R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
The Economic Theory of Exhaustible Resources
Most of the dominant energy sources of today – oil, natural gas,
uranium and coal – are non-renewable or exhaustible. These
resources are formed by geological processes that typically take
millions of years, so we can view these resources for practical
purposes as having a fixed stock of reserves. That is, there is a
finite amount of the mineral in the ground, which once removed
cannot be replaced.
Time plays an important role in determining how to exhaust a
mine having a finite quantity of a resource (say coal in a
coalmine or oil in a oil well). A unit of ore extracted today
means less in total is available for tomorrow. Each period is
different because the stock of the ore remaining is a different
size. We are interested in studying how quickly the mineral
should be extracted – what the flow of production is over time,
and when the stock will be exhausted.
We now study the simplest model of the theory of the mine.
Obviously, decisions regarding the exploitation of mine are
dynamic: they are spread over multiple time periods. A manager
of a mine will have to bother about not only the current profit
but also for future profits. With production using non-renewable
resources, the decision to produce and earn a profit of 0 today
1
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
necessarily precludes the ability to produce and earn profits in
future. To provide the trade off, the manager should apply a
discount rate to future profits since a dollar earned today is
worth more than a dollar earned tomorrow.
The present value (PV) of profit streams, 0, 1,…n, is
calculated as,
PV   0 
1
1 r

2
1  r 2

n
1  r n
The profit streams, 0, 1,…n should be chosen such that PV is
maximized.
In static analysis (involving no effect of time on profits),
classical economic theory states that profit maximization is
achieved by setting the production level where marginal cost
(MC) is equal to the marginal revenue (MR) received. Here the
marginal cost consists of marginal production cost (capital,
labour and materials) of producing the last unit of output. Let us
denote the marginal production cost as MCp, where MC =MCp.
In the case of exhaustible resource, the resource manager must
trade off the opportunity value of selling the resource today
versus the opportunity value of selling it at some further time.
This is precisely the notion captured by the concept of User
Value or User Cost. Thus the user cost in period i (Ui) reflects
2
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
the opportunity value of producing a unit of output in that
period.
Thus for a firm dealing with exhaustible resource production,
MC =MCp + Ui.
What are the properties of user cost? It reflects the opportunity
cost of extracting the resource. Obviously, it depends upon the
resource left at the mine. It tends to reduce as more quantity is
produced. Finally, when the mine is exhausted, user cost is zero.
Note that the term exhaustion of a mine is an economic concept.
Exhaustion does not mean that the mine will no more have any
ore. A mineral deposit is economically exhausted when the
marginal cost of production exceeds the value (price) of the
mineral. If the ore is continued to be extracted beyond this point,
the mine owner will incur loss.
Note that user cost is also called as profit or resource rent or
royalty price in the literature on exhaustible resources.
3
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
User cost
Cost $
Price line
Marginal cost
User cost
Quantity Q
4
Marginal
production
cost
Average
production
cost
User cost
is zero at
exhaustion,
when
MCp>price
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us now consider a mine that has option to exhaust it in two
periods. The firm will maximize the PV of profits, subject to the
condition that the amount extracted in both the periods should
be equal to the total ore available in the mine.
max 
 0 

1
1
1 r
P0Q0  C (Q0 ) 
1
P1Q1  C (Q1 )
1 r
subject to
Qo  Q1  Q
where Pi and Qi represent the price and quantity of mine in
period i, and Q is the total quantity of resource available in the
mine. Note that Q0 and Q1 are the decision variables here, and Q
is a constant.
We can solve this optimization problem analytically using the
Lagrangean method.
5
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Form the Lagrangean,
L  P0Q0  C (Q0 ) 
1
P1Q1  C (Q1 )   Qo  Q1  Q 
1 r
The optimality conditions are given by the following.
L
P0Q0  C (Q0 )
0
 0
Q0
Q0
L
 1   P1Q1  C (Q1 )
0

  0
Q1
1

r

Q



1
L
 0  Q0  Q1  Q1  0

Note that,
P0Q0  C (Q0 )
P0Q0  C (Q0 )


Q0
Q0
Q0
 MR0  MC0p
 U0
Thus, the above optimality conditions lead to,   U 0 
U1
.
1 r
This is the relationship that governs the behaviour of user cost
over time. This means that the user cost must rise at the rate of
interest if the net present value of profits from the resource is to
be maximized.
6
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
The same relationship can be extended for more periods. For n
periods, the relationship is,
U0 
U1
U2
U n 1


2
1  r 1  r 
1  r n 1
Note also that,
Ui
PiQi  C (Qi )
Qi
 i

Qi
 Profit per unit (or, marginal profit) in period i

Hence, the above relation for user cost shows that profitmaximizing pattern of extraction should occur such that the
marginal profit increases at the rate of interest. This relation is
not difficult to understand.
What will happen if the marginal profit (i.e., user cost) increases
slower than the rate of interest? The owner of the mine will try
to extract all the resources from the mine as quickly as
technically feasible, sell it, and invest the money in some other
assets whose value would rise at the rate of interest (e.g., a
savings account). He is better off by doing this.
On the other hand, if marginal profit rises faster than the rate of
interest, the entire stock of ore would be held in the ground until
the last moment in time and then extracted. In this case, the
7
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
mine is worth more unextracted because the rate of return on
holding the ore in the ground exceeds the return on alternative
investments.
Thus, unless the marginal profit of the mine is growing at
exactly the same rate as the value of other assets, extraction will
either be as fast as possible or deferred as long as possible. To
have mineral extraction, hence, the marginal profit must be
growing at the same rate as that of alternative assets.
So far we assumed that the time periods are discrete. The above
formula can also be generalized for the continuous case. It t is
continuous, we can write,
U t   U t 1  r 
or
U t   U t

 rU t
Taking limits on both sides,
lim
 0
U t   U t

 lim rU t
 0
dU
 rU t
dt
whose solution is, U t  U 0e rt .
8
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
It is also possible to solve the above problem assuming
continuous case using the principles of dynamic optimization.
For the continuous case, the optimization problem should be
rewritten as follows.
T
max   q, t e  rt dt
0
such that
T
 q dt  Q
0
This can be solved using Optimal Control Theory or Calculus of
Variations.
Let us now consider a numerical example. Let us use the
discrete version of the model.
Let Q = 150 and r = 12%. Let the demand curve for Period 0 be,
p0 = 50 – 0.5 Q0, with the marginal cost of production to be $2
per unit. Similarly, let the demand curve for Period 1 be, p1 = 60
– 0.2 Q1, with the marginal cost of production to be $3 per unit.
Now, U0 = 50 – 0.5 Q0 – 2 and U1 = 57 – 0.2 Q1. For profit
maximization, U1 = (1.12)*U0 , or, 57 – 0.2 Q1= 1.12(48 – 0.5
9
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Q0). Solving, we have Q0 = 35.21, p0 = 32.40 and U0 = 30.40.
And, Q1 = 114.79, p1 = 37.04 and U1 = 34.04. This is the
optimal
combination
of
outputs.
Consider
some
other
combination, say, Q0 = 30 (which will keep p0 = 35).
Correspondingly, Q1 = 120 and p1 = 36. U0 = 33 = U1 = 33, and
hence, U1 < (1.12)*U0. This will encourage mine owners to
produce more Q0 in the current period till the equilibrium U1 =
(1.12)*U0 is reached.
Let us now consider a situation where the marginal cost of
production is zero. This is a reasonable assumption when we
consider the extraction of oil from oil wells. Once some fixed
costs are incurred, additional oil production can take place with
negligible additional costs. This assumption of zero marginal
costs has much relevance in the literature on economics of
natural resources as most of the studies have concentrated on the
most important natural resource of this century, namely oil.
When MCpi = 0, then Ui = MRi – MCpi = MRi = pi
Thus, whatever we have calculated in the context of user costs
apply directly to prices. Thus, when MCp = 0,
10
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
p0 
p1
p2
p n 1




1  r 1  r 2
1  r n 1
pt  p0e rt
dp
 rp
dt
pt
r
p
This result is often called Hotelling's r-percent rule or
Hotelling's rule.
11
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Dollars
Hotelling's rule
p t =p 0 e rt
Time
12
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Variation of User cost and Price
Dollars
Price
U t = U 0 e rt
Marginal cost
of production
Time
13
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Hotelling's rule is based on three key assumptions:
1. Zero marginal production costs,
2. Long term profit maximization, and
3. Perfectly competitive market.
Hotelling's rule says that prices will tend to rise in a smooth,
predictable manner with the rate of interest. In order to entice oil
producers to hold oil for future periods, they must receive a
return on exactly r per cent per year. Prices cannot rise faster
than r per cent per year since current production would cease in
anticipation of a return greater than r per cent, driving up current
prices, which would induce shifting of production toward the
current period and restore the rule. The reverse logic holds if
prices rise slower than r per cent per year. Producers will raise
current production which will lead to lower price levels, and
thereby induce shifting production to future periods, thus
restoring the equilibrium.
Let us now continue the numerical example we saw earlier. Let
the marginal cost of production be zero. Now, p0 = 50 – 0.5 Q0
and p1 = 60 – 0.2 Q1. For profit maximization, p1 = (1.12)*p0 ,
or, 60 – 0.2 Q1= 1.12 (50 – 0.5 Q0). Solving, we have Q0 = 34.21
and p0 = 32.89. And, Q1 = 115.79 and p1 = 36.84. This is the
optimal
combination
of
outputs.
14
Consider
some
other
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
combination, say, Q0 = 30 (which will keep p0 = 35).
Correspondingly, Q1 = 120 and p1 = 36. Obviously, p1 <
(1.12)*p0. This will encourage mine owners to increase Q0 in the
till the equilibrium p1 = (1.12)*p0 is reached.
15
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Price
60
Choke price, p'
50
40
p0 =
32.8
30
20
10
0
0
q0 =
10
20
30 34.2 40
Quantity mined in Period 0
16
50
60
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Price
80
Choke price, p'
60
40
p1 =
36.8
20
0
0
q1 =
50
100 115.8
Quantity mined in Period 1
17
150
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
At any price, given the demand curve, there is likely to be a
price p' at which no one will be willing to buy more of the
mineral. This price p' is generally called the choke price. For
example choke price is 50 for Period 0 and 60 for Period 1 in
our example above. Choke price occurs because other
competitive resources may become cheaper than the given
resource. For example if oil prices continue to escalate, at one
point of price, cost of using oil to derive energy may become
costlier and other energy technologies such as solar or nuclear
may become more competitive. These technologies are often
called the backstop technologies of oil.
Ideally, the mine owner would seek to have a stock of mineral
go to zero at exactly the point of zero demand. Therefore, Given
a choke price, the planner would seek to have the last unit of
output extracted at p'. To do otherwise deprives society of
maximum benefits.
It is possible to use Hotelling's rule to determine the optimal
extraction path (quantities to be mined over time) for a given
choke price. Here, the time periods needed for full extraction is
an endogenous variable. This is left as an exercise for students.
Hotelling's rule provides a very fundamental relationship for the
price behaviour of exhaustible resources. The paper was
18
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
published in the 1930s. However, there have been claims and
counter claims about the validity of Hotelling's rule in practice.
We shall study a few empirical studies later on. However,
Hotelling's rule is based on very simplistic rules discussed
earlier. Let us now relax some of the assumptions and
experiment how the rule changes.
Expectations
Hotelling's rule is based on a unique set of expectations about
the future, and present supply and demand conditions. With a
certain set of assumptions of future prices, profits, future supply
and, current demand and supply conditions, the initial price is
kept at p0, and it continues to rise according to Hotelling's rule
(at the rate r). But, at time t1, say, the expectations of producers
have changed. This can happen on several counts. Expectations
are heavily influenced by the expected size of the resource base.
What will happen if new resources are identified in large
quantities? In the absence of any change in demand patterns,
current and expected future market prices are reduced. However,
if producers expect a much greater growth of future demand
than previously expected, the prices will be raised.
19
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Effect of expectations on Hotelling's rule
Dollars
High
New
expectations
Low
Old
expectations
t1
Time
20
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
These expectations have played an important role in fixing the
prices of exhaustible resources, especially the world oil prices,
in practice. Note that price of oil has not varied smoothly as
predicted by the Hotelling's rule. Perhaps, expectations have
played a key role in the sudden jumps observed in 1973, 1979
and in 1991.
Draw a figure of actual price of oil by hand.
21
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Apart from expectations, still other explanations on the
movement of prices in practice can be found by relaxing the
assumptions behind the Hotelling's rule. Let us now do this.
An increase in costs of production (extraction costs for oil)
If costs of production are positive, note that prices do not follow
Hotelling's rule, but the user costs will. For a given price with
positive extraction costs, the present profit will reduce. This will
induce producers to produce less, as they do not have the
incentive to produce more. The lower quantity produced in the
current period will, in turn, increase the current price relative to
the zero cost case.
For example, if we assume a constant marginal production cost
of $5 dollars per unit of resource in our latest example
(Hotelling's rule in page 14), we will find that Q0 = 33.42 which
is less than 34.21 for the case of zero marginal cost of
production. Correspondingly p0 = 33.3 (more than 32.89 for the
zero cost case). Also, the ratio of p1 to p0 is 36.7/33.3 = 1.10,
which is lower than the rate of price increase in the zero cost
case (equal to 1 plus the rate of interest, 1.12).
Thus price rise will be slower with positive production costs.
Because production in every time period will now be smaller,
22
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
the lifetime of the mine will now increase. Thus, an increase in
production cost results in lengthening the life of the mine.
Changes in patterns of demand
Changes in demand are a function of the level of income,
technical substitution possibilities, and relative prices. Under
competitive economic conditions, the effect of an increase in
demand will be to increase the price of the resource in all
periods. This will result in slight re-alignment of optimal
quantities to be produced in all periods.
Assuming a higher price elastic demand curve for period 0, say,
p0 = 50 – 0.25Q0, we now have Q0 = 54.2, which is higher than
the earlier lower elastic demand curve. This is because, the
larger production in period 0 will not diminish the prices very
much compared to the case of a lower elastic demand curve, and
hence the mine owners are encouraged to produce more.
Changes in interest rate
Suppose that the rate of return on investing assets alternative to
mineral extraction raises. If interest rate adopted by oil
producers is lower than the rate they could earn by investing in
other assets (market rate), oil producers will tend to shift all the
23
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
production to the present and extract more ore today (compared
to a smaller market rate). This will in turn reduce the current
market price. Thereafter less ore will be extracted so that the
rate of return on the remaining ore rises now at the higher
general interest rate. As an increase in interest rate will tend to
shift production to the present than in the future, the life of the
mine will be reduced.
Let us assume a higher interest rate (r = 20%) in our example.
Continuing with our calculation, we find that new Q0 = 37.5
which is more than 34.21 for the case of 12% interest rate.
Correspondingly p0 = 31.3 (less than 32.89 for 12% rate).
Size of the resource base
If the stock is large enough, an extracted resource is much like a
conventional product. That is, the resource will have zero user
cost and price will be equal to the marginal production cost. As
the stock diminishes and if there are no substitutes in sight, the
fact that the resource is exhaustible becomes important. At his
stage, there is a high user cost.
Consider the picture in the next page. From the time period 0-t,
the resource stock is so large that its price is almost constant
(assuming a constant marginal production cost). At t1, its
24
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
exhaustibility becomes critical and hence its price jumps and
thereafter (price or user cost) begins to rise at the rate of interest.
At t2, new reserves are discovered which drives prices down. At
t3, because of expected high future demand and low reserves,
price suddenly shoots up.
Dollars
Effect of size of resource base
t1
t 2 Time
25
t3
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
We can analyze the effect of time and size of resource base on
resource prices more mathematically. We have,
ut = u0 ert.
Then,
pt = ut + ct,
where ct is the marginal cost of extraction. Assume that the
marginal cost of extraction is constant, i.e., ct = c. Hence,
pt = c + u0 ert.
Let us assume an iso-elastic demand curve, say, pt = qt-, where
 is the constant price elasticity. Thus, qt = pt-1/. We have,

 qt dt  Q
0


  c  u0e rt

 1
dt  Q
0
Assuming unit elasticity of demand ( = 1), we have,

dt
Q
rt
c

u
e
0
0

e  rt dt
   rt
Q
 u0
0 ce

Integrating the LHS, we have,
1  c  u0 
  Q
ln
rc  u0 
 u0 
c
e rQc  1
26
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Hence,
ce rt
pt  c  u0e  c  rQc
e 1
rt
We can make a few observations using this result.
Let there be a huge stock of the resource, i.e., Q is large. In this
case, u0 becomes small and hence pt  c initially. Thus, when
stock of the resource is large, its user cost is small and hence
price equals marginal costs. The exhaustible resource behaves
like a conventional commodity, whose unit cost of production is
c. But, as time increases, the fact that the resource is exhaustible
ce rt
begins to bite. It t is large, rQc
becomes very large and
e 1
hence the contribution of c in determining pt is negligible.
27
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Presence of a backstop fuel
A backstop fuel is a substitute for the conventional exhaustible
resource which, though not cost effective at present, may
become competitive at some price in future. A backstop fuel can
be a renewable energy source such as solar energy or nuclear
fusion that can supply unlimited quantities of energy.
Alternatively, they may be the unconventional crude oil from tar
sands, oil shales, coal etc., which though non-renewable, are
available in such large quantities that their user costs are
effectively zero. Assume that these backstop fuels are infinitely
elastic at the choke price p' and that virtually unlimited supplies
of the backstop fuels are available at the price p'.
At the choke price, the backstop fuel will take over from the
exhaustible resource and the producers will like to exhaust the
mine at the choke price in order to derive maximum benefit
from the mine. We have already seen how p' can fix the life of a
mine and the initial price p0.
Backstop fuels can affect the pricing of exhaustibe reources
today. Assume that solar energy is available as a backstop fuel
for oil at a constant cost of $70 per barrel or oil equivalent, i.e.,
p' = 70. Assume that existing oil reserves are sufficient to menad
the demand in the next thirty years. Then, we can estimate what
28
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
should be the price of oil today (assuming competitive markets
and applicability of Hotelling's rule).
We have, using Hotelling's rule, p0 
p30
.
1  r 30
If r = 10%, p0 = 70/(1.1)30 = $4.01.
If r = 5%, p0 = $16.20
If r = 12%, p0 = $2.34
Calculations such as these are obviously highly subjective and
inexact. However, our calculations indicate that the present oil
prices (around $25-30 per barrel) appear to be well above the
levels implied by a competitive market.
If marginal costs of production increases from nearly zero today
to say $10 per barrel in the thirtieth year, then, Hotelling's rule
modifies as, p0  0 
p30  10
.
1  r 30
If r = 10%, p0 = 60/(1.1)30 = $3.44
If r = 5%, p0 = $13.82
If r = 12%, p0 = $2.00
29
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Deposits of differing quality
Let us assume that there are two different mines of the same
resource with different quality (characterized by differing
production costs). The difference in production costs may be
due to the ore quality or the thickness of the seam, but it can also
be the same quality ore with different distances from a central
market (transport costs will be higher for the mine located far
away from the market).
Consider the deposits with differing costs within a competitive
industry. Mine 1 has c1 extraction costs and has s1 total reserves.
Similarly, mine 2 has c2 extraction costs and has s2 total
reserves. Let c2 > c1. Obviously production begins at mine 1 first
as for any given price mine 1 will enjoy more profits than mine
2. In fact, if the price is less than c2 (but greater than c1), mine 2
will incur loss if it begins production. Hence mine 2 will have to
wait till all the deposits in mine 1 are exhausted.
30
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Dollars
Deposits of differing quality
p 20
c 20
p 10
c 10
User cost for mine 2
User cost for mine 1
0
t1
31
Time
t2
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Suppose that mine 1 gets exhausted attime t1. By then, the price
would have increased to, say p2 and if p2 > c2, mine 2 will begin
production. However, now the price will not follow old path, but
will begin a new path with the user cost for mine 2 at t1, i.e.,
(p2–c2) as the initial user cost, and it will rise at the rate of
interest. Mine 2 will continue production either till it is
physically exhausted or till a choke price is reached (when a
backstop technology will takeover).
Suppose now that the lower grade deposit is available in
unlimited quantities. Then, the lower grade deposit will not
behave like an exhaustible resource, but is rather a backstop
technology, for which user cost is zero. Hence, price equals cost
for this resource. As we have assumed constant costs, the price
of second deposit will be constant at c2.
32
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Dollars
Deposits of differing quality and backstop technology
p 2 = c2
p 10
c 10
User cost for mine 1
0
t1
33
Time
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
It is easy to find out u10 and t1 using the mathematical
framework suggested earlier. Assuming an isoelastic demand
curve with unit elasticity, i.e., pt = 1/qt, we have, total resource
in Mine 1, Q1, is,
t1
dt
rt
0 c1  u10 e
Q1  
Integrating the LHS, we have,
1  c1  u10 e rt
Q1 
ln
rc1  c1  u10 e rt
1
1



and c2  c1  u10 e rt . These two equations contain two unknowns
1
(u10 and t1) and hence can be solved. The following is the result.
u10 
c1
 c2  rQ c
e

c

c
 2 1 
1
1

 1

1 c c 
t1  ln 2 1 
r  u10 
For c2 = 10, c1 =1, Q1 = 50, u10 = 0.0022 and t1 = 69.16.
34
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Introduction of Taxes
Resource based industries are subjected to substantial taxation.
A large part of their profit may be pure rent, and this is
obviously a tempting target for taxation. Some taxes can be
levied on the extractive industry without distortion, and will not
affect the allocative efficiency. Other forms of tax may have
different impact on the economy.
We shall study the extent to which imposition of different taxes
affects the patterns of resource extraction, usually called "bias"
or "distortion" due to the tax.
Profits Tax
Of the many forms of taxation that affects resource depletion, by
far the most widespread is the profits tax or rent tax or royalty
tax or tax on user costs. Let  be the tax rate levied on mineral
profits. Hotelling's rule for two consecutive periods t and t+1 is
now written as,
 pt   c(1   )   pt  1  c(1   )
1 

1  r 
Because rent in each period is taxed exactly the same, the term
(1 – ) cancels from both sides. Thus with profits tax, there is
no way the mine operator can avoid the tax by shifting
35
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
production. Hence, profit tax is neutral or non-distorting to
extraction path. But, the tax does affect future discoveries. As
rent tax reduces the returns from explorations, a higher tax rate
provides a lesser incentive in investing in further exploration.
The above equation shows that there is no change in price or
quantity. Hence, the profit tax is completely absorbed by
producers, and not passed on to consumers.
Royalty
Extractive companies are normally required to pay a royalty to
the government of the country in which they operate. This is
typically a payment on the total revenues, not on profit. Now if
we equalize the present value of user costs, we have,
(1   ) pt   c  (1   ) pt  1  c
1 

1  r 
where  is the royalty tax. Note that, unlike the profit tax, it is
not possible to cancel the term (1 –  ) now. Let us redo our
calculation with  = 20%, r = 12% and c = 5. Using Hotelling's
rule, we can compute the equilibrium quantities (q0 and q1) and
corresponding prices (p0 and p1) as q0 = 33.22; q1 = 116.78; p0 =
33.39; and p1 = 36.64. For the case of zero royalty tax, q0 =
33.42 (we have computed this earlier). Note that there is a
36
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
reduction in q0 compared to zero royalty tax case. Thus, royalty
reduces the production in initial periods. This is because royalty
has an effect analogous to rising costs of extraction. Because
royalty is calculated on revenue, its effect can be reduced by
postponing production to future years, as the royalty on the
present value of rent can then be reduced. Royalty is a fixed
share of revenue. By postponing it to future years, one can
improve the present value of profits. Thus, the ultimate effect of
royalty tax is extension of the life of the mine. Lower amounts
of extraction and higher prices in the initial periods reduce
resource use, indirectly inducing conservation of the resource.
Note the equilibrium price p0 is higher now compared to the
same for zero royalty case (33.29). This means, a part of royalty
tax is passed on to consumers in the form of higher price.
Sales Tax
Suppose that the government announces a constant specific tax 
on the sale of the resource. Then, pt  ut  c   . Equalizing
present value of user costs over to successive periods, we have,
pt  c   
1
 pt 1  c   .
1 r
Note that the effect of sales tax is equivalent to increasing the
cost of production from c to (c + ). As before quantity
37
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
produced in the initial periods will be reduced, rising the prices
up. Thus, part of the tax burden is passed on to consumers in the
form of higher prices. Lower initial quantities induce
conservation, and obviously postpone the time to depletion of
the mine.
Now suppose that the government announces a sales tax
schedule of the form t = 0 ert, i.e., government sets an initial
tax 0 and allows the specific tax to grow at the rate of interest.
Now,
pt  c   t
1
 pt 1  c   t 1 
1 r
1
 pt 1  c   1  t 1

1 r
1 r
1
 pt 1  c    t

1 r

Note that t cancels on the both the sides. Thus this tax
introduces no distortion, as producers cannot avoid the tax bu
shifting production. However, this tax effectively reduces the
user costs, or the value of the unextracted resources. Resource
owners absorb the entire tax.
38
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Effect of Uncertainty
Our discussion so far alluded to the impact of uncertainty on
allocations involving exhaustible resources. Let us now study
the impacts of uncertainty.
Uncertainty arises in many different areas in exhaustible
resource use – stock size r the amount of ore in the ground,
effects of research and development (cost and arrival of
backstop technologies) etc. Before studying the effects of
uncertainty, let us briefly recapitulate the mathematics for
handling uncertainty.
Consider a situation where an individual is facing uncertainty
while making decisions – such as the stock market. Suppose that
he has a chance of receiving FIM 200 with a probability of 0.2,
and FIM 300 with a probability of 0.8. This means that if the
individual is in a similar situation, say, a thousand times, he will
receive FIM 200 two hundred times and FIM 300 eight hundred
times. Thus, on an average, he will get FIM 280 in this situation,
i.e., his expected payoff in this situation is FIM 280. One can
treat this expected payoff as a certainty equivalent of the
uncertain situation. However, an individual will consider an
assured payment of FIM 280 to be more valuable than an
uncertain prospect of FIM 200 with probability 0.2 and FIM 300
39
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
with a probability 0.8. To account for these subjective feelings,
the theory of expected utility has been developed.
The expected utility of the uncertain prospect can be written as
0.2 * U (FIM 200) + 0.8 * U (FIM 300) where U (.) is the
individual's utility of income. This can now be compared with
U (FIM 280).
Normally, most of the individuals are risk-averse. They will
consider the utility of an uncertain income to be smaller than the
utility of its certainty equivalent. That is, for risk-averse people,
0.2 * U (FIM 200) + 0.8 * U (FIM 300) < U (FIM 280)
In other words, for a risk-averse person, the utility function is
concave. A certain income of FIM 280 yields more utility than
the uncertain prospect of FIM 200 with probability of 0.2 and
FIM 300 with probability of 0.8. The distance FIM 280 – FIM Z
is called risk premium required for the individual to be
indifferent between the uncertain prospect and a certain FIM Z.
If the individual is more risk-averse, the concavity of the utility
function becomes larger and the risk premium becomes higher.
40
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Utility
Concave utility function
U (FIM 280)
0.2U (FIM 200)+0.8U (FIM 300)
FIM 200
FIM 280
FIM Z
FIM 300
41
Income
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Coming back to our study on resource extraction, let us first
study the effect of uncertainty in the stock size or the price
behaviour. Suppose that the total stock size is uncertain, but
certain probable values are known. Let the stock size is 100 tons
with probability 0.2 and 150 tons with probability 0.8. By this
we mean that if the producer continues to extract resource from
the mine and if he finds additional resource after extracting 100
tons, then he is certain that the mine will have exactly 50 more
tons.
In this situation, how does the planner arrange a plan of
extraction? Say an arbitrary plan is devised and extraction
proceeds. At the instant when the 100th tone is removed, the
planner can know whether the mine has no further resource or
50 more tons of the resource. Now he faces this certain situation.
If the stock is zero, backstop fuel (if available) takes over and if
50 more tons are remaining, extraction continues till exhaustion
and then back stop technology takes over. Note that the first 100
tons and the next 50 tons (if some mineral is found after 100
tons have been extracted) are known with certain, and hence the
extraction in these cases will follow Hotelling's rule. The actual
problem is the linkage between the two phases.
See the figure in the next page. Note that there is a discontinuity
in price at the end of the first phase. If more ore is found after
42
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
100 tons are extracted, there will be a reduction in user costs
bringing down prices. If no more resource is found, user costs
suddenly increase and price reaches the cost of backstop fuel.
This situation can lead to an externality involving information.
Note that because of the discontinuity, it is possible for
producers to shift production from the first phase to the second,
and gain additional profit. However, by assumption, it is not
possible as we have assumed that the availability of zero or 50
more tons will be known exactly after the first 100 tons have
been extracted. But, in practice, producers do get an idea of the
future availability as they remove ore, and they can modify their
production schedule.
If a person has benefited using this information externality, he
can sign contracts today to deliver ore in future at today's prices.
Such contracts are called contingent contracts.
43
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Effects of uncertainty
Price
Choke Price p'
Time
44
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Societies are normally risk-averse. This means, the utility to
society of an uncertain prospect of 100 tons of ore with 0.2
probability and 150 tons with 0.8 probability will be less than
the utility of its certainty equivalent. In other words,
U (140) > U (100) + {0.2 * U (0) + 0.8 * U (50)}
In effect, the society views the uncertain situation as equivalent
to the availability of lesser mineral.
We know from our previous lectures that when the total value of
the resource becomes lesser, initial quantity decreases and price
increases. That is why we have seen that when positive
extraction costs are introduced or higher distortionary taxes are
introduced,
quantity
produced
initially
reduces
with
a
corresponding increase in initial prices. This fact brings us to an
important conclusion – the presence of uncertainty leads to a
higher price than would be observed under certainty.
45
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Now suppose that a planner has two deposits, one with a certain
stock (say, 120 tons) and another with uncertain stock discussed
earlier. Which deposit will he chose first?
Suppose planner extracts the uncertain deposit first. He would
then know, after mining 100 tons, whether zero or 50 tons
remained in that deposit. Thus a new phase can be designed to
extract 120 + 0 tons or 120 + 50 tons. But, by extracting the
certain deposit first, this information comes very late and is
wasted. Thus, the optimal plan is to exploit the uncertain deposit
first and then have a second phase with 120 + 0 or 120 + 50
tons.
46
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Choke Price p'
Effects of uncertainty
Price
120 + 0
120 + 50
100
Time
47
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Uncertainty and backstop technologies
Fusion as a source of energy is said to have no effective fixed
stock of reserves. If it becomes available, fusion will be able to
replace other energy sources, such as coal, oil, uranium and gas.
At what cost will it become available for commercial use? If
fusion is the backstop energy supply, how will uncertainty about
its long-run cost affect the extraction paths of alternative sources
of energy?
Let us now deal with the situation in which the actual cost
becomes known at the instant the stock of conventional fuels is
exhausted. The date of arrival of the backstop is assumed to be
known and is related to the speed of exhaustion.
Let us assume that a backstop technology will take over once the
available resource is exhausted, but its cost is uncertain. Let the
uncertain cost assume value CH with probability , CL with
probability (1 – ), 0 < < 1. Thus the average cost or certainty
equivalent is CM =  CH + (1 –) CL.
Now, the issue here is setting up the initial prices so that
exhaustion occurs near CM. At the moment exhaustion occurs,
48
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
the actual value of the cost of the backstop is revealed and price
moves to that level.
For normal risk-averse societies,
 UH + (1 –) UL < UM,
which means that the total welfare (measured here by utility)
under certainty with CM is considered to be higher than the total
welfare with uncertainty. As before, this will lead to a higher
initial prices compared to the certainty case.
A more realistic case would have the cost of the backstop
revealed at a known date in the future. At the date cost is
revealed, the problem becomes one of exhausting the remaining
stock along a certainty path – say one of the two branches if
there are two uncertain values for the backstop at the outset (see
figure).
Let t1 be the time at which the cost of backstop is revealed. Note
the discontinuity of the price path at this time. Obviously, if CL
is realized, then price at t1 will drop and rise sharply to reach the
smaller backstop price. On the other hand, if CH is realized,
there will be a jump in initial price, and the backstop price will
be reached at a slower rate. The dotted line indicates the path if
there is no uncertainty and it is known at time = 0 that the cost
of backstop is knows with certainty as CM. Because,  UH + (1 –
49
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
) UL < UM, the initial price p0 will be smaller than the case of
uncertainty.
50
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Effects of uncertainty on the cost of backstop
CH
CM
Price
CL
p0
t1
51
Time
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Uncertain date of arrival of backstop technology
Let TL be the earliest possible date of arrival with probability (1-
) and TH with probability . If early date is realized, price
jumps down at TL, such that it reaches backstop cost at
exhaustion (TX). If the late date is realized, price jumps at TL,
goes above back stop cost p' such that the stock gets exhausted
at TH (=TX), and the price then reduces to the backstop price.
The certainty equivalent date is TM = TH + (1 – ) TL. Under
certainty, exhaustion will be planned for that date. Since TH >
TM, uncertainty has led to a higher initial price than under
certainty path ab, and extends the life of the mine.
52
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Uncertain date of backstop realization of early date
Price
Choke Price p'
p0
TL
Time
53
TX
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Uncertain date of backstop realization of late date
Choke Price p'
Price
b
p0
a
TL
Time
54
TM
T H=TX