Julien Chevallier
email: jchevall@u-paris10.fr

Intertemporal Emissions Trading and Market Power: A Dominant Firm with Competitive Fringe Model

Keywords: C73, L11, Q52 Emissions Trading, Banking, Borrowing, Market Power

In international emissions trading schemes such as the Kyoto Protocol and the European Union Emissions Trading Scheme, the suboptimal negotiation of the cap with respect to total pollution minimization leads us to critically examine the proposition that generous allocation of grandfathered permits by the regulator based on recent emissions might pave the way for dominant positions. Stemming from this politically given market imperfection, this chapter develops a differential Stackelberg game with two types of non cooperative agents: a large potentially dominant agent, and a competitive fringe whose size are exogenously determined. The strategic interactions are modeled on an intra-industry permits markets where agents can freely bank and borrow permits. This chapter contributes to the debate on initial permits allocation and market power by focusing on the effects of allowing banking and borrowing. A documented appraisal on whether or not such provisions should be included is frequently overlooked by the debate to introduce the permits market itself among other environmental regulation tools.

Numerical simulations provide a quantitative illustration of the results obtained.

Introduction

What happens on a tradable permits market when distortions occur as a consequence of the initial allocation? Whereas [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF] contributed first to this debate by demonstrating the non neutrality of permits allocation for an agent able to exert market power2 in a static context and concerning the spatial exchange of permits only, this chapter addresses the critical aspect of initial permits allocation in a dynamic context and concerning intertemporal emissions trading.

Theoretical analyses remain scarce in this domain, even if the properties of banking (i.e. the ability to stock permits for future use) and borrowing (i.e. the ability to borrow permits from future periods) have been detailed. In a continuous time model under certainty, [START_REF] Rubin | A model of Intertemporal Emission Trading, Banking, and Borrowing[END_REF] shows that an intertemporal equilibrium exists on a permits market from the viewpoint of the regulator and the firm, and that banking and borrowing allow firms to smooth emissions. Under uncertainty, [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] shows the permits price may rise at a rate less than the discount rate and new public information may cause jumps in the price and emissions paths, among other major contributions.

This article builds on the intertemporal emissions trading literature with market imperfections. It aims at filling the gap in the literature between the pros and cons of authorizing banking and borrowing in permit trading programs, a topic which is typically not enough debated when deciding to adopt such an environmental regulation system. Against this background, it attempts to shed some light on the ability of a large agent to move dynamic markets when permits are grandfathered. Liski and Montero (2005b) study the effect of market power on the equilibrium of a permits market by introducing a large potentially dominant and a competitive fringe. Based upon two cases, their analysis reveals first that the large agent might manipulate the market by banking allowances when it owns all the stock of permits and second when the fringe receives all the stock of permits, the large agent has an incentive to exchange permits at the competitive price and to build a permits bank for the next period. While previous papers restricted their analysis to banking only3 , both banking and borrowing are allowed without restrictions in a continuous time setting.

The model brings to the regulatory economics literature a realistic description of relationships between agents on a tradable permits market with information asymmetry. As Liski and Montero (2005b) did not impose a particular game structure, I adopt a Stackelberg game structure that allows dealing with strategic interactions between two types of agents: a leader with an informational advantage associated to a large agent, and a follower associated to a competitive fringe.

The market imperfection arises from the free distribution of permits on the basis of past emissions, while the product market is assumed to remain competitive 4 . I explicitly include the Hotelling conditions 5 that must apply if permits are considered as an exhaustible resource.

The chapter unfolds as follows: first, I describe the institutional framework of current permit trading programs, namely the Kyoto Protocol (KP) and the European Union Emissions Trading Scheme (EU ETS); second I develop the Stackelberg game and derive an expression for market power; third I compute numerical simulations for the price distortion condition.

Institutional Framework

In this section, I describe how the model hinges on critical design issues of existing international emissions trading schemes, namely the KP and the EU ETS. I also attempt to provide a balanced picture of the EU ETS and KP market power concerns.

The Kyoto Protocol

The Kyoto Protocol entered into force on February 2005 following the ratification of Iceland, and which aims at reducing the emissions of six greenhouse gases (GHG) considered as the main cause of climate change. Among the Members of Annex B, these agreements include CO 2 emissions reductions for 38 industrialized countries, with a global reduction of CO 2 emissions by 5,2%. 174 countries, Australia being the latest on December 3, 2007, have ratified the Protocol, with the notorious exception of the United States. The first commitment period of the Kyoto Protocol goes from January 1, 2008 to December 31, 2012.

The question of the Kyoto Protocol as an "unfinished business" is often evoked. Very heterogeneous sectors were included under the same regulation, which could be detrimental to find the right method to allocate permits depending on price elasticities between sectors.

The intra-industry structure adopted in this chapter may be seen as a simplification of the KP. Yet it may propose useful policy recommendations when dealing with such an international scheme. In what follows, I focus on the negotiation phase, the special case of Russia and the prospective use of banking and borrowing.

Negotiation phase

4 For the distinction between permits market and industry structure imperfections, see [START_REF] Sartzetakis | Tradeable Emission Permits Regulations in the Presence of Imperfectly Competitive Product Markets: Welfare Implications[END_REF] and [START_REF] Sartzetakis | On the Efficiency of Competitive Markets for Emission Permits[END_REF]. 5 Namely, the exhaustion and terminal conditions.

Since there exists no historical data for carbon emission reduction cost, it may prove particularly difficult to induce a cost-effective allocation of the initial quotas to participating countries. In the context of the KP, the case of countries supplied with allocations in excess of their actual needs has been coined as Hot Air in the literature 6 . The distribution of a large number of permits to Former Soviet Union (FSU) and Eastern Europe countries (Russia, the Ukraine forming two thirds) may be seen indeed as an imperfection of the KP, as those countries were given generous allocations to foster agreement during the first phase (2008embodies the notions of "Burden Sharing" or "common but differentiated responsibilities" 10 attached to the KP, whereby developed countries agree to spend a higher income share on fighting climate change than developing countries 11 .

Those conflicting views undermine the negotiation of the cap, which is fixed at a suboptimal level compared to what would be needed to minimize the total damage to the environment. Greenhouse gases (GHG) emission targets under the KP represent a mere 5% 6 See [START_REF] Baron | Market Power and Market Access in International GHG Emission Trading[END_REF], [START_REF] Burniaux | How important is market power in achieving Kyoto?: An assessment based on the GREEN model[END_REF], [START_REF] Bernard | Russia's Role in the Kyoto Protocol[END_REF], [START_REF] Bohringer | Market Power and Hot Air in International Emissions Trading: the Impact of the US Withdrawal from the Kyoto Protocol[END_REF]Loschel (2003), Holtsmark (2003). 7 One could symmetrically evoke the case of monopsony power whereby large buyers would lower the permit price from its level under perfect competition. Yet outlooks for the KP do not match this perspective. Indeed, market power is more likely to come from sellers than from buyers. 8 Such negotiation with Russia was determinant for the KP to enter into force on February 16, 2005. 9 See [START_REF] Guesnerie | The design of post-Kyoto climate schemes : an introductory analytical assessment[END_REF]. 10 See Muller (2002). 11 Note that the implicit assumptions of the existence of such an Environmental Kuznets Curve (the environment is a superior good and environmental regulation becomes stricter through time at higher levels of GDP per capita) is left out of the debate. reduction below 1990 levels. Now, if early movers like EU countries are willing to ratchet down the cap, little progress can be achieved without luring in major players like the USA, India and China. Thus, many difficulties arise to pierce the "veil of uncertainty" around international negotiation12 . Uncertainty also affects the nature and the size of individual market participants. As [START_REF] Klepper | Trading Hot Air. The Influence of Permit Allocation Rules, Market Power and the US Withdrawal from the Kyoto Protocol[END_REF] put it:"The Kyoto Protocol and its related decisions do not explicitly state who is actually supposed to be trading. Probably we will observe both government and firm trading. Under the former, market power might indeed become a relevant issue". Therefore, the risk of market power is higher if governments are trading large amounts of permits in a centralised manner.

The fact that the creation of a permits market gives some countries the opportunity to draw a financial advantage without a direct environmental gain (i.e. in the absence of effective emissions abatement) may be puzzling. Yet as stated by [START_REF] Maeda | The Emergence of Market Power in Emision RightsMarkets: the Role of Initial Permit Distribution[END_REF], "[this debate] seems misguided because it focuses on the political importance of the issue, rather than addressing it from an economic perspective." That is why in this chapter I investigate how permit price manipulation strategies may entail additional economic costs to achieve the same level of abatement as under perfect competition.

Overall, the hypothesis that generous allocations that broaden the scope of a cap-andtrade program might also give birth to dominant positions shall not be neglected. This leads me to comment in depth the case of Russia.

Will Russia be a net seller of permits?

Russia seems the best example to investigate potential market power within the KP according to [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF]: "Given the collapse of its emissions in the course of its economic transition, Russia is the country with by far the largest potential surplus of emission allowances for sale under the Kyoto international trading mechanisms. It is also generally considered to be the country with the greatest potential for continuing emission-reducing improvements in energy efficiency. Indeed, in the first commitment period under the Kyoto Protocol it could be described as the Saudi Arabia of the emerging carbon market, with the potential to try to manipulate the market through strategic decisions as to when and how it releases it surplus -if there are buyers willing to deal."

Empirical evidence gathered by [START_REF] Grubb | Russian Energy and CO2 Emission Prospects: Evidence from Domestic Analyses and International Comparisons[END_REF], Liski and Montero (2005a) 13 and [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF] suggests Russia will be a net seller of allowances during the first phase of the KP. Different projections for Russian CO 2 emissions and surplus are detailed in Tables 1 and2.

Source

Year [START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF] and [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF].

The key finding in Table 1 is that under all scenarios Russia would meet its Kyoto targets, as its CO 2 emissions projections consistently hit below 1990 levels. The room for interpretation of Table 2 is limited by the wide variation in surplus estimates with a lowest value 14 Scenario with Annex B Trading. 15 Scenario assuming trading without the U.S. 16 International Energy Agency. 17 Cambridge Economic Policy Associates. Scenario with a 2% energy intensity reduction. 18 In million tonnes of carbon equivalent (Mt Ce). 19 Mt CO 2 20 Scenario assuming trading without the U.S. 21 Adapted from [START_REF] Korppoo | Russia and the Kyoto Protocol: Opportunities and Challenges[END_REF]. 22 United Nations Framework Convention on Climate Change. 23 Cambridge Economic Policy Associates. Scenario with a 2% energy intensity reduction. 24 Mt CO 2 of 111 Mt CO 2 found by [START_REF] Decaux | Analysis of Post-Kyoto CO2 Emissions Trading Using Marginal Abatement Curves[END_REF] and, as expressed above, by the current absence of clearly defined international trading rules to monetize such a surplus.

Further projections regarding Russia's own energy demand after the first period of the KP are needed to determine whether Russian industrial firms might actually benefit from their Hot Air. Besides, the treatment of Hot Air needs to be supplemented by considerations about a potential "leakage" 25 of emissions to other regions that are not covered by the KP and by additional allowances from the Clean Development Mechanism or Joint Implementation 26 that might compete with Russian allowances 27 .

Prospective use of banking and borrowing in the KP

This section offers a description of the possible use of banking borrowing in the KP. On the one hand, provisions on banking are explained by [START_REF] Klepper | Trading Hot Air. The Influence of Permit Allocation Rules, Market Power and the US Withdrawal from the Kyoto Protocol[END_REF]: "Assigned Amount Units (AAUs) resulting from the Kyoto commitment can be banked without a time constraint. Credits from Joint Implementation (JI) or Clean Development Mechanism (CDM) can be banked up to a limit of, respectively, 2.5% and 5% of a Party's initial assigned amount.

Sink credits cannot be banked".

On the other hand, implicit provisions on borrowing may be found in the United Nations Framework Convention on Climate Change (UNFCCC ( 2000)) report 28 . As explained by [START_REF] Newell | Managing Permit Markets to Stabilize Prices[END_REF]: "International climate policy discussions have implicitly included borrowing within possible consequences for noncompliance under the Kyoto Protocol, through the payback of excess tons with a penalty (i.e., interest)." This penalty could be fixed to 40% of additional emissions reduction for the next period of the Kyoto Protocol despite uncertainties regarding the enforcement of this particular provision. This question is addressed in depth by [START_REF] Alberola | European Carbon Prices and Banking Restrictions: Evidence from Phase I (2005-2007)[END_REF].

The European Union Emissions Trading Scheme

The European Union Emissions Trading Scheme (EU ETS) has been created on January 1, 2005 to reduce by 8% CO 2 emissions in the European Union by 2012, relative to 1990 emissions levels. The introduction of a tradable permits market has been decided to help Member States in achieving their targets in the Kyoto Protocol. 25 As documented by [START_REF] Decaux | Analysis of Post-Kyoto CO2 Emissions Trading Using Marginal Abatement Curves[END_REF], the net effect of potential market power associated to Hot Air depends on the compensating emissions that might "leak" to regions unconstrained by the KP. 26 Conservative estimates range from a lower bound of 800Mt CO 2 according to [START_REF] Ukerc | Implementing Kyoto -Does It All Add Up[END_REF] to an upper bound of 1,000Mt CO 2 according to Point Carbon. 27 As [START_REF] Baron | Market Power and Market Access in International GHG Emission Trading[END_REF] put it, trading based on projects may reduce the risk of market power or shift it to other regions which already host a large number of CDM projects like China. 28 Paragraph II.XV.

In what follows, I draw comments on two critical aspects of the EU ETS. First, I deal with possible design flaws in the allocation of permits that might pave the way for dominant positions during the first phase. Second, I provide an overview of the prospective use of banking and borrowing.

Over-allocation or relative success?

The EU ETS gently constrains emissions (8% reduction for EU-15) so as to start with a low carbon price. Yet the debate has shifted toward a possible over-allocation of permits during the first phase. The production decisions of private actors are under scrutiny: do permits surpluses constitute a relative success (i.e. firms have reduced their emissions above projected levels) or an imperfection in the design of the system?

The While it is not our goal to comment on the structure of the European carbon market here, it seems interesting to look at the possible surplus those countries were endowed with. It reflects a wide variety of cases among market participants as a bulk of countries (Poland, France, Finland, Denmark, Slovakia, Hungary) was able to build a permits surplus above 15%, while others (UK, Ireland) are short of permits by more than 15%.

The biggest player, Germany does not seem to be in a position to exert market power with less than 5% excess allowances. The surplus of Poland need not be overstated either since the use of 10% of allowances is missing in the CITL. Portugal, Greece, Italy and Austria form a group of countries where the regulator strived to allocate optimally. On the contrary, stricter emissions reductions were enforced in the case of the UK, Ireland and Spain.

31 Apart from the EU ETS, there is a need to be cautious here with the notions of "over allocating" and conversely "under-allocating" permits depending on the country. Their meaning depends on the reference point (business as usual plus some abatement for instance). If other trading schemes implement per capita distribution for instance, it may appear less relevant to talk about "over allocation". each National Allocation Plan (NAP). [START_REF] Godard | L'allocation initiale des quotas d'émission de CO 2 aux entreprises à la lumière de l'analyse économique[END_REF] and [START_REF] Godard | Politique de l'effet de serre: une évaluation du plan francais de quotas de CO 2[END_REF] describe the logic behind the French NAP when allocating shares of recent emissions baselines: non-electric utilities were supplied with their projected need in permits, while electric utilities were more constrained. This situation may be justified by the perceived abatement potential of the electricity industry, but it reveals overall the necessary arbitrages to be made due to sector heterogeneity and a stringent cap.

As long as governments continue to allocate allowances to existing facilities based on historic emissions, the scheme is flawed by a perverse "updating" incentive [START_REF] Grubb | Allocation and Competitiveness in the EU Emissions Trading Scheme: Policy Overview[END_REF]). Firms have indeed an incentive to delay early action in abatement technologies since higher emissions today will be rewarded with bigger allocations in future periods. [START_REF] Ellerman | Over-Allocation or Abatement? A Preliminary Analysis of the EU ETS Based on 2005 Emissions Data[END_REF] provide a first empirical assessment of the EU ETS allocation process based on 2005 emissions data. They estimate a slight over-allocation of 4% during the first period of allocation, while there are strong signs that some emissions abatement measures have occurred.

But the analysis is not straightforward since "a long position is not per se evidence of overallocation". The difference between 2005 allocation and verified emissions suggests too many allowances were allocated, but the benchmark against which this conclusion is reached may be biased by insufficient data reporting on emissions before 2005 and by a lack of comparability at the EU-wide level. Firms may also be long because of differences in marginal abatement costs or in expectations (regarding economic activity, energy prices, etc.) under uncertainty.

Overall A more detailed analysis of the consequences of banning banking from one period to another on the permits price and firms behaviour may be found in [START_REF] Alberola | European Carbon Prices and Banking Restrictions: Evidence from Phase I (2005-2007)[END_REF].

Moreover, preliminary analyses of the 2005-2007 data concerning the extent of the use of banking in the EU ETS may be found in [START_REF] Ellerman | Compliance Behavior in the EU ETS: Cross Border Trading, Banking and Borrowing[END_REF] and [START_REF] Chevallier | Bankable Pollution Permits under Uncertainty and Optimal Risk-Management Rules: Theory and Empirical Evidence[END_REF].

This section provided an overview of two major tradable permits market along with their allocation methodology. It revealed a wide range of opportunities for strategic behaviour in 32 The iREP is monitored by the Minister of the Environment and displays public information at www.pollutionsindustrielles.ecologie.gouv.fr/IREP/index.php 33 Note however that [START_REF] Convery | Market and Price Developments in the European Union Emissions Trading Scheme[END_REF] compute a very low Herfindahl-Hirschman Index for the electricity-generating sector which indicates an unconcentrated market. Thus, market observers are overall less wary of market power concerns in the EU ETS than in the KP, even if the Directive explicitly allows installations to "pool" allowances at the sector level. 34 Permits that were bought cannot be banked. Besides, only "green" firms that have effectively reduced emissions may bank allowances in Poland.

the design of international permit trading regimes. The presence of countries with large permits holdings increases the probability of price manipulation and the risk of efficiency loss in the allocation of abatement efforts between countries. This background information is used as the basis for the modelling of a differential game with hierarchical play in the chapter.

Model

This section details the features of the model. First, I explain the design of the cap-andtrade program. Second, I examine the industry and information structures. Third, I define an intertemporal emissions trading constraint. Fourth, I express the Hotelling conditions. Finally, I explicit the properties of the abatement cost function.

Design of the Cap-and-Trade Program

The regulator sets a cap E on emissions of a given pollutant that corresponds to a specific environmental goal. The fix endowment is therefore exogenous to the model, and may be broken down into individual permits allocation i e mandatory for each agent i.

Agents are further decomposed into two types:

1. agent {i = 1} is a large polluting agent, who is initially allocated a large number of permits;

2. agents {i = 2,. . .,N} aggregate many small polluting agent, who are assumed to be comparatively smaller permits-holders, and belong to the competitive fringe.

An agent may be a country, a firm or a cartel 35 . The competitive market price is determined by fringe agents' abatement costs.

The market imperfection arises during the free permits distribution on the basis of recent emissions 36 . The premise of the paper is that the large agent may be able to exert market power.

I do not include the effects of allowing a safety valve 37 in the model.

Industry Structure

35 For instance, within the KP permits may be exchanged party-to-party, but also firm-to-firm if Annex B members delegate this ability to private actors. In the third case, collusive behaviours may appear either between parties or between firms. (Liski and Montero (2005b)). 36 See [START_REF] Ellerman | Shortage, Inter-period Pricing, and Banking[END_REF] for a review concerning the use of projections, benchmarking and intensity targets. While it is beyond the scope of this paper to study the relative merits of grandfathering and auctioning, theoretical analyses stress the superiority of auctioning as in [START_REF] Jouvet | Optimal Growth with Pollution: How to Use Pollution Permits[END_REF]. In the view of the Public Choice Theory, free allocation of permits may also be seen by some firms as a means to extract more permits as a scarcity rent, and therefore lobbying takes place. But it is also imposes liabilities on firms that will be reflected in their balance sheets. As highlighted by [START_REF] Raymond | Private Rights in Public Resources: Equity and Property Allocation in Market-Based Environmental Policy[END_REF], initial permits allocation reveals social norms embedded by newly created permits. The free distribution of permits may be seen as an entitlement over an environmental resource.

As conceptions evolve and auctioning might become predominant, the question arises whether the probability of achieving a dominant position will increase or decrease. 37 A safety valve may be defined as an hybrid instrument to limit the cost of capping emissions at some target level whereby the regulator offers to sell permits in whatever quantity at a pre-determined price.

This partial equilibrium model features an intra-industry permit market in a single good economy. I tend to neglect the interaction with the output market.

Market power is defined by [START_REF] Burniaux | How important is market power in achieving Kyoto?: An assessment based on the GREEN model[END_REF] as "the capacity of a firm / country to influence the transaction price of traded permits" (referred to as a "cost minimising manipulation"). Thus, I do not address exclusionary manipulation strategies 38 that occur when the dominant agent uses its market power on the permits market to raise entry barriers or exclude agents on the output market.

Information Structure

Intertemporal Emissions Trading

Agents may bank and borrow permits without restrictions. Let B i (t) be the permits bank, with B i (t) > 0 in case of banking, and Bi(t) < 0 in case of borrowing.

Any change in the permits bank is equal to the difference between ) (t e i and e i (t), respectively agent's i permits allocation and his emission level at time t. The banking and borrowing constraint may be written as:

) ( ) ( ) ( t e t e t B i i i - = & (1)
with B i (0) = 0 as an initial condition.

Hotelling Conditions

Notwithstanding differences between a permit and an exhaustible resource 41 , it is assumed in the literature that the Hotelling conditions for exhaustible resources must apply on a permits market. Consequently, the terminal and exhaustion conditions are detailed below. 38 See [START_REF] Misiolek | Exclusionary Manipulation of Markets for Pollution Rights[END_REF]. 39 See [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF] for an overview of differential games. 40 For instance, agents do not incur information costs. 41 According to [START_REF] Liski | On Pollution Permit Banking and Market Power[END_REF], the following differences may be highlighted. First, in a permits market with banking, the market may remain after the exhaustion of the bank; while the market of a non-renewable resource

Terminal Condition

Let [0, T] be the continuous time planning horizon 42 . At time T, cumulated emissions must be equal to the sum of each agent's depollution objective and therefore to the global cap E set by the regulator 43 :

∑ ∫ ∑ = = = = N i i T N i i E e dt t e 1 0 1
) (

(2)

Exhaustion Condition

At time T, there is no more permit in the bank (either stocked or borrowed):

∑ = = N i i T B 1 0 ) ( (3)
Those conditions ensure that agents gradually meet their depollution objective so that the marginal cost of depollution is equalized in present value over the time period, and the permits bank clears in the end.

There is typically a truncation problem at the end of the period:

• if B i (T) > 0, surplus allowances are worthless and agents are wasting permits;

• if B i (T) < 0, agents need to pay a penalty 44 .

Abatement Cost Function

Let C i [e i (t)] be the abatement cost function 45 incurred by agent i in order to comply with his permits allocation i e . C i [e i (t)] is defined on ℜ → ℜ continuous and is of class C 2 [0, T], i.e. twice continuously differentiable. The classical assumption 46 of strictly increasing abatement costs leads C i [e i (t)] to be convex, with

C' i [e i (t)] < 0 and C'' i [e i (t)] > 0. I can set C i [e i (0)] = 0.
Agent's i marginal abatement costs (MAC) are associated with a one-unit reduction from his emission level e i at time t, and are noted -C' i [e i (t)] > 0. At the equilibrium of a permits vanishes after the last unit extraction. Second, permits extraction and storage costs are equal to zero; while those costs are generally positive for a non-renewable resource. Third, the demand for an extra permit usually comes from a derived demand of other firms that also hold permits; while the demand for an extra unit of a non-renewable resource comes more often from a derived demand of another actor (e.g., a consumer). 42 This planning period seems appropriate for a theoretical study of intertemporal emissions trading. Alternative time settings including distinct phases may be found in [START_REF] Montero | Explaining low sulfur dioxide allowance prices: the effect of expectation errors and irreversibility[END_REF], [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] or [START_REF] Ellerman | The Temporal Efficiency of SO 2 Emissions Trading[END_REF], but they reflect the specific requirements of the Acid Rain Program (USA). 43 See also [START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF]. 44 For instance, the penalty is equal to 40€ and 100€ per unit plus a compensating allowance for the first two periods of the EU ETS, and can add up to 40% of additional emissions during the first period of the Kyoto Protocol. 45 Compared to a situation where profits are unconstrained, abatement costs appear in order to meet the emission cap. 46 Stated first by [START_REF] Montgomery | Markets in Licenses and Efficient Pollution Control Programs[END_REF]. The conditions given by [START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF] include the output q(t) where C i [q i (t), e i (t), t] is strongly convex with C' i [q i (t)] > 0 and C'' i [q i (t)] > 0. Properties of non-convex abatement cost functions may be found in [START_REF] Godby | Market Power and Emissions Trading: Theory and Laboratory Results[END_REF]. market in a static framework 47 , price-taking agents adjust emissions until the aggregated MAC is equal to the price P at time t:

)] ( [ ' t e C P i i t - = (4)

Stackelberg Game

In what follows, the dominant agent sets the price at the level which corresponds to the maximization of the difference between revenues from permits sales and its abatement costs. All other agents behave as price takers, i.e. they minimize their abatement and trading costs given the permit price set by the dominant agent. I want to evaluate how the dominant agent will set the permit price higher and abate less (or sell fewer permits) compared to the competitive solution. This differential game with hierarchical play is solved by backward induction.

Fringe Agents' Reaction Function

The first step of the game consists in forming the strategy of the fringe. Fringe agents choose their optimal emissions level according to the possibility to bank and borrow permits in constraint (1). The cost minimization program may be written as follows:

0 )] 0 ( [ 0 ) 0 ( ) ( ) ( ) ( )]} ( ) ( )[ ( )] ( [ { min 0 = = - = - + ∫ - i i i i i i T i i i i

Behaviour of the Dominant Agent

In the second step of the game, I use the first-order conditions of the fringe problem as the constraints in the leader's problem. The large agent adjusts strategically his optimal emissions levels according to its initial allocation 1 e as expressed by (2) and the banking borrowing constraint (1). The cost minimization program for agent {i = 1} is: 47 See [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF].

0 ) 0 ( 0 ) 0 ( ) ( ) ( ) ( ) ( ) ( )]} ( ) ( [ )] ( [ { min 1 1 0 0 2 1 1 1 1 0 1 1 1 1 1 = = + = - = - + ∫ ∫ ∑ ∫ = - C B
] ) ( 1 [ ) ( )] ( [ ' 2 1 1 ∑ = + + = - N i i i t e t P t e C ε (5)
The large agent's MAC is equal to the competitive permits price plus an element of price distortion i ε defined as fringe agents' elasticity:

i i i i i i i dC dC e C e C ' ) ( ' ) ( ' ' = = ε (6)
Thus, permits price manipulation results in higher total abatement costs than under perfect competition. Market power is function of fringe agents' elasticity and of the large agent's number of permits:

∑ = - = = N i i i i t e t e t e 2 1 1 )] ( ) ( [ ) ( ε ε (7)
Due to the convexity assumption, fringe agents' elasticity is negative, and reveals the possibility for the leader to affect negatively fringe agents' behaviour. The large agent's MAC is lower than under perfect competition. Since he enjoys a dominant position and has the ability to influence the permits price, the large agent may be characterized overall as a net gainer and fringe agents as net losers.

In either case where both agents have a net banking or borrowing position in terminal period, the large agent is able to affect negatively fringe agent's MAC through the number of permits he holds in excess of his emissions. This condition holds when both agents clear their permits bank in terminal period, the large agent is still able to affect negatively fringe agent's MAC. I have therefore confirmed the possibility of strategic manipulation for both forms of fringe agents' reaction function.

In the next section, I provide numerical simulations to illustrate these results. 48 See [START_REF] Chevallier | Strategic Manipulation on Emissions Trading Banking Program with Fixed Horizon[END_REF] for the formal resolution.

Numerical Simulations

This section attempts to provide quantitative estimates of the degree of price distortions induced by dominant behaviour on existing emissions trading schemes. More precisely, I use data on marginal abatement cost curves given by [START_REF] Decaux | Analysis of Post-Kyoto CO2 Emissions Trading Using Marginal Abatement Curves[END_REF] using the MIT EPPA model, and [START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF] using the world energy system model POLES (Criqui et al., 1996) to simulate the effect of the price distortion condition isolated in equations ( 5) to ( 7).

On the one hand, the MAC cost curves given by Decaux and Ellerman (1998) have the following functional form:

X X Y β α + = 2 (8)
where Y is the marginal cost of carbon in 1985 US$, X is the amount of abatement in million metric tons of carbon (Mton), α and β are parameters.

On the other hand, [START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF] retain the following specification:

β α ) ( X Y = (9)
where Y is the marginal abatement cost, X is the amount of abatement, α and β are parameters.

Several scenarios may be used in order to derive numerical estimates of the market power condition. In what follows, I simulate the situation where Foreign Soviet Union countries form a cartel, and attempt to distort prices based on the Hot Air created by the political market imperfection at the negotiation stage of the Kyoto Protocol.

The values given for all parameters are re-called in Tables 3 and4. Plugging these values in the price distortion condition (5), I get: 5) to (7).

From Tables 4 and5, I obtain quantitative estimates on the degree of price distortions induced by the Stackelberg game. It is indeed striking to observe that, based on data by Decaux and Ellerman (1998), price distortions and efficiency losses based on the fringe agents' elasticities and the large agent's permit endowment range from 13% to 35% in the context of the Hot Air discussion underlying the Kyoto Protocol. These estimates are even wider based on data by [START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF], ranging from 33% to 229%. Notwithstanding the presence of outliers in our numerical simulations, these results globally illustrate the risk of market power and associated economic inefficiencies due to the initial allocation of emission rights, as pointed out by [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF].

However, the spectre of a large agent achieving a market power position may be averted by a careful design of the cap-and-trade program. As illustrated by the ongoing debate on the EU ETS NAPs, there is a need for further research to assess the best solution to allocate permits efficiently (be it through output-based methods, benchmarking, minimum price auctioning, etc.).

Conclusion

The description of the institutional framework on which the model hinges provided a balanced picture of market power concerns in existing international emissions trading schemes.

As for the Kyoto Protocol, trading rules in the making and the key role played by projections preclude from reaching a definitive conclusion, but our analysis reveals the possibility to move an international permits market in a dynamic context. A global conclusion concerning EU ETS market power concerns gears towards a prudent approach: if some firms have received more permits than projected, they might very well end up with a shortage of permits at the end of the first period because of an increase in emissions. The EU Commission is especially careful during the validation of NAPs about their stringency and to the fact that there will be no ex-post adjustment. Still, there are strong signs of concentration at the installation level. Both schemes allow full banking and borrowing intra-period.

To capture the distortions induced by initial allocation, I introduce a differential game where agents differ in terms of their exogenous permits endowment and adopt a Markovian strategy. The main result consists in a price distortion condition based on fringe agents' elasticity and the large agent's permits endowment that explains how the large agent is able to affect negatively fringe agents' marginal abatement costs. In both cases where agents either compensate their net banking/borrowing positions or clear their permits bank at the end of the period, it is possible to identify net losers (i.e., fringe agents) and a net gainer (i.e., the large agent) as the large agent benefits from a lower marginal abatement cost than under perfect competition.

Since the price set by the dominant agent directly depends on the amount of permits initially allocated to that agent, this paper contributes to the link between distributional aspects and overall efficiency of tradable permits markets. It extends [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF]'s analysis on market power in a dynamic framework and builds upon Liski and Montero (2005b) and [START_REF] Liski | On Pollution Permit Banking and Market Power[END_REF] by modelling strategic interactions after a Stackelberg game and providing a full characterization of effects of unrestricted banking and borrowing. The numerical simulations computed for the price distortion condition based on data by [START_REF] Decaux | Analysis of Post-Kyoto CO2 Emissions Trading Using Marginal Abatement Curves[END_REF] and [START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF] provide a quantitative illustration of the results obtained.

They also reveal the magnitude of the economic inefficiencies induced by the initial allocation of emission rights.

But the spectre of market power need not be raised if the cap-and-trade program appears properly designed. The negotiation process of each NAP at the Member State level is typically an example of a manipulable rule whereby industries may conduct lobbying activities to extract more permits as a monopoly rent. With reference to the debate "rules vs. discretion" in monetary economics, this unhealthy lobbying by major industries calls for further research to ascertain the conditions under which it would be optimal to delegate the determination of the cap and the distribution of permits to an independent agency [START_REF] Helm | Credible Carbon Policy[END_REF], [START_REF] Grubb | Allocation and Competitiveness in the EU Emissions Trading Scheme: Policy Overview[END_REF]).

The model could be extended by the adoption of an intertemporal trading ratio specific to borrowing as discussed by [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF] 49 , allowing for a better grasp of the possibilities offered by intertemporal emissions trading. It may also be interesting to look at another source of heterogeneity between agents, for instance based on their emissions reduction function.

Figure 1 :Figure 3 :

 13 Figure 1: EU ETS National Allocation Plans -Phase 1 (2005-2007) Source: CITL (2007) and CDC (2006)

Figure 4

 4 Figure 4 takes a closer look at the allowances surpluses in percentage of the allocation.

Figure 4 :

 4 Figure 4: Potential for Market Power by Country (in percentage) Source: CITL (2007) and CDC (2006)

I

  model a differential game 39 played in continuous time where all players have the possibility of influencing the rate of change of the permits bank through the choice of their current actions. It is therefore assumed that they adopt a Markovian strategy. The common knowledge includes the fact that all players need to comply to the environmental constraint exogenously set by the regulator. The game unfolds in two steps. First, I derive the follower's reaction function to any action announced by the leader through fringe agents' cost minimization problem at the competitive price. Second, I observe how the leader might exercise market power as large agent integrates the reaction function into his own cost minimization problem, and decides how to adjust his emissions level. I hold all other parameters constant 40 .

  depending on the net banking/borrowing position of fringe agents at the end of the period. See[START_REF] Chevallier | Strategic Manipulation on Emissions Trading Banking Program with Fixed Horizon[END_REF] for the formal resolution of this optimization program.Next, I turn to the large agent's behaviour and to how he integrates the two possible cases of reaction function into his own optimization program.

Table 1 : A Survey of Projections for Russian Carbon Dioxide Emissions.

 1

			of Estimate Percentage of 1990 Levels	Period
	Decaux and Ellerman 14	1998	73	2010
	Russian Energy Strategy	2000	76-93	2012
	Loschel and Zhang 15	2002	83.6	2010
	IEA 16 World Energy Outlook	2004	72	2008-2012
	CEPA 17	2004	75	2008-2012
	Source: adapted from Decaux and Ellerman (1998), Loschel and Zhang (2002) and Korppoo et
	al. (2006).
	Source	Year of Estimate Size of the surplus 18	Period
	Decaux and Ellerman	1998	111 19	2010
	Loschel and Zhang 20	2002	157.8	2010
	Russian Ministry of Economic	2003	408-545	2008-2012
	Development and Trade 21
	Russian Forecast to the	2003	456-913	2008-2012
	UNFCCC 22
	CEPA 23	2004	400	2008-2012
	Klepper-Peterson	2005	410	2010
	Bohringer et al.	2006	246 24	2008-2012

Table 2: A Survey of Projections for Russia's Surplus under the KP. Source: adapted from Decaux and Ellerman (1998),

2 Prospective use of banking and borrowing in the EU ETS

  , we may state that during 2005-2007, allowances distributed have more than covered verified emissions, with a net cumulated surplus of 156 million tons. This surplus has however decreased, going from 83 million tons in 2005 to 37 million tons in 2006, and finally 36 million tons in 2007. Emissions have increased by 0.4% in 2007 compared to 2006, and reached 2,043 million tons with respect to 2,080 million allowances distributed.Market power positions in the EU ETS may not be derived from analyses at the country level only, but rather in conjunction with analyses at the installation level. For French installations subject to the Directive 2003/87/CE, an estimation based on 1,402 installations totalling 185.3 Mt CO 2 taken from the Register for Polluting Emissions (iREP) 32 reveals almost 50% of permits were distributed to four players, and the first ten permits holder sum up to 60% of permits allocation.

Besides,

[START_REF] Convery | The European Carbon Market in Action : Lessons from the First Trading Period[END_REF]

point out a strong concentration of EU ETS installations, with 10% of 4,019 installations in surplus representing 75% of the total surplus (totalling 145 Mt) and 2% of installations representing 2% of the surplus 33 . Each of these big players might exert a dominant position on its own sector if permits are distributed freely based on recent emissions, as modelled in this chapter.

2.2.

Member states have allowed banking and borrowing without restriction within each compliance period. But the possibility to carry over EUAs from 2005-2007 to 2008-2012 has been restricted, even by France and Poland who allowed it to a certain extent in the first place 34 .

Table 3 : Simulated Price Distortions based on data by Decaux and Ellerman (1998).

 3

	Country	α	β	X	MAC'/MAC	i i e e -	P t	Price Distortion
	USA	0.0005 0.0398	571	0.0033	571	127	-7.23%
	JAP	0.0155 1.8160	144	0.0108	144	127	-23.70%
	EU	0.0024 0.1503	308	0.0059	308	127	-13.08%
	OECD	0.0085 -0.0986 171	0.0121	171	127	-26.67%
	EUC	0.0079 0.0486	122	0.0160	122	127	-35.20%
	FSU	0.0023 0.0042	-110 -0.0183	-110	127

Source: adapted from

[START_REF] Decaux | Analysis of Post-Kyoto CO2 Emissions Trading Using Marginal Abatement Curves[END_REF]

. Note: USA stands for United States, JAP for Japan, EU for European Union, OECD for countries members of the Organisation for Economic Co-operation and Development, EUC for Eastern European Countries, and FSU for Foreign Soviet Union Countries. The price distortion is computed using equations (

5

) to (7).

Table 4 : Simulated Price Distortions based on data by Loschel and Zhang (2002).

 4 Source: adapted from[START_REF] Loschel | The Economics and Environmental Implications of US Repudiation of the Kyoto Protocol and the Subsequent Deals in Bonn and Marrakech[END_REF]. Note: AUN stands for Australia and new Zealand, JAP for Japan, EU for European Union, CAN for Canada, and FSU for Foreign Soviet Union Countries. The price distortion is computed using equations (

For an exhaustive literature review on permits trading and market power, see[START_REF] Petrakis | Location decisions of a polluting firm and the time consistency of environmental policy[END_REF].

In fact, no major international agreement on greenhouse gases allowed borrowing to a full extent at that date.

See Kolstad (2005).

Based on the MIT-EPPA database that aggregates FSU countries.

Tendances Carbone is published by the Caisse des Depots, and is available at: www.caissedesdepots.fr

Available at http://ec.europa.eu/environment/ets

concentrate emissions on early periods, which is not socially optimal.

As a final comment, one could say a greater reliance on banking and limited borrowing (i.e. with a specific discounting factor) should be promoted to allow firms to smooth their emissions and take investment decisions in abatement technologies with a better capacity to react to the evolution of the carbon constraint over time.


N Daraoui

P Dufour
email: dufour@lagep.univ-lyon1.fr

H Hammouri

Benefits in using model based predictive control during drying and lyophilisation

Keywords: drying, lyophilisation, model based process control, predictive control, modeling uncertainties

Introduction

The aim of this paper is to show how model based predictive control (MBPC or MPC) techniques may also be used for drying and lyophilisation processes to improve their efficiency in term of production, like in other industrial application domains. For example, it is well known that drying is a highly energy intensive operation and it represents from 10% to 25% of the national industrial energy in the developed world [START_REF] Mujumdar | Guide to industrial drying, principles, equipment and new developments[END_REF] whereas a majority of industrial dryers operate at low energy efficiency, from a disappointing 10% to 60% (this ratio is defined as the theoretical energy required for the drying to the actual energy consumed). Therefore, due to the escalating energy costs and global competition, these performances have to be improved. This can be done using control tools. With process control, we mean control methods where the inputs (manipulated variable) of a processing plant are automatically tuned during the production by designed controllers, such that the controlled variable follows the specified production needs. In industrial applications, MPC is one of the most popular controllers since its birth in the 1970s and after the PID (born in the 1940s), MPC has become the second paradigm in control: in 2003, there were more than 4000 industrial applications of MPC [START_REF] Qin | A survey of industrial model predictive control technology[END_REF], especially in refinery and petrochemicals. In the meantime, control tools are not widely applied in drying [START_REF] Dufour | Control engineering in drying technology: Review and trends[END_REF], in spite of some benefits already obtained. For example [START_REF]Energy saving using model-based predictive control[END_REF], in a beet sugar factory, a MPC helped to reduce the energy costs by 1.2% (18,900 £/year) and to decrease the downstream energy cost by 14,000 £/year. The product yield increased by 0.86% worth 61,600 £/year and off specification production decreased from 11% to 4%. Finally, the payback time (including hardware and development) was 17 months. In this paper, an example of the use of a MPC is shown for a lyophilisation process during the primary stage of the freeze drying process of a Bovin Serum Albumin (BSA) solution. In a user-friendly code (MPC@CB 3 ), simulation results show here the efficiency of the MPC algorithm, especially in term of robustness with respect to uncertainties in one of the model parameters. The MPC allows the controlled temperature to track a specified time-varying trajectory. Any other constrained optimization problem may be specified in MPC@CB, based on measured (or estimated) and modeled variables.

Model predictive control features

Model-based predictive control is a particular class of optimal controller. The idea began in the 1960s and a real interest started to emerge in the 1980s after publication of the first papers. Then, new generations of MPC algorithms were formulated (QDMC, GPC, PFC, DMC+ …). The first main advantage of MPC is that constraints (due to: manipulated variables physical limitations, operating procedures or safety reasons…) may be explicitly specified into this formulation. The second main advantage of MPC is its ability to be used for simple and complex model based processes (time delays, inverse responses, significant nonlinearities, multivariable interaction, modeling uncertainties). Many MPC approaches have been proposed along the past three decades, most of them based on a receding-horizon strategy, i.e., at each current sampling instant k the following actions are taken: the plant measurements are updated for use in the feedback/feedforward control loop; the plant model is used to predict the output response to a hypothetical set of future control sequence; a function including the cost of future control actions and future deviations from a reference behaviour is optimized to give the best future control sequence; the first movement of the optimal control sequence is applied on the process. These operations are repeated at time k+1. However, if the model exhibits a non-linear behavior, a numerical solution technique must be used to solve this optimal problem. The computational effort varies somewhat because some solution methods require only that a feasible (and not necessarily optimal) solution should be found or that only an improvement should be achieved from time step to time step.

MPC for a lyophilisation process: example

Freeze drying, also called, lyophilisation, is a drying process used in biotechnology, food and pharmaceutical industries frequently used to stabilize and preserve the products [START_REF] Pikal | Freeze drying. Encyclopedia of Pharmaceutical Technology[END_REF]. Compared with conventional drying techniques, freeze drying is generally considered to produce the dried product with the highest quality. Until now, a larger use of the freeze drying is limited by its cost and its use remains therefore limited for high value added products, like pharmaceutical products. One very important parameter in the study of the freeze drying is the temperature of the product which must be carefully controlled during the primary and the secondary stages of freeze drying. Generally, dynamic models for freeze-drying predict drying behaviors more accurately than steady models. As a counterpart, complex transport equations are required, which are difficult to solve and are based on variables and parameters difficult to measure accurately experimentally. Dynamics model are preferred for model based control. An example of MPC in lyophilisation is discussed here: the controlled temperature has to track as best as possible the specified time-varying reference trajectory during the primary stage of the freeze drying process of a Bovin Serum Albumin (BSA) solution. To satisfy this objective, a mathematical model, that can predict the behavior of the freeze drying process, is used [START_REF] Sadikoglu | Mathematical modelling of the primary and secondary drying stages of bulk solution freeze-drying in trays: Parameter estimation and model discrimination by comparison of theoretical results with experimental data[END_REF]. During the primary drying, the product is composed of two phases: a dry layer, in which the majority of water is sublimated and a frozen layer. These two areas are separated by a moving interface called the sublimation ice front described by the space coordinate z at z=H(t). This complex model has already been used in our predictive control software (MPC@CB) to tune online the shelve temperatures according to the control objective [START_REF] Daraoui | Optimal operation of sublimation time of the freeze drying process by predictive control: Application of the MPC@CB Software[END_REF]. In term of uncertainties, [START_REF] Hottot | Thermophysical properties of aqueous and frozen states of Water/BSA/Tris systems[END_REF] shown that the heat capacity in the frozen layer Cp 2 is known with relatively large errors (up to 24%). The idea is to show here how the MPC is able to control online the primary stage drying to a desired behavior, in spite of this parameter uncertainty. Top of Fig. 1 shows that, with the same constant open loop control conditions, the measured temperature is different for 2 runs with different values for Cp 2 . In the meantime, the position of the sublimation front also differs (top of Fig. 2). If the (closed-loop) MPC is employed in order to track a specified temperature trajectory, the measured temperature is almost the same for 2 runs with different values for Cp 2 (bottom of Fig. 1): this is a robustness property of this controller. In the meantime, the position of the sublimation front is almost the same in both cases (bottom of Fig. 2).

MPC is therefore of interest to control such processes featuring parameter uncertainties.

Conclusion

Since 1998 [START_REF] Dufour | Control engineering in drying technology: Review and trends[END_REF], the yearly mean publication rate of papers using a first principle model for control has been multiplied by 11! This increase is quiet similar to the increase of the control papers in drying based on optimal control strategies, like MPC. With 60 000 products dried and 100 dryer types used worldwide [START_REF] Mujumdar | Guide to industrial drying, principles, equipment and new developments[END_REF], a real potential exists. With control tools like MPC, it is expected that the industrial drying operation will continue to improve its energy efficiency while enhancing product quality and reducing the negative environmental impact of dryers.

Fig. 1 :

 1 Fig. 1: Controlled temperature in: open loop control (top), MPC control (bottom). No error (cont.) and with modeling error (dash). In closed loop, reference trajectory (dash dot).

Fig. 2 :

 2 Fig. 2: Sublimation front position in: open loop control (top), MPC (closed loop) control (bottom). No modeling error case (cont.) and modeling error case (dash).

1 Work partially supported by the AUF (Agence Universitaire de la Francophonie)