EXHAUSTIBLE RESOURCE EXTRACTION UNDER DEMAND HETEROGENEITY
by
Ujjayant Chakravorty, Darrell Krulce and James Roumasset*
ABSTRACT
This paper develops a general model of exhaustible resource extraction when there are multiple
independent demands and multiple resources and grades. Resources are characterized by constant
unit extraction costs and conversion costs to each demand. We characterize patterns of resource
use over time based on the concept of absolute advantage of resources in specific demands and
across all demands. Conditions under which resources will be used at the beginning of the
planning horizon for all uses and at the end of the planning horizon as an endogenous backstop
are derived. We show that under demand heterogeneity progressive increases in the stock of a
resource (for example, through unexpected, exogenous discoveries) may completely alter the
sequence of extraction. For example, if oil stocks are limited, coal may be the backstop resource.
However, if new discoveries of oil are made, the economy may shift completely to oil. If further
additional discoveries of oil are made, we may revert to using coal in sectors where it has
absolute advantage and preserve oil for later periods. These results are in sharp contrast to models
with one demand that predict increased use of a resource with additional discoveries. The solution
to a simple two-demand two-resource case is characterized.
JEL Classification: D9, Q3, Q4
This Version: February 2002
*Respectively, Department of Economics, Emory University; Xiox Corporation and Department
of Economics, University of Hawaii at Manoa. Address for correspondence: unc@emory.edu.
EXHAUSTIBLE RESOURCE EXTRACTION UNDER DEMAND HETEROGENEITY
1. Introduction
The literature on the extraction of exhaustible resources (Hotelling, 1931; Dasgupta and Heal
1974; Pindyck, 1978) has almost exclusively dealt with the time path of resource extraction and
resource prices under the assumption of a single homogenous demand for the resource. Beginning
with Herfindahl (1967) effort has been made to study the issue of extracting multiple grades of a
resource, focusing in particular on the tinme sequence of extracting different grades (Solow and
Wan, 1976; Kemp and Long, 1980; Lewis, 1982) again under the assumption of a single demand
for the many grades of a resource.
Empirically, however, it is common knowledge among industry observers that the energy sector
of an economy is composed of distinct sub-sectors characterized by use, such as transportation,
electricity, commercial and residential energy, etc. Although there may be resource substitution
between these various end-uses, at least in the short run, it may be plausible to assume that the
demand for each sub-sector can be represented independently. Empirical observation also
suggests that typically there is more than one exhaustible resource (and grade) being extracted
simultaneously for meeting the diverse energy requirements of an economy.
Chakravorty and Krulce (1994) have developed a two demand and two resource model in an
infinite horizon framework where one resource has absolute advantage over the other in both
demands. They have shown that under these specific conditions, it will always be the case that a
more expensive resource will be used for a finite time interval even though the cheaper resource
is not exhausted, violating the well known “least cost principle” of exhaustible resource
economics. However, they did not proceed to develop the full implications of the multiple
demand framework. This paper generalizes their framework in several directions, by considering
an arbitrary number of resources and demands. Each end use is characterized by a downward
sloping demand function and each resource has a constant extraction cost and a fixed use-specific
conversion cost. No assumption is made regarding the absolute advantage of any resource over
other resources as in Chakravorty and Krulce (1994). Solutions of an infinite horizon
maximization problem yield equilibrium relationships for a given resource for a given end use in
terms of the royalty and cost characteristics of the resource.
2
We start by developing general propositions that govern the extraction of resources in this general
framework. In particular, we develop two distinct definitions of absolute advantage. Resources
may have absolute advantage over other resources in a specific demand or in all demands. A costdependent concept of “relative efficiency” of a resource is developed that determines the ordering
of royalties and the order of extraction of resources for any given use. The present paper develops
conditions under which a resource may be exclusively used for all demands at the beginning of
the planning horizon. Similarly conditions under which a resource may be exclusively used at the
end of the planning horizon are described, which results in an “endogenous” backstop
technology. The two resource two demand case is completely characterized under conditions in
which one resource may have absolute advantage in all uses, and when each resource has absolute
advantage in a specific use. It is shown that a weak absolute advantage in each use is likely to
lead to a single resource being used exclusively at the beginning. Comparative dynamics results
are obtained to show that an exogenous addition to reserves of a resource may
In a related piece of work, Gaudet, Moreaux and Salant (2001), henceforth referred to as GMS,
have solved a somewhat analogous problem where solid wastes are transported from urban
centers to spatially distributed landfills. In their model, landfill capacity is exhaustible, and they
are differentiated by transportation costs from each city. There are some similarities between our
problem and theirs, as well as important differences. The similarity is that transportation costs
from city to landfills can be thought of as conversion costs of resources to demands. The key
difference is that resources may be differentiated by class (oil, coal) or by grade (different grades
of oil) while landfills are homogenous except for their location.
The focus of their paper is on developing the general solution in a spatial setting and applying it
to the case of set up costs. In our paper, we focus on the general solution as well as develop
conditions based on exogenous model parameters under which certain patterns of resource
extraction may occur at the beginning and at the end of the planning horizon. In addition, we
completely characterize the two-resource two-demand solution. These results are not part of the
GMS paper. In general, transportation costs and end-use specific conversion costs as in our model
have an important difference. Conversion costs to all demands may be equal for resources of the
same class, and this allows us to make a distinction between resources of the same grade (oil of
different costs) and class (oil and coal) and develop a taxonomy that distinguishes between
resource class and grade. Thus a Herfindahl-induced ordering of resource grades can be
3
developed for each class, as shown in this paper. In later work it may be useful to develop a
general model with both end-use specific transportation and conversion costs.
In what follows, section 2 develops the general Hotelling theory with multiple demands. Section
3 provides the complete solution to the two demand two resource model. Section 4 concludes the
paper by highlighting the usefulness of a multi-demand multi resource approach over the
traditional single demand Hotelling framework.
2. A Model of Multiple Resources and Heterogenous Demands
Consider a finite set of resources R (such as oil, coal, natural gas, etc.) and a finite set of uses for
these resources defined by the set U (such as electricity, heating, transportation, etc.). The
available stock of resource I Є R is qi(t0)>0 which can be extracted at a unit cost of ci ≥0. Demand
for use j Є U is a strictly positive. Bounded, continuous, strictly decreasing function of price,

Dj(p) with
 D ( p )dp  .
j
This last restriction implies a finite consumer surplus and is useful
0
in guaranteeing a solution to the problem. These demands could be assumed to have been derived
from the final demands for that particular use. The resources are not perfectly substitutable
between uses. Some resources may be better suited for particular uses, such as petroleum
products as fuel for automobiles, whereas other contributions may be more problematic, for
example, running an automobile on coal. These differences between resources are summed up
into a conversion cost, vij>0, which is the cost of converting a unit of resource I for demand j.
Resource units are equivalent for particular demands once the demand specific conversion cost
has been expended. We define the net cost of supplying resource I to demand j as wij ≡ ci + vij.
The social planner chooses the quantity of each resource supplied to each demand. We denote by
dij(t) the quantity of resource i supplied to demand j at time t. The problem is to determine the
resource allocation that maximizes the present value of net social benefit. Given a discount rate
r>0, this can be posed as the following optimal control problem:
Choose dij(t) for i Є R and j Є U to maximize
 dij ( t )

 rt
e [
t0
jU
i R
D
0
1
j
( x )dx   wij d ij ( t )] dt
(1)
iR jU
4
subject to
dij(t) ≥ 0, qi(t) ≥ 0 for i Є R and j Є U
(2)
and
qi(t)  -
 d (t) for i Є R.
(3).
ij
j U
The state variable qi(t) is the residual stock of resource i over time. The first bracketed term of (1)
is the standard consumer surplus of the resources and the second term is the producer surplus.
The current value Hamiltonian for the above problem is given by
 d ij ( t )
H 
jU
i R
D
1
j
( x )dx    wij d ij ( t )] dt   i ( t ) d ij ( t )
0
iR jU
iR
jU
where λi(t) ≥0 has the standard interpretation as the royalty of resource i.
The solution is defined in terms of optimal price paths as functions of time. Let the price of the
resource input for demand j be pj( t )  Dj 1 (
 d ( t ) ). The necessary conditions for a solution
ij
iR
are then
qi(t)  -
 d (t) for i Є R.
(4)
ij
j U
i( t )  ri( t ) for i  R
(5)
pj(t) ≤ wij + λi(t) (if < then dij(t)=0) for i Є R and j Є U
(6)
and
lim e  rt i ( t )qi ( t )  0 for i Є R.
(7)
t 
We can now develop the following propositions:
5
PROPOSITION 1. There exists a unique optimal solution to program (1)-(3) and the necessary
conditions (4)-(7) are also sufficient.
PROOF: See the Appendix of Chakravorty and Krulce (1994).
The basic principle of resource use, proved by the following proposition, is that the resource that
is available at the lowest price (net cost plus royalty) is always used for each demand.
PROPOSITION 2. The price (net cost plus royalty) of a resource that is supplied for a given
demand is no more that of any alternative resource.
PROOF: Suppose that daj(t)>0 for some aЄR, jЄU and tЄ(t0,∞). Then from (6) waj+ λa(t) = pj(t) ≤
wij+ + λi(t) for iЄR. Q.E.D.
Consider the resource royalty λi(t). Solving (5) produces the familiar Hotelling equation
i ( t )  i ( t0 )e rt for i Є R
(8)
which states that royalty rises at the rate of interest. Condition (8) also implies that the royalties of
all resources are ordered. Based on this ordering, we write λa < λb to mean λa(t) < λb(t) for all t Є
(t0,∞). It may also be the case that the royalties of two resources are the same. As shown by the
following proposition, this must be the case if two resources ever simultaneously supply the same
demand.
PROPOSITION 3. Two resources simultaneously supplying the same demand have the same
royalty and net cost for that demand.
PROOF. Let daj(t)>0 and dbj(t)>0 for some a,b Є R, j Є U, and t Є I where I  ( t 0 ,  ) is an open
interval. Then from (6),
waj   a ( t )  p j ( t )  wbj  b ( t ) for t Є I.
(9)
Differentiating, we get a ( t )  b ( t ) for t Є I. Then from (5) and (8), λa=λb. That is, the
royalties are the same. Combining with (9) yields waj = wbj. That is, the net costs are equal.
Q.E.D.
Before proceeding, we prove the useful result that all resources approach exhaustion in the limit.
6
LEMMA. lim qi ( t )  0 for i ЄR.
t 
PROOF. Pick a ЄR and suppose that λa(t0)=0. Then from (8), λa(t)=0 and so from (6), p1(t)≤wa1.
Since demand is positive and downward sloping,
m
0  D1 ( wa1 )  D1 ( p1 ( t ))   d i1 ( t ). Thus
i 1
 m

t0 i 1
t0
  d i1 ( t )dt   so there exists b Є R such that  d b1 ( t )dt  . From (4),
q b ( t )    d bj ( t )  d b1 ( t ) and so eventually qb(t) will become negative which contradicts
jU
(2). Thus the supposition is false and so λa(t0)>0. Combining (7) and (8) yields
0  lim e  rt a ( t )qa ( t )  lim e  rt a ( t 0 )e rt qa ( t )  a ( t 0 ) lim qa ( t ) which since λa(t0)>0
t 
t 
t 
implies that lim q a ( t )  0. Since a was arbitrary, then lim qi ( t )  0 for for i ЄR. Q.E.D.
t 
t 
In the standard Hotelling model with a single demand, resource royalties are ordered by cost: the
resource with the lowest cost has the highest royalty. With heterogenous demand, the ordering of
resource royalties is more problematic since there is not necessarily an ordering of costs among
resources. One resource may be cheaper for one demand and more costly for another demand
when compared to other resources. The following definitions relate three different types of cost
orderings that may occur:
DEFINITION. Resource a ЄR has absolute advantage relative to resource b ЄR in use j ЄU if waj
< wbj, some j Є U.
DEFINITION. Resource a ЄR is more efficient (less efficient) than resource b ЄR if waj < wbj (waj
> wbj) for all j Є U.
DEFINITION. Resource a,b ЄR are the same resource class if wbj – waj = k for all j Є U and
some constant k. Furthermore, if k>0 (k=0, k<0) then resource a is a higher grade (same grade,
lower grade) of the resource class than resource b.
7
A resource that is more efficient is strictly cheaper for all demands. Thus efficiency implies
Ricardian absolute advantage in all uses. By resource class we mean intuitively that we can
classify different resources as different types of “coal”, “oil”, “gas”, etc. where these
classifications imply equivalence among demands. The difference between resources within any
class is only cost – higher grade resources have a lower cost and this difference in cost is the
same for all demands. Note that this classification is based on the economic properties of the
resource, not its chemical properties. It may be that resources that are economically the same
have completely different chemical compositions.
The next two propositions generalize the principle of cost-ordered royalties.
PROPOSITION 4. More efficient resources have a larger royalty.
PROOF: Let waj<wbj for resources a,b ЄR and all demands j Є U. Suppose that λa ≤ λb. Then from
(6), pj(t) ≤ waj + λa(t) < wbj + λb(t) for all t Є (t0,∞). Thus dbj(t)=0 for all t Є (t0,∞) and j Є U;
resource b is never extracted for any demand. Since this contradicts the Lemma, the supposition
is false and thus λa > λb. Q.E.D.
PROPOSITION 5. Higher grade resources have a larger royalty.
PROOF: If resource a Є R is a higher grade of the same resource class as resource b Є R, then by
definition, wbj – waj = k >0 for j Є U which implies that waj < wbj for j Є U. Then from
Proposition 4, resource a has a larger royalty than resource b. Q.E.D.
With homogenous demand, the Herfindahl principle states that resources are extracted,
sequentially, in order of cost. The following two propositions generalize this principle to show
that the use of resources is always in order of net cost and that resources are extracted by
decreasing grade.
PROPOSITION 6. Resources are supplied for a given demand in order of increasing net cost.
PROOF. We show that if a resource is supplied for a given demand then a lower net cost resource
will not subsequently be supplied for that demand. Thus resources supplied for a given demand
must be in order of increasing net cost.
Let daj(t1)>0 and waj>wbj for resources a,b Є R and demand j ЄU at time t1 Є (t0,∞). Then from
(6),
8
waj   a ( t1 )  p j ( t1 )  wbj  b ( t1 )
(10)
which since waj > wbj implies that λa < λb. Then from (5), a  b and so the left hand side of
(10) increases more slowly than the right hand. Thus
waj   a ( t )  wbj  b ( t ) for all t Є (t1,∞).
Then from (6), pj(t) ≤ λa(t) + waj < λb(t) + wbj for all t Є (t1,∞) and so dbj(t)=0 for all t Є (t1,∞).
Q.E.D.
Note that Proposition 6 does not say that all resources will be supplied for each demand but that
of those resources that are supplied, their use will be in strict order of increasing net cost. In the
special case of a single demand, resource use is identical to resource extraction and Proposition 6
reduces to the Hefindahl principle.
PROPOSITION 7. Resources of the same resource class are extracted in order of decreasing
grade.
PROOF. We show that if one resource is being extracted, than a higher grade of the same
resource will not be subsequently be extracted. Thus resources of the same class are extracted in
order of decreasing grade.
Let q a ( t1 )  0 and q b ( t1 )  0 for resources a,b Є R at time t1 Є (t0,∞) where resource b is a
higher grade of the same resource class as resource a. By the last inequality, from (4) there exists
c Є U such that dac(t1)>0. Then from (6),
wac  a ( t1 )  pc ( t1 )  wbc  b ( t1 )
(11)
which since wbc<wac, from the definition of higher grade, implies that λa(t1) < λb(t1). Then from
(5), a  b , the left hand side of (11) increases more slowly than the right hand side, and so wac
+ λa(t) < wbc + λb(t) for all t Є (t1,∞), which since wbj – waj is constant for all j Є U (from the
definition of resource class) implies that waj + λa(t) < wbj + λb(t) for all t Є (t1,∞) and j Є U.
9
Combining with (6) yields pj(t) ≤ waj + λa(t) < wbj + λb(t) for all t Є (t1,∞) and j Є U which
implies that dbj(t)=0 for all t Є (t1, ,∞) and j Є U. Then from (4),
q b ( t )    d bj ( t )  0 for all t Є (t1, ,∞). Q.E.D.
jU
Note that if there is a single resource class, all demands can be aggregated into one composite
demand and Proposition 7 reduces to the Herfindahl Principle. Since Proposition 7 demonstrates
that deposits within a resource class will be extracted in strict order of grade, we can aggregate
resource grades and consider the resulting composite resource that has an extraction cost function
that increases with cumulative extraction. This provides a microeconomic foundation for
resources with rising, cumulative extraction cost functions, used widely in the literature (e.g,
Heal, 1974).
Since demand is positive at all prices, there will always be some resource available for each
demand. The following proposition provides a condition under which all resources except one
will be exhausted.
PROPOSITION 8. A resource that is less efficient than all other resources will eventually be
used exclusively for all demands.
PROOF. Let resource a Є R be less efficient than all other resources. Then from Proposition 6, λa
< λi for all i Є R – {a}. Since royalties rise exponentially, there exists a time ta Є (t0,∞) such that
waj + λa(t) < wij + λi(t) for all i Є R – {a}, j Є U, t Є (ta, ∞). Then from (6), dij(t)=0 for all i Є R
– {a}, t Є (ta, ∞) and so daj(t)>0 for j Є U, t Є (ta,∞). Q.E.D.
The least efficient resource in Proposition 8, if it exists, is of course the natural backstop resource.
It is the resource that will eventually be used when all other resources are exhausted. At the other
end of the spectrum, it is interesting to ask if a single resource could be used exclusively for all
demands at time t0. The next proposition demonstrates that if there is a resource that is very cheap
and or very plentiful, it may be used exclusively, i.e., for all uses at the beginning of the
extraction program.
10
PROPOSITION 9. A resource a Є R that is more efficient than all other resources will be used
s j  waj
exclusively at time t0 if
  D ( p ) /( p  w
jU s0  waj
j
aj
)dp  rq a ( t 0 ) where
s0 = min {wij - waj | i Є R - {a}, j Є U}, and
sj = max {wij - waj | i Є R, j Є U.
PROOF. Suppose that dac(t0)=0 for some c Є U. Then since demand is positive, there exists b Є
R such that dbc(t0)>0. Then from (6), wbc  b ( t 0 )  pc  wac  a ( t 0 ) which since wbc – wac ≥
s0 implies that a ( t 0 )  s0  b ( t 0 ) which from (8) yields
 a ( t )   a ( t 0 )e rt  ( s 0  b ( t 0 ))e rt  s 0 e rt  b ( t ) .
(12)
Since resource a is more efficient than any other resources then sj>0 for j Є U. Let
t̂ j  log( s j / s 0 ) / r for j Є U so that
s0ert > sj for t > t̂ j and j Є U.
(13)
Then from (12), (13) and the definition of sj,
waj  a ( t )  waj  so e rt  b ( t )  waj  s j  b ( t )  wbj  b ( t ) for t  t̂ j and j Є U. So
from (6),
daj(t)=0 for t  t̂ j , j  U .
(14)
Let γj(t) = waj + s0ert for j Є U so that
 j ( t 0 )  waj  s 0 , j ( t̂ j )  waj  s j , and  j  ( t )  rs 0 e rt  r(  j ( t )  waj ) for j Є U. (15)
From (12) and since λb(t)≥0,
waj  a ( t )  waj  s0 e rt  b ( t )  waj  s0 e rt   j ( t ) for j Є U.
11
(16)
From (6),
p j ( t )  waj   a ( t )  D j ( waj   a ( t ))  D j ( p j ( t ))  d aj ( t ) for j Є U and
p j ( t )  waj   a ( t )  d aj ( t )  0  D j ( p j ( t )) for j Є U, and therefore
D j ( waj   a ( t ))  d aj ( t ) for j Є U. Thus
s j  waj
 
jU s0  waj
Dj( p )
( p  waj )dp

 j ( t̂ j )

jU  j ( t 0 )
Dj( p )
( p  waj )dp
t̂ j

jU t0
jU t0
t̂ j
r   D j (  ( t ))dt
jU t 0
  D j ( waj  a ( t ))dt  r   d aj ( t )dt  rqa ( t 0 )
where the change of variables follows from (15), the first inequality follows from (16) and that
demand is downward sloping, the second inequality follows from (14) and (17), and the last
equality follows from (14).
Since (18) contradicts the premise, the supposition is false and so dac(t0) > 0. Then since c was
arbitrary, daj(t0) > 0 for j Є U. Q.E.D.
The following result specifies that if two resources have absolute advantage in specific demands,
then they cannot simultaneously be used in the demand where the other resource has the absolute
advantage. This result is helpful in narrowing the set of solutions as we will see in the illustrative
case with two resources and two demands in the next section.
PROPOSITION 10. If resource a (resource b) has absolute advantage over resource b (resource
a) in use j (use k), then the former cannot simultaneously supply the other demand, i.e., resource
a (resource b) cannot be extracted simultaneously for use k (use j), a, b Є R, j, k Є U.
PROOF. Let dak(t)>0, dbj(t)>0 for t Є I where I  ( t 0 ,  ) is an open interval. By the definition of
absolute advantage, waj < wbj and wbk < wak which yields wbk - wbj < wak - waj. Then from
12
Proposition (2), pk ( t )  wak  a ( t )  wbk  b and p j ( t )  wbj  b ( t )  waj   a ( t ) .
Subtracting the inequalities gives b  wbk  ( b  wbj )   a  wak  (  a  waj ) which
implies that wbk  wbj  wak  waj which contradicts the inequality following from the
definition. Q.E.D.
Comparative Dynamics
In order to investigate the comparative dynamics properties of the above problem, we can define
the value function for the above problem in (1)-(3) as

V ( Qi ( t 0 ))   B(( Qi ( t 0 ),t ))dt
(17)
t0
where
 d ij ( t )
B(( Qi ( t 0 ), t )  e
 rt
[
jU
i R
D
1
j
( x )dx   wij d ij ( t )]
0
iR jU
represents the discounted benefits from extraction at any given instant of time t. To obtain results
that relate the change in the resource shadow prices to the change in the aggregate resource
stocks, we need to establish the differentiability of the value function V (( Qi ( t 0 ),t )) . Benveniste
and Scheinkman (1979) have shown that under fairly general conditions, the value function V(∙)
is once differentiable. In particular, we can invoke their Corollary 1, when the optimal control is
piecewise continuous as in our case. It is straightforward to check that (17) satisfies the
assumptions of Corollary 1. This gives the following result:
V ( Qi ( t 0 ),t )) / Q0 ( t 0 )  i ( t 0 )
which implies that the initial shadow price is the derivative of the optimal value function with
respect to the initial stock of the resource. The next step is to establish that the value function is
twice differentiable. In general, our control functions are discontinuous since there may exist
intervals I such that qij(t)=0, t Є I and I  ( t 0 ,  ) . By proposition 6, and because we have only a
finite number of demands and resources, it is obvious that there can only be a finite number of
such intervals. At the switch points between these intervals, the state variables may not be
13
differentiable and the control functions may be discontinuous. These discontinuities together form
a set of measure zero. From Epstein (1978) we know that the Le Chatelier Principle applies, and
V(  ) is a positive semi definite matrix and therefore
i ( Qi ( t 0 ),t 0 )) / Qi ( t 0 )  0 .
The above implies that as the stock of any given resource increases, its shadow price decreases. In
the limit,
lim i ( t )  0 which yields
Qi ( t0 )
lim pij ( t )  wij  lim i  wij . This result
Qi ( t0 )
Qi ( t0 )
allows us in the next section to examine the sensitivity of the solution to exogenous shocks to the
stock of resources.
3. Characterization of the Special Case of Two Demands and Two Resources
We now consider the simplest possible setting, that of two demands and two resources. This
allows us to see how the extraction path changes as the parameters of the examine fix ideas,
following Chakravorty and Krulce (1994), let us assume that the demands are electricity and
transportation, and the resources are oil and coal. There are three cases to consider as follows: (i)
Oil is more efficient than coal; (ii) Coal is more efficient than oil, and finally, (iii) Oil (coal) has
absolute advantage in transportation (electricity).
Cases (i) and (ii) can be characterized by the following proposition which generalizes Proposition
2 of Chakravorty and Krulce (1994) to the case when neither resource is efficient:
PROPOSITION 11(a). Under Assumption 1 and
CC VCT VOE VOT

CC VCE
DT ( p )
dp  rQO ( t o ) ,
p( t )  ( CO  VOE )
oil must be used for both uses at the beginning.
CO VOE VOT VCE
(b). If coal is more abundant than oil, and

CO VOT
DE ( p )
dp  rQC ( t o ) , coal
p( t )  ( CC  VCT )
must be used for both uses at the beginning,
where CO (CC) denote unit extraction costs of oil (coal), and VOE and VOT (VCE and VCT) denote
conversion costs of oil (coal) to electricity and transportation, respectively.
PROOF. We only prove part (a) since the proof of part (b) is similar.
14
Define  E ( t )  pOE ( t )  pCE ( t ) and T ( t )  pOT ( t )  pCT ( t ) . Then
 E ( t )  T ( t )  ( pOE  pOT )  ( pCT  pCE ) = ( vOE  vOT )  ( vCT  vCE ) 
( vCT  vOT )  ( vCE  vOE )  k . From the absolute advantage of coal (oil) in electricity
(transportation), CC  vCE  CO  vOE and CO  vOT  CC  vCT , subtracting one from the
other gives k>0. Thus
 E ( t )  ( O  CO  vOE )  ( C  CC  vCE )  ( O  C )  ( CO  CC )  ( vOE  vCE )
 ( O ( t 0 )  C ( t 0 ))e rt  ( CO  CC )  ( vOE  vCE )  0 since o ( t 0 )  C ( t 0 ) and coal has
absolute advantage in electricity, i.e., wCE > wOE. It is clear that ΦE(t) is continuous and
E ( t )  0 . Now assume that ΦE(t0) >0. Then since ΦE(t) is monotone increasing,
 E ( t )  pOE ( t )  pCE ( t )  0 for all t ε (t0,∞) this implies that qOE(t)≡0. Thus  E ( t 0 )  0
implies from above that O ( t 0 )  C ( t 0 )  ( CC  CO )  ( vCE  vOE ) so that
 E ( t )  ( CC  CO )( e rt  1 )  ( vCE  vOE )( e rt  1 )  ( CC  CO  vCE  vOE )( e rt  1 ) .
Define t N  (log
CC  CO  vCT  vOT
) / r . For t ≥ tN,
CC  CO  vCE  vOE
 E ( t )  ( CC  CO  vCE  vOT )( e rt  1 )  k so that
N
T ( t )  pOT ( t )  pCT ( t )   E  k  0 hence qOT(t) ≡ 0 for t ≥ tN. Let
 ( t )  ( CC  CO  vCE  vOE )e rt  ( CO  vOE ). Then  ( t O )  CC  vCE ,
 ( t N )  CC  vOE  vCT  vOT and  ( t )  r(  ( t )  ( CO  vOE )). Finally,
CC  vOE  vCT vOT

CC  vCE
(t )
N
N
DT ( p )
DT ( p )
DT (  ( t ))
dp  
dp  
 ( t )dt
p( t )  ( CO  vOE )
 ( t )  ( CO  vOE )
 ( t0 ) ( p  ( C O  vOE )
t0
tN
tN

t0
t0
t0
t
r  DT (  ( t ))dt  r  DT ( pOT ( t ))dt  r  qOT ( t )dt  rQ0 ( t 0 ) .
Proposition 11(a) suggests that oil will only be used at the beginning if it is abundant, the
discount rate is high, the unit extraction cost of coal (oil) is high (low) and the demand for
transportation is low. Ceteris paribus, if either resource has weak absolute advantage, then it
increases the likelihood of oil being used exclusively at the beginning. Oil has weak absolute
advantage if vCT is relatively low and vOT is high. Similarly, the absolute advantage of coal over
15
oil is weak if vOE is low and vCE is high. More generally, strong absolute advantage in either
resource leads to specialization and in that case, it is unlikely that a single resource would be used
for all uses at the beginning. Notice that the Chakravorty-Krulce condition (their Proposition 2(b)
emerges as a special case of the above if we substitute vCE = vOE = vOT = 0 and vCT = z.
Fig.1 graphs the φE(t) and φT(t) functions over time. When φE(t)<0 (φT(t)<0), oil has comparative
advantage in electricity (transportation). When they are positive, coal has comparative advantage
in both uses.
When each resource has absolute advantage (case (iii), there are four possible solutions in the two
by two case, as shown in Table 1. However, by Proposition 10, we can immediately exclude the
solutions (a) and (d) since both imply that there will be simultaneous extraction of oil for use in
electricity and coal in transportation. Thus the possible solutions are cases (b) and (c). We have
thus proved the following result:
Proposition 12. When each resource has absolute advantage in a given demand, the two by two
model has two only solutions as given by Table 1(b) and (c).
Effect of Exogenous Discoveries of a Resource
Without loss of generality, consider the interesting case of exogenous, unexpected discoveries of
oil. From the earlier comparative dynamics results, as more oil is discovered, its shadow price
will fall over the entire planning horizon, and will approach zero in the limit, when oil becomes
an “inexhaustible” backstop resource.
Oil thus moves from being “abundant” to being an inexhaustible resource. Figs 2 and 3 show the
solution with different stocks of oil. When oil is abundant, it is used exclusively for all uses at the
beginning of the planning horizon. Then by Proposition 12, coal must be used for both uses at the
end. However, when oil becomes inexhaustible in the limit, coal is preserved for use in electricity
at the beginning of the planning horizon and oil is used for both uses at the end. Thus we can state
the following result:
Proposition 13. As the stock of oil increases without bound, it goes from being used exclusively at
the beginning of the planning horizon to being used exclusively at the end.
16
This result is important because in traditional resource extraction models with one demand, as the
stock of a resource increases through new discoveries, its shadow price falls, leading to increased
extraction at each time period. In a multiple demand framework, this result may be violated.
4. Concluding Remarks
This paper extends the standard theory of exhaustible resources to multiple demands in an infinite
horizon framework. Increasing the dimensionality of demand allows resources to possess absolute
advantage in specific demands and in all demands. Several interesting results obtain. For
example, if two resources simultaneously supply the same demand, they must have the same
royalty and net cost. The Herfindahl Principle of “least cost first” holds weakly under each
individual demand but may not hold under all demands. Conditions under which a resource may
be used at the end or at the beginning are developed for a arbitrary number of resources and
demands. Comparative dynamics results suggest that exogenous increases in the stock of a
resource may result in shifting the use of the resource from the beginning of the planning horizon
to the end, a result that is in sharp contrast to Hotelling models with one demand, where
discoveries increase the use of the resource but do not displace their use over time.
Although this paper deals with the concept of absolute and comparative advantage for exhaustible
resources, the concept may be quite applicable to other areas such as international trade. If the
shadow cost of production for tradable commodities changes over time, transportation costs could
be thought of as conversion costs. In that case, one could apply the above theorems to determine
the shifting pattern of specialization of a country in a multi-country, multi-good world. In the case
of trade in two goods between two countries, the results in this paper may be directly applicable.
17
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18
Table 1. Possible solutions for the case when oil has absolute advantage in transportation and coal
in electricity. Cases (b) and (c) are ruled out by Proposition 10.
E T
E T
E T
E T
O O
O O
C
C
C C
O C
C
O
C
O
O C
C C
C
C
O
O
C C
(a)
(b)
(c)
19
(d)
ΦE(t)
ΦT(t)
t0
t1
t2
k
Fig.1. The premium k represents the absolute advantage of
oil in transportation relative to coal in electricity.
20
time t
pOE
pCE
$
POT
PCT
tr
tS
time t
Fig.2. Extraction profile with abundant but exhaustible reserves of oil:oil is
used exclusively at the beginning and coal at the end
21
pCE
$
pCT
pOE
pOT
E
tS
time t
Fig.3.Extraction profile with inexhaustible reserves of oil: coal is preserved
for electricity while oil is used at the end.
.
22