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Assessing the implementation of the Market

Stability Reserve

Corinne Chaton, Anna Creti, and Maria-Eugenia Sanin,

Working Paper

RR-FiME-17-05

Auguste 2017

Assessing the implementation of the Market

Stability Reserve

Corinne Chaton∗

, Anna Creti†and Mar´ıa-Eugenia Sanin‡

August 2017

Abstract

In October 2015 the European Parliament has established a market

stability reserve (MSR) in the Phase 4 of the EU-ETS, as part of the 2030

framework for climate policies. In this paper we model the EU-ETS in

presence of the Market Stability Reserve (MSR) as it is deﬁned by that

decision and investigate the impact that such a measure has in terms of

permits price, output production and banking strategies. To do so we

build an inter-temporal model in which polluting ﬁrms competing in an

homogeneous good market are price takers in a permits market and face

an uncertain demand. Our main ﬁnding is that the MSR succeeds in in-

creasing the permits’ price correcting an excess supply (and conversely de-

creasing it in case of excess demand). However, when the output demand

is stochastic, the MSR may alter the arbitrage conditions that determine

permits’ prices. In some cases which depend on the extend of the demand

variation, unintended eﬀects on the price pattern appear. This in turns

may adversely aﬀect welfare. Key words: ETS; market stability reserve;

MSR; banking.

JEL Codes: D43, L13, Q2.

We wish to thank participants in the Climate Economics Chair Seminar and in

the workshop ”Sectoral Instruments for Environmental Regulation”, Montreal, April

2016. The usual disclaimers apply.

∗Laboratoire de Finance des March´es de l’Energie (FiME), Paris.

†Universit´e Paris Dauphine, Chaire Economie du Climat and Ecole Polytechnique, Paris.

‡Universit´e d’Evry Val d’Essone and Ecole Polytechnique, Paris.

1

1 Introduction

Tradable emission permits (TEP) can achieve a given pollution reduction tar-

get in a cost-eﬀective manner (Montgomery, 1972) and, in a dynamic perspec-

tive, if these markets have full temporal ﬂexibility (fungibility), ﬁrms can op-

timally allocate abatement eﬀorts across time (Cronshaw and Brown-Kruse,

1996). The attractiveness of TEP regulation in relation to environmental taxes

is that the regulator is not required to have information regarding the produc-

tion and abatement technologies available in the sector under regulation for the

cost-eﬀective equilibrium to arise. Such equilibrium is achieved through the

market mechanism itself. However, there is a consensus on the fact that the

European Emission Trading System (EU-ETS) is not working properly in this

regard. Duncan (2016) analysis is unequivocal: “Right now the ETS is like a car

without an engine, we need to ensure it is ﬁt to do the job it should and drive

emissions reductions in Europe”. In fact, several factors have contributed to the

actual situation, in which the price of allowances is low with a very high surplus

of permits, such as the economic crisis, the introduction of renewables and the

use of Kyoto credits. The fact that the current cost of reducing emissions is low

is not a good news since it suggests that the ETS may fail to induce a transfor-

mation away from fossil fuels. For all these reasons, the market design of the

EU-ETS is being reformed on several issues, such as the speed at which the cap

decreases, carbon leakage amendments, rules about innovation funds. So far, a

step forward has been taken by creating a market stability reserve (MSR), by

the Decision (EU) 2015/1814 of the European Parliament and of the Council.

”The purpose of the MSR is to avoid that the EU carbon market operates with

a large structural surplus of allowances, with the associated risk that this prevents

the EU ETS from delivering the necessary investment signal to deliver on the

EU’s emission reduction target in a cost-eﬃcient manner” (EC 2017). The idea

behind such reform is a ﬂexibility mechanism that allows the supply of permits

to be responsive to fundamental changes in permits demand (like technology

advances or economic shocks). The mechanism works as follows: each year the

EC publishes the number of allowances in circulation and, if the number is higher

or equal than 833 million, 12% are placed in the reserve1(and consequently

withdrawn from next year’s auctions to the electricity sector). Instead, if the

allowances in circulation are below 400 million, or if for six month the price is

more than 3 times the average carbon price during the two preceding years, 100

million are released from the reserve. The number of allowances in circulation is

deﬁned as the number of allowances issued from 2008 (plus international credits

used from 2008) until the year in question minus total emissions since 2008

and minus the number of allowances already in the stability reserve: i.e. ﬁrms

accumulated banking of allowances. The ﬁrst calculation of these allowances has

been released in May 2017 and amounts to 1,693,904,897 allowances. In line with

the agreed MSR rules, no reserve feed is triggered by the indicator published in

1In fact the text says that what is retired is the maximum between 12% of allowances in

circulation and 100 Mt. The proposal of increasing this factor to 24% has been discussed but

not approved yet.

2

2017. The next publication will be made in May 2018. This will result in the

determination of the ﬁrst reserve feed for the period January to August 2019.

Moreover, ”backloaded” allowances (900 million allowances withdrawn from the

market at least until 2019), will be placed in the MSR’s reserve as well as any

remaining allowances not allocated by the end of the current trading phase, that

is 2020.

Several scholars have studied similar ﬂexibility mechanisms that to some

extent are used in the Californian CO2market and the Regional Greenhouse

Gas Initiative (RGGI).2Firstly, Pizer (2002) introduces the idea of a ”safety

valve” which consists in coupling a cap-and-trade system is with a price ceiling.

As long as the allowance price is below the safety-valve price, this hybrid system

acts like cap-and-trade, with emissions ﬁxed but the price left to adjust. Instead,

when the safety-valve price is reached the system behaves like a tax, ﬁxing the

price but leaving emissions to adjust. Later, Philibert (2008) and Burtraw, et

al. (2009) have proposed a symmetric safety valve, also known as a price collar,

which would limit price volatility on both the upside and the downside. Fell and

Moregerstern (2010) extend this kind of analysis by introducing uncertainty and

coupling the collar mechanisms to restrictions on banking and borrowing. They

ﬁnd that adding a price collar to the reserve borrowing proposal can reduce

costs: a price collar can achieve costs almost as low as a tax but with less

emissions variation. The price collar mechanisms outperform their safety valve

counterparts in terms of expected abatement costs at the same level of expected

cumulative emissions.

Traditionally, the literature has analyzed price ﬂexibility measures whereas

the EC has chosen instead to go for a quantity mechanism. Some recent papers

have then analyzed this design. Schopp et al. (2015) show in a computational

model that low EUA prices are observed because current supply exceeds current

demand of the electric industry that use them to hedge emissions associated with

existing 3 to 4 year power contracts. In this view, the MSR is a good solution

since it aﬀects the short-time price without touching to the long-run price signal.

Similarly Trotignon et al. (2016) and Perino and Willner (2016) ﬁnd that the

MSR reduces the short-medium term price, fostering earlier emission reductions.

This is precisely what Zetterberg et al. (2014) criticize, saying that the risk of

price volatility is higher in the presence of the MSR due to the diﬃculty of

predicting hedging needs. There is also a concern that the MSR will not erode

the current surplus quickly enough with an excess supply present until 2028

(Mathews et al., 2015). Salant (2016) suggests that low hedging demand from

the power sector is not compensated by other sectors expecting to buy low now

and sell high later due to the lack of credibility of the survival of the system. In

contrast, Fell’s (2016) simulations ﬁnd that the MSR can decrease price volatility

(but that its performance is very sensitive to parameters). FTL-Lexecon (2017)

suggests that alternative design would improve the performance of the market.

Several results are put to a trial in an experimental setting by Holt and Shobe

2The RGGI covers emissions from the power sector in 9 States of the Unites States of

America (Those states are Connecticut, Delaware, Maine, Maryland, Massachusetts, New

Hampshire, New York, Rhode Island and Vermont) as from January 2009.

3

(2016), who ﬁnd that there is little beneﬁt associated to the MSR but that a

price collar may instead enhance eﬃciency.

The paper closest to ours is Kollenberg and Taschini (2016) who model

the adjustments in permits availability due to the existence of the MSR using

a stochastic partial equilibrium framework. Their model and scope are very

diﬀerent from ours but some of the results are in line: the MSR substitutes

private banking and reduces variability in allowance holdings by withdrawing

(reinjecting) when the surplus is too high (low).

In this paper we consider a polluting sector subject to the EU-ETS in the

presence of the MSR (like for instance the electricity sector). To this end,

we study the MSR impact on banking strategies, allowances price and output

production to assess to which extent private banking is crowded out by this

mechanism. Diﬀerently from Kollenberg and Tashini (2016) we perform such

exercise for diﬀerent designs of the ﬂexibility mechanism. We model a ”ﬁxed”

rule, that is, for an MSR mechanism that is set independently of the banking

already accumulated. This rule is similar to the backloading policy already in

place in the EU ETS. We then compare it with a ”proportional” rule in which the

MSR withdraws a given percentage of the accumulated banking. Furthermore,

we study uncertainty under the form of a shock on the output demand, to

understand whether the MSR actually makes the EU ETS price more responsive

to output changes with respect to no intervention. To our knowledge, this is

the ﬁrst paper that studies to which extent the proposed design of the MSR

interacts with ﬁrms’ market strategies under demand uncertainty. To do so,

we assume that ﬁrms may delay banking as it was an ”option”, waiting for the

MSR to regulate the market. We then calculate ﬁrms’ optimal strategies under

Cournot-like competion, when the regulator modiﬁes the cap, and present a

fully ﬂedged analysis of output pricing and banking behavior.

Our main ﬁnding is that the MSR succeeds in increasing the permits’ price

when there is an excess supply (and conversely decreasing it in case of excess

demand). However, when uncertainty on the output demand is factored in, the

MSR may alter the perfect arbitrage conditions. In some cases which depend on

the extend of the demand variation, dynamic ineﬃciencies in the price pattern

appear. In particular, ﬁrms prefer to delay banking for wider valued of the

demand variation compared to the no intervention case. This in turns may

adversely aﬀect not only producers’ proﬁts, but also consumers’ surplus.

The paper is organized as follows. We ﬁrst explain our modelling strategy

(Section 2), then we develop the model under uncertainty (Section 3). We

introduce the notion of delaying banking. We calculate how backloading and

MSR modify it, including welfare eﬀects (Section 4). Our main results are also

presented by intuitive graphical illustrations. We conclude by pointing out some

policy implications.

4

2 Modelling strategy

2.1 Assumptions and notation

We consider nsymmetric ﬁrms (indexed by i= 1 . . . .n ) that compete in quan-

tities during three periods (t= 0,1,2) where (b−dPn

i=1 qi,t) is the inverse

demand in tand cis the constant marginal costs. One (some) of the inputs

used for production is polluting (eis the polluting intensity of output in t) and

therefore ﬁrms are subject to environmental regulation based on TEP. A regula-

tor ﬁxes a yearly cap on emissions amounting to the pollution reduction target

and sells an equivalent volume of permits in an auction. We denote αtAis the

amount of permits auctioned by the authority each period,3with αt+1 < αt≤1.

Firms are price takers in the TEP market whose price is σt. Firms maximize

inter-temporal proﬁts over three periods (by discounting with an interest rate

denoted r), and decide optimal production qi,t and banking zi,t.4

The regulator also stabilizes the market by setting a supply ﬂexibility mech-

anism: depending on whether there are more (less) permits than those allowed

by a given upper (lower) bound, the regulator will withdraw or inject additional

allowances in the next period.

2.2 Benchmark modelling

The previous assumptions can be summarized as follows: each ﬁrm imaximizes

inter-temporal proﬁts:

Max

qi,t,zi,t

Πi=

2

X

t=0

πi,t

(1 + r)t,(1)

s.t.

n

X

i=1

(eqi,t +zi,t −zi,t−1)≤αtA+xt,(2)

zi,2= 0.(3)

where for each period t:

πi,t = (b−d

n

X

i=1

qi,t)qi,t −cqi,t −σt(eqi,t +zi,t −zi,t−1).(4)

We assume that banked permits are used in the subsequent period and that

at the end of the regulatory period 2 there is no further incentive to bank.

3Notice that we consider the allocation A, the emission intensity e, demand b,dand cost

parameter cas constant all along the regulatory period.

4Cronshaw and Brown-Kruse (1996), as well as Rubin (1996), show that, when the trad-

able emission permits market is competitive and all ﬁrms comply with the environmental

regulation, allowing ﬁrms to save permits for future use increases inter-temporal eﬃciency,

since ﬁrms can optimally allocate pollution abatement eﬀorts across time.

5

Therefore, for every t,zi,t−1represent ﬁrm ibanked permits at the end of t−1

that is unused permits at that date.5

The MSR can take diﬀerent forms. If there is excess demand, that is banking

below Z(in the EC decision 400Mt), the MSR rule leads to an injection of a

permits (in the decision 100Mt). In case of excess supply, measured by banking

exceeding a given threshold Z(in the proposal 800Mt), −ζPn

i=1 zi,t−1is the

amount of permits withdrawn (in the decision 800Mt). Injection of permits is

done in a ﬁxed amount, whereas permits withdrawal is a fraction ζof the gap

between supply and demand, that is unused permits (in the decision ζis 12%).

For sake of comparison between the MSR withdrawing rule and the back-

loading measure,6we also consider the case of a ﬁxed amount of −apermits

withdrawn. Finally, no intervention is needed when unused permits remain

within the corridor deﬁned by Zand Z.

The regulatory intervention will thus be modeled as follows:

xt=

aif Pn

i=1 zi,t−1< Z,

0 if Z≤Pn

i=1 zi,t−1≤Z,

−aif Pn

i=1 zi,t−1> Z,

or

−ζtPn

i=1 zi,t−1if Pn

i=1 zi,t−1> Z.

(5)

with zi,2= 0 ∀i.

The previous cases will be divided in two sub-cases: (i) the case in which the

regulator reinjects aor withdraws an exogenous amount −a, respectively and

that we will call the ”ﬁxed amount rule”; (ii) the case in which the regulator

withdraws a percentage of banking, ζ, that we call from now on the ”proportional

to banking rule”. For simplicity, we will also label these rules as backloading

and MSR respectively.

To assess the functioning of the policy intervention, the ﬁxed withdrawal

rule (or backloading) is formalized as follows:

tFixed withdrawal rule or Backloading

0Pn

i=1(eqi,0+zi,0)≤α0A,

1Pn

i=1(eqi,1+zi,1−zi,0)≤α1A−a,

2Pn

i=1(eqi,2−zi,1)≤α2A−a.

(6)

Equations (6) simply say that, in period 0, total permits auctioned must be

enough to cover total emissions due to production and permits banked for period

5This hypothesis might seem in contrast with the literature on banking (see for instance

Schennach 2000 where banking is dont at each period for the entire regulatory time span).

However, the standard formulation and the one we choose result in the same optimal arbitrage

equation, meaning that inter-temporal eﬃciency holds.

6As a short-term measure to resorb the allowance surplus, the Commission postponed the

auctioning of 900 million allowances until 2019-2020.This backloading of auction volumes does

not reduce the overall number of allowances to be auctioned during phase 3, only the distri-

bution of auctions over the period.The auction volume is reduced by 400 million allowances in

2014; 300 million in 2015 and 200 million in 2016. The backloading was implemented through

an amendment to the EU ETS Auctioning Regulation, which entered into force in 2014.

6

1. Then, in period 1, total permits auctioned considered the exogenous permits

withdrawal must be enough to cover emissions and the net variation in the bank

of permits. Finally, in period 2, auctioned permits (again minus withdrawal)

must be enough to cover emissions considering that all banked permits must be

exhausted.

The proportional withdrawal rule (or MSR) takes the form of a smooth

adjustment of the cap, as described by the following conditions:

tProportional withdrawal rule or MSR

0Pn

i=1(eqi,0+zi,0)≤α0A,

1Pn

i=1(eqi,1+zi,1−zi,0)≤α1A−ζ1Pn

i=1 zi,0,

2Pn

i=1(eqi,2−zi,1)≤α2A−ζ2Pn

i=1 zi,1.

(7)

Indeed the ﬁrst equation is identical for both rules as the MSR does not op-

erate in period 0. Then, the second equation above shows the MSR withdrawal

as a percentage of the banked permits from period 0 and the third equation

models the MSR withdrawal as a percentage of banked permits from period 0

and 1.

To solve the model we apply a two-step-solution (similarly to Chaton et al.

2015):

(i) considering permits price ( σt) as exogenous, we ﬁrst ﬁnd the symmetric

Nash equilibrium in quantities at each period by simply solving the system of

FOCs given by ∂ πzi,t

∂qi,t = 0

qt=b−c−eσt

(1 + n)dt

; (8)

(ii) secondly, we solve the system of FOCs given by ∂Πi

∂zi,t = 0 and the permits

market clearing condition in equation, which gives the inter-temporal arbitrage

condition deﬁning the optimal banking strategies that maximize inter-temporal

proﬁts:

σ0=σt

(1 + r)t.(9)

Finally, solving the system of all equations resulting from (i) and (ii) gives the

equilibrium values. Note that (ii) can be done because ﬁrms are non-strategic

in the permits market. Finally, we check ex post (strict) positivity constraints

and threshold restrictions that deﬁne the functioning of the intervention. In

particular, although the equilibrium can be detailed by the equations below

for any scenario, depending on the speciﬁc total inter-temporal permits supply,

each case will be characterized by diﬀerent constraints on the parameters to

ensure positive quantity and output price, as well as banking. These constraints,

detailed in the Appendix A.2, must be carefully checked when comparing the

diﬀerent cases to assess the impact of the policy.

2.2.1 Fixed amount rule or backloading

Recall that the quantity of permits injected ( xt=a) or withdrawn ( xt=−a)

from the market in t. The benchmark is obtained with Pxt= 0.

7

Firms are constrained by the regulation if the total intertemporal supply of

permits auctioned Γ = A(α0+α1+α2) + x1+x2>0 is lower than the overall

emissions in the 3 periods, when the pollution constraint would not be binding:

Γ<ne

(n+ 1) d×3 (b−c).(10)

Whatever the functioning of the policy intervention under the ﬁxed amount

rule, the constraint (10) must hold. Notice that the regulator modiﬁes the inter-

temporal cap as long as x1+x26= 0, that is the measure is not cap-preserving.

The mainequilibrium valuesfor the ﬁxed amount rule are as follows:

q∗

0=1

RΓ

ne +(R−3) (b−c)

(n+ 1) d,(11)

z∗

0=Aα0

n−1

RΓ

n+(R−3) e(b−c)

(n+ 1) d,(12)

z∗

1=−Aα2+x2

n+1

R (1 + r)2Γ

n−R−3 + r2e(b−c)

(n+ 1) d!,(13)

σ∗

0=1

Re 3(b−c)−(1 + n)dΓ

ne ,(14)

where

R=

2

X

t=0

(1 + r)t.(15)

Note that q∗

1and q∗

2and all other equilibrium variables have similar ex-

pressions (see Appendix A.1). In particular, σ∗

1and σ∗

2are obtained by inter-

temporal arbitrage (equation 9). Moreover, due to the structure of the three-

period model, there is no banking at the ﬁnal stage, that is z∗

2= 0.

Recalling that the total intertemporal permits supply is Γ = A(α0+α1+α2)+

x1+x2, we can easily compute equilibrium values for cap reduction (x1=

x2=−a), with two successive withdrawing periods succeeding each other),

cap increase (x1=x2=a) and compare them with the no intervention case

(x1=x2= 0).7

Comparative statics can also be easily obtained. If the ﬂexibility mecha-

nism operates by reducing the cap, the equilibrium permits price increase. The

mechanism at stake is as follows: banking increases, production decreases and

so does the permits’ demand, explaining the upward shift of the permits’ price.

The opposite occurs when the regulator reinjects permits (or x1=x2=a).

2.2.2 Proportional to banking withdrawal rule or MSR

Recall that ζtis the percentage of banking withdrawn from the market at time

t. Similarly to the ﬁxed amount rule, ﬁrms are constrained by the regulation as

7Notice that the results hold when the cap is adjusted at period 1 only (x1=−aand

x2= 0 or x1=aand x2= 0).

8

long as the total intertemporal permits supply is tight enough:

Γζ<ne

(n+ 1) d×(3 −ζ1(1 −ζ2)−2ζ2)(b−c),(16)

where

Γζ=Aαζ,(17)

αζ=α0(1 −ζ1)(1 −ζ2) + α1(1 −ζ2) + α2.(18)

The main equilibrium values in the proportional rule are deﬁned as

follows (see Appendix A.1 for the other values):

q∗

0,ζ =1

DΓζ

ne +(R−3−ζ2r) (b−c)

(n+ 1) d,(19)

z∗

0,ζ =Aα0

n−1

DΓζ

n+r(3 + r−ζ2)e(b−c)

(n+ 1) d,(20)

z∗

1,ζ =A

D

(1 + r)2(α0(1 −ζ1) + α1)−α2((1 + r) + (1 −ζ1))

n(21)

−r(3 + 2r−(2 + r)ζ1)e(b−c)

D(n+ 1) d,

σ∗

0,ζ =1

De (3 −ζ1(1 −ζ2)−2ζ2)(b−c)−(1 + n)d

ne Γζ,(22)

where

D=R−(1 −ζ2)ζ1−(2 + r)ζ2>0.(23)

Straightforward calculations show that increasing ζ1and/or ζ2, that is the

parameters which deﬁne the withdrawal rate, increase the permits price σ∗

0,ζ

(and by arbitrage, also σ∗

1,ζ and σ∗

2,ζ ) compared to the no intervention case.8

3 Uncertainty on demand and the option to de-

lay banking

In this Section we assume that there is a shock ∆ on the market size at t= 1,

that is, the demand intercept can be bm=b+∆. If ∆ >0 the demand increases,

and conversely, if ∆ <0 there is a recession. The demand variation occurs with

a probability (1−λ) ; we denote the expected demand as Eb=λb+(1−λ)bm.

Such uncertainty is resolved at t= 1, where either bor bmrealizes, until the

second period.

We consider the decision on banking as a partially reversible investment.

Firms could decide not to bank at period 0 and wait for the uncertainty re-

garding the level of demand to be resolved at period 1. This option to wait or

8Notice that by setting ζ1=ζ2= 0 in the equations deﬁning the equilibrium, we get the

no intervention case.

9

opportunity to delay the banking decision, denoted by DB, has a value that

must be considered. Since there is no abatement in our model, ﬁrms bank per-

mits only if they expect them to be more expensive in the future. In order to

evaluate this option to delay we calculate the diﬀerence between the expected

discounted proﬁt when banking is positive at t= 0 (z∗

0>0) denoted by E(Πi)

and the expected discounted proﬁt under the assumption that banking is delayed

to t= 1 (z0= 0) denoted by E(Πi/z0= 0). Therefore, we have:

DB = max(E(Πi/z0= 0) −E(Πi),0).(24)

The diﬀerence between those expected proﬁts give us the expected gain due

to delaying banking, which is considered as sequential investment (like in Majd

and Pindyck, 1987).

We calculate the equilibrium under uncertainty on demand, by maximizing

the expected discounted intertemporal proﬁts. Similarly to the scenario un-

der certainty, ﬁrms are constrained if the total oﬀer of permits is low enough:

APαt<n

n+1 ×3(b−c)e+2∆e(1−λ)

d. This constraint can be expressed in terms

of ∆ = bm−b. The total supply constraint gives a threshold ∆csuch that if

∆>∆c, the permits price is positive at each period:9

∆>∆c=(n+ 1) dA

2ne (1 −λ)

2

X

t=0

αt−3 (b−c)

2 (1 −λ). (25)

The main equilibrium values under uncertainty and z∗

0>0 are as

follows:

bq∗

0=q∗

0+2 (1 −λ) ∆

R(1 + n)d,(26)

bz∗

0=z∗

0+2 (1 −λ) ∆e

R(1 + n)d,(27)

z∗

1,b =z∗

1+2e(1 + r) (1 −λ) ∆

d(1 + n) (2 + r)R,(28)

z∗

1,bm =z∗

1,b +∆er

(n+ 1) (2 + r)d,(29)

bσ∗

0=σ∗

0+2 (1 −λ) ∆

eR .(30)

Notice that these values are calculated without any intervention modifying

the cap, that will be introduced in the next Section. The variables with a

hat represent expected values, whereas the others are realized values (once the

uncertainty is resolved, that is when either bor bmrealizes).10

9In this Section, we assume α0= 1, to simplify the calculations.

10The permits prices values for periods 1 and 2 are: σ∗

1,b =σ∗

1−2(1−λ)∆

e(2+r)R,σ∗

1,bm =σ∗

1,b +

2∆

e(2+r).

10

We also have to check that the arbitrage condition satisﬁed with z0>0 and

z1>0.The following conditions must hold:11

2(1 + r)∆(1 −λ) + rR max(0,∆) ≤

(2 + r)((n+1)dA

ne ((1 + r)2

2

X

t=0

αt−α2R)−(2r+ 3))r(b−c) , (31)

∆>∆0= ∆c+R

2 (1 −λ)(b−c−(n+ 1) dA

ne )(n+ 1) dA

ne ((1 + r)2

2

X

t=0

αt−α2R) .

(32)

Positive quantities imply the following constraint:

∆<∆0=(R−3) (b−c)

2(1 −λ)+(n+ 1) dΓ

2ne (1 −λ).(33)

The value of delaying the banking decision as an option. The pre-

vious equilibrium allow the calculation of the expected inter-temporal proﬁts

when ﬁrm decide to wait until period 1 to bank E(Πi/z0= 0) or when they

don’t (that is E(Πi)). Under these hypotheses, the value of delaying banking

DB is as follows:

DB =A2dΛ

e2n2−((b−c)r(3 + r)−2 (1 −λ) ∆)2

d(1 + n)2(1 + r) (2 + r)R,

where Λ = (Pαt)2−2RPαt+R2α0

(1+r)(2+r)R.

DB is a quadratic function of the shock ∆ (but also of b, c, A and of the

probability of shock λ). To ease the calculations, we set α0= 1.

The equation DB = 0 has two roots, namely ∆0[.] and ∆DB [.]:

∆0= ∆0−(n+ 1) dA

2ne (1 −λ)R, (34)

∆DB =(R−3) (b−c)

2 (1 −λ)+(n+ 1) dA

2ne (1 −λ)(R−

2

X

t=0

αt).(35)

Recall that:

1. for all ∆ <∆0ﬁrms would borrow permits in the ﬁrst period (z0<0),

but this strategy is discarded by the functioning of the EU ETS;

2. for all ∆ >∆0ﬁrms don’t produce in the ﬁrst period (q0= 0) in order

to gain proﬁts when the demand is expected to be high.

Therefore, our analysis is conducted in the interval ∆ ∈[∆0,∆0] . The

graphical illustrations are obtained by using the following values: b= 1.7; c= 1;

e= 1; d= 1; α1= 1; α2= 0.9; α3= 0.6; r= 0.05; λ= 0.5; n= 6; x3= 0;

A= 0.18.

11Note that ∆ = 0 gives the same constraints than in the case without demand uncertainty.

This property also holds for the cases developed afterwards.

11

Straightforward computations show that for all ∆ ∈[∆0,∆DB ], the option to

wait DB is positive. This means that the expected proﬁts when there is no ﬁrst

period banking exceeds the expected proﬁts when banking at t= 0.Therefore

it would not be optimal for the ﬁrms to bank (z0= 0). The discontinuity in

the banking decision implies that the carbon prices are not linked by the inter-

temporal arbitrage equation. Therefore, for all ∆ ∈[∆0,∆DB ] , the equilibrium

carbon price at t= 0 is determined by the fundamentals of the current period

only.12 Moreover, as banking starts at t= 1,carbon prices at t= 1 and t= 2

are arbitraged.

Instead, for all ∆ >∆DB, banking starts as from the ﬁrst period (z∗

0>0) . In

this case, carbon prices follow the inter-temporal arbitrage equation (with bσ∗

0

deﬁned by equation 30

Figure 1. Delaying Banking as an option to wait

Notice that increasing the allocation, the discounting factor Ror the jump

probability λ(see the Figure 2) amplify the interval where the DB is positive,

implying that the interval such that it is optimal to wait is broader, both for

demand increase (∆ >0) and decrease (∆ <0) .

12In this case, σ0=1

e(b−c)−(1+n)dA

ne .

12

Figure 2. The impact of the jump probability on DB

4 Impact of backloading and MSR policies

The main driver of ﬁrms’ choice in their banking strategy is demand uncertainty,

as explained in the previous Section. We now look at the impact of the MSR

and backloading on such strategies. To do so, we study how the intervention

policies modify the DB equation and the interval under which it is optimal for

ﬁrms to delay banking (Subsection 4.1). Then we analyze the impact on prices

and welfare (Subsection 4.2).

To summarize our results, stabilization policies have two eﬀects:

•they modify the demand variation interval under which ﬁrms prefer to

delay banking until the second period;

•the shift of the demand variation thresholds such that banking is delayed

creates inter-temporal ineﬃciencies, in particular on the CO2price.

All the result are analytically proved.13 Whenever useful we illustrate them

graphically. This is for instance the case for the impact of stabilization policies

on consumers’ surplus and ﬁrms’ proﬁts.

4.1 Delaying banking under stabilization policies

To simplify the analysis, we assume that the regulator only intervenes at t= 1.

The cap is either tightened (x1=−a, x2= 0) under backloading,or released

(ζ1>0 , ζ2= 0) under proportional withdrawing (MSR), with respect to the

benchmark case of no intervention.

13The detailed calculations of production, banking and welfare are available by the authors

upon request, under Mathematica ﬁles.

13

4.1.1 Backloading

With x1=−a, the thresholds such that it is optimal for ﬁrms to delay

banking, with positive production in the ﬁrst period, are modiﬁed as follows:

∆0,x = ∆0−d(n+ 1) a

2ne (1 −λ)<∆0,(36)

∆DB

x= ∆DB +d(n+ 1) a

2ne (1 −λ)>∆DB ,(37)

∆0,x = ∆0−(n+ 1) da

2ne (1 −λ)<∆0.(38)

The interval under which it is optimal for ﬁrms to delay banking is wider with

respect to the one without intervention: ﬁrms wait until the demand variation

is stronger (both with demand boom and boost) to start banking (equations 36

and 37). Producing positive quantities in the ﬁrst period shrinks (equation 38).

Moreover, the DB value is modiﬁed as follows:

DBX=DB +da(2A(R−Ptαt) + a)

e2n2(1 + r) (2 + r)R.(39)

In particular DBXlies above DB when both are positive.

Backloading widens the interval over which is optimal to bank, whereas

permits injection has the opposite eﬀects of narrowing the interval of the demand

variation under which ﬁrms do not bank.

Figure 3 illustrates these eﬀects, as well as the injection case for comparison.

The black line represents DB under the benchmark, the red line is DBXwith

a ﬁxed reduction of permits (x1=−0.1, x2= 0), and the dotted line represents

the value of delaying banking under permits injection (x1= 0.1, x2= 0). This

latter case has opposite but symmetric eﬀects with respect to backloading. The

demand variation interval over which ﬁrms bank shrinks.

14

Figure 3. Fixed rules and DB

4.1.2 MSR

We study the impact on the interval variation such that it is optimal to delay

banking under the MSR rule, with positive production in the ﬁrst period. The

critical thresholds are modiﬁed as follows:

∆0,ζ = ∆0,(40)

∆DB

ζ= ∆DB +

dA (n+ 1) ((R−1) P2

0αt−ζ1ζ1)

ne(1 −λ)(R−2ζ1+ζ2

1)>∆OW ,(41)

∆0,ζ = ∆0−dA (n+ 1)

2ne (1 −λ)ζ1<∆0.(42)

The stability mechanism ampliﬁes the values of the demand gap such that

it is optimal to delay banking (equation 41) on the right side, for demand

booms. The left threshold remains the same (equation 40). Positive quantities

are obtained for a threshold value smaller than in the benchmark case (equation

42).

Let ϑ= (b−c)r(3 + r)−2 (1 −λ) ∆; the value of delaying banking writes

as follows:

DBζ=DB +

ζ1(Ad (1 + n)R−P2

0αt−enϑ)

de2n2(1 + n)2R(R−ζ1)2

×(Ad(1 + n)((2R−ζ1)

2

X

0

αt−Rζ1) + enζ1ϑ).(43)

15

The term that adds up to DB is positive. Consequently DBζ> DB when

both are positive, as Figure 4 illustrates. Also notice that the higher the with-

drawal coeﬃcient, the stronger the ampliﬁcation eﬀect.

Figure 4. MSR (under diﬀerent withdrawal coeﬃcients) and DB

4.2 Impact of the stabilization policies on the carbon price

The impact on the CO2price can be calculated in the intervals of demand

variation ∆ .

4.2.1 Backloading

The following eﬀects are at stake:

•If ∆ ∈∆0,x ,∆DB , a backloading policy doesn’t impact the banking

behavior (there is no banking in either case), therefore there is no impact

on the CO 2price in t= 0 . In the following periods, the cap reduction

increases carbon prices. Notice that in the subinterval ∆ ∈[∆0,x ,∆0] a

backloading policy has no impact as borrowing is not allowed.

•If ∆ ∈[∆DB ,∆DB

x], ﬁrms anticipate that the backloading policy will per-

fectly substitute private banking. Therefore they don’t bank in the ﬁrst

period, which insulates the current allocation from the following ones. At

t= 0, the current allocation is higher than its expected inter-temporal

value. As a consequence, over this interval, at t=0, the permits’

16

price is lower than the one without intervention. The proﬁts max-

imising strategy is to produce instead of banking, to saturate the equilib-

rium constraint in the permits’ market.14

•If ∆ ∈[∆DB

x,∆0], ﬁrms bank as from the ﬁrst period. Over this interval,

the inter-temporal permits’ allocation tightens the emission constraint.

Therefore, the backloading policy increases CO2prices for all t.

The following tables detail the above-mentioned eﬀects, by calculating the

diﬀerences between the equilibrium prices after and before the intervention.

∆∈t= 0

[∆0,x ,∆0] 0

∆0 , ∆DB 0

∆DB ,∆DB

x−Ad(1+n)(R−Pαt)−enϑ

e2nR <0

∆DB

x,∆0,xd(1+n)a

e2nR(R−1) >0

∆O,x,∆Oenϑ+d(1+n)(Γ−a)

e2n(2+r)R(1 + r)t−1>0

∆∈t= 1,2

[∆0,x ,∆0]d(1+n)a

e2n(2+r)(1 + r)t−1>0

∆0 , ∆DB d(1+n)a

e2n(2+r)(1 + r)t−1>0

∆DB ,∆DB

xd(1+n)(AR−Γ)−enϑ

e2n(2+r)R(1 + r)t−1>0

∆DB

x,∆0,xd(1+n)a

e2nR (1 + r)t>0

∆O,x,∆Oenϑ+d(1+n)(Γ−a)

e2n(2+r)R(1 + r)t−1>0

Figure 5 illustrates these eﬀects. The red line represents the DB value with

backloading at period t= 1, whereas the benchmark is displayed in black. The

interval over which there is no banking under the policy intervention is enlarged

to focus on the eﬀect of the CO2price decrease at t= 0.

14See equation (2).

17

Figure 5. Backloading and impact on CO2prices

The short term ineﬃciencies in the ﬁrst period are widened in Figure 6.

Figure 6. Carbon price: ineﬃcient decrease

Figures 7.1 and 7.2 display the permits’ price increase at periods 1 and 2,

for all demand realizations (band bm):

18

Figure 7.1 Carbon price increase Figure 7.2 Carbon price increase

4.2.2 MSR

As in the previous case, the MSR (or proportional withdrawing) may create

dynamic ineﬃciencies, depending on the extent of the demand variation:

•if ∆ ∈[∆0,∆D B ], the MSR policy does not impact the CO2price;

•if ∆ ∈[∆DB ,∆DB

ζ], at t= 0 the MSR reduces the CO2price over

the interval where banking is ineﬃciently delayed with respect to the no-

intervention case;

•if ∆ ∈∆D B ,∆0,ζ , the MSR rises CO2prices.

The detailed eﬀects are as follows.

∆∈t= 0

∆0 , ∆DB 0

[∆DB ,∆DB

ζ]−Ad(1+n)(R−Pα)−en((b−c)r(3+r)−2∆(1−λ))

e2nR <0

[∆DB

ζ,∆0,ζ ] (1 + r)t×Ad(1+n)(R−Pα)−en((b−c)r(3+r)−2∆(1−λ))

e2n(R−ζ1)>0

∆∈t= 1,2

∆0 , ∆DB 0

[∆DB ,∆DB

ζ] (1 + r)t×Ad(1+n)(R−Pα)−en((b−c)r(3+r)−2∆(1−λ))

e2nR >0

[∆DB

ζ,∆0,ζ ] (1 + r)t×Ad(1+n)(R−Pα)−en((b−c)r(3+r)−2∆(1−λ))

e2n(R−ζ1)>0

Mixed interventions: proportional withdrawing and ﬁxed injec-

tion. To reﬂect the full design of the EU decision on the stabilization mecha-

nism, we consider the case of a ﬁxed reinjection of permits and a proportional

19

withdrawal. This case does not diﬀer from the previous ones. There are still

demand interval variation in which ﬁrms would bank without intervention but

delay banking when the measure adjusting the cap are operational. Under the

intervals where the optimal strategies diﬀer, the permits’ price decrease.

Figure 8 illustrates this case. The DB value under proportional withdraw-

ing and ﬁxed injection is displayed in red. We compare it both with the no

intervention benchmark (in black) and the case of withdrawing at time t= 1

only (in green).15

Figure 8. Mixed interventions and DB

4.2.3 Stabilization policies and welfare

The calculations above allows to calculate ﬁrms’ proﬁts and consumer welfare.

Over the demand interval variation where there is no banking, ﬁrms’ proﬁts

follow the demand variation as ﬁrms produce up to saturating the permits’

constraint. For higher expected demand variations, ﬁrms start banking at t= 0.

In this case, ﬁrst period production decreases and, due to Cournot competition,

ﬁrms’ proﬁt increases much faster. Banking supports ﬁrms’ proﬁts maximising

strategies which consist in adjusting to the demand increase (banked permits

release the environmental constraint). With respect to the no intervention case,

we see that if the cap becomes more stringent, ﬁrms proﬁt is shifted below the

benchmark case due to higher permits price.

As for the consumers’ surplus, the demand increase dominates all the other

eﬀects, which explains why it increases. High permits price and lower production

15The permits withdrawn and injected do not compensate each other, to ensure that the

policy modiﬁes the overall cap.

20

with respect to the no intervention case explain why consumer surplus lies always

below the no intervention case when the permits’ supply is restricted. At the

value of demand variation such that there is a discontinuity in banking decisions,

consumers’ surplus is discontinuous too, due to a jump in production.

Over the interval of demand variation where banking is delayed due to the

cap modiﬁcation, ﬁrms proﬁts remain almost constant instead of increasing, and

the consumer surplus changes the slope at the point where the banking decision

changes and increases more slowly than in the no intervention case. However,

consumer surplus slightly increases even in the zone of zero banking as demand

increases too.

Figures 9.1 and 9.2 illustrate these eﬀects. The no intervention case is dis-

played in black, the backolading in red. We have added injection (dotted line),

to see mirror eﬀects when the cap increases.

Figure 9.1 Fixed rules: firms’ profit Figure 9.2 Fixed rules: consumers’ surplus

Under the MSR rule, the expected proﬁts and consumer surplus display the

same properties than in the case of the ﬁxed withdrawal. Similar ineﬃciencies

arise at t= 0 when private banking is discouraged.

In Figure 10, we focus on the impact of the increase of the withdrawing pa-

rameter, which is actually under discussion. The benchmark case is represented

by the black line, the case of withdrawal at 12% by the green line and the case

where the parameter is doubled is represented by the dotted line. We easily

see that the higher the withdrawing parameter, the stronger the MSR eﬀect.

The discontinuity in the consumer surplus at the value of ∆ where ﬁrms start

banking is in this case very neat.

21

Figures 10.1 MSR: firms' profits under different

withdrawing rates

Figure 10.2 MSR: consumers' surplus under

different withdrawing rates

5 Conclusions

The MSR, as it is to be implemented, withdraws (or reinjects) a percentage of

allowances in circulation. Contrarily to what is expected, withdrawing permits

does not simply crowds out private banking it aﬀect production decisions, permit

prices and output prices. Most importantly, the stabilization policy interaction

with the uncertainty on output demand generates (under some conditions) an

option to delay banking permits. To the light of our results, we conclude that

policy makers should be aware of eventual unintended eﬀects of the measures

adjusting the cap, as they modify ﬁrms’ banking strategies. As long as banking

is discouraged and it is replaced by some form of supply restriction, the permits

price may not increase as it should. Simulating the eﬀect of the MSR under

demand shocks seems to be a critical factor to ensure the success of the EU ETS

quantitative ﬂexibility mechanisms.

As Burtraw (2015) aﬃrmed: ”Theory and some evidence suggest that, so

far, the MSR will have a limited eﬀect in ﬁxing the problem directly. However,

to the credit of EU regulators, the MSR signals that the doctor has not given up

on the patient. The European Union has a long-term commitment to emissions

trading—the MSR may buy enough time for prices in the ETS to recover as

the economy recovers. If that does not happen, I believe the European Union

may ultimately replace the MSR with a more direct and simpler instrument—a

reserve price in auctions for emissions allowances that will instill a minimum

price in the market”. In the follow up of our work, we would like to compare

the impact of a price ﬂoor on ﬁrms’ banking behavior, or to include investment

decisions (that would impact pollution intensity) in ﬁrms’ strategies.

22

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7 Appendices

Appendix A.1 Equilibrium values

Benchmark modelling

Exogenous case. The other equilibrium values not presented in the text

for shortness are:

q∗

1=1

R(1 + r) Γ

ne +r2(b−c)

(n+ 1) d, (44)

q∗

2=1

R (1 + r)2Γ

ne −R−3 + r2(b−c)

(n+ 1) d!. (45)

Endogenous rule. The other equilibrium values not presented in the text

for shortness are:

q∗

1,ζ =1

D (1 + r) Γζ

ne +r2+r((1 −ζ2)ζ1+ζ2)(b−c)

(n+ 1) d!, (46)

q∗

2,ζ =1

D (1 + r)2Γζ

ne −r(3 + 2r−(2 + r)ζ1) (1 −ζ2) (b−c)

(n+ 1) d!. (47)

Equilibrium under uncertainty

24

Exogenous rule.

σ∗

2,b =σ∗

2−2 (1 + r) (1 −λ) ∆

e(2 + r)R, (48)

σ∗

2,bm =σ∗

2,b +2 (1 + r) ∆

e(2 + r), (49)

=σ∗

2+2 (1 + r) (λ+ (2 + r) (1 + r)) ∆

e(2 + r)R, (50)

q∗

1,b =q∗

1+2 (1 −λ) ∆

(3 + r(3 + r)) (2 + r) (1 + n)d, (51)

q∗

1,bm =q∗

1,b +r∆

(2 + r) (1 + n)d, (52)

q∗

2,b =q∗

2+2 (1 + r) (1 −λ) ∆

R(2 + r) (1 + n)d, (53)

q∗

2,bm =q∗

2,b −r∆

(2 + r) (1 + n)d. (54)

Endogenous rule.

σ∗

2,b,ζ =σ∗

2,ζ −(1 −ζ1) (1 −ζ2) (2 −ζ2) (1 + r) (1 −λ) ∆

e(2 + r−ζ2)D, (55)

σ∗

2,bm,ζ =σ∗

2,b,ζ +(2 −ζ2) (1 + r) ∆

e(2 + r−ζ2), (56)

q∗

1,b,ζ =q∗

1,ζ +(1 −ζ1) (1 −ζ2) (2 −ζ2) (1 −λ) ∆

(1 + n) (2 + r−ζ2)dD , (57)

q∗

1,bm,ζ =q∗

1,b,ζ +r∆

(2 + r−ζ2) (1 + n)d,(58)

q∗

2,b,ζ =q∗

2,ζ +(1 + r) (1 −ζ1) (1 −ζ2) (2 −ζ2) (1 −λ) ∆

(1 + n) (2 + r−ζ2)dD , (59)

q∗

2,bm,ζ =q∗

2,b,ζ −r(1 −ζ2) ∆

(2 + r−ζ2) (1 + n)d.(60)

Appendix A.2 Equilibrium constraints

Exogenous rule.

Withdrawal case. When nz∗

0< Z and nz∗

1< Z , the MSR reinjects

permits. The absence of arbitrage opportunities given by (9) holds if z∗

0>

0, z∗

1>0 , i.e.

A≥ne

(n+ 1) d×(R−3) (b−c)

α0R−P2

t=0 αt

+x1+x2

α1R−P2

t=0 αt

(61)

25

and

A>ne

(n+ 1) d×(2r+ 3) r(b−c)

(1 + r)2P2

t=0 αt−α2R+(2 + r)x2−(1 + r)2x1

(1 + r)2P2

t=0 αt−α2R.(62)

Backloading case. The withdrawal case is deﬁned by x1<0 and x2<0,

if nz∗

0> Z and nz∗

1> Z. The constraints associated to this case are as follows:

A > ne

(n+ 1) d×(R−3) (b−c)

α0R−P2

t=0 αt

+x1+x2

α1R−P2

t=0 αt

+RZ

α1R−P2

t=0 αt

,(63)

A > ne

(n+ 1) d×(2r+ 3) r(b−c)

(1 + r)2P2

t=0 αt−α2R+(2 + r)x2−(1 + r)2x1

(1 + r)2P2

t=0 αt−α2R

+RZ

(1 + r)2P2

t=0 αt−α2R.(64)

Endogenous case. The absence of arbitrage opportunities given by (9) holds

if z0>0, z1>0, i.e.

A≥ne

(n+ 1) d×(3 + r−ζ2)r(b−c)

Dz0

+Z(R−ζ1(1 −ζ2)−(2 + r)ζ2)

Dz0

,(65)

since nz∗

0> Z and

A≥ne

(n+ 1) d

r(3 + 2r−(2 + r)ζ1) (b−c)

Dz1

,(66)

where16

Dz0=(1 + r)2+ (1 + r) (1 −ζ2)α0−(1 −ζ2)α1−α2,(67)

Dz1= (1 + r)2((1 −ζ1)α0+α1)−((1 + r) + (1 −ζ1)) α2.(68)

No intervention case. The constraints such that there is no intervention,

that is Z < nz∗

0< Z and Z < nz∗

1< Z are as follows:

RZ

α1R−P2

t=0 αt

< A −ne

(n+ 1) d×(R−3) (b−c)

α0R−P2

t=0 αt

<RZ

α1R−P2

t=0 αt

,(69)

RZ

(1 + r)2P2

t=0 αt−α2R< A −ne

(n+ 1) d×(2r+ 3) r(b−c)

(1 + r)2P2

t=0 αt−α2R

<RZ

(1 + r)2P2

t=0 αt−α2R.(70)

16Notice that Dz0>0 and for r∈[0,1], 0 ≤α2≤α1≤α0≤1 we have Dz1>0.Given the

structure of the MSR in the proportional rule, (65) does not depend on ζ1and the constraint

(66) is independent from ζ2.

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Finance for Energy Market Research Centre

Institut de Finance de Dauphine, Université Paris-Dauphine

1 place du Maréchal de Lattre de Tassigny

75775 PARIS Cedex 16

www.fime-lab.org