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Key words. Emissions markets, Cap-and-trade schemes, Equilibrium models, Environmental

Finance.

MARKET DESIGN FOR EMISSION TRADING SCHEMES

REN´

E CARMONA ∗, MAX FEHR †, JURI HINZ ‡,AND ARNAUD PORCHET §

Abstract. The main thrust of the paper is the design and the numerical analysis of new cap-

and-trade schemes for the control and the reduction of atmospheric pollution. The tools developed

are intended to help policy makers and regulators understand the pros and the cons of the emissions

markets. We propose a model for an economy where risk neutral ﬁrms produce goods to satisfy an

inelastic demand and are endowed with permits by the regulator in order to oﬀset their pollution

at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions

do so, while those for which it is harder buy permits from those ﬁrms anticipating that they will

not need them, creating a ﬁnancial market for pollution credits. Our model captures most of the

features of the European Union Emissions Trading Scheme. We show existence of an equilibrium and

uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the

optimal production and trading strategies of the ﬁrms. We choose the electricity market in Texas

to illustrate numerically the qualitative properties observed during the implementation of the ﬁrst

phase of the European Union cap-and-trade CO2emissions scheme, comparing the results of cap-

and-trade schemes to the Business As Usual benchmark. In particular, we conﬁrm the presence of

windfall proﬁts criticized by the opponents of these markets. We also demonstrate the shortcomings

of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which despite of

its ease of implementation, leads to smaller windfall proﬁts than the standard scheme.

1. Introduction. Emission trading schemes, also known as cap and trade sys-

tems, have been designed to reduce pollution by introducing appropriate market mech-

anisms. The two most prominent examples of existing cap and trade systems are the

EU-ETS (European Union Emission Trading Scheme) and the US Sulfur Dioxide

Trading System. In such systems, a central authority sets a limit (cap) on the total

amount of pollutant that can be emitted within a pre-determined period. To en-

sure that this target is complied with, a certain number of credits are allocated to

appropriate installations, and a penalty is applied as a charge per unit of pollutant

emitted outside the limits of a given period. Firms may reduce their own pollution

or purchase emission credits from a third party, in order to avoid accruing potential

penalties. The transfer of allowances by trading is considered to be the core principle

leading to the minimization of the costs caused by regulation: companies that can

easily reduce emissions will do so, while those for which it is harder buy credits.

In a cap-and-trade system, the initial allocation (i.e. the total number of al-

lowances issued by the regulator) should be chosen in order for the scheme to reach a

given emissions level. This total initial allocation is indeed the crucial parameter that

the regulator uses as a knob to control the emission level. But while the value of the

total initial allocation is driven by the emissions target, the speciﬁc distribution of

these allowances among the various producers and market participants can be chosen

∗Department of Operations Research and Financial Engineering, Princeton University, Prince-

ton, NJ 08544. Also with the Bendheim Center for Finance and the Applied and Computational

Mathematics Program. (rcarmona@princeton.edu).

†Institute for Operations Research, ETH Zurich, CH-8092 Zurich, Switzerland

(maxfehr@ifor.math.ethz.ch).

‡National University of Singapore, Department of Mathematics, 2 Science Drive, 117543 Singa-

pore (mathj@nus.edu.sg), Partially supported by WBS R-703-000-020-720 / C703000 of the Risk

Management Institute at the National University of Singapore

§Timbre J320, 15 Boulevard Gabriel Pri, 92245 MALAKOFF Cedex, FRANCE,

(arnaud.porchet@gmail.com).

1

2R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

in order to create incentives to design and build cleaner and more eﬃcient production

units.

Naturally, emissions reduction increases the costs of goods whose production

causes those emissions. Part or all of these costs are passed on to the end con-

sumer and substantial windfall proﬁts are likely to occur. Based on an empirical

analysis of power generation proﬁtability in the context of EU-ETS, strong empir-

ical evidence of the existence of such proﬁts is given in [14]. The authors of this

study come to the conclusion that power companies realize substantial proﬁts since

allowances are received for free while they are always priced into electrical power at a

rate that depends upon the emission rate of the marginal production unit: producers

seem to take advantage of the trading scheme to make extra proﬁt. This phenomenon

can even happen in a competitive setting. What follows is a simple illustration in a

deterministic framework.

Let us consider a set of ﬁrms that must satisfy a demand of D= 1 MWh of

electricity at each time t= 0,1,· · · , T −1, and let us assume that there are only

two possible technologies to produce electricity: gas technology which has unit cost

2 $ and emits 1 ton of CO2per MWh, and coal technology which has unit cost 1 $

and emits 2 tons of CO2per MWh. In this simple model, the total capacity of gas

is 1 MWh and coal’s capacity is also 1 MWh. We also suppose that producers face

a penalty π > 1 $ per ton of CO2not oﬀset by credits, and that a total of T−1

credits are distributed to the ﬁrms, allowing them to oﬀset altogether T−1 tons of

CO2. In this situation, we arrive at two conclusions. First, as demand needs to be

met, total emissions will be higher or equal than Ttons, even if all ﬁrms use the clean

technology (gas). Second, ﬁrms are always better oﬀ reducing emissions than paying

the penalty. As a consequence, the optimal generation strategy is to only use the

gas technology and emit Ttons of CO2. At least one ﬁrm has to pay the penalty,

and the price of emission credits is necessarily equal to πat each time. Indeed the

missing credit has a value πfor both the buyer and the seller. The price of electricity

is then 2 + πbecause a marginal decrease in demand will induce a marginal gain in

generation cost and a marginal decrease of the penalty paid. The total proﬁt for the

producers is π(T−1), the penalty paid by the producers to the regulator is π, and the

total cost for the customers is (2 + π)T. Consider now the Business As Usual (BAU)

situation: the demand is met by using coal technology, the price of electricity is 1, the

total proﬁt for producers is 0 and the total cost for the customers is T. In this simple

example the producers cost induced by the trading scheme is T+π: producers must

buy more expensive fuel, so a proﬁt Tis made by the fuel supplier and they have to

pay the penalty π. The increase in fuel price, or switching cost, is a marginal cost

that must factor into the electricity price. The penalty is a ﬁxed cost paid at the end,

but we see that in this trading scheme, this ﬁxed cost is rolled over the entire period

and paid by the customers at each time, inducing a windfall proﬁt for the producers.

This windfall proﬁt is exactly equal to the market value of the T−1 credits: these

credits are given for free by the regulator but their market value is actually funded

by the customer.

Another feature of emissions trading schemes is the risk of non compliance faced

by the producers and the regulator. The EU-ETS was introduced as a way of comply-

ing with the targets set by the Kyoto Protocol. Phase 1 of the Kyoto Protocol sets a

ﬁxed cap for annual emissions of CO2by year 2012 to all industrialized countries that

ratiﬁed the protocol (Annex I countries). This reduction should guarantee on average

a level of emissions of 95 % of what it was in year 1990. All countries are free to

Market Designs for Emissions Trading Schemes 3

adopt the emission reduction policy of their choice, but in case of non-compliance in

2012, they face a penalty (payment of 1.3 emission allowances for each ton not oﬀset

in Phase 1). The EU-ETS was designed to ensure compliance for the whole EU zone.

However, in an uncertain environment, there exists the possibility that the scheme

will fail its goal and that the producers will exceed the ﬁxed cap set at the beginning

of the compliance period. In this case, it is the regulator’s responsibility to com-

ply with the target by buying allowances from other countries or generate additional

allowances by investing in clean projects under the Clean Development Mechanism

(CDM for short) or the Joint Implementation (JI for short) mechanism, or otherwise,

to pay the penalty. The design of emission trading schemes must also address this

question.

In the present work, we give a precise mathematical foundation to the analysis of

emission trading schemes and quantitatively investigate the impact of emission regu-

lation on consumers costs and company’s proﬁts. Based on an equilibrium model for

perfect competition, we show that the action of an emission trading scheme combines

two contrasting aspects. On the one hand, the system reduces pollution at the low-

est cost for the society, as expected. On the other hand, it forces a notable transfer

of wealth from consumers to producers, which in general exceeds the social costs of

pollution reduction.

In a perfect economy where all customers are shareholders, windfall proﬁts are

redistributed, at least partially, by dividends. However, this situation is not the

general case and the impact of regulation on prices should be addressed. There

are several other ways to return part of the windfall proﬁts to the consumers. The

most prominent ones are taxation and charging for the initial allowance distribution.

Beyond the political risks associated with new taxes, we will show that one of the main

disadvantages of this ﬁrst method is its poor control of emissions under stochastic

abatement costs. Concerning auctioning, it is important to notice that, in the ﬁrst

phase of the EU-ETS, individual countries did not have to give away the totality of

their credit allowances for free. They could choose to auction up to 10% of their

total allowances. Strangely enough, except for Denmark, none of them exercised this

option. On the other hand using auctioning as a way to abolish windfall proﬁts,

one looses one of the main features of cap-and-trade schemes, namely the mechanism

which allows to control the incentives to invest in and develop cleaner production

technologies. Indeed, a signiﬁcant reduction of windfall proﬁts through auctioning,

if at all possible, requires that a huge amount or even the total initial allocation

is auctioned. Further it involves a signiﬁcant risk for companies since the capital

invested to procure allowances at the auction may be higher than the income later

recovered from allowances prices.

In this work, we argue that cap-and-trade schemes can work, even in the form

implemented in the ﬁrst phase of EU-ETS, at least as long as allowance distribution is

properly calibrated. Moreover, we prove that it is possible to design modiﬁed emission

trading schemes that overcome these problems. We show how to establish trading

schemes that reduce windfall proﬁts while exhibiting the same emission reduction

performance as the generic cap and trade system used in the ﬁrst implementation

phase of the EU-ETS. These schemes also have the nice feature that a signiﬁcant

amount of the allowances can be allocated as initial allocation to encourage cleaner

technologies.

Despite frequent articles in the popular press and numerous speculative debates

in specialized magazines and talk-shows, the scientiﬁc literature on cap-and-trade

4R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

systems is rather limited. We brieﬂy mention a few related works chosen because of

their relevance to our agenda. The authors of [3] and [9] proposed a market model for

the public good environment introduced by tradable emission credits. Using a static

model for a perfect market with pollution certiﬁcates, [9] shows that there exists a

minimum cost equilibrium for companies facing a given environmental target. The

conceptual basis for dynamic permit trading is, among others, addressed in [2], [15],

[11], [7], [12] and [13]. Meanwhile, the recent work [13] suggests also a continuous-time

model for carbon price formation. Beyond these themes, there exists a vast literature

on several related topics, including equilibrium [1], empirical evidence from already

existing markets [6], [14], and uncertainty and risk [5], [8], [16]. The model we present

below follows the baseline suggested in [4].

We close this introduction with a quick summary of the contents of the paper.

Section 2 gives the details of the mathematical model used to capture the dynamic

features of a cap-and-trade system. We introduce the necessary notation to describe

the production of goods and the proﬁt mechanisms in a competitive economy. Ex-

ogenous demand for goods is modeled by means of adapted stochastic processes. We

assume that demand is inelastic and has to be met exactly. This assumption could

be viewed as unusually restrictive, but we argue that it is quite realistic in the case

of electricity. We also introduce the emissions allowance allocations and the rules of

trading in these allowances.

Section 3 deﬁnes the notion of competitive equilibrium for risk neutral ﬁrms in-

volved in our cap-and-trade scheme. Preliminary work shows that most of the theoret-

ical results of this paper still hold for risk averse ﬁrms if preferences are modeled with

exponential utility. However, in order to avoid muddying the water with unnecessary

technical issues which could distract the reader from the important issues of pollution

abatement, we restrict ourselves to the less technical case of risk neutral ﬁrms. For

the sake of completeness, we solve the equilibrium problem in the Business As Usual

(BAU from now on) case corresponding to the absence of market for emissions per-

mits. In this case, as expected, the prices of goods are given by the standard merit

order pricing typical of deregulated markets. The section closes with the proof of a

couple of enlightening necessary conditions for the existence of an equilibrium in our

model. These mathematical results show that at compliance time, the equilibrium

price of an emission certiﬁcate can only be equal to 0 or to the penalty level chosen

by the regulator. The second important necessary condition shows that in equilib-

rium, the prices of the goods are still given by a merit order pricing provided that

the production costs are adjusted for the cost of emissions. This result is important

as it shows exactly how the price of pollution gets incorporated in the prices of goods

in the presence of a cap-and-trade scheme. The following Section 4 is devoted to the

rigorous proof of the existence of an equilibrium. The proof uses classical functional

analysis results on optimization in inﬁnite dimensional spaces. It follows the lines

of a standard argument based on the analysis of what an informed central planner

(representative agent) would do in order to minimize the social cost of meeting the

demand for goods.

Section 5 is devoted to the analysis of the standard cap-and-trade scheme featured

in the implementation of the ﬁrst phase of the EU-ETS. By comparison with BAU

scenarios, we show that properly chosen levels of penalty and pollution certiﬁcate

allocations lead to desired emissions targets. However, our numerical experiments

on a case study of the electricity market in Texas show the existence of excessive

windfall proﬁts. As explained earlier in our literature review, these proﬁts have been

Market Designs for Emissions Trading Schemes 5

observed in the ﬁrst phase of EU-ETS, giving credibility to the critics of cap-and-

trade systems. Section 6 can be viewed as the main thrust of the paper beyond the

theoretical results proven up to that point. We propose a general framework includ-

ing taxes and subsidies along the standard cap-and-trade schemes. We demonstrate

the shortcomings of the tax systems which suﬀer from poor control of the windfall

proﬁts and unexpected expensive reduction policies when it comes to emissions re-

duction targets under stochastic abatement costs. We concentrate our analysis on

several new alternative cap-and-trade schemes and we show numerically that a rela-

tive allocation scheme can resolve most of the issues with the other schemes. Such a

relative allocation scheme is easy to describe and implement as pollution allowances

are distributed proportionally to production. Even though the number of permits is

random in a relative scheme, and hence cannot be known in advance, its statistical

distribution is well understood as it is merely a scaled version of the distribution of

the demands for goods. Consequently, setting up caps to meet pollution targets is not

much diﬀerent from the standard cap-and-trade schemes. Moreover, the coeﬃcient of

proportionality providing the number of permits is an extra parameter which should

make the calibration more eﬃcient. Indeed, one shows that properly calibrated, the

relative schemes reach the same pollution targets as the standard schemes while at

the same time, they keep social costs and windfall proﬁts in control.

Section 7 gathers more mathematical properties of the generalized cap-and-trade

schemes introduced in the previous section. Our results demonstrate the versatility

and the ﬂexibility of such a generalized framework. It shows that regulators can con-

trol cap-and-trade schemes in order to reach pre-assigned pollution targets with zero

windfall proﬁts and reasonably small social costs, or even to force equilibrium elec-

tricity prices to be equal to target prices. However, because of the level of complexity

of their implementations, it is unlikely that the schemes identiﬁed there will be used

by policy makers or regulators. The paper concludes with Section 8 which reviews

the main results of the paper recasting them in the perspective of the public policy

challenging issues uncovered by the results of the paper.

2. Standard Cap-and-Trade Scheme. In this section we present the elements

of our mathematical analysis. We consider an economy where a set of ﬁrms produce

and supply goods to end-consumers over a period [0, T ]. The production of these goods

is a source of pollutant emissions. In order to reduce this externality, a regulator

distributes emissions allowances to the ﬁrms at time 0, allows them to trade the

allowances on an organized market between times 0 and T, and at the end of this

compliance period, taxes the ﬁrms proportionally to their net cumulative emissions.

In what follows (Ω,F,{Ft, t ∈ {0,1, . . . , T }},P) is a ﬁltered probability space.

We denote by E[.] the expectation operator under probability Pand by Et[.] the

expectation operator conditional to Ft. The σ-ﬁeld Ftrepresents the information

available at time t. We will also make use of the notation Pt(.) := Et[1{.}] for the

conditional probability with respect to Ft.

2.1. Production of Goods. A ﬁnite set Iof ﬁrms produce and sell a set K

of diﬀerent goods at times 0,1, . . . , T −1. Each ﬁrm i∈Ihas access to a set Ji,k

of diﬀerent technologies to produce good k∈K, that are sources of emissions (e.g.

greenhouse gases ). Each technology j∈Ji,k is characterized by:

•a marginal cost e

Ci,j,k

tof producing one unit of good kat time t;

•an emission factor ei,j,k measuring the volume of pollutants emitted per unit

of good kproduced by ﬁrm iwith technology j;

•a production capacity κi,j,k.

6R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

For the sake of notation we introduce the index sets

Mi={(j, k) : k∈K, j ∈Ji,k}, i ∈I ,

M={(i, j, k) : i∈I , k ∈K, j ∈Ji,k}.

In this paper, our main example of produced good is electricity. We make the assump-

tion that the production costs are non-negative, adapted and integrable processes.

At each time 0 ≤t≤T−1, ﬁrm i∈Idecides to produce throughout the period

[t, t + 1) the amount ξi,j,k

tof good k∈K, using the technology j∈Ji,k . Since the

choice of the production level ξi,j,k

tis based only on present and past observations, the

processes ξi,j,k are supposed adapted and, since production cannot exceed capacity,

we require that the inequalities

0≤ξi,j,k

t≤κi,j,k, i ∈I, k ∈K, j ∈Ji,k, t = 0,1,· · · , T −1,(2.1)

hold almost surely. Our market is driven by an exogenous and inelastic demand for

goods. Since electricity production is a signiﬁcant proportion of the emissions covered

by the existing schemes, this inelasticity assumption is reasonable. We denote by Dk

t

the demand at time tfor good k∈K. This demand process is supposed to be adapted

to the ﬁltration {Ft}t. For each good k∈K, we assume that the demand is always

smaller than the total production capacity for this good, namely that:

0≤Dk

t≤X

i∈IX

j∈Ji,k

κi,j,k almost surely, k∈K. (2.2)

This assumption is a natural extension of the assumption of inelasticity of the de-

mand as it will conveniently discard issues such as blackouts which would only be a

distraction given the purposes of the paper.

2.2. Emission Trading. We denote by π∈[0,∞) the penalty per unit of pol-

lutant. For example, in the original design of the European Union Emissions Trading

Scheme (EU-ETS) πwas set to 40€per metric ton of Carbon Dioxyde equivalent

(tCO2e). For each ﬁrm, the net cumulative emission is the amount of emissions which

have not been oﬀset by allowances at the end of the compliance period. It is com-

puted at time Tas the diﬀerence between the total amount of pollutants emitted over

the entire period [0, T ] minus the number of allowances held by the ﬁrm at time T

and redeemed for the purpose of emissions abatement. The net cumulative emission

is this diﬀerence whenever positive, and 0 otherwise.

For the sake of simplicity we assume that the entire period [0, T ] corresponds to

one simple compliance period. In particular, at maturity T, all the ﬁrms have to

cover their emissions by allowances or pay a penalty. Moreover, certiﬁcates become

worthless if not used as we do not allow banking from one phase to the next. So in this

economy, operators of installations that emit pollutants will have two fundamental

choices in order to avoid unwanted penalties: reduce emissions by producing with

cleaner technologies or buy allowances.

At time 0, each ﬁrm i∈Iis given an initial endowment of Λi

0allowances. So if

it were to hold on to this initial allowance endowment until the end, it would be able

to oﬀset up to Λi

0units of emissions, and start paying only if its actual cumulative

emissions exceed that cap level. This is the cap part of a cap-and-trade scheme.

Depending upon their views on the demands for the various products and their risk

appetites, ﬁrms may choose production schedules leading to cumulative emissions in

Market Designs for Emissions Trading Schemes 7

excess of their caps. In order to oﬀset expected penalties, they may engage in buying

allowances from ﬁrms which expect to meet demand with less emissions than their

own cap. This is the trade part of a cap-and-trade schemes.

Remark 1. A ﬁrst generalization of the above allowance distribution scheme

is to reward the ﬁrms with allocations Λi

tat each time t= 0,1,· · · , T −1. Even

though modeling EU-ETS would only require one initial (deterministic) allocation Λi

0

for each ﬁrm, we shall assume that the distribution of pollution permits is given by

adapted stochastic processes {Λi

t}t=0,1,··· ,T −1. Indeed, all the theoretical results proven

in the paper hold for these more general permit allocation processes since existence,

uniqueness and characterization of the equilibrium price processes depend only upon

the total number of emission permits issued during the compliance period, not on the

way the permits are distributed over time and among the various economic agents.

However as we will demonstrate, the statistical properties of social costs and wind-

fall proﬁts depend strongly on the way permits are allocated. The challenge faced by

policy makers is to optimally design these allocation schemes to minimize social costs

while satisfying emissions reduction targets, controlling producers windfall proﬁts and

setting incentives for the development of cleaner production technologies. We shall

concentrate on these issues in Sections 6 and 7.

Allowances are physical in nature, since they are certiﬁcates which can be re-

deemed at time Tto oﬀset measured emissions. But, because of trading, these

certiﬁcates change hands at each time t= 0,1,· · · , T , and they become ﬁnancial

instruments. However in general the allocation of allowances does not take place at a

single timepoint 0. For example, in EU ETS, allowances are allocated in March each

year, while the 5 year compliance period starts in January. Therefore a signiﬁcant

amount of allowances are traded via forward contracts. Because compliance takes

place at time T, and only at that time, we will restrict ourselves to the situation

where trading of emission allowances is done via forward contracts settled at time T.

Remark 2. Because compliance takes place at time T, a simple no-arbitrage ar-

gument implies that the forward and spot allowance prices diﬀer only by a discounting

factor, such that trading allowances or forwards gives the same expected discounted

payoﬀ at time T. Therefore under the equilibrium deﬁnition that will be introduced

in Section 3, considering only forward trading yields no loss of generality. Moreover

allowing trading in forward contracts in our model provides a more ﬂexible setting: it

is more general than considering only spot trading, since it allows for trading pollution

permits even before these allowances are issued and allocated. This turns out to be an

important feature when dealing with general allocation schemes.

We denote by Atthe price at time tof a forward contract guaranteeing delivery of

one allowance certiﬁcate at maturity T. The terminology price at time tis misleading

as there is no exchange of funds at time t.Atis better seen as a strike than a price

in the sense that it is the price (in time Tcurrency) at which the buyer at time tof

the forward contract agrees to purchase the allowance certiﬁcate at time T.

Each ﬁrm can take positions on the forward market, and we denote by θi

tthe

number of forward contracts held by ﬁrm iat the beginning of the time interval

[t, t +1). As usual, θi

t>0 when the ﬁrm is long and θi

t<0 when it is short. We deﬁne

a trading strategy of ﬁrm ias an adapted process {θi

t}t=0,··· ,T . If we denote by fi

t

the quantity of forward contracts bought or sold at time tand throughout the period

[t, t + 1), fibeing an adapted process, the position at time tveriﬁes: θi

t+1 =θi

t+fi

t.

The net cash position resulting from this trading strategy, leading to a net position

8R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

of θi

Tcontracts at time T, is:

RA

T(θ) := −

T

X

t=0

fi

tAt=

T−1

X

t=0

θi

t(At+1 −At)−θi

TAT.(2.3)

We here make the assumption that allowances can be traded until time T, whereas

production of goods is decided at time tfor the whole period [t, t + 1), so that the

last production decision occurs at time T−1. This assumption is reasonable since

production of good is a less ﬂexible process than trading.

2.3. Proﬁts. As we argued earlier, it is natural to work with T-forward al-

lowance contracts because compliance takes place at time T. By consistency, it is

convenient to express all cash ﬂows, position values, ﬁrm wealth, and good values in

time T-currency. As a side fringe beneﬁt, this will avoid discounting in the computa-

tions to come. So we use for num´eraire the price Bt(T) at time tof a Treasury (i.e.

non defaultable) zero coupon bond maturing at T. We denote by {˜

Sk

t}t=0,1,··· ,T the

adapted spot price process of good k∈K, and according to the convention stated

above, we shall ﬁnd it convenient to work at each time twith the T-forward price

Sk

t=˜

Sk

t/Bt(T)

and we skip the dependence in Tfrom the notation of the T-forward price as Tis the

only maturity we are considering.

Hence, a cash ﬂow Xtat time tis equivalently valued as a cash ﬂow Xt/Bt(T) at

maturity T. So if ﬁrm ifollows the production policy ξi={(ξi,j,k

t)k∈K j∈Ji,k }T−1

t=0 its

instantaneous revenues at time tfrom goods production is given by

X

(j,k)∈Mi

(˜

Sk

t−e

Ci,j,k

t)ξi,j,k

t

and its time T-forward value is given by:

X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t

provided we set Ci,j,k

t=e

Ci,j,k

t/Bt(T). The total net gains from producing and selling

goods are thus:

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t.(2.4)

In order to hedge their production decisions, ﬁrms trade on the emissions market

by adjusting their forward positions in allowances. In addition, at maturity T, each

ﬁrm iredeems allowances to cover its emissions and/or pay a penalty. Let

Πi(ξi) :=

T−1

X

t=0 X

(j,k)∈Mi

ei,j,kξi,j,k

t(2.5)

be the actual cumulative emissions of ﬁrm iwhen it uses production strategy ξi.

We also suppose that there exists another source of emissions on which ﬁrm ihas

Market Designs for Emissions Trading Schemes 9

no control, denoted ∆i, and supposed to be an FT-measurable random variable. If

we think of electricity as one of the produced goods for example, the presence of this

uncontrolled source of emissions can easily be explained. Usually electricity producers

are required to hold a reserve margin in order to respond to short time demand changes

and to protect against sudden outages or unexpectedly rapid ramps in demand. When

scheduling their plants it is not yet known how much of this reserve margin will

be used. Therefore in most markets there is an uncertainty on the exact emission

level when a production decision is made. Alternatively, we can see ∆ias a sink

of emissions, accounting for example for the credits gained from Clean Development

Mechanisms or Joint Implementation mechanisms. In this case it can take negative

values. In a ﬁrst reading ∆ican be thought of as being 0 for the sake of simplicity. We

shall see later in the paper that its presence helps characterizing the equilibrium of

the economy and that it is a useful tool for modeling several variations of the model.

Introducing the net amount Γiof allowances that producer i∈Ican use to oﬀset the

scheduled emissions by

Γi= ∆i−

T−1

X

t=0

Λi

t(2.6)

the total penalty paid by ﬁrm iat time Tis:

π(Γi+ Πi(ξi)−θi

T)+.(2.7)

Combining (2.4) and (2.7) together with (2.3), we obtain the expression for the

terminal wealth (proﬁts and losses at time T) of ﬁrm i:

LA,S,i(θi, ξ i) :=

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t

+

T−1

X

t=0

θi

t(At+1 −At)−θi

TAT

−π(Γi+ Πi(ξi)−θi

T)+.(2.8)

To emphasize the mathematical technicalities of the model, we underline the fact

that demands and production costs change with time in a stochastic manner. The

statistical properties of these processes are given exogenously, and are known at time

0 by all ﬁrms. Moreover, we always assume that these processes satisfy the constraints

(2.1) and (2.2) almost surely. Agents adjust their production and trading strategies

in a non-anticipative manner to their observations of the ﬂuctuations in demand and

production costs. In turn, the production and trading strategies ξiand θibecome

respectively adapted stochastic processes on the stochastic base of the demand and

production costs.

3. Market Equilibrium. In this section, we follow the common apprehension

that a realistic market state is described by prices which correspond to a so-called

market equilibrium, a situation, where the demand for each product is covered, all

ﬁnancial positions are in the zero net supply, and each ﬁrm is satisﬁed by its own

strategy. We deﬁne such an equilibrium and provide necessary conditions for its

existence.

10 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

3.1. Deﬁnition of Equilibrium. For any 1 ≤p≤ ∞ and for any normed

vector space F, we introduce the following space of adapted processes:

Lp

t(F) := (Xs)t

s=0;F-valued, kXsk ∈ Lp(Fs), s = 0, . . . , t.(3.1)

We also introduce the spaces of admissible production strategies:

Ui:= n(ξi

t)T−1

t=0 ∈ L∞

T−1(RMi); 0 ≤ξi,j,k

t≤κi,j,k, t = 0, . . . , T −1o,

U:=

ξ∈Y

i∈I

Ui;X

i∈IX

j∈Ji,k

ξi,j,k

t≥Dk

t, k ∈K, t = 0,· · · , T −1

and the spaces of admissible trading strategies:

Vi(A) := (θi

t)T+1

t=1 ,adapted,kRA

T(θi)k ∈ L1(FT)

V(A) := Y

i∈I

Vi(A).

In order to avoid problems with existence of expected values in (2.8), we suppose that

allowance demand and production costs are integrable:

assumption 1.

Γi∈L1,{Ci

t= (Ci,j,k

t)(j,k)∈Mi}T−1

t=0 ∈ L1

T−1(RMi)i∈I, (3.2)

In what follows, we also use a technical assumption on the nature of the uncontrolled

emissions. Even though this assumption is not needed for most of the equilibrium

existence results, it will help us characterize the prices in equilibrium by ruling out

pathological situations. This technical assumption states that up until the end of the

compliance period, there is always uncertainty about the expected pollution level due

to unpredictable events as described in Section 2.3 in the sense that conditionally

on the information available at time T−1, the sum of all the Γi’s has a continuous

distribution. More precisely, we shall assume that

assumption 2. the FT−1-conditional distribution of Pi∈I∆ipossesses almost

surely no point mass, or equivalently, for all FT−1-measurable random variables Z

P(X

i∈I

∆i=Z)= 0 (3.3)

As we already pointed out, this technical assumption will help us reﬁne the statements

of some of the results leading to the equilibriums.

Following the intuition that given price processes A={At}T

t=0 and S={(Sk

t)k∈K}T−1

t=0

each ﬁrm aims at increasing its own wealth by maximizing

(θi, ξi)7→ E[LA,S,i (θi, ξi)],(3.4)

over its admissible investment and production strategies, we are led to deﬁne equilib-

rium in the following way:

Definition 1. The pair of price processes (A∗, S∗)∈ L1

T(R)× L1

T−1(R|K|)are

an equilibrium of the market if for each i∈Ithere exists (θ∗i, ξ∗i)∈ Vi(A∗)× Uisuch

Market Designs for Emissions Trading Schemes 11

that:

(i) All ﬁnancial positions are in zero net supply, i.e.

X

i∈I

θ∗i

t= 0, t = 0, . . . , T (3.5)

(ii) Supply meets demand for each good

X

i∈IX

j∈Ji,k

ξi,j,k

t=Dk

t, k ∈K, t = 0, . . . , T −1 (3.6)

(iii) Each ﬁrm i∈Iis satisﬁed by its own strategy in the sense that

E[LA∗,S∗,i (θ∗i, ξ∗i)] ≥E[LA∗,S ∗,i(θi, ξ i)] for all (θi, ξi)∈ V i(A∗)× U i.(3.7)

3.2. Equilibrium in the Business As Usual Scenario . When the penalty

πis equal to zero, an equilibrium should correspond to the Business As Usual sce-

nario. As we explain below, it is characterized by the classical merit order production

strategy. At time tand for each good k, all the production means of the economy are

ranked by increasing production costs Ci,j,k

t. Demand is met by producing from the

cheapest production means and good k’s equilibrium spot price is the marginal cost

of production of the most expensive production means used to meet demand Dk

t.

More precisely, if (A∗, S∗) is an equilibrium, the optimization problem of ﬁrm iis

sup

(θi,ξi)∈Vi(A∗)×Ui

E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t+

T−1

X

t=0

θi

t(At+1 −At)−θi

TAT

.

Trading and production strategies are thus decoupled from each other and we are left

with a classical competitive equilibrium problem where each ﬁrm maximizes

sup

ξi∈Ui

E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t

,(3.8)

and the equilibrium prices S∗are set so that supply meets demand. The solution

of this equilibrium problem is given by the following linear program for each good

k∈K:

((ξ∗i,j,k

t)j∈Ji,k )i∈I= argmax((ξi,j,k

t)j∈Ji,k )i∈IX

i∈IX

j∈Ji,k

−Ci,j,k

tξi,j,k

t(3.9)

s.t. X

i∈IX

j∈Ji,k

ξi,j,k

t=Dk

t

ξi,j,k

t≤κi,j,k for i∈I, j ∈Ji,k

ξi,j,k

t≥0 for i∈I, j ∈Ji,k .

for all times t, and the associated equilibrium prices are

S∗k

t= max

i∈I, j ∈Ji,k(Ci,j,k

t)1{ξ∗i,j,k

t>0},(3.10)

12 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

This is exactly the merit order pricing mechanism of electricity that can be observed

in most deregulated electricity markets without emission trading scheme. Conversely,

it is easily seen that the above prices together with the above strategies deﬁne an

equilibrium. In Section 4 we will see that even under an emission trading scheme the

dispatching of production among producers is still a merit order-like dispatching with

costs adjusted to take into account the mark-to-market value of emissions.

3.3. Necessary Conditions for the Existence of an Equilibrium. Before

turning to the full characterization of the equilibriums, we present some necessary

conditions that will provide interesting insight.

Proposition 3.1 (Necessary Conditions). Let (A∗, S∗)be an equilibrium and

(θ∗, ξ∗)an associated optimal strategies, then following conditions hold:

(i) Then the allowance price A∗is a bounded martingale with values in [0, π]such that

{A∗

T= 0} ⊇ {Γ + Π(ξ∗)<0},{A∗

T=π} ⊇ {Γ + Π(ξ∗)>0}(3.11)

up to sets of probability zero.

(ii) If moreover Assumption 2 holds, then it follows that A∗is almost surely given by

A∗

t=πE[1{Γ+Π(ξ∗))≥0}|Ft] (3.12)

for all t= 0, . . . , T .

(iii) The spot prices S∗kand the optimal production strategy ξ∗icorrespond to a merit

order-type equilibrium with adjusted costs Ci,j,k

t+ei,j,kA∗

t.

Proof. First let us show that A∗has to be a martingale. This is seen as follows:

if not, there exists a time tand a set A∈Ftof non-zero probability such that

Et[A∗

t+11A]>1AA∗

t(resp. <). Then for each agent i∈Ithe trading strategy given

by ¯

θi

s=θ∗i

sfor all s6=tand ¯

θi

t=θ∗i

t+ 1A(resp ¯

θi

t=θ∗i

t−1A) outperforms the

strategy θ∗i, contradicting the third property of an equilibrium.

To prove (3.11) notice that according to the deﬁnition of the equilibrium, θ∗i

T(ω)

coincides for almost all ω∈Ω with the maximizer of

z→ −A∗

T(ω)z−π(Γi(ω)+Πi(ξ∗i)(ω)−z)+.(3.13)

As a consequence, we obtain A∗

T∈[0, π] almost surely, since if A∗

T(ω)6∈ [0, π] then

there exists no maximizer to (3.13). Further, observe that if A∗

T(ω)∈(0, π] then the

maximizer is less than or equal to Γi(ω) + Πi(ξ∗i)(ω) and if A∗

T(ω)∈[0, π) then the

maximizer is greater than or equal to Γi(ω)+Πi(ξ∗i)(ω). This holds for each ihence

following inclusions are satisﬁed almost surely

{A∗

T∈(0, π]} ⊆ ∩i∈I{θ∗i

T≤Γi+ Πi(ξ∗)} ⊆ {X

i∈I

θ∗i

T≤Γ + Π(ξ∗)}(3.14)

{A∗

T∈[0, π)} ⊆ ∩i∈I{θ∗i

T≥Γi+ Πi(ξ∗)} ⊆ {X

i∈I

θ∗i

T≥Γ + Π(ξ∗)}.(3.15)

That is

{A∗

T∈(0, π]}∩{Γ + Π(ξ∗)<0} ⊆ {X

i∈I

θ∗i

T<0},(3.16)

{A∗

T∈[0, π)}∩{Γ + Π(ξ∗)>0} ⊆ {X

i∈I

θ∗i

T>0}.(3.17)

Market Designs for Emissions Trading Schemes 13

Observe that due to the ﬁrst equilibrium condition Pi∈Iθ∗i

T= 0, the events on the

right hand sides of (3.16) and (3.17) are sets of probability zero which shows that the

inclusions (3.11) hold almost surely. Condition (ii) is a direct consequence of (i) and

Assumption 2.

Finally, the optimization problem of agent ican be written as:

sup

θi∈Ui

E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−ei,j,kA∗

T)ξi,j,k

t

= sup

θi∈Ui

E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−ei,j,kA∗

t)ξi,j,k

t

(3.18)

thanks to the martingale property of A∗. Comparing the above optimization problem

with (3.8), we observe that the equilibrium can be seen as a competitive production

equilibrium with adjusted costs Ci,j,k

t+ei,j,kA∗

t.

This concludes the proof.

The above results provide a better understanding of what a potential equilibrium

should be. The allowance price must always be in [0, π], which is very intuitive since

buying an extra allowance at time twill result in a gain of at most πat time T.

As highlighted in the previous section, the equilibrium in the BAU scenario can be

related to a global cost minimization problem. We shall see in the next section that

the equilibrium in the presence of a trading scheme enjoys the property of social

optimality in the sense that any equilibrium corresponds to the solution of a certain

global optimization problem, where the total pollution is reduced at minimal overall

costs. We call this optimization problem the representative ﬁrm problem. Beyond the

economic interpretation of social-optimality, the importance of the global optimization

problem is that its solution helps calculate the allowance prices in equilibrium. We

now explore this connection in detail.

4. Equilibrium and Global Optimality. In this section, we show rigorously

the existence of an equilibrium as deﬁned in Deﬁnition 1. We do so by re-framing

the problem as an equivalent global optimization problem involving a hypothetical

informed central planner (which we call a representative agent). We prove the equiv-

alence of the two approaches, and as a by-product of the necessary condition proven

in the previous section, we derive the uniqueness of the allowance price process.

4.1. The Representative Agent Problem . For each admissible production

strategy ξ={ξi}i∈I∈ U, the overall production costs are deﬁned as

C(ξ) :=

T−1

X

t=0 X

(i,j,k)∈M

ξi,j,k

tCi,j,k

t.

and the overall cumulated emissions as

Π(ξ) :=

T−1

X

t=0 X

(i,j,k)∈M

ei,j,kξi,j,k

t.(4.1)

14 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Using the notation

Γ := X

i∈I

Γi

for the aggregate uncontrolled emissions and allowance endowments, the total costs

from production and penalty payments can be deﬁned as

G(ξ) := C(ξ) + π(Γ + Π(ξ))+, ξ ∈ U .(4.2)

We introduce the global optimization problem

inf

ξ∈U E[G(ξ)] (4.3)

which corresponds to the objective of an informed central planner trying to minimize

overall expected costs. Recall that ξis admissible if ξ∈ U , i.e. if the demand is met

and the production constraints are satisﬁed. The reason for the introduction of this

global optimization problem is contained in the second necessary condition for the

existence of equilibrium.

Proposition 4.1. If (A∗, S∗)is an equilibrium with associated strategies (θ∗, ξ∗),

then ξ∗is a solution of the global optimization problem (4.3) .

Proof. Obviously, it suﬃces to show that

E(G(ξ∗)) ≤E(G(ξ)) for all ξ∈ U.(4.4)

In order to do so we notice that:

X

i∈I

E[LA∗,S∗,i (θ∗i, ξ∗i)] = ET−1

X

t=0 X

i,j,k∈M

(S∗k

t−Ci,j,k

t)ξ∗i,j,k

t

+

T−1

X

t=0 X

i∈I

θ∗i

t(A∗

t+1 −A∗

t)−X

i∈I

θ∗i

TA∗

T

−πX

i∈I

(Γi+ Πi(ξ∗i)−θ∗i

T)+

=ET−1

X

t=0 X

k∈K

S∗k

tX

i∈IX

j∈Ji,k

ξ∗i,j,k

t−C(ξ∗)

−πX

i∈I

(Γi+ Πi(ξ∗i)−θ∗i

T)+

where we used the fact that in equilibrium, Pi∈Iθ∗i

t= 0 holds for all t= 0, . . . , T

due to condition (i) of Deﬁnition 1. Next we use the convexity inequality

X

i∈I

x+

i≥X

i∈I

xi+

Market Designs for Emissions Trading Schemes 15

and once more the fact that the ﬁnancial positions are in zero net supply to conclude

that

X

i∈I

E[LA∗,S∗,i (θ∗i, ξ∗i)] ≤

T−1

X

t=0 X

k∈K

E[S∗k

tDk

t]−E[C(ξ∗)]

−πEX

i∈I

Γi+X

i∈I

Πi(ξ∗i)+

=

T−1

X

t=0 X

k∈K

E[S∗k

tDk

t]−E[C(ξ∗)] −πE[(Γ + Π(ξ∗))+]

=

T−1

X

t=0 X

k∈K

E[S∗k

tDk

t]−E[G(ξ∗)].

Now, for each ξ∈ U we deﬁne θ(ξ) as

θi

t(ξ) = 0 for all i= 1, . . . , N ,t= 0, . . . , T −1,

θi

T(ξ) = Γi+ Πi(ξi)−Γ + Π(ξ)

|I|.

Repeating the above argument for (θ(ξ), ξ) yields

X

i∈I

E[LA∗,S∗,i (θi(ξ), ξi)] = X

t,k

E[S∗k

tDk

t]−E[G(ξ)].(4.5)

Applying the third property (each agent is satisﬁed with its own strategy) of the

(A∗, S∗) equilibrium to the optimal investment and production strategies (θ∗i, ξ∗i)

and (θi(ξ), ξi) yields

E[G(ξ∗)] ≤X

t,k

E[S∗k

tDk

t]−X

i∈I

E[LA∗,S∗,i (θ∗i, ξ∗i)]

≤X

t,k

E[S∗k

tDk

t]−X

i∈I

E[LA∗,S∗,i (θi(ξ), ξ)] = E[G(ξ)].

This holds for all ξ∈ U completing the proof.

The existence of an optimal ξfor the global optimization problem (4.3) follows

from standard functional analytic arguments.

Proposition 4.2. Under Assumption 1, there exists a solution ξ∈ U of the

global optimal control problem (4.3).

Our proof relies on two simple properties which we state and prove as lemmas for the

sake of clarity. First, we note that L1:= Qi∈IL1

T−1(RM), equipped with the norm

kXk=

T−1

X

t=0 X

(i,j,k)∈M

E[|Xi,j,k

t|]

is a Banach space with dual L∞:= Qi∈IL∞

T−1(RM), the duality form being given by

hX, ξi:=

T−1

X

t=0 X

(i,j,k)∈M

E[Xi,j,k

tξi,j,k

t], X ∈ L1, ξ ∈ L∞.

16 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Next, we consider the weak∗topology σ(L∞,L1) on L∞(see [10]), namely the weakest

topology for which all the linear forms

L∞3ξ7−→ hX, ξi ∈ R,(4.6)

for X∈ L1are continuous.

Lemma 4.3. The real valued function

L∞3ξ7−→ E[G(ξ)] = E[C(ξ)] + πE[(Γ + Π(ξ))+] (4.7)

is lower semi-continuous for the weak∗topology.

Proof. Obviously, the real valued function

L∞3ξ7−→ E[C(ξ)]

is continuous for the weak∗topology since it is of the form ξ7−→ hX, ξifor some

X∈ L1since X=C={Ci,j,k

t}is a ﬁxed element in L1by assumption. So we only

need to prove that the real valued function

L∞3ξ7−→ E[(Γ + Π(ξ))+] (4.8)

is lower semi-continuous. Using the fact that for any integrable random variable X

one has

E[X+] = sup

0≤Y≤1

E[XY ]

one sees that

E[(Γ + Π(ξ))+] = sup

0≤Y≤1

(E[ΓY] + E[YΠ(ξ)])

and hence that the function (4.8) is the supremum of a family of continuous function

since for ﬁxed Y,E[ΓY] is a constant and ξ7−→ E[YΠ(ξ)] is continuous for the weak∗

topology by the very deﬁnition of this topology. Since the supremum of any family

of continuous functions is lower semi-continuous, this concludes the proof that (4.7)

is lower semi-continuous.

Lemma 4.4. For the convex subset Uof L∞it holds that:

(i) Uis norm-closed in L1

(ii) Uis weakly∗closed in L∞.

Proof. (i)If (ξn)n∈Nis a sequence in Uconverging in L1to some random vari-

able ξ, then (ξi,j,k

t,n )n∈Nconverges in mean for each (i, j, k)∈Mand t= 0, . . . , T −

1, and extracting a subsequence if necessary, one concludes that (ξi,j,k

t,n )n∈Nand

(Pi∈IPj∈Ji,k ξi,j,k

t,n )n∈Nconverge almost surely to ξi,j,k

tand Pi∈IPj∈Ji,k ξi,j,k

tre-

spectively, showing that the constraints deﬁning Uare satisﬁed in the limit, implying

that ξ∈ U.

(ii) Since Uis a convex and a norm-closed subset of L1it follows from the Hahn-

Banach Theorem that Uis the intersection of halfspaces Hξ,c ={X∈ L1|hX, ξi ≤ c}

with ξ∈ L∞and c∈Rsuch that U ⊆ H. Since L∞⊆ L1it holds for each of these

halfspaces Hξ,c that ξ∈ L1. Thus we conclude that Hξ,c ∩L∞={X∈ L∞|hX, ξ i ≤ c}

is closed in (L∞, σ(L∞,L1)). Since by deﬁnition it holds that U ⊆ L∞it follows that

Uis given by the intersection of the sets Hξ,c ∩ L∞. Since any intersection of closed

sets is closed we conclude that Uis weakly∗closed in L∞.

Market Designs for Emissions Trading Schemes 17

Proof of Proposition 4.2 Since Uis bounded and weakly∗closed due to Lemma 4.4, it

follows from the Theorem of Banach-Alaoglu that Uis weakly∗compact. Lemma 4.3

concludes the proof since any lower semi-continuous function attains its minimum on

a compact set.

4.2. Relation with the Original Equilibrium Problem . As a consequence

of Assumption 2, for each production policy ξ∈ U , no point masses occur in the

FT−1-conditional distribution of Γ −Π(ξ). Hence, for all t= 0, . . . , T −1 we have:

Pt(Γ + Π(ξ)≥0) = Pt(Γ + Π(ξ)>0).(4.9)

In the next theorem, we show that the value of the conditional probability in (4.9)

characterizes the equilibrium allowance price at time t. To prepare for the proof of

this result, we ﬁrst prove a technical lemma.

Lemma 4.5. Let ξbe any solution of (4.6) whose existence is guaranteed by

Proposition 4.2, then it follows that:

(i) For ﬁxed t∈ {0, . . . , T −1}and any ξ∈ U with ξs=ξsfor all s= 0, . . . , t −1

Et(G(ξ)) ≥Et(G(ξ)) (4.10)

holds almost surely.

(ii) If Assumption 2 is satisﬁed, then for each k∈Kand i, i0∈I,j∈Ji,k, j 0∈Ji0,k

it holds that

{ξi,j,k

t∈[0, κi,j,k)}∩{ξi0,j 0,k

t∈(0, κi0,j0,k ]}

⊆ {Ci,j,k

t+ei,j,kAt≥Ci0,j 0,k

t+ei0,j0,k At}(4.11)

for all t= 0, . . . , T −1where At=πPt(Γ + Π(ξ)≥0).

Proof. (i) The assertion (4.10) is seen by the following argumentation: On the

contrary, one uses the Ft–measurable set

O:= {Et(G(ξ)) < Et(G(ξ)}of positive measure P(O)>0,

to outperform ξby ξ0as

ξ0

s= 1Oξs+ 1Ω\Oξsfor all s= 0, . . . , T −1. (4.12)

Note that since ξand ξ0coincide at times 0, . . . , t −1, this deﬁnition indeed yields an

adapted process ξ0∈ U . With (4.12), we have the decomposition

G(ξ0) = 1OG(ξ)+1Ω\OG(ξ),

which gives a contradiction to the optimality of ξ:

E(G(ξ0)) = E(Et(1OG(ξ)+1Ω\OG(ξ))

=E(1OEt(G(ξ)) + 1Ω\OEt(G(ξ)))

< E(1OEt(G(ξ)) + 1Ω\O Et(G(ξ))) = E(G(ξ)).

(ii) Introduce a deviation from the global optimal strategy ξ. At time t, consider a

shift in production of htunits of the good k∈K, where the agents i∈Iand i0∈I

increase/decrease their outputs from technologies j∈ J i,k,j0∈ J i0,k respectively.

18 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

This results in the new policy ξ+χwhere χ∈Πi∈IUithe deviation vanishes at all

times with the exception of tand

χi,j,k

t=ht, χi0,j0,k

t=−ht,

Consider

D(ξ, λ) = Et(G(ξ+λχ)) −Et(G(ξ))

λ, λ ∈(0,1].

Approaching 0 by λin the countable set (0,1]∩Qwe obtain by dominated convergence

limits for ei,j,k ≤ei0,j0,k0and ei,j,k ≥ei0,j 0,k0as

lim

λ→0D(ξ, λ) = −(Ci,j,k

t−Ci0,j0,k

t)−πPt(Γ + Π(ξ)>0)(ei,j,k −ei0,j0,k )ht

lim

λ→0D(ξ, λ) = −(Ci,j,k

t−Ci0,j0,k

t)−πPt(Γ + Π(ξ)≥0)(ei,j,k −ei0,j0,k )ht.

That is, with Assumption 2 we obtain the limit as

lim

λ→0D(ξ, λ) = −(Ci,j,k

t−Ci0,j0,k

t)−At(ei,j,k −ei0,j0,k )ht.(4.13)

Further, if the production shift is given by

ht= min{κi,j,k −ξi,j,k

t, ξi0,j 0,k

t},

then ξ+λχ ∈ U for all λ∈(0,1] ∩Qwhich, due to (i), yields

D(ξ, λ)(ω)≤0,for all ω∈˜

Ω with P(˜

Ω) = 1.

Passing through the limit λ↓0, we obtain with (4.10) and (4.13)

−(Ci,j,k

t−Ci0,j0,k

t)−Pt(Γ + Π(ξ)>0)(ei,j,k −ei0,j0,k )ht≤0

almost surely. Hence the inclusion

{ht>0} ⊆ {− (Ci,j,k

t−Ci0,j0,k

t)−At(ei,j,k −ei0,j0,k )≤0}

holds almost surely, which is equivalent to (4.11).

We can now turn to the main result of this section.

Theorem 4.6. Under the above assumptions, the following hold:

(i) If ξ∈ U is a solution of the global optimization problem (4.3), then the processes

(A, S)deﬁned by

At=πPt(Γ + Π(ξ)≥0), t = 0, . . . , T (4.14)

and

Sk

t= max

i∈I, j ∈Ji,k(Ci,j,k

t+ei,j,kAt)1{ξi,j,k

t>0}, t = 0, . . . , T −1k∈K, (4.15)

is a market equilibrium (in the sense of Deﬁnition 1), for which the associated pro-

duction strategy is ξ.

(ii) The equilibrium allowance price process is almost surely unique.

Market Designs for Emissions Trading Schemes 19

(iii) For each good k∈K, the price Skis the smallest equilibrium price for good kin

the sense that for any other equilibrium price process S∗k, we have Sk≤S∗kalmost

surely.

Proof. (i) We show that (A, S) so deﬁned forms an equilibrium by an explicit

construction of ﬁrm investment strategies θi∈ Vi(A) such that (θi, ξi) satisﬁes (3.5),

(3.6) and (3.7). Deﬁne

θi

t= 0 for all i= 1, . . . , N ,t= 1, . . . , T −1,

θi

T= Γi+ Πi(ξi)−Γ + Π(ξ)

|I|.

Since conditions (3.5) and (3.6) are obviously fulﬁlled, we focus on (3.7). We ﬁrst

show that E[LA,S,i(θi(ξi), ξ i)] ≥E[LA,S,i(θi, ξi)] for all (θi, ξi)∈ V i(A)× U i, where

θi(ξi) is constant equal to 0 until time T−1 and

θi

T(ξi) := Γi+ Πi(ξi)−Γ + Π(ξ)

|I|.

We have:

E[LA,S,i (θi, ξi)] = E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t−θi

TAT

−π(Γi+ Πi(ξi)−θi

T)+

since Adeﬁned by (4.14) is a bounded martingale. For all ξi∈ Ui, we show that we

can maximize the above quantity by computing the maximum pointwise in θiinside

the expectation. In view of (4.14), when ω∈ {Γ + Π(ξ)<0}we have ω∈AT(ω) = 0

and the maximum of

z7→ −zAT(ω)−π(Γi(ω)+Πi(ξi)(ω)−z)+(4.16)

is attained on each point z∈[Γi(ω) + Πi(ξi)(ω),∞) showing that θi(ξi)(ω) is a

maximizer. On the other hand, when ω∈ {Γ + Π(ξ)≥0}, we have AT(ω) = π, the

maximum of (4.16) is attained on each point z∈(−∞,Γi(ω) + Πi(ξi)(ω)], and once

again, θi(ξi) is a maximizer. Notice for later reference that in both cases, the value

of the maximum of (4.16) is E[−(Γi+ Πi(ξi))AT].

To ﬁnish the proof, we prove that E[LA,S,i(θi, ξ i)] ≥E[LA,S,i(θi(ξi), ξi)] for all ξi∈ U i.

According to the above computation, we have:

E[LA,S,i (θi(ξi), ξi)]

=E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t)ξi,j,k

t−(Γi+ Πi(ξi))AT

=E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−ei,j,kAT)ξi,j,k

t−ΓiAT

=E

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−ei,j,kAt)ξi,j,k

t−ΓiAT

.(4.17)

20 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

We now show that the following inclusions hold almost surely:

{Sk

t−Ci,j,k

t−ei,j,kAt>0} ⊆ {ξi,j,k

t=κi,j,k},(4.18)

{Sk

t−Ci,j,k

t−ei,j,kAt<0} ⊆ {ξi,j,k

t= 0}.(4.19)

Inclusion (4.19) is a direct consequence of Deﬁnition (4.15) of the price process S.

Using this same Deﬁnition (4.15) and Lemma 4.5 we see that:

{Sk

t> Ci,j,k

t+ei,j,kAt}

⊆[

i0∈I,j 0∈Ji0,k

{Ci0,j0,k

t+ei0,j0,k At> Ci,j,k

t+ei,j,kAt}∩{ξi0,j 0,k >0}

⊆[

i0∈I,j 0∈Ji0,k {ξi,j,k

t=κi,j,k}∪{ξi0,j 0,k

t= 0}∩ {ξi0,j0,k >0}

⊆ {ξi,j,k

t=κi,j,k}.

These inclusions allow us to show that E[LA,S,i(θi(ξi), ξi)] ≤E[LA,S ,i (θi, ξi)], thus

completing the proof of (i).

(ii) Proposition 3.1 gives the form of an equilibrium price. Due to Part (i) of

Proposition 3.1 and Proposition 4.1 to prove almost sure uniqueness of the allowance

price process, it is suﬃcient to prove that for any two solutions ˆ

ξ,˜

ξof the global

optimization problem (4.3) we have:

P{Γ + Π(ˆ

ξ)>0}∩{Γ + Π(˜

ξ)>0}[{Γ + Π(ˆ

ξ)<0}∩{Γ + Π(˜

ξ)<0}= 1

(4.20)

We know that these production strategies are solution of the global problem (4.3),

that we rewrite as a linear programming problem:

inf

ξ∈U, Z ∈L1(FT)

Z≥Γ+Π(ξ)−θ0, Z≥0

E[C(ξ) + πZ].(4.21)

Each solution (ξ?, Z?) of (4.21) satisﬁes

Z?= (Γ + Π(ξ?))+(4.22)

almost surely. Assume now that there are two optimal solutions (ˆ

ξ, ˆ

Z) and (˜

ξ, ˜

Z) of

the above linear programming problem. Due to the linearity of (4.21) it follows that

any convex linear combination

(λˆ

ξ+ (1 −λ)˜

ξ, λ ˆ

Z+ (1 −λ)˜

Z) (4.23)

is also a solution to (4.21) for all λ∈[0,1]. In view of (4.22), we conclude that for

each λ∈[0,1]

λ(Γ + Π(ˆ

ξ))++ (1 −λ)(Γ + Π(˜

ξ))+

=λ(Γ + Π(ˆ

ξ)) + (1 −λ)(Γ + Π(˜

ξ))+

holds almost surely. Since the above assertion is obviously violated on

{Γ + Π(ˆ

ξ)<0<Γ + Π(˜

ξ)}∪{Γ + Π(ˆ

ξ)>0>Γ + Π(˜

ξ)}

Market Designs for Emissions Trading Schemes 21

this union must have a probability 0, which together with Assumption 2 yields (4.20).

(iii) Assume on the contrary that there exists an equilibrium price process S∗with

S∗k

t(ω)<¯

Sk

t(ω) for all ω∈B(4.24)

for some t∈ {0,1, . . . , T −1},B∈ Ft,P(B)>0 and k∈K. Let ξ∗be the

corresponding equilibrium strategies. Since equilibrium allowance price ¯

Ais unique

it follows from (4.19) that

{S∗k

t−Ci,j,k

t−ei,j,k ¯

At<0} ⊆ {ξ∗i,j,k

t= 0}

up to sets of probability zero. Consequently we obtain

X

i∈IX

j∈Ji,k

ξ∗i,j,k

t=X

i∈IX

j∈Ji,k

ξ∗i,j,k

t1{S∗k

t≥Ci,j,k

t+ei,j,k ¯

At}(4.25)

≤X

i∈IX

j∈Ji,k

κi,j,k1{S∗k

t≥Ci,j,k

t+ei,j,k ¯

At}

almost surely. Moreover it follows from (4.19) and (4.18) that

X

i∈IX

j∈Ji,k

κi,j,k1{¯

St>Ci,j,k

t+ei,j,k ¯

At}(4.26)

=X

i∈IX

j∈Ji,k

¯

ξi,j,k

t−X

i∈IX

j∈Ji,k

¯

ξi,j,k

t1{¯

Sk

t=Ci,j,k

t+ei,j,k ¯

At}

<X

i∈IX

j∈Ji,k

¯

ξi,j,k

t

holds almost surely. In the last equality we used

X

i∈IX

j∈Ji,k

¯

ξi,j,k

t1{¯

Sk

t=Ci,j,k

t+ei,j,k ¯

At}(ω)>0 for all ω∈Ω

which follows from the deﬁnition of ¯

S. Further due to (4.24) it holds that

X

i∈IX

j∈Ji,k

κi,j,k1{S∗k

t≥Ci,j,k

t+ei,j,k ¯

At}(ω) (4.27)

≤X

i∈IX

j∈Ji,k

κi,j,k1{¯

Sk

t>Ci,j,k

t+ei,j,k ¯

At}(ω) for all ω∈B.

From (4.25), (4.26) and (4.27) we conclude that there exists a C⊆Bwith

X

i∈IX

j∈Ji,k

ξ∗i,j,k

t(ω)<X

i∈IX

j∈Ji,k

¯

ξi,j,k

t(ω) = Dt(ω)

for all ω∈C, which implies that S∗is no equilibrium product price.

Remark 3. On the basis of what is known for merit-order equilibria with discon-

tinuous cost functions, we do not expect uniqueness of the price process S∗k.

Remark 4. In the introduction, we referred to social costs as the costs of regu-

lation, i.e. the pollution reduction costs. We now give a formal deﬁnition of what we

mean by social costs. For each regulatory allocation ((Λi

t)T−1

t=0 )i∈I, and for any choice

22 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

of an equilibrium production schedule ξ∗∈ U, we deﬁne the social costs SC as the

random variable given by the diﬀerence between the production costs G(ξ∗)under this

production schedule and the production costs incurred in the same random scenarios

had we used the BAU equilibrium production schedule. In other words, the social costs

are given by the random variable:

SC =C(ξ∗)−C(ξ∗

BAU ).(4.28)

Notice also that as deﬁned, the social costs do not depend upon the trading strategies

of the individual ﬁrms in the emissions market.

Remark 5. The results of this section were derived under the assumption that

the emission coeﬃcients ei,j,k were constant. However, by mere inspections of the

proofs, the reader will easily convince herself that all the results remain true if these

emission coeﬃcients are instead adapted stochastic processes in L1

T−1(R).

5. Prices and Windfall Proﬁts in the Standard Scheme. The previous

sections were dedicated to the introduction and the mathematical analysis of what we

called the standard emission trading scheme. This cap-and-trade scheme was chosen

because it is representative of the EU-ETS implementation.

In this section, we focus on an economy where one single good is produced. We

choose the example of electricity because the power sector is worldwide one of the

most important sources of green house gases. We study the impact of regulation on

spot prices and producers’ proﬁts. In order to provide insight on the eﬀects of cap-

and-trade legislations, we performed numerical simulations of equilibrium prices and

optimal production schedules by solving the global optimization problem (4.3) using

data from the Texas electricity market. Speciﬁcs about the numerical implementation

are given in the Appendix at the end of the paper. We shall report numerical ﬁndings

from this case study throughout the remainder of the paper.

5.1. A Model for Electricity and Carbon Trading in Texas . To perform

numerical simulations, we chose to focus on the electricity sector in Texas. Texas has

an installed capacity of 81855 MW, mainly split into gas-ﬁred (51489 MW), coal-ﬁred

(23321 MW), and nuclear (9019 MW) power plants. These ﬁgures are based on the

installed capacity in 2007, including also additional nuclear and coal ﬁred power plants

that are planned to come online for the next 7 years. Including upcoming capacity

slightly changes the production stack and leads to more interesting results than using

the actual 2007 installed capacity. Nuclear technology has close to zero emissions, and

it is always running in base-load. The source of emission reduction thus essentially

comes from fuel switching between gas and coal.

So for all practical purposes, our model for Texas can be assume to involve one

good, electricity, produced from two diﬀerent technologies, gas and coal. Stochastic

costs of production are equal to Ci,j,k

t=HjPj

t, where j∈ {g, c},Hjis the heat rate

of technology jand Pj

tis the corresponding fuel price. Dtstands for the electricity

demand from which nuclear capacity has already been subtracted. We set the emission

rates to 0.42 ton/MWh for gas technology (CCGT-like) and 0.95 ton/MWh for coal

technology respectively. These average emission rates have been chosen to give a

faithful representation of Texas’ park of power plants.

The global optimization problem reads:

inf

ξ∈U E

T−1

X

t=0

(Cg

tξg

t+Cc

tξc

t) + π T−1

X

t=0

(egξg

t+ecξc

t)−Γ0!+

Market Designs for Emissions Trading Schemes 23

under the constraint: ξg

t+ξc

t=Dtfor every time t. In the particular case of two

technologies, we can proceed to the change of variable (ξg

t, ξc

t)7−→ (Et,Ct), where

Et=ecξc

t+egξg

tand Ct=Cc

tξc

t+Cg

tξg

t

are respectively the total emission and the cost of production for the period [t, t +

1). Using the constraint that the demand has to be met, we obtain an equivalent

formulation in terms of an emission abatement problem:

min

E≤E ≤E

E

T−1

X

t=0

(Dt(ecFt+Cc

t)−FtEt) + π T−1

X

t=0

Et−Γ0!+

(5.1)

where:

Et=egmin(Dt, κg) + ec(Dt−κg)+

Et=ecmin(Dt, κc) + eg(Dt−κc)+

are respectively the maximal and emissions possible at time t, and

Ft:= Cg

t−Cc

t

ec−eg(5.2)

is the fuel spread per ton of CO2(or abatement cost). The fuel spread Frepresents

the marginal switching cost necessary to decrease emissions by 1 unit. We observe

that the above formulation (5.1) only involves 2 exogenous stochastic processes: D

and F. Finally, we set the aggregated uncontrolled emissions Pi∈I∆iinﬁnitesimally

small to stay in the realm of the assumptions of Theorem 4.6, and solve the global op-

timization problem by stochastic dynamic programming on a 2-dimensional trinomial

tree. Details are given in the appendix at the end of the paper.

5.2. Electricity Prices Under the Standard Scheme. In this subsection, we

discuss the impact of the regulation on electricity prices. We already emphasized that

uniqueness of equilibrium electricity prices was not granted. However, we identiﬁed

the minimal price among all the possible equilibrium prices in Equation (4.15). In

what follows, we focus on this price.

!

!""!

##

Figure 5.1.Histograms of the consumer costs, social costs, windfall proﬁts and penalty pay-

ments under a standard trading scheme scenario.

24 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Equation (4.15) shows two sources of change in the spot price compared to busi-

ness as usual. First, the marginal technology may be diﬀerent: this induces a varia-

tion in marginal cost. This variation is likely to be positive but a negative variation

is possible. Suppose for example that in a BAU scenario, coal is started ﬁrst but

that demand is high enough so that gas is the marginal technology. Suppose that in

the presence of the trading scheme, allowance price is high enough to induce a fuel

switch, so that gas is started ﬁrst. Assume also that demand is high so that coal is

the marginal technology. In this case, the variation in marginal cost can be negative.

The second source of variation is the price of pollution ei,j,kA∗

tfor the marginal tech-

nology. The producers pass through the cost of expected penalties to end-consumers.

This second contribution is always positive and is such that the spot price under the

trading scheme is always greater than the spot price in BAU.

A possible interpretation of formula (4.15) is that the allowance price enters the

electricity price as the price of an additional commodity that is used for power genera-

tion besides fuels. Producing the last inﬁnitesimal unit of electricity at time tinduces

not only costs due to extra fuel consumption, but also increases the emissions by ei,j,k

and hence also the expected penalty at time Tby ei,j,kA∗

t. Consequently these costs

have to be covered by the end-consumers, for the marginal production of product k

to be proﬁtable. Since this amount is passed on to the endconsumer in each timestep

the consumer cost PT−1

t=0 (S∗

t−SBAU

t)Dtare much bigger than the penalty that is

actually paid. As we will see in the following the consumer costs exceed also by far

the social cost of the scheme.

Figure 5.1 quantiﬁes both the penalty payments and the consumer cost and com-

pares them to social costs and windfall proﬁts (as deﬁned in the next section) under

a standard trading scheme for the Texas electricity sector. The penalty and initial

allocation for this example are π= 100$ and θ0= 1.826 ×108allowances respectively.

This allocation corresponds to a reduction target of 10%, i.e. 1.827 ×108t Carbon, to

be reached with 95% probability.

The results depicted in Figure 5.1 illustrate the major critic articulated by some

of the opponents of the cap-and-trade systems. We observe that end consumer costs,

are approximately more than 10 times higher than social costs due to the trading

scheme. Hence the consumers’ burden exceeds by far the the overall reduction costs,

which gives rise for signiﬁcant extra proﬁts for the producers.

5.3. Windfall Proﬁts and Penalty Under the Standard Scheme. As ex-

plained above, the pricing mechanism of the standard emissions trading scheme in-

duces a signiﬁcant wealth transfer from consumers to producers.

Another way of understanding the extra proﬁts made by the producers is to con-

sider the windfall proﬁts deﬁned as follows. In the general framework of a standard

cap-and-trade system with multiple goods introduced earlier, if ξ∗is an optimal pro-

duction strategy associated with the equilibrium (A∗, S∗), we deﬁne the target price

ˆ

Sk

tof good kas:

ˆ

Sk

t:= max

i∈I,j ∈Ji,k Ci,j,k

t1{ξ∗i,j,k

t>0}.(5.3)

This price is the marginal cost under the optimal production schedule without taking

into account the cost of pollution. We then deﬁne the windfall proﬁts of ﬁrm ias:

T−1

X

t=0 X

(j,k)∈Mi

(S∗k

t−ˆ

Sk

t)ξ∗i,j,k

t,

Market Designs for Emissions Trading Schemes 25

and the overall windfall proﬁts as

W P =

T−1

X

t=0 X

k∈K

(S∗k

t−ˆ

Sk

t)Dk

t.(5.4)

These windfall proﬁts measure the proﬁts for the production of goods in excess over

what the proﬁts would have been, had the same dispatching schedule been used, and

the target prices (e.g. the marginal fuel costs) be charged to the end consumers

without the cost of pollution.

Remark 6. Another reasonable deﬁnition of the windfall proﬁts of ﬁrm iwould

be

T−1

X

t=0 X

(j,k)∈Mi

(S∗k

t−ˆ

Sk

t)ξ∗i,j,k

t−πΓi−Πi(ξ∗i)+(5.5)

meaning that the penalty payments due to the scheme are withdrawn from the extra

proﬁts. Since producers decide upon their production strategy and therewith the risk to

pay the penalty, we take the point of view that they should pay the penalty and not the

endconsumer. However as can be seen in Figure 5.1 the penalty payments vanish in

comparison to the windfall proﬁts as deﬁned in (5.4). Hence in practical applications,

both deﬁnitions should give similar results.

Figure 5.1 shows the distribution of windfall proﬁts as computed in the example

of the Texas electricity market chosen for illustration purposes. We observe that the

windfall proﬁts are in average almost 10 times higher than actual abatement costs.

Furthermore it also shows that the costs of expected future penalty passed to the

customers are much higher (4637 times) than the penalty actually paid. This is

consistent with the deterministic example presented in the introduction.

5.4. Incentives for Cleaner Technologies. Using (4.17) we see that the ex-

pected proﬁts and losses of ﬁrm i∈Iin an equilibrium (A∗, S∗) with associated

production schedules ξ∗are given by

E[LA∗,S∗,i (θ∗i, ξ∗i)] = E"(−∆i+

T−1

X

t=0

Λi

t)A∗

T#

+E

T−1

X

t=0 X

(j,k)∈Mi

(S∗k

t−Ci,j,k

t−ei,j,kA∗

t)ξ∗i,j,k

t

.(5.6)

As will be shown in Proposition 7.1, both the equilibrium price processes (A∗, S∗) and

the production strategies ξ∗are preserved under a change of the regulatory allocation

from ((Λi

t)T−1

t=0 )i∈Ito ((˜

Λi

0)T−1

t=0 )i∈Ias long as

X

i∈I

T−1

X

t=0

Λi

t=X

i∈I

T−1

X

t=0

˜

Λi

t

holds almost surely. However, such an adjustment of the allocation changes the ex-

pected proﬁts and losses of producer i∈Iby the amount:

Eh(˜

Λi

0−Λi

0)A∗

Ti.(5.7)

26 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Obviously this gives a relative (relative to Λi

0) expected money transfer of

E"X

i∈I

(˜

Λi

0−Λi

0)+A∗

T#(5.8)

from producers with ˜

Λi

0−Λi

0<0 to producers with ˜

Λi

0−Λi

0>0. If the initial allo-

cation is given depending on the type of production plant it is possible to utilize this

mechanism to increase or decrease the incomes of clean and dirty plants respectively,

i.e. the initial allocation can be used to adjust the incentives to build cleaner plants.

Depending on the speciﬁc market this will often be the main incentive to build clean

plants.

This mechanism is one of the main features of cap and trade schemes and will in

general fail if auctioning is used to abolish windfall proﬁts. First notice that even a

100% auction can not always reduce windfall proﬁts to zero. This becomes obvious

in a market with a lot of nuclear power plants where coal is marginal the whole time.

In such a market the producers of nuclear power make huge windfall proﬁts but since

their emissions are zero they do not need any allowances. Hence the auction can only

cover the windfall proﬁts due to the coal ﬁred plants. Therefore using auctioning to

cut windfall proﬁts a huge amount if not all allowances of the initial allocation should

be auctioned. However in such a case, the regulator looses the instrument to control

above incentives. Therefore in the next section we propose alternative cap and trade

schemes that not only reduce windfall proﬁts to zero in average, but also provide

a considerable amount of allowances that can be used to adjust incentives to build

cleaner plants.

6. Alternative Designs of Emission Trading Schemes. The main objec-

tives of emission trading schemes are both to force the market to reach a certain reduc-

tion target, and at the same time, to give incentives to develop and build cleaner pro-

duction facilities. In view of the shortcomings of the standard cap-and-trade scheme

demonstrated in the last section, we propose alternative designs which fulﬁl both ob-

jectives at low social costs, low windfall proﬁts and hence low costs transfered to the

consumer.

This is possible because the mathematical theory developed in the previous sec-

tions allows us to study emissions reduction policies that are diﬀerent from the stan-

dard EU-ETS scheme.

In the ﬁrst Subsection 6.1 below, we introduce a general (and fairly complex)

cap-and-trade scheme including taxes and subsidies. We argue that the theoretical

results derived earlier in the paper for standard schemes, still hold in this more gen-

eral situation. The remaining of the section is devoted to the identiﬁcation and the

calibration of two of the simplest particular cases of interest. A relative scheme is

introduced in Subsection 6.2 and a tax scheme is introduced in Subsection 6.3. The

ﬁnal Subsection 6.4 provides comparative statics highlighting the diﬀerences between

these schemes on the case study of the Texas electricity market.

6.1. General Market Designs for Emission Trading Schemes. We de-

scribe the new regulator policies by ﬁrst generalizing the allocation procedure. Be-

yond the static allocation Λi

tfor ﬁrm iat time t, the regulator is now allowed to

distribute credits dynamically and proportionally to production. To be more speciﬁc,

at each time 0 ≤t < T , ﬁrm iis provided with an allocation

Λi

t=Xi

t+X

(j,k)∈M(i)

Yi,j,k

tξi,j,k

t,(6.1)

Market Designs for Emissions Trading Schemes 27

where Xiand Yi,j,k are adapted processes in L1

T−1(R). For the sake of generality we

let Yi,j,k

tdepend upon j. However in this case the opportunity to relate the number

of allowances to real emissions is lost.

In addition, the regulator can also tax or subsidize the various ﬁrms by means of

ﬁnancial incentives or disincentives similar to the credit endowments described above.

In this case, the ﬁrms’ proﬁts are lowered at time tby an amount

T Si=Vi

t+X

(j,k)∈M(i)

Zi,j,k

tξi,j,k

t,(6.2)

where Viand Zi,j,k are as before, adapted processes in L1

T−1(R). Remark that Vi

and Zi,j,k stand for a tax when positive and a subsidy when negative. Examples of

positive Zi,j,k include fuel and CO2taxes. The combination of Viand Zi,j,k allows

for the introduction of alternative regulation such as a system of reward/penalty with

respect to a given production (or equivalently emission) target ξi,j,k. By charging the

quantity

T−1

X

t=0 X

(j,k)∈Mi

Zi,j,k

t(ξi,j,k

t−ξi,j,k

t)

corresponding to Vi

t=−P(j,k)∈MiZi,j,k

tξi,j,k

t, the regulator can provide incentives

for ﬁrm ito stay close to a given production or emission strategy.

Under such a generalized cap-and-trade scheme, the terminal wealth (or proﬁts

and losses) of ﬁrm i∈Ireads:

LA,S,i(θi, ξ i) := −

T−1

X

t=0

Vi

t+

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−Zi,j,k

t)ξi,j,k

t

+

T−1

X

t=0

θi

t(At+1 −At)−θi

TAT

−π

∆i+ Πi(ξi)−

T−1

X

t=0

Xi

t+X

(j,k)∈M(i)

Yi,j,k

tξi,j,k

t

−θi

T

+

.(6.3)

Despite the obviously greater generality of the present framework, the proofs of the

results of Theorem 4.6 are suﬃcient to cover the analysis of this broader class of

trading schemes:

Proposition 6.1. If we set

ˆ

Γi:= ∆i−

T−1

X

t=0

Xi

t,ˆei,j,k

t:= ei,j,k −Yi,j,k

t,and ˆ

Ci,j,k

t:= Ci,j,k

t+Zi,j,k

t(6.4)

for a set of adjusted parameters, then the results of Theorem 4.6 hold true in the case

of the the generalized cap-and-trade scheme of this subsection provided we replace the

parameters of Theorem 4.6 by the adjusted parameters so-deﬁned.

Proof. The proof of this proposition follows a straightforward adaptation of the

arguments used in the previous sections and Remark 5 about stochastic emission

factors.

28 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

The present formulation gives a general framework for the analysis of a broader

class of cap-and-trade schemes. We mostly focus on two important particular cases:

1) the case where Zi,j,k

t≥0 varies with iand jwhich represents a fuel or emission

tax scheme, and 2) the case where Zi,j,k

t≤0 only depends on k, which corresponds

to a subsidy for the production of good k.

For an equilibrium (A∗, S∗) of the generalized scheme with associated strategies

(θ∗, ξ∗) it is straightforward to extend the deﬁnition of windfall proﬁts of ﬁrm ias:

GW P i=

T−1

X

t=0 X

(j,k)∈Mi

(S∗k

t−ˆ

Sk

t)ξ∗i,j,k

t−

T−1

X

t=0 Vi

t+X

(j,k)∈Mi

Zi,j,k

tξ∗i,j,k

t,

the overall windfall proﬁts being then deﬁned as

GW P =

T−1

X

t=0 X

k∈K

(S∗k

t−ˆ

Sk

t)Dk

t−

T−1

X

t=0 Vi

t+Dk

tX

i∈I,j ∈Ji,k

Zi,j,k

t.(6.5)

The above discussion suggests that windfall proﬁts could be reduced with a relative

allocation rule constant over time. This motivates the following analysis.

6.2. Cap-and-Trade Schemes with Relative Allowance Allocation. A

positive relative allocation for some product Yi,j,k

t=yk<0 for some k∈Kall i∈I

and those j∈Ji,k can be seen as a subsidy for good kthat is given in the form of

allowances rather than in cash. When producing one unit of good k, the marginal

penalty increases only by (ei,j,k −yk)A∗

trather than by ei,j,kA∗

tas in a standard

scheme. Thus the net marginal overall production costs of the ﬁrms are lower when

compared to the standard scheme. This should result in a decrease of the price of good

k. In the present subsection, we study the simplest generalized cap-and-trade scheme

taking advantage of this mechanism. It applies this mechanism only for production

means Ji,k

marg ⊆Ji,k which can be marginal (e.g. in the case of the electricity markets,

this excludes nuclear plants) and is obtained by setting:

Yi,j,k

t=yk1{j∈Ji,k

marg}∈Rfor all t= 0, . . . , T −1

Xi

0=xi∈R, Xi

t= 0 for all t= 1, . . . , T −1

Vi

t= 0, Zi,j,k

t= 0 for all t= 0, . . . , T −1

for all (i, j, k)∈M.

In what follows, not only do we discuss this relative cap-and-trade scheme, but

we also gain new insight into the standard cap-and-trade scheme by treating it as a

relative cap-and-trade scheme with yk= 0 for all k∈K.

But for any comparison of diﬀerent cap-and-trade schemes to be meaningful, we

need to calibrate their respective parameters to common characteristics. We proceed

to the discussion of such a calibration.

6.2.1. Calibration of the Parameters. The relative scheme has three regula-

tory parameters. Using the notation of this section, they are: 1) the penalty π, 2) the

relative allocation coeﬃcients (yk)k∈K, 3) the total initial allocations x=Pi∈Ixi

given to the ﬁrms i∈I. In this subsection we show, using again the example of

the Texas electricity market, how one should choose these parameters in order to

guarantee an emissions reduction target with given probability while keeping the ex-

pected windfall proﬁts near zero and controlling the social costs to keep them as low

Market Designs for Emissions Trading Schemes 29

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ye

1.7⋅1081.8⋅1081.9⋅1082.0⋅1082.1⋅108

x+E(yeD)

-10

-5

0

5

10

15

20

25

$/MWh

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ye

1.7⋅1081.8⋅1081.9⋅1082.0⋅1082.1⋅108

x+E(yeD)

-10

-5

0

5

10

15

20

25

$/MWh

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ye

1.7⋅1081.8⋅1081.9⋅1082.0⋅1082.1⋅108

x+E(yeD)

1.7⋅108

1.8⋅108

1.8⋅108

1.8⋅108

1.9⋅108

2.0⋅108

2.0⋅108

t CO2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

ye

1.7⋅1081.8⋅1081.9⋅1082.0⋅1082.1⋅108

x+E(yeD)

1.7⋅108

1.8⋅108

1.8⋅108

1.8⋅108

1.9⋅108

2.0⋅108

2.0⋅108

t CO2

Figure 6.1.Windfall proﬁts (left) and 95% percentile of total emissions (right) as functions of

the relative allocation parameter and the expected allocation. Here D=PT−1

t=0 De

tdenotes the total

electricity demand over one compliance period.

as possible. In the particular simulation used to illustrate the strategy, we choose an

emissions reduction target of 1.827 ×108to be reached with probability 95%.

To gain a ﬁrst insight into the numerics, we ﬁx the penalty πat 100$. The left

pane of Figure 6.1 gives the expected windfall proﬁts while the right pane gives the

95% percentile of the total emissions for diﬀerent values of the relative allocation coef-

ﬁcient (ye) and the expected total allocation. It appears that the expected allocation

controls the amount by which carbon emissions are reduced, while the relative alloca-

tion coeﬃcient yecontrols the windfall proﬁts. Designing a cap-and-trade scheme with

zero windfall proﬁts and pre-decided emissions target levels can be done by choosing

the parameters of our relative scheme at the intersection of the zero windfall proﬁt

level set with the 1.827 ×108emission percentile level set. This procedure is depicted

in Figure 6.2. We ﬁnd ye= 0.54 and Pi∈Ixi= 5.4×107.

30 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Figure 6.2.The left pane shows the level sets of the two plots of Figure 6.1. The blue and the

red lines indicate level sets of the windfall proﬁts and the 95% quantile of emissions respectively.

The right pane gives the plots of the overall production costs for electricity for one year as function

of the penalty level for both the absolute and relative schemes. The free regulatory parameters are

chosen to guarantee the desired emissions percentile, and in the case of the relative scheme, such

that the windfall proﬁts are zero.

Since for the standard cap-and-trade scheme the parameter ye= 0 is ﬁxed, we

have one less regulatory parameter. Thus controlling the emissions level by the initial

allocation, we are not able to control the windfall proﬁts. Hence the desired parameter

values are obtained at the intersection of the 1.827 ×108emission percentile level set

with ye= 0. Giving the initial allocation Pi∈Ixi= 1.826 ×108.

Repeating the above procedure for diﬀerent penalties levels gives regulatory set-

tings with diﬀerent production costs for the relative and the standard scheme in Figure

6.2. Obviously for both schemes the social costs are reduced by increasing penalty.

As shown in the right pane of Figure 6.2, this decrease in social costs is signiﬁcant

until the penalty reaches the level π=100$. after that, the social costs stay nearly

the same, becoming independent of πfor larger values of π. Hence, we conclude that

in this setting a penalty of 100$ is a reasonable choice for both the relative and the

standard scheme.

6.3. Emission Taxes. A static tax scheme is a regulation that penalizes the

emission of each ton of carbon by a ﬁxed amount, say z > 0. Formally, it can be

viewed as a generalized scheme for which

Zi,j,k

t=ei,j,kz , V i

t= 0

Yk

t= 0 , Xi

t= 0

for all (i, j, k)∈Mand t= 0, . . . , T −1. Using the results of Proposition 6.1,

we see that in such a tax scheme the prices of goods follow a merit order pricing

rule with eﬀective production costs given by Ci,j,k

t+ei,j,kzfor all (i, j, k )∈Mand

t= 0, . . . , T −1. The earnings under a tax scheme are based on the spread of these

eﬀective production costs. Since this spread does not depend solely on the original

fuel spread, it is in general not clear what windfall proﬁts will be. It is not even clear

if they are negative or positive. To gain some insight on this issue consider a tax of

Market Designs for Emissions Trading Schemes 31

z= 60$ (which is realistic for a 10% reduction target as will be seen below), and

assume that at some point in time, the marginal production costs of coal and gas are

the same while all plants have to run to satisfy the demand. In this case the spread

in eﬀective production cost is (ec−eg)z= 31.8$ and will be earned for each MWh

that is produced with gas. However, in the case of BAU, the earnings are zero. Hence

the windfall proﬁts are 31.8$ per MWh of electricity produced with gas.

In tax schemes the only regulatory control parameter zshould be adjusted in order

to guarantee a speciﬁed reduction target. Thus the windfall proﬁts are automatically

given by the reduction target and can not be adjusted.

6.4. Comparison of the Various Abatement Schemes. We now compare

the characteristics of the standard and the relative cap-and-trade schemes with the

regulatory parameters chosen in the previous subsection.

We ﬁrst consider the windfall proﬁts and the consumer costs. The results are

given in Figure 6.3.

Figure 6.3.Histograms (computed from 500000 simulation scenarios) of the yearly distribution

of windfall proﬁts (left) and consumer costs (right) for the Standard Scheme, a Relative Scheme and

a Tax Scheme.

As expected the relative scheme gives much lower consumer costs than the stan-

dard scheme. This is related to the fact that the windfall proﬁts have a narrow

distribution around zero in the case of the relative scheme, while the windfall proﬁts

of the standard scheme are 10 times higher than the social costs. When compared

to the standard scheme, the only drawback of the relative scheme seems to be the

slightly higher level of social costs which can be observed on the right pane of Figure

6.4. However since this cost increase corresponds to approximately 0.4$ per MWh

it is small in comparison to production costs and thus can be neglected in practice.

Moreover those higher production costs are not just wasted money, they are paid for

higher emission reduction in many scenarios as can be seen on the left pane of Figure

6.4. In particular the relative scheme takes advantage of cheap fuel switches when the

standard scheme cannot reduce emissions anymore. Moreover relative scheme is less

sensitive to weather, since in warm winters less allowances are allocated pushing the

price up. This in turn is responsible for higher emission abatements and consequently

32 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

!

Figure 6.4.Yearly emissions from electricity production (left) for the Standard Scheme, the

Relative Scheme, a Tax Scheme and BAU, and yearly abatement costs (right).

higher abatement costs.

In this example approximately 30% of the allowances are given as initial allocation,

by allocating these to clean plants, further incentives can be set to build cleaner plants.

This seems to be an important advantage of the relative scheme over other mechanisms

such as auctioning and tax.

Next, we study the eﬀect of an emission tax on the Texas electricity market.

Figure 6.4 shows that a pure tax scheme that fulﬁlls the above reduction target of

1.827×108tCO2with 95% probability, is on average, more than twice (2.4×) as expen-

sive as the standard cap and-trade-scheme. In other words, it has a poor emissions

reduction performance. These extra costs are paid for extra emission reductions.

However in contrast with the results in the case of the relative scheme, the aver-

age cost increase per reduced ton of carbon is considerable when we compare it to

the case of the standard scheme. The reason is that a tax is not ﬂexible enough to

control emissions when abatement costs are stochastic. This results in an emission

uncertainty that exceeds even the BAU uncertainty with several orders of magnitude.

Notice moreover that it carries a signiﬁcant risk to reduce nearly no emissions. In

such a scenario the tax corrections for upcoming years will be extremely expensive.

Needless to say a tax scheme induces a huge money transfer from consumers to the

regulator, which as can be seen in Figure 6.4 is even bigger than the costs transferred

to the consumer in a standard cap and trade scheme.

7. More Financial Incentives. One of the main arguments in favor of the

relative schemes studied in the previous section is the fact that they reduce windfall

proﬁts. However, this reduction comes with slightly higher reduction costs than in

the case of the absolute scheme. While this cost increase is negligible in practice, it is

of great theoretical interest to understand how and why one can design schemes that

give exactly zero windfall proﬁts at exactly the same reduction costs as the standard

cap-and-trade scheme. In order to do so, we need to identify the generalized schemes

which are in a one-to-one correspondence with the production policies of the standard

Market Designs for Emissions Trading Schemes 33

scheme. The latter are given by a subclass of generalized schemes for which Zi,j,k

t

and Yi,j,k

tdepend only on k. The terminal wealth of ﬁrm i∈Iunder such a scheme

reads:

LA,S,i(θi, ξ i):=−

T−1

X

t=0

Vi

t+

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−Ci,j,k

t−Zk

t)ξi,j,k

t

+

T−1

X

t=0

θi

t(At+1 −At)−θi

TAT

−π∆i+ Πi(ξi)−

T−1

X

t=0 Xi

t+X

(j,k)∈M(i)

Yk

tξi,j,k

t−θi

T+

.(7.1)

The results of this section will demonstrate the versatility and the ﬂexibility of the

generalized framework introduced in this paper. However, because of the level of

complexity of their implementations, and despite the high degree of control they

provide the regulator with, it is unlikely that the schemes identiﬁed here will be used

by policy makers or regulators.

7.1. Equilibria Equivalence. Our ﬁrst result exhibits a one-to-one correspon-

dence between the equilibria of standard schemes and generalized schemes leading to

proﬁts and losses for ﬁrm iof the type (7.1).

Proposition 7.1. If (A∗, S ∗)is an equilibrium with production strategies ξ∗for

a standard cap-and-trade scheme with adjusted uncontrolled emissions given by

Γi= ∆i−

T−1

X

s=0

(Xi

s+ Ξi

s)for all i∈I , (7.2)

where Ξis a stochastic process in L1

T−1(R)such that

T−1

X

t=0 X

i∈I

Ξi

t=

T−1

X

t=0 X

k∈K

Yk

tDk

t,(7.3)

then the prices (A∗, S†)where

S†k

t=S∗k

t+Zk

t−Yk

tA∗

tfor all k∈K, t = 0, . . . , T −1 (7.4)

deﬁne an equilibrium of the generalized cap-and-trade scheme with the same production

strategies ξ∗. The converse statement also holds.

In particular if

T−1

X

t=0 X

i∈I

Xi

t+X

k∈K

Yk

tDk

t=X

i∈I

Λi

0

where Λi

0is the allocation in standard scheme, there is a one-to-one correspondence

between the generalized scheme and the standard scheme with initial allocation Λi

0.

In particular, the lowest equilibrium product price is given by (7.4) where S∗is

given by Theorem 4.6 for the standard scheme with adjusted uncontrolled emissions

(ˆ

Γi)i∈I. To prove Proposition 7.1, we shall need the following lemma.

34 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Lemma 7.2. Let Abe an integrable martingale, θ, θ0∈ V i(A), ξ i∈ Ui, and S,S0

be two integrable price processes, such that:

θ0i

t=θi

tfor all t≤T−1 (7.5)

θ0i

T=θi

T−

T−1

X

t=0 Ξi

t−X

(j,k)∈M(i)

Yk

tξi,j,k

t(7.6)

S0k

t=Sk

t+Zk

t−Yk

tAtfor all k∈K, t = 0, . . . , T −1

where Zk≤0is a subsidy, then we have:

E[LA,S,i(θi, ξ i)] = E[HA,S0,i(θ0i, ξi)] + E"T−1

X

t=0

(Vi

t−Ξi

tAt)#.(7.7)

Proof. The martingale property of Ayields:

E

θ0i

TAT+π

∆i+ Πi(ξi)−

T−1

X

t=0

(Xi

t+X

(j,k)∈M(i)

Yk

tξi,j,k

t)−θ0i

T

+

=E

θi

TAT+

T−1

X

t=0

−Ξi

t+X

(j,k)∈M(i)

ξi,j,k

tYk

t

E(AT|Ft)

+π ∆i+ Πi(ξi)−

T−1

X

t=0

(Xi

t+ Ξi

t)!+

=E

θi

TAT+

T−1

X

t=0

X

(j,k)∈M(i)

ξi,j,k

tYk

t

At+πΓi+ Πi(ξi)−θi

T+

−E(

T−1

X

t=0

Ξi

tAt))

which proves the desired result.

We can now turn to the proof of Proposition 7.1.

Proof. Let (A∗, S ∗) be equilibrium price processes of a standard scheme with

strategies (ξ∗, θ∗). Let θ0∗be the adjusted optimal strategy as in Lemma 7.2. The

assertion follows by checking that conditions (i) to (iii) of Deﬁnition 1 are fulﬁlled

by the pair of price processes (A∗, S†) and strategies (ξ∗, θ0∗). Since θ∗satisﬁes

the market clearing condition (3.5), so does θ0∗. This proves (i) while condition (ii)

follows directly from (7.3). Moreover, given (θ0, ξi)∈ Vi(A∗)× U i, we deﬁne strategies

θi∈ Vi(A∗) such that (7.5) and (7.6) hold. According to Proposition 3.1 A∗is an

integrable martingale and the result of Lemma 7.2 yields

E[HA∗,S†,i (θ0i, ξi)] = E[LA∗,S ∗,i(θi, ξ i)] −E"T−1

X

t=0

(Vi

t−Ξi

tA∗

t)#

≤E[LA∗,S∗,i (θ∗i, ξ∗i)] −E"T−1

X

t=0

(Vi

t−Ξi

tA∗

t)#(7.8)

=E[HA∗,S†,i (θ0∗i, ξ∗i)]

Market Designs for Emissions Trading Schemes 35

where we used the optimality of the equilibrium strategies (ξ∗, θ∗) of the standard

scheme in (7.8). This holds for all (θ0, ξi)∈ V i(A∗)× U iwhich proves condition (iii).

The converse can be proved in exactly the same way.

Note that not only do allocation prices coincide, but also equilibrium production

strategies: ξ†=ξ∗. Thus the switching costs of the generalized cap-and-trade schemes

are the same as for the standard cap-and-trade schemes with adjusted uncontrolled

emissions (Γi)i∈I.

7.2. Design of Financial Incentives. In this paragraph we discuss the design

of ﬁnancial incentives that adjust the ﬁnancial positions of each ﬁrm i∈Iby

−

T−1

X

t=0

(Vi

t+X

(j,k)∈Mi

ξi,j,k

tZk

t) (7.9)

depending on his production strategy ξi. Obviously, the results of Proposition 7.1

hold in this case. Allowance prices, production strategies, and penalty are identical

in equilibrium, to those of the standard scheme. Electricity price is increased by the

quantity Zk

tat t= 0, . . . , T −1. Hence, as depicted in Figure 7.1 the scheme induces

a money transfer

T−1

X

t=0

(Vi

t+X

(j,k)∈M(i)

ξi,j,k

tZk

t)

from producers to the regulator. In the meantime, the quantity

T−1

X

t=0 X

(j,k)∈M(i)

ξi,j,k

tZk

t

is entirely passed on to the end consumer, so that (Zk

t)T−1

t=0 ≥0 results in a money

transfer from consumers to the regulator.

Remark 7. This explains why it is not trivial to reduce windfall proﬁts (and this

precise discussion of incentives is needed). At ﬁrst sight one could e.g. think that

windfall proﬁts could be reduced by keeping a book on any traded MWh , marginal

technology and carbon price with the objective to charge the producers the windfall

proﬁts and reimburse consumers later. The amount that would be charged from each

producer i∈Iwould then depend on his strategy ξiand the markets product price

process Swould be given by

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−ˆ

Sk

t)ξi,j,k

t.(7.10)

Obviously under such a regulation the demand will never be satisﬁed, and hence there

exists no equilibrium. On the other side if we change the amount to be charged to

T−1

X

t=0 X

(j,k)∈Mi

(Sk

t−ˆ

Sk

t)ξi,j,k

t.(7.11)

it follows from above discussion that the entire amount will be passed on to the end-

consumer. Thus the windfall proﬁts will not be reduced.

36 R. CARMONA, M. FEHR, J. HINZ AND A. PORCHET

Figure 7.1.V > 0gives a money transfer from producers to the regulator, while Z > 0gives a

money transfer from consumer to the regulator. By choosing Vand Zin an appropriate way it is

possible to avoid a money transfer to/from the regulator.

Proposition 7.1 may look esoteric at ﬁrst. However, it happens to be very versatile

a tool when it comes to designing new schemes with required properties. As corollaries

to this proposition, two appropriate adjustments with zero windfall proﬁts are given

in Sections 7.3 and 7.4.

7.3. Zero Windfall Proﬁt Scheme with Tax and Subsidy. In this section

we consider the generalized allocation scheme given by

Xi

t= Λi

01{t=0}

Yk

t= 0

for all i∈I,k∈Kand t= 0, . . . , T −1. We show here how the tax/subsidy system

comprised into the generalized scheme can theoretically lead to zero windfall proﬁts

at equilibrium. This result is a direct corollary of Proposition 7.1.

Corollary 7.3. Consider a generalized cap-and-trade scheme such that for all

i∈Iand k∈K

Vi

t=X

(j,k)∈M(i)

ξi,j,k

t(St−ˆ

Sk

t)for all t= 0, . . . , T −1

Zk

t= 0 for all k∈K, t = 0, . . . , T −1.(7.12)

Then each equilibrium (A∗, S∗)of the standard cap and trade scheme is also an equi-

librium of this generalized scheme. In particular the equilibrium with lowest product

prices is given by (A, S)from Theorem 4.6. For this scheme the windfall proﬁts for

the aggregated producing sector are zero.

Remark 8. The processes S,ˆ

Sand ξthat occur in (7.12) are given by Theo-

rem 4.6 and hence independent of the actually realized equilibrium (A†, S†)and their

strategies (θ†, ξ†). Hence their computation involves necessarily solving the global opti-

mal control problem. However generalizing Proposition 7.1 easier ways could be found

to compute the amount to charge.

To adjust end-consumer costs to a reasonable level, the amount Pi∈IPT−1

t=0 Vi

t

has to be redistributed from the regulator to the end consumers in an appropriate

way. As can be seen in Figure 7.1 this can be omitted for the ﬁnancial incentives

fulﬁlling

T−1

X

t=0 X

i∈I

Vi

t=−

T−1

X

t=0 X

k∈K

Dk

tZk

t.(7.13)

Market Designs for Emissions Trading Schemes 37

Here there is no money transfer from producers to regulators. Such a setting is

discussed in the following Corrolary.

Corollary 7.4. Consider a generalized cap-and-trade scheme such that for all

i∈Iand k∈K

Vi

t=X

(j,k)∈Mi

ξi,j,k

t(Sk

t−ˆ

Sk

t)for all t= 0, . . . , T −1

Zk

t=−(Sk

t−ˆ

Sk

t)for all t= 0, . . . , T −1

where Sdenotes the equilibrium electricity price in the standard scheme, recall Theo-

rem 4.6, and ˆ

Sdenotes the pure merit order price as deﬁned in (5.3). In this setting

each equilibrium (A∗, S∗)of the standard cap-and-trade scheme corresponds to an

equilibrium of the generalized cap-and-trade scheme given by (A∗, S∗−(S−ˆ

S)). In

particular the equilibrium with lowest product price is given by (A, ˆ

S). In this equilib-

rium, each producer realizes null windfall proﬁts.

If a ﬁrm ifollows strategy ξiits net money transfer to the regulator is given by

X

(j,k)∈Mi

(ξi,j,k

t−ξi,j,k