Monopoly Extraction of an Exhaustible Resource with Two
Markets With an Application to Antibiotics
Carolyn Fischer¤
Resources for the Future
Ramanan Laxminarayan
Resources for the Future
February 21, 2002
Abstract
Although much has been written about the implications of monopoly power for the rate of extraction of
natural resources have been explored, the speci c case in which the resource can be sold in two markets with
different elasticities of demand has escaped notice. This analysis is relevant in the case of many resources
– such as natural gas used for power generation and household heating, or oil used for making plastics and
as fuel – and is particularly useful in current debates regarding the simultaneous use of antibiotics (where
their effectiveness is a resource) for growth promotion in cattle and poultry, and for treatment in humans.
1 Introduction
A number of authors have written on the implications of monopoly for the extraction of an exhaustible
resource: Weinstein and Zeckhauser (1975), Stiglitz (1976), Kay and Mirrlees (1975), Lewis (1976), Stiglitz
(1976), Sweeney (1977), Dasgupta and Heal (1979), Tullock (1979), Lewis, Matthews, and Burness (1979),
Eswaran and Lewis (1984), Pindyk (1987), and Gaudet and Lasserre (1988), among others. However, based
on our and other’s review of the literature ((Peterson and Fisher 1977) (Dasgupta and Heal 1979) Devarajan
and Fisher (1981) and Krautkraemer (1998)), there has been no previous analysis of a relatively simple, but
important, case: when the sole owner of a resource can sell to two markets at different prices. Although
only a slight variation on Stiglitz (1976)’s model of constant elasticity of demand with zero extraction costs,
our model does not yield the result that the monopolist extracts at the same rate as the social planner nor
does it con rm Solow (1974)’s suggestion that the monopolist is the conservationist’s best friend. Instead,
a monopolist facing two markets with differing iso-elastic demand schedules extracts more quickly than the
social planner1 .
When the monopolist can discriminate between the two markets, it sets prices so that they rise at the rate
of interest, much like the social planner. However, unlike the planner, the monopolist nds it bene cial to
set a different price in each market, and these prices are such that total quantity sold is higher than under the
social planner. When arbitrage forces the monopolist to sell at the same price in both markets, the monopolist
acts like a social planner in that prices are equal, but these prices differ from those of the social planner, rising
in such a way that they encourage faster consumption of the resource.
These results are not just of academic interest, as many resources have multiple uses. In a number of
cases, resources are sold in segmented markets where arbitrage is dif cult. For example, coal is sold to both
¤ Corresponding author: Resources for the Future, 1616 P St. NW, Washington, DC 20036-1400. E-mail: scher@rff.org. Thanks to
Derek Gurney for research assistance.
1 A similar result where the monopolist extracts too rapidly is provided by Lewis et. al. - they obtain this result both when there are
xed costs that do not vary with the extraction rate, as well as when demand elasticities vary with consumption, instead of time (Lewis
1976).
1
manufacturers and electricity generators natural gas for both electricity generation and household heating
petroleum for combustion and plastics, and antibiotics (where antibiotic effectiveness is the resource) for both
human and animal use.
The basic model and results are presented in Section 2. Policy policy implications of the results are
discussed in Section 3. Section 4 concludes the paper.
2 Model
Suppose a single exhaustible resource is sold in two separate markets with two different elasticities of demand. Let us call these two markets A and B. Total stock of the available resource is K0 . We make the
following assumptions:
1. There is no cost to extracting the resource other than the scarcity value.
2. The elasticity of demand in each market is constant and greater than 1.
3. There is no possibility for arbitrage between markets. The monopolist cannot engage in price discrimination within each market.
The rst two assumptions are identical to those made by Stiglitz (1976). The third is added for our twomarket problem. Under these circumstances, how does a monopolist’s behavior compare with that of a social
planner’s, both with respect to the quantity sold in each market and to total extraction?
2.1
Social Planner’s Problem
We frame the optimal resource allocation problem in terms of a social planner maximizing welfare (de ned as
total consumer surplus, since costs are absent) with respect to quantities sold in markets A and B, denoted by
qtA and qtB , subject to the constraint that total extraction not exceed the available resource stock. The resulting
equilibrium corresponds to the market allocation under perfect competition. The optimization problem is
represented in the following Lagrangean:
ÃZ A
!
µZ
¶
Z
Z B
US =
1
e¡rt
t=0
qt
qt
PA (s)ds +
s=0
s=0
PB (s)ds dt ¡ ¸
1
t=0
(qtA + qtB )dt ¡ K0
(1)
where r denotes the discount rate, PA and PB denote the price of the resource in the two markets, and ¸
is the shadow price of the resource. The optimum is characterized by the rst-order conditions with respect
to resource sales in each market
qtA > 0;
qtB > 0;
and the stock constraint
¸ > 0;
Z
PA (qtA ) = ert ¸;
PB (qtB ) = ert ¸;
1
t=0
(qtA + qtB )dt
(2)
(3)
K0 ;
(4)
where a binding total extraction constraint leads to a positive shadow value of the resource (¸).2
2 The
full complementary slackness conditions are
Z 1
¸ ¸ 0;
(qtA + qtB )dt
K0 ;
¸
t=0
µZ
1
t=0
(qtA + qtB )dt ¡ K0
¶
=0
If some of the stock were left unexploited, then ¸ = 0, but this would imply from 2 and 3 that prices were zero and extraction positive
in all periods, which would violate the constraint of a nite stock. Thus, ¸ > 0.
2
The combination of 2 and 3, in any period t reveal that we know that
Pi (qti;S ) = ert Pi (q0i;S ); i = A; B
PA (qtA;S )
=
PB (qtB;S );
t = [0; 1]
(5)
(6)
First, prices in each market rise over time at the rate of interest. Second, in an equilibrium where the
planner is selling in both markets, those prices must be equal, re ecting equivalent scarcity costs. Consider
price functions with constant elasticity of demand, such as
Pi (qti ) =
µ
qti
¹i
¶¡ ´1
i
where ´ is the elasticity (in absolute value terms), while ¹ represents a shifting parameter, indicating the
relative size of the market. In our examples, we will designate market B as the one with the greater elasticity
(´B > ´A ). Since for these inverse demand functions, price goes to in nity as quantity goes to zero, we
know that to satisfy the rst-order conditions, some amount will be produced and sold in each market in every
period, although those amounts will eventually become in nitessimally small. Solving for the extraction
paths, we get
qtA;S = e¡´Art q0A;S ; qtB;S = e¡´B rt q0B;S :
(7)
Since prices must be equal across the markets at all times including the start, we can solve for the initial
quantity in market B in terms of that in A:
q0B;S = ¹B
Ã
q0A;S
¹A
! ´´B
A
:
(8)
Although prices are the same, the quantities will differ according to the elasticities and the demandshifting parameter. Integrating 7 over time, we can express cumulative extraction for each market as a
function of initial resource sales, the interest rate, and its own market elasticity:
Z
1
t=0
q0i;S
:
r´i
e¡´ i rt q0i;S dt =
(9)
From 8, 9, and the stock constraint, we can solve for q0A;S :
q0A;S
¹
+ B
´A r
´B r
Ã
q0A;S
¹A
! ´´B
A
= K0 :
(10)
With that solution, the entire optimal paths are determined.
2.2
Monopolist’s Problem
The monopolist’s problem is to maximize pro ts (revenues) subject to the total resource stock constraint:
µZ 1
¶
Z 1
¡
¢
UM =
e¡rt PA (qtA )qtA + PB (qtB )qtB dt ¡ ¸
(qtA + qtB )dt ¡ K0
(11)
t=0
t=0
3
Along the pro t-maximizing path of resource sales in each market, the following rst-order conditions must
hold:
qtA > 0;
qtB > 0;
M RA (qtA ) = ert ¸;
M RB (qtB ) = ert ¸;
(12)
(13)
where M Ri denotes marginal revenue in market i. The nal condition is the same stock constraint as 4,
although the actual shadow value may differ. Thus, for the monopolist to sell in both markets, their marginal
revenues must be equal in each period, re ecting equivalent scarcity costs. Marginal revenues in each market
must rise at the rate of interest. With constant elasticity demand, since marginal revenue is a constant times
the price:
Ã
!¡ ´1
i
´ ¡1
´i ¡ 1 qti;M
i;M
M Ri (qt ) =
= i
Pi (qti;M ); i = A; B
´i
¹i
´i
This latter result implies that prices will also be rising at the rate of interest in each market:
Pi (qti;M ) = ert Pi (qti;M ); i = A; B
(14)
Solving for qti;M we then get the same result as 7:
qtA;M = e¡´A rt q0A;M ;
qtB;M = e¡´B rt q0B;M :
However, while this Hotelling rule seems the same for the planner and the monopolist, the relative prices are
different, meaning the paths do diverge. Setting marginal revenues equal to each other, we see the monopolist
does not want prices to be equal if the elasticities are different:
PB (qtB;M ) =
´B
(´A ¡ 1)
PA (qtA;M )
(´B ¡ 1) ´A
(16)
From this result, we can prove three statements comparing monopolist resource use to the planner’s with
respect to prices, quantities, and welfare:
Theorem 1 When assumptions 1-3 hold, the monopolist raises the price in the market with relatively inelastic
demand and lowers it in the market with relatively elastic demand compared to the optimal (planner’s) price.
Proof. Let ´B > ´A > 1. From 16, we nd that PtB;M < PtA;M the monopolist wants a lower price in
the more elastically demanded market for all t. From 12 and 13, monopoly prices in each market must rise
at the rate of interest from 2 and 3, optimal prices are the same in each market and rise at the rate of interest.
Thus, if PtS < PtA;M for any t, it must hold for all t. Suppose now that the price path in market A were
the same as the planner’s, while that in B were lower, implying greater sales in that market in every period.
Over the same horizon, then, total extraction would be greater for the monopolist, thereby violating the stock
constraint. (By the same logic, if PtB;M = PtS and PtA;M > PtS for all t, then the monopolist would extract
less over the same horizon, implying a slack stock constraint, which violates the monopolist’s conditions
for optimality.) Thus, the monopolist’s equilibrium must have higher prices than optimal in market A in all
periods, while prices in market B must be lower than optimal.
Corollary 2 Consumers in the market with relatively inelastic demand obtain less of the resource in each
period and overall with monopoly provision, while those in the market with relatively elastic demand obtain
more of the resource in each period and overall.
Proof. This result falls out of the previous theorem: since
PtA;M > PtS , then qtA;M < qtA;S for all t.
R
R1
R1
1
B;M
B;M
B;S
Since Pt
< PtS , then qt
> qt
for all t. Thus, t=0 qtA;M dt < t=0 qtA;S dt and t=0 qtB;M dt >
R 1 B;S
q dt.
t=0 t
4
Corollary 3 Compared to the social optimum, consumers in the market with relatively elastic demand are
better off under monopoly provision, as is the monopolist, while consumers in the market with relatively
inelastic demand are worse off, as is society overall.
Proof. This result falls out of the previous corollary. Consumer surplus is higher with more consumption
since qtB;M > qtB;S and qtA;M < qtA;S for all t, consumers in B are unambiguously better off with monopoly
provision while those in A are unambiguously worse off. The monopolist is better off by de nition of pro t
maximization, while society overall is worse off by de nition of welfare maximization, since the two paths
differ.
We can show that the price path for relatively inelastic market A gets shifted up, while that of B gets
shifted down, compared to the planner’s path. The dual to this result is that the extraction path for market B
gets shifted up, while that of A gets shifted down, compared to the planner. However, either market could
still be the larger than the other, depending again on the relative demand parameters.
To solve for the complete system of extraction, we must combine the rst-order conditions with the stock
constraint. Cumulative extraction for each market is the same function of initial resource sales, the interest
rate, and its own market elasticity as we saw for the planner problem:
Z 1
q i;M
(17)
e¡´i rt q0i;M dt = 0 :
r´ i
t=0
However, the initial extraction terms will differ for the monopolist. From 16 we solve for initial extraction
in B in terms of that in A:
q0B;M
= ¹B
µ
´B
(´A ¡ 1)
(´B ¡ 1) ´A
¶¡´B Ã
q0A;M
¹A
! ´´B
A
:
(18)
Then, substituting this result with 17 into the stock constraint 4, we arrive at the equation determining the
monopolist’s optimal q0A;M :
! ´´B
Ã
µ
¶
A
q0A;M
¹B
´B
(´A ¡ 1) ¡´B q0A;M
= K0 :
(19)
+
´A r
´B r (´B ¡ 1) ´A
¹A
Recall that ¹ represents the relative size of the market. From 10 and 19, we can also observe that as
¹B ! 0 and one market shrinks to nothing, q0A;M ! q0A;S , and we are back in the Stiglitz world where the
monopolist behaves as the planner in providing to a single market.
2.3
Total Extraction with Two Markets
We have shown that the extraction paths of the planner and the monopolist diverge when facing two markets
with different elasticities. But what does this imply for the path of total extraction and thereby the rate of
depletion of the single resource?
Theorem 4 When assumptions 1-3 hold, total extraction by the monopolist is greater than that of the planner
in the initial period.
Proof. From 9 and 17 we can rewrite the stock constraint 4 as
q0A;j
q B;j
+ 0 = K0
r´ A
r´B
(20)
for j = S; M . Solving for q0B;j , we get
q0B;j = ´B rK0 ¡
5
´B A;j
q ;
´A 0
(21)
which lets us write the total initial extraction Q0 for each actor, j, as
µ
¶
´
Qj0 = ´B rK0 + 1 ¡ B q0A;j :
´A
(22)
Subtracting the planner’s initial extraction from the monopolist’s, we see that the difference is positive:
µ
¶
´B
S
QM
¡
Q
=
1
¡
(q0A;M ¡ q0A;S );
(23)
0
0
´A
since ´B > ´A implies that q0A;M < q0A;S .
Although the monopolist contracts supply in the market with less elastic demand, it expands it in the
market with more elastic demand, with the net effect being an increase in total supply early on.
Corollary 5 The monopolist follows a faster extraction path than the planner.
Proof. The previous theorem states that total extraction is higher for the monopolist initially, but it cannot
be so inde nitely, else the stock constraint would be violated. Thus, at some point the extraction paths must
cross, with the monopolist’s total extraction being less than the planner’s in later periods. Therefore, the
monopolist’s extraction path must be steeper.
In other words, the monopolist will tend to be less “conservationist” than the planner. Furthermore, we
see from 23 that this tilting of the total extraction path must be greater, the more market elasticities diverge.
2.4
Monopolist Extraction with Arbitrage
Does arbitrage between the two markets eliminate the monopolist’s incentive to behave differently than the
planner? If the markets can resell to each other, a single equilibrium price will result. Let Pt1 = PA (qtA ) =
PB (qtB ). Whereas in the segregated markets case we worked with inverse demand, in this case we will
simplify the expressions as a function of the common price Pt1 . With our constant-elasticity demand, we can
write qti = ¹i (Pt1 )¡´i . In other words, at any price the monopolist faces a single, uni ed demand schedule
of
Q1t = ¹A (Pt1 )¡´A + ¹B (Pt1 )¡´ B
which no longer displays constant elasticity. The point elasticity is now
¹́t = ¡
´ ¹ (P 1 )¡´A + ´B ¹B (Pt1 )¡´B
dQ=Q
= A A t ¡´
dP=P
¹A (Pt ) A + ¹B (Pt1 )¡´B
which collapses to a constant as ¹i ! 0 or ´A ! ´B :
The monopolist’s new problem is
µZ
Z 1
1 ¡rt
1
Q(P
)e
dt
¡
¸
max
P
t
t
1
Pt
t=0
1
t=0
Q(Pt1 )dt ¡ K0
The new rst-order condition is
Q(Pt1 )e¡rt + Pt1 Q0 (Pt1 )e¡rt ¡ ¸Q0 (Pt1 ) = 0
which simpli es to
Pt1
µ
¶
1
1¡
= ert ¸:
¹́t
The rate of change in the price path is then
µ
¶
P_t1
¹_´t
1
=r¡
:
Pt1
¹́t ¹́t ¡ 1
6
¶
(24)
Since we assume ´ B > ´A > 1, then ¹́t > 1. Thus, whether the price rises faster or slower than the interest
rate depends on whether the total elastiticity is rising or falling:
µ
¶
P_t1 (´B ¡ ´ A )2 ¹A ¹B (Pt1 )¡(´A +´ B )
<0
¹_t́ = ¡ 1
Pt
(¹A (Pt1 )¡´A + ¹B (Pt1 )¡´B )2
Since the effective elasticity falls as the price rises, in equilibrium Pt1 > ert P01 . In other words, with an
arbitrage constraint, the monopolist will choose a steeper price path than the planner: the monopolist prefers
to lower prices in relatively elastic periods and raise prices in relatively inelastic ones, resulting in lower
initial prices and higher ones later on. A steeper price path over the same time horizon with the same stock
constraint implies the monopolist’s initial price must be lower than the planner’s: P01 < P0S . This in turn
implies that starting consumption is greater in both markets: not only q01 B > q0B;S , but also q01 A > q0A;S .
Thus, total extraction is greater at the beginning, and the monopolist is again less conservationist than the
planner. This result is similar to that of Stiglitz (1976) and others who note that if the elasticity of demand
increases over time, the monopolist picks a more conservationist path relative to the social planner. However,
these increases are modeled as exogenous changes to an elasticity that is otherwise constant with respect to
price within any period. Here, by presenting overall demand as the sum of distinct markets each with constant
elasticities of demand, we arrive at an effective elasticity that is decreasing in price, and thus decreasing over
time. Thus, our example corresponds to the extension by Lewis (1976) and Lewis, Matthews, and Burness
(1979), who consider a stationary demand schedule with elasticity increasing in consumption.
In summary, without arbitrage, we saw that the monopolist had the right (Hotelling) rule for the price
paths, but differed from the social planner with respect to the relative prices between markets. With arbitrage,
the monopolist’s relative prices between markets coincide with the planner’s, but as a consequence the monopolist’s price path is distorted. In both cases, the existence of two markets with different elasticities causes
the monopolist to behave differently from the planner and, overall, to follow a steeper path of depletion. A
caveat to this conclusion comes from Stiglitz (1976) who notes that when the price of a resource is projected
to rise faster than the interest rate, speculators will seize the opportunity and buy the resource, causing its
current price to increase until, with no storage costs, the price is projected to rise no faster than the interest
rate. Thus, when there is arbitrage both between time periods and among markets, the monopolist is further
restricted in its ability to control prices, as speculation may limit increases to the rate of interest. One can
only conclude that the monopolist’s equilibrium price path will rise at least as steeply as that of the social
planner.
2.5
Other Demand Functions
We recognize that the results under constant-elasticity demand may be special. Let us now maintain the
assumption of no extraction costs but relax the assumption about the demand speci cation. We can summarize
previous results more generally:
PtB =PtA
Pti
Pt_i =Pti
Social Planner
1
ert P S
r
³ i ´0
´
B
i
_i ³
´A
¡1
´
´
´
¡1
´
i;M
1
rt
t
t
t
0
t
Monopolist: Segregation
e
P
r
¡
i
i
i
i
A
B
0
´ t ¡1
´ t ´ t ¡1
´ t ´ t ¡1
³ ´0 ´
³
´
¹_´
¹́t
¹́0 ¡1
rt 1
Monopolist: Arbitrage
1
e P0
r ¡ ¹́ t ¹́ 1¡1
¹́ ¡1
¹́
t
0
t
t
The difference in incentives between the planner and the monopolist relies on the difference in elasticities
between the two markets. The difference between the monopolist’s behavior facing segregated markets compared to perfect arbitrage depends on the difference between the change in individual compared to average
demand elasticities.
7
Let us de ne general demand as Q(P ) = q A (P ) + q B (P ).3 Average demand elasticity can be written as
a function of individual elasticities:
µ A
¶
@Q P
@q B P
´ qA + ´B qB
@q
¹́t = ¡
=¡
+
= A
:
@P Q
@P
@P Q
Q
Next, we see how the average elasticity responds to a price increase:
@´A q A @´B q B
(´ ¡ ´B )2 q A q B
@¹́t
=
+
¡ A
:
@P
@P Q
@P Q
P
Q Q
The last term means that the change in the average elasticity will always be less than the average change in
the individual elasticities.
This general result implies that, for other demand functions as well, the monopolist follows a steeper
price path in an arbitrage situation than on average in a stituation where it can discriminate between markets.
Thus, the monopolist follows a more conservationist extraction path when markets are segregated. Compared
to the planner’s path, the monopolist tends to be more conservationist if elasticities are greater than one and
increasing with the price.
3 An Application to the Market for Antibiotics
The results in this paper are particularly useful in illuminating policymaking with respect to patent-protected
antibiotics. Many antibiotics are sold both for treatment of bacterial infections in humans as well as for growth
promotion in animals. The sales of antibiotics for growth promotion in animals is the subject of a contentious
policy debate. Recognizing the impact of antibiotic use in non-human, sub-therapeutic use on the overall
level of bacterial resistance in the environment, many countries in Europe have banned the use of antibiotics
for these purposes. However, antibiotics continue to be used in farm feed and agriculture in the United States.
In fact, almost half the 50 million pounds of U.S.-produced antibiotics is used in farm animals (of which 80
percent is used to help animals grow faster and the rest is used treat disease), and 40,000 pounds are sprayed
on fruit trees (of Technology Assessment 1996). While the animal industry and the pharmaceutical industry
are largely in favor of continued use of antibiotics for growth promotion, this policy is opposed strongly
by the medical profession as well as by many environmental groups who are concerned increasing bacterial
resistance to antibiotics.
Antibiotic effectiveness can be considered an exhaustible resource (Laxminarayan and Brown 2001). Due
to the selective pressure placed by antibiotics on bacteria that are susceptible to them, bacteria, antibiotics
become less effective with use. At rst glance, it may appear that a pharmaceutical rm that own a patent on
an antibiotic would have suf cient inventives to care about the use of antibiotics for growth promotion when
this could potentially impact their market for sales in humans4 . Typically, the growth promotion market
contributes only a small percentage (in the order of 10 to 20%) of revenues, while the market for antibiotics
in humans is far larger. This line of reasoning has been used to justify the use of antibiotics for growth
promotion and to obviate the need for regulatory intervention. To the extent that the5 elasticity of antibiotics
3 For example, let P = P A . Since P B is always a constant ratio of P A either in segregation or arbitrage, we can consider the effect
of this ratio subsumed into the demand function q B (:).
4 For instance, Bayer owns a patent on a number of antibiotics in the class of uoroquinolones of which Cipro oxacin is the most
well-known. Bayer also sells uoroquinolones for growth promotion in poultry. Recently, the Food and Drug Administration (FDA)
has ordered Bayer to stop selling uoroquinolones for use in animals, a decision based on the perception that this use may leading to
increasing uorouinolone resistance in Campylobacter pylori that causes infections in humans. However, Bayer has challenged this
decision and continues to sell in both markets.
5 We can justify this characterization on two grounds. First, there are many substitutes for antibiotic use in growth promotion
(including non-antibiotic inputs such as better on-farm hygiene), unlike for human treatment . Second, the price of antibiotics sold in
humans is typically an order of magnitude greater than that sold for animal use.
8
sold for growth promotion is greater than that for antibiotics sold for treatment in humans, patent-protected
pharmaceuticals may be encouraging the overuse of antibiotics in animals and too quickly depleting the stock
of antibiotic effectiveness.
3.0.1
Other Applications
The difference in extraction between the monopolist and the social planner in this model relies on there being
a difference in elasticities between the two markets. Otherwise the results collapse to those of Stiglitz (1976)
and Sweeney (1977). The practical importance of the model, then, lies in the empirical difference in elasticities between markets. Due to contradictions among studies that attempt to measure them, these elasticties are
dif cult to bound, much less state authoritatively. However, from surveys by Bohi (1981) and Dahl (1993),
one can conclude that the long-run elasticities for natural gas for commerical and for electrical use are 1 and
0.72, suggesting that a monopolist would ration gas for electricity use in order to oversupply commerical use.
Similarly, if the short-run elasticities of oil are indicative of their relative long-run elasticities, a truly effective oil cartel would constrain production of oil for commerical use in favor of overproduction for residential
heating. Of course, in reality, OPEC suffers from defections and natural gas distribution and other natural
monopolies are regulated. Our analysis would suggest that these are fortunate, as, contrary to much of the
literature, our model suggests that the monopolist does not face inherent constraints on its ability to exert
market power. Thus, where the monopolist over a resource goes unregulated, welfare consequences may be
severe.
4
Conclusion
This paper illustrates an important extension to the literature on the extraction of a depletable resource by a
monopoly. We nd that the ability to extract at zero marginal cost and sell in two constant-elasticity markets,
with or without arbitrage between them, leads in either case to a monopolist extracting a resource at a faster
than optimal rate. This result suggests that, contrary to the seminal literature on the topic, there is room
for a monopolist to exert market power in the sale of an exhaustible resource. Indeed, based on surveys of
the elasticities of demand for resources used in energy production, there may be signi cant welfare losses
associated with unregulated monopoly provision.
Needless to say, our assumption of iso-elastic demand, though a “natural rst approximation” Stiglitz
(1976), is restrictive and the assumption of exogeneity of the resource stock needs to be tested. Indeed,
incorporating the ability of resource owners to expand the supply of resource through exploration and innovation may prove a particularly fruitful avenue of research, as models that incorporate an endogenous stock
of resource, without the potential for two markets Gaudet and Lasserre (1988), indicate that the monopolist
extracts too slowly. This suggests that a model incorporating both the potential for discovering new stock of
the resource and selling it in multiple markets may yet vindicate the monopolist.
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