1
Updated 18.02.04
ECON4925 Resource economics, Spring 2004
Olav Bjerkholt:
Lecture notes on the Theory of Non-renewable Resources
2. The Hotelling rule for prices of exhaustible resources
In LN-1 we found that the resource rent increases by rate r, cf. relation (1.4). This
equilibrium law is in the literature commonly referred to as the Hotelling rule. In his
frequently cited work from 1931, discussing the developments of prices in resource
markets, Harold Hotelling arrived at a condition for the market price that is formally
identical to (1.4).
Thus, a main conclusion in Hotelling's study was that in equilibrium the resource rent
(the net price), defined as the difference between the market price of the resource and
marginal extraction costs, must increase at a rate equal to the rate of interest. The
underlying assumption is again that the producers in the market possess exclusive rights
to non-renewable natural resources. The only way of having a return on preserving the
resource stock, is that the net price of the resource increases over time. In order for the
asset market to be in equilibrium, the growth rate for the resource rent must equal the
opportunity cost, i.e. the interest rate or the return on investments. Hotelling (1931) also
showed that the competitive equilibrium path for the net price coincide with the
conditions for optimal allocation of the total stock of the resource, to be shown below.
The Hotelling rule derived from social optimization when unit costs are constant
Formally, the Hotelling rule in its simplest version can be derived in a way that is quite
similar to the solution of Gray's problem. Let S0 denote the total resource stock in the
economy, Rt is the total extraction in the market at time t, U(Rt) is the utility of the
consumption of the resource at time t (in money terms). Extraction is assumed to be carried
out with constant unit costs, b .
The optimal extraction path for the society as a whole is found by maximizing
T
(2.1)
 [U ( R )  bR ]e
t
t
 rt
dt , Rt  0
0
with respect to Rt, over a possibly infinite time horizon [0,T] . Again, the maximization is
constrained by the condition
T
(2.2)
 R dt  S
t
0
0
Like in Gray's problem, the solution to (2.1) and (2.2) can be found by using optimal
control theory. The Hamiltonian is
(2.3)
H (t , St , Rt , t )  [U ( Rt )  bRt ]e  rt  t Rt
2
According to the necessary conditions, the optimal extraction path must fulfil the
following relation:
(2.4)
[U ( Rt )  b ]e  rt  t (=t for Rt  0)
The shadow price of the resource stock at time t, λt, is in the present case, with extraction
costs not depending on the accumulated production, constant. We have H / S  0 . For
positive extraction, (2.4) can then be written as
(2.5)
U ( Rt )  b   e rt
Using that in a market economy, U ( Rt )  pt , it is seen that (2.5) expresses the Hotelling
rule: along the optimal path the marginal net price, which is identical to the resource rent,
should increase at the rate of discount.
The interpretation of the Hotelling condition as a market equilibrium condition is similar
to the case of the resource extracting firm: along the socially optimal extraction path the
owners of the resource stocks are indifferent between extracting and leaving the resource
in the ground. If this arbitrage condition at some point is not fulfilled, some agents will be
able to increase their profits by changing the speed of extraction or by auctioning the
complete resource stock on the market.
In order to determine the complete equilibrium paths for the net price and the level of
extraction, it remains to solve for the initial price p0. We assume here that the demand is
such that there exists a choke price, pmax, for which demand equals zero.The terminal
condition in this case yields
(2.6)
[ p0  b ]e rT  [ p max  b ]
where pmax is the intercept of the demand schedule. This equation, together with the
resource restriction (2.2) in the optimization problem and the demand equation, determine
the initial values for the extraction level and the resource rent.
Consistency between socially optimal depletion and a competitive solution
Hotelling (1931) showed not only that there is a socially optimal depletion profile, but
also that there is a competitive solution consistent with the socially optimal depletion
profile. We approach this by solving the social planning problem for the optimal
depletion profile for n identical natural resource firms and then showing that the
optimality problem for one of them (essentially Gray’s problem) gives optimality
conditions consistent with those of the social planning problem.
We thus assume that there are n identical natural resource firms. The socially optimal rate
of depletion is conceived as the rate that maximizes the gross surplus (consumers’ surplus
plus producers’ surplus) derived from the demand function given as p(.). The amount
depleted (per unit of time) from each firm at time t is Rt while the amount of unextracted
resource in each firm at time t is St. The cost of extraction is given for each firm by the
function b(Rt, St) with bR’ > 0, bR’’ > 0 and bS’ < 0. The rate of discount is, r, the same
for the social planning problem and in the competitive solution.
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T nRt
max  [  p( x)dx  nb( Rt , St )]e rt dt
(2.7)
Rt
0
0
nSt  nRt , S0  S , ST  S , Rt  0
The state variable in this problem is the amount of remaining resource nSt, while the
control variable is the rate of depletion nRt. The Hamiltonian of this problem is – with
adjoined price (shadow price) t - as follows:
nRt
(2.8)
H (t , nSt , nRt , t )  [  p( x)dx  nb( Rt , St )]e  rt  t nRt
0
Assume that St* and Rt* solves the problem. Then it follows from the maximum principle
that Rt* maximizes the Hamiltonian, which implies that when continuity, differentiability
and concavity of the Hamiltonian hold, we have
H
 [ p(nRt )  bR ( Rt , St )]e  rt  t  0
(2.9)
 (nRt )
Furthermore, the rate of change of the shadow price is given by
H
t  
 bS ( Rt , St )e  rt
(2.10)
 (nSt )
Alternatively, for the current value problem the Hamiltonian is
(2.11)
H (t , nSt , nRt , t ) 
C
nRt

p( x)dx  nb( Rt , St )  t nRt
0
The first order condition is now
H C
 p(nRt )  b( Rt , St )  t  0
(2.12)
 (nRt )
While the rate of change of the shadow price in current values is given by
H C
 t  r t  
 bS ( Rt , St )
(2.13)
 (nSt )
Then, the remainder of Hotelling’s proof is to consider the profit maximizing problem of
one of the n firms facing a given price path (in fact the price path resulting from the given
demand curve and the depletion of the n firms). We thus have the problem:
T
(2.14)
max  [ pt Rt  b( Rt , St )]e  rt dt
Rt
0
St   Rt , S0  S , ST  S , Rt  0
The current value Hamiltonian of this problem is
(2.15)
H C (t , St , Rt ,  t )  pt Rt  b( Rt , St )   t Rt
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Assume that St* and Rt* solves the problem. Then it follows from the maximum principle
that Rt* maximizes the Hamiltonian, i.e.
(2.16)
H C
 pt  bR ( Rt , St )   t  0
Rt
Furthermore, the rate of change of the adjoined price in current values is given by
(2.17)
 t  r t  
H C
 bS ( Rt , St )
St
As the price pt is assumed to be p(nRt*), i.e. n identical firms producing the same amount,
these conditions are exactly the same as in the social planning problem and in equilibrium
t is equal to t.
Will the correct equilibrium be reached in the market?
Hotelling thus derived conditions for the existence of equilibrium in markets for
depletable resources. In general, equilibrium theory does not tell how equilibrium is
brought about. Neither does the theory give guidelines for the market development in
cases where prices and quantities for some reason have reached values outside the
equilibrium path. More specifically, it is worth noting that due to the dynamic nature of
the problem, the solution demands significantly more from the market agents than in the
traditional static market equilibrium.
Strictly, market equilibrium in the Hotelling model requires the existence of future
markets. Individual resource owners must have perfect foresight as to the evolution of net
price of the resource, ( pt  b ) . It is not sufficient that the producers have assessed correctly
the rate of growth in returns on different assets (i.e. a Hotelling rule prevails); total
extraction must also satisfy the end point conditions (2.2) and (2.6).
If the initial price is too high, there is too much conservation in early years and a part of the
resource stock will be left in the ground at the time when the demand schedule reaches the
choke price, pmax. If, on the other hand, p0 is too low, there is over-exploitation initially, and
the resource stock will be depleted too early. There is also the possibility that in the latter
case, the pressure against the resource stock, at a higher level of demand than along the
optimal path, will induce the price of the resource to increase at a rate higher than what
follows from the r % rule.
Constant, but different extraction costs
Implicit in the derivation of the simplest Hotelling model above was an assumption of
market consisting of price taking firms, with homogeneous and identical resource stocks
and cost structure. In practice, extraction costs may vary considerably between different
areas and producers. In the more realistic case of heterogeneous producers and stocks, an
interesting question relates to the succession of depletion of the various resource deposits.
Assume that the supply side of the market consists of two resource owners, with unit
costs b1 and b2 , respectively. From the derivation of the Hotelling rule above it should be
clear that in the present case market equilibrium must fulfill the following relations:
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(2.18)
pt  bi  e rt i (i  1, 2)
where λi is the shadow price related to the resource stock of producer i. As before, the strict
inequality in (2.18) prevails for Ri  0 .
With constant unit cost functions, market equilibrium requires sequential extraction of
each resource stock. Assume otherwise that for some time interval we have
simultaneously R1 , R2  0 . This implies that the following equation holds:
(2.19)
b1  e rt 1  b2  e rt 2
 b1  b2  (2  1 )e rt
While the left hand side in the lower part of (2.19) is a constant, the right hand side grows
at a rate equal to r. Thus, by assuming simultaneous extraction for some t, we have
arrived at a contradiction, the premise must be false, and we conclude that given constant,
but different unit costs, market equilibrium requires sequential extraction of each
resource stock.
In a first time interval, [0, T1 ] , the market is supplied by the cheapest resource (say
producer group 1). The corresponding resource rent must increase at the rate of discount. In
the same time period, producer group 2 with the higher costs reaps a return on leaving his
resource stock in the ground that exceeds the interest rate, since we have
(2.20)
pt
pt

r
pt  b2 pt  b1
It should be noted that price path is continuous, but with a "kink" in T1, when producer 2
"takes over" the market. At the same time, the resource rent jumps to a lower level,
before continuing to increase at a rate equal to r. With (constant) higher unit costs of
production, there is no way producer 2 can escape this fall in the resource rent; the best
he can achieve is to postpone extraction until T1.
With a more general cost structure, the result that stocks of different quality should be
extracted sequentially, does not hold, however. If for instance the extraction costs vary
with current production, i.e. bi = bi(Rit ) , the complete market equilibrium path will
generally involve periods of overlapping extraction. To what extent extraction of different
stocks will be simultaneous, depend on the flexibility in the cost structure of the producers
and on the size of the different deposits. Market equilibrium requires that resource rents, i.e.
the difference between the market price and marginal costs, grows at rate r for all extracting
firms. With flexibility in the cost structure, producers may equalize their returns by
operating at different levels of extraction. Formally, the condition for simultaneous
extraction is that the sum of marginal cost and the resource rent should be equalized
between producers, which in the two-deposit case is equivalent to the following relation:
(2.21)
b( R1t )  b( R2t )  (2  1 )e rt
It may be noted that in general extraction of individual resource deposits may take place
at different marginal production costs.
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With variable extraction costs, the interpretation of "high" and "low" cost deposits may
not be unique. From (2.21) it is clear that the level of extraction, in addition to the cost
structure, also depends on the size of the initial resource stock through its shadow price,
 . The larger is the deposit, the lower is the value of the shadow price, and the higher is the
speed of extraction. There will be a tendency that "high cost producers" enter the market
later (at higher prices), or restrict their extraction to a larger extent, than producers with
large stocks and/or low production costs.
Summing up
The equilibrium solution and the price path in the Hotelling model presented above is
remarkably simple. It is perhaps this simplicity that has led many authors and
commentators to completely reject the theory by referring to lack of empirical evidence1.
In particular, it is hard to point at actual resource markets with observed exponential
growth for the net price. Clearly, the relevance of the Hotelling theory should be
questioned. However, it is important to be aware of that since the pioneering works of
Gray and Hotelling the theory of exhaustible resources has been further developed and a
number of complicating elements have been added.
With more realistic assumptions regarding technology and market structure, the classical
and clearcut r%-rule for the development of the resource rent does not survive. One
important case was studied in the previous section, with extraction costs depending on
remaining reserves. Extending the Hotelling model similarly, yields exactly the same
conclusion, namely that the resource rent in equilibrium increases at a rate less than the
rate of interest, cf. equation (1.17).
Moreover, we may extend the market model further, e.g. by assuming that both the cost
function and the demand function shift over time, due to changes in other variables, such
as technological improvements and income growth. Such changes severely complicate
the solution of the model. However, it has been shown that with the demand and cost
functions shifting over time, the equilibrium path for the resource rent may even fall over
some time intervals (see e.g. Farzin (1992)). Finally, in a world involving imperfect
competition and uncertainty, as will be discussed below, the "simple and smooth"
Hotelling results may be significantly affected. Clearly, as a consequence of increased
complexity, it also becomes more difficult to undertake empirical tests of the theory.
Hotelling versus Ricardian rent
Harold Hotelling and David Ricardo are probably the most widely quoted economists
within the field of resource economics. Both were studying markets for natural resources,
and each ended up by concluding that in market equilibrium there will be excess profits or rents - in production. However, the concept of Ricardian rent is principally very
different from the resource or Hotelling rent concept that is involved in the theory of
exhaustible resources. Ricardo did not focus explicitly on resource scarcity. Rather, the
Ricardian view is that natural resources such as land consist of heterogeneous units.
Given that the best quality pieces of land are cultivated first, as demand increases lower
1
An early empirical study of resource markets was Barnett and Morse (1963). Among more recent analysis
are Smith (1979) and Slade (1982).
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quality land is brought into production. Assuming that the market price has to equal unit
costs of the marginal land, the price will exceed costs of all more productive units of the
resource. Thus, the latter will earn Ricardian rent.
The market equilibrium for a good involving Ricardian rent can be illustrated by a supply
curve, which is equal to the marginal costs for the market as a whole. It increases with
increasing volume since increased production implies use of resource units of descending
quality. Market equilibrium is found in the intersection between the supply and demand
curve. On the marginal unit of production, there is no pure profit, while Ricardian rent is
earned on all intramarginal resource units.
Turning to a market for an exhaustible resource, we have already seen that the market
equilibrium may involve some element of Ricardian rent. More specifically, if the
extraction cost function of a resource owner is of the form described by relation (1.15),
Ricardian rent will be present. Hotelling rent or resource rent may come on top of
Ricardian rent. As discussed above, this is due to resource scarcity in a strict sense, so
that there is an absolute limit to accumulated resource extraction. As a result, in
equilibrium the market price equals marginal costs plus the Hotelling rent, q , i.e.
(2.22)
pt 
b ( Rt , S t )
 qt
Rt
In accordance with the results derived above, at the chosen level of extraction the market
price exceeds marginal costs. Thus, in addition to the Ricardian rent indicated production
now requires an additional return, since current extraction inevitably is at the expense of
future production. The resource (Hotelling) rent is the difference between the market
price and marginal costs.
In a competitive market for an ordinary good all producers that supply the market operate
at identical marginal costs. In a market for an exhaustible resource, on the other hand,
marginal extraction costs at the chosen levels of extraction may in general differ, since
individual resource rents vary.
From the discussions above we know that when production costs depend solely on
current extraction, the resource rent in equilibrium will rise over time at a rate equal to
the rate of interest. However, in the more general case where the remaining reserve stock
is included in the cost function, this result is altered; the rate of increase in the resource
rent will in general be lower than the interest rate (see (1.17)). Since  2b / Rt St  0 , the
cost function shifts upwards as the stock is depleted, and the resource rent may therefore
even diminish over time.
The fact that future marginal costs increase as a direct consequence of current production
activity implies that the distinction between Ricardian and Hotelling rents may not be
quite clear cut. By integrating (2.13) we find a decomposition of the resource rent, cf.
Lasserre (1991, p.10):
(2.23)
giving
T
T
t
t
 r ( s t )
 r ( s t )
 (t  r t )e ds   bS ( Rs , Ss )e ds
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(2.24)
t  T e
 r (T  t )
T
  e r (t  s )
t
b( Rs , S s )
ds
S
This relation expresses the resource rent at time t as the sum of the discounted rent at
time T plus the cumulated present value of all future effects on extraction cost of
extracting the marginal unit of the resource at time t. Because the presence of S t in the
cost function should reflect different qualities of the stock, the second component may be
interpreted as a Ricardian component of the (resource) rent, while the (pure) scarcity rent is
contained in the first part of equation (2.24). The implication of adopting this "dynamic"
interpretation of Ricardian rent is that a relatively smaller share of the excess profit at time t
is classified as being of the Hotelling type. Still, common terminology is to denote the
complete value of  t in (2.24) as the Hotelling rent.