would be larger than unity, which does not make
sense.
On the other hand if q for a given value of z, is lower than the corresponding point
on the saddle path, it will imply q < 0 and lead to an exponentially falling q. On this
path q will be converging towards zero. The knife-edge case where z = z∗ and q is below
the saddle path is analyzed in appendix A.3. According to this analysis, this path is not
economically rational, since the economy most likely will end up not using its entire stock
of resources. Thus there will be excess profit for the extraction sector to earn, since the
price of the resource u always will be positive. These arguments suggest that for a given
value of z, q must be such that the economy is located on the saddle path.
In order for the solution of q∗ to be feasible, it has to be positive. Thus the z-value of
q = 0, denoted by z′ has to be larger than z∗. Setting q = 0 in (5.36) yields.
0 = α1− sα + β
· 1
z′− β(g + n+ δ) + εδ
α + β(5.37)
z′ =α(1− s)
β(g + n+ δ) + εδ(5.38)
We analyze the conditions for z′ being larger than z∗.
z′ =α(1− s)
β(g + n+ δ) + εδ>
βs+ αε
β(g + n+ δ) + εδ= z∗ (5.39)
α(1− s) > βs+ αε
(1− β − ε)(1− s) > βs+ αε
1− s− β + βs− ε+ εs > βs+ αε
α− s+ εs > αε
α(1− ε) > s(1− ε)
α > s
Again we end up with the condition that the income share of capital has to be larger than
the savings rate, as for q∗ > 0.
6 Analysis
6.1 Characteristics and long run growth
In this section we intent to give an economic explanation of the theory described above.
We have chosen to divide the analysis between the resource variables and the long term
growth of the model.
Resource price and extraction
The evolution of the growth rate in the resource price and the evolution of the extraction
rate, both depend on the capital-output ratio. In the following we assume that the
economy is located on the saddle path.
The growth rate of the resource price depends negatively on z according to equation
(5.19). The evolution of z is depicted in the phase diagram – if z is less than its steady
state, z will be rising and vice versa. A rising value of the capital-output ratio will thus
lead to a falling growth rate of resource prices, and vice versa. This can be explained
as follows. A rising capital-output ratio means that K is rising faster than the other
input factors, since the production function has CRS. The marginal product of capital
will consequently fall, thus lowering the real rental rate. The growth rate of the resource
price is determined by the real interest rate, and hence a rising capital-output ratio will
lead to a falling growth rate of the resource price. This is the main effect of the capital-
output ratio on the resource price. The growth rate of the resource price will however
not always be falling. As the capital-output ratio converges towards its steady state the
speed of convergence will decline. Furthermore, an increasing capital-output ratio implies
a downward pressure on the extraction rate according to the phase diagram. This leads
to less of the resource being extracted, thereby increasing the marginal product of the
resource. The increase in the marginal product implies an upward pressure on the resource
price, and thereby further stabilizing the decline in the growth rate.
The time path of q depends on the initial value of z, as also seen in the phase diagram.
If the capital-output ratio is below its steady state, the extraction rate will be falling
towards equilibrium, and vice versa. In the following this is explained by the economic
forces of the model. A rising capital-output ratio will as before lead to a relative increase in
capital compared to the other input factors, and thereby decreasing the marginal product
of capital. Hence the real rental rate will decrease, which will imply an altered equilibrium
in the asset markets. The decrease in the real interest rate makes it more profitable to
24 6. ANALYSIS
keep the resources in situ. Ceteris paribus the extraction rate falls since the resource stock
is fixed.
From the paragraphs above, we know that the paths of uuand q depend highly on the
initial value of z. But how can we determine this value? One might argue that in the
very first period z must be very small, since the initial production is positive, while the
capital stock is close to zero. Thereby z would be smaller than its steady state. Imagine
production of consumable goods in the earliest ages, where production was carried out
without any capital. This argument is however rather naive, since it is only a theoretical
situation concerning the initial period.
Empirics of the Danish economy suggest that the capital-output ratio has been nearly
constant for the past 44 years, see appendix A.2. This conclusion is not consistent however,
when considering a longer time period and a broader range of countries. According to
Maddison [4, p. 67] the capital-output ratio has been increasing for the Netherlands,
France, Japan, and the UK and constant for the USA, during the period from 1890 –
1987.
These empirics propose that the capital-output ratio has been increasing in the long
run. This suggests that the convergence towards steady state has been from the left part
of the saddle path. This is supported by the empirically decreasing extraction rate.
Long run growth
An important feature of the Solow model is that it determines the long run growth in
output per worker. We have found the growth rate of output per worker in the exogenous
and the endogenous model. For convenience the growth rates are restated below, and we
have also included the growth rate of the general Solow model [10, pp. 133 - 134].
gyGEN = g (6.1)
gyEXO =βg − εnβ + ε
− ε
β + ε· sE (6.2)
gyENDO =βg − εnβ + ε
− ε
β + ε· α− sβs+ αε
(β(g + n+ δ) + εδ) (6.3)
As Malthus suggested with land and food, the essentiality of a resource in production
can create a growth drag, see [5, chapter 2]. From our model we can confirm that a
nonrenewable resource in production, implies a growth drag on output per worker. It is
however still possible to achieve a positive growth of output per worker, when the output
6.1. CHARACTERISTICS AND LONG RUN GROWTH 25
elasticity of the nonrenewable resource is relatively small. This is seen when comparing
the general Solow model with either the exogenous or the endogenous model.
It is interesting to analyze whether an exogenous or an endogenous extraction rate
will yield the highest growth in income per capita. We compare the exogenous extraction
rate with the endogenous extraction rate.
sE Rα− sβs+ αε
(β(g + n+ δ) + εδ) (6.4)
Table 6.1: Parameter estimates
Parameter α β ε s g n δ sE
Estimate* 0.30 0.60 0.10 0.15 0.02 0.01 0.05 0.0225**
Endogenous var. z q r(uu
)∗EXO
(uu
)∗ENDO
gyGEN gyEXO gyENDOSteady state 2.264 0.066 0.133 0.045 0.083 0.020 0.013 0.006
* Parameter estimates are based on [10].** The exogenous extraction rate is based on the empirical findings from figure 2.3.
For realistic parameter values, we estimate the endogenous extraction rate to be app. 6.6
percent. This is considerably higher than the empirical extraction rate, which was found
to be around 2 – 2.5 percent, c.f. figure 2.3. The empirically chosen value of sE has for
consistency reasons not been corrected for unproved reserves, since the endogenous model
does not account for this. As mentioned before it has though been suggested by Nordhaus
that the extraction rate has been close to 0.5 percent, see [6, pp. 15–16]. We further see,
that the endogenous steady state growth rate of the resource price is almost twice as high
as the exogenous. The larger steady state extraction rate in the endogenous model results
in a faster depletion, thus resources become scarcer, hence pushing the growth rate of
the resource price upwards. Again we find that the model may not explain the actual
evolution of the resource price, given in section »2.2«. However the model now has a more
detailed explanation of the price evolution, cf. the previous paragraph.
When pursuing the highest possible growth rate in output per worker, the exoge-
nous model performs better than the endogenous model. Though it would seem logical,
the optimizing behavior of the extraction sector does not imply a higher growth rate.
From equation (6.2) and (6.3) we find, that the minimal technological growth required
to sustain a nonnegative growth in output per worker is 0.6 percent and 1.2 percent re-
spectively. Again the exogenous model performs better. Hence the endogenous setup has
some mechanism that leads to a larger growth drag.
26 6. ANALYSIS
But what is really causing the extraordinary large growth drag in the optimizing
model? One possible explanation is the long-term profit maximizing behavior of the
extraction sector given by Hotelling’s rule. This profit maximization may not lead to the
highest possible growth rate of output per worker in the long run, since the extraction
sector might not take the growth drag into account. Another explanation could be the
absence of an infinite perfect futures market, determining all future prices of the resource,
as suggested by Stiglitz [9].
6.2 Initial and feasible values
As described in the previous section, the initial values of z and q have a large impact
on the time path of the resource price and the extraction rate. We now directly analyze
the effects of the initial and feasible values. We start this section by finding the relation
between the variables that create a feasible solution. This connection is further used to
analyze the evolution of the economic outcomes. We proceed to determine whether the
results of the exogenous model can be seen as a special case of the endogenous model.
Finally we study the effects of an unexpected increase in the resource reserves.
The initial input factors determine the initial capital-output ratio.
z0 ≡K0
Y0
=K0
Kα0 (A0L0)βEε
0
= K1−α0 (A0L0)−βE−ε0 (6.5)
We isolate the initial value of the energy input E0 as a function of z0, K0, A0 and L0.
E0 = z− 1ε
0 Kβ+εε
0 (A0L0)−βε (6.6)
To find the initial value of the extraction rate q0, we insert E0 from (6.6).
q0 ≡E0
R0
=z− 1ε
0 Kβ+εε
0 (A0L0)−βε
R0
= z− 1ε
0 · Kβ+εε
0
R0(A0L0)βε
(6.7)
This is the band of initial feasible values, which the economy must fulfil. Graphically it
is illustrated as a hyperbola with no intersections with either the z-axis or the q-axis, as
seen in figure 6.1. Economically the feasible band must intersect the saddle path, since
the saddle path denotes the stable time path of the economy.
Since ε ∈ (0, 1) the curvature of the feasible band is higher than the curvature of
the q = 0-locus. For a precise illustration of this, see the mathematical calibration of
the model in appendix A.4. The feasible band intersects the saddle path twice - on the
steep part (close to E) and on the flat part (far from E). The intersection with the flat
6.2. INITIAL AND FEASIBLE VALUES 27
part cannot been seen in the figures, since it occurs for a very large value of z. Based
on the mathematics and our calibration of the model in appendix A.4, we find that the
intersection on the steep part is the only feasible outcome. The reason for this result is,
that the feasible band does not have an additive constant. Hence the flat part of the
feasible band will never be able to intersect the steady state. For this reason we will only
focus on the intersection on the steep part of the feasible band.
Hence there are two outcomes of the initial state, A0 and B0, as shown graphically in
figure 6.1. The initial state could either be higher or lower than the steady state value
of z. We consider the convergence from the point A0 to E to be most realistic, since the
empirics suggest an increasing capital-output ratio.
Figure 6.1: Phase diagram with feasible band
After period zero the values of the initial band move in accordance with the motions of
the model. The band denotes the economically feasible outcomes of the given state of the
economy. As the economy moves along the saddle path towards steady state, so will the
feasible band. Thus the economy will fulfil this condition at each point in time.
qF (z) = z−1ε · K
β+εε
R(AL)βε
(6.8)
Naturally the economy must be at a feasible outcome in steady state, and hence the
steady state is intersected by the feasible band.
28 6. ANALYSIS
Is the exogenous model a special case of the endogenous model?
If the exogenous setup could be modelled and interpreted as a special case of the endoge-
nous setup, the analysis of the model dynamics would be much simpler, as we saw in
chapter »3«. For this to be true, the extraction rate of the endogenous model would have
to be constant at each point in time. One possible solution for this to be true is if the
saddle path is horizontal. This possibility can be rejected, since the saddle path is not
linear and thus not horizontal, due to the fact that the q = 0-locus is a hyperbola.
Another possibility is if the combination of initial values places the economy directly
in the steady state. For this possibility to be true, the initial band would have to intersect
the long run equilibrium E. This will not replicate the exogenous model perfectly, since
the exogenous model has a constant extraction rate for all values for z. In contrast this
pseudo-exogenous model only has a constant extraction rate for z already being in steady
state. In mathematical terms this can be expressed as.
z− 1ε
0 · Kβ+εε
0
R0(A0L0)βε
= q∗ =α− sβs+ αε
(β(g + n+ δ) + εδ) (6.9)
It seems highly unrealistic that the combination of initial values by coincidence will place
the economy in steady state. In addition the empirical data suggest, that the capital-
output ratio is increasing, indicating that steady state has not been reached yet. These
findings argue against this special case.
A shock to the resource reserves
The most applicative shock to analyze is an increase in the nonrenewable resource reserves,
i.e. new findings. We analyze the effects on the extraction rate and the resource price.
As the basis of our analysis, we assume that the economy is in steady state, and hence
graphically located at the point E. It is important to stress that the analysis will be based
on an unexpected increase in the reserves. Hence expectations are not taken into account.
Technically, an increase in the nonrenewable resource reserves is modeled as an increase
in the variable R. This will lead to a decrease in the coefficient of the band of feasible
values. Thus the band will be shifted towards origo. The shift in the feasible band,
will cause the intersection between the saddle path and the feasible band, to be shifted
to the left-hand side of the saddle path. We can be certain of this outcome since the
feasible band does not have a constant term, and this is supported by the calibration
of appendix A.4. The the capital-output ratio will fall and the extraction rate will rise.
6.2. INITIAL AND FEASIBLE VALUES 29
After the adjustment, the capital-output ratio will be increasing and the extraction rate
will decrease as the economy converges towards equilibrium. Graphically this is shown
in figure 6.1, where the economy jumps from the point E to A0 and afterwards converge
back to the steady state, E.
The economic explanation for this is described in the following.
E → A0: In this model the construction of the feasible band will cause the extraction
rate will rise. Hence the relative growth in the extraction is larger than the relative
growth in the resource reserves. A rise in the extraction will imply a larger supply of
the nonrenewable resource. This lowers the resource price and firms will use more of the
resource in production, due to the marginal product condition. The larger production
lowers the capital-output ratio, since capital is predetermined.
Intuitively, one could argue that the amount extracted would be constant or increasing
at a lower relative rate than the reserves, implying that an increase in the resource reserves
would lead to a fall in the extraction rate. This argument can not be supported in this
setup, since it would require an increase in z, which is unrealistic since K is predetermined
and Y will not fall.
A0 → E: The increase in extraction raises the marginal product of capital, and will
thereby increase the real interest rate. According to Hotelling’s rule the growth rate of
the resource price must rise. This lowers the amount of resource used in production, due
to the marginal product condition. This will lead to a lower amount extracted and a
falling extraction rate. As the input of the resource in production falls, total production
will fall and the capital-output ratio will increase.
The increasing capital-output ratio causes the real rental rate to decline, thus lowering
the growth rate of the resource price. Hence the fall in the extraction will decline, thereby
reducing the fall in the extraction rate. This secondary effect will gradually reduce the
size of the first effect causing the economy to converge towards the equilibrium.
Another realistic shock to consider is the invention of new technology in the economy.
The results of the previous analysis can be projected to this case of an unexpected increase
in the technological level. Mathematically, this would again result in an decrease in the
coefficient to z in the feasible band, since A is in the denominator. This results in a
inwards shift in the feasible band, thus implying a decrease in the capital-output ratio
and an increase in the extraction rate. The economic explanation will however differ in
some aspects.
7 Conclusion
In this thesis we have analyzed the endogenization of the extraction rate of a nonrenewable
resource in a Solow model. The transition from exogenous to endogenous extraction
behavior brought the model to a higher degree of general equilibrium.
The analysis showed that an exogenous extraction rate leads to an exponentially in-
creasing price of the nonrenewable resource. Conferring with the empirics provided further
motivation to improve the model by implementing Hotelling’s rule. The endogenization
of the extraction sector led to a steady state extraction rate, which was considerably
higher than the empirical data suggested. Qualitatively, the higher degree of general
equilibrium resulted in more complex and accurate economic explanations. Especially
the link between the extraction sector and the production sector became more realistic.
Our empirics suggested that the extraction rate has been falling and Maddison argued
that the capital-output ratio has been increasing, which combined is in accordance with
our theory. Regarding the long run growth we found that the exogenous model lead to
a higher growth in output per worker. This can be explained by the profit maximization
of the extraction sector, which does not take a growth drag in output per worker into
account.
The main aspect of the endogenous model is a simultaneous convergence of the capital-
output ratio and the extraction rate. These complex dynamics are not compatible with a
constant extraction rate, and thus the exogenous model cannot be analyzed as a special
case. Since we find it to be highly relevant, we analyzed a rise in the resource reserves.
The result was an immediate increase in the extraction rate and a fall in the capital-
output ratio, followed by a convergence back to equilibrium. The structure of the model
gave an unambiguous outcome, and we find the result to be realistic. In general we find
our endogenous model to provide a thorough description of an economy restricted by
a nonrenewable resource in production. In addition, the model is intuitive and can be
justified by conventional economic explanations.
8 Perspective
Throughout the medias of the world, the dependence of natural resources is debated heav-
ily, and the focus has been increasing rapidly during the past two decades. The perspective
of this thesis is that the existence and essentiality of a nonrenewable resource has impor-
tant implications on the economy. Especially the assumptions regarding the extraction
behavior has important qualitative consequences for key variables in the economy.
The analysis of a rise in the resource reserves has a significant perspective, since it is
applicable to many aspects of the extraction behavior in the real world. For the analysis
to be consistent however it is important, that the shock to the economy is unexpected.
Historically there have been recurring findings of new oil reserves e.g. the North Sea oil
findings, and in the long run this will cause the agents to expect new findings. Hence a
rise in the reserves might not have the precise described effects.
Future analysis could incorporate more realistic assumptions on the essentiality of
the resource in production. Explicitly the model could include relations regarding the
possibilities for substitution and the existence of back-stop technologies see [7, p. 477]
e.g. wind and wave power. The analysis could further be expanded to contain more input
factors in production such as human capital or land. Finally it would be interesting to
carry out the analysis in a more complex model, that integrate the utility maximization
of the consumers or contains endogenous technological growth.
Bibliography
[1] Canadian Economics Association. Natural Resource Economics under the Rule of
Hotelling, June 2007.
[2] Harold Hotelling. The economics of exhaustible resources. The Journal of Political
Economy, 39(2):137–175, 1931.
[3] Jeffrey A. Krautkraemer. Nonrenewable resource scarcity. Journal of Economic
Litterature, 36(4):2065–2107, 1998.
[4] Angus Maddison. Dynamic Forces in Capitalist Development: A Long-Run Compar-
ative View. Oxford University Press, first edition, 1991.
[5] Thomas Robert Malthus. An Essay on Population. London: John Murray, first
edition, 1798.
[6] William D. Nordhaus. Lethal model 2: The limits to growth revisited. Brookings
Papers on Economic Activity, 2:1–59, 1992.
[7] Roger Perman, Yue Ma, James McGilveray, and Michael Common. Natural Resource
and Environmental Economics. Pearson, third edition, 2003.
[8] Robert M. Solow. A contribution to the theory of economic growth. Quarterly
Journal of Economics, 70(1):65–94, 1956.
[9] Joseph E. Stiglitz. Growth with exhaustible natural resources: The competitive econ-
omy. Review of Economic Studies, 41(Symposium on the Economics of Exhaustible
Resources):139–152, 1974.
[10] Peter Birch Sørensen and Hans Jørgen Whitta-Jacobsen. Introducing Advanced
Macroeconomics: Growth and Business Cycles. McGraw-Hill, second edition, 2010.
A Appendix
A.1 SAS test output
Figure A.1: OxMetrics output for test of the growth rate in crude oil prices 1862 – 2010
Figure A.2: OxMetrics output for test of the growth rate in crude oil prices 1900 – 2010
Figure A.3: SAS output for test of crude oil prices 1861 – 2010
34 A. APPENDIX
Figure A.4: SAS output for test of crude oil prices 1900 – 2010
Figure A.5: OxMetrics output for test of the extraction rate 1980 – 2008
A.2 Capital-output ratio statistics
Figure A.6: Capital-output ratio for Denmark 1966 – 2010
A.3. ANALYSIS - LAW OF MOTION FOR THE EXTRACTION RATE 35
A.3 Analysis - law of motion for the extraction rate
We want to analyze the evolution of q for the knife-edge case of z being in steady state
(z = z∗). The analysis will be partial, and only focused on the evolution of q. First we
derive the law of motion for q, for z = z∗, by inserting (5.24) into (5.23).
q
q=
1
α + β
(αs− 1
z∗+ β(g + n+ δ) + εδ
)+ q (A.1)
q
q=
1
α + β
(α(s− 1)
β(g + n+ δ) + εδ
βs+ αε+ β(g + n+ δ) + εδ
)+ q
q
q=
1
α + β
(α(s− 1)
βs+ αε+ 1
)(β(g + n+ δ) + εδ) + q
q
q=
1
α + β· αs− α + βs+ αε
βs+ αε(β(g + n+ δ) + εδ) + q
q
q=
1
α + β· αs− α + (1− α− ε)s+ αε
βs+ αε(β(g + n+ δ) + εδ) + q
q
q=
1
α + β· αs− α + s− αs− εs+ αε
βs+ αε(β(g + n+ δ) + εδ) + q
q
q=
1
α + β· s− α− ε(s− α)
βs+ αε(β(g + n+ δ) + εδ) + q
q
q=
1
α + β· (s− α)(1− ε)
βs+ αε(β(g + n+ δ) + εδ)︸ ︷︷ ︸
Ω<0
+q (A.2)
We see that Ω is negative, since we have assumed that α > s. Multiplying by q yields.
q = q2 + Ωq (A.3)
Figure A.7: Law of motion for the extraction rate
36 A. APPENDIX
The law of motion for q, z being in steady state, is illustrated in figure A.7. From the
figure and equation (A.3), it can be seen that q has two stationary points. A stable
stationary point in q = 0, and a unstable stationary point in q = q∗.
We thus see that for q starting above q∗, q will converge towards infinity, while for q
starting below q∗, q will converge towards zero.
In order to obtain an explicit solution for the evolution of q, we solve the first order
non-linear differential equation in q. We use the subscript t to denote the time dependence.
First we see that the differential equation can be solved as a Bernoulli differential equation.
Multiplying by q−2 gives.
qtq−2t = 1 + Ωq−1
t (A.4)
Substituting ht = q−1t , and thus ht = −q−2
t qt. Solving the differential equation in h.
−ht = 1 + Ωht (A.5)
ht + Ωht = −1 (A.6)
ht = C · e−Ω·t − 1
Ω(A.7)
Substituting back for qt in order to find the solution to the differential equation in qt.
qt =1
C · e−Ω·t − 1Ω
(A.8)
From (A.8) we see that qt is highly dependent on the value of Ω, and that the value of C
must make qt positive for an economically reasonable solution to exist. When inserting
the parameter estimates given in table 6.1, we get Ω = −0.06625. We find the value of
C, by inserting the initial conditions of the equation (q0 and t = 0).
q0 =1
C · e−Ω·0 − 1Ω
(A.9)
C =
(1
q0
+1
Ω
)(A.10)
This gives the following evolution of qt, when z is in steady state, which depends on the
value of q0.
qt =1(
1
q0
+1
Ω
)︸ ︷︷ ︸
C
·e−Ω·t − 1Ω
(A.11)
Hence the entire evolution of qt outside the saddle path is determined by the initial value,
q0. If C is negative, but the entire denominator is still positive, since 1Ω
is negative, qt
A.3. ANALYSIS - LAW OF MOTION FOR THE EXTRACTION RATE 37
will be rising as t increases. In contrast a positive value of C makes qt decline as time
increases.
We have now determined the entire time path of qt for z = z∗, and we now want to find
the effect of qt on the evolution of the resource stock Rt. We especially want to examine
the evolution of Rt for qt converging towards zero, that is for C being positive. First we
find an explicit expression for Rt using equation (5.8) and (5.6).
Rt = −qt ·Rt (A.12)
This is a first order linear differential equation with variable coefficient. Solving the
equation yields.
Rt = −qt ·Rt (A.13)
Rt · e∫ t0 qτdτ = −qt ·Rt · e
∫ t0 qτdτ (A.14)∫ T
0
Rt · e∫ t0 qτdτdt = −
∫ T
0
qt ·Rt · e∫ t0 qτdτdt (A.15)
Rewriting the lefthand side by integration by parts.∫ T
0
Rt · e∫ t0 qτdτdt = [Rt · e
∫ t0 qτdτ ]T0 −
∫ T
0
qt ·Rt · e∫ t0 qτdτdt (A.16)
Inserting the expression on the lefthand side.
[Rt · e∫ t0 qτdτ ]T0 −
∫ T
0
qt ·Rt · e∫ t0 qτdτdt = −
∫ T
0
qt ·Rt · e∫ t0 qτdτdt (A.17)
[Rt · e∫ t0 qτdτ ]T0 = 0 (A.18)
RT · e∫ T0 qτdτ = R0 (A.19)
RT = R0 · e−∫ T0 qτdτ (A.20)
We thus see that for qt = q∗, Rt will converge to zero. In contrast when qt is declining, as
we saw with positive value of C, we can not be sure that Rt will converge to zero. Hence
we numerically estimate the evolution of Rt for a declining qt in Maple 15.01. Setting
the parameters according to table 6.1. We randomly set q0 = 0.05 which is less than
q∗ = 0.066 implying a value of C = 4.90566, the given choice of q0 has no implications for
the qualitative results. Furthermore we set the initial stock of recourses R0 = 100 for an
easy percentage interpretation.
First we numerically estimate the integrated q’s, that is the exponent of e from equation
(A.20). We estimate 10.000 and 100.000 periods, and see no change in the integration.
The estimations can be seen in figure A.8.
38 A. APPENDIX
Figure A.8: Estimations of the integration of q - 10.000 and 100.000 repetitions
We interpret the estimations as if t → ∞, which seems reasonable according to the
numerical estimations in figure A.8. Thus inserting the result of figure A.8 into (A.20) to
find the convergence of Rt for t→∞.
limt→∞
Rt ≈ 100 · e−1.380410317 = 25.15 (A.21)
This implies that the resource reserves left after 100.000 periods is app. 25 percent of the
original reserves. Graphically the evolution of Rt for 500 periods is illustrated in figure
A.9.
Figure A.9: Estimation of the evolution of R - 500 periods
We thus see that a value of the extraction rate starting below the saddle path, for the
capital-output ratio being in steady state, results in a case where the economy will not
use its entire stock of nonrenewable resources.
The analysis have only been performed for the knife-edge case where z = z∗. We do
however expect the result to apply asymptotically for values of z different from z∗. The
A.4. CALIBRATION OF THE ENDOGENOUS MODEL 39
reason is, that at some point in time qt will be declining towards zero, which will lead to
a similar conclusion as the one given above. These results will be with a lower end level
of Rt, since qt would be at a higher level for a longer period of time.
The explanation has so far been very formal, and we will now try explain the dynamics
intuitively. It is logical that a constant extraction rate implies that the resource level will
be converging to zero as time goes to infinity. A fast declining extraction rate will in
contrast result in a situation, where the extraction rate is infinitesimally close to zero. At
this point the extraction rate would have an indifferent impact on the resource reserves,
which would thus stagnate at the given level.
In the formal part of the analysis we have chosen not to analyze the case where qt is
rising, that is qt starts above steady state. The solution to this problem is more simple.
An extraction rate rising towards infinity would become larger than unity, which is not
feasible since q ∈ [0, 1] for a feasible solution to exist.
A.4 Calibration of the endogenous model
In order to analyze the location and movements of the band of feasible values, and its
intersection with the saddle path, we perform a calibration of the model. That is we find
the initial values of the variables, using the parameter estimates given in table 6.1 that
satisfies our solution.
First we find the combination that places the economy directly in steady state. Since
all the initial values compose the coefficient to z0, we solely adjust K0 and normalize R0,
A0 and L0 to unity.
q0 = z− 1ε
0 · Kβ+εε
0
R0(A0L0)βε
(A.22)
q∗ = z∗−1ε ·K
β+εε
0 , since R0 = A0 = L0 = 1 (A.23)
K0 =(q∗ · z∗−ε
) εβ+ε (A.24)
Inserting the steady state parametrized values of z and q and the estimated parameters.
K0 =(0.06625 · 2.26415−0.10
) 0.100.60+0.10 = 2.18075 (A.25)
There are many combinations of K0, R0, A0 and L0 that satisfies this solution for the
coefficient to z0. The analysis is indifferent to the different combinations and we will only
focus on the normalized combination. In figure A.10 we have illustrated the calibration.
40 A. APPENDIX
Figure A.10: Calibration
Besides from illustrating that the band of feasible values has a higher curvature than the
q = 0 locus, we also see that the band of initial values intersects the saddle path from
above, on the steep part of the band. This is an important feature of the calibration,
since it is in line with our description in figure 6.1 of the initial band.
In the end of section »6.2« we analyzed an increase in the resource reserves. Based
on the calibration, an increase in the resource reserves moves the intersection between
the saddle path and the band of initial values to the left. This is also in line with the
description given earlier. Based on the calibration, we can say that an increase in R will
imply a decrease in z and an increase in q.
The MS Excel spreadsheet can be downloaded from:
http://hvelplund.net/lars/bsc