ï»¿ WPS6281
Policy Research Working Paper 6281
Emissions Trading with Offset Markets
and Free Quota Allocations
Knut Einar Rosendahl
Jon Strand
The World Bank
Development Research Group
Environment and Energy Team
November 2012
Policy Research Working Paper 6281
Abstract
This paper studies interactions between a â€œpolicy blocâ€™sâ€? either not constrain the offset market whatsoever, or ban
emissions quota market and an offset market where offsets completely. These cases occur when free allocation
emissions offsets can be purchased from a non-policy of quotas is less (very) generous, and the offset market
â€œfringeâ€? of countries (such as for the Clean Development delivers large (small) quota amounts. Governments
Mechanism under the Kyoto Protocol). Policy-bloc firms of policy countries would instead prefer to buy offsets
enjoy free quota allocations, updated according to either directly from the fringe at a price below the policy-bloc
past emissions or past outputs. Both overall abatement quota price. The offset price is then below the marginal
and the allocation of given abatement between the policy damage cost of emissions and the quota price in the
bloc and the fringe are then inefficient. When the policy- policy bloc is above the marginal damage cost. This is also
bloc quota and offset markets are fully integrated, firms inefficient as the policy bloc, acting as a monopsonist,
buying offsets from the fringe, and all quotas and offsets, purchases too few offsets from the fringe.
must be traded at a single price; the policy bloc will
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contacted at jstrand1@worldbank.org..
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Produced by the Research Support Team
Emissions Trading with Offset Markets and Free Quota Allocations
by
Knut Einar Rosendahl*
and
Jon Strand**
Key words: Emission quota markets; offset markets; free quota allocations; updating of
allocation rights; global carbon market.
JEL Classification: H23; Q41; Q54; Q58.
Sector boards: Energy and Mining; Environment
*Senior researcher, Research Department, Statistics Norway, 0033 Oslo, Norway. E-mail:
ker@ssb.no. **Extended Term Consultant, Development Research Group, the World Bank,
Washington DC 20433, USA. E-mail: jstrand1@worldbank.org. We are grateful for valuable
comments from Cathrine Hagem, Ian Parry, Halvor Briseid StorrÃ¸sten and Michael Toman.
Rosendahl acknowledges financial support from the Research Council of Norway. Views
expressed in this paper are those of the authors and should not be attributed to the World Bank,
its management, or member countries.
1. Introduction
Free emissions rights or quotas are a standard feature of most existing emission trading schemes.
In particular, within the EUâ€™s Emissions Trading System (EU ETS) for greenhouse gases
(GHGs), 99.9 % of emissions rights were on average handed out for free to participating entities
from the start (Convery et al., 2008). This share was reduced somewhat in the second period, but
has been well above the minimum requirement of 90 %. From 2013 on, the share is scheduled to
be reduced to below 50%; a more complete phase-out is however not yet on the horizon.
Free allocation of quotas has four major impacts in our context, three negative and one
potentially positive. The first, very well recognized, is that substantial revenue is foregone for
governments (see, e.g., Goulder et al., 1999); this includes the fact that the polluter pays
principle is more or less abandoned. Two further issues have until recently been less recognized,
but are no less detrimental. One is that free allocations may reduce firmsâ€™ incentives to abate.
This follows mainly because current activity (emissions or production) may serve as the basis for
current of future free allocations, thus acting as an effective premium on emissions (directly, or
indirectly as a premium on output). This â€œraises the barâ€? with respect to abatement that is
privately efficient for emitters, since more abatement may reduce the extent of future free
allocations. The other issue, the main focus of this paper, is that free allocations can make offset
markets less efficient. In fact, when firmsâ€™ gains from free allocations are sufficiently high, we
find that policy-country governments may choose to ban the offset market completely. But even
when the offset market is allowed to operate, it will do so inefficiently. The fourth and
potentially positive effect of free allocation is that it may alleviate carbon leakage and improve
the competitiveness of trade-exposed sectors. In addition to the third issue, we will also touch
upon the second and the fourth issue.
Two alternative mechanisms for allocating emissions quotas could here be at play. The first is
based on updating of quota allocations according to past emissions, as these may be taken as an
indication of future quota â€œneedsâ€?. This issue has been treated in the literature, e.g. by BÃ¶hringer
and Lange (2005), Rosendahl (2008), Harstad and Eskeland (2010) and Rosendahl and
StorrÃ¸sten (2011), who have shown that when free allocations are updated in such a fashion,
much of the incentive to abate could be removed from emitters. Furthermore, the quota price will
exceed firmsâ€™ marginal mitigation costs. However, it has also been shown that such allocations at
least in principle can provide a cost-effective solution, in the context of a â€œclosed systemâ€? with
no offset market and identical updating rules and price expectations across emitting firms
(BÃ¶hringer and Lange, 2005). A second type of allocation mechanism entails that free
allocations are based on output of the relevant entities, using then a â€œbenchmarkâ€? emissions
intensity index for the industry (â€œoutput-based allocationâ€?). Under this mechanism a high level of
output will secure a large amount of free allocations (on the presumption that the â€œneedâ€? for free
2
allocations is proportional to output). In this case the distortion to incentives will be in the
direction of too high output, and in consequence also too high emissions. However, for sectors
that are highly exposed to carbon leakage, i.e., increased output and corresponding emissions
abroad, output-based allocation could in fact be superior to no free allocation if the effects on
foreign emissions are also considered (see e.g. BÃ¶hringer et al., 2010). Carbon leakage exposure
is the main argument put forward by the EU for continuing with free allocation in the EU ETS.
Strand and Rosendahl (2012) show that the Clean Development Mechanism (CDM), the key
â€œoffsetâ€? mechanism under the Kyoto protocol, may similarly create incentives for excessive
production.
This paper discusses the effects of free quota allocations in an emission trading system
comprising a â€œpolicy blocâ€? of countries, which faces a â€œfringeâ€? of (non-policy) countries. We
assume that an offset market is established, whereby emissions in policy bloc countries can be
offset through emissions reductions executed in the fringe, and purchased by entities in the
policy bloc. In the context of the Kyoto Protocol (under which the EU ETS is established), the
CDM serves such a role; it will be useful for the reader to have this mechanism in mind in the
following. A main purpose of this paper is to study how free quota allocations to firms in policy-
bloc countries interact with the working of the offset market.
We study two separate cases by which emissions are limited through a market of tradable
emissions quotas in policy bloc countries, and where a fraction of these quotas are given away
for free by governments to emitters. In the first case, we assume that emitters may buy offsets
directly from the fringe, which mimics the current situation with respect to the EU ETS and the
CDM. Further, offsets substitute perfectly with domestic quotas. In this case, the price of offsets
must be equal to the quota price in the policy bloc. 1
In the second case, the quota markets in the policy bloc and fringe are kept apart. There is, as in
the first case, free trading of emissions quotas within the policy bloc. The difference is that, in
this case, free trading with offsets among market participants is not allowed. Instead, offset
purchases from the fringe are made directly by policy countriesâ€™ governments, at an offset price
that may differ from (and is typically lower than) the quota price prevailing in the policy bloc
countries. Strand (2012) has recently shown, in a model with such separate markets (but without
free quota allocations), that it is optimal for the policy bloc to set a lower quota (i.e., offset) price
in the fringe, so that fringe emitters obtain an emissions price that is lower than that obtained for
policy bloc emitters.
1
In our analysis we disregard factors such as uncertainty and restrictions on the use of CDM credits, which may
explain why prices of CDM credits are usually below the quota price in the EU ETS.
3
We first consider a unified quota market with updating of free allocations, free trading of
emissions quotas among firms, and with no fringe and thus no potential offset market. For this
case we briefly replicate, in Section 2, some important results demonstrated in earlier studies (see
above). In Sections 3-4 we extend this model to include an offset market with free trading of
quotas among all market participants, so that the trading price of offsets is the same as the
internal quota price in the policy bloc.
In Section 3 we consider updated allocation of quotas based on emissions. When the quota
allocation is â€œnot too beneficialâ€? to firms, we show that it is optimal for the policy-bloc not to
constrain the use of offsets. Due to the updating rule, the marginal mitigation cost of policy-bloc
firms will be below that of the fringe. This will lead to an excessive share of offsets. However,
we show that the optimal emissions price at the same time is below (the policy blocâ€™s value of)
marginal environmental costs, meaning that there will be inefficiently low volumes of abatement
both in the policy bloc and in the fringe. When considering gradually more generous quota
allocation rules, at some point it is optimal for the policy-bloc to switch, discretely, from free use
of offsets to a solution where the offset market is banned completely. In the latter case the
solution is also clearly inefficient as there is no abatement in the fringe whatsoever.
In Section 4 we assume instead that free allocations are based on firmsâ€™ past outputs. The
solution is also in this case inefficient; for the offset market and also internally for the policy
bloc. When there is no carbon leakage, output of policy bloc firms is always inefficiently high.
This conclusion can however be changed when there is non-negligible leakage. The policy
chosen as optimal by the policy bloc could here as in Section 3 entail inclusion of an offset
market with emission prices below marginal environmental costs (when the implicit output
subsidy is â€œnot too largeâ€?); or it could entail a higher quota price but no offset trading permitted
when the output subsidy is larger.
A basic assumption in Sections 3-4 is that all emissions quotas must be traded at a unified quota
price within the policy bloc and in the offset market. This is, at least in principle, the basic policy
being applied to the CDM today. In Section 5 we relax this assumption. We then instead assume
that an institution set up by the policy-bloc countries has a monopsony on offset purchases from
the fringe, behaving as a non-discriminating monopsonist with no knowledge of project-specific
abatement costs in the fringe. We then show that this institution will set a (unified) offset price
below marginal environmental damage cost. The optimal quota price within the policy bloc is set
such that marginal damage costs equal marginal abatement cost (as in the case of no offset
market); thus the quota price may well exceed marginal damage costs. The consequence could be
a large difference between the internal quota price and the external offset price. Moreover, the
use of offsets will be inefficiently low, but suboptimal from the policy bloc perspective, due to
the coordinated, monopsonistic behavior of the policy countries versus the fringe.
4
2. A Basic Model with Emissions Trading and Updated Allocations
Consider a â€œpolicy blocâ€? of countries which initiates and issues emissions quotas for GHGs, to a
large extent given out for free to emitters within the bloc. It is also possible to purchase
emissions rights from â€œnon-policy countriesâ€? (the â€œfringeâ€?). Assume that the offset market works
perfectly in the sense that all offsets are additional and efficient. In the policy bloc, there are a
given number of firms with aggregate revenue function R1(E1, X1), where E1 and X1 denote
respectively emissions and production from the policy bloc. We have R1Eâ€² > 0 for E1(t) < E10(t)
and R1Xâ€² > 0 for X1(t) < X10(t), where E10(t) and X10(t) are the BaU emissions and production
levels. The probability that any given firm survives to the next period is Î² (same for all firms and
such that firm exits are random events).
Assume that free allocation of emissions rights follow an â€œupdated grandfatherâ€? rule whereby the
number of free quotas awarded to firms with emissions E(t) and production X(t) in period t
equals Î±E(t) + Î³X(t) in period t+1. That is, allocation of quotas is based on firmsâ€™ past emissions
and/or production, where the updating parameters Î± and Î³ are assumed to lie between zero and
unity. In the two first phases of the EU ETS (2005-2012), Î± has been closer to one while Î³ has
been mostly zero. In the third phase (2013-2020), however, Î± is mostly zero except in some
sectors, 2 while Î³ is close to one in exposed sectors but smaller (or zero) in other sectors (cf. the
discussion in Section 4).
Denote the discount factor between periods by Î´, and assume that firms have a potentially
infinite life span. The discounted value of net returns for a representative firm in the policy bloc,
V1(t), can then be expressed as
(1) =
V1 (t ) W1 (t ) + Î²Î´ W1 (t + 1) + Î² 2Î´ 2W1 (t + 2) + ....
where W1(t) denotes net returns in period t. W1(t) is in turn given by 3
(2) W1 (t ) R1 ( E (t ), X (t )) âˆ’ q (t )( E1 (t ) âˆ’ Î± E1 (t âˆ’ 1) âˆ’ Î³ X 1 (t âˆ’ 1)) ,
=
2
Although output-based allocation will be the main allocation rule in the third phase (75-80% of freely allocated
quotas), for several products the allocation will be based on either past energy input or past (process) emissions for
the individual firm. Energy input is closely related to emissions, except for the possibility of fuel switching. For
further discussion see Lecourt (2012).
3
A condition for (2) to hold is E1(t) â‰¥ Î±E1(t-1), which we assume to hold (in particular, it holds in steady state).
5
where q(t) is the quota price in period t. We note that Î±E1(t-1) + Î³X1(t-1) represents the amounts
of free allocations of emissions rights available to the representative firm in period t. This
amount is exogenous to the firm when period t arrives. However, the firm looks ahead to future
periods, in which the payoff will be affected by current emissions through the updating
mechanism. Inserting from (2) into (1) we may write
=V1 (t ) R1 ( E1 (t ), X 1 (t )) âˆ’ q (t )( E1 (t ) âˆ’ Î± E1 (t âˆ’ 1) âˆ’ Î³ X 1 (t âˆ’ 1))
(1a) .
+ Î²Î´ [ R1 ( E1 (t + 1), X 1 (t + 1)) âˆ’ q (t + 1)( E1 (t + 1) âˆ’ Î± E1 (t ) âˆ’ Î³ X 1 (t ))] + Î² 2Î´ 2V1 (t + 2)
The representative firm seeks to maximize V1(t) with respect to current emissions and production
levels, E1(t) and X1(t). This yields (note that E1(t) and X1(t) do not enter in V1(t+2))
dV1 (t )
(3) = =
R1E '( E1 (t ), X 1 (t )) âˆ’ q (t ) + Î±Î²Î´ q (t + 1) 0
dE1 (t )
and
dV1 (t )
(4) = + Î³Î²Î´ q (t + 1) 0
R1 X '( E1 (t ), X 1 (t ))=
dX 1 (t )
and thus
(3a) q (t ) âˆ’ Î±Î²Î´ q (t + 1) .
R1E '( E1 (t ), X 1 (t )) =
(4a) âˆ’Î³Î²Î´ q (t + 1) .
R1 X '( E1 (t ), X 1 (t )) =
Focusing on steady-state cases, with constant revenue functions, constant number of firms (so
that a fraction Î² of all firms are replaced by entering firms in any given period), and a constant
overall emissions cap, we have q(t) = q(t+1) = q. (3a) and (4a) may then be written as
(3b) (1 âˆ’ Î±Î²Î´ )q =
R1E '( E1 (t ), X 1 (t )) = (1 âˆ’ a )q .
(4b) âˆ’Î³Î²Î´ q =
R1 X '( E1 (t ), X 1 (t )) = âˆ’bq .
where a=Î±Î²Î´ and b=Î³Î²Î´. The parenthesis on the right-hand side of (3b) expresses the â€œnet priceâ€?
paid for emissions quotas by policy country firms in a steady state. This price is lower than the
â€œgrossâ€? price q since the free quota allocation is an increasing function of past emissions. The
difference between the â€œnetâ€? and the â€œgrossâ€? price depends on the product of three parameters:
the updating share (Î±); the probability of firm survival to the next period (Î²); and the discount
factor (Î´). All these three parameters might be close to unity; in that case their product will also
6
be relatively close to unity. The effective quota price could then be much lower than the statutory
price, q.
Similarly, the right-hand side of (4b) expresses the implicit subsidy per unit production by an
output-based allocation where Î³>0. 4 Note that this subsidy is proportional to the quota price.
As stated in the introduction, our analysis so far is just a restatement of already known results
from the literature. Our innovation here is to embed it in an emissions trading market where also
offsets are allowed. The main difference between offsets and regular emissions quotas within the
policy bloc in our context is that offsets are not directly affected by the types of incentives that
affect policy bloc firms, as represented by (3b). Instead, for the offset market, the â€œstatutoryâ€?
quota price, q, is also the effective quota price for offsetting units. This implies an asymmetry
between the regular (internal) quota market, and the (external) offset market; with an effective
favoring of emitters operating in the former of these markets.
In the next section we will assume that Î³=0, and focus on the case with emissions-based
allocation (Î±>0). In Section 4 we consider output-based allocation and set Î±=0 (and Î³>0). To
simplify notation, we skip X(t) in the expressions in Section 3.
3. Offset Policies with Emissions-Based Allocation of Quotas
Consider the offset market in the fringe countries. This market has an aggregate revenue function
R2(E2(t)) in period t, and can be viewed as operating on a period-by-period basis. We assume
(conservatively) that all offsets represent real emissions reductions in the offsetting region,
where the comparison benchmark is overall emissions in the absence of offsets. 5 Define this
benchmark by E20(t) in period t, given by
(5) R2 '( E20 (t )) = 0 .
We assume that quotas and offsets can be traded freely by all actors in the carbon markets, both
within the policy bloc, within the fringe, and between the policy bloc and the fringe. Such free
trading implies that there exists a single trading price q for all quotas (including offsets). Fringe
4
To be precise, the right-hand side expresses the implicit tax. However, since this is negative in our case, we have
an implicit subsidy.
5
There are several reasons why not all offsets need to reduce global net emissions. Two are leakage (see Rosendahl
and Strand (2011)), and baseline manipulation and output inflation under â€œrelative baselinesâ€? with incentives to
increase emissions (see Fischer (2005), Germain et al (2007), Strand and Rosendahl (2012)).
7
market participants have no incentives to buy quotas except for resale; this we can ignore here.
In Section 5 we consider alternative assumptions with different quota trading prices within the
policy bloc and in the offset market.
Define next the maximal (potential) supply of offsets from the fringe, for a given offset price q,
by QË† (t ) . This supply corresponds to the difference between the benchmark emissions E (t )
2 20
Ë† (t ) given by
and the emissions level E2
(6) Ë† (t )) = q .
R2 '( E2
As shown below, it might be optimal for the policy bloc to restrict the number of offsets. Let k
denote the share of offset supply from the fringe that is utilized in the policy bloc, and let Q2k
denote the corresponding offset purchases. We then have
(7) =
Q2 k (t ) =
Ë† (t ) k ( E (t ) âˆ’ E
kQ2 20
Ë† (t )) .
2
We assume that the sales of offsets are regulated on a first-come-first-served basis, meaning that
realized offsets are a random draw among all potential offsets (so that any offset supplier has
probability k of successfully selling offsets, and the cost distribution for realized offsets is the
same as for all potential offsets). 6 We show below that it is optimal for policy-bloc country
governments to choose either k = 0 or k = 1, so this potential challenge turns out to be irrelevant.
It is clear that mitigation cannot be overall optimal, under our assumptions. The reason is that
policy bloc firms and fringe firms face different effective mitigation costs, with lower costs for
policy bloc firms than for fringe firms (compare (3b) with (6)). Thus, there exist some firms in
the fringe that mitigate to the level where marginal mitigation cost equals q, whereas no
mitigation options in the policy bloc with marginal cost between (1-a)q and q are realized.
Hence, mitigation through offsets is on average more costly (and inefficiently so) than mitigation
by the policy bloc. This inefficiency can however to some degree be counteracted by reducing
the overall volume of offsets, given by (7), by lowering the â€œpurchase rateâ€? parameter k. If k is
sufficiently small, the overall share of offsets might become â€œoptimalâ€?. Nevertheless, the
distribution of mitigation within the fringe will still remain inefficient, since costs of abatement
in the fringe will generally not be minimized for given abatement.
6
This need not be the case. Since low-cost projects imply more rent to project sponsors, these will have greater
incentives than others to promote their projects thus attracting more attention from policy bloc firms. Rent sharing
between contracting parties, ignored here but studied by Brechet, Meinere and Picard (2011), would also make low
cost projects more directly attractive to policy bloc parties.
8
We now search for optimal combinations of q and k, i.e., the quota price and the share of
potential offsets to be purchased, and study how these depend on the parameter a (=Î±Î²Î´) which
we treat as exogenously given. 7 Equivalently, we could search for optimal combinations of E*
and k, where E* is the emissions cap level in the policy bloc. 8 We have
(8) E* =E1 âˆ’ Q2 k =E1 âˆ’ k ( E20 (t ) âˆ’ E
Ë† (t ))
2
E20 is assumed given and unaffected by the model parameters. 9
The issue of what constitutes an â€œoptimalâ€? offset policy for the policy bloc itself is not obvious.
We postulate the following simple central objective function for the entire policy bloc (defining
now all relevant variables as functions of the target variables q and k):
(9) = R1 ( E1 (q)) âˆ’ cE (q, k ) âˆ’ qk ( E20 âˆ’ E
B1 (q, k ) Ë† (q)) .
2
The first term is simply the aggregate revenue function, the second term accounts for the
environmental damages from global emissions (E), valued at a constant unit cost c, and the last
term represents costs of buying offsets from the fringe. E1 and EË† are both simple functions of q
2
only (from (3b) and (6) respectively). For E we have the following accounting definition:
(10) E1 E20 âˆ’ k ( E20 âˆ’ E
E =+ E1 (1 âˆ’ k ) E20 + kE
Ë† ) =+
2
Ë† .
2
We can now insert into (9) from (10) for E, which yields
(9a) (q, k ) R1 ( E1 (q )) âˆ’ cE1 (q ) âˆ’ (c âˆ’ q )kE
B1= Ë† (q ) âˆ’ [c(1 âˆ’ k ) + qk ]E .
2 20
This expression can be maximized with respect to q and k, yielding the following general first-
order conditions for internal solution:
âˆ‚B1 (q, k ) âˆ‚E âˆ‚E
Ë†
(11) Ë† )= 0
= ( R1 'âˆ’ c) 1 âˆ’ k (c âˆ’ q ) 2 âˆ’ k ( E20 âˆ’ E
âˆ‚q âˆ‚q âˆ‚q
2
7
Î² and Î´ are non-policy parameters, whereas Î± is clearly a policy parameter.
8
For a given combination of E* and k, a corresponding level of q follows (and vice versa for a given combination of
q and k). Furthermore, instead of regulating k directly, which is difficult, the bloc could regulate the total number of
offsets Q2k.
9
This requires that there is no (positive or negative) emissions â€œleakageâ€? from the policy bloc to the fringe. In
Section 4 we return to this issue.
9
âˆ‚B1 (q, k )
(12) = Ë† )=
(c âˆ’ q )( E20 âˆ’ E 0,
âˆ‚k
2
Together, (11) and (12) determine the optimal combination of q and k. Let us first consider how
the optimal k depends on the level of q. From (12) we notice that, since E âˆ’ EË† > 0 , an internal
20 2
solution for k is not feasible unless q = c, in which case any value of k fulfills (11). 10 Without
loss of generality we assume that k is set to zero whenever q = c, as the objective function is
independent of k in this case. If q < c is optimal, k = 1 as B1 is then increasing in k for any k. On
the other hand, if q > c is optimal, k = 0 (B1 is then decreasing in k). The intuition here is the
following: From above we know that E1 only depends on q, i.e., emissions in the policy bloc are
independent of k, for given q. Hence, k only determines how many offsets, or emissions
reductions, the policy bloc purchases from the fringe. Consequently, it is optimal for the policy
bloc to buy offsets from the fringe if and only if the costs of buying offsets (q) are lower than the
damage costs of emissions (c), which then represents the benefits of buying offsets.
What about the optimal level of q? At first glance, one would expect the optimal q to be set equal
to the marginal damage costs c. However, there are two reasons why it may be optimal to deviate
from this standard result, which we come back to below.
From the reasoning above, we must either have q â‰¥ c and k = 0, or q < c and k = 1. Let us
characterize these two potential outcomes of the policy blocâ€™s optimization.
In the first case there is no offset market available for the firms since k = 0. From (11) we then
have the standard optimality condition R1â€² = c. Then there is no inefficiency within the policy
bloc, but not using offsets at all is inefficient since cheap abatement options are foregone. Still, it
may be a second-best solution for the policy bloc. This outcome implies, from (3b) and (11):
c
(13) =q â‰¥c.
1âˆ’ a
Thus, an optimal solution with k = 0 requires that the quota price be set higher than the marginal
damage cost of emissions, c, as long as a > 0. This is just as in BÃ¶hringer and Lange (2005) and
Rosendahl (2008), who show that the quota price is driven up by the updating rule (for a given
emissions constraint). A high quota price makes it too expensive to purchasing offsets, to make
this worthwhile for the policy bloc. Note that when a is relatively close to unity, the mark-up
relative to c could be large.
10
We have assumed linear environmental damage costs, represented by the marginal costs c. If we rather assume
convex damage costs C(E), we would still have q = Câ€™(E) as the only internal solution. However, in this case E and
thus Câ€™(E) is a function of both q and k, and so the choice of k is no longer irrelevant under an internal solution.
10
Consider next the outcome where q < c and k = 1. Naturally, q = 0 cannot be an optimal solution
(cf. (11)), so only internal solutions are feasible here. We may write (11) as follows, inserting
from (3b):
âˆ‚E1 Ë† ï£¶
ï£« âˆ‚E1 âˆ‚E
(11b) âˆ’ aq Ë† (q )) + (c âˆ’ q ) ï£¬
= ( E20 âˆ’ E + 2ï£·
âˆ‚q ï£ âˆ‚q âˆ‚q ï£¸
2
We notice that the LHS and the first term on the RHS are both positive for a, q > 0, whereas the
second term on the RHS is negative for q < c. Thus, we distinguish between the following three
cases:
âˆ‚E1
a) âˆ’aq > E20 âˆ’ E
Ë† (q ) for any q < c. In this case, equation (11b) has no solution when q
âˆ‚q
2
< c, and thus it is optimal to increase q until q â‰¥ c (cf. (11)). But then we are back to the
case with k = 0 and q determined by (13), as described above. The intuition here is that
the ability of the offset market to profitably deliver offsets to the policy bloc (RHS of the
inequality) is relatively small. The solution is then determined out of a concern for the
domestic mitigation market. Notice that the higher a is (e.g., the higher the allocation rate
Î± is), the more likely this case is. Further, without the updating rule (i.e., a = 0) this case
can never occur.
âˆ‚E1
b) âˆ’aq < E20 âˆ’ E
Ë† (q ) for q = c. In this case, equation (11b) must have (at least one)
âˆ‚q
2
internal solution with q < c. 11 The ability of the offset market to profitably deliver offsets
is now greater. This implies that the quota price may be determined more out of a direct
concern for the offset market, and less out of a concern for the domestic mitigation
market in the policy bloc. However, we cannot conclude in general whether or not this
solution with q < c and k = 1 is preferred over the solution with k = 0 and q determined by
(13). The former case utilizes relatively cheap abatement options in the offset market, but
abatement in the policy bloc is far too low as R1â€²(E1) < q < c. In the latter case, mitigation
in the policy bloc is optimized, but none of the abatement options in the offset market are
utilized.
11
If emissions in the two regions are convex (or linear) in q, we can show that the second order condition of (12) is
fulfilled for q â‰¤ c. Thus, there is just one internal solution which is optimal given q â‰¤ c.
11
âˆ‚E1
c) âˆ’aq < E20 âˆ’ EË† (q ) for some q < c (but not for q = c). In this case we may or may not
âˆ‚q
2
have an internal solution of (11b). For instance, the fringe may be quite able to deliver
profitable offsets relative to emissions reductions in the policy bloc at low levels of q, but
not at higher levels of q. The policy bloc does not want the quota price to be too low,
however, due to the environmental concern. Hence, it may be optimal to increase q until
q â‰¥ c, i.e., similar to a). However, we may also have an internal solution similar to b).
To sum up so far, it is optimal for the policy bloc to either ban offsets completely, and let the
quota price be given by (13), or to have no restrictions on offsets, implying R1â€²(E1) < q = R2â€²(E2)
< c. Further, the higher is a and the lower is the offset potential relative to domestic emissions
reductions, the more likely it is that offsets are banned.
Let us investigate case b) further. We first notice that if a = 0, i.e., no updating rule, then q < c
and k = 1 is preferred over the alternative solution q = c and k = 0. The reason is that the latter
outcome is equivalent with q = c and k = 1 (see above). But then we know from the investigation
of case b) that reducing q below c will be beneficial. This case, i.e., without any updating rule,
has already been analyzed in Strand (2012). In other words, without updating it is never optimal
to put any restrictions on offset purchases (given our model assumptions), and the optimal quota
price should be below the marginal damage cost. The latter conclusion is seen by replacing
R1â€²(E1) by q and setting k = 1 in (11). At q = c the RHS is negative as the policy bloc, acting as a
monopsonist, benefits from reducing q due to lower costs of importing quotas.
When a is marginally increased from zero, i.e., if a mild updating rule is introduced, the optimal
level of q will increase. This is shown in the appendix in the case of quadratic mitigation costs.
The reason is that introducing updating increases the deviation between the marginal costs of
abatement, given by (3b), and the marginal costs of emissions, c, for a given level of q. Thus, it
is optimal to increase q even though the policy blocâ€™s costs of buying offsets increase. This
implies that, as long as k = 1, emissions reductions in the fringe, and thus the use of offsets, will
increase. That is, introducing updating will tend to increase the optimal level of offsets at low
levels of a. However, at some point, a further increase in a will tend to reduce the optimal level
of q (cf. the appendix). In other words, then it would be optimal to reduce the use of offsets. The
reason is, as explained above, that the policy bloc ideally would like to see a lower q in the offset
market in order to reduce the costs of buying offsets. This matter becomes relatively more
important when a is high, and hence the optimal q declines. Finally, when a becomes sufficiently
high, it becomes optimal to stop buying offsets whatsoever, i.e., switch from k = 1 to k = 0.
Although the optimal quota price increases when a is increased from zero, abatement in the
policy bloc declines (at least, this is the case with quadratic mitigation costs, cf. the appendix).
12
The reason is that for abatement to stay constant within the policy bloc when a increases, (13)
would need to be fulfilled; while we find that q falls relative to that level. In other words, the
effects of higher a dominate the effects of higher (optimal) q in equation (3b), so that E1
increases. Moreover, global emissions increase, too, when a is increased, as the higher emissions
in the policy bloc (â€œregion 1â€?) will always dominate the (initially) lower emissions in the fringe
(â€œregion 2â€?; see the appendix). This holds only as long as k = 1, however, as when a becomes so
high that k = 0 is optimal, q jumps to the level given by (13). Then emissions in both regions are
insensitive to a further increase in a (which then only affects q, from (13)).
In the appendix we show that, in the case of quadratic mitigation costs, welfare in region 1
decreases monotonically in the level of a as long as offsets are used. Thus, introducing (or
intensifying the level of) updating in a quota system with access to offsets will unambiguously
reduce welfare in the policy bloc. The reason is that updating increases the deviation between the
desired domestic quota price and the desired offset price (if these could be set independently).
Moreover, we find that it becomes optimal to switch to no offsets exactly when global emissions
are the same with and without offsets (see the appendix). It follows that increasing the level of a
will always increase global emissions as long as offsets are used initially. A partial explanation
for this result, which may not hold with other specifications than quadratic mitigation costs, is
that the policy bloc wants to minimize global emissions, and hence has incentives to pick the
alternative where these are lowest.
Figure 1. Emissions in the policy bloc, the fringe and total emissions as a function of a
13
E1 (k=1) E1 (k=0) E2 (k=1) E2 (k=0) E (k=1) E (k=0)
E1 (k 1) E1 (k 0) E2 (k 1) E (k 1) E (k 0)
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
Figure 2. Quota price and welfare in the policy bloc and total welfare as a function of a
14
q (k=1) q (k=0) B1 (k=1) B1 (k=0) B (k=1) B (k=0)
B (k 1) B (k 0) B1 (k 1) B1 (k 0) (k 0) (k 1)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
Figures 1 and 2 illustrate the cases discussed in this section, where we assume identical,
quadratic mitigation costs in the two regions. 12 The thick curves show the outcome of region 1â€™s
optimization as a function of a. In addition, the figures show the (hypothetical) outcome given k
= 1 also when k = 0 would be optimal (see the thin curves). With the chosen parameters, it is
optimal to switch from k = 1 to k = 0 at a = 0.5. From Figure 2 we notice that the quota price is
almost constant up to a = 0.5 (first slightly increasing, then slightly decreasing). Then it jumps
substantially when offsets are no longer utilized. In Figure 1 we see that this implies almost
constant emissions in region 2 up to a = 0.5, whereas emissions in region 1 are steadily
increasing. Hence, global emissions increase, consistent with the analytical findings. At a = 0.5
emissions in region 2 jumps to its BaU-level, whereas emissions in region 1 falls due to the much
higher quota price. As indicated above, the changes in the two regions exactly cancel each other
out so that global emissions neither jump nor fall.
12
Referring to the quadratic model specification in the appendix, parameters are as follows: Î¼jA = Î¼jB = Î¼jC = 1; c
= 0.5.
15
Welfare in region 1 declines as a increases (see Figure 2) until it reaches the welfare level with k
= 0 (in which case it is unaffected by a). We also notice that total welfare (B) decreases. 13
We sum up our main findings in the following proposition:
Proposition 1. Consider a policy bloc with an emissions trading system, free quota allocations
based on firmsâ€™ past-period emissions, and an offset market with free trading among market
participants. Then:
i) If free allocations are sufficiently generous, it is optimal for the policy bloc to ban offsets. If
banning offsets is not optimal, the use of offsets should be unrestricted.
ii) If offsets are used, marginal damage costs strictly exceed the quota price and marginal
abatement costs in the fringe, which strictly exceed marginal abatement costs in the policy bloc.
iii) As long as using offsets is optimal for the policy bloc, increasing the free allocations of
quotas leads to higher emissions and lower welfare in the policy bloc, and higher global
emissions (given that mitigation costs are quadratic).
Proof: Follows from the discussion above, and from the appendix.
4. Offset Policies with Output-Based Allocation of Quotas
In the previous section we assumed that quotas are given for free to domestic firms based only
on their past emissions. As explained in Section 2, the EU ETS is now largely moving towards
output-based allocations of quotas, from 2013 onwards, although for some sectors emissions-
based allocations will still be used (see footnote 2). Output-based allocations are also highly
relevant for other regions such as Australia, New Zealand and California (see e.g. Hood, 2010).
A main justification for switching to output-based allocations is the fear of â€œcarbon leakageâ€?
through the markets of emission-intensive, trade-exposed goods. Such â€œleakageâ€? could take the
form that lower output of such goods in one region, due to unilateral climate policy, leads to
13
Here we have assumed that Region 2 values global emissions by the same price as Region 1, i.e., by c.
16
greater output and emissions in other regions with no or more lenient climate policy. 14 With
output-based allocations, an â€œemission intensity benchmarkâ€? is defined for each product, based
on e.g. an average standard of all or the best firms in the industry. This would, at least in
principle, make this â€œbenchmarkâ€? independent of the emissions of any one given firm. 15
In view of these considerations, we will in this section focus on the case with output-based
allocation, and set Î³>0 and Î±=0. From (3b) we see that absent an offset market, the level of
emissions within the policy bloc is â€œoptimalâ€? in the sense that marginal value of emissions
equals the quota price, which is set equal to marginal damage cost. On the other hand, (4b)
shows that in absence of carbon leakage, output is excessive: Optimality would here entail (net)
marginal value equal to zero. Emissions (from using fossil energy) are then likely to also be
excessive, given that (as is reasonable) output and energy use are complementary.
We now introduce offsets into this alternative model. Instead of (9a) we then have the following
alternative objective function for the policy bloc:
=
(14) Ë† (q, X (q )) âˆ’ [c(1 âˆ’ k ) + qk ]E ( X (q )).
B1 (q, k ) R1 ( E1 (q), X 1 (q )) âˆ’ cE1 (q ) âˆ’ (c âˆ’ q )kE2 1 20 1
Ë† / âˆ‚X < 0 . The first of
Following the discussion above, we assume that âˆ‚X 1 / âˆ‚q < 0 , and âˆ‚E2 1
these derivatives simply expresses that a higher quota price (or emission cost) reduces energy-
intensive output in the policy bloc. The second derivative states that when output is reduced in
energy-intensive sectors in the policy bloc, emissions in the fringe will shift up, presumably as
related industrial activity is shifted to this region. Both conditions are intuitive for emission-
intensive, trade-exposed industries, and are preconditions for having a leakage problem in our
context. The size of âˆ‚X 1 / âˆ‚q determines how sensitive domestic output is to the quota price,
Ë† / âˆ‚X determines â€œleakage exposureâ€? for domestic firms. To simplify the analytics
whereas âˆ‚E2 1
Ë† / âˆ‚X =
below, we assume âˆ‚E2 1 âˆ‚E20 / âˆ‚X 1 , so that the amount of leakage is independent of offset
projects.
Maximizing (14) with respect to q and then reorganizing yields in this case:
14
There is a large literature on emissions leakage (e.g., Hoel, 1996; Rosendahl and Strand, 2011; BÃ¶hringer et al.,
2011). Leakage may occur through the international markets for fossil fuels, and through the markets for emission-
intensive, trade-exposed goods. Here we focus on the latter channel.
15
In the EU ETS, benchmarks for the period 2013-2020 are mainly determined based on the ten per cent least
emission-intensive installations.
17
âˆ‚B (q, k ) âˆ‚E âˆ‚E
Ë† ï£« Ë† ï£¶ âˆ‚X
âˆ‚E
(15) Ë† )âˆ’ï£¬
=(q âˆ’ c) 1 âˆ’ k (c âˆ’ q ) 2 âˆ’ k ( E20 âˆ’ E bq + c 2 ï£· 1 =0
âˆ‚q âˆ‚q âˆ‚q âˆ‚X 1 ï£¸ âˆ‚q
2
ï£
Equation (12) still holds, so we must either have q â‰¥ c and k = 0, or q < c and k = 1.
Reasonably, the value of free allocations is likely related to the costs of leakage exposure; this
may be because the authorities are inclined to compensate firms, in terms of reduced quota costs,
for their loss of competitive position due to being subject to climate policy. Such effects are
captured by the two terms inside the parenthesis of the last term on the RHS: The first is the
expected value of future allocations per unit output today (where b=Î³Î²Î´), i.e., the implicit output
subsidy. The second is the environmental cost of leakage due to a marginal reduction in domestic
output. Assume first that this parenthesis is zero. Then we are back to equation (11) (remember
that R1Eâ€² = q). From the discussion in Section 3 with a =0 we know that the optimal solution is
characterized by q < c and k = 1. This further implies that for the parenthesis to be zero, we must
have b > âˆ’âˆ‚E Ë† / âˆ‚X . Let q* denote the optimal quota price in this case.
2 1
Ë† / âˆ‚X ) is large, it is optimal to reduce the
If the last term in (15) is negative, e.g., because ( âˆ’âˆ‚E2 1
quota price below q*. The reason is that leakage reduces the environmental effectiveness of
climate policy in the policy bloc, and hence the optimal quota price falls.
If the last term is positive, e.g., because firms for some reason are given free quotas even though
leakage exposure is negligible, the optimal quota price is higher than q*. The intuition is that the
free quota allocation stimulates output too much, and so the optimal (second-best) response is to
increase the price of emissions to moderate output. 16 If this effect, represented by the last term of
(15), is big compared to the offset potential, represented by the third term of (15), then it is
optimal to increase q at least up to q = c (the two first terms in (15) are both positive for q < c).
But then we know from above that k = 0, i.e., banning offsets is optimal.
Let us now discuss the sign of this last term in (15), based on the allocation rules of the EU-ETS.
The sign depends on i) the ratio between the quota price q and the marginal environmental costs
c, and ii) how generous the free allocation is (the size of b) compared to the leakage exposure
( âˆ’âˆ‚EË† / âˆ‚X ). We have just concluded that q < c if the term is non-positive, but cannot conclude
2 1
if the term is positive.
What about the magnitude of b? For the most exposed sectors in the EU ETS, which jointly
account for a majority of industry emissions in the EU, Î³ = b/(Î²Î´) is set close to E1/X1 (almost
16
Obviously, the first-best response would be to lower b, but this might be difficult for political reasons.
18
100% compensation at the sector level). This means that, if reductions in domestic output are
replaced one-to-one by foreign output, and emissions intensities are similar inside and outside
the policy bloc, b /( Î²Î´ ) â‰ˆ âˆ’âˆ‚E
Ë† / âˆ‚X . More likely, however, the output replacement is less than
2 1
100%. But since emissions intensities will often be higher in the fringe, it is still difficult to
judge in general whether b is higher or lower than âˆ’âˆ‚EË† / âˆ‚X , at least for the highly exposed
2 1
sectors.
The EU has been criticized for allocating too many free quotas also to sectors that are only
slightly exposed to leakage (see e.g. Martin et al., 2012). This relates both to sectors given 100%
compensation (e.g., fossil fuel extraction), and to the remaining sectors which initially receive
70% compensation. Hence, sectors can probably be found where b exceeds âˆ’âˆ‚E Ë† / âˆ‚X . This
2 1
could be explained by strong industry lobbying groups. Still, unless leakage exposure is
negligible, the sign of the last term in (15) is ambiguous.
To sum up, free output-based allocations to leakage-exposed sectors have ambiguous effects on
the optimal quota price. This price will most likely be below the marginal environmental costs
when an offset market is available. However, if sectors with limited exposure to leakage for
political reasons are granted substantial amounts of free quotas, in proportion to their output, the
preferred strategy for the policy bloc may be to ban offsets completely.
We sum up our main findings from the discussion above, in the following proposition:
Proposition 2. Consider a policy bloc with an emissions trading system, free quota allocations
based on firmsâ€™ past-period output, and an offset market with free trading among market
participants. Then:
i) If allocation is not too generous relative to the leakage exposure, i.e., b â‰¤ âˆ’âˆ‚E
Ë† / âˆ‚X , it is not
2 1
optimal to put any restrictions on the use of offsets.
Ë† / âˆ‚X ), and the offset
ii) If allocation is generous relative to the leakage exposure (b >> âˆ’âˆ‚E2 1
potential E20 âˆ’ E
Ë† is sufficiently limited, it is optimal for the policy bloc to ban the use of offsets.
2
iii) If offsets are used, marginal damage costs strictly exceed the quota price and marginal
abatement costs in the fringe and in the policy bloc (which are all equal).
5. Optimal Offset Policy with Quota Price Discrimination
19
We now revert to the case of emissions-based free allocations in Section 3. We assume that the
quota market is not necessarily unified, and that the quota price can be set at different levels in
the policy bloc and the fringe. One such possibility would be where all trading of offsets is done
by a government agency representing all policy countries, and the offset price could be set lower
in the fringe. Strand (2012) then shows, in a similar model except that free quota allocations are
not considered, that it is optimal for a unified government representing all policy bloc countries
to operate as a monopsonist in the offset market, and set the offset price below the quota price
inside the policy bloc.
We will here model a similar case but instead with updated quota allocations. Assume that the
government representing the policy bloc is required to set one single offset price at which offset
quotas can be purchased from the fringe. While this may seem as a natural limitation to impose
on the policy bloc, it is still not first-best for this bloc. In particular, it precludes price
discrimination in the offset market whereby quotas can be purchased â€œcheaperâ€? from â€œfringeâ€?
firms known to have lower abatement costs. Such price discrimination probably takes place at
least to some degree in the CDM market today. Our assumption is akin to assuming an extreme
version of asymmetric information about abatement costs, where low-cost firms will in general
have incentives to mimic as high-cost. If such mimicking is fully successful, no type revelation
will take place in equilibrium. 17
The problem is now formally similar to above except that we have two quota prices, q1 for the
policy bloc, and q2 for the offset market (in the fringe), instead of just a single price, q. 18 Define
now the policy blocâ€™s objective function in similar fashion to (9a), by
(16) , q2 , k ) R1 ( E1 (q1 )) âˆ’ cE1 (q1 ) âˆ’ (c âˆ’ q2 )kE
B(q1 = Ë† (q ) âˆ’ [c(1 âˆ’ k ) + q k ]E .
2 2 2 20
(16) is here maximized with respect to q1, q2 and k, yielding the following set of first-order
conditions:
âˆ‚B(q1 , q2 , k ) âˆ‚E
(17) ( R1 'âˆ’ c) 1 =
= 0
âˆ‚q1 âˆ‚q1
âˆ‚B âˆ‚E
Ë†
(18) âˆ’k (c âˆ’ q2 ) 2 âˆ’ k ( E20 âˆ’ E
= Ë† (q )) =
0
âˆ‚q2 âˆ‚q2
2 2
17
For an introduction to the game theoretic basis for such an equilibrium, see e.g. Gibbons (1992), Chapter 3.
18
Strand (2012) argued that an offset trading arrangement of this sort could go together with even a carbon tax
within the policy bloc, and be utility enhancing for this bloc.
20
âˆ‚B(q1 , q2 , k )
(19) (c âˆ’ q2 )( E20 âˆ’ E
= Ë† (q )) â‰¥ 0 .
âˆ‚k
2 2
From (17), we immediately find that R1â€² = c: mitigation is optimal in the policy bloc. This
implies that q1, given from (13), exceeds marginal damage cost from emissions as long as a > 0.
Consider next q2. From (18) we now find
(E âˆ’ EË† )
(20) c âˆ’ 20
q2 = 2
0, (19) must hold with inequality, so that k = 1 at the optimal solution. This is
clearly intuitive. The policy bloc now wishes to implement its desired volume of offsets as
19
Price differences exist today between the CDM market and the EU ETS, and within the CDM market, for a
variety of reasons beyond our simple model. Among these are 1) transaction costs and other imperfections in the
offset market; 2) uncertain delivery of effective offsets from the point of view of offset purchasers (not all offsets are
actually achieved and credited); and 3) bilateral bargaining or monopsonistic power by buyers in the offset markets.
21
cheaply as possible. It is obviously cheaper for the bloc to reduce the supply of offsets by
reducing the offset price, than to restrict the number of offsets purchased by having k < 1.
We may sum up these results as follows.
Proposition 3. Consider a policy bloc with a domestic emissions trading system, free quota
allocations based on firmsâ€™ past-period emissions, and an offset market fully controlled by
policy-bloc governments. Then:
i) Within the policy bloc, the equilibrium quota price exceeds firmsâ€™ marginal abatement
costs which in turn equal marginal damage cost, so that abatement is efficient within
the policy bloc.
ii) Offsets are always used.
iii) The equilibrium offset price is lower than marginal damage cost; thus abatement in the
fringe is inefficiently low.
Proof: The results follow from the discussion above.
From the point of view of policy bloc countries, the solution described in this section is
preferable compared to a unified quota market. However, loss of efficiency still occurs due to the
wedge between marginal damage cost and marginal mitigation cost in the fringe, leading to too
little mitigation. On the other hand, the policy bloc now always finds it optimal to utilize the
offset market, which is not necessarily the case under a unified market. Moreover, if offsets are
used under a unified market, the amount of domestic mitigation is too low. Both types of
solutions thus entail inefficiencies, which are quite different. Although price discrimination
obviously is the preferred solution for the policy bloc, we cannot easily say which of the two
types of solutions is preferable from a global efficiency perspective.
It is also ambiguous which solution is more favorable to fringe countries. The solution with
different quota prices in the two regions is clearly more favorable to fringe countries than the
solution with k = 0 in Section 3 above, i.e., when the offset market is not utilized at all by the
policy bloc. Price discrimination can in fact also be more favorable to fringe countries when the
offset market is used in the case without price discrimination; this has been shown in a related
but slightly different model by Strand (2012). 20
20
This depends among other things on whether the amount of offsets purchased is smaller or greater. This however
needs further study, and we refer to the discussion in Strand (2012).
22
The solution possibilities sketched in Section 3, for the case of a unified quota price, are also
easier to understand when viewed in the context of the solution derived here. In particular, when
the offset market is overwhelmingly important compared to the domestic mitigation market, as it
may be under case b) in Section 3 above, the unified solution for q will be close to that found in
(20). The quota price will then be below the marginal damage cost of emissions, and this will
hold also for the domestic mitigation market. By contrast, when the domestic mitigation market
is overwhelmingly important compared to the offset market, as under case a) in Section 3, the
unified solution for q would entail that the quota price be set by (13) and there would be no
offset market.
A final issue is whether the solution derived in this section can be implemented as a
decentralized market solution where emitters in the policy bloc also trade with the offset market.
Any such trading must be subject to a price difference q1 â€“ q2 facing policy bloc actors versus
fringe actors. Focusing on policy bloc actors, the price of all quotas facing these must be the
same, for unified market trading to take place.
In principle, there are at least two ways of implementing such a solution. The first entails that the
government imposes an (excise) tax per quota purchased in the offset market, equal to tq = q1 â€“
q2. 21 Given such a tax, the government needs not impose other restrictions on quota sales (all
possible offsets ought to be realized given this tax).
The other solution, suggested by Castro and Michaelowa (2010), and discussed analytically by
Klemick (2012), is to â€œdiscountâ€? the offsets by giving them less value to purchasers per ton CO2
offset, when compared to abatement in the policy bloc itself. Thus, a very similar solution would
be implemented if the rate at which offset quotas are â€œdiscountedâ€? (or reduced in value relative
to domestic abatement), call it Î», were to equal (q1 â€“ q2)/q1. Put otherwise, when buying one unit
of offset from the fringe, a policy bloc firm is credited with only q2/q1 units of actual offset. One
difference between the two solutions is that, in the first, the government would raise revenue (q1
â€“ q2)Q2, where Q2 is the amount of abatement taking place in the fringe; and this abatement
would all be credited to policy bloc firms. In the second case, the government would raise no
revenue. However, if the government auctions off Î»Q2 quotas, we get exactly the same outcome
with respect to emissions, costs and government revenues as in the first case. 22
21
We thank Ian Parry for suggesting this solution.
22
Note that in the â€?discountâ€? case, policy bloc emissions will be lower than in the â€œtaxâ€? case if the overall cap of the
emissions trading system in the policy bloc is the same in the two cases. By increasing the cap by Î»Q2 units,
emissions will be the same both in the fringe and in the policy bloc. It is straightforward to see that the revenue from
selling Î»Q2 quotas is (q1 â€“ q2)Q2.
23
6. Conclusions
We have studied optimal policies of a â€œpolicy blocâ€? of countries enforcing an emissions trading
system with free quota allocation, combined with an offset market with emissions reductions
purchased from a â€œfringeâ€? of (non-policy) countries. We have considered two separate models.
In the first, there is a unified market for emissions reductions in the policy bloc and the fringe,
allowing market participants to trade emissions quotas in both blocs. In the second, all offsets are
purchased directly from the fringe by a central unit in the policy bloc countries, at an offset price
below the price charged to policy bloc emitters. A key feature of our analysis is that a large share
of emission quotas are given away for free to participating firms, based on their emissions and/or
output in the preceding period. The combined policy-bloc and offset market is then always
inefficient, in both models.
A main reason for inefficiency is that the free emissions rights may raise the equilibrium quota
price to a level above marginal mitigation cost of firms in the policy bloc, but not in the â€œfringeâ€?.
With a unified emissions market with a single quota price, this leads to a (perhaps substantially)
higher marginal abatement cost in the fringe than in the policy bloc. Moreover, purchasing
offsets from the fringe becomes expensive for the policy bloc. When the fringe dominates the
overall quota market, and/or the effect of free allocations on the quota price is not too great, the
optimal solution for the policy bloc is to choose a low quota price. Offsets are then not very
expensive, but the main inefficiency is too little abatement within the policy bloc. When the
fringe becomes less significant, and/or there is a large effect of free allocations on the domestic
quota price, policy-bloc countries rather choose to ban the offset market altogether; the reason
being that offsets are too expensive to be worthwhile. However, if quotas are allocated in
proportion to output and not emissions, and the allocation is not too generous relative to leakage
exposure, it is never optimal to ban the use of offsets.
In the second model, with full government control of offset purchases, the inefficiency instead
takes the form of too little abatement in the fringe, as the policy bloc acts as a monopsonist in
limiting the number of offsets. The internal quota price within the policy bloc may then be higher
than, and the offset price is always lower than, the marginal abatement cost in the policy bloc.
The analysis shows that providing free quota allocations to participating firms on the basis of
updating schemes can be problematic for the functioning of offset markets, in particular when
offsets are to be traded freely. It can even lead policy countries to choose to ban trading in the
offset market entirely. Possible solutions are to eliminate or reduce the value of free allocations
and/or make the updating rules less distortive; to separate the domestic quota market and the
24
offset market, thus not allowing free trading across these markets; or to tax offsets making the
policy blocâ€™s optimal price discrimination solution implementable for the offset market.
25
References:
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27
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28
Appendix: Quadratic Mitigation Costs: Some More Specific Results
In this appendix we assume quadratic mitigation costs, and derive some more specific results for
the model in Sections 2-3. Assume, in particular, Rj(Ej) = Î¼jA + Î¼jBEj â€“ (Î¼jC/2)(Ej)2, with Î¼jA, Î¼jB,
Î¼jC > 0 (j = 1,2). To simplify notation, we assume that the two regions (the policy bloc, and the
fringe) are identical except for their sizes, so that Î¼1A = Î¼2A = Î¼A, Î¼1B = Î¼2B = Î¼B, Î¼1C = Î¼C/h and
Î¼2C = Î¼C/(1-h), with 0 < h < 1. All results shown here also apply in the general case.
First, we have
(A1)-(A2) R1â€²(E1) = Î¼B â€“ Î¼CE1/h, R2â€²(E2) = Î¼B â€“ Î¼CE2/(1-h), E10 = hÎ¼B/Î¼C
and
(A3) E20 = (1-h)Î¼B/Î¼C.
Equation (3b) then gives:
Âµ B âˆ’ (1 âˆ’ a)q
(A4) E1 (a, q ) = h
ÂµC
and hence:
âˆ‚E1 (a, q ) 1âˆ’ a
(A5) = âˆ’ h
âˆ‚q ÂµC
whereas
(A6) Ë† (q ) = q (1 âˆ’ h)
E20 âˆ’ E2
ÂµC
and hence:
Ë†
dE âˆ‚E
Ë† (q) 1
(A7) 2
= 2
âˆ’
= (1 âˆ’ h)
dq âˆ‚q ÂµC
Equation (11b) now gives:
âˆ’(1 âˆ’ a ) q ï£« âˆ’(1 âˆ’ a ) 1 ï£¶ q 1
(A8) âˆ’aq =
h h + (c âˆ’ q) ï£¬ hâˆ’ (1 âˆ’ h) ï£·
= h + (c âˆ’ q) ( ah âˆ’ 1) ,
ÂµC ÂµC ï£ ÂµC ÂµC ï£¸ ÂµC ÂµC
leading to:
29
1 âˆ’ ah
(A9) q= c
a h âˆ’ 2ah âˆ’ h + 2
2
2âˆ’a
(in the special case with h = Â½, this simplifies to q = c)
a âˆ’ 2a + 3
2
We can now examine the effects on q of increasing a:
dq h ( h + ha 2 âˆ’ 2a )
(A10) = c
da ( a 2 h âˆ’ 2ah âˆ’ h + 2 )2
We see that this expression is positive for a sufficiently close to zero, but negative for a
sufficiently close to one. Thus, the optimal quota price will increase when updating is (weakly)
introduced, but may start to decline when a becomes higher (unless it has become optimal to
switch to k = 0).
Ë† obviously follows the opposite direction of q.
What about the effects on Ej of increasing a? E2
For E1 the outcome is less clear as long as q increases in a, as higher a leads to higher emissions
(for a given q).
We total differentiate the expression for E1(a,q) above, using the expression for q:
dE1 âˆ‚E1 (a, q ) âˆ‚E1 (a, q) dq q 1âˆ’ a h + ha 2 âˆ’ 2a
= + = hâˆ’ hc
da âˆ‚a âˆ‚q da ÂµC ÂµC ( ha 2 âˆ’ 2ha âˆ’ h + 2 )2
ï£® ï£¹
h ï£¯ 1 âˆ’ ah h + ha 2 âˆ’ 2a
= c âˆ’ (1 âˆ’ a ) c ï£º
ÂµC ï£¯ ha 2 âˆ’ 2ha âˆ’ h + 2 ( ) ï£º
2
ï£° ha 2
âˆ’ 2 ha âˆ’ h + 2 ï£»
(A11)
ï£° ha âˆ’ 2ha âˆ’ h + 2 âˆ’ h a + 2h a + h a âˆ’ 2ha âˆ’ h + h a âˆ’ h a + h a + 2ha âˆ’ 2ha ï£¹
hc ï£® 2 2 3 2 2 2 2 2 2 2 2 3 2
ï£»
=
ÂµC ( ha 2 âˆ’ 2ha âˆ’ h + 2 )
2
(1 âˆ’ h ) ( 2 + h âˆ’ ha 2 âˆ’ 2ha )
= hc > 0.
ÂµC ( 2 âˆ’ h + ha 2 âˆ’ 2ha )
2
Thus, domestic emissions will always increase when a is increased (as long as k = 1). To find the
effect on global emissions, we differentiate with respect to A, and find:
30
Ë† dq (1 âˆ’ h ) ( 2 + h âˆ’ ha 2 âˆ’ 2ha )
dE dE1 dE 1 âˆ’ h h ( h + ha âˆ’ 2a )
2
= + 2
= hc âˆ’ c
ÂµC ( 2 âˆ’ h + ha 2 âˆ’ 2ha ) ÂµC ( 2 âˆ’ h + ha 2 âˆ’ 2ha )2
2
da da dq da
(1 âˆ’ h)hc
=(A12) ï£° 2 + h âˆ’ ha âˆ’ 2ha âˆ’ h âˆ’ ha + 2a ï£¹
ï£® 2 2
ï£»
ÂµC ( 2 âˆ’ h + ha 2 âˆ’ 2ha )
2
2(1 âˆ’ h)hc
= ï£°1 + a âˆ’ ha âˆ’ ha ï£¹
ï£® 2
ï£» > 0.
ÂµC ( 2 âˆ’ h + ha âˆ’ 2ha )
2 2
Hence, global emissions will unambiguously increase when a is increased as long as offsets are
used.
We can also show, given a = 0, that global emissions are lower if offsets are used than if they are
not. Updating will then increase global emissions, regardless of the optimality of offsets.
Furthermore, we can show that welfare is decreasing in the level of a as long as offsets are used
(k = 1), cf. equation (9):
dB ï£® Âµ ï£¹ dE dE Ë†
Ë† ï£¹ dq
= ï£¯ Âµ B âˆ’ C E1 âˆ’ c ï£º 1 âˆ’ [ c âˆ’ q ] 2 âˆ’ ï£® ï£° E20 âˆ’ E2ï£»
da ï£° h ï£» da da da
ï£® Âµ ï£¹ dE ï£® dEË† q(1 âˆ’ h) ï£¹ dq
(A13) = ï£¯ Âµ B âˆ’ C E1 âˆ’ c ï£º 1 âˆ’ ï£¯( c âˆ’ q ) 2 + ï£º
ï£° h ï£» da ï£° dq ÂµC ï£» da
dE ï£® (1 âˆ’ h )( c âˆ’ 2q ) ï£¹ dq
= [(1 âˆ’ a)q âˆ’ c ] 1 + ï£¯ ï£º
da ï£° ÂµC ï£» da
The first term in the last line of (A13) is negative due to the 1.o.c. in (3b). The bracket in the
second term is also negative, as q > 2c (cf. the expression of q above). Thus, since q increases in
a for sufficiently low levels of a, we know that B must decrease when a is increased starting at
zero, and as long as dq/da is positive. Furthermore, consider a level of a where dq/da is negative.
Assume that a is increased from a1 to a2, so that q(a1) > q(a2). Consider the suboptimal pair a1
and q(a2). We know that B(a1,q(a1)) is higher than B(a1,q(a2)), since q(a1) is the optimal quota
price given a1. What about the comparison between B(a1,q(a2)) and B(a2,q(a2))? Obviously,
emissions in region 2 are the same in these two cases, and thus the two last terms in (10a) are
identical. Moreover, emissions in region 1 are lowest when a is lowest (given the same q). Since
R1â€²(E1) < c whenever offsets are used, B must be highest when E1 is lowest. Thus, B(a1,q(a2)) >
B(a2,q(a2)). But then we have shown that B is decreasing in a also when q is decreasing in a. A
higher level of a will therefore unambiguously reduce welfare whenever offsets are used.
31
Finally, we can show that it is optimal to switch to no offsets exactly when global emissions are
the same with and without offsets. To see this, we first calculate the level of a where global
emissions are identical with and without offsets. Then we must have:
= k 1=
E âˆ’E
k 0 Ë† = q (1 âˆ’ h)
= E20 âˆ’ E
1 1 2
ÂµC
Âµ B âˆ’ (1 âˆ’ a )q Âµ âˆ’c q
hâˆ’ B h = (1 âˆ’ h)
ÂµC ÂµC ÂµC
q [1 âˆ’ h + (1 âˆ’ a )h ] =ch
(A14)
1 âˆ’ ah
c [1 âˆ’ h + (1 âˆ’ a )h ] = ch
a h âˆ’ 2ah âˆ’ h + 2
2
1 âˆ’ 2ha + h 2 a 2= h 2 a 2 âˆ’ 2h 2 a âˆ’ h 2 + 2h
1âˆ’ h
a=
2h
We denote this level of a as Ã¢. It follows straightforwardly that:
(A15) ï€¤ ) = 2h c
q(a
h +1
We next calculate the difference in welfare with and without offsets when global emissions are
the same:
32
=k 1=k 0
B âˆ’B = R= =
1 ( E1 ) âˆ’ R1 ( E1
k 1 k 0
) âˆ’ q E20 âˆ’ E
Ë†
2( )
= ÂµB
=
( E1k 1 =
âˆ’ E1k 0
2h (
)âˆ’ Âµ C=
(E ) âˆ’(E )
1
k 1 2 =
1
k 0 2
) âˆ’ q Âµq (1 âˆ’ h)
C
q
= ( ÂµB âˆ’ q ) (1 âˆ’ h) âˆ’
ÂµC
ÂµC ï£® Âµ B 2 ÂµC 2 h 2 + c 2 ÂµC 2 h 2 + q 2 ÂµC 2 (1 âˆ’ h) 2 âˆ’ 2cÂµ B ÂµC 2 h 2 + 2q Âµ B ÂµC 2 h(1 âˆ’ h) âˆ’ 2cq ÂµC 2 h(1 âˆ’ h)
ï£¯
2h ï£° ÂµC 4
Âµ B 2 h 2 âˆ’ 2cÂµ B h 2 + c 2 h 2 ï£¹
âˆ’ ï£º
ÂµC 2 ï£»
1
= ï£® 2 Âµ B hq âˆ’ 2 Âµ B h 2 q âˆ’ 2hq 2 + 2h 2 q 2 âˆ’ Âµ B 2 h 2 âˆ’ c 2 h 2 âˆ’ q 2 (1 âˆ’ h) 2
2 h ÂµC ï£°
+ 2cÂµ B h 2 âˆ’ 2q Âµ B h(1 âˆ’ h) + 2cqh(1 âˆ’ h) + Âµ B 2 h 2 âˆ’ 2cÂµ B h 2 + c 2 h 2 ï£¹
ï£»
ï£« 2h ï£¶
q ( âˆ’q + h 2 q + 2ch(1 âˆ’ h) ) q ï£¬ (âˆ’1 + h 2 ) c + 2ch(1 âˆ’ h) ï£·
h +1
=(A16) = ï£ = ï£¸ 0
2 h ÂµC 2 h ÂµC
Thus, we have shown that, when global emissions are the same with and without offsets, welfare
is also the same with and without offsets. As welfare is monotonically decreasing in the level of
a, and global emissions are monotonically increasing in the level of a, it must be optimal to
switch to no offsets exactly when global emissions are the same in the two cases.
33