The rise and fall of bioenergy

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Sub-Categories	EnvironmentKraft og energi
Year	2020
Author(s)	Michael Hoel
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Journal of Environmental Economics and Management 101 (2020) 102314	
Contents lists available atScienceDirect
Journal of Environmental Economics and Management
journal homepage:www.elsevier.com/locate/jeem
The rise and fall of bioenergy	☆
Michael Hoel
Department of Economics, University of Oslo, P.O. Box 1095, Blindern, N-0317, Oslo, Norway
article info
Article history:
Received 6 April 2019
Revised 24 January 2020
Accepted 19 February 2020
Available online 29 February 2020
JEL classiﬁcation:
Q42
Q48
Q54
Q58
Keywords:
Bioenergy
Renewable energy
Climate policy
Carbon tax
Second best
Subsidies
abstract
If bioenergy has a less negative impact on the climate than fossil energy, it may be optimal
to have a signiﬁcant increase in the use of bioenergy over time. Due to the difference in the
waytheclimateisaffectedbythetwotypesofenergy,thefuturetimepathoftheuseof
bioenergy may be non-monotonic: It may be optimal to ﬁrst have an increase in its use, and
later a reduction. Optimal taxes/subsidies are derived both for the ﬁrst-best case and for the
case of a constraint on the size of the fossil tax.
©2020 The Author. Published by Elsevier Inc. This is an open access article under the CC BY
license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
How far can bioenergy contribute to the mitigation of the climate change problem? Consuming energy from fossil fuels has
led to one of the greatest challenges humans have ever faced. Nearly every country in the world has signed the Paris Agreement,
intending to reduce global emissions. At the same time, the world needs energy, and scholars as well as policy makers point
to bioenergy as a possible solution. This paper provides an analysis of whether, when, and how bioenergy can contribute. The
results are surprising as well as provocative.
Policies to promote the use of bioenergy are often justiﬁed by an argument that bioenergy is ”climate neutral”, and that the
production and use of bioenergy gives a reduction in the use of fossil energy. There are two problems with these arguments. First,
recent contributions have questioned whether the production and use of bioenergy is climate neutral in the sense that it has no
climate impact. Moreover, even if the production and use of bioenergy had no climate impact, the argument for subsidizing it
is questionable. It is widely recognized among economists that a price on carbon emissions, through a carbon tax or a price on
tradeable emission permits, is the most important policy instrument to reduce such emissions. Standard economic reasoning
also implies that in the absence of other market failures, an appropriately set carbon price is theonlyinstrument needed to
☆I gratefully acknowledge comments and suggestions from participants at conferences and seminars in Athens, Geilo, Munich, Oslo, Tilburg and Umeå. While
carrying out this research I have been associated with CREE -Oslo Centre forResearch on Environmentally friendly Energy. CREE is supported by the Research
Council of Norway.
E-mail address:mihoel@econ.uio.no.
https://doi.org/10.1016/j.jeem.2020.102314
0095-0696/
©2020 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 2
achieve an eﬃcient climate policy. If bioenergy had no direct climate impact, it should therefore be neither taxed nor subsidized.
A possible reason for subsidizing bioenergy could be that tax on fossil energy is ”too low”, i.e. lower than the Pigovian rate (equal
to the marginal environmental cost of carbon emissions).
To study these issues, this paper considers a model with three types of energy; fossil energy, bioenergy and non-carbon
energy. The main focus of the paper is on the interaction between fossil energy and bioenergy, which are assumed to be perfect
substitutes. The climate effects of the two types of energy are discussed in Section2. Unlike fossil energy, bioenergy is climate
neutral in the sense that it is possible to have a constant positive use of bioenergy without this giving any change over time in
the carbon concentration in the atmosphere. This implies that in a long-run steady state, we will (under conditions speciﬁed in
section3) have positive bioenergy productionand zero fossil energy production.
Although bioenergy is climate neutral in the sense given above, it is shown in Section2.2that bioenergy has a negative
climate impact in the sense that for any given time path of fossil energy, the carbon in the atmosphere is higher the higher is the
level of the time path of the production of bioenergy (constant or varying over time). Hence, both fossil energy and bioenergy
have a negative climate impact, but the dynamics of these impacts differ. In section3the ﬁrst-best optimum is derived, and it
is shown that fossil energy production is zero in the long run, while the production of bioenergy and non-carbon energy are
positive and constant. Due to the difference in how the climate is affected by fossil energy and bioenergy, the time path of
bioenergy production towards its steady-state level may be non-monotonic. Particular attention is given to the case in which
marginal climate costs have an inverse L property: zero (or small) for a carbon content in the atmosphere below some threshold,
and rapidly rising after the threshold is reached. In this case bioenergy production will ﬁrst rise, and later decline towards its
steady-state level. This result holds also more generally, as long as the marginal climate cost function is suﬃciently convex in
the carbon stock.
To achieve the ﬁrst-best optimum, bioenergy must have a non-negative tax. In the limiting case of zero marginal climate
costs for low levels of carbon in the atmosphere the tax will be zero initially, but will eventually become positive.
As mentioned above, the tax on fossil energy could for some reason be constrained to be lower than its optimal value. The
implications of such a constraint for bioenergy production and bioenergy policies are studied in section4. In particular, it is
shown that in this case it may be optimal to subsidize bioenergy for all or some of the time. For the inverse L climate cost
function, it is optimal to subsidize bioenergy as long as carbonin the atmosphere is below the threshold. Moreover, during this
phase the subsidy is increasing over time. However, once the threshold is reached the subsidy will start to decline. In the long
run the tax on bioenergy may be negative (i.e. a subsidy) or positive, depending on parameters of the model.
Section5summarizes and brieﬂy discusses the main ﬁnding of the analysis.
1.1. Related literature
While the present study focuses on taxes and subsidies as policy instruments, there are several studies on the effects of a
renewable portfolio standard for bioenergy. A renewable portfolio standard in this case means that a certain politically deter-
mined minimum percentage of the total energy use must come from bioenergy. Such a renewable portfolio standard is equiv-
alent to a revenue neutral combination of a tax on fossil energy and a subsidy to bioenergy, see e.g.Amundsen and Mortensen
(2001)andEggert and Greaker (2012). A combination of a tax on fossil energy and a subsidy may be (second-best) optimal if
there is a constraint on the tax on fossil energy (see Section4), but this combination will generally not be revenue neutral.
De Gorter and Just (2009)ﬁnd that a renewable fuel standard may lead to a decrease in the total use of fuel (oil and biofuel).
This happens if the elasticity of biofuels supply is lower than the elasticity of oil supply. The effect of the policy is in this case
not only to replace oil with biofuels, but also to reduce total consumption of transport fuels, which by itself will reduce climate
costs. This result is reversed if the elasticity of biofuels supply is higher than the elasticity of oil supply, or if a biofuels subsidy
is imposed rather than a renewable fuel standard.
If a renewable fuel standard is in place, adding a subsidy or a tax rebate for biofuels will increase climate costs, even if there
were no negative climate effects from bioenergy. The reason is that if the ratio between fossil energy and bioenergy is ﬁxed
due to the renewable portfolio standard, a subsidy to bioenergy works as an implicit support to oil and, hence, GHG emissions
increase (seeDeGorter and Just, 2010). The effects of combining several policies related to bioenergy have also been studied by
Eggert and Greaker (2012)andLapan and Moschini (2012).
Lapan and Moschini (2012)compare a renewable fuel standard with a subsidy to biofuels, and ﬁnd that the former welfare
dominates the latter. This result follows directly from the fact that a renewable fuel standard is identical to a revenue neutral
combination of a tax on oil and a subsidy to biofuels, since there is an emission externality from the use of fossil fuels.
There are several studies on the interaction between renewable and nonrenewable energy. In particular,Tahvonen and Salo
(2001)show that energy production may evolve from renewables to nonrenewables and back to renewables. However, in this
study there is no consideration of climate or other environmental impacts of producing the two types of energy.Chakravorty et
al. (2008b)analyze the transition from coal to clean energy via natural gas, taking climate effects into consideration. This has
some resemblance to the present study where I consider bioenergy instead of natural gas. However, the way bioenergy affects
the climate is very different from natural gas, as will be shown in Section2.
There are some contributions in the literature that analyze bioenergy policies in models with dynamic oil supply, see e.g.
Chakravorty et al. (2008a),Chakravorty and Hubert (2013),Grafton et al. (2012),andFischer and Salant (2012). However, none
of these studies includes negative climate effects from the use of bioenergy, which seems to be crucial when assessing the effect
of bioenergy policies on climate costs.

3 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
time
xt
xSt
time
Fig. 1.Atmospheric carbon from fossil energy.
Climate effects from bioenergy are included in the analysis ofGreaker et al. (2014), where cumulative oil extraction depends
on the policies used. An important result is that whereas a non-declining tax on oil or a non-declining subsidy to biofuel reduces
cumulative oil extraction, a blending share has no effect on cumulative oil extraction (as long as the implied share of fossil is
bounded away from zero). Such a blending mandate may nevertheless contribute to lower GHG emissions, as this policy will
increase consumer prices and hence reduce the total fuel use at any time.
While most studies of bioenergy consider bioenergy from fuel crops, an alternative is to use the harvest from standing forest
to produce bioenergy.Hoel and Sletten (2016)study climate policies for this type of bioenergy when it is assumed to be a perfect
substitute for fossil energy (as assumed in the present study). However, unlike the present study they assume that the relative
climate impact of the two types of energy areconstant over time. Moreover, their main focus is on the long-run steady state,
where there is no production of fossil energy.
2. The climate effects of fossil energy and bioenergy
The climate is affected by the amount of carbon in the atmosphere, and burning of fossil fuels (i.e. the use of fossil energy)
adds carbon to the atmosphere. As we shall see below, this is true also for bioenergy, although the mechanism is somewhat
different. For both types of energy, the model I use gives a verycrude and simple description of how the atmospheric carbon
is affected. The following quotation fromMillner (2015)mighthelptojustifythesomewhatdrasticsimpliﬁcationsIuse:”Like
many economic models, our is a fable in which reality is pushed to absurdly simplistic extremes in order to cleanly illustrate a
mechanism that may help explain real world policy outcomes.”
2.1. Atmospheric carbon from fossil energy
Using carbon energy gives an immediate release of carbon to the atmosphere. Over time, some of this carbon is transferred to
the ocean and other sinks. However, a signiﬁcant portion (about 25% according to e.g.Archer, 2005) remains in the atmosphere
for ever (or at least for thousands of years).Farzin and Tahvonen (1996)showed how this can be modeled by splitting the carbon
emissions in two, with a standard decay rate for the ﬁrst but zerodecay for the second. In the present paper I only consider the
carbon released that remains in the atmosphere for ever. This is thus a special case of the model just mentioned, with immediate
decay of the ﬁrst part of the emissions.
1
In my model,x	(t)measures the yearly production of fossil energy, andS	x(t)measures the stock of carbon in the atmosphere
caused by the production of fossil energy. I measure units of fossil fuel and carbon in the atmosphere in the same units, e.g. tons
of carbon, implying
̇Sx(t)=	x(t). The dynamics of carbon in the atmosphere from fossil energy production is illustrated inFig. 1.
In this ﬁgure we ﬁrst have constant production, then production jumps up to a higher level, and ﬁnally it drops to low level.
The important point (to be contrasted with bioenergy production below) is that as long as fossil energy production is positive,
carbon in the atmosphere grows; it is only the rate of growth that depends on the size of fossil energy production.
1This simpliﬁcation is of no importance for the long-run steady state, but the details of the dynamics towards the steady state would be slightly different if I
had taken the partial depreciation into account.

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 4
time
t
ySt
time
Fig. 2.Atmospheric carbon from bioenergy.
2.2. Atmospheric carbon from bioenergy
The most important source of greenhouse gas emissions from bioenergy is related to land use changes. As long as bioenergy
production is constant, the balance of carbon in the atmosphere and in biomass and soils will remain constant: The yearly
release of carbon from using bioenergy will be exactly matched by regrowth of biomass. However, achangein the production
of bioenergy will change this balance. As pointed out by e. g.Fargione et al. (2008), converting forests, peatlands, savannas, or
grasslands to increase the production of crop-based bioenergywill give a large immediate release of carbon from plant biomass
and soils to the atmosphere. Similarly a more intensive use of forests to increase bioenergy production will reduce the carbon
content stored in the forests.
These mechanisms are discussed in more detail in the next two sub-sections. Based on this discussion, I model the relation-
ship between bioenergy production and the atmospheric carbon caused by this production as follows.
Lety
(t)measure the yearly production of bioenergy, andS	y(t)measure the stock of carbon in the atmosphere caused by the
production of bioenergy. The relationship between the two is in the model simply assumed to beS
y(t)=	������y(t).Therelationship
is illustrated by the heavily drawn curve inFig. 2, with a similar time pattern of energy production as for fossil energy production
inFig. 1. The level of carbon in the atmosphere depends on the size of bioenergy production, but as long as bioenergy production
is constant, carbon in the atmosphere is also constant. This is in sharp contrast to the case of fossil fuel illustrated inFig. 1,where
carbon in the atmosphere was growing for any positive production level.
In the formal model, and inFig. 2,anincreaseiny
(t)from one constant levely	1to a new constant levely	2will immediately
increaseS
y(t)from	������y1to	������y2for the reasons given above (and elaborated upon in the next two subsections).	2Similarly, a
reductiony
(t)from one constant levely	2to a new constant levely	3will immediately reduceS	y(t)from	������y2to������y3.Inpractise,
the latter comes about by reconverting land used for bioenergy production to forest land, and from less intensive harvesting of
standing forests, so the volume, and hence stored carbon, of such forests increases.
In reality, the change in the amount of carbon to a new long-run level will not be immediate. Hence, the adjustment is likely
to look more like the case illustrated by the dashed lines inFig. 2. In particular, the regrowth of biomass after a reduction in
the production of bioenergy will not be immediate. Our simpliﬁed way of modeling this may nevertheless be justiﬁed for two
reasons. First, even if full regrowth may take quite a long time, a signiﬁcant part of the increased long-run carbon stored in
the biomass may occur within a reasonable timeframe. Second,in our model it is never optimal to have a downward jump in
bioenergy production fromy
2toy	3as illustrated inFig. 2. If bioenergy production declines, it will be a gradual decline, with a
gradual regrowth of biomass.
The next two subsections give a more detailed justiﬁcation for the relationshipS
y(t)=	������y(t). Readers who prefer to move
directly to the model and the analysis may skip these two subsections and go directly to Section3.
2.2.1. Bioenergy from agricultural crops
In many cases bioenergy crops are grown on existing agricultural land, so that there is little or no direct effect on carbon
emissions. However, increasing the production of bioenergy crops at the expense of food production raises other concerns: Food
2The three horizontal lines in the upper part ofFig. 2correspond to the valuesy	1,y2andy	3.

5 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
F
Fh
F	F	F
h
h
gF
Fig. 3.Forests as a carbon sink.
prices may increase, with negative consequences for low-income households world wide (see e.g.Chakravorty et al., 2008a;
Chakravorty et al., 2009). Moreover, reduced land for food production may cause indirect land use changes: The increased value
of land used for food production may lead to forest clearing to convert land for food production. This forest clearing will have
the immediate effect of releasing carbon to the atmosphere, and thus increase climate costs. BothSearchinger et al. (2008)and
Lapola et al. (2010)analyze indirect land-use change, and show that the effect upon emissions may be of great signiﬁcance.
In my model I treat direct land-use change and indirect land-use change together. In particular, I assume that each unit of
bioenergy production, denotedy, requires
������units of land, and that each unit of land converted to bioenergy production will
reduce the carbon sequestered on this land by an amount
������.3Increasing the yearly production of bioenergy by one unit hence
gives a one-off increase in the amount of carbon in the atmosphere by
������������≡	������units.
The denomination ofyis energy output per year. The denomination of
������is land units pery, and the denomination of	������is
tons of carbon per land unit. Hence, the denomination of
������is tons of carbon pery, i.e. tons of carbon/(energy output per year).
As mentioned above, the output of fossil fuel is measured in tons of carbon. Moreover, I choose units of bioenergy so that one
unit of bioenergy is equally useful to the user as one unit of fossil energy. Since fossil energy is measured in tons of carbon,
bioenergy output is also measured in tons of carbon. Hence, energy output per year has the denomination tons of carbon per
year. It follows that the denomination of
������is tons of carbon/(tons of carbon per year), i.e. ”years”. Using this approach,Greaker
et al. (2014)show that estimates in the literature of
������and	������suggest that	������typically will be in the range of about 9–24 years.
Fargione et al. (2008)have used the term ”biofuel carbon dept” to describe how many years it takes for carbon in the atmo-
sphere to be reduced following a constant increase in biofuel production combined with an equivalent reduction in fossil fuel
production. In Section3.1it is shown that the parameter
������measures this carbon dept.
In addition to greenhouse gas emissions caused by land use changes, bioenergy may have climate impacts due to the use
of fertilizer when growing biofuels crops (Crutzen et al., 2008), and the use of fossil energy in the harvesting and processing of
biofuels (Macedo et al., 2008). Even if these emissions related to the production and transportation of bioenergy are positive,
one must be careful about what conclusions one draws from this. There are, after all, emissions related to practically all eco-
nomic activity. It is only if the emissions related to production and transportation of bioenergy are larger per unit of general
resources used (labor, capital etc.) than the emissions relatedto these resources being used elsewhere in the economy that one
can conclude that increased use of bioenergy contributes to increased climate costs. The focus of the present study is on land
use changes, and other possible climate effects of bioenergy production are therefore ignored.
2.2.2. Forest based bioenergy
In the reasoning above it has been implicitly assumed that bioenergy is based on fuel crops. An alternative to converting
grazing land or forest land into land for growing suitable crops for bioenergy production is to use the harvests from standing
forests to produce bioenergy. However, wood-based bioenergy from standing forests is not unproblematic from a climatic point
of view. The carbon stored in the forest is highest when there is little or no harvest from the forest, see e.g.van Kooten et al.
(1995), or more recently,Hoel et al. (2014)and the literature cited there. Hence, increasing the harvest from a forest in order to
produce more bioenergy may conﬂict with the direct beneﬁt of the forest as a sink of carbon.
Wood-based bioenergy may take many forms, including e.g. raw ﬁrewood, processed charcoals, and pellets. The possibility
of producing liquid bioenergy from cellulosic biomass may also be a promising alternativeto using fuel crops, see e.g.Hill et
al. (2006). To the extent that bioenergy is produced from residuals that otherwise would gradually rot and release carbon, the
climate impact of this type of bioenergy is limited. If increasedproduction of bioenergy implies increased harvesting from the
forest, the time path of the carbon in the forest biomass will be shifted downwards, giving an unambiguously negative climate
effect (i.e. increased climate costs). This is illustrated inFig. 3,whereg
(F)is a standard biological growth function showing the
growth of the forest as a function of its volumeF(measuredintonsofcarbon).Withharvestinghthe net growth is hence
g
(F)−	h.
3As explained in Section2.1, Ionly consider the carbon that remains in the atmosphere for ever. Hence,	������represents the amount of carbon from land
convertion that remains in the atmosphere for ever.

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 6
If yearly harvesting increases from a constant valueh	∗to a new constant valueh	∗∗, the forest volume will eventually decline
fromF	∗=g−1(h∗)toF	∗∗	=g−1(h∗∗). Assume that one unit increased bioenergy production implies	������units increased harvest,
implying that one unit permanent increase of bioenergy production will give a permanent reduction of the volume of the forest
equal to
−������(g′)−1(h). Moreover, assume that a fraction	������of the carbon released from the forest is permanently added to the
carbon in the atmosphere. Then one unit permanent increase ofbioenergy production will give an immediate and permanent
increase in the carbon in the atmosphere equal to
−������������	(g′)−1(h)≡������.Alsointhiscase,	������is measured in years, and in this case it
will depend strongly on how intensively the forest initially is harvested. If the initial harvesthis close to the maximal sustainable
yield value (i.e. close to maxg
(F)),−g′will be small and hence	−(	g′)−1and	������will be large.
3. The optimal use of bioenergy
3.1. The model
Letxandydenote the production and use of fossil energy and bioenergy, respectively. I assume theyare perfect substitutes so
the gross beneﬁt of using these two types of energy isu
(x+y), which is strictly increasing and concave. In some of the reasoning
I assume thatu	′(0)is ﬁnite, although this is not a crucial assumption.
Although the main focus of the present paper is bioenergy and its interaction with fossil energy, the model also implicitly
gives some insight into the optimal use of energy that has no climate impact (hydro, wind, solar, nuclear). An obvious interpre-
tation of the functionu
(x+y)is that this third type of energy is an imperfect substitute forx	+y.Letzdenote the output of this
non-carbon energy and let its cost be
������(z).Thenu	(x+y)is deﬁned asu	(x+y)=	max	z[U(x+y,z)−	������(z)],whereUis the gross
beneﬁt of energy use. Assume that
������′>0,������′′>0andU	(x+y)z	<0. The optimal value ofzis given byU	z(x+y,z)=	������′(z),and
the assumptions above imply that this value ofzis declining inx
+y.
Production costs for fossil energy and bioenergy are given by the strictly convex cost functionsc
(x)andb	(y), respectively.	4I
disregard a resource constraint on the production of fossil energy. An obvious interpretation is that the resource reserves are so
large that they due to climate considerations do not imply a binding constraint on the production of carbon energy.
The model does not explicitly include any upper limit to how much bioenergy that may be produced per year. However, if
such a limit exists it can be implicitly included by letting the marginal costb
′(y)become ”very large” for values ofyclose to this
limit.
To insure an interior solution in the absence of climate costs I assumec
′(0)=	b′(0)<u′(0), although the equationc	′(0)=
b′(0)is not important for the main results.	5Climate costs are given by an increasing and convex function of the total amount of
carbon in the atmosphere,D
(S). Unless otherwise stated, I assume that this function is strictly convex.
The objective function in the social optimization problem is standard:
W
=∫
∞
0	e−rt[u(x(t)+	y(t)) −	c(x(t)) −	b(y(t)) −	D(S(t))]dt(1)
whereris the discount rate. In addition to the non-negativity constraints onxandy, we have the following constraints
(explained in section2):
S
(t)=	Sx(t)+	Sy(t)	(2)
̇Sx(t)=	x(t)	(3)
S
y(t)=	������y(t)	(4)
As mentioned above, we have chosen units of fossil fuel and carbon in the atmosphere in the same units, e.g. tons of carbon,
while the units of bioenergy are such that one unit of bioenergy is equally useful to the user as one unit of fossil energy.
Thedifferencebetween(3)and(4)is a key feature of the present model. This difference corresponds to the difference
betweenFigs. 1 and 2. Formally, the structure of the model corresponds to a model where there is both a stock pollutant (S
x)
and a ﬂow pollutant (S
y), and where the environmental damage is from the sum of these two pollutants.
It follows from(3)that one unit increase in bioenergy production gives a permanent increase of carbon in the atmosphere
equal to
������.From(4)it is clear that this is the same amount of reduced carbon in the atmosphere that follows from	������years of
one unit reduced fossil fuel production. Hence,
������is a measure of the ”carbon dept” as used buFargione et al. (2008)and others.
It is straightforward to solve this problem using optimal control theory, as done inAppendix. Two important conditions are
(ignoring the time references and considering an interior solution)
u
′(x+y)=	c′(x)+	������	(5)
4All three cost functions above should be interpreted as foregone consumption of a numeraire good due to resources, including land, used to produce energy
instead of this numeraire good.
5If e.g.c	′(0)<b′(0), it might be optimal to have an initial period of zero bioenergy production before a phase with simultaneous use of fossil energy and
bioenergy.

7 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
u′(x+y)=	b′(y)+	������	(6)
with straightforward interpretations: For both types of energy, the marginal beneﬁt (which may be interpreted as the consumer
price of energy) should equal marginal production costs plus terms representing the marginal climate costs of the two energy
types.
We show inAppendixthat
������(t)=	∫
∞
0	e−r������D′(S(t+������))d������	(7)
This well-known formula for the climate cost of fossil energy, often called the social cost of carbon, has a straightforward inter-
pretation: The climate cost of fossil energy at timetis equal to the discounted value of the marginal climate cost caused at all
future dates from one unit of fossil use at timet.
FromAppendixwe also ﬁnd
������(t)=	������D′(S(t))	(8)
The climate cost
������of bioenergy follows directly from assumption(4): Using (and producing) one unit more of bioenergy releases
������units of carbon into the atmosphere, giving a climate cost equal to	������D′(S). Clearly this climate cost is always positive (or at
least non-negative, but zero for anytwhenD	′(S(t)=	0).
3.2. Policy
Climate costs are externalities that are not taken into consideration in an unregulated economy. Using Pigovian taxes that
are equal to the climate costs, the ﬁrst-best optimum will be achieved in a market economy within the context of my model.
In reality, it will be considerably more diﬃcult to achieve the ﬁrst-best optimum. One reason for this is that the parameter
������
in reality will differ considerably between different types of bioenergy.	6This suggests heterogeneous tax rates across different
types of bioenergy. However, in practise it likely to be diﬃcult to obtain accurate information about the parameter
������for all types
of bioenergy.
An alternative, and in principle better, instrument to regulate the production of bioenergy follows from standard principles:
The social optimum may be achieved by setting a Pigovian tax onall net carbon emissions to the atmosphere. This tax should be
equal to the climate cost given by(7), and should be applied both to the emissions from fossil energy use and tonetemissions
from using bioenergy (gross emissions minus growth of the forests or bioenergy crops; i.e. a tax on gross emissions and a subsidy
to growth). With such a tax scheme the ﬁrst-best outcome would in principle be achieved (see e.g.Tahvonen, 1995,andHoel
and Sletten, 2016, for a further discussion). However, in practice the government lacks detailed and veriﬁable information about
the net carbon ﬂows from bioenergy at the micro level (i.e. levelof the individual farmer or forest owner), and it will therefore
in practice not be possible to reach the ﬁrst-best solution using only a tax on net carbon emission. Even without detailed and
veriﬁable information about volumes and growth at themicrolevel, the regulator may have reasonably good data of volumes
and growth at the aggregate level, and hence be able to calculate an average
������, and use this to calculate a tax on bioenergy.
As mentioned above, although this gives a ﬁrst-best outcome in my simple model, this policy will in reality give a sub-optimal
outcome.
In the rest of the paper I disregard the heterogeneity of
������across different types of bioenergy, and hence assume that the ﬁrst
best may be achieved by appropriate energy taxes. In Section41consider the possibility of constraints on the taxes so that the
ﬁrst best nevertheless may not be achieved, and calculate second-best policies.
InAppendixit is shown that the optimal value ofS
(t)is non-declining, implying that the value ofD	′is non-declining. From
(7)this implies that
������(t)≥D′(S(t))
r
It follows that the tax
������=������D′on bioenergy is always lower than the tax on fossil energy providedr	������<	1. This seems a reason-
able assumption: If e.g.
������=10 (see section2.2.1) and the interest rate isr	=0.02, we haver	������=0.2. With these numbers, the
tax on bioenergy should therefore be much lower than the tax on fossil energy.
Without more assumptions about the functionD
(S)and the parameter	������it is not possible to say much about the relative
sizes or relative growth rates of the two energy taxes (
������and	������), at least outside of steady state.
6������could even be negative for some types of bioenergy - e.g. growing bioenergy crops or trees on formery desert areas.

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 8
3.3. Steady state
AssumingD
′
′>0, there are two possible types of steady states. In theﬁrst there is positive production of bioenergy, i.e.
y	∗>0. This steady state follows directly from the equations in the previous subsection. For	̇
������=̇Sx=0wehave
x
∗=0(9)
u
′(x∗)=	c′(0)+	������∗	(10)
u
′(y∗)=	b′(y∗)+	������D′(S∗x+������y∗)	(11)
������∗=1
rD	′(S∗x+������y∗)	(12)
In the long run, the optimal fossil tax is constant and exactly so high that fossil energy production is zero. This level of the
fossil tax is achieved by cumulative fossil energy production giving so much carbon in the atmosphere that discounted future
damages are equal to this tax rate. Bioenergy will be produced at the level that is privately optimal given a tax equal to
������D′.The
ratio between this tax and the carbon tax is hence
������r.
Sincec	′(0)<b′(y∗)fory	∗>0, these equations can only be valid if	������D′<������	∗,i.e.ifr	������<	1. As mentioned above, this seems a
reasonable assumption, and if this inequality holds we hence have a steady state with positive bioenergy production.
InAppendixI show that if the initial value ofS
xis suﬃciently small, this steady state is reached asymptotically with mono-
tonically rising values of
������(t),Sx(t),andS	(t).
Ifr
������>	1 the production of both fossil energy and bioenergy will be zero in the steady state. Equations(9)-(12)remain valid,
except thaty∗=0, and	=is replaced by	<in(11).Noticethatevenify	∗=0, it may be optimal to havey	(t)>0forsomet.To
see this, assumey
(t)=	0forallt. With positive production of fossil energy	������(t)will be positive and rising forD	′
′>0(see(7)).
However, it may well be the case thatD	′is small or even zero for low values ofS	x, implying that we may have	������(t)<������	(t)for
someteven ifr
������>	1. But if this is the case it follows immediately from(5) and (6)thatx	(t)>0andy	(t)=	0 cannot be optimal.
This contradiction proves that even ifr
������>	1, it may be optimal to have positive production of bioenergy for somet,although
this production must go to zero in the long run.
In the remainder of this paper I restrict myself to the case ofr
������<	1. I now give an analysis both of the steady state and the
dynamics for two limiting cases.
3.4. Linear climate costs
If the climate cost function is linear,D
′is independent ofS. It follows from(7)that the fossil tax in this case is constant and
given by
������=D′∕r, and the optimal tax on bioenergy is also constant, equal to	������D′. The ratio between the tax on bioenergy and
fossil energy is
������r, as in the steady-state situation for the general climate cost function.
With a linear climate cost function, and hence constant energy taxes, production of fossil energy and bioenergy are constant
over time. Provided fossil energy production is positive, this means that the stock of carbon in the atmosphere is continuously
increasing. However, if there is a resource constraint on the production of fossil energy the equilibrium would be different. In
addition to production costs and the fossil tax there would now be a term reﬂecting the resource scarcity, rising at the rate of
interest in the standard Hotelling manner. With this modiﬁcation of the model, fossil fuel production would gradually decline
and eventually reach zero. From(6)it is clear that with a constant tax
������on bioenergy, the production of bioenergy would
increase monotonically over time as a response to the decline in fossil energy production. Once the fossil energy resource is
depleted, the production of bioenergy will be constant.
3.5. A binding threshold on carbon in the atmosphere
Linear climate costs do not seem particularly realistic. The opposite limiting case seems to be more in line with the climate
goals that were agreed upon in the Paris agreement. These goals might be interpreted as what we might call an ”inverse L
″cost
function, with negligible costs for carbon concentrations (or temperature) below some threshold, and ”inﬁnitely” high costs
above the threshold. Hence, we therefor consider this case below. Assume now thatD
′=0forS	<SandD	′=”∞”forS	>S.
(Our results would not be changed much if we instead assumed a small and constant marginal cost for values ofSbelow the
threshold
S.). The interpretation ofD	′for this case is that it is an endogenous shadow price associated with the constraintS	≤S.
ProvidedS
x(0)is small enough, it follows from the monotonicity result above that the solution will have two phases: First
(beforet
1inFig. 4)S	(t)<Sand rising, and then a phase withS	(t)=	S.7
7t1must be ﬁnite: If not, we would haveS	(t)<Sfor allt, implying that the shadow priceD	′=0forallt. But then we would be in the situation described in
Section3.4,i.e.	̇S(t)positive and constant, hence violatingS	(t)<Sfor allt.

9 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
x
time
y
xy
t	t
y
xy
Fig. 4.Time paths of energy use.
Consider ﬁrst the phase whereS	(t)<S. In this phase the carbon tax	������(t)rises at the rate of interest, while the bioenergy tax
������=������D′is zero. This gives (from(5) and (6)) a decliningx	(t),arisingy	(t), and a decliningx	(t)+	y(t), as illustrated inFig. 4.
Sincexis positive andyis increasing during the phase beforet
1,S(t)must be rising, eventually reaching its upper limit	S.
Thetimepointt
1is the time point at whichS	(t)reaches	S. From this time point onwards the constraintS	≤Sis strictly binding,
making the shadow priceD	′positive. This increase inD	′and hence	������makesy	(t)decline (see(6)). As long asx	(t)is positive,y	(t)
must be declining, sinceS	x(t)+	������y(t)is constant during this phase. This phase lasts tillx	(t)has declined all the way to zero, i.e.
till untilt
2inFig. 4.Aftert	2we have the steady-state outcome	(x,y)=(	0,y∗).8
The reason we get ”the rise and fall of bioenergy” as illustrated inFig. 4is the following. As long as carbon in the atmosphere
is below the threshold, there is no negative climate impact of using bioenergy, and since its marginal utility increases as fossil
energy production declines, the production of bioenergy increases. However, once the threshold
Sis reached, the production
of bioenergy must be reduced to ”make room for” the carbon released from the fossil energy use. In practise, this comes about
by reconverting land used for bioenergy production to forest land, and from less intensive harvesting of standing forests, so the
volume, and hence stored carbon, of such forests increases.
3.6. Generalizing the rise and fall result
If the ”rise and fall” result derived above was a knife-edge result that was valid only for a strict inverse L climate cost function,
it would perhaps not be very interesting. However, as I now will show, the result is much more general.
Assuming an interior solution, it is straightforward to derive from(5), (6)and(8)that
̇y=	1
N
[(u″−c″)������D″̇S−u″̇
������]	(13)
where
N
=u″(−b″)+	u″(−c″)+	c″b″>0(14)
Using(2)-(4)this implies
[N−(	u″−c″)������2D″]̇y=(	u″−c″)������D″x−u″̇
������
The term in front of	̇yis positive. We hence have the following result:
̇y⋛0 depending onD	″⋚	u″
(u″−c″)������̇
������x(15)
Both fractions in(15)are positive, and in the second fraction both the numerator and denominator approach zero as the steady
state is approached.
Consider a climate cost functionD
(S)that has the property thatD	′is moderately rising inSfor small values ofS,butthatD	′
begins to rise much more rapidly inSasSbecomes larger. In other words, we haveD	′
′”small” for low values ofSbut ”large” for
high values ofS.From(15)it follows that we in this case will get the ”rise and fall” property of the previous section: Optimal
production of bioenergy will initially be rising, but as the carbon stock in the atmosphere becomes suﬃciently large, it will be
optimal to reduce bioenergy production.
8In reality this stady-state phase is reached only asymptotically; i.e.t	2=”∞”. Small modiﬁcations of the model, e.g. an arbitrary small ﬁxed cost of fossil fuel
extraction, would maket
2ﬁnite.

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 10
4. Second-best policy
Assume that the energy taxes are
������x(t)and	������y(t), so that an interior market equilibrium is given by
u
′(x(t)+	y(t)) =	c′(x(t)) +	������x(t)	(16)
u
′(x(t)+	y(t)) =	b′(y(t)) +	������y(t)	(17)
As discussed previously, this equilibrium will coincide with the social optimum if the taxes
������x(t)and	������y(t)are equal to the	������(t)
and	������(t)given by(7) and (8). Assume now that, for whatever reason,	9������x(t)is exogenously set at a level satisfying
������x(t)<������	(t)=	∫
∞
0	e−r������D′(S(t+������))d������	(18)
In other words, there is an exogenous carbon tax that is lower than the social cost of carbon.
For a given time path ofy
(t),(16) and (17)determine the time path ofx	(t)(and of	������y(t)). We write this relationship as
x
=x(y,������x). Inserting this into our welfare function(1)hence gives us (ignoring time references):
W
=∫
∞
0	e−r������[u(x(y,������x)+	y)−	c(x(y,������x)) −	b(y)−	D(S)]
As before,Wis maximized subject to the constraints(2)-(4).InAppendixI show that the time pathy	(t)that solves this opti-
mization problem is achieved by the following second-best tax on bioenergy:
������y(t)=	������D′(S)−	k[������(t)−	������x(t)]	(19)
where
k
=−	������x(y,������x)
������y(20)
From(16) and (17)it follows thatk
=	u″
u″−c″,implyingthatk	∈(	0,1).
To interpret(19),itisusefultorewriteitas
������y=������D′(S)−	k������+k������x. The ﬁrst term in this expression,	������D′(S),measuresthe
direct climate cost of producing and using bioenergy. The second term,
−k������, represents the reduced climate costs from fossil
energy caused by the increased use of bioenergy. The ﬁnal term,k
������x, represent a non-environmental indirect cost of increasing
the use of bioenergy: By increasing the use of bioenergy the use of fossil energy declines. This reduction gives a social cost (ignor-
ing the climate effect, since this is taken care of by the second term), since the marginal beneﬁt of using fossil energy exceeds the
marginal cost of supplying it. This difference is due to the carbon tax, since the carbon tax is identical to the difference between
the user and producer price.
The reason why the second-best tax on bioenergy is lower than the ﬁrst-best, is that the production and use of bioenergy
affects the production of fossil energy negatively. The same is true for non-carbon energy: With our assumptions an increase in
the production of non-carbon energy will reduce the production of fossil energy. Hence, non-carbon energy should be subsidized
in the second-best optimum. Formally, it is straightforward to extend the analysis inAppendixand show that the second-best
optimal tax
������zon non-carbon energy is
������z(t)=−	h[������(t)−	������x(t)]
where
h
=−	������x(y,z,������x)
������z	=−	U(x+y)z
c″−U(x+y)(x+y)∈(	0,1)
For	������x<������	it is hence optimal to subsidize non-carbon energy (	������z<0). The interpretation is the same as that given above for
������y.
4.1. Steady state
Without any assumptions about the exogenous carbon tax
������x(t), we cannot know whether a steady state exists. Hence, we
assume that
������x(t)in the long run is equal to (or approaches asymptotically) a constant value	������∗x. If this is the case andD	′
′>0
there is a steady state given by
x
∗=0(21)
u
′(x∗)=	c′(0)+	������∗x	(22)
9I brieﬂy discuss some possible reasons in Section4.3.

11 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
u′(y∗)=	b′(y∗)+	������∗y	(23)
������∗=1
rD	′(S∗x+������y∗)	(24)
������∗y=r������������	∗−k[������∗−������∗x]	(25)
Sincec
′(0)<b′(y∗)fory	∗>0, these equations can only be valid if	������∗y<������	∗x. From the equations above it is clear that this inequal-
ity is identical tor
������<	k+(	1−k)������∗x∕������∗.For	������∗x<������	∗the r.h.s. of this inequality is smaller than 1. Hence the condition fory	∗>0
is stricter in the second-best case than in the ﬁrst-best optimum, and is stricter the larger is the relative fossil tax distortion (i.e.
the smaller is
������∗x∕������∗). In the remainder of this section1assume that the inequalityr	������<	k+(	1−k)������∗x∕������∗holds. In other words,
I only consider cases where bioenergy production is positive in a steady state.
From the equations above we see that the sign of
������∗yis ambiguous. It is positive (i.e. we have a tax on bioenergy in the
long run) ifr
������>	k(1−������∗x∕������∗), and negative (i.e. a subsidy to bioenergy in the long run) ifr	������<	k(1−������∗x∕������∗).Inotherwords,
with a small carbon tax distortion (
������∗x∕������∗close to 1) bioenergy should be taxed in the long run (as in the ﬁrst-best case), while
bioenergy should be subsidized in the long run if the tax distortion is suﬃciently high (
������∗x∕������∗close to 0).
I assumed above that
������x(t)in the long run was equal to (or approaches asymptotically) a constant value	������∗x.Withoutmore
assumptions on the time path of
������x(t)it is of course not possible to say anything about the dynamics toward the steady state.
In the next section1shall nevertheless brieﬂy consider the dynamics of the second-best equilibrium for the case of an inverse L
damage function and a particularassumption about the time path of
������x(t).
4.2. A binding threshold on carbon in the atmosphere and a constant parameter of tax distortion
Assume that the exogenous carbon tax develops in a manner that makes the variable
������x(t)������(t),whichisameasureofthetax
distortion, constant over time, denoted
������. As in the ﬁrst-best optimum, the equilibrium will have two phases. FirstS	(t)<S,and
then a phase withS
(t)=	S.
Consider ﬁrst the phase whereS
(t)<S. In this phase the fossil tax	������������	(t)rises at the rate of interest. SinceD	′is zero in this
phase, it follows from(19)that
������y=−	k(1−������)������. Hence, bioenergy is subsidized, and the subsidy is increasing over time. This
gives a decliningx
(t)and a risingy	(t), as in the ﬁrst-best case.
WhenS
(t)reaches its upper limit	S,y	(t)is declining forx	(t)>0, sinceS	x(t)+	������y(t)is constant during this phase. This is
achieved through a gradual reduction in the subsidy to bioenergy (asD	′increases), which might even eventually turn into a tax.
4.3. Why second best?
We assumed above that the fossil tax was exogenous and belowthelevelgivingthesocialoptimum.Thistypeofassumption
is quite often made in various studies of second-best climate policies. One reason for such a constraint on the fossil tax could be
distributional concerns. Even if the government intends to fully recycle the revenues from a carbon tax, each voter may focus
on the visible tax increase and not trust that the revenue from the carbon tax will be recycled in a way compensating him or
her. Moreover, some persons will be hurt more by the carbon tax than others; this will typically be those who consume more
than the average share of fossil fuels due to their current preferences or earlier investments (e.g. a large house with a long
commuting distance). On the production side, some industries will bear a disproportionately high share of the total costs from
the carbon tax. Consumers with a high use of fossil fuels as well as workers and owners in such high emission sectors will often
be successful in lobbying against a carbon tax.
In contrast, the costs of subsidizing renewable energy are likely to be less visible to the typical voter and also be more evenly
shared by everyone in the economy. Sectors in the economy producing renewable energy or inputs to this production will gain
from a subsidy to renewable energy, and might thus engage in lobbying for the use of such subsidies instead of a carbon tax.
These arguments suggest that it might be easier to obtain political support for a subsidy to renewable energy than for a carbon
tax.
Since fossil energy production in the long run is zero, there is no revenue from the fossil energy tax in the long run. The
only purpose of the tax in the long run is to make it prohibitively expensive to produce fossil energy, and hence avoid any fossil
energy production. In such a situation, it can obviously not be the fossil tax itself, and the distributional issues related to the
revenue from it, that raises political concerns. There may however be distributional concerns related to the user cost of energy
(be it fossil or bio), and political objections to high energy prices. From(23)it immediately follows that the long-run energy
priceu
′(y∗)is higher the higher is the fossil tax	������∗x. Hence, there are political arguments for a constraint on the fossil tax even if
there are no tax revenues collected and redistributed.

M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314 12
5. Concluding remarks
In recent years it has become clear that the production of bioenergy in most cases will have a negative direct climate impact.
This suggests that contrary to what is the case in many countries, bioenergy ought to be taxed, and certainly not subsidized.
Ideally, all ﬂows of carbon to the atmosphere should be taxed at the same rate, and this same rate should be used to subsidize
ﬂows of carbon from the atmosphere. For fossil energy this simply implies a uniform carbon tax, as in this case there is no
ﬂow of carbon from the atmosphere to the biosphere. However, for bioenergy there are generally ﬂows of carbon between
the biosphere and the atmosphere going in both direction. The ideal tax/subsidy scheme for bioenergy is likely to be diﬃcult or
impossible to achieve in practise. In the context of our simple model, the ﬁrst-best social optimum may nevertheless be achieved
by appropriate taxes on both types of energy. Due to the differences in dynamics of the climate impact of bioenergy and fossil
energy, the optimal taxes of the two types of energy may have very different time paths.
If policy makers for some reason are not willing to tax carbon energy as high as the climate considerations require, it may be
optimal to subsidize bioenergy. However, even if such a subsidy is (second-best) optimal in the short run, it may be optimal to
eventually reduce the subsidy and perhaps tax bioenergy in the long run.
The difference in the dynamics of the climate impacts of bioenergy and fossil energy have an important implication for the
dynamics of the production of the two types of energy. Both in the ﬁrst-best and second-best case it may be optimal to have an
initially rising, but later declining, production of bioenergy.
The present study has completely ignored the possibility ofCCS (Carbon Capture and Storage). An interesting extension
would be to include the possibility of CCS, both in a ﬁrst-best and second-best setting.
Appendix. Derivation of the optimal solution
The ﬁrst-best optimum
The current value Hamiltonian corresponding to the maximization ofWsubject to(2)-(4)and an exogenous value ofS
x(0)is
(ignoring the time references)
H
=u(x+y)−	c(x)−	b(y)−	D(Sx+������y)+(−	������)x(26)
The necessary (and suﬃcient, sinceHis concave inx
,y,Sx) conditions for the optimum are
u
′(x+y)−	c′(x)−	������≤0[=	0forx	>0]	(27)
u
′(x+y)−	b′(y)−	������D′(S)≤0[=	0fory	>0]	(28)
̇
������=r������−D′(S)	(29)
Lim
t→∞e−rt������(t)=	0(30)
For an interior solution it follows from(27) and (28)and the properties of the functions that
x
=x(������−,Sx+)	(31)
y
=y(������+,Sx−)	(32)
It follows that
S
=Sx+������y(������,Sx)=	S(������+,Sx+)	(33)
To see thatSis increasing inS
x, assume the opposite. Then it would follow from(27) and (28)that an increase inS	xfor given	������
would makeyhigher or unchanged, and hence increaseS. This contradiction proves the last sign in(33).
The phase diagram inFig. 5follows from(3)and(29).From(3)and(31)it is clear that the line forx
(������,Sx)=	0isupward
sloping, and that	̇Sxis zero above this line and positive below.
When	̇
������=0 it follows from(29)that
������=1
rD	′(Sx+������y(������,Sx))
Whenx	=0 it follows from(28)thatyis independent of	������, and sinceSandD	′is higher the higher isS	x,thecurvefor	̇
������=0must
be upward sloping above the line forx
(������,Sx)=	0. The slope below this line is not obvious, but is of no importance. In any case
we must have	̇
������=r������−D′(S(������,Sx))>0 to the left of the curve giving	̇
������=0.
If the initial value ofS
xis higher than the steady-state valueS	∗x, the optimal solution is trivial: We should havex	=0forallt,
andyconstant and determined by(28)for allt.
If the initial value ofS
xis lower than the steady-state valueS	∗x,asinFig. 5, the dynamics are as illustrated:S	x(t)and	������(t),and
hence alsoS
(t), rise monotonically and asymptoticallytowards their steady-state values.

13 M. Hoel / Journal of Environmental Economics and Management 101 (2020) 102314
Fig. 5.Phase diagram forS	xand	������.
The second-best optimum
Choosingy
(t)to maximize our objective function(26)gives the following Hamiltonian (ignoring the time references):
H
=u(x(y,������x)+	y)−	c(x(y,������x)) −	b(y)−	D(Sx+������y)+(−	������)x(y,������x)	(34)
The conditions for an interior optimum are
u
′(x+y)=	b′(y)+	������D′(S)+
[
(������+c′−u′)������x(y,������x)
������y
]
(35)
̇
������=r������−D′(S)	(36)
Lim
t→∞e−rt������(t)=	0(37)
The costate variable
������(t)may as in the ﬁrst-best case be interpreted as the social cost of carbon, and is as before given by(7).
Using the deﬁnition(20)ofk, it follows from(16)that(35)may be rewritten as
u
′(x+y)=	b′(y)+	������D′(S)−	k[������(t)−	������x(t)]
Using(17), it immediately follows that the second-best optimal tax on bioenergy is given by(19).
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