American Economic Review 2008, 98:3, 1128–1144
http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.3.1128
Ordering the Extraction of Polluting Nonrenewable Resources
By Ujjayant Chakravorty, Michel Moreaux, and Mabel Tidball*
Energy markets are often characterized by the use of multiple nonrenewable resources, such
as coal and natural gas, with different pollution characteristics. The same resource may have
different qualities, as when high- and low-sulfur coal is used to generate electricity. Multiple
resources are used in providing transportation services, such as cars running on gasoline or plugin hybrids that may be fueled by electricity produced from coal.
When these nonrenewable resources contribute differentially to pollution, what is the optimal sequence of extraction over time? Harold Hotelling (1931) developed the classical theory of
extraction of a nonrenewable resource. Orris C. Herfindahl (1967) extended the Hotelling model
by considering many resources with different unit costs of extraction and proposed the “least
cost first” principle: extraction must be ordered by cost, with the cheapest resource used first.
Others have examined whether Herfindahl’s model remains valid in a variety of situations: in a
general equilibrium setting (see Murray Kemp and Ngo V. Long 1980; Tracy R. Lewis 1982), in
the presence of setup costs (Gérard Gaudet, Moreaux, and Stephen W. Salant 2001), under heterogeneous demands (Chakravorty and Darrell L. Krulce 1994), and when the extraction rate is
constrained (Jean-Pierre Amigues et al. 1998; Stephen P. Holland 2003).
In these studies following Hotelling and Herfindahl, resources were differentiated by cost
alone. In this paper, we abstract from cost considerations and focus on how the sequence of
extraction may be affected when resources are differentiated only by their pollution characteristics. We consider two resources, one more polluting than the other. Without loss of generality,
consider coal to be the dirty resource and natural gas the clean one. Both may be used to generate
electricity. Environmental regulation is imposed through a cap on the stock of pollution. This
may be a stylized approximation of an international agreement such as the Kyoto Protocol, which
aims to stabilize the concentration of greenhouse gases in the atmosphere.
With heterogeneity in pollution, the results are nonintuitive and do not depend on the emission rates of the resources. When the economy is already at its allowable stock of pollution, the
clean natural gas is used first and use of the dirty coal is postponed to the future. This is what
one would expect, armed with insights from the Herfindahl model. However, when the economy
starts from below the ceiling and accumulates pollution, coal may be used first and use of the
clean natural gas is postponed. The optimal strategy is to benefit from natural dilution by building the stock of pollution as quickly as possible. This is done by burning coal rather than natural
* Chakravorty: Department of Marketing, Business Economics and Law, and Department of Economics, University
of Alberta, Edmonton, Alberta, Canada T6G 2R6 (email: [email protected]); Moreaux: IDEI and LERNA,
University of Toulouse I (Sciences Sociales), 21 allée de Brienne, 31000 Toulouse, France (e-mail: mmichel@cict.
fr); Tidball: INRA-LAMETA, 2 Place Viala 34060–Montpellier cedex 1, France (e-mail: [email protected].
fr). We would like to thank Minhnhat Ho and Valérie Nowaczyk for excellent research assistance and two anonymous
referees of this journal for suggestions that substantially improved this paper. Seminar audiences at the Université
Catholique de Louvain (CORE), Georgia Tech, Iowa State, UCF, Montreal Resource Economics Workshop, World
Congress of Environmental Economics at Kyoto, the EAERE 2007 Conference in Thessaloniki. and SURED 2006 in
Ascona provided valuable comments.
Regulation in the form of a ceiling on the stock of pollution has also been used by Chakravorty, Bertrand Magne,
and Moreaux (2006) although they do not consider multiple nonrenewable resources. Their focus is on the effect of
nonstationary demand on the build-up of pollution and use of a clean backstop resource.
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gas. This phenomenon is counterintuitive to the spirit of Herfindahl—the inferior resource is
used first. Only when natural gas is abundant is it used before coal.
We also obtain a “preference reversal” over resources. That is, coal may be used for a period of
time, then natural gas, and finally coal for another time period. Such phenomena with complete
switching between resources does not occur in models with multiple nonrenewable resources.
Several papers report the joint extraction of a low- and a high-cost resource under cost heterogeneity (Kemp and Long 1980; Chakravorty and Krulce 1994), capacity constraints (Amigues et
al. 1998; Holland 2003), but not a reversal. We show that it may be efficient to extract the dirty
resource first, then the clean resource only until exhaustion, and, finally, the dirty resource.
The pattern of extraction is highly sensitive to the initial endowments of the two resources. If
the stock of coal is relatively low, then the Hotelling rents of the two resources are exactly equal
and regulation is never binding. If, however, coal is abundant, it has a lower Hotelling rent than
natural gas. With abundant resources, extraction paths have a turnpike feature in which both
resources are jointly extracted at the maximum allowed level. We show that all paths must pass
through this turnpike.
Stylized facts suggest that our planet has abundant supplies of coal but limited amounts of
natural gas. Our results imply that if the economy is below the regulated stock of emissions, then
it is optimal to use the more polluting resource first and get to the ceiling. Once the ceiling is
achieved, we must use the cleaner fuel first. From a policy point of view, we thus obtain a perverse result: imposing a cap on the stock of pollution may trigger a race to the ceiling by burning coal. These results suggest that although there is pressure today to switch to cleaner energy,
relying on abundant polluting resources such as coal is not unreasonable, given positive discount
rates and the expected decrease in the standard of living once the threshold level of pollution is
achieved. Section I extends the textbook Hotelling model with two costless but polluting resources.
Section II develops intuition by considering the model with a single polluting resource. Sections
III and IV characterize the sequence of extraction with two resources. Section V concludes the
paper.
I. The Hotelling Model with Two Polluting Resources
Consider an economy in which energy consumption at any time is given by q and the corresponding gross surplus u is twice continuously differentiable, strictly increasing and strictly
concave over the interval 30, q– 2 with q– . 0 and constant over the interval 3q–, `2 with u 1q 2 5 u– for
q $ q–. This implies that limq c q– u91q 2 5 0. To avoid a corner solution, let limq T 0 u91q 2 5 1`.
Denote the marginal surplus by p 1q 2 ; u91q 2 and the corresponding demand function by d 1 p 2.
Welfare W is the sum of the gross surplus discounted at some constant rate r . 0 given by W 5
`
e0 u 1q 2 e2rtdt. Since u 1q 2 # u–, this integral is well defined.
Strictly speaking, in our model both resources have zero extraction costs; thus, Herfindahl’s model predicts indifference in the ordering of extraction. In a model with cost differences, however, Herfindahl predicted that the “good,”
or the low-cost, resource must be used first. In our model with differences in emissions between resources, the “bad”
resource, coal, may be used first. In this sense, the spirit of Herfindahl is violated. We thank an anonymous referee for
making this observation.
In a parallel effort, Edwin van der Werf and Sjak Smulders (forthcoming) develop a model to examine the extraction of heterogeneous resources when the flow of emissions (not the stock, as in our case) is constrained. In their model,
resources are imperfect substitutes. Their findings suggest that the economy may use more of the clean resource before
the constraint is binding in order to use more of the dirty resource during the period when the constraint binds. It may
be useful at a later stage to develop a more general framework in which the effect of different types of regulation (stock
and flow constraints) on the extraction path is better understood.
To keep the notation simple, we avoid writing the time argument t whenever the context is clear.
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We consider two nonrenewable resources indexed by i 5 1, 2, which are perfect substitutes in
demand. Each resource is characterized by the vector 5ui, X 0 i 6, where ui is the pollution generated
by burning one unit of the resource i, and X 0 i is its given initial stock. Let Xi be the residual stock
at time t and xi the extraction rate. Then,
· Xi 5 2xi , i 5 1, 2.
(1) Let the cost of extraction of both resources be zero. Without loss of generality, let us assume
that resource 1 (say, natural gas) is cleaner than resource 2 (coal), 0 , u1 , u2. Let X 0 5 X 01 1
X 02 be the total initial stock, X ; X1 1 X2 the total residual stock at time t, and x ; x1 1 x2 the
aggregate extraction at t.
As in most Hotelling models, we assume an abundant renewable backstop resource (e.g., solar
energy) that is nonpolluting. Its unit cost is given by cr . 0. Let y be the rate of extraction of the
backstop resource. Once coal and natural gas are exhausted, the price of the renewable resource
is equal to cr and its consumption is determined by the solution to the equation u91y 2 5 cr .
Burning of the fossil fuels increases the aggregate stock of pollution denoted by Z 1t 2. This
may be the level of carbon in the atmosphere. We assume that there is a natural decay at the
constant rate a . 0, so that
(2)
·
–
Z 5 a ui xi 2 aZ, Z 102 5 Z 0 # Z given,
i
–
where Z 102 5 Z 0, and Z is the regulated limit on the stock of pollution. This may be exogenously
imposed by some regulatory authority or the outcome of a negotiated international agreement. It
may also approximate a specific form of a damage function that equals zero at stock levels below
–
Z but imposes prohibitive damages beyond that threshold value. The inequality in (2) suggests
that the initial level of pollution is below the ceiling.
–
Suppose only resource i is being used when the stock of pollution equals Z over a time period.
–
–
–
–
Let x i be this maximum allowed extraction rate. By (2), x i 5 aZ / ui, and p i , the corresponding
price of energy, is given by p–i 5 u91x–i 2. Since natural gas is less polluting than coal, more gas
could be used at the ceiling, so x–1 . x–2 and p–1 , p–2 . We assume that cr . p–2 . That is, when
the ceiling is binding, it is cheaper to use the polluting resource, if available, than the clean
renewable.
The constrained social planner maximizes the net surplus by choosing extraction rates of
natural gas and coal and the backstop resource at each time t as follows:
(3)
Max
5x i, i51, 2; y 6
2rt
3 eu a a xi 1 yb 2 cr yf e dt,
`
0
i
This natural decay aZ is an increasing function of the stock Z. The linear form is chosen for analytical convenience. Naturally occurring “carbon sinks,” such as sea water and plants during their growth process, remove carbon
from the atmosphere. According to the Global Change Research Information Office (GCRIO 1997), were it not for
human induced carbon emissions, natural additions of carbon in the atmosphere would be approximately balanced by
the action of these carbon sinks. The assumption that total decay of pollution depends positively on the stock of carbon
Z is important for our results, as discussed later in the paper.
– For example, a policy goal of 550 parts per million of carbon in the atmosphere may correspond to a specific value
of Z .
–
This is a social planner who maximizes welfare subject to the regulated level Z.
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1131
–
–
subject to (1) and (2), Xi0, Z 0, and Z given, with Z 2 Z 1t 2 $ 0. The corresponding current value
Lagrangian can be written as
–
L 5 u a a xi 1 yb 2 cr y 2 a li xi 1 m c a ui xi 2 aZd 1 n 3Z 2 Z 4 1 a gi xi 1 gr y,
i
i
i
i
–
where n is the multiplier attached to Z 2 Z $ 0, li is the Hotelling rent of resource i, m is the
shadow price of the pollution stock, and gi and gr are the multipliers for the nonnegativity constraints xi $ 0 and y $ 0. The first-order conditions are
(4) (5) u9 a a xi 1 yb 5 li 2 ui m 2 gi, i 5 1, 2, and
i
u9 a a xi 1 yb 5 cr 2 gr .
i
The dynamics of the costate variables are given by
(6) (7)
· l i 1t 2 5 rli which gives li 5 l0i ert, i 5 1, 2, and
m· 1t 2 5 1r 1 a 2 m 1 n,
where we write l0i for li 102. The complementary slackness conditions are
(8)
gi $ 0, xi $ 0, gi xi 5 0, i 5 1, 2
(9)
gr $ 0, y $ 0, gr y 5 0, and
–
–
n $ 0, Z 2 Z $ 0, n 1Z 2 Z 2 5 0.
(10)
Finally, the transversality conditions can be stated as
(11)
(12)
lim e2rtli 1t 2 Xi 1t 2 5 l0i lim Xi 1t 2 5 0, i 5 1, 2, and
t c` t c`
lim e2rtm 1t 2 Z 1t 2 5 0.
t c`
Condition (6) implies that the scarcity rent of coal and gas must rise at the rate of discount.
Define the price of resource i as its (full) marginal cost at time t, given by pi 5 li 2 ui m. It has
two components: the scarcity rent and the externality cost, the extraction cost being zero. Since
–
m is the shadow price of the pollution stock, it is negative or zero. If Z , Z over some initial time
0 1r1a 2 t
period, then by (10), n 5 0 and m 1t 2 5 m e
, where m 102 is written as m0. During this period,
the shadow price of pollution grows at an exponential rate given by the sum of the discount
rate and the dilution rate of the pollution stock. If regulation is never binding, then the stock of
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p­ ollution is never constrained in the model, which then becomes a pure Hotelling program. In
that case, the initial shadow price m0 is zero; hence, m ; 0.
It is convenient to split the analysis into two parts. In Section II we build intuition by considering the simple case of only one nonrenewable resource. In Sections III and IV, we characterize
the solution with two resources.
II. Extraction with Only One Resource
To understand how environmental regulation affects extraction, assume that there is only a
single resource i, which may be coal or natural gas. We adopt a recursive approach. First we
–
assume that the initial pollution stock is already at the ceiling, Z 0 5 Z. Next we discuss how to
–
–
0
go from an initial stock lower than the ceiling 1Z , Z 2 to the ceiling Z.
A. The Initial Stock of Pollution Is at the Ceiling
Since extraction is costless, the pure Hotelling (i.e., without binding regulation) resource
price pi 1t 2 is equal to the scarcity rent and is completely determined by its initial value l0i . The
Hotelling price is given by pi 1t 2 ; l0i ert. The initial scarcity rent is then a decreasing function of
the initial stock, X 0i . Let T be the time at which this price equals the cost of the backstop cr .
–
Since Z 0 5 Z , the lowest value of the scarcity rent such that regulation is nonbinding is given
0
–
by l i 5 pi . At scarcity rents lower than this level, initial resource extraction will lead to a pol–
lution stock higher than the regulated level Z, and the ceiling constraint will be binding over a
nonzero time period. If regulation is nonbinding, the resource price rises at the rate of discount
starting from some l0i greater than p–i . Define this cut-off extraction period as DHi , where the
superscript H stands for the Hotelling path. Let the cumulative extraction be given by X Hi . Thus,
X iH is the highest stock of resource i that generates a pure Hotelling path unconstrained by environmental regulation. Since p–1 , p–2 , the aggregate stock of coal that could be used up over this
Hotelling path must be lower than the corresponding stock of natural gas, X1H . X2H.
If the initial resource stock is higher than this maximal Hotelling stock, Xi0 . XiH, then at the
beginning, extraction must be limited to x–i ; otherwise the pollution stock will exceed the ceiling
(see Figure 1). When the resource stock declines to XiH, the Hotelling path begins. The duration
of the first phase is then given by Di 5 1Xi0 2 XiH 2/ x–i . Until time Di , the price is p–i . Since regulation is binding, the shadow cost of pollution m 1t 2 is strictly negative and (4) yields p–i 5 l0i ert 2
mui . The gap between the resource price p–i and the shadow price l0i ert in the figure is the externality cost per unit of the resource i, given by 2mui . Beyond Di , the ceiling is no longer binding;
hence, m ; 0 and both resource price and scarcity rents are equal, as shown. Note that the growth
rate of the marginal utility of consumption given by (4) is constant when the ceiling is binding,
but grows at the discount rate r when the ceiling is no longer binding.
–
This also holds if from some time t the constraint Z 2 Z is no longer tight and will not be any time in the future.
If, however, at some time t the ceiling does not bind but will do so in the future, then m , 0 and grows in absolute
value at the rate r 1 a. Because Hotelling scarcity forces the price of resources to rise over time, the quantity extracted
diminishes. So even though there may be a period when the ceiling binds, eventually there comes a point in time beyond
which the ceiling never binds. So beyond some point in time, the shadow price of pollution m must become zero and
stay there.
DiH
3 ln cr 2 ln pi 4
Then, DHi 5
and XiH 5 3 d 1 –p iert 2 dt.
r
0
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1133
L0i e R t
cr
p–i
MUi
L0i
0
Di
Di DHi
UJNF
Figure 1. Prices with One Abundant Resource
B. The Path from the Initial Stock of Pollution to the Ceiling
–
Now Z 0 , Z. Let l0i be the initial scarcity rent. Since it increases over time, emissions decline.
If emissions at scarcity rent l0i are lower than the natural dilution rate, then future emissions will
also be lower; hence, the stock of pollution will permanently decline. If initial emissions are
higher than the dilution rate, however, then the stock of pollution Z 1t 2 will increase, reach a peak
value, then decline steadily. The lower the value of l0i , the higher is the peak value of Z 1t 2. Thus,
there exists some value of the initial scarcity rent l0i 1Z 02 such that the peak Z 1t 2 exactly equals
–
the regulated ceiling Z, i.e., when l0i ert 5 p–i .10 In a Hotelling path, the pollution stock must be
off the ceiling both before and after this instant. Thus, l0i 1Z 02 is the lowest possible scarcity rent
that depends upon the initial stock Z 0. A lower initial rent will mean higher emissions, so the
path will be constrained by regulation and will no longer be Hotelling. A higher rent will imply
that the pollution stock will always be strictly below the ceiling. This scarcity rent l0i 1Z 02 must
increase with the initial stock of pollution Z 0. Let the cumulative extraction until the ceiling is
reached be given by XiH 1Z 02 , and D0i be the time it takes to get to the ceiling.11, 12 Then, for a given
Z 0, we must have X1H . X2H, since more of the clean fuel can be used on the way to the ceiling.
This also means that in this Hotelling model, the resource price (hence scarcity rent) for gas
must be lower than for coal 1l01 , l022 , since more gas can be used at each period. Let D Hi be the
time duration during which resource i is extracted along a Hotelling path. Gas prices are lower;
hence, it will take longer for it to reach the backstop price, so that D H1 . D H2 . For an initial stock
of resource i lower than the critical level XiH, the stock does not reach the ceiling at all.
When the stock exceeds XiH, the scarcity rent is lower and regulation binds over a nonzero time
–
period. The pollution stock rises from Z 0 to Z . The resource is then extracted at the maximum x–i H
until the Hotelling stock Xi remains and extraction declines gradually to zero. The price at the
ceiling is a constant p–i . Suppose it takes time D0i to go to the ceiling, and the phase at the ceiling
At this instant, x–i units of resource are extracted.
All variables are now a function of the initial stock of pollution Z 0, so we sometimes avoid writing it explicitly.
Variables such as XiH are defined for an initial
– stock and therefore are different from the case in Section IIA when the
initial stock of pollution was the constant Z.
12
0
When Di equals zero, the initial stock is at the ceiling, as in Section IIA.
10
11
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is of duration Di . In the first phase, the price of the resource rises from a lower level to p–i , and the
shadow price of pollution m is nonzero. Beyond time D0i 1 Di , the ceiling never binds again, so
the value of m is zero until infinity and extraction is pure Hotelling.
III. Extraction with Both Resources: Starting from the Ceiling
With two resources, we adopt the same approach. In this section, we assume that the initial
stock of pollution is already at the ceiling. Section IV extends this analysis to pollution stocks
starting from below the ceiling.
The problem now becomes a little more complex. We plan to show that starting from the ceiling, when the aggregate stock of the two resources is small, both coal and natural gas must have
the same scarcity rent and extraction is “pure” Hotelling, i.e., regulation is never binding. When
the stock of coal is small but natural gas is abundant, both resources have the same scarcity rent
but the price paths are non-Hotelling. When coal is abundant, resource rents are not equal and
the price paths are not Hotelling.
Consider that both resources are available, Xi0 . 0, i 5 1, 2. If X 0 # X2H, i.e., the aggregate
stock of the two resources is lower than the minimum of the two critical stocks, the solution must
still be the Hotelling path with initial extraction lower than the maximum allowed at the ceiling,
x–2 . The scarcity rents of the two resources are equal, and their extraction rates a matter of indifference.13 Regulation is never active, except possibly at the initial instant.
This is shown in 3X 01 , X 02 4 space in Figure 2, where each point represents an initial endowment of the two resources. Points A and B denote the stock X2H on each axis. In zone I, which is
the triangle bounded by the axes and the straight line AB with slope 21, the aggregate stock is
always lower than X2H and endowments in this zone yield a Hotelling solution in which only the
aggregate extraction is determinate, but not its composition.
Now let X2H , X 0 , X1H. The aggregate stock is higher than the maximum Hotelling-induced
stock of coal, but lower than that of natural gas. As X 0 approaches X1H, given a stock of natural
gas, we search for the maximum endowment of coal that generates a Hotelling price path, i.e.,
one unconstrained by regulation. In the limit, the pollution constraint can bind only at discrete
points in time but not over an interval of nonzero duration. Consider the extraction sequence in
which gas is consumed initially, followed by coal. In the first interval, as price increases because
of scarcity, extraction of gas falls below the critical x–1 so that the pollution stock declines from
–
–
the initial level of Z . When we switch from gas to coal, the stock is strictly lower than Z .
H
Therefore, we could extract an aggregate stock of coal strictly larger than X2 and still not violate the ceiling. Natural gas is consumed first to bring down the stock of pollution; then coal is
–
extracted at rates higher than x–2 to restore the stock of pollution to Z . Exactly at that instant, the
H
residual coal stock must equal X2 and the Hotelling path follows as before. Aggregate coal use
in this case is strictly larger than X2H.
There may be other such Hotelling paths along which both resources are consumed simultane–
ously while maintaining
the stock of pollution at less than or equal to Z , and again letting it rise
– to the ceiling Z , followed by a phase during which only coal is consumed. A polar case of such a
path is one in which both resources are used while the stock of pollution is maintained exactly at
13
The two resource stocks can be aggregated into a single stock. The extraction rates must satisfy x 1t 2 5 d 1l0 ert 2
D 1X 0 2
and e0
xi dt 5 Xi0, i 5 1, 2, where l0 is the common value of the initial scarcity rent of the two resources and
0 D1X 2 the duration of this extraction period.
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X2
1135
E
F
G
F
X
H
1
;POF *7
;POF 7
A
–0
X2
A
X2H
C
D
C
D
;POF **
;POF ***
;POF *
C
0
B
B
–
X 10
X2H
H
1
X
X1
–
–
Figure 2. Extraction Sequence Depends on Initial Endowments 1A9: X 01 1 X 02 5 X1H2
–
its maximal level Z in the first phase, followed by exclusive use of coal.14 The locus of maximal
resource stocks that lead to a Hotelling path is shown as curve AA9 in Figure 2.
– –
–
Point A and A9 represent the resource endowment vectors 10, X2H 2 and 1X 01, X 02,2 such that X 01
–0
H
1 X 2 5 X1 . As we travel on AA9 toward A9, we increase the stock of natural gas and the corresponding maximal stock of coal that can be used to remain in the Hotelling path. The path AA9
can not cross the 45˚ line through X1H because the aggregate stock would exceed the maximum
stock of natural gas compatible with the Hotelling path. Because the stock increases along AA9,
the common initial scarcity rent declines from p–2 at A to p–1 at A9.15
The shaded region zone II in Figure 2 denotes the set of initial stocks that generates a Hotelling
solution when the aggregate stock is higher than X2H . Each point on AA9 corresponds to a unique
extraction path.16 Extraction proceeds along the locus A9A toward A. Once A is reached, all natural gas is exhausted and the stock of coal equals X2H. In the next phase, only coal is extracted.
If the set of initial stocks is in zone II but strictly below the curve AA9, the amount of coal
available is less than the maximal amount required to preserve the Hotelling path, and there may
be some flexibility in sequencing the use of the two resources, since they are perfect substitutes
on any Hotelling path not constrained by regulation. For instance, optimal extraction beginning
at point C may involve use of both resources at a maximal rate while staying at the ceiling, followed by the exclusive use of coal, and finally the residual stock of natural gas. This is shown by
14
In Web Appendix A available at http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.3.1128, we show that this
property is satisfied by the path that maximizes the consumption of coal for any given stock of natural gas such that
0
X [ 1X2H, X1H 4 .
15
Let D12 denote the interval when the two resources are consumed jointly. From A to A9, D12 increases from zero to
1 ln p–2 2 ln p–12 / r.
16
This is shown in Web Appendix A.
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the path CC9C0O, where the segment CC9 is the translation of AA9 through point C. Since we are
in the strict interior of zone II, however, regulation does not bind; hence, some natural gas extraction at the beginning may be substituted by a compensating amount of coal later in time.17
The union of zones I and II is the area OAA9B9, the set of all solutions that are pure Hotelling,
i.e., unaffected by environmental regulation.
A. Only Natural Gas Is Abundant
–
This case is depicted by zone III in Figure 2, where X 0 . X1H and X 02 , X 02.18 The pattern
of resource use is a variation of the one described for Figure 1. Consider an initial vector of
stocks at point D. Since natural gas is abundant, it is used first at the maximal level x–i until D9 is
reached. The resource price is constant at –p1 . The aggregate stock at D9 equals X1H. From here,
the extraction sequence is similar to the pure Hotelling paths originating in zone II. The scarcity
rents of the two resources are equal and their common initial value is given by l0 5 –p1e2rD1 5
H
cr e2r 1D1 1 D1 2 , where D1 and D1H 15 3ln cr 2 ln p–1 4 /r 2 are the durations of the first and second
phases, respectively. The shadow cost of pollution m is nonzero until D9, and zero beyond. As we
see below, this is the only non-Hotelling path with equal scarcity rents for the two resources.
B. When Coal Is Abundant
If coal, the more polluting resource, is abundant, extraction is constrained because of environmental regulation. The scarcity rents of the two resources are no longer equal. It is easier to first
discuss the case when both resources are abundant (see zone IV in Figure 2), before considering
abundant coal with limited stocks of natural gas, the clean resource (zone V).
–
–
Endowments in zone IV are defined by X 01 . X 01 and X 02 . X 02, such as point E in Figure 2. As
–
–
in zone III, natural gas is used at the maximal rate x 1 until its stock is reduced to X 01 at point F9.
–0 –
0
The duration of this phase equals D1 5 1X 1 2 X 12 / x 1 . The corresponding price and scarcity rent
paths are shown in Figure 3. The vertical distance between a price path and its scarcity rent in the
graph represents the cost of the externality 2mui. During this first phase, the price of energy p–1 is
lower than that of coal, given by p2 5 l02 ert 2 mu2 . It is constant until F9 (Figure 2), when both
resource prices are equal and a period of joint use begins. Extraction proceeds along a path F9F
that is a vertical translation of A9A until the price of energy reaches p–2 at F. Denote this period
of joint use by D12.
When both resources are consumed jointly, xi . 0 and l i 5 0, i 5 1, 2. From (4), l1 2 mu1
5 l2 2 mu2, which implies that l1 2 l2 5 2m 1u2 2 u12; hence, m 5 21l10 2 l20 2 ert / 1u2 2 u12.
Thus, l10 . l20 . Moreover, p1 1t 2 5 p2 1t 2 5 ert 1l10 u2 2 l20 u12 / 1u2 2 u12. Prices rise at an exponential rate. Aggregate supply must equal demand, x1 1 x2 5 d 1l0 ert 2 , and the pollution stock stays
at the ceiling. From (2), u1 x1 1 u2 x2 5 u1 x–1 1 u2 x– 2 so that
(13) 17
x1 5
u 2 3 d 1 l0ert 2 2 x 2 4
u 1 3 x 1 2 d 1 l0ert 2 4
, x2 5
.
u2 2 u1
u2 2 u1
See Web Appendix B for further characterization of paths starting from zone II.
Detailed characterization of the paths originating from zone III and from zones IV and V defined below are given
in Web Appendix C.
18
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p
1137
L01 e Rt
L02 e Rt
cr
p1
p–2
p1p2
p2
p–1
M0
L01
#JOEJOH$FJMJOH
L02
/BUVSBMHBT
$1
$1 $12
#PUIDPBMBOEHBT
$PBM
$1 $12$2 T
$PBM 3FOFXBCMF
t
Figure 3. Prices when Both Resources are Abundant:
Starting from the Ceiling 1Note: pi, 5 li 2 mui, i 5 1, 22
· · Differentiating gives x·1 1t 2 5 3u2 / 1u2 2 u1 2 4 d 1l0 ert 2 , 0 and x·2 1t 2 5 23u1 / 1u2 2 u1 2 4 d 1l0 ert 2 .
0. The extraction rates move in opposite directions.
Natural gas is exhausted at location F. The cleaner resource has a higher scarcity rent, which is
to be expected given that both resources have the same (zero) extraction cost. Coal is used at the
maximum level (see Figure 3). Its price p–2 is constant until time D1 1 D12 1 D 2. This is followed
by the ­terminal phase when regulation is nonbinding and the price of coal follows a Hotelling
path p 1t 2 5 l2 5 l02 ert until it is exhausted and the backstop is used. The stock of coal at the
starting location E (in Figure 2) determines the precise path F9F, each of which is a vertical
translation of A9A.
–
Now consider zone V with smaller endowments of natural gas, specifically X 01 # X 01. The
first period of exclusive natural gas use in zone IV disappears completely. If the inequality is
strict, as in point G in Figure 2, extraction begins with both resources and a common price
in the interval 3 p–1 , p–2 4 that depends upon the location of G on the curve FF9. The higher the
endowment of natural gas, the closer is G to F9, and the closer is the initial price to p–1 . Joint
extraction proceeds along the curve GF until all gas is exhausted at F. The remaining phases
are as in zone IV.
– –
The demarcation of the different zones depends on Z . If Z is higher, point A moves up in Figure
2 and B9 moves to the right, enlarging the set of endowments that result in a Hotelling path (zones
–
I and II). On the other hand, if Z approaches zero, the Hotelling set shrinks toward the origin
until, in the limit, none of the resources may be used. The pollution content of the resources also
determines the set of Hotelling paths. If coal is more polluting and gas cleaner, point A will move
down and B9 will shift to the right, leading to a flat and elongated Hotelling set.
The main insight when regulation is binding at the beginning is that excess natural gas must
be used first. When both resources are used, natural gas extraction declines and that of coal
increases. This “turnpike” feature with joint use allows for a smooth transition from the clean
to the polluting resource until only coal is used. Loosely speaking, the order of extraction is
according to pollution content. The more polluting resource is used later. This trend is driven by
time preference. Using the cleaner resource allows for higher current extraction rates and higher
profits earlier in time. As we see below, however, this tendency need not hold when the initial
pollution stock is below the regulated level.
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IV. Extraction with Both Resources: Starting from Below the Ceiling
For the stock to build up to the ceiling, there must be a period when emissions are strictly
greater than the maximum allowed at the ceiling. We adopt the same approach as before, differentiating between resource endowments that lead to an unconstrained Hotelling path and
those that do not. However, the analysis in this section has an added dimension, the initial stock
of pollution Z 0. This pollution stock, together with the endowment of resources, determines the
approach to the ceiling. Now all variables are also a function of Z 0.
As in Section III, if Hotelling paths were to hold, for initial aggregate stocks lower than X2H,
regulation will not bind, so which resource is extracted is a matter of indifference. This solution
is shown by stocks in zone I in Figure 4, which is a generalization of Figure 2 for a given Z 0. This
new zone I is strictly larger than the one in Section III, since it must include additional resource
stocks that take the initial stock of pollution to the ceiling.
For aggregate stocks X 0 [ 3X2H, X1H 4 , we may obtain pure Hotelling paths as in Section II,
provided the stock of coal is not too large. The Hotelling set corresponding to zone II in Figure 2
is given by figure KLL9MN, where K 5 1X2H, 02 , L 5 10, X2H 2 , L9 is located vertically above A9, M
– –
5 1 X1H 2 X 02, X 02 2 , and N 5 1 X1H, 02. These vectors are all functions of the initial stock of pollu–
0
tion Z . As Z 0 approaches Z, the curve LL9 approaches AA9 and the segment L9M collapses to the
vertex A9. The new zone II reduces to the one in Figure 2.
Consider an initial endowment F 5 1F1, F22 located on the AA9 curve. From Section III, we
already know the resource use profile beginning from F where the pollution stock is at the ceiling. With some abuse of notation, let the current resource price at F be given by p 1F2 , which must
lie in the interval 1 p–2, p–1 2. Suppose only coal is used to go to the ceiling. As before, let D0 denote
the time taken from an initial endowment F9 to reach the ceiling at F. Then the price path beginning from F9 is shown in Figure 5.19 Since all paths are Hotelling in the new zone II, regulation
does not bind, i.e., m 1t 2 is identically equal to zero everywhere. Both scarcity rents are equal, as
shown in the figure. Intuitively, natural gas has the same marginal value as coal, since being a
cleaner fuel confers no additional benefit.
The curve LL9 in Figure 4 is the locus of all such points F9 that map to points F lying on the
curve AA9. However, LL9 is not a vertical translation of AA9. Since p 1A92 5 p–1 , p–2 5 p 1A2 and
both L and L9 are points with the same stock of pollution Z 0, resource prices must be lower starting from L9 relative to L since L9 has a bigger endowment. This will lead to higher coal use and
higher emissions starting from L9 than from L. Thus the vertical distance F9F is smaller, the closer
F is to A9. From the ceiling at F, extraction follows the sequence described in Section II.20
Points on the segment L9M (in Figure 4) can be explained in the same fashion. At vertex M
with aggregate endowment X1H, only natural gas is used during the transition to the ceiling at
point A9. Since this is again a Hotelling path, both resource rents are equal. Consider an intermediate point J on the path A9M, where, by construction, the stock of pollution is below the ceiling.
Starting from M, only gas is extracted to arrive at J. But starting from J9, only coal is extracted.
Since both these paths must start from pollution stock Z 0, follow a Hotelling path, and arrive at
the same stock of resources and pollution, more natural gas needs to be used than coal. Thus,
traveling to J will take longer from M than from J9. This explains why point J9 is located to the
left of the extension of the line NM, which is the locus of points X 01 1 X 02 5 X1H.
0
–
t
This path solves Z 1D 02 5 Z, where Z 1t 2 5 Z 0 e2at 1 u2 e0 d 1p 1F2 e2r 1D 2t2 2 e2a 1t2t2 d t, and F92 5 F2 1
dt 2 , where F9 5 1F91, F922 5 1F1, F922 .
e
0
This is a Hotelling path with l0 5 l10 5 l02 5 p 1F2 e2rD and m 5 0, y 5 0 for t [ 30, D 0 4 . To show that the extraction path F9FAO is optimal, it is easy to check that all the necessary conditions are satisfied. For t $ D 0, the analysis
from Section III applies.
19
0
D0
d 1p 1F2 e2r 1D 2t 2
0
20
Chakravorty et al.: Extraction of Polluting Resources
VOL. 98 NO. 3
1139
X2
4UPDLPGDPBM
X1HZ 0
H
2
L
X Z –0
X2
0
X2H
L
F
J
M
A
F
Q
J
A
;POF ***
;POF **
G
G
;POF *
B
0
H
2
X
K
H
2
B
X Z 0
H
1
X
4UPDLPGOBUVSBMHBT
N
H
1
X Z 0
X1
Figure 4. Hotelling Paths Starting from Below the Ceiling
In the area of zone II above the line AA9M, coal is used first. For example, from F9, only coal is
used until the ceiling is reached at F followed by simultaneous extraction. For points located on
the line J9J, coal is used first to point J, followed by natural gas, until the ceiling is attained at A9.
Why will coal, the more polluting resource, be used before natural gas on the way to the
ceiling? Consider two instants of time t1 and t2 with t1 , t2. Since the two resources are perfect
substitutes and their scarcity rents are equal in zone II, suppose one unit of coal is used at t1
and a unit of gas later at t2. Then, at some time t . t2, the increment to the pollution stock will
be 3u2 e2a 1t22t12 1 u1 4 e2a 1t2t22 . Now, consider the alternative—using a unit of gas at t1 and coal
later at t2. The addition to pollution is 3u1e2a 1t22t12 1 u24 e2a 1t2t22 . Subtracting the latter from the
former gives 1u2 2 u12 1e2a 1t2t22 2 12 e2a 1t2t22 , 0. When coal is used first, the end pollution stock
is smaller! Thus, if both coal and gas must be consumed before the ceiling is achieved, coal is
used first because it generates more “free” dilution of the pollution stock. This policy allows for
an overall higher consumption of energy.
A. Only Gas Is Abundant
–
Here, the initial coal stock is less than X 02, while the aggregate stock is larger than X1H, shown
as zone III in Figure 4. Consider a starting point G. Natural gas is used until the pollution stock
reaches the ceiling. The ceiling is achieved at point G9, before arriving at the frontier A9B9.21
21
If the stock of gas is sufficiently large, the point G9 may lie to the right of the line segment MN.
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THE AMERICAN ECONOMIC REVIEW
$
L0 e R t
cr
p–2
p F
L0
$FJMJOH
0
$0
$0 $12
$0 $12 $H2
UJNF
Figure 5. Hotelling Price Paths in Zone II (l1 5 l2 5 l2
Next, gas is extracted at the maximum level x–1 until the stocks reach the boundary A9B9 of zone
II from Section II. The remaining path is Hotelling. Shadow prices of the two resources are still
equal.
B. When Coal Is Abundant
There are three alternative paths to the ceiling with abundant coal, given by a stock located
higher than the line LL9MQ shown in Figure 6. These are demarcated by zones IV, V, and VI. It is
easier to discuss them in reverse order, starting with zone VI. Coal is abundant with only a small
–
stock of natural gas lower than X 01. We know from Section III that once the ceiling is attained,
both resources are used until natural gas is exhausted. Consider a point R where the stock of pollution is exactly at the ceiling. Then R must be on some path HH9 that is parallel to AA9 and the
sequence of extraction from R is well defined.
Let p 1R2 be the price of energy at R and D0 be the time at which the ceiling is achieved exactly
at R. Since both resources are used beginning from this location, equation (4) implies that p 1R2
5 p 1D02 5 li 1D02 2 m 1D02 ui , i 5 1, 2 with p 1R2 [ 1 p–1, p–2 2. Since u2 . u1, we have l1 1D02 .
0
l2 1D02. Before D0 the ceiling is nonbinding, so m 1t 2 5 m 1D02 e2 1 r1a 2 1D 2t 2 . Then, for t , D0, 1l1 1t 2
0
0
2 m 1t 2 u1 2 2 1l2 1t 2 2 m 1t 2 u2 2 5 1l1 1D02 2 l2 1D02 2 e2r 1D 2t2 2 m 1D02 1u1 2 u22 e2 1 r1a 2 1D 2t 2 . 0.
That is, before location R, the marginal cost of coal is lower than that of gas. Thus, only coal is
used until regulation binds.
There exists a vector of stocks R9 (see Figure 6) with an initial stock of pollution Z 0 such that
coal is used until R. Once at R, both resources are used jointly along the curve H9H as in Sec­
tion III. Conversely, any starting vector R9 in zone VI will imply an exclusive use of coal until the
ceiling is reached. Because natural gas is not abundant, it is relatively expensive, hence coal is
used on the “fast track” to the ceiling. As there is a small stock of natural gas, it is more efficient
to use it to relax the consumption constraint at the ceiling than for slowing down the growth of
the pollution stock before the ceiling is achieved.
–
Now, consider a higher stock of natural gas so that X 01 $ X 01, as shown by endowments in
zone V (see Figure 6). Then, as in zone VI, we use coal at the beginning, but because there is a
higher endowment of natural gas, some of it is used to get to the ceiling. From an initial vector
Chakravorty et al.: Extraction of Polluting Resources
VOL. 98 NO. 3
1141
X2
;POF 7
R
H
1
X Z 0
H
4UPDLPGDPBM
H
S
R
S
;POF 7*
S
H
;POF *7
V
L
L
X2HZ 0
–0
X2
A
Q
M
A
X2H
;POF **
;POF ***
;POF *
B
0
X2H
K
–0
X1
H
2
X Z 0
B
H
1
X
4UPDLPGOBUVSBMHBT
N
X1HZ 0
X1
Figure 6. Approach to the Ceiling with Abundant Resources
S, coal is used until location S9; then natural gas is used, S0, when the ceiling is reached. Beyond
this location, gas continues to be used at its maximum level –x1, and extraction proceeds as in zone
IV in Section III.
Why not only coal to the ceiling as in zone VI? With increased gas resources, there is competition between coal and gas. The benefit of coal use on the access to the ceiling is that it allows for
increased pollution and, therefore, increased dilution, which is costless. With higher reserves of
gas, however, its scarcity rent (hence cost) is lower and it is less polluting, so it makes sense to
use some gas exclusively at the beginning of the ceiling period and additional gas resources to
ensure continuity of the price path as the pollution stock approaches the ceiling. These prices and
the associated scarcity rents are shown in Figure 7. Coal is cheaper at first, followed by natural
gas, until the ceiling is reached at time D0.22
Finally, consider zone IV where both resources are abundant (see Figure 6). As the endowment of natural gas increases, its scarcity rent decreases. Even though the scarcity rent of gas is
higher than coal, large stocks of gas will imply that its marginal cost is lower than for coal, l01 2
m0 u1 , l02 2 m0 u2. During the initial phase, gas is used exclusively all the way to the ceiling.23
22
To prove that this is the only possible sequence, we can show that the price of coal is strictly higher than that of
natural gas and the differential increases as we move recursively from D 0 1 D1 to D 0 (see Figure 7). Then coal cannot be
used in the preceding interval and gas must be used. Consider the price differential p2 1t 2 2 p1 1t 2 at time t [ 30, D14 so
that D 0 1 D1 2 t [ 3D 0, D 0 1 D1 4 , i.e., t is measured from D 0 1 D1. In this interval by (4), p–1 5 l1 2 mu1 which gives
0
m 1t 2 5 1l1 2 p–12/u1. We have p2 1D 0 1 D1 2 t2 2 p–1 5 l2 1D 0 1 D1 2 t2 2 l1 1 1D 0 1 D1 2 t2 2 p–12 u2 / u1 2 p–1 5 l02 er 1D
0
1 D1 2 t2
1 u2 3 p–1 2 l01 er 1D 1 D1 2 t2 4 / u1 2 p–1 which after some manipulation yields p2 2 p–1 5 3 1u2 2 u12 p–1 / u14 31 2 e2rt 4 ,
so that d / dt 1p2 2 p–12 5 3 1u2 2 u12 p–1 / u14 3re2rt 4 . 0.
23
The case of natural gas use followed by coal to the ceiling can be eliminated by arguments made earlier for
endowments in zone V.
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THE AMERICAN ECONOMIC REVIEW
L01 e R t
L02 e R t
cr
p1
p–2
p2
p1 p2
p–1
p2
p1
L01
#JOEJOH$FJMJOH
L02
$PBM
(BT
#PUIDPBMBOEHBT
$PBM
t
3FOFXBCMF
Figure 7. Optimal Price Paths for Endowments in Zone V
Note: pi 5 li 2 mui, i 5 1, 2.
This is followed by maximal use of gas until the boundary of zone IV. The extraction path is
shown starting from an initial endowment V in Figure 6.
The main results of this paper can be summarized as follows:
PROPOSITION 1: In a Hotelling model in which extraction is costless but resources are differentiated by their emission rates, the order of resource extraction cannot be predicted by the
pollution characteristics of the resources:
(a) When the initial stock of pollution is at the regulated level, the clean resource is used first.
When the initial stock of the clean resource is small, both resources may be used in combination until the clean resource is exhausted;
(b)However, when the initial stock of pollution is below the regulated level, the clean resource
may be used first only if it is abundant. Otherwise, the dirty resource is used first.24
V. Concluding Remarks
This paper extends Hotelling theory to resources when they are differentiated by their pollution characteristics. Herfindahl showed that when there is cost heterogeneity among resources,
extraction follows the “least cost first” principle. We show that when resources are differentiated
by pollution, the ordering of extraction is somewhat surprising and not easily predicted by the
emission rates of the resources. When the economy starts with the regulated level of pollution,
the clean resource is used first. When, however, the economy starts from a lower level of pollution, it may use the dirty resource first and build the pollution stock as quickly as possible. In this
manner, it benefits from natural dilution of the pollution stock, which is “free.” However, if the
stock of the clean resource is large, it may be used from the beginning.
24
The proposed extraction order can be implemented by means of a Pigouvian tax on emissions or, equivalently, a
market for carbon permits. Resource prices given by pi 1t 2 will equal the scarcity rent plus the social externality cost.
VOL. 98 NO. 3
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1143
We show that in a model with pollution heterogeneity, a single resource may be used over two
disjoint intervals. Coal may be used exclusively at the beginning, followed by exclusive use of
gas, and again coal. Such “vacillation” in the use of resources does not arise in models of cost
differentiation.
Unlike in the literature following Hotelling, the sequence of extraction in this pollution model
depends strongly upon the initial endowment of the resources. A common feature of extraction
when both resources are abundant is a turnpike property, in which the two resources are used
jointly at their maximal levels until the clean resource is completely depleted. Which resource
will be used to get to the ceiling depends on their relative abundance.
These results have implications for the order of extraction when an economy has to meet environmental goals. For instance, a stated aim of the Kyoto Treaty is to stabilize the atmospheric
concentration of carbon at approximately 550 parts per million. Currently, it is about 380 parts
per million. Since reserves of cleaner fuels (such as crude oil and natural gas) are limited, while
polluting fuels such as coal are abundant, our results suggest that coal should be used exclusively to get to the ceiling. This seems counterintuitive, but by getting to the ceiling as quickly
as ­possible, we also get costless dilution of a larger part of the pollution stock. Once the target
ceiling is achieved, we may use both resources jointly for a period until all gas reserves are
exhausted. Finally, only coal will be used until exhaustion and transition to the backstop.
An important assumption in the model is that both resources are homogenous with respect
to cost. Given that Herfindahl focused on cost heterogeneity among resources, it may be useful
to speculate on how cost heterogeneity interacts with pollution heterogeneity. To some extent,
the abundance of resources works as a proxy for cost, since a higher stock reduces the scarcity
rent and therefore the relevant opportunity cost of the resource. If the extraction cost of coal is
low relative to natural gas, it is likely to be used first. Then coal may be used before natural gas
for two reasons: its low cost and the free dilution of the pollution stock. If, however, natural gas
is cheaper, then we may obtain a sequence of extraction similar to the case with abundant gas
resources. Gas may be used first, followed by coal. The model becomes less tractable with nonzero extraction costs, and the solution depends on the interplay of various parameters, including
the extraction cost differentials between the two resources, pollution characteristics, and initial
stocks of the two resources. A generalized Herfindahl model with pollution is left for future
work.
To keep the analysis tractable, we use a very specific form of environmental regulation, which
is perfectly inelastic at some exogenously fixed level of pollution. However, more realistic damage functions may be used in later work to explore how the sequence of extraction may be sensitive to the specification of damages. If damages were strictly positive at all levels of the pollution
stock, we may not get these sharp transitions between resources. In our framework, when the
pollution stock is below the ceiling, the choice of resources is determined by the future costs of
pollution. With a damage function that is positive at all levels of the stock, one must also consider current environmental damage. For example, burning coal to get to the ceiling will lead to
contemporaneous pollution damages. In that case, the benefit from free dilution may have to be
compared to the cost of the externality. The precise nature of the solution may be highly sensitive
to the specification of the damage function, including its higher-order derivatives. If damages
were to rise gradually, then the analysis would depend on the precise nature of the damages,
as in Olli Tahvonen (1997), who finds that the extraction path may follow an inverted U-shape.
Continuity of the damage function with, say, a low level of damages below the threshold level
–
Z and a high level of damages beyond this level will lead to a solution that is quite similar to the
one discussed in the paper. Thus, we implicitly address the case when damages rise dramatically
beyond some threshold. Our analysis is more pertinent when damages are catastrophic beyond
some threshold level.
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THE AMERICAN ECONOMIC REVIEW
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In our model, the energy demand function is stationary. Thus, energy prices increase and consumption decreases given initial endowments of the resources, except for time intervals when both
are constant. However, as pointed out by a referee, historically the extraction of nonrenewable
resources has been associated with a continuously growing standard of living, while Hotelling
models such as ours predict reductions in consumption. Productivity growth can be modeled by
assuming exogenous shifts in the demand for energy, as has been done by Chakravorty, Magne,
and Moreaux (2006) with only one polluting resource. With increasing demand, price paths have
the same qualitative properties as in the stationary case, but consumption may increase over an
initial period, provided demand shifts are sufficiently large. When the ceiling constraint binds,
energy consumption is constrained by the natural dilution of carbon, not by increases in demand.
With demand or productivity growth, however, the opportunity cost of the pollution stock constraint is now higher, and that may trigger a larger use of the polluting resource before the ceiling
is attained, so that more of the clean resource could be used at the ceiling.
For reasons of tractability, the model presented here does not incorporate some important
features of the energy market, such as the uncertainty in the arrival date of the backstop technology and reduction in costs from endogenous research and development. These issues need to be
studied in future research.
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This article has been cited by:
1. Sjak Smulders, Edwin Van Der Werf. 2008. Climate policy and the optimal extraction of
high- and low-carbon fossil fuels. Canadian Journal of Economics/Revue canadienne d'économique 41:4,
1421-1444. [CrossRef]