Unilateral Deposit Policy and the Green Paradox

Abstract

This paper deals with possible foreign reactions to unilateral deposit policies against global warming. It differentiates between demand side and supply side reactions as well as between intra- and intertemporal shifts of greenhouse gas emissions. Using Ritter & Schopf (2014) two-period general-equilibrium model, we change the policy instrument from an emissions trading scheme to a deposit preserving system. In this system, the deposits that are most costly but still worthwhile to extract are preserved in each period. We find that purchasing additional deposits in the second period reduces early and total emissions. By contrast, leasing additional deposits in the first period leads to higher future emissions and can lead to higher total emissions, which can induce the strong green paradox. However, the weak green paradox does not occur. Finally, we analyze under which conditions the one or the other policy is more effective and discuss which policy could be more cost effective in reducing climate damages.

Full text

Unilateral Deposit Policy and the Green Paradox
Mark Schopf
October 22, 2019
Abstract This paper deals with possible foreign reactions to unilateral deposit
policies against global warming. It differentiates between demand side and supply
side reactions as well as between intra- and intertemporal shifts of greenhouse gas
emissions. Using Ritter & Schopf’s (2014) two-period general-equilibrium model, we
change the policy instrument from an emissions trading scheme to a deposit preserv-
ing system. In this system, the deposits that are most costly but still worthwhile to
extract are preserved in each period. We find that purchasing additional deposits in
the second period reduces early and total emissions. By contrast, leasing additional
deposits in the first period can lead to higher total emissions, which can induce the
strong green paradox. However, the weak green paradox does not occur. Finally,
we analyze under which conditions the one or the other policy is more effective and
discuss which policy could be more cost effective in reducing climate damages.
Keywords Carbon Leakage ·Deposit Policy ·General Equilibrium ·Green Paradox
·Nonrenewable Resources ·Supply Side Climate Policy
JEL Classification D50 ·Q30 ·Q31 ·Q38 ·Q54 ·Q58
I would like to thank Thomas Eichner, Daniel Nachtigall, R¨udiger Pethig, Achim Voss and participants
of the AUR ¨
O Workshop 2017 in Basel and the IIPF Annual Congress 2019 in Glasgow for helpful comments.
Financial support from the German Research Foundation (DFG grant numbers EI 847/1-2 and PE 212/9-2)
is gratefully acknowledged.
University of Hagen, Department of Economics, Universit¨atsstr. 41, 58097 Hagen, Germany, email:
mark.schopf@fernuni-hagen.de.
1

1 Introduction
The President of the Republic of Kiribati called for a global moratorium on new coal mines
to reduce greenhouse gas emissions at the United Nations Conference on Climate Change in
Paris (Tong 2015). This call is scientifically supported (Grantham Research Institute 2015,
Australia Institute 2015), but politically only addressed individually and temporarily by the
People’s Republic of China (State Council 2016). In this paper, we investigate whether such
unilateral deposit policies against global warming can have unintended consequences. We
contribute to the literature by applying a dynamic general-equilibrium model and analyzing
marginal changes in the prevailing unilateral deposit policy.
It is well known that unilateral demand side policies against global warming can cause
intra- and intertemporal shifts of greenhouse gas emissions. An intratemporal shift is known
as carbon leakage.1Sinn (2008) refers to an intertemporal shift that steepens the carbon
extraction path as green paradox. This green paradox occurs if, e.g., the probability of
developing a cheaper backstop is increased (Strand 2007) or if a carbon tax increases in real
terms over time and the carbon price is not bounded from above (Sinn 2008).2
One possibility to avoid these phenomena could be to apply supply side policies. Sinn
(2008) suggests to tax capital income to lower the real interest rate and to flatten the carbon
extraction path. Ritter et al. (2019) find that a unilateral capital income tax leads to less
domestic capital demand and thus to a lower interest rate, so that foreign capital demand
increases (intratemporal carbon leakage), but extraction shifts into the future (reversed
green paradox).3Flattening the carbon extraction path is good but not good enough.
Allen et al. (2009) state that global warming depends first and foremost on total emissions
1For an overview on the channels of carbon leakage, see van der Werf & Di Maria (2012, Section 5.1).
2If the carbon price is bounded from above, Hoel (2012) demonstrates that a carbon tax must increase
more than in real terms over time for this green paradox to occur. With learning-by-doing in the renewable
energy sector, Nachtigall & R¨ubbelke (2016) find that any future carbon tax reduces present extraction as
long as the slope of the marginal extraction cost curve is sufficiently flat.
3If trade-related income effects are sufficiently weak, Eichner & Pethig (2015b) find that a present uni-
lateral consumption-based carbon tax shifts domestic commodity demand into the future, so that the future
commodity price increases and foreign commodity supply shifts into the future (negative intratemporal car-
bon leakage). Since the interest rate is normalized to zero and the present commodity price is normalized
to one, the increase in the future commodity price could be interpreted as a decrease in the real interest
rate. If trade-related income effects are sufficiently strong, these results can be reversed (green paradox).
2

and not on their temporal distribution. However, global warming occurs the earlier the
faster these emissions occur. Along these lines, Gerlagh (2011) refers to an increase in
early emissions as weak green paradox and to an increase in the cumulative and discounted
climate damages as strong green paradox.
The literature on demand side policies finds that these policies can lead to an increase
in total emissions. In a dynamic general-equilibrium model with stock-dependent marginal
extraction costs, Ritter & Schopf (2014) find that a tighter present unilateral carbon cap can
lead to an increase in early and total emissions if the intertemporal elasticity of substitution
in consumption is low and if future carbon demand and supply are relatively inelastic. With
a tighter future unilateral carbon cap, the weak and the strong green paradox can occur,
but total foreign emissions can also decrease (negative cumulative carbon leakage). In a
dynamic general-equilibrium model without extraction costs, van der Meijden et al. (2015)
find that a higher future carbon tax can lead to an increase in the present carbon price if
capital-related income effects are sufficiently strong. In this case, early emissions decrease
but with exploration costs, total emissions increase because a higher present carbon price
leads to more exploration investment.4
Again, supply side policies could be a way to avoid an increase in total emissions. However,
although van der Ploeg (2016) confirms the above result that a capital income tax reduces
present extraction, he also finds that cumulative extraction increases.5Maybe policies
directly aimed at reducing carbon supply are more effective. Examples in the relevant
literature, which we discuss below, are carbon supply taxes and deposit policies. The
former reduce domestic supply, while the latter reduce foreign supply, either temporarily by
leasing foreign deposits, or permanently by purchasing them. However, there are no dynamic
general-equilibrium models concerning these supply side policies. Since demand side policies
can lead to an increase in total emissions in precisely these models, it is necessary to apply
them to investigate the effectiveness of, e.g., deposit policies.
In this paper, we use Ritter & Schopf’s (2014) dynamic general-equilibrium model and
4In a partial-equilibrium Hotelling model with two types of fossil fuels having different constant marginal
extraction costs and different emissions factors, Fischer & Salant (2017) cannot rule out an increase in total
emissions due to a higher unilateral carbon tax.
5With different types of fossil fuels, he also finds that a future carbon tax can lead to an increase in
total emissions if coal is a gross substitute for oil. By contrast, Michielsen (2014) then finds that a future
renewable energy subsidy reduces total emissions.
3

change the policy instrument from an emissions trading scheme to a deposit preserving
system. In this system, the deposits that are most costly but still worthwhile to extract are
preserved in each period. We find that purchasing additional deposits reduces emissions in
each period. By contrast, leasing additional deposits leads to higher future emissions and
can lead to higher total emissions, which can induce the strong green paradox. However,
early emissions decline, so that the weak green paradox does not occur. Finally, we find
that purchasing additional deposits is more effective than leasing additional deposits if the
intertemporal elasticity of substitution in consumption is low.
There are only few papers dealing with policies directly aimed at reducing carbon supply.
Bohm (1993) and Hoel (1994) suggest to combine demand and supply side policies against
global warming to avoid intratemporal carbon leakage. In a static partial-equilibrium
model, Hoel (1994) finds that it is unilaterally optimal to use both, carbon consumption
and production taxes (or subsidies). Hagem & Storrøsten (2019) confirm this result in a
dynamic partial-equilibrium model, and Eichner & Pethig (2015a) confirm it in a dynamic
general-equilibrium model without extraction costs. In Asheim’s (2013, Section 5) partial-
equilibrium Hotelling model without extraction costs, in which one country owns the entire
capital stock and the other country owns the entire resource stock, the first-best can be
implemented by unilateral carbon supply taxes or unilateral deposit policies.6
Closer to our model is Hoel (2014), who uses a partial-equilibrium Hotelling model with
stock-dependent marginal extraction costs and finds that purchasing and preserving any
deposits reduces present and cumulative extraction.7We follow Hoel (2014) in so far as
we do not derive the unilaterally optimal policy, but analyze changes in the prevailing
policy. However, we design the deposit preserving system in a way that it could generally
implement the first-best. If deposits can be traded for preservation and for extraction, and
there is Coasian bargaining on the deposit market, Harstad (2012) demonstrates that the
first-best is implemented despite possible strategic action. By contrast, Eichner & Pethig
(2017a) (Eichner & Pethig 2017b) find that the first-best need not be (is not) implemented
if deposits can be traded for preservation only due to strategic action on the fuel market
6The first-best cannot be implemented by unilateral demand side polices because the capital stock is
perfectly mobile and relocates to the non-abating country.
7However, with emissions from the extraction process, he finds that early emissions can increase by
purchasing and preserving the deposits that are least costly to extract (and have the lowest emissions
factors).
4

(deposit market). However, without strategic action, the first-best is also implemented in
Eichner & Pethig (2017a,b) by purchasing and preserving some of the deposits that are
most costly to extract.
Eichner et al. (2019) demonstrate that this result does not hold if the cumulative and
discounted climate damages depend on the amount and on the timing of emissions. Then,
some of the deposits that are most costly to extract in each period must be preserved in that
period to implement the first-best without strategic action. Thus, we take some (potentially
none) of the deposits that are most costly but still worthwhile to extract in the first period
to be leased and in the second period to be purchased as starting point of our analysis.
The leased deposits will then be given back and extracted in the second period. The policy
instrument of the abating country is either to lease or to purchase additional deposits that
are marginally less costly to extract than those that were leased or purchased in the first
place, respectively. Thereby, we follow Eichner & Pethig (2017b) in so far as we assume a
uniform deposit price in each period.
The outline of the paper is as follows. Section 2 introduces the model. Section 3 and
Section 4 present the results of purchasing and leasing additional deposits, respectively.
Section 5 analyzes which policy is more effective and discusses which policy could be more
cost effective. Finally, Section 6 concludes.
2 The Model
The model and its notation follow Ritter & Schopf (2014). One country (i=F) exports
fossil fuel, and imports and consumes a produced commodity, which is also used as the only
input in the fossil fuel extraction process. The other two countries (i=A, N) import fossil
fuel, which is used as the only input in the commodity production process, and export and
consume the commodity. The two periods represent the time up to the medium term (t= 1)
and the time up to the very long term (t= 2). The abating country (i=A) constrains fossil
fuel demand via an emissions trading scheme and fossil fuel supply via a deposit preserving
system, while the non-abating country (i=N) does not constrain fossil fuel consumption
at all. We follow Harstad (2012, pp. 85), Eichner & Pethig (2017a, p. 79) and Eichner
& Pethig (2017b, p. 52) in so far as we assume a continuum of deposits that are ordered
according to their marginal extraction costs. In the initial equilibrium, all deposits in the
interval [eeF t, eF t ] are already preserved in each period, where eeFt denotes extraction and eF t
5

pe1
pe1
eF1
pe2
pe2
dF2
p' dpe2
eF2
0102
F1F2
e2
p'
e1
eF2
'F2
e'
A1
B1
eF1
'
eF2
B2C2
F1
E2
D2
δ eF1
=eF2
==eF1
e'
A2
Figure 1: Purchasing additional deposits in a simple linear partial-equilibrium model.
denotes fossil fuel supply, i.e. the sum of extraction and preserved deposits. Thereby, the
respective deposits of the first period are leased and, thus, temporarily preserved, whereas
the respective deposits of the second period are purchased and, thus, permanently preserved.
We then analyze how a marginal increase in the amount of leased or purchased deposits,
eF t :=eFt −eeF t ≥0, affects extraction and fossil fuel supply.8Before we introduce the
model in detail, we clarify the design of the deposit preserving system and the functioning
of one policy instrument graphically.
Figure 1 illustrates an increase in the amount of purchased deposits in a simple linear
partial-equilibrium model.9In the initial equilibrium, the price, extraction and fossil fuel
supply are pet,eeF t and eF t in each period, respectively. By purchasing additional deposits
(B2C2), second-period extraction would decline by deF2if second-period fossil fuel supply
did not change. However, this potential decrease leads to an upward pressure on the second-
period price, dpe2, and thus to an increase in fossil fuel supply (A2→C2). Thereby, the
user cost of supply increase, so that the first-period supply curve shifts upwards, whereby
extraction declines in period 1 (A1→B1). This, in turn, leads to lower second-period
8This marginal analysis allows us to define an additional deposit by its marginal extraction cost.
9In Figure 1, there are no trades in deposits in the initial equilibrium. The first-period fossil fuel supply
cost is 1
2e2
F1and the (discounted) second-period fossil fuel supply cost is δ[1
2(eeF1+eF2)2−1
2ee2
F1], so that
pe1=eF1+δeF2and δpe2=δ(eeF1+eF2).
6

marginal extraction cost, so that the second-period supply curve shifts downwards, which
raises extraction and fossil fuel supply in period 2 (B2→D2and C2→E2). In the new
equilibrium, the price, extraction and fossil fuel supply become p0
et,ee0
F t and e0
F t in each
period, respectively. The price increases and extraction declines in each period, fossil fuel
supply declines in period 1 and increases in period 2, and cumulative fossil fuel supply in-
creases, so that there is positive cumulative carbon leakage. However, cumulative extraction
declines, so that there is no green paradox in the simple linear partial-equilibrium model.
In the remainder of the paper, we analyze whether these qualitative results remain valid in
a general-equilibrium model with a more general extraction cost function.
In what follows, we introduce the model in detail. Thereby, we start with the properties of
the fossil fuel supply costs and the optimization problem of the fossil fuel extractor, continue
with the optimization problem of the commodity producers and that of the households, and
close with the properties of the climate damage function.
The analysis is limited to cases in which cumulative fossil fuel supply is strictly less than
the physical stock. In each period, the material fossil fuel supply cost (XEt) depends on
that period’s fossil fuel supply (eF t ).10 In period 2, it additionally depends on first-period
extraction (eeF1).11 In period 1, the marginal physical supply cost (XE1
eF1) is assumed to be
positive and increasing in first-period fossil fuel supply. In period 2, the marginal physical
supply cost (XE2
eF2) and the physical user cost of supply (XE2
eF1) are assumed to be positive
and increasing in both, first-period and second-period fossil fuel supply.12 We assume the
cumulative fossil fuel supply cost, the material fossil fuel supply costs weighted by the
commodity prices (pxt), to be the higher the less balanced the fossil fuel supply path is.13
10We use the terms “material” and “physical” cost for the amount of commodity input needed for the
respective fossil fuel supply. The actual fossil fuel supply cost are equal to these “material” or “physical”
cost times the respective commodity price.
11The leased deposits will be given back and extracted in period 2, so that second-period fossil fuel
supply does not depend on first-period fossil fuel supply, but on first-period extraction.
12The physical user cost of supply estimate how much more material is needed to supply one more unit
of fossil fuel in period 2 if one more unit of fossil fuel is extracted in period 1.
13 Note that condition (3) is sufficient for our results to hold. For the necessary condition, see footnote
21. If the cumulative material supply cost depends on a weighted sum of first-period extraction and second-
period fossil fuel supply, i.e. XE1=X(eF1) and XE2=X(eeF1+ψeF2)−X(eeF1) with X000 ≥0, condition
(3) becomes px1
px2X00(eF1)−X00 (eeF1)≥0. This is fulfilled in a partial-equilibrium model, where px1
px2=1
δ≥1.
Furthermore, it is fulfilled in the numerical example of Appendix A.6 for all parameter values considered.
7

Formally, this can be represented as follows:
XE1:=XE1(eF1) and XE2:=XE2(eeF1, eF2),(1)
XEt
eF t , XE2
eF1, XEt
eF teF t , X E2
eF1eF1, XE2
eF1eF2>0,(2)
px1
px2
XE1
eF1eF1+XE2
eF1eF1XE2
eF2eF2≥XE2
eF1eF22
.(3)
To determine the fossil fuel prices, we need to know how much would be extracted if
nothing were preserved. In this case, the fossil fuel extractor maximizes her intertemporal
profit (ΠF), consisting of output revenues (peteF t) and input costs (pxt XEt ), with respect to
fossil fuel supply. Then, the fossil fuel price is equal to the marginal fossil fuel supply cost
(pxtXE t
eF t ) in each period plus the user cost of supply (px2XE2
eF1) in period 1:
ΠF:=X
thpeteF t −pxtXEt i,(4)
pe1=px1XE1
eF1+px2XE2
eF1and pe2=px2XE2
eF2.(5)
However, some of the deposits that are worthwhile to extract could be preserved in each
period, so that the extractor’s commodity demand (xEt) does not depend on the material
fossil fuel supply cost (XEt ), but on the actual material extraction cost ( e
XEt ) in each period:
xE1=e
XE1:=XE1(eeF1) and xE2=e
XE2:=XE2(eeF1,eeF2).(6)
With a deposit preserving system, the intertemporal profit of the fossil fuel extractor is
equal to revenues from extraction (peteeF t) and from leasing and selling deposits (pzteF t )
minus costs of extraction (pxt e
XEt ). Taking eeFt =eF t −eF t into account and maximizing
with respect to deposit supply yields the deposit price, which is equal to the opportunity
cost of the preserved deposit that is most costly to extract in each period:
ΠF:=X
thpeteeF t +pzteF t −pxt e
XEt i,(7)
pz1=pe1−px1e
XE1
eF1−px2e
XE2
eF1and pz2=pe2−px2e
XE2
eF2.(8)
We close the fossil fuel supply side with the price elasticities of supply for fossil fuel:
ηF1,1:=px1XE1
eF1+px2XE2
eF1
px1eF1XE1
eF1eF1+px2eF1XE2
eF1eF1
and ηF2,1:=px1XE1
eF1+px2XE2
eF1
px2eF2XE2
eF1eF2
,
ηF1,2:=px2XE2
eF2
px2eF1XE2
eF1eF2
and ηF2,2:=px2XE2
eF2
px2eF2XE2
eF2eF2
,
(9)
8

where ηF s,t :=deFs
dpet ·pet
eF s >0 is for s6=tthe intertemporal and for s=tthe intratemporal
price elasticity of supply for fossil fuel in period s.14
In each country and period, the commodity production functions (XAt, XN t) determine
the commodity supply of the commodity producers (xs
At, xs
Nt) and depend on that period’s
fossil fuel demand (eAt, eN t ). They are assumed to be increasing and strictly concave.
The intertemporal profit functions of the commodity producers (ΠA,ΠN) consist of output
revenues (pxtXAt, pxtXNt), input costs (pet eAt, pet eN t), and, for the commodity producer in
the abating country, emission trading costs (πteAt). Both types of costs depend on fossil fuel
demand (eAt, eN t ). The input costs additionally depend on fossil fuel prices (pet ), whereas the
emission trading costs additionally depend on permit prices (πt). The commodity producers
maximize their intertemporal profits with respect to fossil fuel demand. Formally, this can
be represented as follows:
xs
At =XAt :=XAt(eAt ),(10)
xs
Nt =XNt :=XN t(eN t ),(11)
ΠA:=X
thpxtXAt −(pet +πt)eAt i,(12)
ΠN:=X
thpxtXN t −pet eN ti,(13)
px1XA1
eA1=pe1+π1and px2XA2
eA2=pe2+π2,(14)
px1XN1
eN1=pe1and px2XN2
eN2=pe2.(15)
Since the abating country constrains fossil fuel demand, only the price elasticities of
demand for fossil fuel of the commodity producer in the non-abating country are different
from zero:
ηN1:=px1XN1
eN1
px1eN1XN1
eN1eN1
<0 and ηN2:=px2XN2
eN2
px2eN2XN2
eN2eN2
<0.(16)
The intertemporal utility functions of the households (U) depend on first-period and
second-period commodity consumption (xi1, xi2) in each country. They are assumed to
be identical and their intertemporal elasticity of substitution (σ:= 1/(1 + b)) to be con-
stant. Cumulative consumption expenses (px1xi1+px2xi2) are covered by the maximized
intertemporal profits (Πi∗) in each country plus emission trading revenues (π1eA1+π2eA2)
14Note that condition (3) is equivalent to the product of the intertemporal price elasticities of supply for
fossil fuel (ηF1,2ηF2,1) being greater than or equal to that of the intratemporal price elasticities of supply
for fossil fuel (ηF1,1ηF2,2).
9

and deposit leasing and purchasing costs (pz1eF1+pz2eF2) in the abating country. The
relative commodity demand of the households (xi1/xi2) depends on the relative commodity
price (px1/px2) and is identical in each country. Formally, this can be represented as follows:
U(xi1, xi2) = α1x−b
i1+α2x−b
i2−h
b, i =A, N, F, (17)
X
t
pxtxAt = ΠA∗+X
thπteAt −pzteF tiand X
t
pxtxit = Πi∗, i =N, F, (18)
xi1
xi2
=α1px2
α2px1σ
, i =A, N, F, (19)
where α1, α2, h > 0.
In equilibrium, extraction is equal to fossil fuel demand of the commodity producers
(eAt +eNt) in each period, and commodity supply is equal to commodity demand of the
households (xAt +xNt +xF t) plus that of the fossil fuel extractor (xE t) in each period:
eeF t =eF t −eF t =eAt +eN t ,(20)
xs
At +xs
Nt =xAt +xNt +xF t +xE t.(21)
The abating country decides when to preserve and trade how many deposits and emissions,
respectively. Since the intertemporal utility functions of the households are identical and
homothetic, the deposit and permit prices do not change the demand and supply decisions
on the commodity and fossil fuel markets. Furthermore, the abating country increases
the preserved deposits, which changes commodity demand and fossil fuel supply of the
fossil fuel extractor, but does not reduce its traded emissions. Thus, neither the deposit
prices nor the entire emissions trading scheme are distorting.15 The emissions trading
scheme being not distorting implies that we could switch to a two-country model by setting
eAt =xs
At =xAt = 0 and letting the “non-abating” country purchase and lease deposits
without changing our qualitative results below. However, we stick to the three-country
model for two reasons. First, if just one country imported fossil fuel, this country could
effectively and cost-efficiently reduce fossil fuel demand via an emissions trading scheme.
Thus, it would not be necessary to reduce fossil fuel supply via a deposit preserving system.
Second, we want to compare under which conditions the green paradoxes arise with policies
15If there were no conventional deposit markets with uniform deposit prices, but there would be bilat-
eral trades in deposits, see Harstad (2012), Eichner & Pethig (2017a), Eichner et al. (2019), the income
distribution could change, but fossil fuel and commodity prices and quantities would be the same in each
period.
10

aimed at reducing carbon demand and supply. Thus, we remain as close as possible to
Ritter & Schopf’s (2014) model.
Finally, changes of first-period and total emissions are weighted with the following climate
damage function:
D(eeF1,eeFΣ) = c1eed
F1+c2eed
FΣι
d,(22)
dD(eeF1,eeFΣ)R0⇔deeF1+λdeeFΣR0,(23)
where c1, c2, ι > 0, eeFΣ:=eeF1+eeF2=eFΣ−eF1−eF2is cumulative extraction, eFΣ:=
eF1+eF2is cumulative fossil fuel supply, and λ:=c2
c1·eeFΣ
eeF1d−1>0 is the relative weight
attached to changes in total emissions. In what follows, we analyze whether the weak green
paradox, i.e. an increase in first-period emissions deeF1>0, or the strong green paradox,
i.e. an increase in the climate damages dD > 0, can occur due to marginal changes in the
prevailing unilateral deposit policy.
3 Purchasing Additional Deposits
Purchasing additional deposits means that deposits that would otherwise be extracted in
period 2 are either purchased in period 1 and announced to be preserved in both periods or
announced to be purchased in period 2 and preserved in that period.16 The main results of
this policy will be characterized in Proposition 1: Purchasing additional deposits (deF2>0)
causes negative intertemporal carbon leakage (deF1/deF2<0), so that the weak green
paradox does not occur (deeF1/deF2<0). Total emissions also decline (deeFΣ/deF2<0), so
that the strong green paradox does not occur (dD/ deF2<0). Negative cumulative carbon
leakage is possible (deFΣ/deF2<0).
The solution strategy for the comparative statics is as follows: We start with analyzing
the changes on the commodity market, proceed with observing the effects on the fossil fuel
market, and close by combining our results. On the former market, purchasing additional
deposits affects the second-period commodity price via changes in extraction:17
dpx2=px2
σΘ1deeF1−Θ2deeF2,(24)
16We assume the respective announcement to be binding.
17See Appendix A.2, equation (A.23). Throughout the rest of the paper the commodity in period 1 is
chosen as numeraire.
11

where Θ1:=XN1
eN1−
e
XE1
eF1
xA1+xN1+xF1+
e
XE2
eF1
xA2+xN2+xF2and Θ2:=XN2
eN2−
e
XE2
eF2
xA2+xN2+xF2.
Changes in extraction affect commodity supply, because fossil fuel is the only input in
the commodity production process, and the extractor’s commodity demand, because the
commodity is the only input in the fossil fuel extraction process. Thereby, changes in ex-
traction also affect the households’ commodity consumption and thus the commodity price.
Θ2is the relative change in the households’ second-period commodity consumption induced
by a marginal change in second-period extraction. If second-period extraction increased,
the additional commodity supply would outweigh the extractor’s additional commodity
demand, because the real fossil fuel price (pet /pxt) or the marginal productivity of fossil
fuel (XNt
eNt ) exceeds the marginal physical extraction cost ( e
XEt
eF t ) with a deposit preserv-
ing system. The larger the relative change in the households’ second-period commodity
consumption, the more abundant the commodity becomes in period 2 and the stronger
the second-period commodity price declines due to an increase in second-period extraction.
The first term in Θ1is the first-period pendant to Θ2. The second term reflects the relative
change in the household’s second-period commodity consumption due to a marginal change
in first-period extraction. If first-period extraction increased, the extractor’s second-period
commodity demand would increase due to the physical user cost of extraction ( e
XE2
eF1), so
that the commodity would become scarcer in period 2 and the second-period commodity
price would increase.18 Finally, the smaller the intertemporal elasticity of substitution in
consumption, the stronger the commodity price reacts to changes in extraction.
On the fossil fuel market, purchasing additional deposits affects first-period extraction:19
deeF1=−µ1deF2−γ1XE2
eF1
XE1
eF1eF1
dpx2,(25)
where µ1:=pe2px2XE2
eF1eF2eN1
|ηN1|
Γ0>0 is the intertemporal effectiveness of the energy market
channel, γ1:=XE1
eF1eF1eN1
|ηN1|[pe2+px2XE2
eF2eF2eN2
|ηN2|]
Γ0>0, and Γ0>0 is defined in Appendix
A.4.
The first term reflects the intertemporal energy market channel. By purchasing additional
deposits, the second-period fossil fuel price would rise if the second-period commodity price
did not change, so that second-period fossil fuel supply would increase. Thereby, the phys-
ical user cost of supply would increase and first-period fossil fuel supply would decline.
18The physical user cost of extraction estimate how much more material is needed to extract one more
unit of fossil fuel in period 2 if one more unit of fossil fuel is extracted in period 1.
19See Appendix A.1, equation (A.15).
12

Thus, first-period extraction is not directly affected but indirectly reduced due to negative
intertemporal carbon leakage. The second term reflects the intertemporal terms of trade
channel. In period 2, the equilibrium on the fossil fuel market does not depend on the
second-period commodity price, because this price neither affects real inverse second-period
fossil fuel demand (XN2
eN2) nor supply (XE2
eF2). However, in period 1, the user cost of supply
depends on the second-period commodity price, and if this price decreased, ceteris paribus,
first-period fossil fuel supply and thus first-period extraction would increase.
Additionally, cumulative extraction is affected:20
deeFΣ=−µΣdeF2−γΣXE2
eF1
XE1
eF1eF1
dpx2,(26)
where µΣ:=Γ0−pe2Γ2
Γ0> µ1is the aggregate effectiveness of the energy market channel,
γΣ:=XE1
eF1eF1eN1
|ηN1|Γ1
Γ0< γ1and Γ1,Γ2are defined in Appendix A.4.21
The first term reflects the aggregate energy market channel. Since second-period fossil
fuel supply would increase if the second-period commodity price did not change, µΣcan
be smaller than unity. However, µΣ> µ1, so that total emissions would decline sharper
than first-period emissions if the second-period commodity price did not change.22 The
second term reflects the aggregate terms of trade channel. If the second-period commodity
price decreased, ceteris paribus, first-period extraction would increase, which would raise
the marginal physical extraction cost and thus reduce fossil fuel supply and extraction in
period 2, so that γΣ< γ1.
The aggregate terms of trade channel is negative if γΣ<0. This is the case if the sum of
the reciprocals of the intratemporal price semi-elasticities of demand and supply for fossil
fuel in period 2 1
eN2
|ηN2|+1
eF2ηF2,2is smaller than the reciprocal of the intertemporal price
semi-elasticity of supply for fossil fuel in period 1 1
eF1ηF1,2. In this case, the demand and
supply reactions in the second period are relatively strong and the feedback to the first
period is relatively weak.
How strong extraction changes due to purchasing additional deposits crucially depends on
the magnitude of the intertemporal elasticity of substitution in consumption. For example,
20See Appendix A.1, equation (A.17).
21 The necessary condition for our results to hold is µΣ> µ1, which is weaker than condition (3) and
guarantees, i.a., that second-period fossil fuel supply increases if the abating country purchases additional
deposits. See Appendix A.4, equation (A.39).
22See Appendix A.4.
13

if it is very high (σ→ ∞), the commodity price in period 2 hardly changes, so that the
terms of trade channels nearly disappear, see equation (24). Then, there would be negative
intertemporal carbon leakage (µ1>0) and there could be positive cumulative carbon leakage
(µΣ<1), but first-period and total emissions would decline, see equations (25) and (26).
Nevertheless, combining the results from the commodity market with those from the fossil
fuel market, we can infer the following proposition without knowing anything about the
magnitude of the intertemporal elasticity of substitution in consumption:23
Proposition 1.If the abating country purchases additional deposits (deF2>0),
first-period fossil fuel supply decreases (negative intertemporal carbon leakage), second-
period fossil fuel supply increases, and cumulative fossil fuel supply increases if γΣ<0
(positive cumulative carbon leakage),
emissions decline in both periods, so that neither the weak nor the strong green paradox
does occur,
the second-period commodity price rises if and only if (µΣ−µ1)Θ2> µ1Θ1, the real
fossil fuel price (pet/pxt)rises in both periods, and the second-period fossil fuel price
rises if the second-period commodity price rises.
Proof. See Appendix A.3. 
In period 2, the equilibrium on the fossil fuel market does not depend on the commodity
price in that period, so that widening the wedge between demand and supply by purchas-
ing additional deposits increases second-period fossil fuel supply but reduces second-period
extraction. Thereby, the user cost of supply increases and first-period fossil fuel supply
and thus first-period extraction decreases. Since a lower second-period commodity price is
always accompanied by lower first-period extraction if second-period extraction decreases,
see equation (24), and a higher second-period commodity price leads to higher user cost of
supply and thus to lower first-period extraction, see equation (25), the emissions decline in
both periods and neither the weak nor the strong green paradox occurs.
Extraction declines in both periods, so that the second-period commodity price rises
(falls) if the commodity becomes relatively scarcer in the second (first) period. This is the
case if the relative change in the household’s commodity consumption is large in period 2
23Note that in this and all other propositions, the prices (pet , pxt ) and quantities (eF t, eF t, eAt , eN t) are
evaluated before marginal changes in the prevailing unilateral deposit policy.
14

(Θ2↑) (in period 1 (Θ1↑)) and if the decrease in extraction induced by a marginal change
in purchased deposits is strong in period 2 (µΣ−µ1↑) (in period 1 (µ1↑)). Finally, the real
fossil fuel price (pet/pxt ) or the marginal productivity of fossil fuel (XN t
eNt ) increases because
extraction declines in both periods. Thus, the second-period fossil fuel price definitely rises
if the second-period commodity price rises.
Positive cumulative carbon leakage occurs if γΣ<0. The first reason is that γΣbeing
negative implies µΣbeing smaller than unity, so that cumulative fossil fuel supply would
increase if the second-period commodity price did not change.24 The second reason is that
the aggregate energy market channel outweighs the aggregate terms of trade channel if γΣis
negative, so that cumulative fossil fuel supply increases even if the second-period commodity
price falls.25 Finally, note that µΣbeing larger than unity is necessary and sufficient for
negative cumulative carbon leakage in a partial-equilibrium model, but neither necessary nor
sufficient in a general-equilibrium model, which we demonstrate in the following proposition:
Proposition 2.If the abating country purchases additional deposits (deF2>0), cumu-
lative fossil fuel supply decreases (negative cumulative carbon leakage)if and only if γΣ>0
and
1−µΣ<

pe1
px2XE2
eF1
eF2ηF2,1
σ+eN2|ηN2|
eF2ηF2,2
Θ2

−1
·(µΣ−µ1)Θ2−µ1Θ1.
Proof. Inserting the definitions of Γ1, Γ2,µ1and µΣinto equation (A.30) for deF1= 0
and rearranging yields the inequality above. 
Since the first term in brackets is positive and the second term in brackets is positive if
and only if the second-period commodity price rises,26 µΣbeing larger (smaller) than unity
and bpx2being positive (negative) is sufficient for negative (positive) cumulative carbon
leakage. µΣis larger than unity if the sum of the reciprocals of the intratemporal price
semi-elasticities of demand and supply for fossil fuel in period 1 1
eN1
|ηN1|+1
eF1ηF1,1is
smaller than the reciprocal of the intertemporal price semi-elasticity of supply for fossil fuel
in period 2 1
eF2ηF2,1. In this case, the demand and supply reactions in the first period
are relatively strong and the feedback to the second period is relatively weak. If µΣis
24See Appendix A.4.
25See Appendix A.3, equation (A.30).
26See Proposition 1.
15

smaller (larger) than unity and bpx2is positive (negative), the cumulative fossil fuel supply
decreases if the increase (decrease) in the second-period commodity price is sufficiently
strong (weak). The lower the intertemporal elasticity of substitution in consumption, the
stronger the second-period commodity price changes due to purchasing additional deposits.
In contrast to tightening an emissions cap in period 2 (as in Ritter & Schopf 2014,
Section 4), purchasing additional deposits always reduces first-period and total emissions.
On the fossil fuel market, both policies lead to lower extraction in period 2. In case of
the demand side policy, in which extraction is equal to fossil fuel supply, this leads to
lower physical user cost of supply and thus to higher extraction in period 1. In case of
the supply side policy, in which the purchased deposits drive a wedge between demand and
supply, the physical user cost of supply increase, so that extraction declines in period 1.
On the commodity market, a change in the second-period commodity price can prevent the
weak green paradox and affects the condition for the strong green paradox in case of the
demand side policy. In case of the supply side policy, such a change affects the condition
for negative cumulative carbon leakage. With both policies, an increase (decrease) in the
second-period commodity price leads to lower (higher) first-period emissions and, if γΣ>
0, to lower (higher) total emissions.27 Then, a partial-equilibrium model underestimates
(overestimates) the effectiveness of “green” policies.28
In contrast to the simple linear partial-equilibrium model from Figure 1, negative cumu-
lative carbon leakage is possible and occurs if, e.g., µΣis equal to (larger than) unity and the
second-period commodity price declines (stays constant). Thus, this phenomenon hinges
on the general-equilibrium model and on the more general extraction cost function. Except
for negative cumulative carbon leakage being possible or not, the qualitative results are the
same in the simple linear partial-equilibrium model and in the convex general-equilibrium
model: Purchasing additional deposits leads to lower first-period and higher second-period
fossil fuel supply, less emissions in both periods and higher (real) fossil fuel prices.
27See Appendix A.1, equations (A.15) and (A.17).
28Things get more complicated if γΣ<0 because then first-period and total emissions move into opposite
directions due to a change in the second-period commodity price and it depends on the climate damage
function whether an increase or a decrease in this price is preferable.
16

4 Leasing Additional Deposits
Leasing additional deposits means that deposits that would otherwise be extracted in period
1 are preserved in that period and extracted in period 2. The main results of this policy
will be characterized in Proposition 3: Leasing additional deposits (deF1>0) can cause
positive intratemporal carbon leakage (deF1/deF1>0), but the weak green paradox does
not occur (deeF1/deF1<0). Total emissions can increase (deeFΣ/deF1>0), so that the
strong green paradox can occur (dD/ deF1>0). Negative cumulative carbon leakage is
possible (deFΣ/deF1<0).
On the commodity market, leasing additional deposits affects the second-period commod-
ity price via changes in extraction according to equation (24). On the fossil fuel market,
leasing additional deposits affects first-period extraction:29
deeF1=−γ1deF1−γ1XE2
eF1
XE1
eF1eF1
dpx2.(27)
The first term reflects the intratemporal energy market channel. By leasing additional
deposits, the wedge between demand and supply increases, which leads to higher fossil fuel
supply and lower extraction in period 1. Thereby, the second-period marginal physical
supply cost falls, so that fossil fuel supply and extraction increase in period 2. This leads
to higher physical user cost of supply and thus to lower first-period fossil fuel supply and
extraction. The second term reflects the intratemporal terms of trade channel, which differs
from that in the previous section because the change in the second-period commodity price
differs.
Additionally, cumulative extraction is affected:30
deeFΣ=−γΣdeF1−γΣXE2
eF1
XE1
eF1eF1
dpx2.(28)
The aggregate energy market channel and the aggregate terms of trade channel are neg-
ative if γΣ<0. For the energy market channel, this would mean that the increase in
second-period extraction due to the lower second-period marginal physical supply cost out-
weighs the decrease in first-period extraction due to the larger wedge between demand and
supply and the higher physical user cost of supply. For the terms of trade channel, the
reasoning is similar. If, e.g., the second-period commodity price declines, first-period ex-
traction increases, so that the second-period marginal physical supply cost increases and
29See Appendix A.1, equation (A.15).
30See Appendix A.1, equation (A.17).
17

second-period extraction declines. With a negative aggregate terms of trade channel, cu-
mulative extraction would then decline.
Combining the results from the commodity market with those from the fossil fuel market,
we can infer the following proposition without knowing anything about the magnitude of
the intertemporal elasticity of substitution in consumption:
Proposition 3.If the abating country leases additional deposits (deF1>0),
first-period fossil fuel supply increases if eF1ηF1,1eF2ηF2,2≥eN1|ηN1|eN2|ηN2|(positive
intratemporal carbon leakage), second-period fossil fuel supply increases, and cumula-
tive fossil fuel supply increases if first-period fossil fuel supply increases or if eF2ηF2,2≥
eF1ηF1,2(positive cumulative carbon leakage),
emissions decline in period 1and increase in period 2, so that the weak green paradox
does not occur,
total emissions increase if and only if γΣ<0, and the strong green paradox occurs if
and only if γ1+λγΣ<0,
the second-period commodity price falls, the real fossil fuel price (pet/pxt)rises in
period 1and falls in period 2, and the second-period fossil fuel price falls.
Proof. See Appendix A.5. 
In period 1, fossil fuel supply can only decline if eF1ηF1,1eF2ηF2,2< eN1|ηN1|eN2|ηN2|.
In this case, widening the wedge between demand and supply strongly reduces (raises)
demand and only weakly raises (reduces) supply in the first (second) period, so that the
above mentioned increase in the physical user cost of supply can eventually lead to lower
fossil fuel supply in period 1. First-period extraction always declines, so that second-period
fossil fuel supply and thus extraction always increase. Thus, cumulative fossil fuel supply
can only decline if it declines in period 1, and it definitely increases if eF2ηF2,2≥eF1ηF1,2.
In this case, the supply reaction is relatively strong in the second period and the feedback
to the first period is relatively weak.
Cumulative extraction increases if γΣ<0. Then, the strong green paradox occurs if
the increase in climate damages due to higher total emissions outweighs the respective
decline due to lower first-period emissions. Extraction declines in period 1 and increases
in period 2, so that the commodity becomes relatively scarcer in period 1 and the second-
period commodity price falls. For the same reason, the real fossil fuel price (pet /pxt) or the
18

eF2
eF1
0102
F1
pe1
px1
pe2
px2
p'
dpe1
e'
F2
e'
F1F1
p'
pe1
px1
'
pe2
px2
eF1
F2
'
A1
C1
D1
A2
B2
E1
=F2eF2
=
pe2
px2
=
pe1
px1
B1
Figure 2: Leasing additional deposits with perfect cumulative carbon leakage.
marginal productivity of fossil fuel (XNt
eNt ) increases in period 1 and declines in period 2.
Since the second-period commodity price and the real second-period fossil fuel price always
decline, the second-period fossil fuel price always falls.
Figure 2 illustrates an increase in the amount of leased deposits with perfect cumulative
carbon leakage (γΣ= 0). The initial equilibrium on the fossil fuel market is denoted
without dashes. By leasing additional deposits (B1C1), first-period extraction declines
(A1→B1). Thereby, the second-period marginal physical supply cost declines (XE2
eF2),
so that the second-period supply curve shifts downwards and turns clockwise, which raises
fossil fuel supply and extraction in period 2 (A2→B2). This, in turn, leads to higher
physical user cost of supply (XE2
eF1), so that the first-period supply curve shifts upwards and
turns anticlockwise, whereby extraction and fossil fuel supply decline in period 1 (B1→D1
and C1→E1). The new equilibrium on the fossil fuel market without a change in the
second-period commodity price is denoted by dashes. However, the second-period commod-
ity price falls, so that the user cost of supply declines, which leads to an upward pressure
on first-period and thus to a downward pressure on second-period extraction. Nevertheless,
extraction declines in period 1, increases in period 2, and with perfect cumulative carbon
leakage, cumulative extraction stays constant.
To get a better understanding for when the strong green paradox occurs, we now consider
19

a more specific model. Suppose that the cumulative material supply cost depends on a
weighted sum of first-period extraction and second-period fossil fuel supply:
XE1=X(eF1) and XE2=X(eeF1+ψeF2)−X(eeF1).(29)
The idea behind ψ6= 1 is that it could be cheaper to extract a specific unit of fossil fuel in
period 2 than in period 1 due to technological progress, so that ψ≤1.
Proposition 4.If the abating country leases additional deposits (deF1>0) and the
material extraction costs are given by equation (29), cumulative fossil fuel supply increases,
total emissions increase if and only if
(1 −ψ)eN2|ηN2|>X0(eeF1+ψeF2)
X00(eeF1+ψeF2),
and the strong green paradox occurs if and only if
λ
1 + λ−ψeN2|ηN2|>X0(eeF1+ψeF2)
X00(eeF1+ψeF2).
Proof. Inserting equation (29) into the respective conditions in Proposition 3 and re-
arranging yields the inequalities above. 
The conditions in Proposition 4 are weakened if second-period fossil fuel demand is rela-
tively elastic and second-period fossil fuel supply is relatively inelastic. Then, an increase
in second-period fossil fuel supply induced by a decrease in first-period extraction leads to
a strong increase in second-period extraction. However, total emissions can only increase
if ψ < 1, so that the second-period fossil fuel supply increases more than first-period ex-
traction declines. In this case, the strong green paradox can occur if the relative weight
attached to changes in second-period emissions λ/(1 + λ) exceeds ψ. In Appendix A.6, we
conduct a numerical example which confirms the intuition that leasing additional deposits
raises cumulative extraction and induces the strong green paradox if ψis sufficiently low.
In contrast to tightening an emissions cap in period 1 (as in Ritter & Schopf 2014,
Section 3), leasing additional deposits always reduces first-period emissions. The condi-
tions for an increase in total emissions and the occurrence of the strong green paradox are
equivalent if the second-period commodity price does not change (σ→ ∞). In case of
the demand and the supply side policy, first-period extraction then declines by the same
amount, depending on the elasticities of fossil fuel demand and supply in period 1, so that
second-period marginal extraction cost declines and second-period extraction increases by
20

the same amount, depending on the respective elasticities in period 2. If the intertemporal
elasticity of substitution in consumption is finite, the second-period commodity price de-
clines in both cases, leading to higher extraction in period 1 and thus to lower extraction
in period 2. This can lead to the weak green paradox and affects the condition for the
strong green paradox in case of the demand side policy. In case of the supply side policy,
the magnitude of the intertemporal elasticity of substitution in consumption does not alter
the qualitative results.
In contrast to the simple linear partial-equilibrium model, total emissions can increase
if second-period fossil fuel demand is relatively elastic and second-period fossil fuel supply
is relatively inelastic. Then, the strong green paradox can occur if climate damages from
second-period emissions are relatively important. Both phenomena do not hinge on the
general-equilibrium model, but on the more general extraction cost function. In the simple
linear partial-equilibrium model and in the convex general-equilibrium model, emissions
decline and the (real) fossil fuel price rises in period 1, while emissions increase and the
(real) fossil fuel price falls in period 2 by leasing additional deposits.
5 Effectiveness of Leasing or Purchasing Additional Deposits
In the previous two sections, we found that both policies reduce first-period emissions, but
that leasing (purchasing) additional deposits increases (reduces) second-period emissions.
Nevertheless, leasing additional deposits can be more effective than purchasing additional
deposits as long as it does not lead to the strong green paradox. In this section, we analyze
under which conditions this is the case. Furthermore, we discuss which of the policies could
be more cost effective.
Concerning the effectiveness, we can infer the following proposition:
Proposition 5.Purchasing additional deposits reduces first-period emissions more than
leasing additional deposits if µ1≥γ1. Else, purchasing additional deposits reduces first-
period emissions more than leasing additional deposits if and only if
σ < px2XE2
eF1eN2|ηN2|Θ2
pe1
·ηF2,1
ηF2,2
·µ1
γ1−µ1
.
Purchasing additional deposits reduces total emissions more than leasing additional deposits
if µΣ≥γΣ. Else, purchasing additional deposits reduces total emissions more than leasing
21

additional deposits if and only if
σ < px2XE2
eF1eN2|ηN2|PtΘt
pe1
·ηF2,1
ηF2,2
·µ1
γΣ−µΣ
.
Purchasing additional deposits being more effective in reducing first-period emissions is suf-
ficient for purchasing additional deposits being more effective in reducing total emissions.
Proof. Inserting the definitions of ηF2,1,ηF2,2,µ1and γ1into equation (A.27) for
deF1= 0 and deF2= 0, respectively, and rearranging yields expressions for deeF1/deF2
and deeF1/deF1, respectively. If µ1≥γ1, then deeF1/deF2>deeF1/deF1. If µ1< γ1, then
deeF1/deF2>deeF1/deF1if and only if the first inequality in the proposition is fulfilled.
Inserting the definitions of ηF2,1,ηF2,2,µ1,γΣand µΣinto equation (A.31), the second in-
equality in the proposition follows along the same lines. The last sentence in the proposition
follows from γ1−µ1> γΣ−µΣ.
It turns out that the respective energy market channel being more effective in case of
purchasing additional deposits than in case of leasing additional deposits is sufficient for
the former policy being more effective than the latter in reducing first-period and total
emissions. If this is not the case, purchasing additional deposits is still more effective than
leasing additional deposits if the intertemporal elasticity of substitution in consumption is
low, so that the change in the second-period commodity price is relatively large. The reason
is as follows: An increase (decrease) in the second-period commodity price leads to lower
(higher) first-period emissions, and it leads to lower (higher) total emissions if γΣ>0. The
following relation demonstrates that this increase (decrease) is stronger (weaker) in case of
purchasing additional deposits than in case of leasing additional deposits if γ1> µ1:31
bpx2
deF2Rbpx2
deF1
⇔(µΣ−µ1)
| {z }
>0
Θ2−µ1Θ1R(γΣ−γ1)
| {z }
<0
Θ2−γ1Θ1.(30)
Thus, the respective terms of trade channel speaks out in favor of purchasing additional
deposits, and this channel is the more important the lower the intertemporal elasticity
of substitution in consumption. Since leasing (purchasing) additional deposits increases
(reduces) second-period emissions, purchasing additional deposits being more effective in
reducing first-period emissions is sufficient for purchasing additional deposits being more
effective in reducing total emissions.
31Inserting the definitions of γ1,γΣ,µ1and µΣinto equation (A.32) for deF1= 0 and deF2= 0,
respectively, and rearranging yields equation (30).
22

If the policy that is more effective is also cheaper, it is of course also more cost effective
than the other policy. Whether the one or the other policy is cheaper depends on the
opportunity cost of the preserved deposit that is most costly to extract in the respective
period.32 This cost is increasing in the respective fossil fuel price, declining in the respective
marginal extraction cost and declining in the user cost of extraction in case of leasing
additional deposits. Thus, it is the higher the higher the amount of preserved deposits in
the respective period. This speaks out in favor of a combined policy as long as leasing
additional deposits is effective at all.
6 Concluding Remarks
In this paper, we use Ritter & Schopf’s (2014) model and change the policy instrument
from an emissions trading scheme to a deposit preserving system. We do not derive the
unilaterally optimal deposit policy, but check whether preserving additional deposits can
cause the green paradoxes. However, we design the deposit preserving system in a way
that it could generally implement the first-best given that climate damage is a function of
first-period and total emissions. That is, we take some (potentially none) of the deposits
that are most costly but still worthwhile to extract to be leased in the first period and to
be purchased in the second period as starting point of our analysis.
Purchasing additional deposits reduces emissions in both periods and can, in the con-
vex general-equilibrium model, lead to negative cumulative carbon leakage. By contrast,
tightening a second-period emissions cap in a comparable model can lead to higher total
emissions and to the green paradoxes (Ritter & Schopf 2014, Section 4). The demand side
policy reduces extraction in period 2, which leads to lower user cost of supply and thus to
higher extraction in period 1. The supply side policy increases the wedge between second-
period extraction and supply, so that second-period extraction declines, but second-period
fossil fuel supply increases, which leads to higher user cost of supply and thus to lower
extraction in period 1.
Leasing additional deposits leads to lower emissions in the first period and to higher emis-
sions in the second period. In the convex general-equilibrium model, total foreign emissions
can decline, but total emissions can also increase, which facilitates the strong green paradox.
In contrast to purchasing additional deposits, the intertemporal supply reaction does not
32See equation (8).
23

depend on the respective period’s supply, which increases in both approaches, but on the re-
spective period’s extraction, because the additional deposits are only temporarily preserved.
This also explains why the condition for an increase in total emissions is equivalent to that
in case of tightening a first-period emissions cap if the second-period commodity price does
not change (Ritter & Schopf 2014, Section 3): The supply side policy temporarily reduces
supply, while the demand side policy temporarily reduces demand.
Purchasing additional deposits is more effective than leasing additional deposits in reduc-
ing first-period and total emissions if the respective energy market channel is more effective
or if the intertemporal elasticity of substitution in consumption is low. In the latter case,
the respective terms of trade channel is more effective if the abating country purchases ad-
ditional deposits than if it leases additional deposits. However, the respective deposit price
is increasing in the respective amount of preserved deposits, which speaks out in favor of a
combined policy.
In conclusion, the first-best can only be implemented if some deposits are leased and pur-
chased. Leasing additional deposits can lead to higher climate damages, while purchasing
additional deposits is unambiguously good for the climate. However, there are credibility
problems involved when deposits are announced to be permanently preserved. Thus, with-
out full information and perfect future markets, unilateral deposit policies are no panacea
against global warming. However, without global climate agreement, they are most likely
components of second-best policies.
24

A Appendix
A.1 The Fossil Fuel Market
Throughout the appendix the commodity in period 1 is chosen as numeraire. Rearranging (5),
(15) and (20) yields:
pe1−XE1
eF1−px2XE2
eF1= 0,(A.1)
pe2−px2XE2
eF2= 0,(A.2)
XN1
eN1−pe1= 0,(A.3)
px2XN2
eN2−pe2= 0,(A.4)
eeF t −eAt −eNt =eF t −eFt −eAt −eNt = 0.(A.5)
Total differentiating (A.1) to (A.5) yields:
dpe1−XE2
eF1dpx2−XE1
eF1eF1deF1−px2XE2
eF1eF1deeF1−px2XE2
eF1eF2deF2= 0,(A.6)
dpe2−XE2
eF2dpx2−px2XE2
eF2eF1deeF1−px2XE2
eF2eF2deF2= 0,(A.7)
deN1−eN1ηN1bpe1= 0,(A.8)
deN2−eN2ηN2[bpe2−bpx2]=0,(A.9)
deeF t −deAt −deNt = deF t −deFt −deAt −deNt = 0,(A.10)
where ηNt :=XNt
eNt
eNt XN t
eNt eN t
<0.
Inserting (A.8) and (A.9) in (A.10) and afterwards inserting in (A.6) and (A.7) yields:
dpe1−XE2
eF1dpx2−XE1
eF1eF1deF1−[XE1
eF1eF1+px2XE2
eF1eF1][deA1+eN1ηN1bpe1] (A.11)
−px2XE2
eF1eF2[deF2+ deA2+eN2ηN2[bpe2−bpx2]] = 0,
dpe2−XE2
eF2dpx2−px2XE2
eF2eF1[deA1+eN1ηN1bpe1] (A.12)
−px2XE2
eF2eF2[deF2+ deA2+eN2ηN2[bpe2−bpx2]] = 0.
Inserting (A.11) in (A.12) and rearranging yields:
bpe1=XE1
eF1eF1[Γ1−px2XE2
eF1eF2eN2ηN2]
Γ0
deF1−Γ0−pe1[Γ1−px2XE2
eF1eF2eN2ηN2]
eN1ηN1Γ0
deA1(A.13)
+pe2px2XE2
eF1eF2
Γ0
(deF2+ deA2) + px2XE2
eF1[Γ1−px2XE2
eF1eF2eN2ηN2]
Γ0bpx2,
bpe2=XE1
eF1eF1px2XE2
eF1eF2eN1ηN1
Γ0
deF1+pe1px2XE2
eF1eF2
Γ0
deA1(A.14)
−Γ0−pe2[Γ2−px2XE2
eF1eF2eN1ηN1]
eN2ηN2Γ0
(deF2+ deA2) + Γ0+px2XE2
eF1px2XE2
eF1eF2eN1ηN1
Γ0bpx2,
where Γ0, Γ1, and Γ2are defined in Appendix A.4.
25

Inserting (A.8) and (A.9) in (A.10) and afterwards inserting (A.13) and (A.14) and rearranging
yields:
deF1=Γ0+XE1
eF1eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]
Γ0
deF1+pe1[Γ1−px2XE2
eF1eF2eN2ηN2]
Γ0
deA1
(A.15)
+pe2px2XE2
eF1eF2eN1ηN1
Γ0
(deF2+ deA2) + XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]
Γ0
dpx2,
deF2=XE1
eF1eF1eN1ηN1px2XE2
eF1eF2eN2ηN2
Γ0
deF1+pe1px2XE2
eF1eF2eN2ηN2
Γ0
deA1(A.16)
+pe2[Γ2−px2XE2
eF1eF2eN1ηN1]
Γ0
(deF2+ deA2) + XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2
Γ0
dpx2.
Adding (A.15) and (A.16) yields:
deFΣ=Γ0+XE1
eF1eF1eN1ηN1Γ1
Γ0
deF1+pe1Γ1
Γ0
deA1+pe2Γ2
Γ0
(deF2+ deA2) + XE2
eF1eN1ηN1Γ1
Γ0
dpx2.
(A.17)
A.2 The Commodity Market
The relative commodity demand of A, N, F and Eis given by:
qd=Pxi1
Pxi2
=xA1+xN1+xF1+xE1
xA2+xN2+xF2+xE2
, i =A, N, F, E. (A.18)
Inserting (6), (10), (11), (21) and (19) in (A.18) yields:
qd=α1px2
α2σ
−α1px2
α2σe
XE2
XA2+XN2+e
XE1
XA2+XN2.(A.19)
Total differentiation of (A.19) and afterwards inserting (5), (6), (10), (11), (14), (15), (21), (19)
and (A.10) yields:
dqd=α1px2
α2σ
σbpx2−α1px2
α2σ
σbpx2e
XE2
XA2+XN2(A.20)
−α1px2
α2σde
XE2[XA2+XN2]−e
XE2[dXA2+ dXN2]
[XA2+XN2]2
+de
XE1[XA2+XN2]−e
XE1[dXA2+ dXN2]
[XA2+XN2]2
=xA1+xN1+xF1
xs
A2+xs
N2
σbpx2+e
XE1
eF1−e
XE2
eF1·xA1+xN1+xF1
xA2+xN2+xF2
xs
A2+xs
N2
deeF1
−
xs
A1+xs
N1
xs
A2+xs
N2−xA1+xN1+xF1
xA2+xN2+xF2
xs
A2+xs
N2π2
px2
deA2+pe2
px2
deeF2−e
XE2
eF2·xA1+xN1+xF1
xA2+xN2+xF2
xs
A2+xs
N2
deeF2.
The relative commodity supply of Aand Nis given by:
qs=Pxs
i1
Pxs
i2
=XA1+XN1
XA2+XN2, i =A, N. (A.21)
26

Total differentiation of (A.21) and afterwards inserting (10), (11), (14), (15) and (A.10) yields:
dqs=[XA1
eA1deA1+XN1
eN1deN1][XA2+XN2]−[XA1+XN1][XA2
eA2deA2+XN2
eN2deN2]
[XA2+XN2]2(A.22)
=π1
xs
A2+xs
N2
deA1−
π2
px2·xs
A1+xs
N1
xs
A2+xs
N2
xs
A2+xs
N2
deA2+pe1
xs
A2+xs
N2
deeF1−
pe2
px2·xs
A1+xs
N1
xs
A2+xs
N2
xs
A2+xs
N2
deeF2.
Equating (A.20) and (A.22) and rearranging yields:
bpx2=1
σπ1
xA1+xN1+xF1
deA1−
π2
px2
xA2+xN2+xF2
deA2+ Θ1deeF1−Θ2deeF2,(A.23)
where Θ1:=pe1−
e
XE1
eF1
xA1+xN1+xF1+
e
XE2
eF1
xA2+xN2+xF2and Θ2:=
pe2
px2−
e
XE2
eF2
xA2+xN2+xF2.
A.3 The Combined Market
Inserting (A.23) in (A.15) for deA1= deA2= 0 and rearranging yields:
deF1=−px2XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]Θ2
σΓ0−px2XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]Θ1
deF2(A.24)
+ deF1+σXE1
eF1eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]
σΓ0−px2XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]Θ1
deF1
+σpe2px2XE2
eF1eF2eN1ηN1+px2XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]Θ2
σΓ0−px2XE2
eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]Θ1
deF2.
Inserting (A.23) in (A.16) for deA1= deA2= 0 and rearranging yields:
deF2=px2XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2Θ1
σΓ0+px2XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2Θ2
deF1(A.25)
+[σXE1
eF1eF1eN1ηN1−px2XE2
eF1eN1ηN1Θ1]px2XE2
eF1eF2eN2ηN2
σΓ0+px2XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2Θ2
deF1
+σpe2[Γ2−px2XE2
eF1eF2eN1ηN1] + px2XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2Θ2
σΓ0+px2XE2
eF1eN1ηN1px2XE2
eF1eF2eN2ηN2Θ2
deF2.
Inserting (A.25) in (A.24) and rearranging yields:
deF1= deF1+σXE1
eF1eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.26)
+σpe2px2XE2
eF1eF2eN1ηN1−px2XE2
eF1eN1ηN1px2XE2
eF2eF2eN2ηN2Θ2
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Inserting (A.26) in deeF1= deF1−deF1yields:
deeF1=σXE1
eF1eF1eN1ηN1[Γ1−px2XE2
eF1eF2eN2ηN2]
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.27)
+σpe2px2XE2
eF1eF2eN1ηN1−px2XE2
eF1eN1ηN1px2XE2
eF2eF2eN2ηN2Θ2
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Inserting (A.24) in (A.25) and rearranging yields:
deF2=σXE1
eF1eF1eN1ηN1px2XE2
eF1eF2eN2ηN2
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.28)
27

+σpe2[Γ2−px2XE2
eF1eF2eN1ηN1]−px2XE2
eF1eN1ηN1[pe2Θ1−px2XE2
eF1eF2eN2ηN2Θ2]
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Inserting (A.28) in deeF2= deF2−deF2yields:
deeF2=σXE1
eF1eF1eN1ηN1px2XE2
eF1eF2eN2ηN2
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.29)
−σ[Γ0−pe2[Γ2−px2XE2
eF1eF2eN1ηN1]] + px2XE2
eF1eN1ηN1px2XE2
eF2eF2eN2ηN2Θ1
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Adding (A.26) and (A.28) yields:
deFΣ= deF1+σXE1
eF1eF1eN1ηN1Γ1
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.30)
+σpe2Γ2+px2XE2
eF1eN1ηN1[Γ1Θ2−pe2PtΘt]
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Adding (A.27) and (A.29) yields:
deeFΣ=σXE1
eF1eF1eN1ηN1Γ1
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.31)
−σ[Γ0−pe2Γ2] + px2XE2
eF1eN1ηN1px2XE2
eF2eF2eN2ηN2PtΘt
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Inserting (A.27) and (A.29) in (A.23) for deA1= deA2= 0 yields:
bpx2=XE1
eF1eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF1(A.32)
+[Γ0−pe2Γ2]Θ2+pe2px2XE2
eF1eF2eN1ηN1PtΘt
σΓ0−px2XE2
eF1eN1ηN1[Γ1Θ1−px2XE2
eF1eF2eN2ηN2PtΘt]deF2.
Inserting (A.10) in (A.8) and (A.9) for deA1= deA2= 0 and rearranging yields:
bpe1=deeF1
eN1ηN1
,(A.33)
bpe2−bpx2=deeF2
eN2ηN2
.(A.34)
A.4 The Gammas
Γ0= [pe1−[XE1
eF1eF1+px2XE2
eF1eF1]eN1ηN1][pe2−px2XE2
eF2eF2eN2ηN2] (A.35)
−px2XE2
eF1eF2eN1ηN1px2XE2
eF1eF2eN2ηN2
=pe2eN2|ηN2|pe1eN1|ηN1|
eF1ηF1,2eF2ηF2,1
·
 eF1ηF1,2
eN2|ηN2|+eF1ηF1,2
eF2ηF2,2!· eF2ηF2,1
eN1|ηN1|+eF2ηF2,1
eF1ηF1,1!−1
,
Γ1=pe2−[px2XE2
eF2eF2−px2XE2
eF1eF2]eN2ηN2(A.36)
=pe2eN2|ηN2|
eF1ηF1,2
· eF1ηF1,2
eN2|ηN2|+eF1ηF1,2
eF2ηF2,2
−1!,
Γ2=pe1−[XE1
eF1eF1+px2XE2
eF1eF1−px2XE2
eF1eF2]eN1ηN1(A.37)
28

=pe1eN1|ηN1|
eF2ηF2,1
· eF2ηF2,1
eN1|ηN1|+eF2ηF2,1
eF1ηF1,1
−1!.
Γ0>0 because eF1ηF1,2
eF2ηF2,2·eF2ηF2,1
eF1ηF1,1≥1, see footnote 14. For the same reason, Γ1≤0 implies Γ2>0
and Γ2≤0 implies Γ1>0. Furthermore,
Γ0> pe1[Γ1−px2XE2
eF1eF2eN2ηN2]
⇔pe2eN2|ηN2|pe1eN1|ηN1|
eF1ηF1,2eF2ηF2,1
·
 eF1ηF1,2
eN2|ηN2|+eF1ηF1,2
eF2ηF2,2!·eF2ηF2,1
eF1ηF1,1
−1
>0,(A.38)
Γ0> pe2[Γ2−px2XE2
eF1eF2eN1ηN1]
⇔pe2eN2|ηN2|pe1eN1|ηN1|
eF1ηF1,2eF2ηF2,1
·
eF1ηF1,2
eF2ηF2,2
· eF2ηF2,1
eN1|ηN1|+eF2ηF2,1
eF1ηF1,1!−1
>0,(A.39)
which holds for the same reason as above, so that Γ0> pe1Γ1and Γ0> pe2Γ2.
A.5 Proof of Proposition 3
Leasing additional deposits increases first-period and cumulative fossil fuel supply if and only if
(A.26) and (A.30) are positive, respectively. Inserting the definitions of Γ0, Γ1, and ηF s,t in (A.26)
and (A.30) for deF2= 0 and rearranging yields:
deF1
deF1R0
⇔1
σ
px2XE2
eF1
pe1
 1
eN2|ηN2|+1
eF2ηF2,2!Θ1+1
eF1ηF1,2
Θ2
(A.40)
+1
eF1ηF1,2eF2ηF2,1
·
eF1ηF1,2
eF2ηF2,2
· eF2ηF2,1
eN1|ηN1|+eF2ηF2,1
eF1ηF1,1!−1
+px2XE2
eF1eF1
pe1
1
eN2|ηN2|
+1
eN1|ηN1|eN2|ηN2|−XE1
eF1eF1
pe1
1
eF2ηF2,2R0,
deFΣ
deF1R0
⇔1
σ
px2XE2
eF1
pe1
 1
eN2|ηN2|+1
eF2ηF2,2!Θ1+1
eF1ηF1,2
Θ2
(A.41)
+1
eF1ηF1,2eF2ηF2,1
·
eF1ηF1,2
eF2ηF2,2
· eF2ηF2,1
eN1|ηN1|+eF2ηF2,1
eF1ηF1,1!−1
+px2XE2
eF1eF1
pe1
1
eN2|ηN2|
+1
eN1|ηN1|eN2|ηN2|+XE1
eF1eF1
pe1 1
eF1ηF1,2
−1
eF2ηF2,2!R0.
The first two lines of (A.40) and (A.41) are positive. The third line of (A.40) is positive if
eF1ηF1,1eF2ηF2,2≥pe1/XE1
eF1eF1eF2ηF2,2≥eN1|ηN1|eN2|ηN2|. The third line of (A.41) is positive
if (A.40) is positive or if eF2ηF2,2≥eF1ηF1,2. For the remainder of the proof, see Appendix A.3.
29

A.6 Numerical Example
Equating (A.19) and (A.21) and rearranging yields:
XA1+XN1−e
XE1
XA2+XN2−e
XE2−α1
α2σ
pσ
x2= 0.(A.42)
(A.1) to (A.5) and (A.44) constitute a system of seven equations in seven unknowns, eN1,eN2,
eF1,eF2,px2,pe1and pe2, which can be reduced to a system of two equations in two unknowns,
eF1and eF2:
XN2
eN2(eF2−eF2−eA2)−XE2
eF2(eF1−eF1, eF2)=0,(A.43)
XA1(eA1) + XN1(eF1−eF1−eA1)−e
XE1(eF1−eF1)
XA2(eA2) + XN2(eF2−eF2−eA2)−e
XE2(eF1−eF1, eF2−eF2)
−α1
α2σ XN1
eN1(eF1−eF1−eA1)−XE1
eF1(eF1)
XE2
eF1(eF1−eF1, eF2)!σ
= 0.(A.44)
We use the following material fossil fuel supply cost and commodity production functions:
XE1=eω
F1and XE2= (eF1−eF1+ψeF2)ω−eω
F1,(A.45)
XAt =eθ
At and XNt = (eF t −eF t −eAt )θ,(A.46)
and use the following parameter values: α1/α2= 1, σ=θ= 1/2 and ω= 2. Finally, we set the
traded emissions to 50% of the respective emissions without any policy. Then, eF1and eF2just
depend on eF1,eF2and ψ.
Next, we set eF2= 0 and calculate eF1(eF1) and eF2(eF1) for ψ= 0.3,0.4,0.5,0.6,0.7. Thereby,
we must check whether the emissions trading scheme is still binding, i.e. by symmetry eA1≤(eF1−
eF1)/2 and eA2≤eF2/2, and whether the budget is still adhered, i.e. XA1(eA1) + px2XA2(eA2)−
pe1eA1−pe2eA2−pz1eF1≥0. It turns out that the latter inequality is the most restrictive
constraint, and that it is violated for values of eF1above ∼0.4, depending on the values of ψ.
Finally, since px2<1 for eF2= 0 and ψ= 0.3,0.4,0.5,0.6,0.7, condition (3) is fulfilled according
to footnote 13.
Figure 3 shows that leasing additional deposits reduces first-period extraction, raises second-period
extraction and raises cumulative fossil fuel supply, which is consistent with Propositions 3 and
4. The user cost of extraction and the second-period marginal extraction cost are increasing in
ψ, which explains why extraction and supply are decreasing in ψ. Figure 4 shows that leasing
additional deposits raises cumulative extraction for ψ= 0.3,0.4, and reduces cumulative extraction
for ψ= 0.6,0.7. For ψ= 0.5, leasing additional deposits reduces (raises) cumulative extraction
for values of eF1below (above) ∼0.32. Finally, figure 5 shows that the minimum value of λfor
the strong green paradox to occur is increasing in ψ. Figures 4 and 5 confirm the intuition that
leasing additional deposits raises cumulative extraction and induces the strong green paradox if ψ
is sufficiently low.
30

Figure 3: Effect of leasing deposits on first-period extraction, second-period extraction and
cumulative fossil fuel supply from left to right for ψ= 0.3,0.4,0.5,0.6,0.7 from
top to bottom.
Figure 4: Effect of leasing deposits on cumulative extraction for ψ= 0.3,0.4,0.5,0.6,0.7
from left to right and from top to bottom.
Figure 5: Minimum value of λdependent on eF1for the strong green paradox to occur for
ψ= 0.3,0.4,0.5 from left to right.
31

References
Allen, Myles R., David J. Frame, Chris Huntingford, Chris D. Jones, Jason A. Lowe, Malte Mein-
shausen & Nicolai Meinshausen (2009), ‘Warming Caused by Cumulative Carbon Emissions
towards the Trillionth Tonne’, Nature 458(7242), 1163–1166.
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