Jan Angenendt
Environmental Taxes on Exhaustible Resources
ISBN: 978-3-8366-3868-5
Herstellung: Diplomica® Verlag GmbH, Hamburg, 2010
Zugl. Freie Universität Berlin, Berlin, Deutschland, Diplomarbeit, 2009
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i
A
BSTRACT
The link between greenhouse gases and global warming is scientifically well established
nowadays. The burning of fossil fuels causes a large part of the worlds greenhouse gas
emissions. For example, the burning of a ton of coal leads to the release of a certain amount
of carbon dioxide into the atmosphere where it adds to the presently already increased stock
of greenhouse gases. Consequently any further burning of coal is exacerbating the problem
of climate change. This externality calls for political intervention on resource markets and
the question arises which policies should be implemented. In this thesis, the multitude of
optimal solutions taxation offers is derived.
Any analysis of possible policy options on this topic has to consider that fossil fuels are
no normal goods. Their total supply over time is determined only by nature. The owner of a
resource deposit earns profit by extracting a given stock in time. Correspondingly, the value
of the resource deposits is being determined by the discounted stream of future profits that
can be expected from selling the stock (which can be affected by future changes in regula-
tion). This implies that a regulator should pay special attention to the reactions of the supply
side of fossil resource markets to his policies, but it also allows him to use a broad range of
regulation policies regarding taxation. The clue is to set different incentives to extract the
resource at different points in time. The resulting multiplicity of policy options can be used
to correct different kinds of market failure. Sinn summarized the relationship between polit-
ical intervention and supply side reactions against the background of climate change in his
theory of the green paradox. It states that a lenient gradually tightening environmental policy
leads to the counterproductive effect of falling resource prices and an increase in resource
extraction in the present and the near future. This effect and the insight that gradually relax-
ing measures set the right incentives led Sinn to recommend falling tax rates as the optimal
regulation policy to slow down climate change.
Another aspect that threatens the future profits of resource owners is the development of
substitutes to fossil resources, for example alternative methods of energy production. The
availability of those technologies sets an upper limit to the market price of fossil fuels and
leads to a faster depletion of the (economically extractable) resource deposits in finite time.
The faster the development of such technologies progresses, the faster fossil resources are
extracted resulting in an increased speed of climate change. But the (now finite) period of
resource use can be prolonged by the implementation of falling tax rates.
The geographical spread of fossil resources favors the development of oligopolistic or
monopolistic market structures. Abstracting from any other distortions, market power alone
would be a reason for the regulation of resource markets due to the deadweight loss in con-
sumers rent it causes. In the context of global warming, market power on fossil fuel markets
has to be reconveived. Because fossil resources are being extracted too fast without politi-
cal intervention and market power slows down the extraction of exhaustible resources, the
welfare loss due to higher resource prices might be offset by gains due to lower climate-
change-related damages. It will be shown that also in the context of a backstop technology,
market power still decelerates resource extraction. Furthermore (as market power can be
corrected by a whole family of tax/subsidy-schemes), it is possible to correct the effects of
market power by using rising strict taxes and to minimize the costs of regulation if extraction
costs are sufficiently low. This finding supports the presumption that the necessary incen-
tives for the resource owners to postpone extraction of their stocks could possibly be less
strong than under perfect competition. Tax rates could be less falling in time which might
facilitate the political implementation of an optimal policy.
ii
Concerning the optimal taxation of the externality, the regulator needs to be able to cal-
culate the discounted stream of future damages resulting from the burning of an additional
unit of fossil fuels. In a theoretical model, the resulting path of the optimal carbon tax then
depends on the design of the damage function and the question whether stock or flow exter-
nalities are being taxed. In any case, taxation is an especially powerful instrument that can
induce resource owners to follow the optimal extraction path. A whole family of optimal tax
paths exists that can correct the influence of several sources of distortions at the same time;
the optimal tax schemes differ in the splitting of the resource rent between resource owners
and the regulator, but all follow the same movement. In this thesis, a family of optimal tax
schemes for a resource path that is deviating from the optimal path due to market power and
externality is being derived examplarily for the capability of resource taxation to induce
any optimal extraction path desired by the regulator.
iii
C
ONTENTS
Abstract
i
List of variables
v
1.
Introduction
1
1.1.
Economic importance of fossil fuels
1
1.2.
A first theoretical approach to the problem
4
1.2.1.
Resource extraction under perfect competition
4
1.2.2.
Optimal resource extraction and externality
5
1.3.
Introducing environmental policy
7
1.3.1.
Constant cash-flow tax
7
1.3.2.
Constant ad valorem tax
7
1.3.3.
Variable tax rates
8
1.3.4.
Other political measures
9
1.4.
First conclusion and the next steps
9
2.
Backstop Technologies
11
2.1.
Carbon-free energy production
11
2.2.
Hotelling time path and backstop technology
12
2.2.1.
A simple model of resource extraction
12
2.2.2.
Incorporating a backstop technology
13
2.2.3.
Price and extraction paths
15
2.3.
Political intervention and backstop technology
16
2.3.1.
Falling tax rates
16
2.3.2.
Constant tax rates
18
3.
Market Power
21
3.1.
Resource extraction on imperfect markets
22
3.2.
Backstop and market power
23
3.2.1.
Backstop supplied by an Oligopoly (M )
23
3.2.2.
Backstop supplied under perfect competition (M P C)
25
3.3.
Correcting Market Power: Multiple tax paths
29
3.3.1.
Normative and Positive Paths
29
3.3.2.
Efficiency-inducing tax/subsidy-schemes
30
3.3.3.
Definition of a family of tax/subsidy-schemes
33
4.
Optimal time path of a carbon tax
36
4.1.
Stock externality
37
4.1.1.
Without backstop technology
37
4.1.2.
Optimal taxation
38
4.1.3.
With backstop technology
40
4.1.4.
No decay of greenhouse gases (
= 0)
43
4.2.
Flow Externality
46
4.2.1.
Dynamic stock of pollution
46
4.2.2.
No decay of greenhouse gases (
= 0)
49
5.
Optimal Taxation: Stock Externality and Market Power
50
5.1.
Optimal extraction path
50
5.2.
Positive extraction path
51
5.3.
Optimal taxation of the oligopoly
52
5.4.
Optimal taxation and presence of a backstop technology
53
5.5.
A family of taxes correcting market power and externality
53
6.
Conclusion
55
iv
6.1.
The topics covered in this thesis
56
6.2.
Criticism and Outlook
58
Appendix
61
Appendix A.
Introduction
61
A.1.
Derivation of (1.2.3)
61
A.2.
Derivation of (1.3.1)
61
A.3.
Derivation of (1.3.5)
62
Appendix B.
Backstop
63
B.1.
Numerical example: Levy model and falling tax rate
63
B.2.
Exogenous demand reduction
65
B.3.
Scarcity rent and backstop
67
Appendix C.
Market Power
68
C.1.
Case M
68
C.2.
Case M P C
69
C.2.1.
Two phases of extraction
69
C.2.2.
Resource extraction
72
C.3.
Multiple tax paths
73
C.3.1.
Derivation of (3.3.24)
73
C.3.2.
Solutions to (3.3.24) ((3.3.26))
74
Appendix D.
Optimal time path of a carbon tax
74
D.1.
Derivation of...
74
D.1.1.
Equation (4.1.10)
74
D.1.2.
Equation (4.1.12)
75
D.1.3.
Equations (4.1.11)
76
D.1.4.
Equations (4.1.13), (4.1.14) and (4.1.15)
76
D.2.
Characteristics of extraction and carbon tax path
77
D.3.
Gradual transition to backstop technology under taxation
79
D.4.
Numerical example: Optimal carbon tax path (
= 0)
80
D.5.
Derivation of. . .
83
D.5.1.
Equation (4.2.10)
84
D.5.2.
Equation (4.2.12)
84
Appendix E.
Optimal Taxation: Stock Externality and Market Power
85
E.1.
Derivation of. . .
85
E.1.1.
Equation (5.1.10)
85
E.1.2.
Equation (5.2.5)
85
E.1.3.
Equation (5.3.1)
85
E.1.4.
Equation (5.3.5)
86
E.2.
Multiple optimal tax paths (equation (5.5.7))
87
References
v
List of Figures
vii
Attachments
xi
v
L
IST OF VARIABLES
To facilitate reading, all variables used and their meanings throughout this thesis are listed
below.
Variable
Meaning
i
Market interest rate (constant)
Price elasticity of demand (constant)
n
Number of firms operating in a market
S
0
,S
(t)
Initial resource stock at t
= 0 and stock in period t
R
(t)
Resource demand/extraction at time t
P
(t)
Resource price at time t
X
(t)
Demand for energy produced by backstop technology at time t
b
Backstop price (constant)
Rate of decrease of backstop price
A
(t)
Cumulative resource extraction at time t
Z
(t)
Stock of atmospheric pollution at time t
D
(Z(t)) Damage caused by the atmospheric carbon stock Z at time t
d
(
Z
(t)) Damage caused by the change of the carbon stock Z at time t
g
(A)
Extraction costs (if assumed as stock-dependent)
c
Extraction costs (if assumed as constant)
Rate of decay of the atmospheric carbon stock
(t)
Ad valorem or cash-flow tax rate at time t
(t)
After tax return for an ad valorem or cash-flow tax at time t
(t)
Costate variable of cumulative extraction
µ
(t)
Costate variable of the pollution stock for stock-dependent damage function
Costate variable of the pollution stock for flow-dependent damage function
(t)
Scarcity rent at time t
(t)
Absolute stock-dependent tax level at time t
(t)
Absolute flow-dependent tax level at time t
Superscript denoting optimal (time paths of) variables
P
(t)
Optimal resource price at time t
M
Subscript denoting (pure) monopoly or oligopoly
M P C
Subscript denoting a mixed market form: Resource owned by a
monopoly/oligopoly, backstop supplied under perfect competition
P C
Subscript denoting (pure) perfect competition (only used if necessary to
avoid confusion if different market forms are being considered)
Ol
Subscript denoting oligopolistic competition
¯
T
Time of exhaustion of the resource stock (if depleted in finite time)
T
b
Time at which resource price reaches the backstop price
X
Time-derivative of a variable X
^
X
Growth rate of a variable X with ^
X
=
X
X
(t)
Profit function in period t (assumed to be concave)
1
1. I
NTRODUCTION
1.1. Economic importance of fossil fuels. Fossil fuels have been mankind's major sources
for energy production for a long time. Societies started to use coal on a large scale from
the height of the age of industrialization onwards. The link between economic growth and
physical inputs as fossil energy has been weakened in recent decades to some extent as
knowledge became more and more the important driver of growth and energy efficiency has
increased [Smulders, 1995, p. 165f.]. Nonetheless, they remained to be an important deter-
minant for economic activity until today. Fossil fuels are used for nearly 70% of the world's
electricity production and it cannot be expected that this will change on the medium-run (see
figure 1.1). Among this nearly 70% fossil fuel share in world electricity production, gas and
coal account together for more than 90% (see figure 1.1). The time when mankind started
F
IGURE
1.1. World electricity production by fuel and Relativ importance of
fossil fuels, data source [Doman et al., 2009]. Projections from 2006 onwards
(own illustration)
using fossil fuels in energy production on a larger scale was also the period of time during
which the foundations of today's increased atmospheric greenhouse gas stock was laid (see
figure 1.2). Due to increasing emissions, this stock of greenhouse gases has been growing
2
ever since and fossil resources remained a main reason for the emission of gases as carbon-
dioxide. In recent years the debate about global warming has been a dominant topic in
political discussion. Because of their well-known polluting nature, the acceptance of energy
generation by using "dirty" fuels has been shrinking in many (especially richer) countries
and the willingness to support political action to encourage the research and development
for alternative methods of energy production has increased. Environmental policy has be-
come tougher in many countries. For example, Germany has considerably increased petrol
and energy taxes since 1999, when the so called "ecotax" was introduced. Energy generated
by alternative methods of power generation, like solar or wind energy, is being subsidized
significantly in Germany through feed-in-tariffs. On a multinational level, the EU introduced
an emissions trading system in the context of the Kyoto process, which was started in 2005,
and even banned light bulbs from Europe's supermarket shelves from this year on. These
measures stand exemplarily for the general trend to a tighter environmental legislation. But
F
IGURE
1.2. Annual temperature deviation from the 25 year average of
1951-1975 (data source: [Lugina et al., 2006]) and Global CO2 Emissions
from Fossil-Fuel Burning, Cement Manufacture and Gas Flaring (1820-2006)
(data source: [Boden and Marland, 2009]) (own illustration)
political measures taken so far by national (or regional) governments mostly aimed at low-
ering the demand for fossil resources. The underlying assumptions can be pointed out by a
3
simple example of everyday-life Europe's light bulbs: Using energy saving bulbs instead
of standard ones lowers a household's energy consumption. If all households do the same,
an economy's total energy consumption declines and the lower demand for energy would
then translate into less demand for energy generated by, e.g., coal plants, which could lower
the demand for coal. Lower demand for coal should finally lead to less coal being burnt and
to a reduction in carbon emissions.
At first sight, this simplified argumentation sounds logical. But, as figure 1.3 proves,
it cannot be said that this policy approach has been effective. Obviously the market for
exhaustible resources is not solely determined by the demand side. What escaped most
people's attention so far: The supply side plays a special role in this case. Fossil fuels aren't
being produced like normal goods for which production ceases at some point in time when
the product can't be sold profitably anymore. Exhaustible resources are extracted from a
given total stock whose size is determined by nature accordingly their total supply over time
is also naturally determined. The problem the owners of exhaustible resource deposits face
is how to allocate the depletion of their stocks over time. At the same time it is intiuitively
clear that the value of a resource deposit is determined by the present value of all future sales
that can be expected [Solow, 1974, p. 2].
F
IGURE
1.3. World per capita carbon emissions, data source [Doman et al.,
2009]. Projections from 2006 onwards (own illustration)
Keeping this special nature of natural resources in mind, Sinn [Sinn, 2008] has given an
interesting explanation for the finding that global environmental policy has not been effec-
tive. He expounds the effects of a gradually tightening environmental policy as a threat to
the suppliers of fossil fuels. The resource owners have to fear about their future earnings
as the present value of their resource deposits is negatively affected by a foreseeable lower
demand in the future. For example, the profits that can be expected from selling a barrel of
oil in 20 years crucially depend on world oil demand at this point in time. If it can be seen
that the future world oil demand will be lower than originally expected, the owner of an oil
4
well will adopt to the new situation and increase oil extraction in the present and the near
future. The present oil price now seems more favorable compared to the (now lower) future
prices. Consequently, more oil will be produced today and carbon emissions are going to
rise as well. Sinn's argumentation is the starting point to discuss further issues concerning
environmental taxes on exhaustible resources.
1.2. A first theoretical approach to the problem. The following subsections shall help
to explain the nature of market failure on a perfectly competitive fossil resource market
due to greenhouse gas emissions and how (simplified) political intervention affects resource
extraction.
1.2.1. Resource extraction under perfect competition. A resource owner possesses an initial
resource stock of S
0
. Cumulative extraction at t
0
is zero. The accessibility of the resource
stock varies and the easiest accessible deposits are depleted first. Stock-dependent extrac-
tion costs g
(A
t
) are the lowest in t
0
and are afterwards rising as a function of cumulative
extraction.
1
The stock variables S
(t) and A(t) are developing according to. . .
(1.2.1)
S
(t) = S
0
-
t
0
R
(t) dt = S
0
- A(t)
On a competitive market, every resource owner aims at maximizing his discounted flow of
profits from extracting the resource subject to the exhaustibility restriction. Assuming a
concave profit function, the resource owners' maximization problem can be described by
2
(as in [Sinn, 2008, page 389]):
(1.2.2)
max
R(t)
= max
R(t)
0
e
-it
[P (t)R(t) - g(A(t))R(t)] dt
Solving the problem leads to a positive extraction path described by. . .
3
(1.2.3)
P
P - g
(A)
= i
A thought experiment explained by Sinn [Sinn, 2008, p. 366] illustrates the logic behind this
condition. Consider a resource owner who decides whether to extract a unit of his resource
stock today or tomorrow (t
0
or t
1
). By extracting a unit of the resource today, he realizes
a net profit of P
(0) - g(A) that can be invested at the capital market interest rate of i.
Alternatively, he can extract the same unit of resource in the following period and earn a net
profit of P
(0) - g(A) +
P , where
P is the change in price compared to the previous period.
i
[P ((0)) - g(A)] = P (1) - g(A) with P (1) = P (0) +
P
(1)
(1.2.4)
1
This is equivalent to extraction costs declining in S: g
(S) < 0
2
The market interest rate is given by i
3
See appendix A.1 for the complete maximization problem
5
In order to be indifferent between extraction today or in the next period, both alternatives
must yield the same return. Accordingly, the above equation (1.2.4) leads to the same positive
optimality condition (1.2.3).
1.2.2. Optimal resource extraction and externality. There is a normative counterpart to the
positive optimality equation. Before it can be derived, the normative model that it is based
on has to be specified. In this thesis I use a model by Hoel and Kverndokk presented in [Hoel
and Kverndokk, 1996, p. 117ff.]. The model is specified as follows:
· Utility is given by the area under the demand curve: U (R) =
R
0
P
(u) du
· As the utility function is defined by the total willingness to pay, marginal utility
equals the consumer price: U
(R) = P (R)
· Stock-dependent extraction costs g(A) are constantly rising by at least > 0 (that is
g
(A) ,
is a small number). In this case, no initial resource stock S
0
needs to
(but can) be defined.
· Z
t
is the pollution stock in the atmosphere at time t. For now it is developing accord-
ing to
Z
t
= R
t
. For this specification we only need one state variable
4
· The atmospheric pollution causes social damages described by the strictly convex
damage function D
(Z) (D (Z) > 0 and D (Z) > 0, Z > 0).
5
The maximization problem of the benevolent and fully informed social planner is:
max
R
0
e
-iu
[U (R(u)) - g(A(u))R(u) - D(A(u))] du
(1.2.5)
subject to:
A
t
=
Z
t
= R
t
and R
t
0
The current value Hamiltonian and the necessary conditions solving this problem are given
by:
H
= U (R
t
) - g(A
t
)R
t
- D(A
t
) +
t
R
t
(1.2.6)
H
R
t
= U (R
t
) - g(A
t
) +
t
= 0
(1.2.7)
-
H
A
t
= - i = g (A
t
)R
t
+ D A(
t
)
(1.2.8)
The transversality condition for the state variable A
(t) is lim
t
e
-it
t
= 0. As rep-
resents the shadow costs (scarcity rent/user cost + future damages) of cumulative resource
extraction it can assumed to be negative. From (1.2.7) and (1.2.8) it can be seen that without
environmental damage D
(A), the normative counterpart to (1.2.3) would be given by. . .
(1.2.9)
U
(R)
R
U
(R) - g(A)
= i
4
As Z
t
= A
t
. I'm using this assumption to reproduce the result of Sinn in [Sinn, 2008, p. 374].
5
This term is equivalent to
(S) (production function of the resource in situ) in [Sinn, 2008]. Sinn uses
a similarly, but not identically specified model that does not distinguish between resource and pollution stock
and thus would not allow for a dynamic stock of pollution.
6
. . . which would be (as U
(R) = P (R)) equivalent to the result on a perfectly competitive
resource market. This condition changes if the resource is polluting to:
(1.2.10)
U
(R)
R
+ D (A)
U
(R) - g(A)
= i
. . . which implies a lower rate of price increase as D
(A) > 0. Again this condition can also
be derived by a thought experiment (as in [Sinn, 2008, p. 372f.]). If i equals the rate of time
preference, then extraction of a unit of the resource stock today yields a higher discounted
utility than extraction tomorrow. But, extraction of a resource unit causes marginal damage
D
(A) in the following period.
· U (R) - g(A) -
D (A)
1+i
If the resource unit is being extracted in the next period, it yields a higher marginal utility
as it has been more scarce in the previous period. But as this utility will be realized in the
future, it has to be discounted by the rate of time preference.
·
U (R)-g(A)+U (R)
R
1+i
If the society shall be indifferent between both alternatives, they have to yield the same
present net utility. Equalizing the return of both alternatives also leads to (1.2.10).
Equation (1.2.10) can be written as
6
. . .
(1.2.11)
^
P
= i[1 -
g
(A)
P
] -
D
(A)
P
Comparing this path with the positive optimal path (given by (1.2.3) after simple rearrang-
ing). . .
(1.2.12)
^
P
= i[1 -
g
(A)
P
]
. . . one can see that the normative price path is less steep than the positive one, implying that
on a competitive resource market (under the specific assumptions made) the resource would
be extracted too fast (see figure 1.4)
7
.
After incorporating environmental damage in the model, it is no longer guaranteed that
positive and normative extraction paths coincide. In the above case it is clear that the resource
is extracted too fast and the regulator needs to implement political measures that induce a
reduction in the extraction rate of the private sector. The normative extraction path takes
environmental damage due to the stock of pollution in the atmosphere into account. Without
taxation or other kinds of intervention, this damage is not being internalized by the resource
extracting firms.
6
Logarithmic differentiation of (1.2.7) and using (1.2.8) as well as U
(R) = P yields (1.2.11)
7
The extraction path in the R-S-diagram can be derived by
dR
dS
= -
R
R
= -(
^
R
^
P
) ^
P
= ^
P as
=
^
R
^
P
[Sinn,
2008, p. 374]
7
F
IGURE
1.4. Normative vs. market extraction path (as in [Sinn, 2008, p.
375]). The market extraction path is too steep.
1.3. Introducing environmental policy. As market and normative solutions do not always
coincide, it is necessary to consider the effects of environmental policy on the positive extrac-
tion path. As they can be used to explain effects of almost any kind of political intervention,
we will focus on analyzing the effects of ad valorem and cash-flow taxes.
1.3.1. Constant cash-flow tax. A constant cash-flow tax with the tax rate reduces the net
profit of the resource owner in any period by a given percentage rate. After tax net profit is
given by
(1 - )(P - g(A)) = (P - g(A)). Applying the thought experiment according
to Sinn [Sinn, 2008, p. 377f.] again: Extraction in t
0
and investing proceeds at the capital
market yields in the following period t
1
:
[P - g(A)](1 + i). Postponing extraction in t
0
and
extracting the unit in t
1
yields:
[P - g(A)] +
P
8
. As both alternatives must yield the same
net profit, the cash-flow tax is obviously neutral. It is levied at a constant rate on net profit
in all periods. The market solution is again: i
=
P
P -g(A)
the same solution as in the case
without taxation.
1.3.2. Constant ad valorem tax. Introducing an ad valorem sales tax on resource extraction
also reduces the sales margin, but if price or extraction costs change in time, the reduction
in sales margin may vary in time as well. The difference between price and extraction costs
declines from P - g
(A) to (1 - )P - g(A) = P - g(A). The new cash-flow stream is
given by RP - g
(A)R or RP -
g(A)R
[Sinn, 2008, p. 378]. Consequently, the new positive
8
Including
P in the taxed cash-flow leads to a different solution than Sinn's; that is i
=
P
P -g
. It could be
argued that
P is the net marginal price increase.
8
optimality condition can be described by
9
:
(1.3.1)
i
=
P
P -
g(A)
If the ad valorem tax is a "real" tax (
0 < < 1), the denominator of equation (1.3.1) is
smaller than without taxation (as if an increase in extraction costs), so the price change in the
numerator also has to be smaller to satisfy the equality with the interest rate. Accordingly, the
change in quantity is also lower than without taxation and the extraction path consequently
is less steep, presuming significant extraction costs. If extraction costs are close to zero, then
ad valorem and cash-flow tax are equivalent.
1.3.3. Variable tax rates. It is realistic to assume that the tax rate won't remain constant
until the (economically extractable) resource stock is exhausted. First, let the cash-flow tax
rate rise in time at a constant rate of ^
according to the function. . .
(1.3.2)
(t) =
(0)e
^
t
The cash-flow tax loses its neutrality immediately. If the tax rate is rising (
^
>
0, that is
^
<
0), the resource owner's share of rent is constantly falling. The resource owner takes the
rising tax rate into account and chooses a different extraction and price path for the resource.
His new optimality condition is given by [Sinn, 2008, p. 379]
10
:
(1.3.3)
i - ^
=
P
P - g
(A)
As one can see from formula (1.3.3), a tax rate rising in time (^
<
0) gives an incentive to
extract and sell more of the resource in the near future and less when the tax rate has risen to
a higher level (in equation (1.3.3),
P is larger after implementation of the tax).
Now consider an ad valorem tax with a non-constant tax-rate. In a paper by Long and
Sinn [Long and Sinn, 1985] it is shown that tax neutrality now depends on growth of total
tax load P . Neutrality is given if the total tax load is rising at the rate of interest, that is:
(1.3.4)
^
+ ^
P
= i
In this formula, ^
P is the time path of the resource price in a world without taxation and in
case of a neutral tax the growth rate of the price must be equivalent to the rate in a world
without taxation. Equation (1.3.4) can also be written as [Sinn, 2008, p. 379]
11
. . .
(1.3.5)
^
= i
g
( ~
A
)
P
( ~
R, t
)
. . . where ~
R and ~
A are the time paths of resource extraction over time and cumulative extrac-
tion in a world without taxation. If extraction costs are positive, the price growth is smaller
9
For a formally derivation of equation (1.3.1), see appendix A.2
10
Thought experiment to derive equation (1.3.3):
(1 + i)(P - g(A)) = (1 + ^)(P - g(A)) +
P
11
see appendix for derivation of equation (1.3.5) (appendix A.3, page 62)
9
than without extraction costs (smaller than i: ^
P
= i(1 -
g( ~
A)
P ( ~
R,t)
), so that if g( ~
A
) > 0, ^
P < i).
This implies that according to (1.3.4) there is some room for a rising ad valorem tax rate
without giving any incentives to the resource owners to extract the resource faster. Thus,
the neutral ad valorem tax rate is growing in time, while a constant tax leeds to a more
conservative behavior of the resource owners
12
.
We conclude that for both cash-flow as well as ad valorem tax, there exists a growth rate
that leads to more extraction in the present and the near future. For the cash-flow tax, a
growth rate of
^
>
0 is already sufficient. For an ad valorem tax, the tax rate has to grow
faster than i
g( ~
A)
P ( ~
R,t)
.
1.3.4. Other political measures. The considerations for the ad valorem tax are valid for
any kind of demand reducing political measures [Sinn, 2008, p. 381]. Those measures
range from subsidizing the insulation of homes, the subsidisation of alternative methods of
energy production to stricter emission standards for cars. The demand reduction leads to a
downward shift of the demand curve P
(R, t). The resulting "price wedge" can be measured
by comparing the demand curves before and after political intervention (see figure 1.5). The
price wedge can be written similar as in [Sinn, 2008, p. 381]:
(R, t) denotes the downward shift of the demand curve at different points in time. Then
( ~
R, t
) is the "price wedge" between the new demand curve and the old demand curve (case
without political intervention). The new demand curve can then be described as:
(1.3.6)
P
new
(R
new
, t
) = (1 - ( ~
R, t
))P ( ~
R, t
) = ( ~
R, t
)P ( ~
R, t
)
If resource owners expect
^
( ~
R, t
) i[
g( ~
A)
P ( ~
R,t)
], then they will increase extraction in the present
and the near future and the pace of global warming is being accelerated.
1.4. First conclusion and the next steps. The simplified model in section 1.2 leads to the
conclusion that competitive extraction needs to be slowed down and section 1.3 briefly intro-
duces the effects of political intervention on the positive extraction path. Therefore it should
be clear now that under the assumptions made in section 1.2, falling tax rates, rising subsi-
dies or any demand reducing measures relaxing in time would be appropriate to induce the
optimal extraction path. This is what Sinn called the "green paradox" [Sinn, 2008, p. 377]:
A gradually tightening environmental policy leads to counterproductive effects if resource
extraction is concerned.
Resource economics most likely is not the branch of economics that receives the most
public and political attention. But Sinn's argumentation gained some publicity, as it puts all
established approaches of environmental policy into question. One of the comments made
on Sinn's proposal so far was recently published by the German environmental minister
13
.
12
From (1.3.5) it is easy to see that, without extraction costs, a neutral ad valorem tax is equivalent to neutral
a cash-flow tax and requires
^ = 0.
13
See attachments (in German): [Sinn, 2009a], [Gabriel, 2009], [Sinn, 2009b]
10
F
IGURE
1.5. Demand reducing measures and their translation into an ad val-
orem tax (own illustration).
(R
old
, t
) is the "price wedge" between old and
new demand curve at t.
Mr. Gabriel criticizes Sinn's line of argument for the (simplifying) assumption it is based on.
Among other things, he argues that the world's fossil resource stocks would not necessarily
have to be exhausted before the world becomes independent of fossil fuels. Furthermore he
claims that extraction costs for some fossil fuel reserves are uncompetitively high espe-
cially against the background of recent advances in alternative energy production. In a reply
to Gabriel's criticism, Sinn stresses that the largest part of the prices for fossil resources
from some deposits is largely made up by rents and that for some uses it won't be possible
to replace fossil resources in the near future completely. Steps towards alternative methods
of energy production as the recently by German companies announced project to import so-
lar energy from Northern Africa (Desertec) would lead to even faster extraction in the near
future.
In any case, it is in accordance with reality to take the effects of substitutes on fossil re-
source extraction and taxation into account. This will be done in section 2 (and partially also
in the other sections 3, 4 and 5). As resource markets tend to be not perfectly competitive,
the issue of market power will be discussed in section 3 and in section 3.3 the implications
resulting from market power for optimal taxation will be discussed. In section 4 subsection
4.1 deals with an optimal carbon tax for damages caused by dynamic stock pollution (with
and without substitute) and 4.2 with the taxation of flow externalities. Section 5 derives pos-
sible carbon tax paths under the presence of both market power and stock externality, while
section 6 concludes.
11
2. B
ACKSTOP
T
ECHNOLOGIES
A backstop technology is a perfect substitute for any fossil fuel
14
. The backstop technology
is non-polluting and abundantly available.
2.1. Carbon-free energy production. Are such technologies available to replace oil, gas
and coal? Concerning energy production for most purposes, the answer most likely will
be yes. But as for example oil is also used as an input in the production of plastics and
a hybrid engine for airplanes is not yet available, it probably might take a long time until
fossil resources will be fully replaced by renewable resources. Nonetheless, the considera-
tion of a scenario in which the resource owners' profits are threatened by the existence of
a backstop-technology is essential for analyzing the effectiveness of environmental policy
against a realistic background. To be able to abstract from any discussion, we will focus
on energy production, where different carbon-free technologies are already being employed
today
15
. Projections by the United States Energy Information Administration (EIA) [Do-
man et al., 2009] indicate that the worldwide importance of possible backstop technologies
(nuclear power and renewable sources) will remain nearly constant at around 35% of total
electricity production until 2030. The EIA does not expect prices of fossil fuels to reach
levels that are sufficiently high to induce a change of the world's main energy sources from
fossil to renewable fuels (see figure 2.1).
A backstop technology will not be used on a large scale if it doesn't become a cheap
alternative to fossil fuels or the other way round as long as fossil resources are less costly
than the use of the alternative technology, demanders will prefer carbon fuels. The difference
in market prices between both fuels is a real economical value for them. The lower the price
of the backstop technology, the less valuable are fossil fuels. At the same time, the profits
from resource extraction have to be large enough to cover extraction costs. That is why a
decrease in the backstop price can affect the size of the economically extractable resource
stock.
In the following the theoretical results of incorporating a backstop technology in a re-
source extraction model will be derived and the political implications of these results will be
discussed.
14
The term "backstop technology" was introduced by William Nordhaus [Nordhaus et al., 1973, p. 532]
15
Concerning energy production, Nordhaus names fusion reactors as a possible future backstop technology
that could fully replace any other sources of energy generation [Nordhaus et al., 1973, p. 532] including all
fossil fuels. Other options available today might be standard nuclear power or renewable energy sources as
solar or wind power, or even for some countries tidal and wave energy [Whittington, 2002, p. 1657 ff.].
In the present, the most important backstop technology used in energy generation is nuclear power. Nuclear
power generating capacities are being extended in many countries around the world as it is a carbon-free and
reliable method of producing electricity. Even Sweden a country that once was on the forefront of nuclear
phase-out is now refitting its nuclear power generation capacities.
12
F
IGURE
2.1. Importance of "Backstop Technologies" in worldwide energy
production and Oil price projection (reference case), data source [Doman
et al., 2009]. Projections from 2006 onwards (own illustration)
2.2. Hotelling time path and backstop technology. A model published by Levy [Levy,
2000] demonstrates how the existence of a backstop technology influences resource extrac-
tion.
2.2.1. A simple model of resource extraction. Concerning exhaustible resources the Hotelling-
rule implies that the spot price of a resource P is rising at a rate equal to the market discount
rate i [Hotelling, 1931, page 141]. Extraction costs are left out in the following. That is. . .
P
(t)
P
(t)
= i
(2.2.1)
or (by integration): P
(t) = P
0
e
it
13
The resource owners maximize their discounted cash-flows under perfect competition in the
finite
planning horizon
[0, T ]:
(2.2.2)
max
R(t)
T
0
e
-it
P
(t)R(t) dt = P
0
S
0
At the endpoint T , the whole resource stock S
0
has to be exhausted. Isoelastic demand is
given by R
= P
-
. This leads to the following market clearing condition (2.2.3):
(2.2.3)
T
0
R
(t) dt =
T
0
[P
0
e
it
]
-
dt = S
0
Rearranging of equation (2.2.3) gives the initial spot price as a function of the initial stock,
the planning horizon, the elasticity of demand and the discount rate:
(2.2.4)
P
0
= [
1 - e
-iT
iS
0
]
1
A higher initial resource stock S
0
and a higher interest rate i imply a lower initial spot price
P
0
. After the initial spot price P
0
has been determined, it is possible to formulate the spot
price path as. . .
(2.2.5)
P
(t) = [
1 - e
-iT
iS
0
]
1
e
it
. . . and equation (2.2.5) can be used to determine the time path of market demand (given by
P
(t)
-
):
(2.2.6)
R
(t) = [
1 - e
-iT
iS
0
]
-1
e
-iT
= e
-iT
iS
0
1 - e
-iT
As can be seen from (2.2.6), the extraction path is the steeper the higher the initial resource
stock and the interest rate.
2.2.2. Incorporating a backstop technology. Now suppose that the price of the backstop-
technology declines as described in the following equation:
(2.2.7)
b
(t) = b
0
e
-t
The resource becomes obsolete at a point in time ¯
T T at which the economy switches
to the backstop technology. At this point, the price of the backstop-technology has to be
identical with the price of the resource (see figure 2.2). That is. . .
(2.2.8)
P
( ¯
T
) = b( ¯
T
). . . or . . . P
0
e
it
= b
0
e
-t
Knowing this we can derive the point in time ¯
T :
(2.2.9)
¯
T
=
1
i
+
ln
b
0
P
0
The higher the initial price of the backstop technology and the lower the initial price of
the fossil resource, the later in time the economy switches from fossil fuels to the backstop
technology. A higher interest rate i and a higher rate of decline of the costs of the backstop
technology lead to an earlier adoption of the backstop-technology.
14
If a regulator wants to extend the period of resource use and slow down resource extrac-
tion, the price of the backstop should be kept at a high level. Similarly, as falling tax rates,
this result is counterintuitive, as resource extraction and thus climate change is being ac-
celerated if the alternative methods of energy production become more competitive. In this
context, can be interpreted as a variable depending on the rate of investment into the de-
velopment of the backstop technology. The less the economy invests into the development
of the backstop-technology, the lower is the decline in backstop-price b
(t) and the later the
economy stops using fossil fuels.
It is important to note that the resource stock is being depleted in any case in finite time if a
backstop technology is available as there is an endpoint for the resource price path specified
(which is the maximum resource price)
16
. Even if the resource would be constantly sold at
the (now assumed to be constant) price of the backstop b, then the time until depletion of the
resource stock would be finite and given by ¯
T
=
b
-
S
0
. But the resource is being depleted at
an earlier point in time, as the resource price has to rise at the rate of interest to maximize
the discounted stream of profits to the resource owner, which implies a price path below
the constant b for t < ¯
T (see figure 2.3). There is no asymptotic depletion of resource
stocks which is often the result of models of resource extraction without backstop, even if
the planning horizon is infinite.
F
IGURE
2.2. Price paths of backstop technology and fossil resource (as in
[Levy, 2000, p. 4]). Resource price is rising at the rate of interest, while
backstop price is falling at .
16
The whole resource stock S
0
could not be sold in finite time under the presence of a backstop if the price
was equal to the case with no backstop. The differences between marginal profit, resource price and extraction
in both cases must have the same sign during the whole period of resource extraction (until the stock under
presence of a backstop has been exhausted at ¯
T , [Hoel, 1978, p. 32]). The resource price must be lower t < ¯
T
if a backstop technology is available (also see, e.g. [Chakravorty et al., 2007, p. 17])
15
F
IGURE
2.3. Backstop technology in a basic model (no technological
change, figure as in [Perman et al., 2003]). The resource stock is depleted
when the resource price reaches the backstop price.
2.2.3. Price and extraction paths. Knowing that under the presence of a backstop technol-
ogy a resource deposit is (possibly
17
) being exhausted at an earlier point in time than without
backstop technology ( ¯
T < T ), the resource's extraction and price path under the presence of
a backstop technology have to be looked at. The market clearing condition (2.2.3) can now
be written (using (2.2.9)) as:
(2.2.10)
1
i+
ln
b0
P0
0
P
(t)
-
=
1
i+
ln
b0
P0
0
[P
0
e
it
]
-
= S
0
17
Depending on the length of the planning horizon. Under the presence of a backstop technology, the
resource deposit is being exhausted at least as fast as without backstop. If, e.g., the planning horizon is infinite,
then the resource is being depleted faster under the presence of the backstop than in any case without backstop.
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- , 2009, Environmental Taxes on Exhaustible Resources, Hamburg, Bedey Media GmbH, https://www.diplom.de/document/227388
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