Estelle Malavolti
email: estelle.malavolti@enac.fr

Marion Podesta
email: marion.podesta@enac.fr

Inclusion of the aviation sector into the Emission Trading Scheme: an economic analysis

Keywords: JEL classification: L51, L93, Q53 Air transport, Emissions Trading Scheme, Regulation

The air transport sector is going to enter the European Trading Scheme in 2012. The regulation of CO2 emissions is costly for airlines and modifies the organization of their market. Our paper proposes an economic analysis in which the regulation but also CO2 emissions of airlines are modeled. We show that, in a perfect competition setting, the difference between passengers carried without regulation and when the regulation is considered is negative for the best-performing planes. However, for the less efficient aircraft, the implementation of the regulation entails a reduction of airlines activity, and therefore a low level of carbon emissions.

Introduction

Since the Kyoto Protocol in 1997, several governments have made significant progress toward improving air quality. In this sense, the air transport sector in Europe will be fully included in the European Union Emissions Trading Scheme (EU-ETS), in 2012. Despite the relatively low level of greenhouse gas emissions (only 3% of the total European emissions), this sector has known a rapid growth until recently: from 1990 to 2005, the EU aviation emissions increased by 87% and it is expected to double from now to 2020 (See Commission Staff working document 2006 and EU directive 2003/87/EC). On top of this, the air transport sector is also responsible for other releases like nitrogen oxides, water vapor or noise, which effects are not easy to account. Nevertheless, a regulation of these external effects is to be expected. It is thus important to evaluate which impacts a regulation may have on the market and its organization.

The environmental literature defines pollution as an externality which is not taken into account by the market. Several instruments have been put forward (taxation, subsidies, norms or allowances trading) resulting in State intervention, and the literature deeply analyzes the efficiency of these different tools (see for instance [START_REF] Myles | Public Economics[END_REF]or Salanié, 2003). Our work builds on this general literature focusing on the specificities of the air transport sector to model the impact of an environmental regulation. The environment problems have raised several questions. For instance, Portney (2005) makes a review of the existing regulations and tries to evaluate what will be the regulations of the future. Among the economic tools used to regulate, the taxation is the means which has the most received attention. [START_REF] Barthold | Issues in the Design of environmental Excise Taxes[END_REF] presents the different taxes used for environmental regulation and their efficiency to regulate emissions.

The European Commission would include aviation sector in the EU-ETS from 2012 in relation to the airlines' carbon emissions. The aim of carbon markets is to provide incentives to reduce CO2 emissions. The environmental problems have raised several questions. [START_REF] Scheelhaase | Emissions trading for international aviation-an estimation of the economic impact on selected European airlines[END_REF] focus on financial impacts to include aviation in the ETS on airlines. They show that the financial impact on airlines subject to the ETS is relatively moderate. Since the regulation concerns all flights departing from and arriving at EU airports, [START_REF] Scheelhaase | The inclusion of Aviation into the EU emission trading Scheme -Impact on competition between European and non-European network airlines[END_REF], under a model-based empirical estimations, analyze how the EU directive affects competition between european and non-european airlines and if competition distortions are likely to appear. They compute fuel consumption, CO2 emissions and number of allowances to know the difference between allowances allocated free of charge and the total amount of them needed by the airlines (with a price assumed to e20 per ton of CO2 and two scenarios considered for emissions growth). Moreover, regarding consumers' surplus, they show that the impact on prices is relatively moderate.

On an macro-economic view, [START_REF] Anger | Including aviation in the European emissions trading scheme: Impacts on the industry, CO2 emissions and macroeconomic activity in the EU[END_REF] shows that there is not expected to have a negative impact of the EU-ETS on economic growth in the UE or to reduce the UE's competitiveness relative to the rest of the world. However concerning the impact of EU-ETS on the carbon emissions, the results are ambiguous. [START_REF] Mayor | The impact of the UK aviation tax on carbon dioxide emissions and visitor numbers[END_REF] find a negative impact on the carbon emissions whereas [START_REF] Anger | Including aviation in the European emissions trading scheme: Impacts on the industry, CO2 emissions and macroeconomic activity in the EU[END_REF] concludes in a positive way. It seem difficult to have an economic model which combine the regulation of EU-ETS, the financial impact on airlines and the level of carbon emissions.

In this sense, we built an economic model which offers a precise framework of nonetheless the regulation but also of emissions of airlines. The regulation system includes two different elements: the first element concerns free allowances that will be given to airlines according to their current activity. The second element is the payment of rights to pollute on the CO2 market. There is a strategic stake in the setting of the "free of charge" quotas, since their number depends on the activity of the airlines: with this system, the airlines will receive a number of rights to pollute proportional to their activity. On the other hand, the rights to pollute, i.e. the internalization of the pollution, will represent an additional cost, which will be higher, the higher the activity. As a consequence we have paid very much attention to the modelization both of the regulation and of the production of emissions. We model the emissions as a joint product of the airlines activity that comes from the use of the fuel. Models of joint production are presented for instance by [START_REF] Baumgartner | Joint Production, Externalities and the Regulation of Production Networks[END_REF][START_REF] Baumgartner | Joint production[END_REF]. Hence a particular attention is given to the estimation of the fuel cost function, which is calibrated using real data. We chose to model the EU-ETS as an increase of the variable cost, which means that it can be similar to a tax (or a subsidy) on the airline activity, because the regulation is designed as such. A particular attention is given to the study of the use of the fuel by airlines because of its direct relationship with CO2 emissions (See the IPCC report 1999 for an evaluation of the impact of aviation on global atmosphere and the EU Directive 2007/589/EC for the determination of a precise coefficient linking fuel consumption and CO2 emissions). [START_REF] Harris | An Economic Model of U.S. Airline Operating Expenses[END_REF] makes an exhaustive analysis of the US airlines operational costs. [START_REF] Miyoshi | The carbon emissions of selected airlines and aircraft types in three geographic markets[END_REF] focus on the evaluation of the carbon emissions of airlines. They propose an original methodology to compute these emissions. They distinguish between short, medium and long hauls and they take into account the load factor, however in our paper we model it. Therefore, we use our method to evaluate the emission,1 but the contribution of our paper is more on the economic modelling of the EU-ETS consequences on airlines and therefore on their strategies. Our paper is also directly related to papers such as Viera et al. (2007), in which the authors emphasized the importance of having several instruments to reach efficient result. By focusing on the current regulation and trying to describe it as close as possible from the reality, we propose a more positive view. [START_REF] Hofer | The environmental effects of airline carbon emissions taxation in the US[END_REF] try to reach the same aim with the taxation in the US, however no economic model is presented in their paper. Finally, our model is the extension of the paper of [START_REF] Albers | Will the EU-ETS instigate airline network reconfigurations?[END_REF] and Anger and Kholer (2010) which try to evaluate the impact of the EU-ETS on airlines. Again, no economic model is presented in this literature.

We will focus on the regulation of CO2 in EU and try to model it as faithful to reality as possible, the following section presents the system of European Union Emissions Trading Scheme. In section 3, we introduce a modelisation of the EU-ETS system and we show stylised facts on some financial consequences for airlines. This section ends with the equilibrium of the economic model in which we derive the results and give interpretations. The final section presents some concluding remarks.

On COemission trading scheme in EU

The European Union Emissions Trading Scheme (EU-ETS), created in 2003 (EC, 2003) and implemented in 2005, is the first international trading system for carbon dioxide (CO2) emissions in the world. The aim of carbon markets is to provide incentives to reduce CO2 emissions. This system was implementable between european countries under several measures. Allocation is a unique feature of cap-and-trade systems. Indeed, a critical issue in dealing with climate change is deciding who has a right to emit carbon dioxide (CO2), under what conditions, and to what extent those emissions are limited. The EU-ETS is the first instance of creating explicit rights to emit CO2 and distributing these rights among sub-national entities. One of the main measures in the EU-ETS is to provide free allowances. The cap for these free allowances in the aviation sector will be limited to 97% of an annual rate of reference from 2012 and this limit will be restricted to 95% from the period 2013 to 2020. Moreover, the system was able to take advantage of the experience of the United States in the field of the acid rains. The US's success with sulphur dioxide emissions trading provides to European economists insights to apply to the European situation and provides to, in the Member States and the Commission, a body of literature and individual experiences to learn from. The EU-ETS program was divided into two phases, a pilot phase (2005-07) and a second or "Kyoto" phase to ensure a quality program.

The allocation methodologies applied by the 25/27 participating nations were remarkably similar. Four choices seem particularly interesting:

-Auctioning was only little used. One of the most striking features in the EU allocation process was that most Member States chose not to take advantage of the Directive's provision allowing states to auction up to 5% of allowances in Phase I and 10% in Phase II.

-Strong reliance on recent historical emissions. The disparity between advocacy and practice was in no aspect greater than for benchmarking. Benchmarking was strongly advocated however little used, which is a striking difference from US practice.

-Expected shortage was allocated to the power sector. Another distinctive feature in the EU allocation process was that the power sector was compelled to bear almost the entirety of the emissions reduction burden. When a Member state was short on allowances, this shortage was almost entirely allocated to the power sector.

-Highly novel new entrant/closure provisions. All Member States have set up reserves for new entrants, and most require closed facilities to forfeit post-closure allowances, even though there are significant differences between the specific Member State choices.

The first lessons of the pilot phase are that the European Commission has harmonized allocation rules across Member States and has tightened the carbon constraint in Phase II. Moreover, free allocation does not necessarily lead to "windfall profits" and first studies, as presented in the next section, show that the increase of costs due to CO2 regulation affect consumers' price but firms do not put the total cost on prices. Finally, new entrant/closure provisions provided perverse incentives. Indeed, the main effect of these provisions was to preserve pre-policy incentives to invest in polluting technology.

Concerning the quantity of allowances exchanged, these quantities in 2005 was relatively low at 262 Mt. Trades increased nearly fourfold by 2006, when 809 Mt were exchanged. The maturation of the market was confirmed in 2007, when almost 1 500 Mt were traded. These transactions are made with an average price of e22 per tonne in 2005, therefore the allowance transactions totaled e5.97 billion during the year. This total increased to e15.2 billion in 2006 before reaching e24.1 billion in 2007.

Once quantities exchanged are estimated, the EU allowance (EUA) price can be fixed. This price is determined by the equilibrium between supply and demand. Between the phase I and II, EUA price tends to decrease nearly to zero due to surplus saved on the period 2005-2007. Phase I allowance prices' fell under 1 e/tCO2 in February 2007 and ended 2007 at 0.02 et/CO2. Finally, The EUA prices for 2008-2012 remain stable and reach a peak at e25.

Therefore, the collapse of the first period carbon price has not jeopardized the expansion of the trading scheme. This is probably one of the most impressive results of this first trial period: all the big industrial and financial partners now accept that carbon is no longer free in Europe and that the carbon emissions will continue to be costly in the future.

Finally, concerning the level of carbon emissions, we can conclude that a modest amount of abatement occurred in 2005-2006. In a preliminary but detailed analysis of this data, Ellerman and Buchner (2010) concluded that a reasonable estimate of the reduction in CO2 emissions attributable to the EU-ETS lies between 50 and 100 Mt for each year, or between 2.5% and 5% from what emissions would have been without the EU-ETS.

The EU-ETS system is the first carbon market and is now implementable in the avi-ation sector. To know how is the impact of this regulation on the airlines, consumers' surplus or CO2 emission, we present the economic model.

3 Model Settings

The Emission Trading Scheme regulation

Airlines are responsible for externalities such as pollution (NOx, CO2, noise...) congestion and accidents. These externalities have social costs which differ from individual costs. For instance, the air transport sector produces CO2 emissions while the Society would prefer a lower level of these emissions. The levels of production of CO2 differ because the air transport sector is producing CO2 at a zero cost while the Society values negatively the emissions of CO2. The Society would be willing to pay to reduce the level of CO2 emissions. The problem is that the market failed to take into account the social costs and finally, a public intervention is needed because the right level of production is not reached. The economic literature suggests different instruments like norms or emission quotas (regulation in quantity) or taxes or subsidies (regulation in prices) to consider social costs (as pollution). The European Commission has chosen to create a market for pollution in order to make firms buy the rights to pollute. This market for allowances is thus used to regulate the CO2 emissions. The principle is simple: The regulator sets a maximum quantity of CO2 emissions tolerated by the Society for a given period (generally a year) and firms exchange rights to pollute. On this market, there will be firms from different sectors, sectors which pollute and thus have to buy for their emissions and sectors which receive rights because they pollute less than what they are entitled to emit. Airlines will be included in the European cap-and-trade system starting 2012. 2 They thus will have to pay for their CO2 emissions and follow a certain path in terms of reduction of their CO2 emissions. The principle of the regulation is to give part of the total emissions of a given airline for free and to let it buy on a market the rights to pollute for the other part of its emissions. As it is costly, one expects airlines to take measures to reduce their emissions. 85% of the allowances will be given for free. The EU defined a benchmark of emissions which corresponds to the average of the emissions calculated for the period 2004 to 2006. The target of the regulation is then to reduce the emissions with respect to this average level by 3% for the year 2012 and by 5% beyond 2012. The CO2 emissions are depending directly on airlines activity: the more activity, the more they have to pay. It is thus important to define precisely a measure of the activity of the airlines. The indicator which we choose as relevant is a weight and distance indicator. Indeed the CO2 emissions depends proportionnaly on the quantity of fuel used. This quantity itself is sensitive to the distance covered together with the weight carried. Therefore the relevant measure is expressed in tons times kilometers. Let us define W i as the activity of airline i expressed in tons.km (one passenger with his luggage stands for 100kg) during a given time period (typically one year). Airline i will thus receive a share of its yearly activity RW i for free, where R is defined as follows: R = rλc ET S r is the target of reduction of the regulator. It equals 0.97 * 0.85 for the year 2012 and 0.95 * 0.85 beyond 2012.

λ is a constant which represents the tons of CO2 emitted by burning one liter of fuel. It equals 2.52 * 10 -3 tons of CO2/l.

We define c ET S as the average consumption of fuel (expressed in liters of fuel per tons.km) for the benchmark period 2004-2006.

R is thus the benchmark used by the regulator to set up the emissions it authorizes for a given period. R is constant for all the airlines. Their emissions rights are adapted with respect to their activity (W i ).

As a consequence, each airline i has to buy exactly λW i (c i -rc ET S ) allowances, where c i corresponds to its average consumption of liters of fuel per tons kilometers over the given period. 3 The number of allowances asked to each airline relies on the efficiency of the airline. Indeed, it is not only relying on the activity of the airlines, which would have pushed airlines to increase their activity. The allocation also depends on the distance between the average fuel consumption of the airline and the benchmark chosen by the ETS, which is calculated over all the airlines for the period 2004-2006. If airline i is more efficient than the benchmark, which means that its fuel consumption is lower than the average consumption, airline i receives allowances which will be sold on the market. On the contrary, if the airline fuel consumption level is higher than the benchmark, airline i has to buy allowances. At this stage, the number of allowances depends also on the global activity of the airline, which creates a leverage effect. The cost of the emissions of an inefficient airline which operates a lot will be very high. These costs will result in adjustments for airlines in terms of technical progress or different strategies. Indeed, the renewal of the fleet, the modernization of the equipment, the optimization of the planes' trajectories, as well as the reorganization of the network, the reallocation of the planes are the different actions an airline can take.

Stylised Facts

Once the system of regulation described, the first question which seems interesting is the evaluation of the cost of this regulation. Indeed, the airlines will go from a situation in which they were not paying for their pollution to a situation in which they will have to pay for their emissions if not sufficiently efficient. This will necessarily modify the market organization, adding costs that will be probably passed on to consumers. In this section, we aim at giving a first evaluation of the cost or benefit of the ETS regulation for some of the airlines for which we have the CO2 emissions for 2008 (source: carbon disclosure project).

Airlines performance

Airlines RTK000 ( 2008 The first column concerns the Revenue Ton Kilometers which stands for the total carried mass (passengers and luggage, cargo and freight) on the network of a given airline. This variable represents the activity of the airline since the fuel consumption which explains the CO2 emissions is highly related to the weight carried over the network. These figures come from the ENAC database. The RTK include the whole network for a given carrier. However some of the airlines have part of their network outside the perimeter of the regulation. This is for instance the case of BA or AF-KLM. The figures are probably too high (and thus the cost or benefit probably too optimistic), since the emissions are considered for Europe. We are nevertheless not able to obtain exactly the figures corresponding to the relevant area of the ETS. From the figures on the emissions, we can derive the fuel consumption for the current year (2008). The (technical) coefficient used is denoted λ which value is 2.52 × 10 -3 tons of CO2/liter. The average consumption of fuel c avg expressed in l/tons × km is then easy to compute. The average consumption is used to compare to the EU benchmark in order to determine whether the regulation is a cost or a benefit. If c avg is lower than c ET S , the airline will receive allowances to be sold on the market. Its profits will be higher due to its efficiency with respect to the other airlines.

Exogeneous data

c ET S (l/RTK) 0.4724 benchmark = 85% × 95%c ET S 0.3814 λ (T C02 /L) 0.00252 CO2 price p a e13
The benchmark of the European Commission is not available yet. We decided to rely on Airbus analysis of the fuel consumption for the year 2007, i.e. c ET S = 0.4724l/RT K. The price of the allowances chosen is the average one for 2009, i.e. 13e. This price is relatively low, due to the worldwide recession for 2008-2009 and a slowdown in the demand, which relax constraints in the energy market. The entry of the airlines on the market for allowances together with an expected economic upturn suggest to interpret our result as a lower bound for the costs/benefits for airlines. In the first column, we define an efficiency measure based on the distance between the average fuel consumption and the benchmark. We then can estimate the total cost (or benefit) of the regulation by multiplying by the price of allowances. We add a column to measure the cost with respect to one passenger km carried, considering that the whole activity (RTK) are explained by passengers carriage. 4AF-KLM is the only airline to have an average consumption just below the benchmark. It will thus receive more free allowances than needed to cover its emissions and will be able to sell them on the market. With an allowance price of 13e, it will receive 8.3 Million euros from the sale. However, we have considered the whole activity of AF-KLM (world wide) which is higher than the one for the european market. For instance, if we make the assumption that the RTK for Europe represents 90% of the total RTK, the difference between the average consumption and the ETS benchmark would amount +0.033l/tkm, which means a cost instead of a benefit of 28,27 Million euros. This remark suggests the importance of the reliability and precision of the RTK figures together with the CO2 emissions. This suggests as well that an economic model focusing on the exact modelling of the ETS and activity of the airlines would help explaining the driving forces at play.

Easyjet and British Airways are poorly efficient and will have to pay for this lack of efficiency. The amount is very important for BA since its activity is very much higher with respect to Easyjet. It represents for instance 2.7% of the total revenue of BA. 5 Iberia is close to the ETS benchmark, even though a little above, which means that it will have to pay for allowances.

If we make the assumption that the average length of a flight is 1000km for Easyjet, the additional cost for this trip is thus 15e, which is quite important. For BA, taking a 6000km trip would add 113e to the operating costs, which is very important. For Iberia, taking the same distance would lead to an additional cost of 21e, which is far less. This discrepancy is problematic since it will create among the airlines very important competitiveness problems. For an economist point of view, it is very interesting to find the reasons of why there so many differences among the carriers. However, what remains is that the truthfulness of the figures has to be ensured because the impacts are very substantial.

In order to make further comparisons, we compute the same tables using the figures from RDC report 6 AF-KLM and Lufthansa are pretty efficient and receive allowances. However the area of the activity considered (RTK) are overestimated because it includes other traffic than simply european traffic. BA and Iberia are less efficient apparently, even though we are not able to explain why. The recent merger between the two carriers and the restructuration will probably help improving these figures.

The diversity of the figures according to the years considered so as the different sources suggest that a deeper analysis is needed. Building an economic model in which different levers are present will help to identify the problem and the solutions adopted by airlines.

The Cost Function

The regulation of the CO2 emissions means additional costs for airlines. These costs are related to their activity because of the regulation itself and because the regulation is associated to their fuel consumption. Hence, the relevant costs to be taken into account are the variable costs. There are several variable costs to be considered like labour costs, fuel costs, maintenance... depending on the time horizon considered (annual or infra-annual). One important element of these last years is the fuel consumption cost. Indeed, it has increased from 10% in 1994 of the total operating costs to 33% in 2008. 7 The air transport sector is also well known for the importance of its fixed costs, like the investment in planes, like administrative costs (to develop a network) or marketing costs. The fixed costs occur in the decision of launching an activity or entering a new market. Once engaged, what matters to determine firms strategies are the variable costs. In our economic model we will consider only variable costs and analyze the impact of the regulation on the determination of airlines activity: will they carry more or less passengers? according to which strategies? how will they modify their network?

The total costs function of the economic model is composed of three parts: the fixed costs, the variable costs concerning the fuel consumption and the variable costs due to the regulation. Let us define:

T C i (q i , d i ) = F C i + C F,i (q i , d i ) + X i (q i , d i ),
where F C i represents the fixed costs, C F,i (q i , d i ) is the total fuel cost of airline i and X i (q i , d i ) is the emissions cost of airline i. The activity of the airline is a combination of the tons carried, denoted q i and the expanse of the network, denoted d i for airline i.

At first, fixed costs are normalized to zero for each airline because we consider the situation of an airline which has its own network. Fixed costs will be reintroduced to consider the decision to extend a network or even maintain a given route afterwards. The economic model aims to show how the current activity of airlines is impacted by the new regulation.

As we focus on the regulation and its financial impact, we restrict our attention to the sole variable costs which are directly related to the regulation itself, i.e. the fuel costs. The fuel costs are depending on the price of the fuel denoted p F which is given on the crude oil spot market. It increased continuously since 1994 to 2008 and now is more stable, due to the world recession. Nonetheless this price is structurally going to raise again because of non-renewable resources and increasing world demand, putting pressure on airlines for a better use of the kerozen. The other two parameters which determine the fuel costs, are the expanse of the network which expressed as a distance variable denoted d i , and the activity on a network measured by the number of passengers carried expressed in tons, q i . 8 Over a network, the fuel consumption is not homogeneous. Indeed, for a given route, the take-off is the phase of time which consumes more fuel. Differences also exist with respect to the distance of a given route: a long haul flight uses less fuel than a short haul flight per ton km. One reason is technical: as the take-off phase consumes more, it can be less amortized if the flight is short. Moreover, on a long haul flight, it is easier to optimize the fuel consumption as the cruise phase is longer. We have chosen to focus on the problem of capacity. Indeed, the airline supply on a given network is not fully adjustable because of the given capacity of aircraft. For instance if an airline owns 2 planes of equal capacity, let say 100 passengers for each, if the demand is 90 then only one plane is needed and the load factor is 90%. However if demand is 110, then the airline has to put one other plane and the load factor decreases drastically (110/200=55%). This has to be taken into account in the costs and especially in the fuel costs since there is a certain amount of fuel to be carried to fly with a zero mass (when the plane is empty) and the marginal cost of carrying one more passenger is also increasing since more power is needed to carry a higher mass. We thus choose the following fonctionnal form to represent the cost function:

C F,i (q i , d i ) = kC f uel,i ( q i k , d i ),
where q i is the activity of airline i (in tons carried), and k is an integer, d i is the scope of the network. This function is defined for all (k -1)q p < q ≤ kq p , where q p stands for the maximum seats offered for one plane (we assume that the fleet is composed of homogeneous planes of equal maximum capacity measured in tons). The total fuel cost is thus a piecewise function, increasing in q i . The fuel cost function is also increasing with the km covered d i . Besides we define:

C f uel,i (q i , d i ) = C 0 (d i )(1 + a) q i
where C 0 (d i ) represents the minimum cost to be supported if one plane is used. It corresponds to the consumption cost for the route with all the carried staff and cargo but without passenger, on a given distance d i . a is an efficiency measure of the aircraft fuel consumption. It corresponds to the fuel consumed for one passenger added in the plane. a is assumed to be small and can be checked empirically. We assume besides that the whole influence of parameter d i is captured in C 0 and will use several values of this parameter to calibrate the model. For sake of simplicity, we use a Taylor series development and will use the following functionnal form 9 :

C f uel,i (q i , d i ) = p F F 0 (d i )(1 + aq i + a 2 q 2 i 2
).

8 A passenger with his luggage amounts 100kg. 9 For a detailed analysis, see appendix 1.

For instance, if the plane (with a capacity of 270 seats) is such that the minimum quantity of fuel needed to travel is F 0 = 68000l, knowing that the average fuel price for 2009 is p F = 0.4e/l, and taking the overconsumption due to one additional passenger to be equal to a = 0.0055. X(q i , d i ) allows to take into account the cost due to the regulation of the ETS in aviation. This function is built in order to measure the additional cost incurred by the regulation. This cost depends on the activity and on the fuel consumption. It is in fact a measure of the distance between the emissions of the airline on the route expressed as a function of the fuel consumption and the emissions tolerated by the regulator, which we introduced in the previous section. The average fuel consumption of airline i, for a given network, is thus c i = C F (q i ,d i ) d i q i p F and the regulation cost can thus be defined as follows:

X(q i , d i ) = p a ( λC F (q i , d i ) p F -Rq i d i ),
where p a is the price of an allowance in the market. Airlines are price taker, they do not have any influence on it. p F is the price of the fuel, given on the market, R equals λrc ET S as defined in the previous section. X(.) can be positive or negative depending on whether the airline has a fuel consumption respectively above or below the benchmark. If the spot price of the emission rights p a is higher, the costs of the regulation becomes more important if the airline has to buy rights. The airlines have thus an interest in forecasting properly the price of the allowances. The two instruments of the regulator to set a certain level of R are the target coefficient, r which is supposed to decrease over the years and the benchmark evaluation of the fuel consumption of airlines. If the price of the fuel p F is higher, then the X(.) function is less likely to be positive. Indeed, a low fuel price relaxes the constraint of being more efficient in using fuel. If the fuel costs are higher, then the costs of the regulation are also higher. It means that a less efficient firm will support higher costs for the regualtion of its emissions. Eventually, if the activity of the airline increases, then the fuel costs increase but the free allowances as well. There is a trade-off between increase of the fuel cost and free allowances, and the problem for the airlines is if the fuel costs increase at a higher rate than the activity.

Equilibrium

We first analyse a situation in which airlines are competing à la Bertrand on a given city pair. This means that parameter d is exogenously given. Firms are then price takers and their goods are perfect substitutes for consumers. The maximization program is thus

M axΠ(q) {q} = pq -T C(q) s.t. T C(q) = F C + C F (q) + X(q)
The first order conditions give

p = δC F (q) δq + δX(q) δq
The airlines equalize what selling one ticket (or pricing one additional ton carried) yields to what it costs at the margin, i.e. the supplementary fuel consumption needed to carry this additional passenger (or ton) plus the additional regulation cost (if the carrier is less efficient than the average) or minus the benefit (if the carrier is more efficient than the average). Taking into account the definition of function X(.), the expression writes:

p = (1 + p a p F λ) δC F (q) δq -R
This equation corresponds to the supply of each (identical) airline. The demand is assumed to be linear10 :

p = A -Bq,
where A and B are positive. A corresponds to the willingness to pay of consumers.

Parameter A(d) corresponds to the maximum price consumers want to pay for a given network (it can be not only an origin-destination but an aggregate network). It stands for their willingness to pay and is a representation of their preferences. This maximum price is increasing with the scope of the network. We do have typically very few information on the value of this parameter. Some airlines run surveys in order to learn about this value. However the information is quite difficult to estimate. B is a measure of the sensitivity of the variation of the price to a variation of the quantity. It is a very important parameter for the airline because while choosing a quantity to supply (or equally a price to fix), B is a measure of what reduction in the price is needed to attract one more passenger. We make the assumption that demand is influenced by the distance parameter through A and consider that the consumer's reaction to a modification of the price is not infuenced directly by the destination chosen. More precisely, it means that the willingness to pay for a trip from Paris to San Francisco is assumed to be higher than the one for a Paris-Madrid. However, the sensitivity to the price is kept constant. The equilibrium quantity is then11

q P C = k(A -aF 0 (p F + p A λ) + c ET S dp A rλ)) (Bk + a 2 F 0 p F + a 2 F 0 p A λ) .
The quantity at the equilibrium depends on different sets of parameters. The first set of parameters includes technical variables, like k, F 0 and a. These parameters are linked to the technical efficiency in using fuel and capacity of the airline. Hence, they allow to take into account the technical progress that airlines may incorporate into their production process. For instance when an airline decides to renew a plane, both coefficient F 0 and a are modified to take into account a more efficient fuel consumption. Parameter k changes to take into account both the change in capacity and the induced modification of the load factor. The regulation levers are the price of an allowance p a , the average consumption c ET S used as benchmark and the target r. A strict regulation will be for instance a higher price of allowances, induced by a lower quantity of emissions rights decided by the European Commission. It also corresponds to a lower cap set r. Even though the regulator cannot play with all the instruments at the same horizon, it is still interesting to analyze the impact of a variation of the different parameters. Indeed, r is fixed till 2020 by the law and c ET S is computed according to the past activity of airlines. However, the price of allowances p a can be decided on a year basis. Finally, parameters A and B influence the demand and therefore the choice of equilibrium. For instance a positive demand shock would translate into a higher A (maximum of demand). Parameter B measures the sensitivity to a variation of the quantity of the variation of the price. The more sensitive the consumers' demand, the lower the ability of the airline to set a higher price at equilibrium.

If the airline chooses a more efficient plane with a lower F 0 , the number of passengers carried at the equilibrium is higher. More activity is reached due to a better use of fuel. Coefficient a plays in the same direction as well: a lower a means a higher efficiency which enables the airline to carry more passengers. When the capacity of the plane k is increased, the quantity is increased at the equilibrium to take into account the cost of a having a "bad" load factor. A more stringent regulation resulting in an increase of the price of allowances p a leads to a lower activity for the airline. If the average fuel consumption c ET S decreases, the quantity decreases as well at the equilibrium. The effect of the target r plays in the same direction: if the regulator decides to strengthen the regulation (lower r), then the impact on the quantity is negative and less passengers are carried at the equilibrium. The price of the fuel has the same impact in our model than the price of an allowance: the higher the fuel price, the less the quantity. It is important to note that in our model, a partial taxation of the fuel price would have the same kind of effect than the ETS system. If demand increases, for example with a higher A, then the quantity at the equilibrium increases. If demand is less sensitive to the price, the airline is then able to set a higher price for the same quantity or to increase the quantity at the same price. Finally, d stands for the distance of the destination considered or more generally the scope of the network of a given airline. Parameter d has an influence on the choice of parameter F 0 because every type of plane will not be able to operate every destination. Without regulation, the airline programme is the following:

M axΠ(q) {q} = pq -T C(q) s.t. T C(q) = F C + C F (q)
The solution of this programme offers an interesting benchmark to evaluate formally the impact of the regulation on the activity. After computation, the solution is:

q P CN = k(A -aF 0 p F ) (Bk + a 2 F 0 p F ) .
Proposition 1 The regulation can have a positive impact on the activity of the airlines. The effect depends on the parameters.

Proof. We compute the difference between the quantities with regulation and without regulation. This difference is

q P CN -q P C = kp a λ(aBF 0 k -Bc ET S dkr + a 2 F 0 (A -c ET S dp F r)) (Bk + a 2 F 0 p F )(Bk + a 2 F 0 (p F + p a λ))
The denominator is always positive, for all acceptable values of the parameters, thus the sign depends only on

(aBF 0 k -Bc ET S dkr + a 2 F 0 (A -c ET S dp F r)).
Note that this expression is not depending on the price of the allowances. On the contrary, it depends on the price of the fuel which has a negative impact on the difference. Indeed, an increase of the price of the fuel has more influence on the quantity without the regulation, it decreases faster. The technical parameters (a and F 0 ) are more likely to have a negative impact: if the plane chosen is less efficient (higher a and or F 0 ) then the difference is positive. Finally, if the regulation is less strict (higher benchmark c ET S and or higher target r) the difference is more likely to be negative, since the quantity with regulation increases, while the other quantity is not impacted. We cannot say much more since there are too many parameters in the model. To go further we have to specify more and explore the relationship between these parameters. This is what we do in the following section. Whether to know if the quantity of equilibrium is lower in case of regulation is not fully trivial. It is not clear whether this quantity is lower or higher to the quantity with regulation since the ETS can be either a cost or a benefit for the airlines. The regulation has been set in order to limit the CO2 emissions, which leads, if no technical progress is incorporated or simply available, to a direct decrease in the activity of the airlines. This result is thus all the more counterintuitive. However, it is due to the design of the regulation, which induces airlines to increase their activity in order to cover the cost of the regulation. Finally, results depend on the parameters values and we take several examples to illustrate the situation. They are presented in the following section.

Scenarii

The first parameters we set are the regulation, the price of the allowances and the price of the fuel.

Exogeneous data c ET S (l/RTK) 0.4724 r = 85% × 95% 0.8075 λ (t C02 /L) 0.00252 CO2 price p a e13 Fuel price p F e0. [START_REF] Anger | Including aviation emissions in the EU ETS: Much ado about nothing? A review[END_REF]

/l

The other parameters we have described (technical, demand) are not completely independent of the distance parameter which is considered as exogenous in this model. Indeed, on a long haul trip, the plane chosen will have different technical characteristics (F 0 and a) than on a short haul. The plane is typically larger when the distance is higher, which goes with a higher 'fixed' consumption of fuel i.e. with a zero load factor (higher F 0 ) and with a lower marginal consumption of fuel (lower a). For the moment, we cannot give a functionnal form to model the influence of the distance on the technical characteristics. We thus chose to distinguish two cases: a long haul trip, with a given distance and evaluate for what pair of technical parameters (a, F 0 ) the quantity with regulation is higher. Moreover, the demand is also influenced by the distance because the willingness to pay for a long trip will be higher than for a short trip. For this reason, we decide to examine two opposite cases: the first situation corresponds to a long haul flight, the second to a rather short haul flight.

Long haul scenario: Paris-San Francisco example

The distance between Paris and San Francisco is 9000km, thus we set d = 9000. The demand function is defined for a one way trip. It is the residual demand, given all the other costs especially variable costs (cabin crew, maintenance, catering...), thus the maximum demand is lower than the one addressed to the company12 . We chose to set A = 8000k. Parameter k intervenes since the demand is defined by plane. k represents the number of planes used at the equilibrium to carry the optimal number of passengers and cargo, the maximum demand has thus to be multiplied by k to be homogenous. For simplicity, we set k = 1.

We chose to take two values for the sensitivity of the demand to the price. We consider first a demand which is rather inelastic, i.e. B 1 = 0.8. We then consider a rather elastic demand B 2 = 1.2.

Lemma 1

The regulation leads to more activity when the airlines use efficient airplane. For instance, for an elastic demand, B 2 = 1.2, the equation gives

q P CN -q P C ≥ 0 ⇔ F 0 ≥
2746.53 0.8a + 6626.73a 2 The following figure illustrates this situation. Results are robust to the assumption of inelastic demand. For example, if the airlines use a A340 to operate the route Paris-San Francisco, the technical characteristics are: (F 0 = 68000, a = 0.0055). Then the difference is positive: the airlines reduce their activity due to the regulation.

Short haul scenario: Paris-Madrid example

Let us now set d = 1300. The corresponding maximum demand is lower than for the long haul trip, around e500 per ton km, thus A = 500k. For simplicity, we present only the case for which k = 1. Both situations, elastic demand and inelastic demand are analyzed, but we present the results for an elastic demand, since results are consistent with an inelastic demand.

Lemma 2

The regulation leads to more activity when the airlines use efficient airplane. For instance, for an elastic demand, B 2 = 1.2, the equation gives

q P CN -q P C ≥ 0 ⇔ F 0 ≥ 396.722 0.8a + 301.639a 2
The following figure illustrates this situation. In this section, we study a duopoly situation where two airlines compete on a given network. We consider that distance is exogeneous, however in the sequel we will remove this assumption. Now consumers have the choice among different airlines. Each consumer can choose between two options of consumption according to their preferences. The preferences of consumers depend of intrinsic characteristics. Both airlines offer a journey, and consumers choose either one firm or another one depending of their preferences.

In the economic literature, there exist two types of differentiation. 13 The goods can be horizontally differentiated or vertically differentiated. The horizontal differentiation means that consumers have differences of opinion, even if prices are the same, some consumers prefer buying from firm 1 and some from firm 2. Whereas, vertical differentiation expresses that goods have different quality, if prices are the same then every consumer prefers buying from the higher quality firm and latter holds onto some consumers even if its price is higher than its competitors. For instance, horizontal differentiation corresponds to the situation when an identical good is sold in different areas of a town (but at the same price), all consumers located in one area prefer to buy in their own area whereas to buy from other retailers. The horizontal differentiation is associated to spatial models introduced by Hotelling in 1929. The vertical differentiation corresponds to the case where there are some variants of a good, for instance high quality variants ("luxury") and standard quality. Sold at a same price, all consumers prefer to buy the luxury variant. The competition between firms when goods are horizontally or vertically differentiated is different and entails specific consequences in each case. 14In our model we consider that both airlines offer an horizontally differentiated good, and consumers purchase either from firm i or firm j, with i, j = 1, 2. Each consumer consumes one unit of one variant between the set of possible outcomes.

The introduction of competition allows us to better understand the reality of aviation sector.

• Expliquer la demande: comment on modélise la concurrence We still consider that consumers' demand is linear and given by: P i (q i , d) = A(d) -Bq i + Eq j where i, j = 1, 2 with i = j. As in the previous section, P i (q i , d) is the consumers' demand for firm i. The parameter A(d) corresponds to the maximum consumers' willingness to pay for a given network. Quantities q i and q j are imperfectly substitutes for consumers and the relative degree of substitutability is given by B/E.

Duopoly with regulation

The airlines' costs are the same than previously and symmetric for each firm. The profit function is the same for both firms, therefore we can write the maximisation program of an airline i as:

M axΠ i (q i , d) {q i ,q j } = P i (q i , d)q i -T C i (q i , d) s.t. P i (q i , d) = A(d) -Bq i + Eq j T C i (q i , d) = F C i + C F i (q i , d) + X(q i , d)
with i, j = 1, 2. The first-order condition is the same than in monopoly context but now demand takes into account competition:

P ′ i (q i , d) + P i (q i , d) = C ′ F i (q i , d) + X ′ (q i , d)
The maximization program gives us the quantity for airline i (and it is symmetric for airline j):

q * i = k (A + p a rλc ET S d -F 0 (d)a(p F + p a λ)) k(2B -E) + F 0 (d)a 2 (p F + p a λ)
As in the perfect competition situation, this quantity depends on the different sets of parameters. There are technical variables: k, F 0 (d) and a which refer directly to fuel consumption and capacity of the airline. For the regulation constraints, p a is the price of allowances fixed by the market, c ET S is the average consumption and r the target decided by european authorities. Finally, parameters A(d), B and E influence the demand and the level of equilibrium. The equilibrium profit is given in the appendix. We can compare the sum of quantities in duopoly with the monopolist quantity:

(

2 i=1 q * i -q E ) = k (A + p a rλc ET S d -F 0 (d)a(p F + p a λ)) k(2 B + E) + F 0 (d)a 2 (p F + p a λ ) (k(2B -E) + F 0 (d)a 2 (p F + p a λ)) (2 Bk + F 0 (d)a 2 (p F + p a λ))
The total quantity in duopoly is more important than the one in monopoly situation. As the monopoly is a price maker, it restricts its quantity to charge a higher price. The competition entails higher quantities and this quantity effect leads to decrease prices.

Duopoly without regulation

When we consider the case where European Comission do not regulate aviation sector, we can rewrite the maximization program without cost of regulation:

M axΠ i (q i , d) {q i ,q j } = P i (q i , d)q i -T C i (q i , d) s.t. P i (q i , d) = A(d) -Bq i + Eq j T C i (q i , d) = F C i + C F i (q i , d)
At the equilibrium, quantity is given by:

q 0 i = k (A -p F F 0 (d)a) k(2B -E) + p F F 0 (d)a 2
We can compare the situation with and without regulation to show its impact on quantities:

(q * i -q 0 i ) = - kp a λ k(2B -E)(F 0 (d)a -rc ET S d) + F 0 (d)a 2 (A -rc ET S dp F ) (k(2B -E) + F 0 a 2 (p F + p a λ) (k(2B -E) + p F F 0 (d)a 2 )
This difference depends on the sets of parameters defined previously. To be continued...

Conclusion

At the beginning of the application, the EU will create an ex-ante benchmark "so as to ensure that allocation takes place in a manner that provides incentives for reductions in greenhouse gas emissions and energy efficient techniques, by taking account of the most efficient techniques, substitutes, alternative production processes, efficient energy recovery of waste gases, use of biomass and capture and storage of CO2, where such facilities are available, and shall not provide incentives to increase emissions" [source: European Directive]. Our economic analysis shows that the introduction of the ETS system tends to increase the airlines activity for the more efficient aircraft: more passengers are carried at the equilibrium. This result is due to the particular shape of the ETS cost function X, which we have modelled as closely as possible from the real system. Now, airlines cope with the necessity to improve their load factor and their consumption per tonne.kilometer. However, for all less efficient planes, the introduction of the regulation tends to decrease the airlines' activity and therefore to reduce the greenhouse gas emissions. For future research, we will endogenize the expanse of the network in a two-stage game since this variable is very important in the definition of the parameters, especially the technical ones.

A Fuel cost function

We detail the fuel cost as a function of d, the distance of the route, q, the tons carried and p F , the fuel price. The function C F (q, d, p F ) is defined as the fuel cost consumption for a given aircraft and a given route expressed in e. To design this function, we need to define variables and functions C 0 (d, p F ), a, q p , which qualify an aircraft characteristics and performances.

• C 0 (d, p F ) is the consumption cost for the route with all the carried staff and stuff but without any passenger (with a load-factor equivalent to 0).

C 0 (d, p F ) = F 0 (d) × p F ,
where F 0 (d) is the fuel consumption for the route with all the carried staff and stuff but without any passenger (with a load-factor equivalent to 0). This function is a non-linear function of variable d, the distance covered. F 0 (d) is expressed in liters so as C 0 (d).

• a is a parameter modeling the over-consumption due to an additional passenger (marginal cost). We assume that a is constant.

• q p is a parameter modeling the aircraft capacity of passengers expressed in tons.

Besides, we introduce C f uel (q, d) which represents the fuel cost function of the aircraft carrying q ∈ [0, q p ]. We precise that we have C f uel (q, d) = F f uel (q, d) × p F with F f uel (q, d) the total fuel consumption.

With a given aircraft, one cannot carry more passengers than the initial capacity. If q > q p , the airline needs to use more than one aircraft or to use several times the same. If total demand is q and q > q p , the airline must use at least E[q/q p ] + 1 aircraft to carry the passengers where E[q/q p ] is the integer part of q/q p . We will consider the airline uses exactly E[q/q p ]+1 aircraft, especially because of the fixed costs, with the same load-factor equivalent to q (E[q/qp]+1)qp . We define C f uel (q, d) as follows:

C f uel (q, d) = C 0 (d)(1 + a) q ,
with a > 0 and close to 0 15 . When a passenger is added to the flight, the airline needs more fuel because of two physical reasons. Firstly, more passengers means more on-board mass (100kg per passenger), so that the aircraft mass increases and it needs more power to fly. The airline has to take more fuel on board. Secondly, in aviation, we need to board fuel to the propulsion needs. Indeed, because of the fuel mass increases, the aircraft consumes fuel to transport fuel. These two physical reasons illustrate the shape of C f uel (q, d). Moreover, parameter a is positive but close to 0. Indeed, the experience shows that additional passenger makes the necessary mass-fuel to increase however not tremendously because of the significant aircraft mass alone. 16 The parameters which differentiate two aircraft with the same capacity are C 0 (d) and a. C 0 (d) and a are given by the aircraft manufacturers and the fuel market. Hence, an airline's fuel cost results of two factors : the price of fuel and the fuel efficiency. For sake of simplicity, the shape of C f uel (q, d) is approximated using the Taylor series, as a > 0 and close to 0, and with 0 < q ≤ q p :

(1 + a) q = 1 +

n i=1 1 i! ( i-1 j=0
((qj))a i ) + o(a i ) up to the second order, it is equivalent to:

(1 + a) q = 1 + aq + a 2 q(q -1) 2! + o(a 2 ).

Thereby, the cost function writes for q ≤ q p :

C f uel (q, d) = C 0 (d)(1 + a(1 - a 2 )q + a2 2 q2)
which we can simplify the expression (since a small and a ≃ a(1 -a 2 )). Therefore we obtain:

C f uel (q, d) = C 0 (d)(1 + aq + a 2 q2 2 ).
Let us define function C F (q, d) which is the total fuel consumption over a network. C F (q, d) is actually a piecewise function since there exists a discontinuity in the capacity of aircraft. When the airline needs to use an additional plane, it might not be full and thus the cost is more than the double of the cost of one full plane. The total fuel cost, ∀k > 0 and k integer number ∀q ∈ [(k -1)q p , kq p ], can be rewritten as:

C F (q, d) = k × C f uel ( q k , d).
For example in the model, if q ∈ [q p , 2q p ] thus C F (q, d) = 2 × C f uel (q/2, d). Total costs of engaging an additional plane (if not full) are increasing with the activity and distance.

B Equilibrium quantity B.1 Linear function

We compute the first and second-order conditions to find the optimum of the profit function: 16 Notice that in the model we consider a flight without wind and in the standard flight conditions.

28 max {q} Π(q) = max {q} P (q, d)q -T C(q, d) = max {q} P (q, d)q -F C -C F (q, d) -X(q, d).

As C F (q) is a piecewise function equivalent to kC f uel ( q k , d) on the interval [(k -1)q p , kq p ] with k integer, we consider:

∃k ≥ 0 q E ∈ [(k -1)q p , kq p ].
The first order condition gives:

∂Π(q) ∂q = A(d) -2Bq -C 0 (d)k( a k + a 2 q k 2 )(1 + p a p F ) + Rd = 0.
The second order condition is such that:

∂ 2 Π(q) ∂q 2 = -2B - a 2 C 0 (d) k (1 + p a p F ) ≤ 0.
As pa p F ≥ 0, a 2 ≥ 0, C 0 (d) ≥ 0, k > 0, B > 0 then ∂ 2 (q) ∂q 2 < 0, which means that the program is concave.

At the optimum, marginal revenue equals marginal cost, i.e. P (q, d) + qP ′ (q, d) = C F (q, d) with

A(d) -2Bq = C 0 (d)k( a k + a 2 q 2k )(1 + p a p F λ) -Rd.
At the optimum, when q E ∈ [(k -1)q p , kq p ] we have: The equilibrium quantity is thus q E when it is positive, which introduces a constraint

q E ≥ 0 ⇒ A(d) + rλc ET S d ≥ aF 0 (d)(p F + p a λ)
This condition is fulfilled if the market is sufficiently wide (high A(d)) so that the airline can have space to launch the activity. It corresponds for example to a sufficiently large network (high d). If the condition is not satisfied, then the equilibrium quantity is 0. The benchmark situation corresponds to the non regulation case in which the airline maximizes its profit: M axΠ(q) {q} = P (q, d)q -T C(q, d) s.t. P (q, d) = A(d) -Bq T C(q, d) = F C + C F (q, d)

The first order condition gives: P ′ (q, d)q + P (q, d) = C ′ F (q, d)

which leads after simplifications to:

q 0 = k(A(d) -aF 0 (d)p F ) 2Bk + a 2 F 0 (d)p F ,
where d is given.

B.2 Duopoly

Equilibrium profit with regulation is given by: Π i (q i , d) =

  q E = k(A(d) + Rd -aC 0 (d)(1 + pa p F λ)) 2Bk + a 2 C 0 (d)(1 + pa p F λ)When we introduce R = λc ET S r, we haveq E = k(A(d) + λc ET S rd -aF 0 (d)(p F + p a λ) 2Bk + a 2 F 0 (d)(p F + p a λ)

a r 2 λ 2 c 2 ET S d 2 + p 3 a r 2 λ 3 c 2 ET S d 2 ) 3 F + p 3 a λ 3 ) +k 2 F

 222332 2B -E) + F 0 (d)a 2 (p F + p a λ)) 2 +kF 0 (d)a 2 ((p F + p a λ)(A 2 + 2Ap a rλc ET S d) + p F p 2 -kF 0 (d)a 3 (2Ap 2 a λ 2 ) +kF 2 0 (d)a 3 ((p F + 2p a λ)(-2λp a rc ET S dp F -2Ap F ) -2p 3 a rλ 3 c ET S d) -kF 3 0 (d)a 4 ((p F + p a λ)3p F p a λ + p 0 (d)a((p F + p a λ)(-4B(A + λp a rc ET S d))) -k 2 F 2 0 (d)a 2 ((p F + p a λ) 2 (6B -4E)) +2k 2 B(λp a rc ET S d (2A + λp a rc ET S d) + A 2 ) -k 3 F 0 (d)((p F + p a λ)(8B 2 + 2E 2 -8BE))• Parametre important (2B-C) qui doit être positif: qu'est ce que ça veut dire?Equilibrium profit without regulation is the following:2B -E) + p F F 0 (d)a 2 ) -2k 3 p F F 0 (d)(E(4B + E)) + 2k 2 p 2 F F 2 0 (d)a 2 (-3B + 2E) -k(p F F 0 (d)a 2 (p 2 F F 2 0 (d)a 2 + A 2 -2Ap F F 0 (d)a)) +2k 2 AB(A -2ap F F 0 (d))

  ) Emissions2008 (tons) Fuel cons.(l) c avg (l/RTK)

	Easyjet	4012513	4307000	1709126984	0.4259
	Air France-KLM	29284393	27506144	10915136508	0.3727
	British Airways	15228809	16840627	6682788492	0.4388
	Iberia	5910594	5839469	2317249603	0.3920

  for the CO2 emissions of 2009 (exactly from march 2009 to march 2010). These figures are an approximation realized by RDC according to the network, the fleet, the number of flights... Our results are presented in the following table:Before interpreting this table, two remarks have to be made: at first, the emissions of Easyjet are very different in 2009. The evolutions of carbon emissions between 2008 and 2009 are given in the third column with the variable ev. We observe an increase of 80%, which cannot be explained neither by technical reasons, nor economical reasons. This figure has thus to be taken into account with much caution, especially because it will translate into huge costs. The same remark can be made in a lower range however, to the figure concerning Iberia. There is a difference of 48% between the emissions of CO2 in 2008 and 2009, with no reason apparently. The data on the CO2 emissions are not easy to obtain and what airlines have declared does not seem to correspond to what RDC model has computed. For instance, Easyjet seems to be poorly efficient in 2009 with respect to 2008. Nothing can explain this change: its fleet is very young, its network has not changed and its load factor is high.

			Airlines performance
	Airlines	RTK000 (2009) Emi.2008 (tons) Emi. 2009, tons (ev.) Fuel cons.(l) c avg (l/RTK)
	Easyjet	3722220	4307000	7753817 (80%)	3076911508	0.8266
	AF-KLM	28770277	27506144	26407231 (-4%)	10479059921	0.3642
	BA	14520041	16840627	18645484 (11%)	7399001587	0.5096
	Iberia	5481678	5839469	8768755 (48%)	3479664682	0.6348
	Lufthansa	21313185	na	19093956	7576966667	0.3555
			Extra consumption for airlines
		Airlines Efficiency measure (%) Cost per ton km Total cost in e
		Easyjet	0.4452 (+117%)	0.01458	54269968
	Air France-KLM	-0.0172 (-4.5%)	-0.000562	-16168896
	British Airways	0.1282 (+34%)	0.0042	60984172
		Iberia	0.2534 (+66%)	0.0083	45505514
		Lufthansa	-0.0259 (-7%)	0.00085	-18116207

See in appendix A.

For a detailed description of the legislation, see http://www.euractiv.com/en/climate-change/aviationemissions-trading/article-139728.

The average consumption of fuel is computed as follows, for airline i, which operates J routes ci = conso of fuel for all J routes j W ij , with j = 1, ..., J.

One passenger amounts 100kg.

We can not easily compare with operating results since they are negative in 2009.

European Emission Report. Quarter 1-2010 in RDC Aviation.

Source: AEA and EIA forecasts.

The demand is linear to make the model more tractable but results are robust to a log-linear shape of the demand.

The second order condition is satisfied by the assumption made on the cost function: increasing at an increasing rate.

Our simulations are however robust to a large range of maximum demand and elasticity. We tried to make the examples as realistic as possible though. The simulations files are available upon request.

This distinction is given by[START_REF] Lancaster | A new approach to consumer theory[END_REF].

For more details, seeGabszewicz and Thisse (1979, 1980) and[START_REF] Shaked | Natural oligopolies[END_REF].

See ENAC study for the fuel consumption in the air transport sector.


Etienne Billette De Villemeur

Kévin Guittet
email: guittet@recherche.enac.fr.

Optimal Structure of Air Transportation Services when Environmental Costs are taken into account

Keywords: J. E. L. Classification Numbers: D42, H21, H23, L51, L93 air transportation, environmental costs, airport charges, fuel tax

The consequences of environmental taxation on the supply of airtransportation services by a profit-maximizing monopolist are examined. We consider both a passenger related tax and an aircraft related tax, highlight their respective impact on prices and frequency and derive their optimal combination. It is shown that there is a trade-off between the recovery of environmental damages and the distortions created by the tax. If tax can be negative. i.e. if the air transportation industry can be subsidized, it is possible to decentralize the optimal situation. We establish the optimal taxation system in a more realistic setting where tax are to be positive and the ATM costs have to be recouped. To conclude we question the opportunity and the ability of (direct) regulatory mechanisms to address these issues in substitution and/or in addition to (the indirect action of) a fuel tax.

Introduction

Air-transportation markets have known a huge growth over the last decades. There is an increasing concern that this positive traffic growth may conflict with environmental objectives in general and the Kyoto agreements in particular. As air transport now stands for about 4 % of the world's total energy consumption, and given that airport pricing schemes and ATC operators scarcely take externalities into account, there is much to gain from deriving socially optimum taxes. Accordingly, our aim in this paper is to understand how air-transportation services are to be structured to minimize their environmental costs or conversely, how environmental objectives are to be embodied in the design of airport charges and fuel taxes as to minimize their negative impact on the profitability of the industry.

Our study is oriented to the European market, where, despite deregulation, many connections are still operated by a single operator. We thus examine the consequences of environmental taxation on the supply of air-transportation services in a setting where these services are supplied by a (profit-maximizing) monopolist. We however depart from generic works on the taxation of a monopoly (e.g. Barnet (1980)) by endogeneizing the frequency of flights, hence capturing a major specificity of the air-transportation market.

Environmental taxes in the field of air-transportation are considered by [START_REF] Nero | Hub and spoke networks and the inclusion of environmental costs on airport pricing[END_REF]. A single compagny is allowed to choose the price as well as the number of flights, but the main parameter under interest is the network structure. The environmental externality is not explicitely modelled, its effects being described by the concept of "footprints of pollution", but not fully captured 1 . Accordingly, the effect of taxes is studied on the price and frequency choices, the network structure and the profit of the firm, while welfare considerations are not performed. [START_REF] Carlsson | Environmental Charges in Airline Markets[END_REF] build on this article and integrates explicitely the environmental externality in the expression of the welfare. He computes the optimal "frequency-related" tax in the monopoly case as well as in the duopoly case. Though these recent studies on environmental externalities have focussed on the specific impact of hubbing, we consider here a single leg and rule out competition to specifically address the question of the optimal combination of a "passenger-related" and a "frequency-related" tax. We believe that such schemes may allow to gain some insight into the implementation of Ramsey pricing schemes, as derived in the work of [START_REF] Dissler | Modulated airport charges against airport congestion : an economic way of regulation[END_REF], Zhang and[START_REF] Zhang | Concession revenue and optimal airport pricing[END_REF][START_REF] Zhang | Airport charges, economic growth, and cost recovery[END_REF], Carrlson (2003), [START_REF] Pels | The economics of airport congestion pricing[END_REF] and others.

The model

General setting

Our study is oriented to the European market, where, despite deregulation, many connections are still operated by a single operator. In this paper, we thus consider single city-pair operated by a monopolist. The supply of air-transportation services is characterised by the ticket price p and the flight frequency f . It induces a demand2 of journey X (p, f ).

As in Billette de Villemeur ( 2004), from which we borrow the notations, the firms has to bear fixed costs F and operational costs. The latter are directly related to the frequency of connections and the size of the aircrafts. A one-way flight with an aircraft of capacity K translates into operational costs C (K) on the route under scrutiny. Since operational costs are supposed to be proportional to the number of flights, their total amount is thus f C (K). Observe that, if the company adjust in the long run the capacity K of the aircraft to the total traffic observed X, the relation K = X/f holds. As a result operational costs may rewrite f C (X/f ) .We shall assume this is the case all along the paper. The framework may however easily be adapted to situation in which aircrafts are not used at full capacity.

Following the structural form of the operational costs of the airline, we assume that the environmental costs are roughly proportional to the number of flights and writes f E (K) , where K is the capacity of the operated aircraft. Similarily the other costs, namely the costs for the airports and the air traffic control costs are denoted f G (K) .

The airline is subject to tax payments. These taxes denoted by τ x and τ f bears on the number X of passengers and on the number of flights f respectively. Since we essentially address a normative issue, namely the optimal structure of taxation, all coordination problems are washed away by assuming that a single regulator sets these taxes. When considering policy issues, it should taken into account that, in practice, part of these taxes are decided at the airport level. Recall that, on top of these taxes that bears directly on transportation services, part of the operational costs are due to fuel taxation which is not modelled explicitely here.

First best allocation

Before to address the tax issues and in order to obtain a reference point, we now consider the problem of a planer that aims at maximizing the social welfare. The latter writes

W (X, f ) = S(X, f ) -f C(K) -f E(K) -f G (K) , (1)
where S (X, f ) stands for the consumer aggregate (gross) surplus. Typically, it might be convenient to decompose S (X, f ) as the difference between the gross utility U (X) and the time-costs X νT (f ) of the X journeys, where ν denotes the "value of time" and T (f ) the travel time. However, unless usefull for the interpretation we will stack to the general form as to allow the comparison with other results from the litterature that are not compatible with this decomposition. Remark also that the emissions E, although being part of social welfare, do not enter as an argument of the function S (., .) . This is so because we assume that, despite consumers value environmental quality, they do not take it into account in their travelling decisions.

Assuming quasi-concavity of the objective function, the optimality conditions of the problem are given by :

∂ X S(X, f ) = C ′ (K) + E ′ (K) + G ′ (K) , (2)
and

∂ f S(X, f ) = (C(K) -KC ′ (K)) + (E(K) -KE ′ (K)) + (G (K) -KG ′ (K)) . (3)
The demand for services X (p, f ) is directly related to the consumers' net surplus. More precisely,

X (p, f ) = arg max {S(X, f ) -pX} . (4)
It follows that, at equilibrium, the price exactly equates the marginal benefit of a trip for the marginal consumer:

p = ∂ X S(X, f ). (5)
Consequently, equation ( 2) rewrites

p = C ′ (K) + E ′ (K) + G ′ (K) . (6)
In other words, the ticket price should equate its total marginal cost, namely the sum of the marginal operational cost, the marginal environmental costs and the marginal cost for the airport and the traffic control authority. Observe that, although not explicit in the displayed equation ( 6) , the frequency f plays an important role in the determination of the equilibrium demand and price. To see that, consider the decomposition S (X, f ) = U (X) -νXT (f ) introduced above. Equation ( 5) rewrites

p = p + νT (f ) = U ′ (X) .
In other words, for any pair (p, f ) , the demand X is such that the marginal utility of a journey is exactly equal to the generalised price p beared by the users i.e. the sum of the ticket-price p and the time-costs νT (f ) expressed in monetary units. More generally, an increase of f can be shown to have a twofold impact on consumers. First, it increases their gross benefits as a higher frequency is nothing but a higher quality of services. Second, if this increase in frequency does not induce congestion hence further delays, it also decreases their travel time T (f ) , reinforcing further the induced benefits.

The marginal impact of f on the different costs is also twofold, with two opposite effects. On the one hand, it increases the costs since all of them are proportional to the number of flights. On the other hand, however, this first and obviously dominant effect is mitigated by the induced (long-term) decrease of the capacity K = X/f. It follows that the equation (3) can be rewritten as:

f ∂ ∂f S(X, f ) X = C(K) K -C ′ (K) + E(K) K -E ′ (K) + G (K) K -G ′ (K) . (7)
In other words, the marginal benefit of an increase in frequency as computed on a per-passenger basis, namely ∂ f [S(X, f )/X] , is directly proportional to the difference between average and marginal costs, for all sources of costs.

The optimal allocation as characterised by equations ( 6) and ( 7) does not appear to be a realistic situation since it does not require the consumers to pay for all the costs induced by the transportation system. If there are increasing returns to scale in the various dimensions of the transportation system3 , which appear to be a reasonable assumption, a marginal pricing scheme charges passengers below the average costs inflicted to society. The implementation of the first-best allocation builds necessarily on (possibly implicit) tranfers. 4 Increasing returns to scale in the environmental dimension means that, on a per-passenger basis, bigger aircrafts pollute less than (relativly) smaller aircrafts. A consequence is that E ′ (K) < E (K) /K; Hence at the first-best passengers are not required to compensate for the whole environmental externality they generate. Remark also that the implementation of the first-best allocation may rely on the existence of subsidies since the price may not be sufficient to cover the costs of the firm, the airport and the ATC services. Indeed, the overall profit of the industry is expressed as

π T = -f [(C(K) -KC ′ (K)) + (G(K) -KG ′ (K))] -F + E ′ (K) . (8)
Interestingly enough, the higher the magnitude of the environmental exernalities as measured by E ′ (K), the higher the price p hence the lower the necessary subsidies (if any) for the industry to financially break-even. Finally, note also that, whenever necessary, these subsidies are not necessarily directed to the firm. If transportation services are charged according to the first-best rule (6), whether the price p will allow the firm to break-even depends on the magnitude of both environmental, airport and ATC costs. Again, the higher these marginal costs, the higher the prices hence the more likely the firm will recoup operational costs C (k) and fixed costs F . Airport and ATC services would however run a deficit. Previous considerations are made by considering the case of a social planner that is able to control the whole industry and get access to subsidies whenever it appears to be desirable. We now shift to a more realistic situation where (i) the air transportation firm is a for profit compagny, (ii) airport and ATC services are independant from independant from the firm and (iii) the industry as a whole may not be subsidized. Airport and ATC services are charged to the firm by the means of two fees, τ x that is a per-passenger tax and τ f that is a per-aircraft tax. We assume that Airport and ATC services are state owned bodies so that taxes are set as to maximize the social welfare.

Implementing the first-best with a profit-maximising firm

Before we address the second-best problem mentioned above, we uncover the behaviour of a profit-maximizing firm in this environment. Given τ x and τ f , the profit of the firm writes

π(p, f ) = (p -τ x ) X (p, f ) -f (C (K) + τ f ) -F, (9)
where K stands for the capacity of the operated aircraft that is assumed to ajust in the long term 5 to write K = X/f . Assuming the problem to be convex, the profit-maximising pair (p, f ) is fully characterised by the system of F.O.C:

X (p, f ) + [p -C ′ (K) -τ x ] ∂X ∂p = 0, ( 10
) [p -C ′ (K) -τ x ] ∂X ∂f -(C (K) + τ f ) + X f C ′ (K) = 0. ( 11
)
After some manipulations, the system can be rewriten as:

p -C ′ (K) -τ x p = 1 ǫ Xp , (12)
-X ∂X ∂f / ∂X ∂p = C (K) + τ f -KC ′ (K) , (13)
where

ǫ Xp = p X (p, f ) -∂X ∂p
is the (absolute value of the) price-elasticity of demand. Both equations deserve a few comments. Equation ( 12) comes out as the "standard" profit-maximising markup in presence of taxation. Given the frequency f and the consequent long-term capacity K = X/f, the marginal cost for the firm of offering services to one additional passenger is C ′ (K) + τ x . The per-passenger earning are thus p -C ′ (K) -τ x and the benefits of a price increase have to be balanced with the resulting loss of passengers. Interestingly enough, this rule is not altered by the possibly complex interaction of frequency and demand. In order to interpret equation (13) , it is usefull to consider the implications of the consummer program as displayed in equation [START_REF] Carlsson | Airport Marginal Cost Pricing: Discussion and Application to Swedish Airports[END_REF] . By differentiating equation [START_REF] Dissler | Modulated airport charges against airport congestion : an economic way of regulation[END_REF] with respect to p and to f, and combining both expressions, one may indeed establish that:

- ∂X ∂f / ∂X ∂p = ∂ ∂f [∂ X S(X, f )] = ∂p ∂f .
In other word, the ratio measures the marginal impact of frequency on the equilibrium price. As a result equation ( 13) establishes that the frequency is thus set in such a way that the marginal benefits of an increase in the number of flights exactly equates the resulting net cost increase, namely the total cost of operating an aircraft C (K) + τ f minus the downshift due to the capacity adjustment effect KC ′ (K) . In order to make the link with the standard "value of time" concept, one can somewhat specify further the model to write the surplus S(X, f ) as the difference between the gross utility U (X) and the time costs X νT (f ) . In this specific case equation (13) rewrites

νT ′ (f ) = 1 X [C (K) + τ f -KC ′ (K)]
which makes plain the just invoqued argument.

Of interest is the fact that the profit-maximizing goal of the firm does not a priori forbid the government to reach the first-best allocation exhibited in the previous section. This can be done by setting the taxes τ x and τ f at the respective level:

τ x = E ′ (K) + G ′ (K) - C ′ (K) + E ′ (K) + G ′ (K) ǫ Xp ( 14
)
τ f = (r -1) [C(K) -KC ′ (K)] +r [E(K) -KE ′ (K) + G (K) -KG ′ (K)] ( 15
)
where

r = ∂ ∂f ∂S(X, f ) ∂X / ∂ ∂f S(X, f ) X
is the ratio of the marginal value of an increase in frequency for the marginal consumer over the marginal value of an increase in frequency for the average consumer.

Again, equations ( 14) and ( 15) merits some comments. Equation ( 14) highlights the existence of two origins of divergence between the price set by the firm and the price displayed in equation ( 6) that would implement the first-best allocation. First, the firm bears neither the environmental costs nor the airport and ATC costs. This explains the first two components of the tax that aim to induce the firm to internalize both cost sources. Second, since the firm is profit-maximising, it makes a positive markup while marginal pricing would be socially optimal. This explains the last term on the right handside of equation ( 14) . Notice that the tax τ x can be negative or positive. In order to reduce environmental damages and if the firms is required to pay for the airport and ATC services, the firm should obviously be taxed. This is the first effect. However, since a monopoly makes use of its market power to produce less than the socially desired quantity, implementing the first-best requires the firm to be subsidized. This is the second effect, a classical result in optimal tax theory. The magnitude of the tax (or subsidy) will depend on the intensity of both effects. However, if ǫ Xp ≥ 1, which has to be the case for the monopoly pricing formula [START_REF] Zhang | Airport charges, economic growth, and cost recovery[END_REF] to make sense, it is clear that taking into account environmental externalities, as well as airport and ATC services tends to decrease the extend to which air-transportation firms should be otherwise subsidized.

In order to interpret equation ( 15), two points have to be kept in mind. First, as already stated by [START_REF] Spence | Monopoly, quality and regulation[END_REF], there is a "divergence between private and social benefits". Indeed firms'decisions are based upon marginal values whereas socially optimum decisions are based on the average one. This explains the introduction of the factor r in the formula. Second, as already stated above, the firm ignores the impact of its decisions on environment and the burden it imposes on airport and ATC facilities. As a result, although part of the repercussion of frequency on consumers'welfare is taken into account by the firm (because it impacts on its profits), it should be given the correct incentives to take these costs into account. A reasonable assumption on r is that this factor is smaller than one 6 . That is, the marginal consumer can be expected to be the one with the lowest value of time, or at least with a value below the average one. By focusing on the marginal consumer, the firm under-estimates the value of frequency for travellers. In other words, the firm tends to over-estimate the costs of an increase in frequency. As a result, the tax τ f should be smaller than the social cost of a marginal increase in frequency. Whether it should be positive or negative depends again on the relative magnitude of the costs. If transportations services were inducing no costs but the operational costs beared by the firms, the firm should be subsidized in order to contemplate the observation of the socially optimum frequencies. However, if environmental costs as well as airport and ATC costs are taken into account, the socially optimal level of frequency comes out lower and the amount of subsidies becomes lower up to the point were it becomes positive, hence a tax.

Second best

They are several reasons for not implementing the first best. First, it may be politically difficult to justify the subsidization of the airline. Hence, a possible problem to consider could be the setting of optimal levels of taxes under nonnegativity constraints, either for both taxes, or globally. But the main reason is that the implementation of the optimal taxes, even in the "good" case of strong environmental externalities, may not allow the operator to cover its costs. The problem of the cost recovery is indeed the good one, as air navigation service providers (as well as many airports) are precisely constrainded to balance their budget in european countries. Practically, air traffic control operators set their tarifs proportionally to a unit rate which is computed in order to balance their budget.

The second-best problem then consists in the maximization of

W (X, f ) = S(X, f ) -f C t (K) -f E(K) , s.t.
i) the airline maximises its profit, ii) the operator balances its budget.

(

) 16
Condition i) is precisely described by the first-order optimality conditions of the airline ( 10) and [START_REF] Zhang | Concession revenue and optimal airport pricing[END_REF], while condition ii) writes

τ x X + τ f f ≥ F c + f C c (K) . (17)
Notice that the budget balance constraint (17) is writen in such a way that positive profit is allowed for the regulator. Indeed, it would be questionnable to impose the equality constraint if the first best achieves the full recovery of the ATC costs. In this case, depollution operations (...) could be financed.

We may now express the problem faced by the air control operator. The Lagrangian writes as follows

L(X, f, τ x , τ f , λ 1 , λ 2 , β) = W (X, f ) + λ 1 X [p + X ∂ X p -C ′ m (K) -τ x ] + λ 2 f [X ∂ f p -(C m (K) -KC ′ m (K)) -τ f ] + β [τ x X + τ f f -F c -f C c (K)] ,
(18) where the λ i 's are the Lagrangian multipliers related to the airline profit maximization conditions, while β is associated with the budget constraint of the ANSP. Notice that constraints are expressed in such a way that Lagrangian multipliers are homogeneous to each other. The optimality equations with respect to the taxes give (-λ 1 + β)X = 0 ,

(-λ 2 + β)f = 0 (20)

Assuming that operating the line is profitable, and hence that X = 0 and f = 0, we get λ 1 = λ 2 = β. The optimality equations in X and f then writes

ǫ + β(ǫ -3) ǫ p + βf ∂p ∂f + β X 2 ∂ 2 p ∂X 2 + f X ∂ 2 p ∂X∂f = E ′ (K) + (1 + β)C ′ t (K) , (21)
(r + 2β)X ∂p ∂f + β X 2 ∂ 2 p ∂X∂f + f X ∂ 2 p ∂f 2 = (1 + β) [C t (K) -KC ′ t (K)]
+ [E(K) -KE ′ (K)] .

(22) Further computations using the first order conditions yield the expression of the taxes variations around the first best taxes τ *

x and τ * f (though these ones have to be evaluated for the second best quantities f and X). We get

r + 2β β τ f -τ * f = 1 r [(r -2)∂ 2 (C t (K) + E(K)) -r∂ 2 E(K)] -X 2 ∂ 2 Xf p + f X∂ 2 f f p , (23)
ǫ + β(ǫ -3)

β(ǫ -1) (τ x -τ * x ) = 1 ǫ [3∂ 1 (C t (K) + E(K)) -ǫ∂ 1 E(K)] - 1 K ∂ 2 (C t (K) + E(K)) r + τ f -τ * f -X 2 ∂ 2 XX p + f X∂ 2 f X p , (24)
where, by commodity, the following notations are adopted

∂ 1 C = C ′ (K) , ∂ 2 C = C(K) -KC ′ (K) . (25)
Given the sign of the budget constraint, the Kuhn and Tucker optimality conditions ensure that β is non-negative. If β is null, then we recover the first best taxes, an obvious result since this condition reflects the fact that the budget constraint is satisfied at the social optimum. If β is strictly positive, then we observe that none of the taxes will in general maintain its first-best value. Taking appart the second derivatives of the price appearing in these equations, we observe that the "frequency-related" tax is likely (say for r small enough and keeping in mind that the tax τ * f in (23) has to be computed on the second best values of (X, f )) to be smaller than the first best one.

However, its is true that estimating the marginal environmental costs is an hard task : part of the pollutants is emitted in altitude, and is thus not taken into account in studies related to urban pollution, though contributing to some extent to global warming and acid rains...

This demand can be derived in many different ways. If U (X) denotes the gross utility derived by the representative agent from X journeys and ν denotes its value of time, it writesX (p, f ) = arg max X S (X) -p + ν 2f X ,where 1/ (2f ) stands for the average waiting-time of the representative consumer. This representation however abstract from the fact that the utility U (X) derived from a journey is usually related to the value of time of the agent. Our model is however more general than that and we shall use these specification for illustrative purpose only.

The operational, the environmental and the airport and control dimensions.

Transfers are implicit if the only costs that are not covered by the ticket are the environmental costs. Despite there are no monetary transfers, the whole society has to bear the negative externalities caused by the sole users of the air-transportation services. Transfers are explicit if the firm, the airports or the ATM services receive subsidies.

Nero and Black [6] suggest this may even be achieved in the short term thanks to an "active and competitive rental service".

This is indeed the case if a CES utility function is used (e.g. in[START_REF] Brueckner | Airport Congestion When Carriers Have Market Power[END_REF] and[START_REF] Dissler | Congestion aroportuaire et tarification[END_REF]).