int ral links in cap and trade schem es - dipot.ulb.ac.be · ties in future periods (dixit and...

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Page 1: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

 

 

 

Int

50, F.D.

ertempo

SBS‐E

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UL Roosevelt A

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AuEM, ECARES,

CARES wo

ECARES LB - CP 114/Ave., B-1050ww.ecares.o

 

s in Cap‐

 

 urélie Slecht, Université L

     

 

 

rking pap

 

 

/04 0 Brussels Borg

and‐Trad

ten Libre de Brux

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BELGIUM

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Page 2: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

Intertemporal links in cap-and-trade schemes

Aurelie Slechten1

Universite libre de Bruxelles (ECARES and SBS-EM)

Abstract

I study the effects of intertemporal emission permits trading in a cap-and-tradescheme when firms’ abatement investments have long-term effects. In a two-periodgeneral equilibrium model, firms make trading and investment decisions in eachperiod to meet their caps. I compare equilibrium abatement levels and permitprices, with and without intertemporal trading. Intertemporal trading may reducetotal abatement investments over the scheme. Without intertemporal trading,some investments in period 1 are entirely driven by second-period abatement needs;in this case, intertemporal trading may reduce investments in period 1 as some aresubstituted by intertemporal permit trading. Descriptive evidence from the EUEmissions Trading System (ETS) illustrates this potential effect. I also show thatif the number of permits issued by the regulator is not equal to the socially optimallevel of emissions, then banning intertemporal trading may reduce the social costthanks to the long-term properties of investments.

Keywords: cap-and-trade schemes, emission trading, abatements, investment,banking, borrowingJEL classification: Q50; Q58; D92

Email address: [email protected] (Aurelie Slechten)1ECARES (Universite libre de Bruxelles), Avenue F.D. Roosevelt 50, CP139, B-1050, Bel-

gium; Phone: +32 2 650 38 55

Preprint submitted to Elsevier September 19, 2011

Page 3: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

1. Introduction

In recent years, governments have shown increasing interest in the use of cap-and-trade schemes for pollution control because such schemes are considered a costeffective strategy for controlling environmental pollutants compared to the usualcommand-and-control policies. Economists identify permit trading between firmsas a source of potential cost savings. Indeed, Montgomery (1972) proves thatin a static framework, overall pollution abatement costs are minimized becausefirms with high abatement costs may purchase permits from companies with lowabatement costs so that all participants face the same marginal costs.2

Intertemporal permit trading is another source of cost savings. I define in-tertemporal permit trading as banking permits for future use or borrowing per-mits from future allocations.3 In theory, in a dynamic setting, if firms can tradetheir permits intertemporally, they will be able to equalize their marginal costs ofabatements across periods.

However, in practice, regulators often place limits on intertemporal trading.Intertemporal trading was indeed not allowed between the first two phases of theEuropean Emission Trading Scheme (EU ETS). The European Commission chosegenerous permit allocations in the first phase in order to allow firms to achievetheir emission cap at a reasonable cost. However, the Commission feared thatpermit banking would prevent the EU from meeting its Kyoto targets by 2012.In addition, by allowing firms to borrow permits, the EU ran the risk that firmswould default on their permits, compromising their emission reduction goal. Thereis thus a priori a tension between flexibility and political objectives.

In this paper, I first aim at exploring how the additional intertemporal linkresulting from investments in abatement partially helps to ease this tension. In-deed, investments in abatement often have long-term effects and can thus providesome intertemporal flexibility in terms of emission reductions when intertemporaltrading is not allowed. For example, in the EU ETS, the most widely used tech-niques to reduce carbon dioxide (CO2) emissions are efficiency improvements (forexample, heat recovery and thermal insulation) that affect a firm’s core productionprocess and that have lasting effects (Faure et al., 2008).

The second objective of the paper is to study the interactions between these twointertemporal links (abatements with long-term effects and intertemporal trading).I analyze how firms use these long-term abatements depending on whether or not

2This result of static cost effectiveness relies on many simplifying assumptions. Relaxing oneof these assumptions may prevent permit markets from achieving cost effectiveness: regulationof firms that trade permits (Bohi and Burtraw, 1992), market power (Hahn, 1985 and Eshel,2005) or technological linkages between permit and output markets (Misiolek and Elder, 1989).

3See Rubin (1996).

2

Page 4: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

the intertemporal trading of permits is allowed.I build a two-period general equilibrium model in which firms are perfectly

competitive and emit some pollutant. The amount of pollutant emitted deter-mines firms’ demand for permits. The emissions cap, the supply of permits, is setexogenously by a regulator and is known before the opening of the market. I solvethe model with and without intertemporal permit trading. In both cases, firmshave to choose the optimal compliance strategy: the optimal purchase of permitsand the optimal level of investment in abatement technologies.4 The permit priceis determined endogenously by market-clearing conditions. The long-term effectsof investments are twofold. First, investments in abatement remain useful beyondthe period during which they occur. Second, to account for learning-by-doing(Jaffe and Stavins, 1994) and the possibility of diminished abatement opportuni-ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period2 to depend on period 1 investment levels.

I first show that if the regulator does not allow firms to trade permits in-tertemporally, investment in abatement provides firms with some intertemporalflexibility. Indeed, in the case of high future abatement needs, firms reduce theiremissions below the first-period cap (i.e. firms over-invest in abatement in period1). This over-investment is entirely driven by second-period abatement needs andleads to an excess of permits in period 1. For this result to hold, two aspects ofinvestment are essential: investments in abatement must have long-term effectsand must be characterized by learning-by-doing.

Secondly, I show that allowing intertemporal permit trading may reduce thetotal investments in abatement over the scheme—an immediate consequence of myfirst result. If intertemporal trading is allowed, any excess of permits in period1 (due to over-investment) can be banked and used for compliance in period 2,reducing abatement needs.

Thirdly, I show that a scheme with intertemporal trading does not necessarilyimply earlier investments in abatement compared to a scheme without intertempo-ral trading. Intuitively, if intertemporal trading is allowed and if abatement needsin period 2 are high compared to those in period 1, firms will have an incentive toincrease their investments in period 1 because they can bank permits and use themto cover high future abatement needs. However, for such second-period abatementneeds, the effect of intertemporal trading on first-period investments will dependon the level of over-investment in period 1. For very high second-period abate-ment needs, over-investment in the absence of intertemporal trading is so high

4In this model, I ignore the output market. As in Chao and Wilson (1993), I consider onlythe net demand for emissions after the permits for fuel substitution or for output reduction havealready been accounted for.

3

Page 5: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

that part of period 1 investments, which were driven by second-period needs, canbe substituted by banked permits.

Finally, while intertemporal permit trading is cost effective, it is not necessarilyoptimal from the point of view of the society. I show that, if the number of permitsissued by the regulator is higher than the socially optimal level of emissions, ban-ning intertemporal trading may reduce the social cost of pollution control thanksto over-investment.

Investments in abatement and intertemporal trading have largely been studiedseparately in the literature. Models of intertemporal trading (Rubin, 1996 ; Klingand Rubin, 1997 ; Schennach, 2000; Yates and Cronshaw, 2001 or Seifert et al.,2008) have assumed that the effects of investments last for only one period. Inthese models, banking and investment in abatement can never be substitutes. Ifintertemporal trading is not allowed, firms never invest beyond the required levelin period 1. Such investments would be useless for period 1 and would not allowfirms to meet their future emission caps. By contrast, in my model, investmentshave long-term effects. As a result, even if intertemporal trading is not allowed,investment is used not only for immediate compliance but also for minimizationof future compliance cost.

Investments in abatement technologies have been studied by Insley (2003),Xepapadeas (1999) as well as by Chao and Wilson (1993) in partial equilibriummodels. Although these papers are motivated by cap-and-trade schemes, prices ordemand functions in these models are exogenous, so it is impossible to study theinteractions between the permit market and investment decisions. Zhao (2003)studies long-term investments in abatement in a general equilibrium model. Hisfocus is on the effect of the level of cost uncertainty on investment (see also Bal-dursson and von der Fehr, 2004). However, in his model, firms are not allowed totrade permits intertemporally, so there is no consideration of the role of investmentas a substitute to intertemporal trading.

The remainder of the paper is organized as follows. Section 2 lays out thetwo-period model with and without intertemporal trading. Section 3 comparesthe two cases and determines the main effects of allowing intertemporal trading.In section 4, I present a short welfare analysis of intertemporal trading. Section 5illustrates the results of the model using descriptive evidence from the EU ETS. Iconclude in section 6.

2. A two-period model

2.1. The Setting

In this section, I introduce a simple dynamic model to analyze the firms’ be-havior with and without intertemporal permit trading.

4

Page 6: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

There is a continuum of risk neutral firms emitting some pollutant. The reg-ulator decides to limit these emissions by implementing a cap-and-trade schemeover two periods (1 and 2). Firms are heterogenous in terms of their emissions.For each firm, indexed by k ∈ [0, 1], the business-as-usual emissions in periods 1and 2 are noted e1k and e2k. For future reference, I define the average emissionsas5 ∫ 1

0

etkdk = εt t = 1, 2

I also define the idiosyncratic component of emissions as εtk = etk − εt. Firm k’semissions thus have two components: εt is the component common to all firmsand εtk is the idiosyncratic component which differs for each individual firm.

Each firm is allocated ntk permits at each period t = 1, 2. A single permitallows the firm to emit one unit of the pollutant. The aggregate cap is the sum ofallocations for all firms:

∫ 1

0ntkdk = Nt for all t.6

At each period, the individually required abatement, atk = etk − ntk, is thedifference between the business-as-usual level of emissions and the initial allocationfor each k. It represents the effort imposed upon each individual firm by theregulator. At the aggregate level, the required abatement is given by

∫ 1

0atkdk =

At = εt − Nt. It is the total amount by which emissions have to be reduced ineach period. Thereafter, I assume that the total aggregate required abatement(A1 + A2) is strictly positive because the objective of the regulator is pollutionreduction.

Firms have different compliance options: (1) permit trading, (2) investment inabatement technology and (3) intertemporal permit trading

First, they can buy permits from other firms: ytk represents a purchase ofpermits by firm k at period t (if ytk < 0, it is a sale of permits). The permit pricept is endogenously determined through market-clearing.

Second, firms can invest in an abatement technology that has long-term effects.I assume that the investment costs in periods 1 and 2 are the same for all firmsand are respectively7

5Since there is a continuum of firms represented by the interval [0, 1], average emissionscorrespond to aggregate emissions.

6Aggregate variables are represented by capital letters while individual variables are repre-sented by lower case letters.

7It is possible to introduce some heterogeneity in the cost of investment. However it does notchange the main results of the model.

5

Page 7: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

C(i1k) =1

2i21k

C(i2k, i1k) =1

2i2k(i2k + 2θi1k)

In period 1, firm k invests in abatement at a cost C(i1k) to reduce its emissionsin both periods by i1k. In period 2, firm k can invest again in abatement at acost C(i2k, i1k) in such a way that its emissions in period 2 are e2k − i1k − i2k.

8

Aggregate investments in periods 1 and 2 are given by It =∫ 1

0itkdk for t = 1, 2.

I model investments in this way to capture two long-term effects of investment.On the one hand, investments made in each period increase the ”stock of abatementcapital”: investments made in period 1 reduce emissions in periods 1 and 2.

On the other hand, the model allows for investment opportunities in period 2 todepend on the investment made in period 1. This is because the initial investmentcan be expanded in period 2, so that the marginal cost of abatement in period2 will depend on period 1 investments. In Dixit and Pindyck (1999), it becomesmore expensive to invest in abatement the more stock of abatement capital thefirm already has.9 The model also allows for ”learning-by-doing”. As explainedin Jaffe and Stavins (1994), once a firm has found a supplier for an abatementtechnology or has learnt how to fit this technology into its production process,it is less expensive to expand this technology in the firm. To take this form oflearning into account, I introduce the parameter θ ∈ [0, 1] in my model. If θ = 1,there is no learning in the investment process and the investment cost in period2 is simply increasing in the level of investment in period 1, just as in Dixit andPindyck (1999). When θ is lower than 1, the investment process is characterizedby ”learning-by-doing”: the investment cost is still increasing in the first-periodinvestment but it increases by a smaller amount thanks to the firm’s experiencewith this technology. Therefore, in my model, there is learning-by-doing when,given any striclty positive investment level in period 1, the marginal cost of anadditional unit of investment in period 1 is larger than the marginal cost of thefirst unit of investment in period 2, or when θ ∈ [0, 1).

The third compliance option is intertemporal trading: firms can borrow permitsfrom their allocation in period 2 or can bank permits to use them in period 2. Tostudy the effects of intertemporal trading on investments in abatement, I analyzetwo different cases. In the benchmark case, the banking and borrowing of permits

8As in Zhao (2003), I set the depreciation rate of capital equal to zero.9This reflects the assumption that companies first use their cheapest abatement opportuni-

ties when reducing emissions. The more they have to abate the more expensive the remainingabatement opportunities will be.

6

Page 8: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

are not allowed. In the intertemporal trading case, both banking and borrowingare allowed.

For expositional simplicity, I ignore discounting, which does not qualitativelyaffect any of the results. I will come back on the effects of discounting in section3, when I analyze the results of the model.

2.2. The benchmark case: intertemporal trading not allowedWhen intertemporal trading is not allowed, firms minimize compliance costs

by choosing the number of permits they sell or buy in the market and the level ofinvestment in abatement.10

In period 2, each firm k solves the following problem

mini2k,y2k

[p2y2k +1

2i2k(i2k + 2θi?1k)]

s.t. e2k − i2k − i?1k ≤ n2k + y2k (1)

i2k ≥ 0

where i?1k denotes the investment in abatement of firm k in period 1.In addition, market clearing in period 2 entails that sales and purchases of

permits balance one another, i.e. ∫ 1

0

y2kdk = 0

The Lagrangian for problem (1) is given by

L(i2k, y2k, λ2k) = [p2y2k +1

2i2k(i2k + 2θi?1k)] + λ2k(e2k − i2k − i?1k − n2k − y2k)

where λ2k is the Lagrange multiplier. The first order conditions are

∂L

∂y2k= p2 − λ2k = 0 (2)

i2k∂L

∂i2k= i2k(i2k + θi?1k − λ2k) = 0; i2k ≥ 0; i2k + θi?1k − λ2k ≥ 0

λ2k(e2k − i2k − i?1k − n2k − y2k) = 0; λ2k ≥ 0; (3)

e2k − i2k − i?1k − n2k − y2k ≤ 0

10For the benchmark case, I rule out the case θ = 1 to avoid undetermined aggregate invest-ments in abatement at equilibrium when A1 < A2. Aggregate investments in period 1 must atleast cover the aggregate required abatement (I1 ≥ A1). If A1 < A2, as first-period investmentsare also useful in period 2, firms can decide to invest more than A1 in period 1 in order to reducecosts in period 2. However, as θ = 1, the marginal cost of the first unit of investment in period 2is equal to the marginal cost of the last unit of investment in period 1. Firms are thus indifferentbetween investing in period 1 or in period 2 and the equilibrium level of investments in period 1is given by: A1 ≤ I1 ≤ A2.

7

Page 9: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

Because there is a single clearing price p2, λ2k = λ2 for all k: at the equilibrium,the compliance constraint is binding for all firms or for none. λ2 can be interpretedas the marginal shadow value of an additional permit in the market and, by (2),it is equal to the permit price. Equation (3) states that at the end of period 2,either p2 is equal to zero because there is a surplus of permits in the market or p2is positive and the compliance constraint is binding for all firms.

If the compliance constraint in (1) is binding for all firms then the aggregateinvestment I2 should be exactly equal to the net required abatement (A2 − I?1 ).As this aggregate investment in period 2 I2 should either be positive or zero, thecompliance constraint in (1) is binding only if A2 ≥ I?1 . In this case, the solutionfor all k is given by

i?2k = p?2 − θi?1ky?2k = a2k − (1− θ)i?1k − p?2 (4)

p?2 = A2 − (1− θ)I?1

and I?2 = A2 − I?1 , the aggregate net required abatement in period 2.If A2 < I?1 , investment in period 1 is high enough to cover the required abate-

ment in period 2. The period 2 equilibrium is given by

λ?2k = p?2 = 0, i?2k = 0 (5)

Turning now to period 1, firm k solves the following problem:11

mini1k,y1k

p1y1k + C(i1k) + p?2y?2k + C(i?2k, i1k)

s.t. e1k − i1k ≤ n1k + y1k (6)

As before, market clearing entails that∫ 1

0

y1kdk = 0

where p?2, i?2k and y?2k are described either by (4) or (5) depending on the value of I1

(the aggregate investment in period 1) with respect to A2 (the aggregate requiredabatement in period 2).

Due to the compliance constraint in period 1, the aggregate investment inperiod 1 should at least cover the aggregate required abatement in that period (A1).

11I do not impose a positivity constraint on i1k because such a constraint is automaticallysatisfied as long as one of the two aggregate required abatements is positive.

8

Page 10: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

Therefore if A1 > A2, firms know for sure that the compliance constraint in period2 is not binding (the solution in period 2 is given by (5)) because I1 ≥ A1 > A2.In this case, problem (6) simplifies to

mini1k,y1k

p1y1k + C(i1k)

s.t. e1k − i1k ≤ n1k + y1k

And the solution to this problem is simply

pB1 = iB1k = A1 (7)

yB1k = a1k − A1

where index B refers to the solution in the benchmark case.If A1 ≤ A2, firms will never choose an investment in period 1 such that the

aggregate investment I1 is strictly higher than the aggregate required abatement inperiod 2, A2. Indeed, after period 2, the market stops and any abatement beyondthe required level is useless and costly. Solution for period 2 is thus given by (4)and problem (6) simplifies to

mini1k,yi1

p1y1k + C(i1k) + p?2(a2k − (1− θ)i1k − p?2) +1

2(p?22 − θ2i21k)

s.t. e1k − i1k ≤ n1k + y1k

The first order conditions are quite similar to the second-period problem:

∂L

∂y1k= p1 − λ1k = 0

∂L

∂i1k= i1k − p?2(1− θ)− θ2i1k − λ1k = 0 (8)

λ1k(e1k − i1k − n1k − y1k) = 0; λ1k ≥ 0; e1k − i1k − n1k − y1k ≤ 0

As before, the compliance constraint in period 1 is binding for all firms orfor none and λ1k = λ1 for all k. If the compliance constraint in period 1 is notbinding for all k (λ1 = 0), or equivalently if all firms invest more than the requiredabatement in period 1, firm k’s investment in abatement is described by the secondequation of (8)

i1k =p?2(1− θ)

1− θ2

where p?2 = A2−(1−θ)I1. Investment i1k thus only depends on aggregate variables.This implies that firms invest equally in period 1 and that i1k = I1 (since there isa continuum of firms).

9

Page 11: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

In this case, yB1k is undetermined and the period 1 equilibrium solution for allk is given by

iB1k =1

2A2

pB1 = λB1 = 0 (9)

This solution is valid as long as firms invest sufficiently to cover the aggregaterequired abatement, or equivalently as long as IB1 > A1. The condition for thissolution to be valid is thus: A1 <

12A2.

If the compliance constraint is binding in period 1 for all k, then the aggregateinvestment in period 1 just covers the aggregate required abatement in that period:I1 = A1. The solution for each k is

iB1k = A1

pB1 = λB1 = (2A1 − A2)(1− θ) (10)

yB1k = a1k − A1

For this solution to be a minimum for the firms’ problem, λB1 must be positive.This is the case if A1 ≥ 1

2A2.

Using these solutions ((9), (10) and (7)), I can now provide an expression forthe period 2 price and investment in terms of the exogenous variables

iB2k =

{min

{12A2; A2 − A1

}if A1 ≤ A2

0 if A1 > A2

pB2 =

{min

{12A2; A2 − A1

}+ θmax

{12A2, A1

}if A1 ≤ A2

0 if A1 > A2

yB2k =

{a2k − A2 if A1 ≤ A2

undetermined if A1 > A2

To sum up, the form of the solution to the firm’s problem depends on the valueof the first-period aggregate required abatement relatively to the second-periodaggregate required abatement, as illustrated by Figure 1.

A first general property of the solution is that the investments i1k and i2k areentirely subject to the aggregate variables A1 and A2 while for the cases withoutindeterminacy, sales and purchases of permits are given by

ytk = atk − At ∀ k, for t = 1, 2

Since there is a continuum of firms, At can be interpreted as the average requiredabatement (the difference between average emissions and average allocation). This

10

Page 12: Int ral Links in Cap and Trade Schem es - dipot.ulb.ac.be · ties in future periods (Dixit and Pindyck, 1999), I allow investment costs in period 2 to depend on period 1 investment

Figure 1: Solution of the benchmark case

!

A1

!

i1k

=1

2A2

i2k

=1

2A2

!

p1

= 0

p2

=1

2A2(1+ ")

!

i1k

= A1

i2k

= A2" A

1

!

p1

= (2A1" A

2)(1"#)

p2

= A2" (1"#)A

1

!

i1k

= A1

i2k

= 0

!

p1

= A1

p2

= 0!

1

2A2

!

A2

average required abatement is the common component of the individually requiredabatement atk. The idiosyncratic component of the individually required abate-ment is thus (atk − At). Firms use the permit market to cover this idiosyncraticpart of their abatement needs and they use investment in abatement to cover theaverage required abatement. This is why they invest equally in each period. It isthe application of the equi-marginal principle: firms trade permits until they havethe same marginal costs of investment in abatement.

I summarize the main result of this section in the following proposition.

Proposition 2.1. If investments have long-term effects, equilibrium investmentsand prices in period 1 depend on aggregate required abatements in both periods.More specifically, when the required abatement in period 2 is relatively high (A1 <12A2), firms over-invest in abatement in period 1 (i.e. they invest more than the

required level in period 1). Otherwise, firms just invest the required level, A1, andthe price in period 1 is positive. Formally, for θ ∈ [0, 1),

iB1k = max{A1,1

2A2}

pB1 =

{max{0, (2A1 − A2)(1− θ)} if A1 ≤ A2

A1 if A1 > A2

To understand the intuition behind this result, let us first assume that there isonly a compliance constraint in period 2 (A2 being the aggregate required abate-ment). As investments have long-term effects, investment in period 1 remainsuseful beyond period 1 and can be used to cover period 2 emissions. Firms havethen two options to cover their second-period required abatement: investment inperiod 1 or in period 2. As costs of investment in both periods are convex, firmsminimize their compliance costs by smoothing out their investment across periods:they choose

iB1k = iB2k =1

2A2 (11)

11

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whatever the value of θ ∈ [0, 1).However, there is still a compliance constraint in period 1. This constraint in

period 1 will be satisfied by the solution (11) as long as A1 <12A2. For higher values

of A1 relatively to A2, firms must at least invest A1 and they cannot implementthe solution (11).

Therefore, with a high level of required abatement in period 2 (A1 <12A2) and

with learning-by-doing, some of the investments of period 1 are driven entirely bysecond-period abatement needs. Note that this is inefficient because any period 1investment beyond the required level of abatement is ”useless”: firms do not getany credit from abating emissions beyond their target and the equilibrium pricein period 1 is equal to zero.

2.3. Intertemporal trading allowed

I now examine the case where firms can borrow permits from their second-period allocation or bank their period 1 unused permits to use them in period 2.Intertemporal trading thus gives firms an additional tool to minimize compliancecosts. The firm’s objective function is the same as in the benchmark case, butthe constraints in periods 1 and 2 are modified. At the end of period 2, the totalnumber of permits must be greater or equal than the total level of emissions overboth periods, for all k ∈ [0, 1]

(e1k − i?1k) + (e2k − i?1k − i2k) ≤ n1k + y?1k + n2k + y2k (12)

and in period 1, firm k’s borrowing capacity is not unlimited: it can only borrowpermits from its own second-period allocation (n2k). For all k ∈ [0, 1]

e1k − i1k ≤ n2k + n1k + y1k

As before, I impose that the permit market clears in each period. The res-olution of the optimization problem in the intertemporal trading case is similarto the resolution of the benchmark case. Therefore, I only provide the final re-sults. Details of the resolution can be found in Appendix A. The next propositionsummarizes the main results:

Proposition 2.2. Assume

ε1 < min

{(2− θ)(3− 3θ)

, 1

}ε2 +N1 +N2 (13)

12

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then

pT1 = pT2 =

{1−θ2(5−4θ)

(A1 + A2) if θ ∈ [0, 12]

0 if θ ∈ (12, 1)

iT1k = min

{(2− θ)(5− 4θ)

(A1 + A2),1

2(A1 + A2)

}(14)

iT2k = max

{1− 2θ

(5− 4θ)(A1 + A2), 0

}yT2k + yT1k = a1k + a2k − (A1 + A2)

i.e., when intertemporal trading is allowed, equilibrium prices and investments inboth periods are proportional to the total required abatement.Moreover, prices are equal and the level of θ determines how firms share out theirinvestments in abatement between periods 1 and 2.

As in the benchmark case, investments are used to cover average requiredabatements (and thus only depend on aggregate variables) while the permit marketcovers idiosyncratic components of individually required abatements.

Condition (13) guarantees that the borrowing constraint in period 1 does notbind. As long as firms do not borrow more permits than their second-periodallocation (condition (13) holds), constraint (12) is the only binding constraint.Intertemporal trading allows firms to redistribute their initial allocation acrossperiods. The firms’ problem is then how to share out investments in abatementbetween both periods in order to minimize their costs to comply with the totalrequired abatement (A1 +A2). The distribution of abatement needs across periodsdoes not affect investments and prices.

In contrast to the benchmark case, the investment strategy depends on pa-rameter θ. For θ ∈ [0, 1

2], the higher the θ, the higher the firms’ investments

in abatement in period 1 because an increase in θ corresponds to an increase inthe cost of investment in period 2. If θ ∈ (1

2, 1) (low level of learning-by-doing),

firms choose the corner solution iT2k = 0. They prefer not to delay investment inlong-term abatement until period 2.12

As in models with short-term investments, the price in period 2 is equal to theprice in period 1. If an individual firm observes that p1 ≤ p2, it will bank permitsbecause buying a permit in period 1 and saving it for use in period 2 is cheaperthan buying a permit in period 2. Since all firms behave like this, the price ofpermits in period 1 rises while the price in period 2 falls until prices become equal.

12Such a corner solution is due to the two-period setting. By increasing the number of periods,the interval of θ that implies a corner solution becomes smaller and smaller.

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If firms observe that p1 > p2, they will borrow permits. As a consequence, theprice in period 1 falls while the price in period 2 rises.

When aggregate emissions in period 1 are very high (compared to emissions inperiod 2 and allocations in both periods), firms may be willing to borrow morepermits than their second-period allocation in order to invest less in period 1. Itis the case when emissions in period 1 are higher than the RHS of condition (13).Then, each firm borrows the maximum number of permits n2k and the aggregateinvestment in period 1 is simply the difference between aggregate emissions inperiod 1 and the sum of allocations N1 and N2. As firms would like to investless in period 1 by borrowing more permits from period 2, the price of permits inperiod 1 is always higher than the price in period 2.

3. Comparison of both cases

In this section, I analyze how intertemporal trading affects the equilibriumprices, the firms’ investments and the aggregate emission reductions compared tothe benchmark case.

3.1. Effects of intertemporal trading on prices

Using solutions derived in the previous section, I get the following propositionconcerning the price path with and without intertemporal trading.

Proposition 3.1. The price path.

• With intertemporal trading the price path is flat: p1 = p2 (as long as theborrowing constraint in period 1 does not bind)

• Without intertemporal trading, the price path depends on the required abate-ment path (A1, A2): p1 ≤ p2 if A1 ≤ x2(θ)A2

where x2(θ) = min{

2−θ3−3θ

, 1}

The price path is flat in the intertemporal trading case because, as explainedearlier, firms can freely transfer permits from one period to another. Withoutintertemporal trading, the price path depends on the required abatement path,(A1, A2). For example, an increase of A1 relatively to A2 implies an increase in thedemand for permits in period 1 relatively to the demand in period 2. Therefore,the price in period 1 increases relatively to the price in period 2.

The cut-off value, x2(θ), depends positively on θ: a lower level of learning-by-doing (higher θ) implies higher investment costs in period 2 relatively to investmentcosts in period 1. Therefore, the price in period 2 is more often higher than theprice in period 1.

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Moreover, x2(θ) varies from 2/3 to 1. If we had had a model in which invest-ments had short-term effects (i.e. i1k reduces emissions only in period 1), then thecondition for an increasing price path (p1 < p2) would have simply been A1 < A2:the price path is increasing when the second period is more demanding in terms ofabatement than the first one. In my model with long-term investments, if θ < 1/2,then the cut-off value is lower than one and it could happen that p1 > p2, evenwhen period 2 is more demanding than period 1 (x2(θ)A2 < A1 < A2). Such asituation is the result of the interaction between (i) a lot of learning-by-doing inthe investment process and (ii) the fact that period 1 investments, even if theyare not directly driven by period 2 abatement needs, reduce emissions in period 2.The combination of these two facts entails that the marginal cost of investment inperiod 2 is reduced and that, even if A1 < A2, the price in period 2 is lower thanthe price in period 1.

3.2. Effects of intertemporal trading on investments

In a scheme without intertemporal trading, the firms’ investment strategy(iB1k, i

B2k) depends on how abatement effort is distributed across periods. I de-

fine two cut-off values

x1(θ) = max

{1− 2θ

2(2− θ), 0

}= over-investment cut-off (15)

x2(θ) = min

{2− θ3− 3θ

, 1

}= banking/borrowing cut-off (16)

with x1(θ) <12< x2(θ) for all θ ∈ [0, 1). The result is summarized in Figure 2.

Figure 2: Effect of intertemporal trading on firm k’s investments

!

A1

!

i1k" and i

2k"

!

p1 < p2 " firms want to bank permits!

i1k" and i

2k#

!

i1k" and i

2k#

!

p1 > p2 " firms want to

borrow permits

!

i1k

B>> A

1

!

i1k

B" A

1

Large period 1 over-investment

Low period 1 over-investment

In the benchmark case:

Large excess in the supply of permits

Low excess in the supply of permits

!

x2(")A

2

!

x1(")A

2

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Proposition 3.2. Investments in abatement

• If the required abatement path is such that firms borrow permits (A1 >x2(θ)A2), the effect of intertemporal trading is always a reduction of theinvestment in period 1, compensated if necessary by an increase in the in-vestment in period 2;

• If the required abatement path is such that firms bank permits (A1 < x2(θ)A2),the effect of intertemporal trading is a reduction of the investment in period2 and an increase in investments in period 1. However, if period 2 is verydemanding compared to period 1 (A1 < x1(θ)A2), firms reduces their invest-ments in period 1 while still reducing their investment in period 2. In thiscase, banking and over-investment are substitutes.

Proof Results obtained by comparing solutions (9), (10), (7) and (14). �

In existing models studying intertemporal trading, investments always haveshort-term effects, so that without intertemporal trading, firms’ investments ineach period are simply the required abatement in that period. The effect of al-lowing intertemporal trading thus only depends on whether firms bank or borrowpermits. Permit banking always induces earlier investments in abatement whilepermit borrowing delays investments .

In my model, the decision to bank or to borrow permits relies on the cut-offvalue x2(θ). Indeed, for values of A1 larger (resp. lower) than x2(θ)A2, we haveseen that in the benchmark case, p1 > p2 (resp. p1 < p2), and firms are thuswilling to borrow (resp. bank) permits.

As in existing models, permit borrowing always delays investments because thelong-term effects of investments go only in one direction (firms invest in period 1for period 2). By contrast, permit banking can induce earlier or later investmentsdepending on the value of A1 with respect to x1(θ)A2. It is different from existingmodels because with long-term investments, first-period investments lead to areduction in period 2 abatement needs. In the benchmark case, the reduction inthe period 2 abatement needs is equal to the investment in period 1 (i1k). In theintertemporal trading case, this reduction is equal to the investment in period 1plus the amount of banking (which is the difference between investment in period1 and required abatement A1).

In the benchmark case, firms can over-invest in period 1. This over-investmentis entirely driven by second-period abatement needs and leads to an excess in thesupply of permits in period 1. By allowing intertemporal trading, this excess in thesupply of permits can be banked for period 2 and is not useless anymore. Then,firms have an incentive to invest more in period 1 in order to use this bankingpossibility.

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However, for very low abatement needs in period 1 (A1 < x1(θ)A2), the excessin the supply of permits is very large. As a result, the number of permits thatcan potentially be banked is so large that it is possible for firms to reduce theirinvestments in period 1 while still reducing their abatement needs in period 2. Byreducing their investment in period 1, firms reduce the number of banked permits.Nevertheless, the sum of banked permits and investment in period 1 leads to areduction of second-period abatement needs that is still higher than the reductionresulting from over-investment in the benchmark case. In fact, firms substitutepart of their investments in period 1 by banked permits and this allows firms toreduce their compliance costs in both periods.

The cut-off value x1(θ) determines what can be considered as a large over-investment in the benchmark case, given the level of learning-by-doing. Withintertemporal trading, the investment decision depends on the level of learning-by-doing (the higher the θ, the higher the level of first-period investment) while inthe benchmark case, as long as A1 < (1/2)A2, investment in abatement in period1 is constant. Thereby, the higher the θ, the less often an over-investment in thebenchmark case will be regarded as being too large.

3.3. Effects of intertemporal trading on aggregate emission reductions

In contrast to existing models, due to the long-term effects of investmentsin abatement, intertemporal trading also has an impact on the total aggregateemission reductions over both periods: 2IT,B1 + IT,B2 .

Proposition 3.3. Aggregate emission reductionsTotal aggregate emission reductions are strictly lower when intertemporal trading

is allowed for all A1 /∈ [12A2, A2]. They are the same otherwise.

Proof Assume θ ∈ [0, 1). For any required abatement path (A1, A2), whenintertemporal trading is allowed, the aggregate emission reduction is given by:2IT1 + IT2 = A1 + A2. In the benchmark case, the aggregate emission reductionsare given by:

• If A1 < (1/2)A2 ⇒ 2IB1 + IB2 = (3/2)A2 > A1 + A2;

• If (1/2)A2 ≤ A1 ≤ A2 ⇒ 2IB1 + IB2 = A1 + A2;

• If A1 > A2 ⇒ 2IB1 + IB2 = 2A1 > A1 + A2 �

The effect of intertemporal trading on aggregate emission reductions has twodifferent causes which are specific to the long-term nature of investments. ForA1 < (1/2)A2, the higher aggregate emission reductions in the benchmark caseare due to over-investment in period 1. This over-investment in period 1 leads

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to an excess in the period 1 supply of permits which is completely useless. Whenintertemporal trading is allowed, this excess in the supply of permits can be bankedfor period 2, reducing abatement needs.

For A1 > A2, the higher aggregate emission reductions in the benchmark caseare due to the irreversibility of investments in abatement. In the first periodof the benchmark case, firms have to invest at least the required level, A1. Asinvestments have long-term effects, if A1 > A2, there is an excessive emissionreduction in period 2 which is completely useless. Borrowing allows firms to avoidsuch excessive reductions and thus decreases the total emission reductions.

3.4. Effects of discounting

Up until now, I presented the model ignoring future profit discounting. InAppendix B, I show the results of the model for a discount factor δ = 1

1+r∈ [0, 1].

The intuition and the main results, i.e. that firms over-invest in period 1 and thatbanking and over-investment can be substitutes, continue to hold. Investmentstrategies in the benchmark and intertemporal trading cases are now given by(B.1) and (B.2) in Appendix B.

The main difference is that a higher discount factor increases investment in-centives in period 1 relatively to period 2. This effect is the same as in Kling andRubin (1997). When firms use a higher discount factor, they take future abate-ment costs more into account and are willing to invest earlier to reduce these futureabatement costs.

For the rest, discounting affects the exact values of the cut-offs between casesx1 and x2 (equations (B.3) and (B.4) in Appendix B) but not the economic in-terpretation of the cases. x1 and x2 are both increasing in δ. Concerning thebanking/borrowing cut-off x2, it is quite intuitive: a higher discount factor impliesmore banking (see Kling and Rubin, 1997). The intuition for the over-investmentcut-off x1 is more complicated. With intertemporal trading, firms can bank theirsurplus of permits for use in period 2. Therefore, in order to achieve the same levelof cost reduction in period 2, when intertemporal trading is allowed, investmentsin period 1 need to increase by a lower amount than when there is no intertemporaltrading. Consequently, when firms use a higher discount factor, an over-investmentin period 1 of the benchmark case is more often considered as being too large.

4. Welfare analysis

In this section, I first examine the cost effectiveness of cap-and-trade schemeswith intertemporal trading. Then, I derive what is the socially optimal level ofabatement that a regulator should require in order to minimize the environmentaldamages from emissions and the firms’ investment costs.

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4.1. Cost Effectiveness of intertemporal trading

In the context of short-term abatements, Cronshaw and Kruse (1996) andRubin (1996) demonstrate that intertemporal trading is cost effective, i.e. firmswill allocate aggregate emissions between time periods to achieve the lowest presentvalue of abatement costs. This result can be easily extended to my model withlong-term investments in abatement. In my model, the compliance cost of firm kat time t is given by the following equation:

gtk = ptytk + C(itk, it−1,k)

And the total compliance cost of the scheme is obtained by summing individualcompliance costs for each period:∫ 1

0

(g1k + g2k)dk =1

2I21 +

1

2I2(I2 + θI1) (17)

From the solution to firms’ minimization problem, we know that I1 ≡ I1(A1, A2)and I2 ≡ I2(A1, A2). The total compliance cost (17) can thus be written asG(A1, A2). Define GB(A1, A2) and GT (A1, A2) as the total compliance costs inthe benchmark and the intertemporal trading case, respectively.

First, assume that the total required abatement is A1 + A2. In Appendix C, Ishow that for all θ ∈ [0, 1) and for all A1 6= x2(θ)A2, G

B > GT . If A1 = x2(θ)A2,the allocation of permits is such that, in the intertemporal trading case, firmswill neither borrow nor bank permits. Firms’ investments are thus equal to theinvestments in the benchmark case and GB = GT .

Second, due to the long-term effects of these investments, a scheme withoutintertemporal trading can lead to higher aggregate emission reductions (due toover-investment in period 1 or irreversibility) for the same total required abate-ment. However, these higher aggregate emission reductions are not cost effective.A scheme with intertemporal trading can reach the same level of aggregate emis-sion reductions at a lower cost than a scheme without banking or borrowing. Theformal proof is given in Appendix C.

The intuition behind these two results is quite simple. Allowing intertemporaltrading gives firms a greater number of options to meet their emissions targets:firms are able to achieve temporal as well as spatial cost savings.

4.2. Intertemporal trading from the perspective of the society

From the perspective of the society, the regulator’s objective is to minimizethe firms’ costs of investment in abatement and the environmental damages fromemissions. The firms’ investment costs G(A1, A2) are given by equations (C.1) and

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(C.2) in the benchmark and the intertemporal trading case, respectively. I assumethat environmental damages are quadratic and are given by the following function:

D(E) =d

2E2

where E = (ε1 − I1) + (ε2 − I1 − I2) is the cumulative stock of emissions at theend of period 2. In this paper, I am interested in cap-and-trade schemes designedto address climate change. CO2 emissions are the main cause of climate change.The intensity of the potential damage depends upon the accumulation of CO2 inthe atmosphere. The long-term effect of CO2 emissions on climate is due to therelatively long lifetime of this gas.13

Taking into account firms’ investment behavior, the regulator chooses the re-quired abatement (A1, A2) that minimizes the social cost S(A1, A2) of the cap-and-trade scheme:

S(A1, A2) = G(A1, A2) +D(E)

The social optimum (see Appendix D for details) is a scheme with intertemporaltrading that requires a total aggregate abatement of

Aopt = Aopt1 + Aopt2 =

{d(5−4θ)

1−θ2+d(5−4θ)(ε1 + ε2) if θ ≤ 1

24d

1+4d(ε1 + ε2) if θ > 1

2

Since intertemporal trading is allowed, only the total required abatement Aopt

matters. If intertemporal trading is not allowed, there exists only one requiredabatement path (Amin1 , Amin2 ) that achieves the same total emission reductions,Aopt, at the minimum social cost:{

Amin1 = 2−θ3−3θ

Amin2 if θ ≤ 12

Amin1 = 2d1+4d

(ε1 + ε2) and Amin2 < Amin1 if θ > 1

2

Any other required abatement path implies lower total emission reductions andhigher social cost than in the intertemporal trading case.

13In this paper, I concentrate on environmental damages occuring in the long run. Only thestock of emissions accumulated at the end of the scheme matters. However, sometimes total envi-ronmental damages over a period of time depend not only on the quantity of emissions, but alsoon when these emissions occur. Since intertemporal trading allows firms to transfer permits fromone period to another without taking into account the social damages, allowing intertemporaltrading can reduce social welfare (Kling and Rubin,1997; Leiby and Rubin, 2001). In the presenceof uncertainty and asymmetry of information between firms and the government, intertemporaltrading can be welfare improving: Feng and Zhao (2006) study the relative efficiency of bankingregimes, when the government set a total cap on emissions and an intertemporal trading ratio.This relative efficiency depends on the slope of the benefit and damage functions. Even if thesemodels consider only short-term abatements, the intuition can be applied to my model withlong-term abatements.

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4.3. Aopt is not chosen by the regulator

In practice, the regulator does not often implement the social optimum. Inmost cases, the regulator might issue more permits than the socially optimal levelof emissions. Assume a cap-and-trade scheme with intertemporal trading in whichthe total required abatement is A = (A1 + A2) < Aopt. With such a scheme, thetotal emission reductions are lower than the social optimum Aopt and the socialcost is higher.

From proposition 3.3, we know that, without intertemporal trading, total emis-sion reductions are higher than the total required abatement in two cases: over-investment in period 1 to cover high future abatement needs and over-investmentin period 2 due to irreversibility of investment. Therefore, to achieve a totalemission reduction of Aopt with a total required abatement of A, the regulatormust design a scheme without intertemporal trading and with a specific requiredabatement path: either A1 < (1/2)A2 (over-investment in period 1) or A1 > A2

(over-investment in period 2).

Proposition 4.1. If the number of permits issued by the regulator is such that A <Aopt, a scheme without intertemporal trading can implement the socially optimallevel of required abatement.

• If θ ≤ 1/8, the regulator chooses a required abatement path such that A1 <(1/2)A2

• If θ > 1/8, the regulator chooses a required abatement path such that A1 > A2

Proof See Appendix E �

The preferred required abatement path depends on the value of θ. Damages arethe same under both paths ( A1 < (1/2)A2 or A1 > A2) because both lead to atotal emission reduction equivalent to the socially optimal abatement. However,compliance costs will differ because the timing of investments is not the same. Witha low level of learning-by-doing (θ > 1/8), delaying investment is not desirable,and it is better to encourage firms to invest in period 1. If learning-by-doing issufficiently high (θ ≤ 1/8), splitting investment between periods 1 and 2 becomesbeneficial.

By banning intertemporal trading, the regulator uses the long-term propertiesof investment to reach higher total emission reductions and lower environmentaldamages. However, these higher reductions have a cost: firms’ compliance costsare higher than in a scheme with intertemporal trading and with the same requiredabatement A.

If A < Aopt, banning intertemporal trading and choosing a specific requiredabatement path (either A1 < (1/2)A2 or A1 > A2) is socially desirable if this banreduces the social cost compared to a scheme with intertemporal trading,

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In Appendix E, equations (E.1) and (E.2) give the condition under which ban-ning intertemporal trading is socially desirable. There is a trade-off between costlyover-investments and damaging over-emissions. The resolution of this trade-offdepends on the parameters of the model: the marginal damage d, the level oflearning-by-doing θ, the total aggregate emissions (ε1 + ε2) and the difference be-tween the socially optimal and the chosen required abatement.

5. Application to the EU ETS

So far we have seen that constraints on intertemporal trading can increaseinvestments in abatement during the first period of the cap-and-trade schemebecause investments have long-term effects so that some first-period investmentsare driven by second-period targets. This section provides a new perspective onthe debate concerning the over-compliance observed at the end of the first phaseof the EU ETS: is this over-compliance due to over-allocation or over-investment?

As mentioned previously, intertemporal trading was not allowed between thefirst and second phases. Though any unused permit at the end of this first phasecould not be banked and was thus completely useless, one observed over-complianceat the end of the first phase. Firms were left with unused permits since verifiedemissions were 2.3% lower than the total allocation of permits.14

Over-allocation is the usual explanation for this over-compliance observed atthe end of the first phase: Member States over-estimated business-as-usual emis-sions and thus allocated too many permits to firms in the first phase. Already in2004, Ecofys analyzed the available preliminary NAPs of all EU countries and sug-gested that the caps for Phase I were lenient. Moreover they estimate that somecountries (such as the Netherlands) gave more allowances than Ecofys-estimatedbusiness-as-usual emissions. Their conclusion was that no real abatement effortswould be required in the first phase.15 The first release of verified emissions dataconfirmed these predictions: some European countries announced that their emis-sions were lower than the allocations to firms, inducing a sharp drop in the priceof permits in April 2006.

However, despite this over-allocation, some abatements occurred during thefirst phase. Forming a good estimate of these abatements is difficult because weneed to estimate what the CO2 emissions would have been without the EU ETS(business-as-usual emissions). Ellerman and Buchner (2008) use historical baselineemissions to estimate these business-as-usual emissions.16 They estimate that some

14see Anderson et al. (2009).15Analysis of NAPs for the EU ETS, Ecofys, August 2004.16Baseline emissions are firms’ emissions data collected in 2004 by Member States to establish

an initial point for determining the emissions expectations for the first phase.

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over-allocation occurred but estimated aggregate abatement due to the EU ETSstill ranges between 7% and 8% of business-as-usual emissions for 2005 and 2006.However, Ellerman and Buchner (2008) raise some doubts about the accuracy ofthese baseline emissions and using some adjustments, they finally find that theabatement over the first two years of the pilot phase should represent about 4%of the business-as-usual estimate. These results suggest that over-allocation isonly part of the explanation. Abatement also plays a role in explaining the over-compliance at the end of the first phase.

My model suggests an explanation for the role played by abatement in thefirst phase. Despite some over-allocation in the first phase, firms over-invested inabatement and some of these investments were driven by expected high second-phase abatement needs because investments in abatement have ”long-term effects”.

Indeed, with the over-allocation issue in the first phase, it was likely thatthe European Commission would be much stricter concerning the second-periodallocation plans. In 2006, the relationship between the cap for the first phaseand the estimated cap for the second phase (the Kyoto Protocol requirements)varied substantially across countries, but on average, EU-15 States had to reducetheir emissions caps from 6.8% from their current levels to meet their requirementsunder the Kyoto Protocol.17

Moreover, in the EU ETS, besides fuel switching, the most commonly usedtechniques to reduce CO2 emissions are efficiency improvements (for example,heat recovery and thermal insulation) that affect the core production process andthat have lasting effects (Faure et al., 2008). These efficiency investments implyadopting existing technologies, which are characterized by learning-by-doing. In-deed, it is by no means costless to learn how a technological improvement fits intoone’s firm or to learn about reliable suppliers (Jaffe and Stavins, 1994). Once thesesuppliers are known, it is less costly to increase the investment in the technology.

As a consequence, if banking between the first and the second phase of the EUETS had been allowed, it would have acted as a substitute to long-term investmentsin abatement in the first phase. Then, these long-term investments would havebeen lower. Indeed, banking the excess in the supply of permits at the end of thefirst phase would have acted as a cheaper substitute to the long-term investments.

6. Conclusion

The object of this paper has been to examine the effects of intertemporal trad-ing on the level of prices and investments in abatement when these investments

17Source: Congressional Research Service Report, Climate Change: The European Union’sEmissions Trading System (EU ETS), July 2006.

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had long-term effects. It is an important question since many abatement tech-niques have long-term effects and can thus be used as a tool for intertemporal costminimization.

To examine the interaction between long-term investments and intertemporaltrading, I use a simple two-period model in which prices are determined endoge-nously and investments in abatement have long-term effects. Long-term effects ofinvestments are twofold: an investment made in period 1 also reduces emissionsin period 2 and the marginal cost of abatement in period 2 depends on period 1investment level.

In existing models with short-term investments, intertemporal trading does notaffect the total emission reduction over the scheme. It only changes the timing ofinvestments, i.e. permit banking always implies earlier investments in abatement.Considering investments in abatement with long-term effects leads to completelydifferent results. Because investments in abatement have long-term effects, theyare used in period 1 as a means to reduce future compliance costs when intertem-poral trading is not allowed. As shown in the benchmark case, some of period 1investments are entirely driven by second-period abatement needs leading to anover-investment in period 1 (i.e. to invest beyond the required level when anyabatement beyond this level is useless in period 1). Once intertemporal tradingis allowed, abatement beyond the required level is no longer pointless in period1. Permits can be banked for use in period 2, reducing abatement needs in thatperiod. The overall effect of allowing intertemporal trading is a decrease in theaggregate emission reductions. Moreover, permit banking does not necessarily im-ply higher investments in period 1. For a very low level of required abatementin period 1, allowing intertemporal trading reduces investments in period 1. Theexcess in the supply of permits due to over-investment is so high that firms cansubstitute a part of their investments in period 1 by banked permits.

In the welfare analysis, I have shown that intertemporal permit trading iscost effective. However, it is not necessarily socially desirable. Schemes with in-tertemporal permit trading imply the lowest social cost (compliance costs plusenvironmental damages) only when the number of permits issued by the regula-tor is equal to the socially optimal level of emissions. If the number of permitsissued is higher, then banning intertemporal trading may reduce the social cost.The over-investment, generated by the long-term properties of investments, implieshigher emission reductions than the required abatement chosen by the regulator.Such higher reductions lead to lower environmental damages, but higher compli-ance costs for firms. There is thus a trade-off between costly over-investment anddamaging over-emissions.

On the basis of my model, I was able to provide another explanation to theover-compliance in the first phase of the EU ETS. The usual explanation for this

24

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over-compliance is an incorrect estimation of business-as-usual emissions and anover-allocation of permits in the first phase. I have shown that another possiblereason for such over-compliance is that firms over-invest in period 1 to take intoaccount the more stringent allocation in the second phase of the EU ETS. As aconsequence, if banking had been allowed in the EU ETS, it is likely that firmswould have reduced their abatements because banking would have acted as acheaper substitute to long-term investments.

Appendix A. Intertemporal trading case

In period 2, each firm k solves the following problem

mini2k,y2k

[p2y2k +1

2i2k(i2k + 2θi?1k)]

s.t. (e1k − i?1k) + (e2k − i?1k − i2k) ≤ n1k + y?1k + n2k + y2k (A.1)

i2k ≥ 0

with the market-clearing condition:∫ 1

0y2kdk = 0.

As for the benchmark case, the compliance constraint in period 2 is bindingfor all firms or for none. If A1 +A2 ≥ 2I?1 , period 1 long-term investments do notcover the total required abatement, thus the constraint is binding for all firms andthe solution for each k is given by

i?2k = p?2 − θi?1kp?2 = A1 + A2 − (2− θ)I?1 (A.2)

y?2k = a1k + a2k − y?1k − p?2 − (2− θ)i?1k

If A1 +A2 < 2I?1 , the compliance constraint is not binding, y?2k is undeterminedand for all k

p?2 = i?2k = 0

Then, in period 1, each firm k solves

mini1k,y1k

p1y1k +1

2i21k + p?2y

?2k +

1

2i?2k(i

?2k + 2θi1k)

s.t. e1k − i1k ≤ n2k + n1k + y1k (A.3)

i1k ≥ 0

with the market clearing condition:∫ 1

0y1kdk = 0.

Firms will never choose a level of investment in period 1 such that the constraintin period 2 does not bind or such that A1 + A2 < 2I?1 . p?2, y

?2k and i?2k in problem

(A.3) can thus be replaced by (A.2). Two solutions for this problem are possible

25

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depending whether or not the constraint on borrowing in period 1 is binding forall firms or for none.

If this constraint on borrowing in period 1 does not bind, the solution to thefirm’s intertemporal problem can be written as18

pT1 = pT2 =1− θ2

(5− 4θ)(A1 + A2)

iT1k =(2− θ)(5− 4θ)

(A1 + A2)

iT2k =(1− 2θ)

(5− 4θ)(A1 + A2) (A.4)

yT2k + yT1k = a1k + a2k − (A1 + A2)

where index T refers to the optimal solution in the intertemporal trading case.Note that if θ ∈ (1

2, 1], the solution above states that firms choose iT1k such that

there is an excessive investment at the end of period 2: 2IT1 ≥ A1 +A2. Since themarket stops at the end of period 2 and iT2k must be positive, such an excessiveinvestment is completely useless. If θ ∈ (1

2, 1], firms choose the corner solution, in

which they only invest in period 1

pT1 = pT2 = iT2k = 0

iT1k =1

2(A1 + A2) (A.5)

yT1k + yT2k = a1k + a2k − (A1 + A2)

Remember that solutions (A.4) and (A.5) were obtained under the assumptionthat the constraint on borrowing in period 1 was not binding. These solutions arevalid as long as:

e1k − i1k ≤ n2k + n1k + y1k ∀k ∈ [0, 1]

Or, equivalently

ε1 < min{ 2− θ3− 3θ

, 1}ε2 +N1 +N2 (A.6)

18Remember that we assume A1 +A2 > 0.

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If condition (A.6) is not satisfied, I get the following solution:

iT1k = A1 −N2

pT1 = pT2 + λT1k = (2− 2θ)A1 − (3− 3θ)N2 − (1− θ)A2

yT1k = (a1k − n2k)− (A1 −N2)

iT2k = max{0, −A1 + A2 + 2N2}pT2 = max{0, A1 + A2 − (2− θ)(A1 −N2)}yT2k = e2k − ε2

For this solution to be a minimum to problem (A.3), the Lagrange multiplier, λT1k,must be positive. It is the case if ε1 > min{ 2−θ

3−3θ, 1}ε2 +N1 +N2

Appendix B. Discounting future costs.

To highlight the main intuitions of the model, I ignored discounting in the mainpart of the paper. The results generalize when discounting is taken into account.Assume firms discount future profits with a discount factor δ ∈ [0, 1].

Appendix B.1. The Benchmark case

In period 2, each firm k solves problem (1). In period 1, firms discount futureprofits, so that firm k’s problem is given by:

mini1k,y1k

p1y1k +1

2i21k + δ[p?2y

?i2 +

1

2i?2k(i

?2k + 2θi1k)]

s.t. e1k − i1k ≤ n1k + y1k

with the market-clearing condition:∫ 1

0y1kdk = 0. The solution to this problem

depends on the value of A1 with respect to A2:

iB1k = max

{δ(1− θ)

1 + δ(1− 2θ)A2, A1

}iB2k =

{min

{1−θδ

1+δ(1−2θ)A2; A2 − A1

}if A1 ≤ A2

0 if A1 > A2

(B.1)

pB1 =

{min {0, (1 + δ(1− 2θ))A1 − δ(1− θ)A2} if A1 ≤ A2

A1 if A1 > A2

pB2 =

{min

{1−θδ

1+δ(1−2θ)A2; A2 − A1

}+ θmax

{δ(1−θ)

1+δ(1−2θ)A2, A1

}if A1 ≤ A2

0 if A1 > A2

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yB1k =

{undetermined if A1 ≤ δ(1−θ)

1+δ(1−2θ)A2

a1k − A1 if A1 >δ(1−θ)

1+δ(1−2θ)A2

yB2k =

{a2k − A2 if A1 ≤ A2

undetermined if A1 > A2

The main difference is that the investment strategy now depends on θ. Rememberthat the only objective of over-investment is covering future abatement needs, sincebanking is not allowed. An increase in θ implies that over-investment in period1 is more costly. Then, an increase in θ reduces the incentives to over-invest inperiod 1 but, for all θ ∈ [0, 1), the incentives to over-invest are still present, aslong as δ > 0.

Appendix B.2. The intertemporal trading case

In period 2, each firm k solves problem (A.1). In period 1, firm k solves thefollowing problem:

mini1k,y1k

p1y1k +1

2i21k + δ[p?2y

?2k +

1

2i?2k(i

?2k + 2θi1k)]

s.t. e1k − i1k ≤ n2k + n1k + y1k

i1k ≥ 0

If the constraint on borrowing is not binding, the form of the solution depends onthe two parameters, θ and δ:

iT1k =

{δ(2−θ)

1+δ(4−4θ)(A1 + A2) if θδ ≤ 1

212(A1 + A2) if θδ > 1

2

iT2k =

{1−2θδ

1+δ(4−4θ)(A1 + A2) if θδ ≤ 1

2

0 if θδ > 12

(B.2)

pT1 = δpT2 =

{δ(1−θ2δ)1+δ(4−4θ)

(A1 + A2) if θδ ≤ 12

0 if θδ > 12

yT1k + yT2k = a1k + a2k − (A1 + A2)

Appendix B.3. Effect of intertemporal trading on prices and investments

By comparing results with and without intertemporal trading (solutions (B.1)and (B.2)), I can get the same type of figure as figure 2, with two cut-off values,x1 and x2 that depend on the discount factor δ:

x1(θ, δ) =

{ −1+2δ−3δθ+2δθ2

2−θ+2δ−5δθ+2δθ2if θδ ≤ 1

2δ−1

1+δ−2δθif θδ > 1

2

(B.3)

x2(θ, δ) =

{δ(2−θ)

1+δ(2−3θ)if θδ ≤ 1

2

1 if θδ > 12

(B.4)

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Appendix C. Cost effectiveness of intertemporal trading

If intertemporal trading is not allowed, the total compliance cost is given by:

GB(A1, A2) =

1+θ4

(A2)2 if A1 <

12A2

(1− θ)A21 + 1

2A2

2 − (1− θ)A1A2 if 12A2 ≤ A1 < A2

12(A1)

2 if A1 ≥ A2

(C.1)

If intertemporal trading is allowed and if the constraint on borrowing does notbind, the total compliance cost is given by:19

GT (A1, A2) =

{1−θ2

2(5−4θ)(A1 + A2)

2 if θ ∈ [0, 12]

18(A1 + A2)

2 if θ ∈ (12, 1]

(C.2)

For a given total required abatement A1 + A2, comparing (C.1) and (C.2)implies that

• GT (A1, A2) < GB(A1, A2) for all A1 6= x2(θ)A2,

• GT (A1, A2) = GB(A1, A2) for A1 = x2(θ)A2.

Following proposition 3.3, the total emission reduction without intertemporaltrading is higher than the total reduction with intertemporal trading in two cases:A1 < (1/2)A2 and A1 > A2. Assume the required abatements chosen by theregulator in a scheme without intertemporal trading are given by A?1 and A?2.

1. For A?1 < (1/2)A?2, the total emission reduction is 2IB1 + IB2 = 1.5A?2 and thetotal compliance cost is GB(A?1, A

?2) = 1+θ

4(A?2)

2

Now consider a scheme with intertemporal trading in which aggregate re-quired abatement chosen by the regulator are given by A??1 and A??2 .To achieve a total reduction equivalent to 1.5A?2, the regulator has to set thetotal aggregate required abatements such that A??1 + A??2 = 1.5A?2.

• For θ ∈ [0, 12], GT (A??1 , A

??2 ) ≤ GB(A?1, A

?2)

⇐⇒ 9(1−θ2)8(5−4θ)

(A?2)2 ≤ 1+θ

4(A?2)

2

⇐⇒ 9(1−θ)2(5−4θ)

≤ 1 which is the case for any value of θ ∈ [0, 12]

• For θ ∈ (12, 1), GT (A??1 , A

??2 ) ≤ GB(A?1, A

?2)

⇐⇒ 932

(A?2)2 ≤ 1+θ

4(A?2)

2 which is the case for any value of θ ∈ (12, 1).

19I assume that the regulator fixes the required abatement such that the constraint on bor-rowing in period 1 is not binding.

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2. For A?1 ≥ A?2, the total emission reduction is 2IB1 + IB2 = 2A?1 and the totalcompliance cost is GB(A?1, A

?2) = 1

2(A?1)

2

With intertemporal trading, to achieve a total reduction equivalent to 2A?1,the total aggregate required abatement should be such that A??1 +A??2 = 2A?1.

• For θ ∈ [0, 12], GT (A??1 , A

??2 ) ≤ GB(A?1, A

?2)

⇐⇒ 2(1−θ2)(5−4θ)

(A?1)2 ≤ 1

2(A?1)

2

⇐⇒ 0 ≤ 4− 4θ + 4θ2 which is the case for any value of θ ∈ [0, 12]

• For θ ∈ (12, 1), GT (A??1 , A

??2 ) ≤ GB(A?1, A

?2)

⇐⇒ 48(A?1)

2 ≤ 12(A?1)

2. In fact for any value of θ ∈ (12, 1), GT (A??1 , A

??2 ) =

GB(A?1, A?2). �

Appendix D. The Social Optimum

If intertemporal trading is allowed, firms’ investment behavior is given by (14),for a given aggregate required abatement. Knowing the firms’ behavior, the regu-lator minimizes the following social cost:

minA1,A2

ST (A1, A2) =

{1−θ2

2(5−4θ)(A1 + A2)

2 + d2[ε1 + ε2 − (A1 + A2)]

2 if θ ≤ 12

18(A1 + A2)

2 + d2[ε1 + ε2 − (A1 + A2)]

2 if θ > 12

The socially optimal required abatement is given by:

Aopt = Aopt1 + Aopt2 =

{d(5−4θ)

1−θ2+d(5−4θ)(ε1 + ε2) if θ ≤ 1

24d

1+4d(ε1 + ε2) if θ > 1

2

For this socially optimal abatement, the social cost is given by:

ST (Aopt) =

{d2

((1−θ2)

1−θ2+d(5−4θ)

)(ε1 + ε2)

2 if θ ≤ 12

d2

(1

1+4d

)(ε1 + ε2)

2 if θ > 12

If intertemporal trading is not allowed, firms’ investment behavior is given by(9), (10) and (7) and the regulator faces the following social cost:

SB(A1, A2) =

1+θ4A2

2 + d2[ε1 + ε2 − 3

2A2]

2 if A1 ≤ 12A2

(1− θ)A21 + 1

2A2

2 − (1− θ)A1A2 + d2[ε1 + ε2 − (A1 + A2)]

2 if 12A2 ≤ A1 ≤ A2

12A2

1 + d2[ε1 + ε2 − 2A1]

2 if A1 ≥ A2

The required abatement path (Amin1 , Amin2 ) leading to the lowest social cost is{Amin1 = 2−θ

3−3θAmin2 if θ ≤ 1

2

Amin1 = 2d1+4d

(ε1 + ε2) and Amin2 < Amin1 if θ > 1

2

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For this particular path, the total emission reduction is (2IB1 + IB2 ) = Aopt, thesocially optimal required abatement with intertemporal trading, and ST (Aopt1 , Aopt2 ) =SB(Amin1 , Amin2 ). For any other path (A1, A2), the following relationship holds:ST (Aopt1 , Aopt2 ) < SB(A1, A2).

As a result, the social optimum for this economy is a cap-and-trade schemewith intertemporal trading and that requires an aggregate abatement of Aopt.

Appendix E. Proof of proposition 4.1

Denote At, the required abatement chosen by the regulator for period t. Assumethat

A = A1 + A2 < Aopt = Aopt1 + Aopt2

Clearly, if intertemporal trading is allowed, total emission reduction will be lowerwith the required abatement A than with Aopt and,

ST (A1, A2) > ST (Aopt1 , Aopt2 )

By banning intertemporal trading and choosing a specific required abatementpath (A1, A2), the regulator can encourage firms to invest so that the total emissionreduction is equivalent to the socially optimal one, even if A1 + A2 = A < Aopt.Table E.1 gives the two possible paths achieving this objective.

Table E.1: Required abatement paths leading to the socially optimal required abatement

A1 < (1/2)A2 A1 > A2

Required Abatement A2 = (2/3)Aopt A1 = (1/2)Aopt

Path A1 = A− A2 A2 = A− A1

Total Emission Reduction (3/2)A2 = Aopt 2A1 = Aopt

Social Cost[1+θ9

+ d2

](Aopt)2 + d

2(ε1 + ε2)

2[18

+ d2

](Aopt)2 + d

2(ε1 + ε2)

2

SB(A1, A2) −d(ε1 + ε2)Aopt −d(ε1 + ε2)A

opt

These two paths imply the same total emission reduction (Aopt), but the com-pliance costs are different, so that social costs are not equivalent:

S(A1, A2)B;A1<(1/2)A2 − S(A1, A2)

B;A1>A2 =

(1 + θ

9− 1

8

)(Aopt1 + Aopt2 )2

And,

• If θ ≤ 1/8, the regulator prefers a path such that A1 < (1/2)A2 because

S(A1, A2)B;A1<(1/2)A2 − S(A1, A2)

B;A1>A2 ≤ 0

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• If θ > 1/8, the regulator prefers a path such that A1 > A2 because

S(A1, A2)B;A1<(1/2)A2 − S(A1, A2)

B;A1>A2 > 0

If A < Aopt, a cap-and-trade scheme without intertemporal trading and witha specific required abatement path (either A1 < (1/2)A2 or A1 > A2) is sociallydesirable if and only if

• If θ ≤ 1/8, ST (A1, A2)− S(A1, A2)B;A1<(1/2)A2 ≥ 0

• If θ > 1/8, ST (A1, A2)− S(A1, A2)B;A1>A2 ≥ 0

Or equivalently,

• If θ ≤ 1/8,

d(ε1 + ε2)[(Aopt1 + Aopt2 )− (A1 + A2)]︸ ︷︷ ︸

>0

−[

1 + θ

9+d

2

](Aopt1 + Aopt2 )2 +

[1− θ2

2(5− 4θ)+d

2

](A1 + A2)

2︸ ︷︷ ︸<0

≥ 0

(E.1)

• If θ > 1/8,

d(ε1 + ε2)[(Aopt1 + Aopt2 )− (A1 + A2)]︸ ︷︷ ︸

>0

−[

1 + θ

9+d

2

](Aopt1 + Aopt2 )2 +

[1

8+d

2

](A1 + A2)

2︸ ︷︷ ︸<0

≥ 0(E.2)

Acknowledgments : Part of the research for this paper was done during a re-search stay at the Toulouse School of Economics, for which the author benefittedfrom financial support from ERC grant MaDEM and the FRS-FNRS. I am espe-cially grateful to Estelle Cantillon, for her very helpful comments and discussions.I am also indebted to Marjorie Gassner, Nicolas Gothelf, Alice McCathie, the par-ticipants of the Micro workshop at ECARES and the participants of the ENTERJamboree 2010 in Toulouse for their comments. I also thank Brigit Bednar-Friedland Sonia Schwartz for their comments.

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