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Updated 7 March 2004
ECON/SØK500 Natural resource economics, Spring 2003
Olav Bjerkholt:
Lecture notes on the Theory of Nonrenewable Resources
4. Uncertainty in resource markets.
The theory of nonrenewable resources has been presented until now in these notes within
a framework where all elements that affect market equilibrium are known with certainty.
This is a drastic simplification. In reality, several of the central elements and parameters
of the models described above are highly uncertain. The initial stocks of the resource are
in general not known, but the knowledge and the estimates of reserves can usually be
increased by exploration. Similarly, the cost structure of neither resource extraction
activities nor potential back-stop technologies are known at an initial point of time.
Rather, technology typically change over time in an, for the individual agent,
unpredictable way. The rate of discount, reflecting the returns on financial investments
may be regarded as uncertain and varying over time, and the same applies to tax schemes
and other conditions determining the profitability of resource stocks.
Finally, from the discussion of different market structures above, it is clear that individual
agents may face severe uncertainty due to various forms of market imperfections that are
inherent in resource markets. From the viewpoint of an individual resource owner such
market failures and various forms of imperfect information have the important
implication that there may be significant uncertainty regarding future prices of the
resource.
Before turning to a closer discussion of some specific cases of uncertainty related to
resource markets, it may be useful to point at some general aspects and conclusions
derived from the theory of economic behaviour under uncertainty.
When economists analyse uncertainty, probabilities are typically attached to uncertain
events, although the probability distribution may exist only in the mind of the economic
agent1. On the other hand, when the uncertainty related to mechanisms or events means
lack of any information, the traditional probabilistic framework cannot be applied.
Instead, the information problem and market imperfections may call for some kind of
game theoretical analysis. In the following, we shall follow the common procedure and
assume throughout that a set of probabilities can be ascribed to an uncertain prospect.
Another important distinction is between exogenous and endogenous risk. The latter
denotes types of uncertainty that stem from lack of information, incomplete markets or
other kinds of imperfections prevailing in the economic system. By assumption, these are
problems that are related to human behaviour and the actual way of organizing the
economy. Exogenous risks, on the other hand, cannot be escaped, and may for example
be caused by physical phenomena. For an individual agent, the distinction between
exogenous and endogenous risk may not be very important. He simply realizes that future
prices are uncertain! Whether that is due to (exogenous) risk related to an uncertain
1
In the literature risk is used for this kind of uncertainty.
2
physical stock of the resource, or due to imperfect markets, is irrelevant. On the other
hand, in a Hotelling "world", i.e. when analysing market equilibrium, the resource price
is not a primary source of uncertainty, although the resulting (endogenous) price path is
surely stochastic.
From the works of Arrow-Debreu we have learned that under ideal conditions,
uncertainty (risk) simply implies that one new dimension is added to the analytical model
describing market equilibrium. Conceptually, the uncertainty is taken care of by
introducing contingent goods. This means that each market transaction is made
contingent on which of the "states" for the uncertain outcomes that materializes. The
commodity prices are accordingly defined as contingent prices.
It should be noticed that when there is a complete set of contingent prices, there are no
market-induced price uncertainty (cf. previous point). In other words: with perfect
markets, the behaviour of economic agents implies that all endogenous risk are removed
through diversification, so that a Pareto optimal allocation of resources is achieved. On
the other hand, in the absence of perfect contingent markets, there will be endogenous
uncertainty "left" that the society is not able to eliminate through insurance, stock
markets or other hedging arrangements.
In the absence of a complete set of forward and contingent markets, decisions have to be
based on assessments of probability distributions. The common decision rule is then
based on the von Neumann-Morgenstern theorem of expected utility, see e.g. Arrow
(1965). Briefly, this says that given certain axioms, the optimal choice between uncertain
prospects is equivalently obtained by maximizing expected utility. Formally, let the

vector y = ( y1 ,..., y N ) denote the set of possible outcomes for a stochastic variable y , with
a corresponding set of probabilities ~ = (  1 ,..., N ) . The expected value of y is
y  1  i yi .
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The von Neumann-Morgenstern theorem then states that a preference ordering of different
values of the prospect y is equivalent to maximizing expected utility, expressed as
N
(4.1)
U  E[u ( y )]    i u ( yi )
i 1
Given this decision rule, the form of the utility function expresses the attitude towards
risk held by the decision maker. If the utility is (strictly) concave, the agent is said to be
risk averse, since he in this case will prefer a certain outcome equal to the expected value
of y rather than the uncertain prospect. Formally, concavity of the utility function implies
that
N
(4.2)
u ( y )    i  u ( yi )
i 1
If, on the other hand, the utility function is convex, the decision maker prefers the

uncertain prospect y rather than the expected value y .
Intuitively, risk aversion implies that the behaviour of economic agents becomes more
"cautious". Applied to the problem of extracting a nonrenewable resource, the expected
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effect is a slower speed of extraction. As we shall see below, this is also a common result
under a wide class of circumstances.
Assuming risk aversion, a certainty equivalent is defined as a value of y yielding a utility
that exactly matches expected utility, i.e.
N
(4.3)
u ( y )    i  u ( yi )
i 1
With a concave utility function, we will in general have y < y .
A common functional form for the utility function used in economic uncertainty analysis
is the exponential function, expressed as
(4.4)
u ( y)  A  B  e  y
where A, B and  are constants. (A and B just represent an arbitrary linear transformation
and are irrelevant.) The parameter  measures the degree of (absolute) risk aversion2. The
exponential utility is analytically convenient, since, given that the probability distribution for
y is normal, it can be shown that
(4.5)
1
E[u ( y )]  u ( E[ y ]    var( y ))
2
This means that the problem of maximizing expected utility can be transformed to a
deterministic problem of optimizing with respect to the expectation of y and its variance.
From (4.3) it should be clear that the argument of the utility function on the right hand side
of (4.5) is a certainty equivalent, so that with an exponential utility function,
(4.6)
1
y  E[ y ]    var( y )
2
Within a dynamic framework, a key element in decision making under uncertainty is
flexibility. The basic idea is common sense: since usually information and knowledge of
uncertain events increase over time, plans should not be decided once and for all. Hence,
there is a positive value related to keeping options open, that have to balanced against the
potential loss of income of not initiating a project immediately. In particular, the aspect of
flexibility becomes important when e.g. investment decisions are irreversible.
Analytically, the aspect of flexibility implies that optimal decisions are formulated in
terms of strategies. These are rules of the kind "do this if that happens". Information is
utilized as soon as it arrives, and based on the most recent knowledge, actions may be
taken and plans revised.
The aspect of flexibility has a strong appeal when discussing issues of management of
exhaustible resources. For many natural resources, investment decisions are irrevocable,
or only reversible at very high costs. The option value related to the fact that the amount
of information increases over time may therefore be significant. In the literature, see e.g.
2
This is generally defined as -u (y)/u (y) .
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Dasgupta and Heal (1979), it is demonstrated that when investments are irreversible,
there is a tendency towards overinvesting initially when random variables are replaced by
expected values. Below, this effect is briefly discussed within a stochastic version of
Gray's model of optimal timing of extraction of an individual mine.
Resource extraction and price uncertainty.
How are individual resource owners affected by the presence of price uncertainty?
Throughout we have assumed the individual producer to be a price taker. We start by
presenting a simple static model of producer behaviour, with the extraction technology
represented by a general cost function b( R), (b  0, b  0) . The output price, p , is
stochastic with expected value and variance equal to  and  2 , respectively. Let
furthermore the producer's net income be denoted by y . We assume that the producer is risk
averse, and that this attitude is adequately represented by the exponential utility function
(4.4) above. This means that the analysis can be performed by replacing the uncertain net
revenue y by its certainty equivalent, defined as
(4.7)
1
y    R  b( R )    R 2   2
2
The last term in (4.7) should be interpreted as a risk premium, and is seen to be
proportional to the variance of the uncertain income, R 2   2 .
The optimal level of production is found as the solution to the equation dy/dR = 0 ,
yielding the following condition:
(4.8)
  b( R)     2  R
(4.8) is a rather straightforward modification of the condition for profit maximization that
price should equal marginal costs. In the present case of price uncertainty, a (marginal)
risk premium should be added to marginal production costs, and the resulting total should
then equal the expected output price. Compared to the traditional deterministic case, the
marginal cost schedule is shifted upwards. As is rather intuitive, with a stochastic output
price, and given that the producer is risk averse, the production level is reduced compared
to a situation where there is no uncertainty (and the output price has its expected value).
Is this result replicated in a dynamic model describing the behaviour of a resource
extracting firm? Broadly speaking the answer is yes, in the sense that under a wide class
of circumstances uncertainty regarding future prices implies that the production profile is
adjusted so that extraction is restricted in an initial period, compared to the full certainty
case. If that is the case, price uncertainty also implies a higher (expected) scarcity rent of
the resource stock.
However, as noted above, within a dynamic setting the behaviour under uncertainty is
fundamentally different from that of a static model: When there is uncertainty regarding
future events, decisions should be described in terms of strategies. In the following we
seek to give some insight in how the dynamic equilibrium of a resource extraction firm is
affected by price uncertainty.
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Generalizing the Gray model as set out above to cover the case of uncertainty in the
output price, the problem of the resource owner is now to maximize expected discounted
profits, i.e.
T
(4.9)
Max E{ [ pt Rt  b( Rt )]e  rt dt}
Rt
0
again under the restriction that accumulated extraction cannot exceed the initial resource
stock, S 0 . W assume that while the resource price "today" is p , the resource owner knows
that at a future date, T  , the price is either pU with probability  U , or p L with probability
 L  (1   U ) . Analytically, the solution to the maximization problem is found by dynamic
programming, i.e. working backwards from complete exhaustion at some (uncertain) end
point.
The equilibrium can be discussed by using the properties of the (maximized) value
function of the corresponding deterministic maximization problem. For a given (certain)
price p , a given stock S0 and a given horizon T the value function related to the problem
(4.9) is defined as
T
(4.10)
V ( S0 , p, T )  Max  [ pt  Rt  b( Rt )]e  rt dt
Rt
0
The problem facing the producer is basically twofold: Firstly, he must decide how much
of the total resource stock ( S  ) to deplete before reaching T  . Secondly, the rate of
extraction in each period has to be determined. Regarding the latter issue, we first make the
observation that within each sub-period, the resource rent has to increase at rate equal to the
rate of interest, r .3 What remains is to determine the optimal level of resource extraction up
to T  , i.e. S  . This problem can be solved by restating the optimization problem (4.9) as to
maximizing with respect to S  the following expression:
(4.11)
V ( S , p, T )  e  rT 

i  L ,U
i
V ( S0  S , p i , )
The first order condition of the maximization problem is
(4.12)
V ( S0  S , pi , )
V ( S , p, T )
 e rT    i
S 
( S0  S )
i  L ,U
The left hand side of this relation has the interpretation as the shadow price of the
resource stock extracted through the period [0, T ] , which is equivalent to the initial
resource rent of S  . Multiplying on both sides with e rT  yields on the left hand side the
"spot price" for the resource rent at T  . Similarly, the term V ( S0  S , p i , ) /  ( S0  S ) on
In the period preceding T  , there is no uncertainty regarding the current price or its growth rate.
Accordingly, the resource rent will have to match the return on financial assets, as we have analysed earlier.
After T  , the uncertainty regarding the market price is revealed, and the same argument can be applied.
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the right hand side is the resource rent at time T  of the resource remaining stock. By
transformation, the optimality condition (4.12) can therefore be written as
(4.13)
p  b( RT  )   U  ( pU  b( RT  ))   L  ( p L  b( RT  ))
This equation expresses that the rent on the resource that is extracted just before reaching
T  must equal the expected rent on marginal resource unit just after the information about
the uncertain price is revealed. (4.13) is thus a zero arbitrage condition at T  , so that there is
no "jump" in (expected) resource rents between the two periods of extraction.4
The optimal extraction program is such that before reaching T  , the resource rent
increases at the rate of interest. The end point of this rent trajectory is determined by the zero
arbitrage condition (4.13). At T  , the new price, either pU or p L is observed, the market
price jumps to its new level, and the extraction program continues with an exponentially
increasing resource rent. As before, the initial level of extraction, and thus the initial rent, is
determined by the condition that accumulated production must just equal the initial stock.
It should be noted that the extraction program is described in terms of strategies.
Alternative paths can be "calculated" by the producer at time zero. However, the actual
decision on the level of extraction after T  is delayed until the uncertainty regarding the
final price is removed.
Hartwick and Yeung (1985) shows that for a given stock, a resource extracting firm
prefers a stochastic price with mean E[ p ]  p rather than facing a certain price p .
Formally, this follows from the fact that the value function (4.10) is strictly convex in p.5
This implies that when the output price fluctuates around the average price p , positive
deviations more than outweigh negative deviations of similar size. As a consequence of a
higher expected return, the resource owner will seek to allocate a larger share of the total
stock to be extracted after T  under price uncertainty compared to when the resource can be
sold at the risk free price p . Accordingly, a smaller resource stock is left for extraction in
the period up to T  . Thus, the resource becomes more "scarce", implying a higher initial
rent than a corresponding extraction program under certainty.
In the simple static model above, the important conclusion was that price uncertainty
combined with risk aversion induced the producer to restrict production. In the present
dynamic model of a resource extracting firm, a similar conclusion is reached, but without
4
Equation (4.13) is a basic condition for Arrow-Debreu contingent commodity prices. ( p  b( xT  )) is
U
the resource rent at T  conditional on the new price being equal to p . Similarly, ( p  b( xT  )) is
U
L
interpreted as the rent occurring at T  contingent on the price p to materialize. In a general setting involving
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contingent goods, the price of a commodity "delivered" at time T  equals the sum of the contingent prices of
the different "states", reflecting that transactions at any date are constrained by the common information
available at that time.
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Define
p as a linear combination of p 1 and p 2 , i.e. p   p1  (1   ) p2 . A function V(p) is strictly
convex in p if and only if V(p) < V( p 1 )+(1 -  )V( p 2 ) .
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assuming risk aversion. As illustrated, a risk neutral resource owner facing price
uncertainy will seek to slow down initial extraction.
As in the deterministic case, the optimal solution in the stochastic version of the Gray
model generally implies that resource extraction is extended over some period of time.
Above we presented briefly the special case of constant unit costs of extraction, and
concluded that in such a case the producer (if possible) will choose to empty the deposit
completely at a specific point in time. Assuming an exponentially increasing price,
extraction will take place when the price path reaches the "trigger price" p* given by
equation (1.3). Adding price uncertainty to this analytical framework, the problem is stated
in a similar way. Another way of attacking this problem is by making the assumption that
the output price of the resource develops according to a Wiener process, i.e.
(4.14)
dpt
  pt    pt  dz
dt
The last term in (4.14) accounts for the stochastic variations around an exponential trend
(growing at rate  ). The objective of the resource owner is to maximize expected value of
the stock. This is a so-called optimal stopping problem. The producer observes the
development in the market price, and "stops waiting" - carries out extraction - when the
fluctuating price reaches a sufficiently high level - the "trigger price". Let this be denoted by
*
p̂ . It can be shown that pˆ > p ; the trigger price under uncertainty is higher than in the
deterministic case. Again, uncertainty has the implication that the expected value of a
project increases compared to a case with no fluctuations in prices. As a consequence,
projects may be postponed in order to reap a possibly higher profit.
The latter model emphasizes the importance of meeting uncertainty with flexible
decisions, i.e. strategies. The example may be used as a demonstration of the option value
related to uncertain, irreversible projects. Assume that the price at a given moment is p0 ,
and that this is a break-even price. By waiting for a higher price, the expected value of the
resource may increase, since with the price process in (4.14) the producers know that at
some point in time, the price will increase above today’s level. There is a positive option
value from postponing extraction if the (expected) increased value of the stock exceeds the
loss from discounting. At the trigger price p̂ , the loss of interests from further
postponement just equals the expected profit gain. At that point, extraction is carried out.
The size of the resource stock is uncertain.
The case of uncertainty regarding the size of the initial deposit may be analysed in a way
similar to that of price uncertainty. However, as pointed out by Hartwick (1989) there are
important differences between the two problems. Let us again simplify and assume that
there are two "states of the world", i.e. two possible sizes for the resource stock. The
resource stock S is either S 1 and S 2 ( S 1 < S 2 ), with probabilities  1 and  2 respectively.
In the further discussion it is essential to distinguish between different cases regarding how
and when information about the uncertain stock is gained.
i)
The producer observes the actual stock size at a given time, say T .
ii)
The information is revealed when accumulated extraction has reached S 1 .
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iii)
The resource owner can get knowledge about the actual size by investing in
exploration.
With information arriving at an exogenous date ( i), the equilibrium conditions are
derived in a way quite similar to the case of output price uncertainty. The problem for the
resource owner is to decide how much to extract between the initial time and T ; in the
sub-periods before and after this point the resource rent has to increase at rate equal to r .
Again, the overall equilibrium implies equality between expected resource rents at T , i.e.
with equilibrium conditions analogous to equation (4.13).
A comparison of this solution with the optimal extraction program under complete
certainty, the conclusion is again that uncertainty motivates the producer to cut down on
the initial extraction, thus yielding a higher initial resource rent. Without providing any
rigorous proof, this may be realized by noting the following points:
First, it can be shown, see Aslaksen and Roland (1983), that the value function (4.10) is
concave in the initial resource stock. This implies that Jensen's inequality prevails:
(4.15)
E[V (S )]  V (S )
where S   1  S1   2  S 2 is the expected value (size) of the resource stock.
Second, we can apply a general result for the optimal allocation of a given resource stock
S 0 = S 1 + S 2 . Let T 1 denote the time when S1 is completely depleted along the optimal
program, and V ( S 2 ) the optimal value function for the resource stock extracted in the
period after T 1 . Hoel (1982) has shown that if for some reason there is a negative shift in
V( S 2 ) , it is optimal for the producer to restrict extraction so that T 1 increases.
Finally, from Jensen’s inequality we know that E[V( S 2 )] < V( S 2 ) , stressing the crucial
point: In the present model, the introduction of stock size uncertainty can be regarded as a
negative shift in V( S 2 ) .
We then can conclude that uncertainty about the size of the resource stock implies an
initially higher rent and slower rate of extraction (compared to the certainty case where
S0  S   1S1   2 S ).
In ii) by assumption the resource owner gains no information about the size of the initial
stock until the accumulated production equals the minimum size of the stock, S 1 . The
problem facing the producer therefore includes choosing the optimal depletion period for
S 1 . Given the optimal choice, in this first period the extraction program follows the now
well known pattern, with rent increasing at the rate of interest. At T 1 the producer observes
whether the deposit is actually emptied, or that the stock size actually was S 2 . In the former
case, production falls to zero. Alternatively, there is still S 2 - S 1 of the resource left in the
ground, and the producer faces a traditional deterministic Gray problem.
To sketch the optimal solution, let V ( S1 , T ) denote the maximum value function for the
period up to T , while V (S2  S1 , ) is the maximum value function for the rest of the
lifetime of the deposit contingent on the initial stock being S 2 . If S 1 is all there is, the
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resource value in the second period is zero. The optimal depletion time, T , is found by
maximizing the total discounted expected value of the resource, i.e.
(4.16)
Max{V (S1 , T )  e rT (1  0   2 V (S2  S1 , )}
T
The solution to the optimization problem thus has to satisfy the following necessary
condition:
(4.17)
V ( S1 , T )
 re rT { 1  0   2  V ( S2  S1 , )}
T
The left hand side of (4.17) is the change in the value of S 1 (measured at time zero)
obtained by extending the period of depletion marginally. The right hand side is the interest
rate multiplied by the expected value of the remaining reserves, discounted to time zero. In
equilibrium, these two "returns" have to balance.
(4.17) is a generalization of the end point condition in the corresponding deterministic
model of a resource extraction firm. As shown above in equation (1.5), in the traditional
Gray model the optimal extraction horizon is determined by the condition that marginal
costs equals average cost. It can be shown that equation (4.17) can be transformed to
(4.18)
[ p  b( RT )]  RT  r   2 V (S2  S1 , )
From this equation, it is seen that in the present model the "terminal" extraction rate of
S 1 exceeds terminal production in a model where the resource stock is limited to S 1 with
certainty. Since in the extraction program both cases is characterized by an exponentially
increasing resource rent, it may be concluded that the extraction path in the present
stochastic model exceeds that of a certain S 1 . Naturally, this reflects that the producer
realizes that, after all, there is a probability  2 for the resource stock actually being S 2 . On
the other hand, by using the same kind of arguments as above, it can be shown that initial
extraction is restricted compared to the level chosen when the initial stock size equals the
average level, S =  1 S 1 +  2 S 2 .
In iii), the resource owner can reveal the uncertainty by investing, say a given amount k .
Thus, the determination of time for gaining the information about the stock size is again part
of the optimization problem. The solution of this model involves both aspects that are
focused in the previous two cases. More precisely, the problem is to determine
simultaneously the optimal time, T , to "sink" k and the accumulated extraction up to this
point. In the sub-periods ahead of and after T , rent is increasing at rate r . The time to
invest, T and the corresponding optimal accumulated production, are determined by
equations analogous to (4.13) and (4.18) respectively. This latter model is discussed in more
detail in Dasgupta and Heal (1979). A comparison of the optimal contingent extraction
program with corresponding deterministic models, leads to the intuitively reasonable
conclusions: In a model involving uncertainty, initial extraction is higher than when the
problem is to manage a certain S 1 , but lower compared to the case with initial stock equal to
the average size,  1 S 1 + 2 S 2 .