The green paradox of the economics of exhaustible resources
Robert D. Cairns n
Department of Economics and Cireq, McGill University, Montreal, QC Canada
H I G H L I G H T S
 The green paradox is a direct application of Hotelling′s rule from the economics of exhaustible resources.
 Hotelling′s analysis was a profound contribution to economic thought but evidence for it is weak.
 Hotelling-style analysis assumes incorrectly that production can be rearranged at will among time periods.
 Technological and geological features of oil production make the prediction of the green paradox unlikely.
a r t i c l e i n f o
Article history:
Received 27 January 2013
Received in revised form
14 October 2013
Accepted 16 October 2013
Available online 15 November 2013
Keywords:
Green paradox
Hotelling′s rule
Constrained output
Investment
Natural decline
Decline curve
a b s t r a c t
The green paradox states that an increasing tax on emissions of carbon dioxide, consonant with the
expected increase in their marginal damages, may induce oil producers to shift their production toward
the present and thereby to exacerbate the problem of climatic change. The model is based on Hotelling
models of resource use that do not take the natural and technical features of oil production into account.
Natural features include the decline of production through time according to a decline curve. Technical
features include the requirement to sink investment in productive capacity. A model of a profitmaximizing
firm indicates that, if these features are taken into account, the prediction of the green
paradox is unlikely.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction: The meaning of the green paradox
In discussions of appropriate policy responses to climatic
change, the role of fossil fuels, especially oil, takes centre stage.
There is a current sense of urgency to begin to reduce consumption
of these fuels. A method favoured by some economists is a tax
on emissions of carbon dioxide, in essence on oil use. Since the
marginal harm inflicted by emissions is expected to increase over
several decades, a proposal consistent with much of environmental
economics is that the tax should be announced as increasing
through time, in step with the marginal damages.
Suppose that a global tax on fossil fuels were implemented, and
that governments worldwide could commit to the future schedule
of taxes deemed appropriate to balance the costs and the benefits
of oil use. Would this development be salutary in the context of
climatic change?
One theoretic development holds that it may not. The green
paradox states that dynamic influences may thwart the intent of
the tax by giving producers an incentive to shift production
toward the present. It would thereby cause an increase in damages
in the short and medium terms.
Oil is an exhaustible resource. The economics of exhaustible
resources is expressed through Hotelling′s rule. In its simplest
form the rule states that in equilibrium the net price, the price net
of marginal costs and marginal taxes, rises at the rate of interest.
The argument for the green paradox is a direct application of the
rule, which prescribes the optimal timing of the extraction and use
of exhaustible resources. By changing the relative net values of a
unit of oil at different future dates (as compared to the original
equilibrium without the tax) the tax may induce producers “to tilt”
their production toward the present. Greater emissions in the
present and medium term may be induced. Since there is a fixed
quantity of fossil fuels in the earth, the greater emissions come at
the expense of emissions in the long-term future. (In the simplest
models there is a one-to-one shift in production.) By then, other
means to attack the climate problem may be available. Paradoxically,
the tax, intended to help to solve the problem of climatic
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http://dx.doi.org/10.1016/j.enpol.2013.10.047
n
Tel.: þ1 514 398 4400; fax: þ1 514 398 4938.
E-mail address: robert.cairns@mcgill.ca
Energy Policy 65 (2014) 78–85
change in the short and medium runs, may exacerbate it, and yet
provide only limited relief in the long run.
The green paradox merits attention from environmental economists
because the theoretic issue is recast by climate change; it
becomes the timing of a tax instead of the equity and apparent
efficiency of having the “polluter pay” directly for the marginal
damages caused. The policy issue is whether the tilt toward
the present, described in theory, can be expected to play out in
practice.
The present paper expresses doubts about the analysis that
gives rise to the green paradox. These doubts are the product of
doubts about the applicability of Hotelling′s rule. The analysis
draws attention to features of extraction such as sunk investments
and production constraints, which are neglected in Hotelling
models. The paper begins with a brief explanation of Hotelling
models. Then it reviews the application of Hotelling analysis to
effects of the tax on flows. Later, it interprets a survey of the
empirical analysis related to Hotelling′s rule. Finally, it considers
technological and natural features of oil production. These features
make the oil industry more complicated than envisaged in
Hotelling models. Definitive answers are not possible, but indications
are that the effects of a carbon tax may more likely be
conventional than paradoxical.
2. Hotelling models
Hotelling′s (1931) model of the economics of exhaustible resources
is a profound contribution to economic thought. It provides five
major insights:
 Exhaustible resources are a form of capital.
 The price of the resource is determined in a dynamic equilibrium
that regulates both the flow of the resource to market
and the holding of the resource as an asset.
 The timing of decisions is of central significance and warrants
careful analysis.
 Resources are subject to the usual market failures, viz. monopoly,
externality and informational asymmetry.
 Exhaustibility in itself does not entail a special form of market
failure. In particular, competitive markets are not subject to a
myopic inability to allocate an exhaustible resource in way that
efficiently balances the interests of the present and the future.
In the Hotelling model, units of the resource are viewed as
being available to society for extraction at any time, at a known
cost that depends on the quantity extracted and possibly other
factors. A striking analytic result of the model is Hotelling′s rule:
under certain assumptions, in a dynamic, competitive equilibrium
the price net of marginal cost of an exhaustible resource rises at
the rate of interest. (Under other assumptions the rule is more
complicated.) The rule can be proved through optimal-control
analysis and is mathematically incontrovertible.
The economic reasoning behind the rule is even more striking.
Consider what is called herein a type-one Hotelling model of an
exhaustible resource, in which extraction costs depend only on the
quantity of the resource that is currently being extracted. In this
case, the extraction cost of q40 units of the resource is given by
some function c(q). This function is assumed to be increasing,
so that extracting more units at a given time costs more, and to be
convex, so that it becomes ever costlier to extract additional units.
As argued by Solow (1974), the owner of a resource who wishes
to maximize net present value is led to re-arrange extraction such
that what is earned by the marginal unit in each time period is
equal in present-value terms. If the marginal unit at one time gains
less than at another time, present value can be increased by
reallocating output from the period with the lower gain to the one
with the higher gain. In symbols, let p(t) represent the price
at time t, r represent the prevailing rate of interest and
μðtÞ ¼ ½dcðqÞ=dqŠt represent the marginal cost of production. Suppose
that a proposed path of extraction is such that, for times t and
s during production
Dðt; sÞ ¼
pðtÞÀμðtÞ
ð1þrÞt À
pðsÞÀμðsÞ
ð1þrÞs 40:
Then a unit of production can be re-allocated from time s to
time t, at a net gain of D(t,s).
The re-allocation can be done repeatedly, so long as an
inequality holds, at a net gain each time. Ultimately, a constant,
λ say, is determined such that for all times t and s during which
extraction takes place
pðtÞÀμðtÞ
ð1þrÞt ¼ λ ¼
pðsÞÀμðsÞ
ð1þrÞs :
Another re-arrangement yields that
pðtÞÀμðtÞ ¼ λð1þrÞt
ð1Þ
the net price (price net of marginal cost) rises at the rate of
interest. Eq. (1) expresses Hotelling′s rule. The discussion stresses
that Hotelling′s rule is an arbitrage condition for the use of an
asset, through the allocation of extraction, over different periods
of time.
Often it is assumed that the marginal cost is constant, so that
for some number β, c(q)¼βq. In this case, for any value of q or t,
μ(t)¼β. The assumption allows for developing sharp mathematical
results, including the early insights of the green paradox by
Sinclair (1992) and Ulph and Ulph (1994), as well as some morerecent
ones.
A remarkable feature of Hotelling′s original paper is that he
also considered what may be called a type-two model. In type-two
models, cost depends on the total available reserve, Q say, as well
as current output, and is written C(q,Q). The properties of this cost
function are that it is an increasing, convex function of output q
and a decreasing, convex function of available reserves Q. (Costs
are lower if reserves are higher.) Also, C(0,Q)¼0 for any value of Q.
A type-two model delivers less sharp theoretic results than a typeone
model: There is still arbitrage among marginal units of the
resource but the influence of the remaining reserve on cost yields
a more complicated expression of Hotelling′s rule. The rule is
expressed in terms of the discounted sum, over the future of
production from the resource, of the increases in cost that arise
because current production affects future costs through depleting
total reserves. Though harder to work with theoretically, type-two
models are considered to be more realistic.
The function C(q,Q) has been a workhorse of empirical research
in economics since the late 1970s. Several theoretic analyses have
also utilized it. In the main, however, theorists have resorted to the
simpler function βq. The same observations are true of the green
paradox: Although some authors have used the function C(q,Q) in
theoretic work, the simpler function is the foundation of the more
striking conclusions.
A key point is that either function implies that at any time any
level of extraction q is possible if one is willing to incur the current
marginal cost. (If cost is βq, an unbounded level of output can be
had at the constant marginal cost β). Consistently with the nature
of arbitrage in Hotelling models, output can be shifted at will over
time. There is no impediment to tilting output toward the present.
3. The green paradox
Even though the analysis of extraction with cost C(q,Q) is more
complicated than with cost βq, Sinn (2008) deftly uses arbitrage
R.D. Cairns / Energy Policy 65 (2014) 78–85 79
between adjacent periods (times t and tþ1) to make the argument
for the green paradox. Let C(q,Q)¼g(Q)q, where g0
ðQÞo0 and
q¼ ÀdQ/dt. This case is special but only slightly so: it assumes
constant marginal cost at any point in time, but increasing costs as
the total stock Q decreases through extraction. The proceeds from
a single unit of resource extracted at time t and invested for one
period are p(t)Àg(Q) of principal and r[p(t)Àg(Q)] of interest. If
instead that unit is not extracted until tþ1, and also if the change
in Q over the period (which is Àq) is neglected as being
quite small compared to Q, the owner gains p(tþ1)Àg(Q).
Let the change in price over the period be represented by
ΔpðtÞ  p ðtþ1ÞÀpðtÞ. Arbitrage renders the gains in the two
periods equal, so that
½pðtÞÀgðQފð1þrÞ ¼ pðtÞþΔpðtÞÀgðQÞ:
Simple algebra yields a single-period generalization of Hotelling′s
rule:
ΔpðtÞ
pðtÞÀgðQÞ
¼ r ð2Þ
Condition (2) holds on an equilibrium path for each pair of
adjacent time periods and so links all periods.
If a tax on emissions is imposed and it rises through time,
say being equal to the discounted value of the damages due to the
marginal unit of emission, the result may not be what is expected
by proponents of the tax. To illustrate what may happen while
using minimal algebra, Sinn uses a special type of tax. Let the tax
τ(t) be such that the net price received by the firm (net of the tax
as well as marginal cost), [1Àτ(t)][p(t)Àg(Q)], is reduced through
time at rate δ:
1ÀτðtÞ ¼
½1Àτð0ފ
ð1þδÞt : ð3Þ
The arbitrage condition is that
½1Àτð0ފ
ð1þδÞt ½pðtÞÀgðQފð1þrÞ ¼
½1Àτð0ފ
ð1þδÞt þ1
½pðtÞþΔpðtÞÀgðQފ:
In this situation the modified rule becomes
ΔpðtÞ
pðtÞÀgðQÞ
¼ rþδð1þrÞ: ð4Þ
Sinn applies the following economic reasoning to this condition
for different values of δ. When δ¼0, so that the tax is constant,
condition (4) reduces to condition (2) throughout the period of
production. Therefore, the tax is neutral (does not change the path
of extraction and hence does not change the accumulation of
carbon in the atmosphere). For given values of the other variables
in condition (4), the change in price, Δp, is greater when δ40
than when δ¼0. (Price grows faster when the tax is increasing.)
Sinn (2008: 374ff) makes technical assumptions that assure that
the reserves are eventually completely exhausted in both scenarios,
with and without the tax. When the reserves are exhausted
(and hence output is zero), the price must be the intercept of
demand. Since the price rises faster with the increasing tax, and
since each scenario ends up with an equal total quantity extracted,
the price with a tax must start out lower. In equilibrium at the
current time t, therefore, output must be higher with the tax.
The economics behind the result is that the rising tax changes
the relative gains to the producer over the life of the reserve,
making present extraction comparatively more attractive at the
margin. With greater present extraction, the industry is in equilibrium
at a lower present price. Lower prices also prevail in the
near future.
Consequently, global warming is exacerbated in the present.
The green paradox is that a policy designed to tax the emission of
carbon in a way that is strongly related to damages caused may
lead to greater current emissions. Some have suggested that a
better rule may be to have an initially high but deceasing tax
(so that δ is negative). Such a tax would encourage the producer to
shift output toward the future. The form of the tax, then, is central
to its effects.
Sinn observes that other forms of policy, such as a subsidy
to greener forms of energy, would be subject to a green paradox
as well. His analysis, as summarized above, is a benchmark for
understanding the phenomenon.1
4. Evidence of Hotelling′s rule
For a mathematically incontrovertible result, Hotelling’s rule
has been subject to much controversy. Practitioners, whose conscious,
rational decisions are supposed to implement the rule,
flatly and (to this author’s knowledge unanimously) deny its
relevance. Strong challenges have come from academia.
An erudite review of empirical research on Hotelling’s rule
finds that the evidence does “not necessarily invalidate the
conceptual message of the Hotelling Rule” (Livernois, 2009,
pp. 37–38). The finding arises from a conviction “that mining
firms think not just of the present but about the future and that
they wish to maximize the value of their assets”.
Thus, Hotelling’s analytic framework of maximizing net present
value (“the value of their assets”) seems to be borne out. In the
statements of the green paradox, however, as well as in the
foundations of the economics of exhaustible resources, the thrust
of Hotelling’s message is not limited to the notion that extracting
firms maximize net present value. This notion is an initial working
hypothesis about conduct in any industry.2
Hotelling’s rule, his
analytic result, is the message that is specific to exhaustible resources
and is used by resource economists as the basis for thinking about
the dynamics of resource prices (Livernois, 2009, p. 22).
Elsewhere, Livernois (2009, p. 22) describes Hotelling’s rule as
“a condition of inter-temporal arbitrage that ensures that the last
unit extracted in any time period earns the same return (in present
value terms)”. As Solow’s (1974) definitive treatment notes, the
arbitrage consists of being able to move production of units of
output freely among time periods in response to changes in prices
and interest rates. Livernois (2009, pp. 37–38) finds that “overall
one cannot conclude that the Hotelling Rule has been a significant
force”, and that “other factors, notably technological change,
revisions to expectations regarding the resource base, and market
structure, have had a more significant influence on the evolution
of prices”. Again, comparable influences affect all industries, not
just exhaustible-resource industries.
Livernois’s review suggests that, in spite of econometricians’
efforts to control for the other influences, the arbitrage condition
has not received broad support from empirical research. Instead,
the evidence suggests that the arbitrage which underlies the
predictions of Hotelling theory may not be being realized. Is there
an explanation for what appears to be a troubling departure from
an incontrovertible mathematical result?
5. Technological models
The key assumption, usually glossed over, is that output can be
re-arranged as desired. It implies that the output decision is made
1
Several other estimable economists have studied the paradox within the
Hotelling canon but under somewhat different assumptions from Sinn’s and have
qualified his results.
2
It is not a good hypothesis about behaviour of the national oil companies that
are responsible for much of the world’s output of oil. See Cairns and Calfucura
(2012). However, it is accepted herein for the sake of argument.
R.D. Cairns / Energy Policy 65 (2014) 78–8580
period-by-period, with no restriction other than exhaustibility.
The sole physical limit to the level of output in any period is held
to be the level of remaining reserves. All costs are variable. As a
result, Hotelling’s rule is, in principle, observable; econometricians
can try to tease evidence for (or against) it from the data.
The function C(q,Q) of Hotelling models, however, is not a valid
representation of the technology of oil production, let alone the
simpler βq. In reality, the potential to produce is obtained by a
high up-front cost in drilling wells that determines an output
capacity, as reasoned by Campbell (1980) in introducing what may
be called a technological analysis of non-renewable resources. Once
capacity is installed, it is usually not changed for a significant
period of time. Campbell’s ideas were advanced by Crabbé (1982)
and Lasserre (1985).
In the oil industry, after the period of capacity production,
which may be short, the productivity of a well decreases through
what is known as natural decline. Natural decline is the technological
counterpart to the assumption in type-two models that costs
increase as a result of decreases in available reserves. Other than
for “ramping up” production, for maintenance, or for production
problems, the level of output is restricted by capacity and by
natural factors that produce what is known as a decline curve
(Cairns, 2009).
In essence, Hotelling models of both types one and two are
short-run models. They neglect the decision to commit capital to
oil production. They neglect the facts that the technology does not
allow for unlimited output and that there are natural limits to
production at any time. The option to shift production unimpeded
from one point in time to another, the free arbitrage basic to
Hotelling analysis, is not available.
Moreover, the limits or constraints have a shadow value that is
confounded with the shadow value that gives rise to the r-percent
rule in the net price. (The two shadow values sum to the net price.)
The r-percent rule is not observable in principle.
Where empirical evidence of the path of price net of marginal
cost has been sought, it has been found that this variable rises
more slowly than at the rate of interest. The type-two model
predicts that the net price rises at less than the interest rate.
But comparatively few sectorial models incorporating behaviour at
individual reserves have used even a type-two view of these
markets. One, by Cairns and Quyen (1998, p. 181) for mineral
exploration, argues that the modified rule for the rate of change of
resource rent “involves a complicated convex sum, with endogenous
weights, of the deposit-specific terms” so that the cost
function C(q,Q), involving aggregate magnitudes q and Q, does
not generate the correct aggregate rule.
For analysis of the effects of policy, a long-run analysis is
required. Technological models are reserve-based rather than
sector-based, analysing conditions of production at the individual
reserve. Unlike in most Hotelling models, decisions are disaggregated.
The behaviour of the sector is an aggregate of the behaviour
of individual producers. As a result, the industry cannot be fully
represented in formal, analytical models. Numerical analysis or
less formal, economic logic may be required to provide bridges and
extensions.
Cairns and Davis (2001) observe that oil is produced from
underground reservoirs that are under great pressure from the
contained oil and gas. A well drilled into the formation, like a pin
prick into a balloon, allows the pressure to be released, and with it
the oil and gas. In some cases natural drive is sufficient to produce
the oil, but at declining rates. In others, if pressure is not great
enough, pumps are installed at the surface to lift the oil to the
surface. For what is known as secondary production, wells can be
installed at the periphery of the reservoir to inject water or gas
that will drive the hydrocarbons toward the producing well. In all
cases, the valuable natural product is oil (and gas). The scarce
natural or artificial instrument of production is pressure. Technology
allows for the augmentation of pressure in various ways.
The approach can be used to build more comprehensive
views of the sector. First, a short-run model can depict a firm’s
behaviour once investment has been made. Second, one can model
investments in development as contingent upon that behaviour.
Finally, exploration can be viewed as looking forward to succeeding
decisions.
6. Appropriate short-run analysis: the decline curve
An initial analysis is that of a producing property: all investments
have been made and operation is in the short run. This
discussion is the counterpart at the level of the reserve to a
Hotelling analysis at the level of the industry. Following Adelman
(1990, 1993) and Cairns and Davis (1998, 2001) let the oil be
driven to the surface by pressure, P(t), which is subject to natural
decline at rate a, according to the equation
dPðtÞ=dt ¼ ÀaqðtÞ: ð5Þ
For some constant π, output depends on pressure according to
the inequality
qðtÞrπPðtÞ: ð6Þ
As is argued by Adelman and by Cairns and Davis, the growth in
net price is typically less than the rate of interest. Producing at less
than πP(t) thus reduces net present value. Consequently,
qðtÞ ¼ qð0ÞeÀat
: ð7Þ
Eq. (7) is the decline curve of the reservoir.
Suppose that the reserve has been prepared for exploitation at
time t¼0. The (recoverable) reserve, initially available for production,
is
Rð0Þ ¼
Z 1
0
qðtÞdt ¼
qð0Þ
a
:
Let the net price (price net of marginal cost) be represented by
ν(t) and grow at rate γor. Then the value of the reserve is given by
V0 ¼
Z 1
0
½νð0Þeγt
Š½qð0ÞeÀat
ŠeÀ rt
dt ¼ νð0ÞRð0Þ
a
aþrÀγ
:
Since γor, the value V0 is strictly less than ν(0)R(0), which is
the value given by a condition derived from Hotelling’s rule called
the Hotelling Valuation Principle (Miller and Upton, 1985).3
Production in this model is fully determined by geological
features and not economic choices. The formal optimization of
its value is trivial. The economic meaning of the result, however, is
not trivial. Since the reserve is extracted over time, with the net
price per unit rising at less than the interest rate, the value must
be lower than given by the Hotelling valuation principle. Adelman’s
perspective on the oil industry indicates that, at an
individual reservoir and hence for the whole industry, a prediction
of the Hotelling model cannot hold.
The mathematical reason is the shadow value of pressure. It
arises from constraint (6). In an abstract sense, pressure is a scarce
resource because an increase in pressure would allow the operator
to extract more quickly and thereby to increase net present value.
As mentioned above, a portion of the net price is attributable
to pressure. The results hold, with minor modifications, under
3
Adelman (1990) observes that a rule of thumb in the oil industry is that at
any time t during a reserve’s producing life, it is valued at about one half of net
price applied to remaining reserves:Vt ¼ νðtÞRðtÞ a= aþrÀγ
À ÁÀ Á
% 1
2 νðtÞRðtÞ. The
approximation arises when γ % 0 and a % r, and has been called Adelman’s rule
(Cairns and Davis, 2001).
R.D. Cairns / Energy Policy 65 (2014) 78–85 81
conditions of uncertainty as well (Davis and Cairns, 1999;
Thompson, 2001).
The reason that the Hotelling rule does not hold is that the output
from a producing reservoir is constrained. Output can be reduced but
not increased. There is no change in output as a result of a tax (unless
the tax is so high as to put the firm out of business).
In sum, at any time at any producing property in the oil
industry, output is constrained by a technological–geological constraint.
The short-term increase in output assumed in the green
paradox cannot be obtained from currently active reserves without
some increase in investment.
There is no reason to believe in a green paradox if one bases the
analysis on current abilities to increase supplies in response to a
change in the time profile of net prices resulting from a tax. If
anything, the tax may be expected to result in a reduction in
production, and a reduction in emissions of greenhouse gases, since
the tax may render some producing properties no longer profitable.
7. The development of known reserves
The discussion thus far has abstracted from some of the key
decisions that are made by producers. In practice, some reserves
are known but “held on the shelf” for later development. A more
sophisticated analysis admits the possibility that the aggregate
production constraint can be circumvented, to an extent, by
investment at new reserves.
Development of a known reserve involves two major decisions,
namely the timing of investment (often called the order of entry in
resource economics) and its level. There is an option to develop
immediately or to postpone development. If a given reserve is
developed at a proposed time S, it has a present value as of that date.
That present value depends on conditions in the market at S and
throughout the projected productive lifetime of the reserve. Optimal
development takes place when the present value of a development
project, taken as of the current date t, is maximized with respect to
SZt. The ordering criterion is that the rate of increase of the current
net present value of the project falls to the rate of interest at the
optimal strike time (Cairns and Davis, 2007; Davis and Cairns, 2012).
Let the net present value of a reserve at the present date trS be
represented by V(a(S),S). Then the optimal value of S is the solution to
the equation, ðdVðaðSÞ; SÞÞ=dS ¼ 0. Usually, however, conditions are
stated to hold as of the time decisions are made. Let WðaðSÞ; SÞ ¼
erS
VðaðSÞ; SÞ denote the forward (as of time S) net present value of the
project. The rule is that, at the optimal start-up time, the rate of
increase of the forward net present value has fallen to the rate of
interest:
1
WðaðSÞ; SÞ
dWðaðSÞ; SÞ
dS
¼ r: ð8Þ
The order of entry is determined, then, by a formula that takes
into account the conditions throughout the exploitation of the
reserve and not just the net price at a given instant.
A level of investment can be chosen for each possible value of
the start-up date S. In a simple model, the choice of level can be
represented as a choice of the value of a in Eq. (5) (Adelman, 1990;
Smith, 2012). Once made, the investment “locks in” the maximal
rate and, in practice, the actual rate of production in the short run.
Consequently, instead of there being an economically meaningful
choice of the timing of the production of individual units of
a stock from the given reserve as portrayed in Hotelling models,
timing is expressed through the choice of the date of the development
investment. Any change arising from a tax depends on
both the timing (S) and level (a) of investment. The effect of the
imposition of a tax on the “tilt” of production is given by the level
of investment in the reserve and by the decline curve.
Consider the extension of the analysis of the tax introduced
in Eq. (3) to a reserve in which such an investment is made.
Postulating this simple form of tax again facilitates the analysis, in
this case in drawing the distinction between Hotelling and
technological models in their implications for the timing of
extraction of oil and the emission of carbon dioxide.
The analysis is easier in continuous time. In the expression of the
path of the tax through time, the factor ð1ÀτÞ=ð1þδÞt
gives way to
ð1ÀτÞeÀ δt
. Because the net price at reserve i depends on costs, which
are specific to the reserve, the net price and its rate of growth are
specific to the reserve. Let the net price at time t be represented very
simply by νieγit
and the discounted net price (net of the tax as well as
marginal cost) to the producer by νið1ÀτÞ eðγi À δÀ rÞt
.
For simplicity, the analysis abstracts from the possibility of an
initial period of capacity production by combining the constraints
of capacity and natural decline into the variable ai. At producing
property i having a remaining, recoverable reserve Rit, the level of
output at time t is given by qit ¼ aiRit. Because of natural decline,
output at time s4t is given by qis ¼ aiRis ¼ aiRiteÀaiðsÀ tÞ
¼
qiteÀ aiðs ÀtÞ
. The cost of investment is also important. Smith
(2012) states that, for some parameter ki that is specific to the
reserve, the investment cost for (onshore) development is
approximately equal to kia
5=3
i .
Now let the subscripts i be suppressed to reduce clutter. Under
the above assumptions the net present value of the initial reserve
R at the present date trS is
VðaðSÞ; SÞ ¼
Z 1
S
aReÀ aðt À SÞ
νð1ÀτÞeðγ ÀδÞt
eÀ rt
dt Àka5=3
eÀ rS
¼
aRνð1ÀτÞ
aþrþδÀγ
eÀðδÀγÞS
Àka5=3
!
eÀ rS
:
Even under the strong assumptions of this model, including
that of the simple tax from Eq. (3) as modified, the comparativestatics
analysis is inconclusive. It involves finding conditions on
the values of parameters, applicable to individual reserves, for
which complicated expressions have a particular sign. It is not
possible to obtain a definitive solution for the whole industry.
One suggestive contrast with results from Hotelling models can
be obtained. The optimal level of investment is such that, for the
chosen value of S,
∂Vða; SÞ
∂a
¼
Rνð1ÀτÞðrþδÀγÞ
ðaþrþδÀγÞ2
eÀ ðδÀ γÞS
À
5
3
ka2=3
" #
eÀ rS
r0; ð9Þ
with equality holding when a40. It is easily checked that V(a,S)
40 when there is equality in Eq. (9). The positive value V(a,S) is
the resource rent for the reserve under study, its scarcity value.
It is a present value of the reserve, optimized over a and S, as
opposed to the current value per unit that is found in Hotelling
models.4
Using the fact that there is equality in condition (9) when a40
yields that
∂Vða; SÞ
∂S
¼
ÀaðrþδÀγÞRνð1ÀτÞ
aþrþδÀγ
eÀðδÀγÞS
þrka5=3
!
eÀrS
¼ aðaþrþδÀγÞ
ÀaðrþδÀγÞRνð1ÀτÞ
ðaþrþδÀγÞ2
eÀðγ ÀδÞS
"
þ
r
aþrþδÀγ
ka2=3
!
eÀrS
¼ À
5
3
ka5=3
eÀrS
ðaþδÞÀðγÀ
2
5
rÞ
!
:
4
It is all the rent from the reserve. Rent due to quality of the reserve
(“Hotelling” and “Ricardian”) is optimized in the choice of S (rate of growth of
net present value falls to r) and of scale in the choice of a.
R.D. Cairns / Energy Policy 65 (2014) 78–8582
If δoγÀ2
5 r, then ∂V=∂S40. In this case the reserve is never
exploited because its value continues to increase indefinitely. This
result is an artefact of the simplicity of the assumptions: in a
dynamic equilibrium of the industry, the net price would not
continue to rise at the constant rate γ as assumed but price would
adjust so that the reserve would eventually be exploited.
In an internal solution (one in which S4t, a40 and ∂V=∂S ¼ 0),
a ¼ ðγÀ2
5 rÞÀδ40. The optimal value of a decreases as the value of δ
increases until δ ¼ γÀ2
5 r. It is plausible that the timing of investment
is such that the reserve is developed sooner when there is a tax. For
example, if γÀ2
5 r404γÀ2
5 rÀδ there is an internal solution for the
reserve when there is no tax (S4t when δ¼0) but a corner solution
when there is a tax (S¼t when δ40). The timing of investment is
affected in the direction predicted by the green paradox. At the time
that investment takes place, however, its level, a, is reduced. As a
result, output is reduced throughout the life of the reserve according
to the decline curve. This effect counters that of the timing of
investment in the time path of oil production.
For those reserves which, with no tax, are under development
or being studied for development (for which δ¼0 and γÀ2
5 ro0),
the imposition of the tax has no effect on timing: S¼t. In this case,
∂2
V=∂a2
À Á
o0 and ∂2
V=∂δ∂a
À Á
o0, so that da=dδ
À Á
o0: as the
carbon tax increases, the level of investment a and the initial
output aR from the reserve decrease.5
The prediction can be said to
be contrary to that of the green paradox.6
In this model, ν40 and hence νð1ÀτÞeÀδt
40. The tax of
Eq. (3) operates on the net price ν but leaves the net cash flow
always positive, so that a developed reserve is always fully
exploited. Moreover, there is always a value of a40 for which
V(a,S)40. A reserve is always developed.
Rather than being imposed on the net price, a carbon tax in
reality would more likely be imposed on emissions, broadly
speaking on output.7
The discounted, instantaneous net cash flow
to a producer facing tax per unit output of τ0
eδ0
t
at time t is
aRðνeγt
À τ0
eδ0
t
ÞeÀ aðt ÀSÞ
eÀ rt
. This tax is akin to a royalty that
increases at rate δ0
. For a marginal reserve, the imposition of the
tax may reduce the net present value to less than zero, so that the
reserve is not developed at all. Moreover, production at a developed
reserve ceases at the time s when νeγs
À τ0
eδ0
s
¼ 0. Cessation
is a drag on investment.
For other reserves, each of ν and V(a,S) may pass from negative to
positive at some future time S. Other things being equal, the time at
which the net price (net of tax as well as current cost) becomes
positive is pushed forward (toward the future) as compared to the case
with no tax (with τ0
¼ 0). The determination of the optimal start-up
date S is delicate because it obeys an r-percent rule, Eq. (8), that
depends on the dynamic equilibrium of the oil market.8
This type of tax is much more difficult to study analytically than
the one studied above. The tax could reduce net present value
more significantly, having greater effects on the chosen level
of a, or leading to the abandonment of otherwise viable prospects.
It is difficult to predict the start-up date. Simulations of the sort
performed by Smith may be more revealing than trying to pursue
the analytic approach. The assumption that other things are equal,
however, is almost surely not valid. Any simulation would be even
more difficult than Smith’s since the effects of the tax on the
equilibrium level of γ, involving the decisions of all current and
future producers, would have to be determined rather than simply
assumed. The results for individual reserves would again depend
on the values assumed for the various parameters. Much that
matters is not easily modelled analytically or simulated. It is
noteworthy that Smith considers his sophisticated, subtle analysis
to be “highly simplified”.
Some economists may object that the model does not envisage
an attempt to reduce the green paradox through tax design or
even through a deliberate choice of δ or τ. In a Hotelling model it is
relatively easy to fold any form of tax into a maximization of social
welfare and thereby to define an optimal tax. In this way a carbon
tax can be made socially efficient. (It still may have a paradoxical
effect on the timing of emissions.) In an aggregate technological
model, an optimal tax would have to be
 optimized specifically for given reserve, not a “carbon tax” at all
but an optimally varying royalty, or else
 optimized across and common to all of the reserves (developed,
undeveloped and undiscovered), and not optimized for any one
of them.
Sinn’s simple analysis of the green paradox, though not a
welfare analysis, makes the vital observation that the effects of a
policy, especially a dynamic policy, may be contrary to predicted
effects. His strategy of studying a given, non-optimized tax makes
sense in discussing the effects of a carbon tax, both at a given
reserve and in the aggregate.
8. Undiscovered reserves and other forms of capital
Exploration is a response to an incentive provided by a positive
expected net present value of discoveries (Cairns and Quyen,
1998). Exploration cost is a “set-up cost” (Hartwick et al., 1986),
as opposed to an investment in productive capacity (Campbell,
1980). It involves a sunk cost but no constraint on arbitrage of the
output from a reserve that has been “set up”.
An emissions tax would shift the distribution of net present
values from investments in exploration in any mineral province
“to the left” (toward lower values). A tax increasing continually
through time would entail a continual shift to the left. A net
increase or decrease in exploration may result. As predicted by the
green paradox, there may be a “black-gold rush” to realize the
value of exploration provinces earlier than in the original equilibrium.
There may also be a holding back of exploration in the face
of lower returns to the investment. Smith (2012) finds that a
decline in expected net present value due to a tax reduces the
number of wells an exploring firm is willing to drill in a given area.
Any current increase in exploration depends on whether
equipment (e.g. drilling rigs) and professionals are available. In
equilibrium, spare capacity is low. After all, investments are made
5
Also, da/dτo0.
6
Another source of “tilt” may be investment in secondary or tertiary
(enhanced) recovery. Smith (2012) models the additional investment in increasing
the decline factor to ae
4a during the productive life of the reserve, at time Se
4S.
(See Amit, 1986 for a discussion of the determination of Se
.) Smith demonstrates
numerically how the level and timing of such investments complicate the analysis
of the effects of taxes. One might encounter conditions under which the tax would
induce a decrease in Se
. The likelihood of inducing a reduction of ae
remains.
7
The tax is on emissions. For any particular reserve the tax rate depends on
properties of the reserve (loosely, on “carbon content”) but for purposes of analysis
of the exploitation of any particular reserve the tax can be assumed in a first
instance to be proportional to output.
8
Nystad (1985, 1987, 1988) outlines models with both natural decline and
investment. The 1987 paper considers the possibility that long-run recovery from
oil in place is a decreasing function of the choice of the initial rate of extraction. The
effect would provide a drag on investment that would depend on its magnitude. If
this effect were to hold, a tax such as the one modelled by Sinn, by reducing initial
investment, could lead to an increase in the recovery of oil, and in emissions, in the
long term. An increasing royalty per unit would have a more complicated effect
because the reserve would be abandoned once the net price dropped to zero.
Mentioned, but not modelled, is the maximum efficient rate of production. If the rate
(footnote continued)
of production exceeds the maximum efficient rate, m say, the level of recovery of
reserves in place decreases severely. It would provide a stronger disincentive to
investment beyond a¼m.
R.D. Cairns / Energy Policy 65 (2014) 78–85 83
to reap quasi rents, not to sit idle in anticipation of a possible tax
and the consequent reduction of both scarcity and quasi rents. The
response to the tax would have to include changes in investment
sunk in exploration capacity. Because there is a long lead time
from the start of exploration through development to production,
any change due to a tax is at least a decade forward.
Investments in knowledge – in basic and applied research in
exploration, development and extraction (set-up and capacity
costs) and in the training of professionals (comparable to capacity
costs) – are also sunk and have long lead times before coming
into service. Because investments in knowledge are likely to be
embodied in new vintages of capital, there is only a limited effect
of such investments on wells currently in service or under
development. The returns to these investments are likely to be
reduced as compared to those in the original equilibrium before
the tax. Prospective professionals, especially more promising
minds, may shy away from training in an industry that is expected
to be subject to increasing taxation, reduced rents, and societally
mandated attempts to develop substitutes for its product.
The tax would not have an immediate effect on capacity in
refining and transportation. It is difficult to perceive an incentive
to increase investment in these activities in a way that would help
to facilitate the realization of the green paradox.
9. Synopsis
The green paradox predicts oil producers’ response to a carbon
tax that increases through time in step with marginal social
damages. The oil market is in a dynamic equilibrium that balances
at the margin the contributions to firms’ discounted net cash flows
from the outputs of all future periods. A new tax affects the
balance. If the tax increases through time, there is a greater effect
on later than earlier marginal flows. Whatever the model, there is
an incentive to change the timing of the relevant decision
(extraction in a Hotelling model or investments in exploration
and development in the present model). The reallocation across
time is comparable to that of reallocation in a static model in
response to a tax that varies across jurisdictions.
In a Hotelling model the reallocation is unrestricted. The
industry changes the “tilt” of its output (the slope of the path of
output through time) by increasing current production.
In a technological model, the extent to which the industry can
act upon the incentive is limited. Contrary to a Hotelling model,
it is not possible to increase production from reserves that are
currently in production: the capacities are fixed by previous
investment. Moreover, for development investments that are
currently in progress (those that will be realized in the next 5 or
so years), the incentive cannot move production toward the
present because it is already being implemented in the present.
For the great bulk of production in the near term, then, the
tendency of the tax is not to increase but, if anything, to decrease
production. Responding to the incentive requires irreversible
investments at new reserves.
New investments are made continually to respond to increases
in demand and to replace production that decreases through
natural decline. The response may be to shift toward the present
the development of some known but undeveloped reserves, as
well as exploration for new reserves. This response is subject to its
own constraints and may or may not be important in reality.
The tax also provides an incentive to reduce the level of
development investment at those reserves. The investment affects
the variable a, which is set at time S. A reduction in a means that
the reserve is exploited more slowly (with a gentler decline, falling
exogenously as aeÀ aðt ÀSÞ
R), rather than more quickly and at an
endogenous rate as predicted in a Hotelling model.
Unlike in Sinn’s model, there are two hurdles, exploration and
development expenditures. The maximized value of V net of these
expenditures must be positive. Looking forward from the present,
there are incentives to hasten exploration but also to reduce
the expenditure (current and future) when any region is being
explored. Consequently, near-term discoveries may rise or fall, but
there is a medium-term tendency for less to be discovered and
hence exploited. Also, exploration takes time; any inducement to
earlier exploration can lead to an increase in production only one
to two decades forward, not immediately as predicted by the green
paradox. In reality, exploration is constrained by the availability of
personnel and equipment. With exploration and development,
less may be eventually discovered and extracted, but the commitment
of these sunk costs may be moved toward the present.
If the technology of extraction is not modelled, the sole
investment hurdle is a set-up cost. There is no constraint on
extraction once the decision to make the set-up cost is made.
There may be lower levels of reserves, but for what is discovered,
the same pattern of extraction is seen as in Sinn’s model.
In a technological model, a further effect works against the
green paradox. There is no change in the extraction pattern for
reserves that are in production. The tax reduces new investment
levels, represented by the variable a. The reduction in a, with the
consequent “flattening” of the decline curve, acts “to tilt” output in
the opposite direction to that predicted by the green paradox. The
tilt of the decline curve is a constraint, not a choice. As it may with
reductions in output alleged to be an exertion of market power
(Cairns and Calfucura, 2012), natural decline may appear to the
naked eye, as well as to the naked econometric analysis, to be
consistent with a Hotelling response.
In short, Hotelling models of the green paradox are oversimplified.
There are in reality two incentives that work in
opposite directions, in complicated ways. The resultant of the
two depends on assumptions. If the prediction of Hotelling models
is right and the equilibrium price does fall in the near to medium
term, then a price-taking firm has an even greater incentive to
reduce exploration and development than under the assumption,
made above, that the price path is not affected. The microresponse
of producers appears to be inconsistent with the macroeffect on
price, and hence on output and emissions.
10. Conclusion
Hotelling’s rule is a result from a simple model that teaches
that non-renewable resources are a form of capital and that they
should be analysed as such. His important insight concerning farsighted
decision making reigns as a foundation of non-renewable
resource economics.
Hotelling may reign but he does not rule. Models in his
tradition assume free allocation of resources over time. The rule
is an arbitrage condition relating the values of net price over the
productive life of the reserve. Empirical evidence suggests that
allocation is subtler than in the Hotelling model. The operative
constraint in oil industry is that allocation over time is capped in
one of a number of ways, so that arbitrage among periods is
constrained. Calculations and comparisons are not simply of
current costs at different time periods but of commitments,
especially sunk costs, predicated on the entire future of operations.
At each instant in Hotelling models, decisions are made about
the level of flow of units of the resource. Technological models
break qualitatively from type-one and -two Hotelling models
concerning the form of decisions in the oil industry. Decisions
about flows are atrophied. Extraction requires a combination of a
discovered reserve with fixed capital. The fundamental decisions
R.D. Cairns / Energy Policy 65 (2014) 78–8584
are about the timing and level of investments in exploration and
development.
The present analysis is still highly simplified. Features of
several aspects of oil production must be stitched together if one
wishes to begin to analyse the dynamics of the industry in a way
that is relevant to policy. Some of these features have been pointed
to herein. Some are becoming more fully understood. Each, to have
credibility for policy analysis, requires long and deep research.
It is not possible to demonstrate a green paradox given the
current limitations of mathematical analysis of a complicated,
multi-faceted industry. The weight of many influences discussed
in the present paper is inimical to its predictions. The paradox
does not have an adequate foundation in the conditions of
production in the oil industry to affect policy or the timing of
policy respecting climatic change.
Acknowledgements
Thanks to Suzi Kerr, Amos Zemel and two referees for helpful
comments.
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