metadata: authors: 0: author: Rosendahl, Knut Einar
1: author: Strand, Jon
count: World
docna: 0: docna: Emissions trading with offset markets and
            free quota allocations
docty: Policy Research Working Paper
subtopic: Climate Change Mitigation and Green House Gases,Human Rights,Energy Demand,Energy and Environment,Energy and Mining,Global Environment
teratopic: Environment,Macroeconomics and Economic Growth,Energy
abstracts: cdata!: This paper studies interactions between
            a "policy bloc's" emissions quota market and
            an offset market where emissions offsets can be purchased
            from a non-policy "fringe" of countries (such as
            for the Clean Development Mechanism under the Kyoto
            Protocol). Policy-bloc firms enjoy free quota allocations,
            updated according to either past emissions or past outputs.
            Both overall abatement and the allocation of given abatement
            between the policy bloc and the fringe are then inefficient.
            When the policy-bloc quota and offset markets are fully
            integrated, firms buying offsets from the fringe, and all
            quotas and offsets, must be traded at a single price; the
            policy bloc will either not constrain the offset market
            whatsoever, or ban offsets completely. These cases occur
            when free allocation of quotas is less (very) generous, and
            the offset market delivers large (small) quota amounts.
            Governments of policy countries would instead prefer to buy
            offsets directly from the fringe at a price below the
            policy-bloc quota price. The offset price is then below the
            marginal damage cost of emissions and the quota price in the
            policy bloc is above the marginal damage cost. This is also
            inefficient as the policy bloc, acting as a monopsonist,
            purchases too few offsets from the fringe.
repnme: repnme: Emissions trading with offset markets and
            free quota allocations
display_title: Emissions trading with offset
            markets and free quota allocations
listing_relative_url: /research/2012/11/17014680/emissions-trading-offset-markets-free-quota-allocations
topicv3: Carbon Tax m1326372 1845,Energy and Extractives m1326595 507,Climate Change Mitigation m1327079 579,Environment, Natural Resources and Blue
            Economy m1327391 566,Discrimination m1505707 3379
publishtoextweb_dt: 2012-11-01T00:00:00Z
historic_topic: Environment,Macroeconomics and Economic Growth,Energy
no_of_pages: 35
model_extractions: 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 “policy bloc’s” emissions quota market and an offset market where emissions offsets can be purchased from a non-policy “fringe” of countries (such as for the Clean Development Mechanism under the Kyoto Protocol). Policy-bloc firms enjoy free quota allocations, updated according to either past emissions or past outputs. Both overall abatement and the allocation of given abatement between the policy bloc and the fringe are then inefficient. When the policybloc quota and offset markets are fully integrated, firms buying offsets from the fringe, and all quotas and offsets, must be traded at a single price; the policy bloc will either not constrain the offset market whatsoever, or ban offsets completely. These cases occur when free allocation of quotas is less (very) generous, and the offset market delivers large (small) quota amounts. Governments of policy countries would instead prefer to buy offsets directly from the fringe at a price below the policy-bloc quota price. The offset price is then _below_ the marginal damage cost of emissions and the quota price in the policy bloc is _above_ the marginal damage cost. This is also inefficient as the policy bloc, acting as a monopsonist, purchases too few offsets from the fringe.

This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at jstrand1@worldbank.org..

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ 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: <u>ker@ssb.no. **Extended Term Consultant, Development Research Group, the World Bank,</u> Washington DC 20433, USA. E-mail: jstrand1@worldbank.org. We are grateful for valuable comments from Cathrine Hagem, Ian Parry, Halvor Briseid Storrøsten 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 EU’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 firms’ 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 “raises the bar” 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 firms’ 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 “needs”. This issue has been treated in the literature, e.g. by Böhringer and Lange (2005), Rosendahl (2008), Harstad and Eskeland (2010) and Rosendahl and Storrøsten (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 firms’ 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 “closed system” with no offset market and identical updating rules and price expectations across emitting firms (Böhringer and Lange, 2005). A second type of allocation mechanism entails that _free allocations are based on output of the relevant entities_ , using then a “benchmark” emissions intensity index for the industry (“output-based allocation”). Under this mechanism a high level of output will secure a large amount of free allocations (on the presumption that the “need” 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. Böhringer 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 “offset” 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 “policy bloc” of countries, which faces a “fringe” 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 policybloc 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.<sup>1</sup> 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 countries’ 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 “not too beneficial” 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 bloc’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 firms’ 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 “not too large”); 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 “policy bloc” 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 “non-policy countries” (the “fringe”). 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 _R_ 1( _E_ 1, _X_ 1), where _E_ 1 and _X_ 1 denote ′ respectively emissions and production from the policy bloc. We have _R_ 1 _E_ > 0 for _E_ 1( _t_ ) < _E_ 10( _t_ ) ′ and _R_ 1 _X_ > 0 for _X_ 1( _t_ ) < _X_ 10( _t_ ), where _E_ 10( _t_ ) and _X_ 10( _t_ ) are the BaU emissions and production levels. The probability that any given firm survives to the next period is _β_ (same for all firms and such that firm exits are random events).

Assume that free allocation of emissions rights follow an “updated grandfather” rule whereby the number of free quotas awarded to firms with emissions _E_ ( _t_ ) and production _X_ ( _t_ ) in period _t_ equals _αE_ ( _t_ ) + _γX_ ( _t_ ) in period _t_ +1. That is, allocation of quotas is based on firms’ past emissions and/or production, where the updating parameters _α_ and _γ_ are assumed to lie between zero and unity. In the two first phases of the EU ETS (2005-2012), _α_ has been closer to one while _γ_ has been mostly zero. In the third phase (2013-2020), however, _α_ is mostly zero except in some sectors,<sup>2</sup> while _γ_ 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 _δ_ , and assume that firms have a potentially infinite life span. The discounted value of net returns for a representative firm in the policy bloc, _V_ 1( _t_ ), can then be expressed as where _W1_ ( _t_ ) denotes net returns in period _t_ . _W_ 1( _t_ ) is in turn given by<sup>3</sup> 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 _E_ 1( _t_ ) ≥ _αE_ 1( _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 _αE_ 1( _t_ -1) + _γX_ 1( _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 (1a) The representative firm seeks to maximize _V_ 1( _t_ ) with respect to current emissions and production levels, _E_ 1( _t_ ) and _X_ 1( _t_ ). This yields (note that _E_ 1( _t_ ) and _X_ 1( _t_ ) do not enter in _V_ 1( _t_ +2)) and and thus Focusing on steady-state cases, with constant revenue functions, constant number of firms (so that a fraction _β_ 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 where _a_ = _αβδ_ and _b_ = _γβδ_ . The parenthesis on the right-hand side of (3b) expresses the “net price” paid for emissions quotas by policy country firms in a steady state. This price is lower than the “gross” price _q_ since the free quota allocation is an increasing function of past emissions. The difference between the “net” and the “gross” price depends on the product of three parameters: the updating share ( _α_ ); the probability of firm survival to the next period ( _β_ ); and the discount factor ( _δ_ ). 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 _γ_ >0.<sup>4</sup> 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 “statutory” 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 _γ_ =0, and focus on the case with emissions-based allocation ( _α_ >0). In Section 4 we consider output-based allocation and set _α_ =0 (and _γ_ >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 _R_ 2( _E_ 2( _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.<sup>5</sup> Define this benchmark by _E_ 20( _t_ ) in period _t_ , given by 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 “relative baselines” 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 _Q_<sup>ˆ</sup> 2<sup>(</sup> _t_<sup>)</sup> . This supply corresponds to the difference between the benchmark emissions _E_ 20<sup>(</sup> _t_<sup>)</sup> and the emissions level _E_<sup>ˆ</sup> 2<sup>(</sup> _t_<sup>)</sup> given by 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 _Q_ 2 _k_ denote the corresponding offset purchases. We then have 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).<sup>6</sup> 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 “purchase rate” parameter _k_ . If _k_ is sufficiently small, the overall share of offsets might become “optimal”. 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_ (= _αβδ_ ) which we treat as exogenously given.<sup>7</sup> Equivalently, we could search for optimal combinations of _E_ * and _k_ , where _E_ * is the emissions cap level in the policy bloc.<sup>8</sup> We have _E_ 20 is assumed given and unaffected by the model parameters.<sup>9</sup> The issue of what constitutes an “optimal” 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_ ): 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. _E_ 1 and<sup>_E_ˆ</sup> 2<sup>are both simple functions of</sup><sup>_q_</sup> only (from (3b) and (6) respectively). For _E_ we have the following accounting definition: We can now insert into (9) from (10) for _E_ , which yields This expression can be maximized with respect to _q_ and _k_ , yielding the following general firstorder conditions for internal solution:

> 7 _β_ and _δ_ are non-policy parameters, whereas _α_ 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 “leakage” from the policy bloc to the fringe. In Section 4 we return to this issue.

9
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_ 20 − _E_<sup>ˆ</sup> 2 > 0 , an internal solution for _k_ is not feasible unless _q_ = _c_ , in which case any value of _k_ fulfills (11).<sup>10</sup> 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 _B_ 1 is then increasing in _k_ for any _k._ On the other hand, if _q_ > _c_ is optimal, _k_ = 0 ( _B_ 1 is then decreasing in _k_ ). The intuition here is the following: From above we know that _E_ 1 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_ ≥ _c_ and _k_ = 0, or _q_ < _c_ and _k_ = 1. Let us characterize these two potential outcomes of the policy bloc’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 _R_ 1 = _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): 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 Böhringer 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_ = _C_ ’( _E_ ) as the only internal solution. However, in this case _E_ and thus _C_ ’( _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): 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:

- < _c_ , and thus it is optimal to increase _q_ until _q_ ≥ _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 _α_ is), the more likely this case is. Further, without the updating rule (i.e., _a_ = 0) this case can never occur.

- b) − _aq_ ∂∂ _Eq_ <u>1</u> < _E_ 20 − _E_<sup>ˆ</sup> 2<sup>(</sup> _q_<sup>)</sup> for _q_ = _c_ . In this case, equation (11b) must have (at least one) internal solution with _q_ < _c_ .<sup>11</sup> 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 _R_ 1 ( _E_ 1) < _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_ ≤ _c_ . Thus, there is just one internal solution which is optimal given _q_ ≤ _c_ .

11
- c) − _aq_ ∂∂ _Eq_ <u>1</u> < _E_ 20 − _E_<sup>ˆ</sup> 2<sup>(</sup> _q_<sup>)</sup> for some _q_ < _c_ (but not for _q_ = _c_ ). In this case we may or may not

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_ ≥ _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 _R_ 1 ( _E_ 1) < _q_ = _R_ 2 ( _E_ 2) < _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 ′ _R_ 1 ( _E_ 1) 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 bloc’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 _E_ 1 increases. Moreover, global emissions increase, too, when _a_ is increased, as the higher emissions in the policy bloc (“region 1”) will always dominate the (initially) lower emissions in the fringe (“region 2”; 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_

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<!-- Start of picture text --> E1 (k=1) E1 (k=0) E2 (k=1) E2 (k=0) E (k=1) E (k=0)<br>E1 (k 1) E1 (k 0) E2 (k 1) E (k 1) E (k 0)<br>1.75<br>1.5<br>1.25<br>1<br>0.75<br>0.5<br>0.25<br>0<br>0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br>a<br><!-- End of picture text -->

**Figure 2.** Quota price and welfare in the policy bloc and total welfare as a function of _a_

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<!-- Start of picture text --> q (k=1) q (k=0) B1 (k=1) B1 (k=0) B (k=1) B (k=0)<br>B (k 1) B (k 0) B1 (k 1) B1 (k 0) (k 0) (k 1)<br>1.6<br>1.4<br>1.2<br>1<br>0.8<br>0.6<br>0.4<br>0.2<br>0<br>0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br>a<br><!-- End of picture text --> Figures 1 and 2 illustrate the cases discussed in this section, where we assume identical, quadratic mitigation costs in the two regions.<sup>12</sup> The thick curves show the outcome of region 1’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: _μjA_ = _μjB_ = _μjC_ = 1; _c_ = 0.5.

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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.<sup>13</sup> 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 firms’ 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 emissionsbased 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 “carbon leakage” through the markets of emission-intensive, trade-exposed goods. Such “leakage” 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_ .

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greater output and emissions in other regions with no or more lenient climate policy.<sup>14</sup> With output-based allocations, an “emission intensity benchmark” 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 “benchmark” independent of the emissions of any one given firm.<sup>15</sup> In view of these considerations, we will in this section focus on the case with output-based allocation, and set _γ_ >0 and _α_ =0. From (3b) we see that absent an offset market, the level of emissions within the policy bloc is “optimal” 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: ˆ (14) _B_ 1( _q_ , _k_ ) = _R_ 1( _E_ 1( _q_ ), _X_ 1( _q_ )) − _cE_ 1( _q_ ) − ( _c_ − _q_ ) _kE_ 2 ( _q_ , _X_ 1( _q_ )) − [ _c_ (1 − _k_ ) + _qk_ ] _E_ 20 ( _X_ 1( _q_ )).

ˆ Following the discussion above, we assume that ∂ _X_ 1<sup>/</sup> ∂< _q_ 0 , and ∂ _E_ 2 / ∂ _X_ 1 < 0 . The first of these derivatives simply expresses that a higher quota price (or emission cost) reduces energyintensive 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 emissionintensive, trade-exposed industries, and are preconditions for having a leakage problem in our context. The size of ∂ _X_ 1<sup>/</sup> ∂ _q_ determines how sensitive domestic output is to the quota price, ˆ whereas ∂ _E_ 2 / ∂ _X_ 1 determines “leakage exposure” for domestic firms. To simplify the analytics ˆ below, we assume ∂ _E_ 2 / ∂ _X_ 1 = ∂ _E_ 20 / ∂ _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; Böhringer et al., 2011). Leakage may occur through the international markets for fossil fuels, and through the markets for emissionintensive, 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
Equation (12) still holds, so we must either have _q_ ≥ _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_ = _γβδ_ ), 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 _R_ 1E _′_ = _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 ˆ have _b_ > −∂ _E_ 2 / ∂ _X_ 1 . Let _q_<sup>_*_</sup> denote the optimal quota price in this case.

ˆ If the last term in (15) is negative, e.g., because ( −∂ _E_ 2 / ∂ _X_ 1 ) is large, it is optimal to reduce the quota price below _q_<sup>_*_</sup> . 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_<sup>_*_</sup> . 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.<sup>16</sup> 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 ˆ ( −∂ _E_ 2 / ∂ _X_ 1 ). We have just concluded that _q_ < _c_ if the term is non-positive, but cannot conclude 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, _γ_ = _b_ /( _βδ_ ) is set close to _E_ 1/ _X_ 1 (almost

> 16 Obviously, the first-best response would be to lower _b_ , but this might be difficult for political reasons.

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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 ˆ the policy bloc, _b_ /( βδ ) ≈−∂ _E_ 2 / ∂ _X_ 1 . More likely, however, the output replacement is less than 100%. But since emissions intensities will often be higher in the fringe, it is still difficult to ˆ judge in general whether _b_ is higher or lower than −∂ _E_ 2 / ∂ _X_ 1 , at least for the highly exposed 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 ˆ 70% compensation. Hence, sectors can probably be found where _b_ exceeds −∂ _E_ 2 / ∂ _X_ 1 . This 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 firms’ past-period output, and an offset market with free trading among market participants. Then:_

ˆ _i) If allocation is not too generous relative to the leakage exposure, i.e.,_ b _≤_ −∂ _E_ 2 / ∂ _X_ 1 _, it is not optimal to put any restrictions on the use of offsets._ ˆ _ii) If allocation is generous relative to the leakage exposure (_ b _>>_ −∂ _E_ 2 / ∂ _X_ 1 _), and the offset potential E_ 20 − _E_<sup>ˆ</sup> 2 _is sufficiently limited, it is optimal for the policy bloc to ban the use of offsets._ _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**

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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 “cheaper” from “fringe” 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.<sup>17</sup> The problem is now formally similar to above except that we have two quota prices, _q_ 1 for the policy bloc, and _q_ 2 for the offset market (in the fringe), instead of just a single price, _q_ .<sup>18</sup> Define now the policy bloc’s objective function in similar fashion to (9a), by (16) is here maximized with respect to _q_ 1, _q_ 2 and _k_ , yielding the following set of first-order conditions:

> 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.

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′ From (17), we immediately find that _R_ 1 = _c_ : mitigation is optimal in the policy bloc. This implies that _q_ 1, given from (13), exceeds marginal damage cost from emissions as long as _a_ > 0. Consider next _q_ 2. From (18) we now find We see that the optimal quota price, at which quotas are purchased from the offset market, is below marginal damage cost of emissions. This is a standard monopsony solution where a unified policy bloc government (acting as a monopsonist in the offset market) trades off an environmental efficiency aspect (“too little” mitigation) against a fiscal cost aspect (expenditure on the purchase of offset quotas). It results in too little mitigation through offsets, but the gain for the policy bloc is that quota expenditures are reduced.

The main difference between mitigation policy towards domestic firms versus firms in fringe countries is that government payments to domestic firms are part of net social welfare for the policy bloc. Obviously, this is not the case for payments going to fringe country firms. The government thus has an incentive to limit the latter kind of payments, which it does in (constrained) optimal fashion by acting as a monopsonist.

The quota price facing policy bloc countries, _q_ 1, is here _greater than_ marginal damage cost; while the quota price facing firms in fringe countries, _q_ 2, is _lower than_ this damage. The difference between the internal and external quota prices, _q_ 1 and _q_ 2, can here be substantial. In our model, quota prices can differ only because private actors are not allowed to trade in the offset markets. Admittedly, it is unrealistic to assume that governments-sponsored purchases of offsets will lead to such strong discrimination in disfavor of offset sellers; and that offsets are always purchased at a given price. Our analysis must be viewed as a first cut at this issue, so that further research is warranted.<sup>19</sup> Using that _c_ – _q_ 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.

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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 firms’ past-period emissions, and an offset market fully controlled by policy-bloc governments. Then:_

- _i) Within the policy bloc, the equilibrium quota price exceeds firms’ 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).<sup>20</sup>

> 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).

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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 _q_ 1 – _q_ 2 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_ = _q_ 1 – _q_ 2.<sup>21</sup> 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 “discount” 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 “discounted” (or reduced in value relative to domestic abatement), call it _λ_ , were to equal ( _q_ 1 – _q_ 2)/ _q_ 1. 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 ( _q_ 1 – _q_ 2) _Q_ 2, where _Q_ 2 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 _λQ_ 2 quotas, we get exactly the same outcome with respect to emissions, costs and government revenues as in the first case.<sup>22</sup> 21 We thank Ian Parry for suggesting this solution.

22 Note that in the ”discount” case, policy bloc emissions will be lower than in the “tax” 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 _λQ_ 2 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 _λQ_ 2 quotas is ( _q_ 1 – _q_ 2) _Q_ 2.

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# **6. Conclusions**

We have studied optimal policies of a “policy bloc” of countries enforcing an emissions trading system with free quota allocation, combined with an offset market with emissions reductions purchased from a “fringe” 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 “fringe”. 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 bloc’s optimal price discrimination solution implementable for the offset market.

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# **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_ ) = _μjA_ + _μjBEj_ – ( _μjC_ /2)( _Ej_ )<sup>2</sup> , with _μjA_ , _μjB_ , _μjC_ > 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 _μ1A_ = _μ2A_ = _μA_ , _μ1B_ = _μ2B_ = _μB_ , _μ1C_ = _μC_ / _h_ and _μ2C_ = _μC_ /(1- _h_ ), with 0 < _h_ < 1. All results shown here also apply in the general case.

First, we have ′ ′ (A1)-(A2) _R1_ ( _E1_ ) = _μB_ – _μCE1_ / _h_ , _R2_ ( _E2_ ) = _μB_ – _μCE2_ /(1- _h_ ), _E_ 10 = _hμB_ / _μC_ and (A3) _E2_ 0 = (1- _h_ ) _μB_ / _μC_ .

Equation (3b) then gives: Equation (11b) now gives: leading to: 29
2 − _a_ (in the special case with _h_ = 1⁄2, this simplifies to _q_ = _a_ 2 − 2 _a_ + 3 _c_ ) We can now examine the effects on _q_ of increasing _a_ : (A10) 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).

What about the effects on _Ej_ of increasing _a_ ?<sup>_E_ˆ</sup> 2<sup>obviously follows the opposite direction of</sup><sup>_q_.</sup> For _E_ 1 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 _E_ 1( _a_ , _q_ ) above, using the expression for _q_ : 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
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): 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_ > 2 _c_ (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 _a_ 1 to _a_ 2, so that _q_ ( _a_ 1) > _q_ ( _a_ 2). Consider the suboptimal pair _a_ 1 and _q_ ( _a_ 2). We know that _B_ ( _a_ 1, _q_ ( _a_ 1)) is higher than _B_ ( _a_ 1, _q_ ( _a_ 2)), since _q_ ( _a_ 1) is the optimal quota price given _a_ 1. What about the comparison between _B_ ( _a_ 1, _q_ ( _a_ 2)) and _B_ ( _a_ 2, _q_ ( _a_ 2))? 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 ′ _R_ 1 ( _E_ 1) < _c_ whenever offsets are used, _B_ must be highest when _E_ 1 is lowest. Thus, _B_ ( _a_ 1, _q_ ( _a_ 2)) > _B_ ( _a_ 2, _q_ ( _a_ 2)). 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.

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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: We denote this level of _a_ as _â_ . It follows straightforwardly that: We next calculate the difference in welfare with and without offsets when global emissions are the same: 32
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.

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