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Strategic Climate Policy with Offsets and Incomplete Abatement : Carbon Taxes Versus Cap-and-Trade
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Journal of Environmental Economics and Management
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© 2013 The World Bank
Strategic Climate Policy with Offsets and Incomplete Abatement: Carbon
Taxes versus Cap-and-Trade*
Jon Strand
Development Research Group, Environment and Energy Team
The World Bank
Washington DC 20433, USA
Jstrand1@worldbank.org
and
Department of Economics, University of Oslo
Journal of Environmental Economics and Management, forthcoming, 2013
Abstract:
This paper provides a first analysis of a “policy bloc” of fossil fuel importers which implements
an optimal coordinated climate policy, faces a (non-policy) fringe of other fuel importers, and a
bloc of exporters, and purchases offset from the fringe. We compare a carbon tax and a cap-and-
trade scheme for the policy bloc, which in either case is accompanied by an efficient offset
mechanism for reducing emissions in the fringe. The policy bloc is then shown to prefer a tax
over a cap, since only a tax leads to a lower fuel export price and by more when the policy bloc
is larger. Offsets are also more favorable to the policy bloc under a tax than under a cap. The
optimal offset price under a carbon tax is below the tax rate, while under a cap and free quota
trading the offset price must equal the quota price. The domestic carbon and offset prices are
both higher under a tax than under a cap when the policy bloc is small. When the policy bloc is
larger, the offset price can be higher under a cap. Fringe countries gain by mitigation in the
policy bloc, and more under a carbon tax since the fuel import price is lower.
JEL Classification: Q31; Q38; Q54; Q58; H23
Key words: Climate policy; carbon taxes, cap-and-trade schemes; carbon emissions; strategic
trade policy.
*I wish to thank, without implicating, Lawrence Goulder, Michael Toman, Ian Parry, Sauleh
Siddiqui, and seminar participants at several institutions, plus two referees and an editor of this
journal for helpful comments to previous versions. I in particular thank the arrangers of and
participants at the Environmental Protection and Sustainability Forum at the University of
Exeter, where the paper was first presented. The views expressed in this paper are those of the
author only and should not be taken to represent those of the World Bank, its management, or
member countries.
1
1. Introduction
Today, only countries under Annex 1 of the Kyoto Protocol who have extended the validity of
the Protocol up to 2020, at KOP18 in Doha, in December 2012, have policies which include
formal climate policy targets. These countries might at later stages be joined by other high-
income countries (including Canada, Japan and the U.S.), and perhaps also by some major
emerging economies (among which China and South Africa have already signaled a willingness
to impose greenhouse gas (GHG) pricing in the relatively near future). What seems not
achievable, anytime soon, is a set of comprehensive and coordinated climate policies for all
GHG emitters globally.
The countries with formal climate policies have agreed to binding emissions caps for the period
2013-2020; but these countries comprise less than 20 percent of global GHG emissions. Current
policy includes two “offset” schemes. The most important of these is the Clean Development
Mechanism (CDM), whereby abatement of carbon emissions, to comply with the overall cap, can
be purchased from countries that do not have a climate policy. 1 An objective of the CDM is to
make it easier (and less costly) for emitters in the policy countries to abide by their emissions
caps.
A climate policy could, alternatively, take the form of a carbon tax. No comprehensive carbon
tax policy is so far used or seriously contemplated. 2 Most observers see little difference between
a climate policy involving a carbon tax, and a cap-and-trade (c-a-t) scheme with (expected)
1
The other scheme is Joint Implementation which involves some Annex B countries purchasing offsets from other
countries within this bloc.
2
A few smaller nations, including the Scandinavian countries already from the early 1990s, have enacted unilateral
and relatively comprehensive carbon taxes. But these countries constitute a too small fraction of global emissions to
matter globally.
2
emissions as under a tax. Some differences are still widely recognized; they all speak in favor of
a tax over a cap. 3
This paper focuses on differences between taxes and caps that have so far been less widely
discussed. First, carbon taxes and c-a-t work differently when policy countries are net fossil fuel
importers, and exporters behave strategically. Secondly, offset schemes may work differently
under the two policies. Both differences, it is shown, tend to favor taxes over c-a-t for the
countries implementing (or benefiting from) a climate policy.
In my model it is assumed that all countries can be split into two main groups, importers and
exporters of fossil fuels. Most countries belong to the first group, including virtually all countries
that may wish to establish a climate policy. The exporting group is smaller, notably the OPEC
countries, and Russia including some previous Soviet republics. Importers consist of a “policy
bloc” which pursues a climate policy (those Annex 1 countries that have extended the Kyoto
Protocol); and a “fringe” with no policy (the “rest of the world”). The latter group acts without
coordination, each perceiving no market power in fuel markets.
The producer and policy blocs are each assumed to coordinate their policies fully within their
bloc, but not across blocs. The solution concept is static non-cooperative Nash Equilibrium (NE)
in simultaneous strategies, for both models treated (in sections 2 and 3 respectively), in focusing
3
Under uncertainty the effects differ as only emissions vary under a tax, and only the emissions price under a cap.
From Weitzman (1974), when uncertainty takes the form that benefits (in terms of reduced climate change) of
mitigation policy are less uncertain than costs in the short run (which, arguably, is the case in practice), a tax
solution is preferred. See also Hoel and Karp (2001, 2002), Pizer (2002), and Karp and Costello (2004) for dynamic
analyses. Secondly, the government’s ability to recuperate income may be greater under a tax as many or most
emissions permits are handed out for free to emitters under a cap. A third, politically important, difference is in
terms of transparency of gains and losses to different affected parties. Under c-a-t it is easier to obscure these
distributional implications. This may be a political reason why many countries seem to opt for c-a-t solutions,
despite of the drawbacks pointed out.
3
on short-run demand and supply relations. In both models the exporter sets a fuel export tax. In
the first model, the policy bloc sets a carbon tax and offset price; and in the second model the
policy bloc sets an emissions cap. Dynamic issues, in particular the exhaustibility of fossil fuels
and the dynamics of climate change, are not studied. 4 Sinn’s (2008) “green paradox” argument
that carbon pricing could lead to increased emissions in the short run, is also not addressed. 5
The policy bloc establishes an offset scheme for inducing abatement in the fringe, with two
potential motivations. First, overall emissions can thus be further reduced, perhaps more cheaply
than through mitigation in policy countries alone. Secondly, reduced fuel demand in the fringe
may help reduce aggregate fuel demand and thus the fuel export price. Under c-a-t, I assume that
the market for quotas is competitive with the same trading price for the policy bloc and fringe.
Fringe country emitters are then paid an amount per abated emissions equal to the quota price
facing policy country emitters. Under a carbon tax, it is less obvious how an offset market should
be modeled. I assume that offsets are purchased from fringe countries by the policy bloc at a
given offset price, set by this bloc, which clears the offset market in fringe countries. This offset
price could, in principle, be either higher or lower than the carbon tax charged to policy bloc
emitters. Importantly also, we assume no informational problems in implementing offsets and
that all offsets are “additional”.
4
This requires a dynamic model for a more complete analysis. A large literature here exists; see e g Bergstrom
(1982), Karp (1984), Karp and Newbery (1991), Wirl (1994), Rubio and Escriche (2001), Salo and Tahvonen
(2001), Rubio (2005), Liski and Tahvonen (2004); and more recently Karp and Zhang (2010) and Wirl (2012).
5
The profile of future carbon taxes is here important. When the future carbon tax is expected to increase rapidly, an
increase in the general level of carbon taxation could induce Sinn’s paradox by raising emissions in the short run.
Also, if climate policy partly implies supporting development of backstops for replacing fossil fuels, emissions may
be worsened in the short run; see e g Strand (2007), Hoel (2010), and Ploeg and Withagen (2013).
4
The preference for taxes over c-a-t for fuel importers has been shown in earlier papers by Berger,
Fimreite, Golombek and Hoel (1992), and Berg, Kverndokk and Rosendahl (1997). Strand
(2011) considers two fuels, one imported (oil) and one produced by consumer countries.
Importers’ oil demand is then somewhat elastic under a cap, allowing some rent extraction by
importers. A tax here still dominates a cap for fuel importers. A possible objection to the current
model is its static nature; this issue is elaborated further in the final section, with reference to
follow-up work.
2. Model 1: The Policy Bloc Sets a Carbon Tax
2.1 Basics
Assume the following aggregate utility function related to fossil-fuel consumption for countries
with a climate policy (the “policy bloc”):
1γ 2
(1) W1 = R1 − R1 − pR1 − hcR − q1∆RF ,
2h
where γ > 0, and h identifies the relative size of the policy bloc in total fuel demand, while the
complement, 1-h, represents the relative size of the fringe. The policy bloc and fringe are
assumed to be identical apart from their relative “sizes”, represented by h. An increase in h
implies a proportional increase in both equilibrium fuel demand for given import prices, and in
the externality cost experienced from GHG emissions. An interpretation is that all individuals’
preferences are identical, and h represents the share of the global fuel-demanding population that
resides in the policy bloc. While not by itself very realistic (and see comments at the end of
section 5), this assumption facilitates the analysis of changes in relative sizes of policy bloc and
fringe, a highly policy-relevant issue.
5
p is the fossil fuel import price. R1 is the fossil-fuel consumption for the policy bloc, R is global
fossil-fuel consumption, while hc represents the climate externality cost per unit of global fossil
fuel consumption for policy bloc countries. Equation (1), and other demand and supply
functions, take “linear-quadratic” forms standard in the literature (and as in related work by
Strand (2011), Karp, Siddiqui and Strand (2013); and Wirl (2012)). We assume (with little loss
of generality) that fossil-fuel importers produce no fuels, and that producer countries consume no
fossil fuels and export all their production.
Let RF denote fuel consumption in the fringe. The last term in (1) represents an assumption that
the countries in the policy bloc are able to induce a reduction of the fringe’s fossil fuel
consumption and thus carbon emissions, below the “business-as-usual” level, through a subsidy
(or “offset price”) q1 to those units of fossil fuel consumption in the fringe with the smallest net
productive yield (implying that abatement in the fringe is efficient). This term gives the net
outlay by the policy bloc, related to incentive payments from the policy bloc to the fringe, given
that all offsets in the fringe are purchased at price q1, posted by the policy bloc. Such payments
represent a mechanism under a carbon tax scheme that corresponds closely to an offset market
under a c-a-t scheme (such as the CDM). q1 should however be interpreted somewhat differently
from the trading price under a c-a-t scheme. In particular, it need not be identical to the domestic
tax on emissions. Under c-a-t, market arbitrage will ensure identical carbon trading prices for all
carbon units, which does not necessarily hold in the tax case.
In model 1 the policy bloc uses two instruments. First, it sets an excise tax, t1, per unit of the
imported fossil fuel. This leaves the consumer fuel price in these countries at p+t1. Fossil fuels
are imported and consumed by many small competitive agents. The public demanding fossil
fuels in this group of countries maximizes
6
1γ 2
(2) V1 = R1 − R1 − ( p + t1 ) R1
2h
with respect to R1, yielding the demand level
1 − p − t1
(3) R1 = h .
γ
γ/h is the inverse demand sensitivity of fossil fuels with respect to price in the policy bloc.
The “fringe” of fuel-importing countries with no climate policy (with subscripts F) has aggregate
utility function (given no transfers from the policy bloc)
1 γ
(4) WF 0= RF 0 − RF 0 2 − pRF 0 − (1 − h)cR ,
2 1− h
where subscript 0 denotes no transfers. (1-h)c is the climate-related externality of global fossil
fuel consumption for the fringe. These countries in aggregate behave competitively. In the
absence of transfers these countries would maximize
1 γ
(5) RF 0 −
VF 0 = RF 0 2 − pRF 0
2 1− h
with respect to RF0, yielding the first-order condition
(1 − h)(1 − p )
(6) RF 0 = ,
γ
where γ/(1-h) is the inverse demand sensitivity for the fringe. The slope of the (global) aggregate
demand function is 1/γ.
7
The second instrument of the policy bloc is to pay a subsidy q1 per unit of offsets (“foregone
fossil fuel consumption”) in the fringe; i. e., the difference between the fuel consumption that
would have materialized had it not been for this subsidy, and actual fuel consumption in the
fringe. 6 The fringe thus reduces its fossil fuel consumption below the “benchmark” (6). The fuel
price at the margin for the fringe equals p+q1, where the policy bloc makes up this difference
through a subsidy to fringe fuel consumers. The induced fuel consumption in the fringe is then
1− h
(7) =
RF (1 − p − q1 ) .
γ
The subsidy is assumed to be paid only on the amount of fuel consumption avoided in the fringe
by the incentive payment, called ΔRF = RF0 – RF, given by
1− h
(8) ∆RF = q1 .
γ
Aggregate fossil-fuel demand, from both blocs combined, is now
1 − p − t1 1 − p − q1 1 − p − ht1 − (1 − h)q1
(9) R = R1 + RF = h + (1 − h) = .
γ γ γ
Assume a single (unified) producer country or region with aggregate utility function
(10) W2 =Π 2 + s1 R − c2 R ,
where П2 is net profit of its petroleum producers, s1R is excise tax revenue for fuel exporting
countries, while c2R denotes negative emissions externalities for the exporter bloc. Individual
fuel exporters are price takes with profit functions
6
This requires full additionality; see comments in section 5.
8
1
(11) Π 2 = ( p − s1 ) R − p0 R − φ R 2 ,
2
where p0 is a lower bound on marginal fuel extraction cost. Maximizing (11) with respect to R
yields the fossil-fuel supply function
(12) p = p0 + s1 + φ R .
φ (> 0) represents the (inverse) supply sensitivity of petroleum output.
The externality cost of one unit of carbon emissions is hc for the policy bloc, and (1-h)c for the
fringe. Individual fringe countries are small and ignore this factor in their own decisions. The
global externality cost per fossil fuel unit equals c+c2, which would correspond to a Pigou tax
imposed by a benevolent global regulator, given that markets are otherwise competitive.
Solving (9) and (12) for R and p as functions of the tax parameters t1, q1 and s yields
1 − p0 − s1 − ht1 − (1 − h)q1
(13) R= ,
γ +φ
γ φ
(14) =p ( p0 + s1 ) + (1 − ht1 − (1 − h)q1 ) .
γ +φ γ +φ
We derive fuel demand for each bloc as functions of s, t1 and q1, as follows:
h
(15) =R1 [γ (1 − p0 − s1 ) − (γ + (1 − h)φ )t1 + (1 − h)φ q1 ]
γ (γ + φ )
1− h
(16) =RF [γ (1 − p0 − s1 ) + φ ht1 − (γ + φ h)q1 ] .
γ (γ + φ )
9
A higher q1 increases R1 (but lowers R), since p is reduced thus incentivizing higher policy bloc
fuel demand. Indeed, this is the basic purpose for the policy bloc of “subsidizing offsets” in the
fringe.
2.2 The policy bloc solution
An authority representing the entire policy bloc sets t1 and q1 to maximize W1 in (1), considering
its own fuel demand response (15), the aggregate fuel demand response (13), and the export
price response (14), to changes in t1 and q1; while the exporter tax, s1, is taken as exogenous. The
solution concept is (static) non-cooperative Nash Equilibrium (NE) where the policy bloc sets
carbon tax and offset price, and the exporter bloc sets a fuel export tax; each taking the other’s
strategy variable(s) as exogenous.
Appendix 1 now shows the following result.
Proposition 1: The static NE solution in t1 and q1 for the policy bloc is given by the following
solutions, expressed as functions of s1:
2hγ
(17) =t1 [φ (1 − p0 − s1 ) + (γ + φ )c]
2(γ + φ ) − h(1 + h)φ 2
2
hγ 1
2 [
=
(18) q1 φ (1
= − p0 − s1 ) + (γ + φ )c ] t1 .
2(γ + φ ) − h(1 + h)φ
2
2
q1 < t1 is here a general result. More specifically, our ratio of one-to-two follows from the linear-
quadratic structure of the model and is not general. When h is small, t1 and q1 are both small: the
policy bloc then neither charges high domestic carbon taxes nor induces much offsets in the
fringe, for two separate reasons. First, with low h, the climate externality for the policy bloc, hc,
10
and the related “Pigou” tax, is small. Secondly, market power of the policy bloc in the fossil fuel
market is then small.
The reason why the offset incentive price, q1, is lower than the domestic carbon tax t1 in the
policy bloc, is that offset payments go to foreigners, which reduces the attractiveness of foreign
offsets, relative to domestic mitigation via the carbon tax t1. The policy bloc acts as a
monopsonistic purchaser of offsets from fringe countries (which act non-cooperatively), and
limits its offset purchases to maximize its net return from such purchases.
When the offset price instead is exogenous (and not necessarily optimal), we have
h
(19) =t1 [γ (γ + φ )c + γφ (1 − p0 − s1 ) + (1 − h)φ 2 q1 ] .
(γ + φ ) 2 − h 2φ 2
When q1 is higher (corresponding to a larger volume of offsets), t1 is set higher in response.
Intuitively, more offsets in the fringe leads to a lower fuel export price which makes a higher
carbon tax in the policy bloc advantageous.
2.3 The exporter solution
The exporter bloc maximizes bloc welfare, W2, with respect to its fuel export tax s1, given the
supply function (12) from its individual producers, and the price relation (14). Still invoking the
NE concept, t1 and q1 are taken as given by the exporter.
I now solve for s1, p and R, as functions of the carbon tax t1 and offset subsidy rate q1 set by the
policy bloc. The following result emerges, shown in Appendix A.
Proposition 2: The static NE solution for s1 is the following function of t1 and q1:
γ γ +φ
(20) =s1 (1 − p0 − ht1 − (1 − h)q1 ) + c2 .
2γ + φ 2γ + φ
11
As expected, s1 is reduced in response to increased t1 and q1. This effect is stronger for t1 (q1)
when h > (<) ½, so that the fraction of the policy bloc in total fuel demand is greater (smaller)
than one half. Interestingly, when offsets are used (q1>0), the exporter’s fuel tax is lower.
2.4 Overall Nash Equilibrium
Simultaneously solving (17), (18) and (20) for t1, q1 and s1 gives the following result.
Proposition 3: The simultaneous, static, NE for the non-cooperative tax-setting game between
the policy bloc and the fuel exporting bloc is characterized by
2hγ
(21) =t1 [φ (1 − p0 − c2 ) + (2γ + φ
= )c] 2q1
D1
(22) =s1
1
D1
{[2γ (γ + φ ) − h(1 + h)γφ ](1 − p0 ) + [2(γ + φ ) 2 − h(1 + h)φ 2 ]c2 − h(1 + h)γ 2 c} ,
where
D1 = (γ + φ )[2(2γ + φ ) − h(1 + h)φ ] .
To interpret these expressions we rely on simulations based on simplifying parametric
assumptions; and comparative-static results; all presented in section 4.
Two further features are considered. First, the “incentivized emissions” level describes net
emissions as an outcome of incentive mechanisms applied (t1, and q1), both in the policy bloc
and in the fringe. This emission level corresponds notionally to a cap discussed under model 2 in
the next section. It can be defined by
1 1
(23) = R1 − ∆RF=
RI 1 h(1 − p ) − (1 + h)t1 .
γ 2
12
This magnitude is simulated in section 4, in some parametric cases. In all cases studied, RI1 > 0
(optimal emissions in the policy bloc always exceed emissions offset by the fringe).
The second feature is abatement in the policy bloc versus fringe (in terms of reducing R), via the
tax and offset policies applied. From (13) (noting that t1 = 2q1), R is reduced more (less) in the
policy bloc than in the fringe given that h > (<) 1/3. Thus in particular, when h = 1/3 it is optimal
for the policy bloc to implement equally much abatement in each of the two blocs.
3. Model 2: The Policy Bloc Uses Cap-and-Trade
3.1 Basics
In the second model, the policy bloc sets a cap on its emissions, still taking fringe demand as
exogenous. This cap can be achieved in part through offsets purchased from the fringe by the
policy bloc, which give room for higher emissions within the policy bloc for a given cap. Call
the cap RP, and the amount of offsets RFP. Emissions by the policy bloc, R1, are then given by RP
+ RFP (offsets allow for policy-bloc emissions above the cap). Denote fringe emissions by RF,
and fringe emissions in the (counterfactual, but here still well defined) case with no offsets by
RF0; then RF = RF0 – RFP. Total emissions, R, are given alternatively as R1 + RF, or RP + RF0. I
assume free trading of emissions rights within the policy bloc at a single quota price t2 (applying
parallel symbols with the tax case). As a condition for offset market equilibrium in the quota
market, emission offsets need to be purchased from the fringe at price t2: domestic emitters in the
policy bloc must be indifferent between abating one unit of emissions, and purchasing one unit
of offsets whereby abatement is avoided.
The equilibrium concept is simultaneous NE where the policy bloc determines the quantity of
fuel demand and the exporter sets the fuel export tax. In calculating its optimal fuel demand
13
(including any offsets), the policy bloc takes the export tax as given, as in model 1. The main
difference from model 1 is that the exporter, in setting its optimal export tax, takes net fuel
demand by the policy bloc (incorporating any demand reduction induced in the fringe by the
offset policy), and not the carbon price, as given.
I put two constraints on the number of allowable offsets, RFP. First, RFP must be non-negative.
Secondly, RFP cannot exceed abatement in the fringe, given a uniform carbon price t2 enforced in
the fringe. A carbon price no greater than t2 would then implement the offset quota RF0. As under
model 1, I assume efficient offsets: for emissions that are offset through incentive payments from
the policy bloc to the fringe, the mitigation cost is lower than for any one unit of residual
emissions (where offsets are not taking place). 7 We can treat the strategy of the policy bloc as
setting the quota trading price of emission rights (or tax) within the bloc, which is dual to the
quantity solution. The basic strategy of the fringe is also the same in this case as under policy.
A consequence is that the amount of offsets in the fringe, to be financed by the policy bloc, is
still given by (8). The offset price is however different here. Use of c-a-t (and with no “quota
discounts” for offsets as discussed in the final section) requires the offset price in the fringe to
equal the domestic quota price in the policy bloc countries (equivalent to the tax t2) as a
condition for market clearing in the quota market. In the last expression in (1), q2 is then replaced
by the domestic trading price in the policy bloc, t2. No similar constraint on the offset trading
price was imposed in model 1, where the policy bloc implemented offsets directly via transfer
payments to fringe countries, and the offset trading price could be set freely.
3.2 Importer solution
7
This is not an obvious outcome, since the unit incentive pay by policy bloc emitters to the fringe, t2, is generally
higher than the carbon price that would otherwise implement the actual offsets taking place in the fringe.
14
(10)-(16) from model 1 are still valid. The fuel demand functions of the policy bloc and the
fringe, as viewed by each, are still given by (3) and (6), where t in (3) is interpreted as the quota
price within the c-a-t scheme in the policy bloc. Policy-bloc fuel demanders (and emitters) still
maximize profits given the fuel price including the quota price, serving as a “tax” within the
policy bloc. The strategy of the policy bloc itself can also be viewed as very similar to model 1.
This is because the tax and c-a-t solutions are formally identical for fuel consuming countries
facing a given fuel import price, under full certainty and with full auctioning of emissions
quotas. (15) can be interpreted, alternatively, as the condition for optimal energy demand R1, or
for optimal quota price t, in either case taking the fuel import price, p, from (12), and s as
exogenous.
Proposition 4: When the importer bloc uses a c-a-t policy with optimal offsets and assuming free
offset trading, the constrained optimal quota price, t2, expressed in terms of the exporter tax, s2,
equals the offset price and is given by
h 2γ
(24) =t2 {φ (1 − p0 −
= s2 ) + (γ + φ )c} q2 .
(γ + φ ) 2 − h 2φ 2
Moreover, the solution entails
(25) p + t2 =
m
where m is the marginal productivity of energy use in both the policy bloc and the fringe.
Proof: See the appendix.
The appendix demonstrates that the constrained optimal solution for the importer takes the form
of a corner solution where a maximum number of offsets is utilized for the given offset price,
15
which equals the carbon trading price within the c-a-t scheme in the policy bloc. This result has a
simple intuitive explanation: For a given cap (which effectively constrains energy consumption
within the policy bloc) and given that offsets and quotas are sold at the same price, the policy
bloc wishes to consume as much fossil energy as possible which means that it uses offsets to the
maximum extent (for given offset price). This is different from Section 2, where the offset price
was independently optimized, and set lower than that tax. The offset amount was then also lower
for any given (policy bloc-internal) carbon price.
Comparing (24) to (17), t2 < t1 (for any given export tax s); but the difference is small when h is
close to one. When h is low, by contrast, the difference is greater; and t2/t1 tends to zero as h goes
to zero.
All net offset market rent is here captured by fringe emitters. This follows from the assumption
of perfect competition and free arbitrage in the offset market, so that all units in that market
(whether domestic in the policy bloc or purchased from the fringe) need to be traded at a uniform
price.
Offsets are in general more costly to the policy bloc in this case than under model 1 (where they
could be bought at a “discount” relative to the domestic carbon tax t1); here they must be paid at
full cost t2. But this also serves to reduce the internal carbon price within the policy bloc, t2,
below t1 in model 1.
3.3 Exporter solution
For fuel exporters, c-a-t is more dramatically different from a carbon tax solution for the policy
bloc. Exporters no longer face an importer tax, but instead a cap by the policy bloc, in amount
RP. Instead of (9), the exporter faces the aggregate fuel demand function
16
1− p
(26) R = R1 + RF = RP + RF 0 = RP + (1 − h)
γ
RP, the emissions cap set by the policy bloc, includes possible offsets purchased by the policy
bloc from the fringe, so that actual policy bloc emissions may exceed RP by the amount of
offsets. The exporter now takes RP as fixed, and only the baseline demand by the fringe, RF0, as
variable. We have the following result, shown in the appendix:
Proposition 5: When the policy bloc chooses a c-a-t solution, the optimal strategy of the exporter
bloc is to set its export tax s according to
γ (1 − h)(γ + φ )
=
(27) s2 (1 − p0 − ht2 − (1 − h)q2 ) + c2 .
(2 − h)γ + (1 − h)φ (2 − h)γ + (1 − h)φ
Comparing (27) to (20), we find s2 > s1 (from model 1) for any given t2 and q2. The exporter is
(much) more aggressive in setting its export excise tax when the importer chooses a c-a-t policy,
than when it chooses a tax policy. The difference is greater when h (the share of the policy bloc
among all fuel demanders) is larger. Simulations, discussed in section 4 below, also indicate that
s2 > s1 more generally.
The export price, p, can be expressed as
γ + (1 − h)φ (1 − h)γ
(28) =p − p0 (1 − p0 − ht2 − (1 − h)q2 ) + c2
(2 − h)γ + (1 − h)φ (2 − h)γ + (1 − h)φ
Comparing to the carbon tax case, p is greater here for any given t (where t = t1 is the carbon tax
in model 1, and t = t2 the quota price in model 2). We find:
17
dp γ2 γ + (1 − h)φ d (ht2 )
=
(29) (1 − p0 − ht2 − (1 − h)q2 − c2 ) −
dh [γ + (1 − h)(γ + φ )]2
γ + (1 − h)(γ + φ ) dh
This expression is always positive for given ht2 (so that t2 falls proportionately). But it is also
positive when ht2 increases in h, provided that the first term dominates the second. This is always
so when h is initially small; t2 is then also small (from (30) below); and d(ht2) must consequently
be small. We thus find that when h is small at the outset, the export price always increases when
the policy bloc comprises a larger fraction of total fuel demand (h increases). This is
diametrically opposite to the conclusion under model 1, where the policy bloc used a carbon tax.
We find, in the simulations in section 4 below, that p can increase in h, also for larger h values
(when c is low).
It is important to stress that the behavior of the policy bloc is formally identical in the two
models (except for the constraint t2 = q2 only in model 2); despite the fact that the policy bloc
sets a tax in the first, and a cap in the second model. It occurs because the two problems,
maximizing with respect to the cap, and to the tax, yield the same result for given behavior of the
exporter. It follows from duality of the tax and cap solutions: both implement the same allocation
under competitive conditions. The difference between the two models lies in the response of a
non-competitive exporter to the chosen policy bloc strategy, tax or cap.
3.4 Overall Equilibrium
Overall equilibrium is found by solving (24) and (27) for t2 and s2 (noting that q2 = t2).
Proposition 6: Given that the importer bloc uses a c-a-t policy with free offset trading, and the
exporter sets an optimal fuel export tax, the static NE (t2, s2) combination is given by
18
h 2γ
(30) =t2 {[(2 − h)γ + (1 − h)φ ]c + (1 − h)φ (1 − p0 − c2 )}
D2
(1 − h)[(γ + φ ) 2 − h 2φ 2 ]c2 + γ [γ + (1 − h 2 )φ ](1 − p0 ) − h 2γ 2 c
(31) s2 =
D2
where
D2 = (γ + φ )[2γ + (1 − h 2 )φ ] − h[(γ + φ ) 2 − h 2φ 2 ] .
t2 here tends to c as h tends to one. t2 is always rising in h for low h, but could fall or rise for
larger h; this is found in comparative static results, and in simulations in section 4 below.
s2 = s1 from model 1 for h = 0 and no climate policy. When h > 0, s2 > s1 and more so when h is
higher. The exporter adopts a more aggressive taxation strategy the higher is h, since the fuel
demand elasticity faced by the exporter is lower (as less of fuel demand is variable).
Consider implications of the overall solution for the optimal “cap” to be set, analogously to the
amount of “incentivized emissions” (from (23)) under model 1. The optimal cap, RC2, is
1
(32) = R1 − ∆RF=
RC 2 [h(1 − p ) − t2 ] .
γ
Simulations in section 4 show that RC2 > 0 independent of h. This is similar to what was found in
model 1. The intuition is also here that for low h (where, conceivably, the cap could be negative)
the emissions price is too low to really matter in terms of emissions reductions.
Compare also here the amounts of abatement taking place in the policy bloc versus in the fringe.
From (13), (15) and (16) and inserting q2 = t2 we now simply have
19
h
(33) =
R1 = hR
(1 − s − t2 )
2
1− h
(34) RF = (1 − s − t2 ) = (1 − h) R .
2
Thus in this case fuel demand in the policy bloc and fringe are proportional to bloc size. This is
simply a consequence of fuel consumption being efficiently allocated across fuel-consuming
countries in this case, from (25). 8
4.Comparative Statics with Simulations
I will now discuss some key comparative-static results given changes in two key exogenous
parameters, h (the share of fuel importers with a climate policy) and c (the global externality of
GHG emissions for fuel importers). 9 These are illustrated by model simulations in a simple
numerical example where p0 = c2 = 0 (the exporter’s fuel supply function has intercept at zero,
and no negative climate impacts on the exporter), and γ = φ = 1 (demand and supply functions
for fossil fuels are equally sloped). 10 In figures 1-8, all model 1 variables are in blue (except q1
which is in green), and all model 2 variables are in red. Figures 1-3 illustrate solutions as
functions of h (the fraction of the fuel demand market represented by the policy bloc), for three
alternative values of c: = 0 (no climate concern); = ¼ (“medium” climate concern); and = ½
(“high” climate concern). Five variables are shown in figures 1-3: clockwise from upper left t
and q (the carbon tax and offset price); s (the exporter fuel tax); p (fuel export price); and R
8
This is an idealized model where the offset markets are assumed to function perfectly; in particular, all units of
excess emissions in the fringe are perfectly offset using the offset price t2. This is clearly not realistic; see the
discussion of this issue in the conclusion below.
9
I thank Sauleh Siddiqui for invaluable help in creating the simulations and figures.
10
The exact comparative-static results are not reproduced; they can be obtained from the author upon request.
20
(consumed amount of the resource); all as functions of h (the policy bloc as share of fuel-
demanding countries).
When c = 0 (figure 1), everything is driven by strategic concerns. As noted, q1 = ½ t1. Both are
higher than q2 (= t2). While t1 and q2 (= t2) differ also for small h, the difference is greater for
larger h. 11 The carbon price of the policy bloc is dramatically higher under a carbon tax than
under c-a-t when h is high. While the carbon tax increases strongly in h, the quota price in the c-
a-t case also increases in h up to a certain point, but is reduced when h increases further. Two
factors give opposite effects on t2: a higher h makes the policy bloc more collusive and more
aggressive in its pricing; but a higher h also makes the exporter (much) more aggressive which
reduces the scope for rent extraction by the policy bloc. Interestingly, the policy bloc’s carbon
price is always positive even as there is no climate concern in this case.
----- Figure 1 in about here -----
Fossil fuel consumption drops in h, only slightly in the tax case, and more dramatically in the cap
case, as the exporter price then increases drastically. The tax case is “good” for importers as the
import price is substantially reduced when h increases.
Figures 2 and 3 (“intermediate” and “high” climate concern) differ more from figure 1 as h
grows. This is because a climate concern of the policy bloc affects policy very little when h is
small, but much more when h is high: the carbon tax and quota price are then also much higher.
In particular, the quota price under c-a-t rises uniformly in h. q2 now (slightly) exceeds q1 for
high h (greater than about 0.65 for c = ¼; and greater than about 0.6 for c = ½). This more
11
As noted, however, the model is less suitable for describing what happens under a cap solution for high h values.
21
aggressive carbon pricing strategy of the policy bloc leads to a greater reduction in both the
import price, and total fossil fuel consumption, when h increases.
----- Figures 2 and 3 in about here -----
Generally, t1 > t2 for all values of h and c. This difference is however not uniformly greater when
h is higher. Two main factors explain this. First, s2 > s1 always, and the difference is greater
when h is higher. The importer’s optimal response is to set t1 > t2 (when the exporter tax, and
thus the export price, is lower). 12 The second factor is related to the functioning of the offset
market. In the tax case, under model 1, the domestic carbon tax in the policy bloc is independent
of the offset price (effective within the fringe); there is no direct effect of the offset market on the
domestic carbon tax. In model 2, by contrast, the offset carbon price must equal the carbon price
within the policy bloc. This puts downward pressure on the carbon price when the policy bloc is
small and the fringe is large (h small), and the offset market is a large share of total abatement.
When the fringe is small (h large), by contrast, this factor is less important (as the offset market
is also less important). There is then less downward pressure on the carbon price from an offset
market under the cap.
These two factors work in opposite directions with respect to Δt = t1 - t2 when h increases. When
c = 0, Δt increases strongly in h. When c = ¼, Δt increases but more slowly over a large range for
h. When c = ½, Δt is reduced for higher h.
12
In the limit as h tends to unity, under a cap the NE solution in this model entails the exporter setting the export
price at its maximal level choking off demand. The carbon quota price is then equal to zero. This is an unrealistic
economic model; see Strand (2010) for elaboration and discussion of alternative equilibrium concepts.
22
As noted, t1 = 2q1, while t2 = q2. Still, q2 < q1 for low h and/or low c; but q2 > q1 when both h and
c are high. When h is low, the dominating factor is the “drag” (toward low carbon prices) from
the constraint t2 = q2 under c-a-t. When h is high, by contrast, the offset market is small and
pricing in that market means little for efficiency within the policy bloc. When c is high (and the
quota price high under c-a-t for high h), q2 = t2 then implies q2 > q1 in such cases.
The more general comparative-static results show that t1 rises, while both s1 and R in model 1
fall, when both h and c increase. 13 The same holds when c changes in model 2. Effects of h on t1
and s2 as h are slightly more complex. When c is small, dt2/dh > 0 always when h is small; and
dt2/dh < 0 always when h is large. When c is high, t2 rises uniformly in h. s2 always rises in h
when c is small, and always falls in h when c is high and h already high. This confirms the main
results from the simulations in figures 1-3.
Figure 4 shows “incentivized quotas” RI1 in (23), and RC2 in (32) for the simulated example.
Under my numerical example these two expressions are found as
h 1
(23a) RI 1= 1 − s1 − (3 − h)t1
2 2
(32a) RC 2 = h(1 − s2 − R) − t2 .
RC2 corresponds to the optimal quota (accounting for offsets) set by the policy bloc in model 2.
RI1 has a similar interpretation in model 1 (except that a government-managed offset scheme is
here assumed instead of a free offset trading scheme). “Optimal quotas” are similar in the two
models for low h (slightly greater in model 2); but are much smaller in model 2 for high h.
13
Analytical results can be obtained from the author upon request.
23
----- Figure 4 in about here -----
Figures 5-7 show welfare levels for the policy bloc, fringe and exporter bloc under models 1-2.
Most strikingly, for both policy bloc and fringe, utility is everywhere higher in model 1 than in
model 2. As is easily verified, this result holds generally and does not rely on the chosen
parameter values. For the policy bloc, the difference in outcome under the two models grows
with h, as seen from the figures. In a relevant example, h = ¼ (as for Annex B under the Kyoto
Protocol), the difference in utility for the policy bloc is small, and it matters little to the policy
bloc whether a tax or a c-a-t solution is chosen. When the policy bloc is larger, the difference can
be large, and the choice of policy regime a major concern for the policy bloc. 14 Exporter welfare
is greater with c-a-t then with carbon taxes, for h values up to a maximum point. For even higher
h values, the aggression in exporter fuel price setting (as fuel demand becomes less elastic)
backfires, as fuel output tends to zero when h tends to one.
----- Figures 5-7 in about here -----
We finally simulate net welfare of fringe countries due to offsets (disregarding climate effects),
expressed by BF, which takes the form
1− h 2
(35) BF = q ,
2
where q is the generic offset price, given by t1/2 from (21) under model 1, and by t2 from (30)
under model 2. Simulations for three alternative values of c (= 0, ¼, and ½) are shown in figure
8. For moderate values of h (< 0.6) the welfare gain to the fringe from an offset market is
14
For such c values or higher, utilities tend to be negative in the model, for both fuel-consuming blocs. This is due
to the large negative externalities from emissions. It would then have been better to have no fossil fuels available at
all. This is not realistic, but follows from my assumption that the choke price of fossil fuels is quite low (= 1 and
thus only twice the externality value for c = 0.5), and the demand function linear.
24
everywhere greater under a tax than under c-a-t. This mirrors results for q from figures 1-3:
wherever q1 > (<) q2 welfare gains from offsets are greater (smaller) under model 1.
----- Figure 8 in about here -----
5. Conclusions and Final Comments
This paper has analyzed a carbon tax versus an emissions cap with free quota trading (c-a-t) as
alternative climate policy strategies for a fossil fuel-importing “climate policy bloc” facing a
fuel-importing group of countries (a “fringe”), with no climate policy, and a fuel-exporting bloc
which sets its fuel export tax optimally. The optimal carbon price in the policy bloc is influenced
by both a climate (“Pigouvian”) motive, and a strategic motive whereby the policy bloc
influences the exporter’s fuel price through its tax or cap. A positive carbon tax leads to a lower
fuel import price, which benefits all fuel importers including the fringe. The tax is set higher
when the policy bloc is larger, for two reasons: the “Pigou” element is then greater; and the
strategic element, whereby the tax reduces overall fuel demand and the fuel export price, is
greater.
I find that a carbon tax is always preferred over a c-a-t policy by fuel importers, mainly because
fuel exporters charge a lower fuel price under a carbon tax than under c-a-t. Under c-a-t, once a
cap has been set, overall fuel demand is less sensitive to the fuel export price than under a carbon
tax. This gives a monopolistic exporter an incentive to set its export price higher under c-a-t than
under a carbon tax, and this hurts all importers. Having an offset market is also more
advantageous to the policy bloc under a carbon tax in my model, because the offset price can be
set lower than the tax, while unified trading makes price differentiation between domestic quotas
and offsets infeasible under c-a-t. The fringe also fares better under a carbon tax, and more so
25
when the policy bloc is larger and the fringe smaller. A small fringe benefits more from being a
“free rider” on a (relatively high) carbon tax set by the larger policy bloc, which pushes the fuel
export price down.
To my knowledge this paper represents the first attempt in the literature to analyze optimal offset
policies in the context of a global model of strategic GHG mitigation policy. Being a first
attempt, the analysis is stylized. Offset markets are assumed to be fully efficient, with the “best”
projects always implemented, and all offsets additional. My assumption that when the fringe
faces an offset price equal to q, the mitigation in the fringe is the same as it would be when
facing this level of carbon tax, is not realistic. Several problems with the CDM have been
uncovered, including lack of additionality (many projects would have been implemented even
without CDM financing; see Hagem (1996), Fischer (2005); Flues, Michaelowa and Michaelowa
(2010)), manipulation of baselines (Wirl, Huber and Walker (1998), Fischer (2005), Strand and
Rosendahl (2012)), and leakage (Rosendahl and Strand (2011)); which all limit the global
abatement effects of CDM projects.
Offset markets are assumed to be designed differently in the two cases. In the carbon tax case
offsets are purchased directly by a central authority, with a single offset price that may differ
from the carbon tax. In the c-a-t case, by contrast, the offset mechanism is market-based and of a
standard CDM type. In my view these are logical ways of organizing an offset market in the two
cases; but they are not the only ways. In one sense, my model can be seen as “stacked against”
the c-a-t solution and in favor of the tax solution by allowing the policy bloc to price
discriminate in the offset market only in the tax case. One might alternatively consider ways in
which to differentiate the “domestic” and “foreign” markets under c-a-t, allowing for different
prices in the two sub-markets also with private trading. Some authors, including Castro and
26
Michaelowa (2010), and Klemick (2012), have discussed offset “discounts” (the purchaser of an
offset can increase its emissions by only a fraction of the purchase), which might be preferable
when offsets are not fully additional. Bargaining over net returns from individual CDM projects
might also be relevant. With project bargaining, market arbitrage will no longer hold for offset
quotas, which will be priced lower, as discussed by Bréchet, Ménière and Picard (2011) who
stress that a greater bargaining power to project sponsors makes offsets more attractive for the
policy bloc. One may also question whether a totally separate sub-market for offsets (managed e
g by governments) can provide the same degree of efficiency as an integrated, private, domestic
and foreign c-a-t market. The wider implications of alternative offset market mechanisms should
remain as a priority topic for future research, given that non-policy countries continue to play
important roles for global GHG mitigation. In realistic cases, offsets might then be found to
represent a less attractive option for a policy bloc, so that their optimal volume is smaller.
However, given a large fringe, offsets might still play a significant role.
Considering the overwhelming evidence in favor of tax over cap solutions, in this and most other
modeling contexts, a natural question is, why are c-a-t solutions at all used? I will not enter a
deep discussion, only note that in various policy contexts there are strong biases against solutions
that put a direct price (or tax) on energy resources, and in favor of more roundabout and indirect,
and less efficient, solutions. Heavy political economy issues often seem to lie behind: in the U.S.
and many other countries, biases against taxes in general; in the EU, succumbing to political
pressure from industry lobbies to spare energy-intensive sectors from any burden of the EU-ETS
(demanding c-a-t systems with high levels of free allocations); and in many other countries
strong populist pressure not to tax (but often rather subsidize) fuels and electricity. Hopefully,
this analysis can serve as an additional reminder that energy tax solutions can be highly
27
beneficial, and where one key additional argument (relative to standard ones) is offered, namely
the effects on fuel price setting and thus terms of trade which, independently of other factors,
make tax solutions unambiguously favorable for fuel importers.
My model is highly simplified and could be changed or expanded in other ways than those
already discussed. I here only briefly mention a few possibilities.
A) Static analysis, while fossil energy extraction and climate change are both inherently dynamic
processes. A key question is whether main results, such as preference of a tax over a cap in
climate policy for fuel importers, carries over to a dynamic context. Some ongoing research
indicates that they do. Wirl (2012) has recently shown that taxes are chosen over caps in a
corresponding dynamic model of two monolithic blocs but no fringe. 15 Work is also in progress
on the (analytically more intriguing) dynamic case with a fuel importing fringe. See Karp,
Siddiqui and Strand (2013), with results pointing in the same direction. 16
B) No fuel production in consuming countries, and no fuel consumption in fuel-producing
countries. Changing this would eliminate some extreme cases (including non-existence of
positive resource extraction under c-a-t for h = 1), but would otherwise leave main results intact.
C) Only one fuel. Strand (2011) considers two differentiated fuels in a model of two blocs only
(exporters and importers). This changes conclusions slightly, by making the ex post fuel demand
function more elastic under c-a-t. While this increases the attractiveness of the c-a-t solution, the
basic preference for taxes still remains.
15
Wirl (2012) shows that taxes are dominant dynamic instruments also for exporters, as we have assumed here.
16
The examples simulated in this paper show that the exporter always chooses a tax; and that the policy bloc in
response chooses a tax for at least a long initial period (50-100 years). Note that neither Wirl (2012) nor Karp,
Siddiqui and Strand (2013) include offset markets. A related, much earlier unpublished paper is Karp (1988).
28
D) A monopolistic fuel exporter dictates the fuel export price; and there is no fringe of
competitive fuel suppliers. Starts of analysis incorporating competitive fringe fuel supply are
found in Keutiben (2010), and Karp, Siddiqui and Strand (2013).
E) All fuel-importing regions are equally averse to climate change; and have equal utility loss
(relative to population size) per unit of carbon emissions. With different loss parameters,
countries’ incentives to join the policy bloc will generally vary among countries. The climate
impact factor for the policy bloc, here hc, will then also be a more complex function of h.
F) Climate costs are linear in total emissions. More plausibly, climate costs are strictly convex in
emissions. With a quadratic cost function we would get an additional quadratic term in (1),
reducing the optimal t1 and t2; and more so the larger is the policy bloc (and h). In other respects,
however, little would change. Also, since climate is a slow-moving variable, convexity is likely
not to have a serious impact on policy when considering only relatively short periods (such as a
year). In dynamic long-run models (Wirl (2012) and Karp, Siddiqui and Strand (2013))
convexity plays a larger role, making policy more restrictive over time.
29
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Simulations: Figures 1-8
Figure 1: Carbon prices, export tax, import price and fuel demand, as
functions of h for c = 0
Models 1&2 c =0 Models 1&2 c =0
Carbon Tax Export Tax
0.4 1
0.3 0.8
t,q (Green)
0.2 0.6
s
1
0.1 0.4
0 0.2
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0 Models 1&2 c =0
Aggregate Demand Import Price
0.4 1
0.3
0.8
0.2
R
p
0.6
0.1
0 0.4
0 0.5 1 0 0.5 1
h Model 1 Model 2 qh
1
33
Figure 2: Carbon prices, export tax, import price and fuel demand, as
functions of h for c = 0.25
Models 1&2 c =0.25 Models 1&2 c =0.25
Carbon Tax Export Tax
0.8 0.8
0.6 0.6
t,q (Green)
0.4 0.4
s
1
0.2 0.2
0 0
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0.25 Models 1&2 c =0.25
Aggregate Demand Import Price
0.4 1
0.3 0.8
0.2 0.6
R
p
0.1 0.4
0 0.2
0 0.5 1 0 0.5 1
h Model 1 Model 2 q1h
34
Figure 3: Carbon prices, export tax, import price and fuel demand, as
functions of h for c = 0.5
Models 1&2 c =0.5 Models 1&2 c =0.5
Carbon Tax Export Tax
0.8 0.8
0.6 0.6
t,q (Green)
0.4 0.4
s
1
0.2 0.2
0 0
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0.5 Models 1&2 c =0.5
Aggregate Demand Import Price
0.4 1
0.3 0.8
0.2 0.6
R
p
0.1 0.4
0 0.2
0 0.5 1 0 0.5 1
h Model 1 Model 2 qh
1
35
Figure 4: Size of “cap quota” under carbon tax and c-a-t scheme, as function
of h, for different c values
Models 1&2 Models 1&2
c=0 c = 0.25
0.4 0.2
0.3 0.15
C2
C2
R ,R
R ,R
0.2 0.1
I1
I1
0.1 0.05
0 0
0 0.5 1 0 0.5 1
h h
Models 1&2
c = 0.5
0.2
0.15
Model 1 (RI1)
C2
R ,R
0.1 Model 2 (RC2)
I1
0.05
0
0 0.5 1
h
36
Figure 5: Welfare of (importer) policy bloc, fringe, and exporter bloc, as
functions of h, for c = 0
Models 1&2 c =0 Models 1&2 c =0
Policy Bloc Fringe
0.1 0.06
Utility (W )
Utility (W )
F
1
0.04
0.05
0.02
0 0
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0
Exporter Bloc
0.2
0.15
Utility (W )
2
Model 1
0.1 Model 2
0.05
0
0 0.5 1
h
37
Figure 6: Welfare of importer policy bloc, fringe, and exporter bloc, as
functions of h, for c = 0.25
Models 1&2 c =0.25 Models 1&2 c =0.25
Policy Bloc Fringe
0.06 0.02
0.04
Utility (W )
Utility (W )
F
1
0
0.02
-0.02
0
-0.02 -0.04
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0.25
Exporter Bloc
0.2
0.15
Utility (W )
2
Model 1
0.1 Model 2
0.05
0
0 0.5 1
h
38
Figure 7: Welfare of importer policy bloc, fringe, and exporter bloc, as
functions of h, for c = 0.5
Models 1&2 c =0.5 Models 1&2 c =0.5
Policy Bloc Fringe
0.05 0.05
0
Utility (W )
Utility (W )
F
1
0 -0.05
-0.1
-0.05 -0.15
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0.5
Exporter Bloc
0.2
0.15
Utility (W )
2
Model 1
0.1 Model 2
0.05
0
0 0.5 1
h
39
Figure 8: Welfare effect to fringe of offset market under models 1 and 2, for
alternative values of h and c
Models 1&2 c =0 Models 1&2 c =0.25
-3 -3
x 10 Benefit to Fringe x 10 Benefit to Fringe
1 4
Offset Benefit
Offset Benefit
3
0.5 2
1
0 0
0 0.5 1 0 0.5 1
h h
Models 1&2 c =0.5
Benefit to Fringe
0.01
Offset Benefit
Model 1
0.005 Model 2
0
0 0.5 1
h
40
Appendix: Analytical results and proofs
Proof of Proposition 1:
Maximizing (1) with respect to t1 and q1, given (13)-(15), yields the following set of first-order
conditions for the policy bloc:
dW1 ∂R ∂p ∂R
(A1) =(1 − γ 1 R1 − p ) 1 − R1 − hc =0
dt1 ∂t1 ∂t1 ∂t1
dW1 ∂R ∂p ∂R 1− h
(A2) =(1 − γ 1 R1 − p ) 1 − R1 − hc −2 q =0,
dq1 ∂q1 ∂q1 ∂q1 γ 1
where we recognize from (3) that
1 − γ 1 R1 − p =
t1 .
(A1)-(A2) solve simultaneously for t1 and q1, and with the respective partial derivatives are
found from (13)-(15). Together this yields (17)-(18). Q.E.D.
Proof of Proposition 2:
The first-order condition for the exporter bloc is
dW2 1 γ
(A3) = ( p − p0 − φ R − c2 ) − + R = 0,
ds γ +φ γ +φ
which yields:
p − p0 − c2
(A4) R= .
γ +φ
(13), (14) and (A4) together yield the desired solution, (20). Q.E.D.
Proof of Proposition 4:
Differentiating (1) with respect to t and q in this case gives the following set of equations:
dW1 ∂R ∂p ∂R (1 − h)
(A5) =(1 − γ 1 R1 − p ) 1 − R1 − hc − q2 =0.
dt2 ∂t2 ∂t2 ∂t2 γ
dW1 ∂R ∂p ∂R 1 − h
(A6) =(1 − γ 1 R1 − p ) 1 − R1 − hc − t =0.
dq2 ∂q2 ∂q2 ∂q2 γ 2
41
As before (13)-(15) must be invoked to find partial derivatives. The system (A5)-(A6) constitutes
a saddle-point solution, where the partial derivative with respect to t2 provides a partial
maximum, and the partial with respect to q2 provides a partial (local) minimum, with solution q2
= 0. Thus (A5), but not (A6), can be invoked to find an optimal solution. (A5) takes the form
1− h hγ
(A7) t2 =− q2 + {φ (1 − p0 − s) + (γ + φ )c}
h (γ + φ ) 2 − h 2φ 2
(A7) solves alone for t2, observing also q2 ≤ t2 (q2 cannot exceed t2 since all realized offsets must
have cost less than or equal to t2). Optimality then requires that q2 (= marginal cost of energy
abatement in the fringe) be set at its highest possible level = t2. Setting q2 = t2 we find (24).
(25) is found realizing that both the policy bloc and fringe face a marginal energy cost of p + t2,
which must be the marginal energy productivity in both regions. Q.E.D.
Proof of Proposition 5:
We solve (12) and (26) for p and R to yield
1
(A8) =p [γ ( p0 + s ) + γφ RP + (1 − h)φ ]
γ + (1 − h)φ
γ 1− h
(A9) =R RP + (1 − p0 − s )
γ + (1 − h)φ γ + (1 − h)φ
The exporter takes (A8)-(A9) and RP as given, and faces the following responses to changes in s:
∂p γ ∂R 1− h
(A10) = , = − .
∂s γ + (1 − h)φ ∂s γ + (1 − h)φ
This yields the following condition for the exporter’s optimal strategy in this case:
dW2 1− h γ
(A11) = ( p − p0 − φ R − c2 ) − + R = 0,
ds γ + (1 − h)φ γ + (1 − h)φ
with the corresponding optimal condition on R:
1− h
(A12) =R ( p − p0 − c2 ) .
γ + (1 − h)φ
The exporter considers RP as exogenous. RP is still set optimally by the policy bloc, as part of the
market equilibrium, with t2 as the equilibrium emissions quota price. R and p are still determined
by (13)-(14). We then find s, p and R as functions of t2 and q2, and in particular, (27). Q.E.D.
42