Julien Chevallier
email: <jchevall@u

A Differential Game of Intertemporal Emissions Trading with Market Power

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 paper develops a differential Stackelberg game with two types of noncooperative agents: a large potentially dominant agent and a competitive fringe whose size are exogenously determined. The strategic interactions are modelled on an intra-industry permits markets where agents can freely bank and borrow permits.

This paper 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. Results are presented under perfect information.

INTRODUCTION

1 Introduction

What happens on a tradable permits market when distortions occur as a consequence of the inital 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 power1 in a static context and concerning the spatial exchange of permits only, this paper 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 & 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 only2 , 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 & Montero (2005b) did not impose a particular game structure, I adopt a Stackelberg game structure that allows to deal 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 competitive3 . I explicitly include the Hotelling conditions4 that must apply if permits are considered as an exhaustible resource.

The paper unfolds as follows: first, I describe the institutional environment of current permit trading programs, namely the Kyoto Protocol (KP) and the European Union Emissions Trading Scheme (EU ETS); second I derive in a benchmark case an expression for market power; third I develop the differential Stackelberg game.

Description of the Institutional Environment

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 question of the Kyoto Protocol as an "unfinished business" is often evoked. Very heterogenous 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 paper 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

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 5 . 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 4 of the KP, as those countries were given generous allocations to foster agreement during the first phase (2008)(2009)(2010)(2011)(2012). Market power concerns arise as industrial firms may benefit from the gap between their initial permits allocation (based on 1990 production levels) and their real emission needs in 2008 (after a period of recession), and the use of these permits surpluses remains unclear. If a pure monopoly emerges, a single seller may price its ouput at a higher level than its marginal cost of production. Under international emissions trading, the case of relatively large buyers or cartels exerting market power sounds more relevant 6 .

This situation emerged as a conflict between the internal and the external consistency of the permits market:

• the internal consistency refers to the situation where agents freely receive or bid for permits according to their real needs. The regulator may be interested however in distributing more permits to a country than strictly needed (according to business as usual emissions or a benchmark for instance) in order to ensure participation to the permits market 7 . As a consequence, one agent may achieve a dominant position which in turns threatens the efficiency of the permits market itself.

• the external consistency of the permits market is linked to the broader debate of climate change as the purchase of a "global public good" 8 . This altruistic view embodies the notions of "Burden Sharing" or "common but differentiated responsibilities" 9 attached to the KP, whereby developed countries agree to spend a higher income share on fighting climate change than developing countries 10 .

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 represents a mere 5% 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 6 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.

7 Such negotiation with Russia was determinant for the KP to enter into force on 16/02/05

8 See [START_REF] Guesnerie | The design of post-Kyoto climate schemes : an introductory analytical assessment[END_REF]. 9 See [START_REF] Muller | Equity in Climate Change: The Great Divide[END_REF].

10 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.

and China. Thus, many difficulties arise to pierce the "veil of uncertainty" around international negotiation 11 .

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] 12 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 Rights Markets: the Role of Initial Permit Distribution[END_REF] 13 , "[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 paper 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-and-trade program might also give birth to dominant positions shall not be neglected. This leads me to comment in depth the case of Russia.

2.1.2 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] 14 : "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 & Montero (2005a) 15 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.

The key finding in 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 of 111 M tCO 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 neeeded 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" 16 of emissions to other regions that are not covered by the KP and by additional allowances from the Clean Development Mechanism or Joint Implementation 17 that might compete with Russian allowances 18 .

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 explicited 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] 19 : "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 can not 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 20 . As explained by [START_REF] Newell | Managing Permit Markets to Stabilize Prices[END_REF] 21 : "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 adressed in depth by [START_REF] Alberola | Banking Borrowing in the EU ETS: An Econometric Appraisal of the 2005-2007 Intertemporal Market[END_REF]. 16 As documented by Decaux & Ellerman (1998) (pp.15-16), 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.

17 Conservative estimates range from a lower bound of 800M tCO2 according to UKERC (2006) to an upper bound of 1000M tCO2 according to Point Carbon.

18 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.

19 p.295 20 paragraph II.XV 21 p.149

The European Union Emissions Trading Scheme

Here 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? Figure 1 represents the repartition of 2005 European Union Allowance Units (EUAs in million tonnes of CO 2 ) among countries, where Germany, Poland, Italy, UK and Spain stand out as the most important actors by totalling about two thirds of the total allowances. Data is taken from CDC (2006)22 and the Community Independent Transaction Log administered by the European Commission23 . 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.

Figure 2 depicts the 2005 reported emissions and, if any, the size of the surplus. The sum of the two bars is equivalent to the 2005 allocation of permits for a given country. Its main finding lies in the fact that most countries seemed to favor generous allocations during the first phase of the EU ETS24 . Surpluses also reflect in a limited amount reserves for new entrants, which are included in the data used.

Figure 3 takes a closer look at the allowances surpluses in percentage It reflects a wide variety of cases among market participants as a bulk of countries (Poland, France, Finland, Denmark, Slovaquia, 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.

How could one explain those contrasting patterns in actual emissions for EU ETS participants in 2005? Part of the answer may be found in the decision making process within each National Allocation Plan (NAP). [START_REF] Godard | L'allocation initiale des quotas d'émission de CO2 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 CO2[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], p.8). 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] Buchner | 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-allocatation 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 over-allocation" 25 . 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.

Due to a lack of data availability from 2004 onwards, it appears difficult to provide a more precise characterization of the room for market power at the sectoral level. For French installations subject to the Directive 2003/87/CE, an estimation based on 1,402 installations totalling 185,3 M tCO 2 taken from 3 THE MODEL 12 the Register for Polluting Emissions (iREP)26 reveals almost 50% of permits were distributed to four players, and the first ten permits holder sum up to 60% of permits allocation.

Ongoing research by De Perthuis et al. 27 points out a strong concentration of EU ETS installations, with 10% of 4019 installations in surplus representing 75% of the total surplus (totalling 145 Mt) and 2% of installations representing 2% of the surplus.

Each of this big players might exert a dominant position on its own sector if permits are distributed freely based on recent emissions, as modelled in this paper.

Prospective use of banking and borrowing in the EU ETS

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 place28 . 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 | Banking Borrowing in the EU ETS: An Econometric Appraisal of the 2005-2007 Intertemporal Market[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 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 paper.

The Model

This section details the features of the model. First, I explain the design of the cap-and-trade 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 e i 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 either a country, a firm or a cartel 29 . 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 30 . 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 31 in the model.

Industry Structure

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] 32 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 29 For instance, within the KP permits may be exchanged party-to-party, but also firmto-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 & Montero (2005b)) 30 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.

31 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.

32 p.2

3 THE MODEL 14 manipulation strategies 33 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

I model a differential game 34 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 35 .

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 B i (t) < 0 in case of borrowing.

Any change in the permits bank is equal to the difference between e i (t) 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:

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

Hotelling Conditions

Notwithstanding differences between a permit and an exhaustible resource 36 , it is assumed in the literature that the Hotelling conditions for exhaustible 33 See [START_REF] Misiolek | Exclusionary Manipulation of Markets for Pollution Rights[END_REF].

34 See [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF] for an overview of differential games. 35 For instance, agents do not incur information costs.

36 According to Liski & Montero (2006) (p.3), 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 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).

THE MODEL

15

resources must apply on a permits market. Consequently, the terminal and exhaustion conditions are detailed below.

Terminal Condition

Let [0, T ] be the continuous time planning horizon37 . 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 regulator38 :

T 0 N i=1 e i (t)dt = N i=1 e i = E
(2)

Exhaustion Condition

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

N i=1 B i (T ) = 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 penalty39 .

Abatement Cost Function

Let C i [e i (t)] be the abatement cost function 40 incurred by agent i in order to comply with his permits allocation e i . C i [e i (t)] is defined on in continuous and is of class C 2 [0; T ], i.e. twice continuously differentiable. The 16 classical assumption 41 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 market in a static framework 42 , price-taking agents adjust emissions until the aggregated MAC is equal to the price P at time t:

P t = -C i [e i (t)] (4)
Thus, at the equilibrium, there is no free-lunch for price-taking agents.

Benchmark Case

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. I develop a benchmark case where by assumption the large agent receives all the stock of permits. Both types of agent need to comply to the environmental constraint by adjusting their emissions level and trading permits. In this setting, fringe agents' emissions come from trading with the large agent. The expression of market power may be derived straightforward when the large agent directly integrates the competitive price into his maximization program.

Optimization program

The large agent minimizes its abatement costs with respect to total emissions and behaves as follows:

                       min e 1 ,e 2 ,...,e N T 0 e -rt C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)] dt T 0 e 1 (t) dt = E - T 0 N i=2 e i (t) dt P t = -C i [e i (t)] ∀i = 2, . . . , N [e 1 (t) -e 1 (t)] = N i=2 e i (t)
41 Stated first by [START_REF] Montgomery | Markets in Licenses and Efficient Pollution Control Programs[END_REF]. The conditions given by Leiby & Rubin (2001) include the output q(t) where Ci[qi(t), ei(t), t] is strongly convex with C i [qi(t)] > 0 and C i [qi(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].

42 See [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF].

where the expression [e 1 (t) -e 1 (t)] represents the number of permits bought (> 0) or sold (< 0). I replace the value of P t and [e 1 (t) -e 1 (t)] in the objective function and form the Lagrangean with e 1 (t) and e i (t) as control variables, and λ(t) as a multiplier:

L = T 0 e -rt C 1 [e 1 (t)] -C i [e i (t)] N i=2 e i (t) dt +λ(t) E - T 0 e 1 (t)dt - T 0 N i=2 e i (t)dt
The first-order conditions are: t)] in the second equation and rearranging terms, I get:

∂L ∂e 1 (t) = C 1 [e 1 (t)] -λ(t) = 0 ∂L ∂e i (t) = -C i [e i (t)] N i=2 e i (t) -C i [e i (t)] -λ(t) = 0 Replacing λ(t) = C 1 [e 1 (
-C i [e i (t)] N i=2 e i (t) -C i [e i (t)] -C 1 [e 1 (t)] = 0 -C 1 [e 1 (t)] = C i [e i (t)] N i=2 e i (t) + C i [e i (t)] -C 1 [e 1 (t)] = C i [e i (t)] 1 + C i C i N i=2 e i (t) -C 1 [e 1 (t)] = P (t) 1 + ε i N i=2 e i (t)
The large agent's MAC is equal to the competitive permits price plus an element of price distortion ε i defined as fringe agents' elasticity:

ε i = C i (e i ) C i (e i ) = dC i de i dC i de i = dC i de i de i dC i = dC i dC i
Thus, permits price manipulation results in higher total abatement costs than under perfect competition. Next, I explicit this market power condition.

FRINGE AGENTS' REACTION FUNCTION

18

Market Power Condition

In this setting, market power is function of fringe agents' elasticity and of the large agent's number of permits:

ε i N i=2 e i (t) = ε i [e 1 (t) -e 1 (t)] (5)
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.

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:

           min e i T 0 e -rt {C i [e i (t)] + P (t) [e i (t) -e i (t)]} dt Ḃi (t) = e i (t) -e i (t) B i (0) = 0 C i [e i (0)] = 0
I write the corresponding current-value Hamiltonian and first-order optimality conditions:

H(B i (t), e i (t), λ(t), t) = {C i [e i (t)] + P (t) [e i (t) -e i (t)]} -λ(t)[e i (t) -e i (t)] ∂H ∂e i (t) = 0 : P (t) = -C i [e i (t)] + λ(t) (6) Ḃi (t) = ∂H ∂λ(t) = 0 : Ḃi (t) = e i (t) -e i (t) (7) λ(t) -rλ(t) = - ∂H ∂B i (t) = 0, λ(T )B i (T ) = 0 (8)
It can be infered from (8) that λ(t) = λ(0)e rt . The transversality condition ( 8) is required to meet the exhaustion condition (3), i.e. to reflect the idea that the bank has no scrap value at the end of the period. I can distinguish several cases:

• if B i (T ) > 0 or B i (T ) < 0, then λ(T ) = 0, λ(t) = 0: when fringe agents have a net banking or borrowing position at the end of the period, the reaction function is equal to the static equilibrium condition (4) where fringe agents equalize their MAC with the permits price. As stated earlier, surplus allowances are worthless and agents need to pay a penalty in case of net borrowing;

• if λ(T ) > 0, then B i (T ) = 0 and λ(t) > 0: when the constraint on λ is binding, this implies a positive shadow price for units in the bank that is cleared. The reaction function is equal to (6);

• if λ(T ) < 0, then B i (T ) = 0 and λ(t) < 0: this would imply a negative shadow price for units in the bank when it is cleared. Since this result is not in accordance with the specific purpose of a cap-and-trade program, I do not further comment this case.

I have therefore highlighted two possible forms of fringe agents' reaction function depending on a net banking/borrowing position or to a positive value of the co-state variable.

In the second step of the game, 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.

6 Behaviour of the Large Agent

Optimization program

The large agent adjusts strategically his optimal emissions levels according to its initial allocation e 1 as expressed by (2) and the banking borrowing constraint (1). The cost minimization program for agent [i = 1] is:

                     min e 1 T 0 e -rt {C 1 [e 1 (t)] + P t [e 1 (t) -e 1 (t)]} dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B 1 (0) = 0 C 1 [e 1 (0)] = 0
6 BEHAVIOUR OF THE LARGE AGENT 20

First case

Replacing P t by (4), the large agent's optimization program becomes:

                     min e 1 T 0 e -rt C 1 [e 1 (t)] -C i [e i (t)] [e 1 (t) -e 1 (t)] dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B 1 (0) = 0 C 1 [e 1 (0)] = 0
Assuming fringe agents are homogenous, I write N i=2 e i (t) = (N -1)e i (t) and replace the emissions constraint (2) into the objective function:

           min e 1 T 0 e -rt C 1 [e 1 (t)] -C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] dt Ḃ1 (t) = e 1 (t) -e 1 (t) B 1 (0) = 0 C 1 [e 1 (0)] = 0
I form the corresponding current-value Hamiltonian with e 1 (t) as a control variable, B 1 (t) as a state variable, and µ(t) as a co-state variable:

H(B 1 (t), e 1 (t), µ(t), t) = C 1 [e 1 (t)] -C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] + µ(t)[e 1 (t) -e 1 (t)]
Assuming the existence of an interior solution, necessary optimality conditions include:

∂H ∂e 1 (t) = 0 : C 1 [e 1 (t)] + 1 N -1 C i E -e 1 (t) N -1 [e 1 (t) -e 1 (t)] -C i E -e 1 (t) N -1 -µ(t) = 0 (9) Ḃ1 (t) = ∂H ∂µ(t) = 0 : Ḃ1 (t) = e 1 (t) -e 1 (t) (10) μ(t) -rµ(t) = - ∂H ∂B 1 (t) = 0, µ(T )B 1 (T ) = 0 (11)
With reference to the transversality condition (11), I conduct the same analysis as in the previous section:

BEHAVIOUR OF THE LARGE AGENT

21

• if B 1 (T ) > 0, then µ(T ) = 0, µ(t) = 0 and B i (T ) < 0: net banking by the large agent at the end of the period is compensated by fringe agents's net borrowing;

• if B 1 (T ) < 0, then µ(T ) = 0, µ(t) = 0 and B i (T ) > 0: net borrowing by the large agent in terminal period is compensated by fringe agents' net banking.

Both cases of net banking B 1 (T ) > 0 or borrowing B 1 (T ) < 0 by the large agent in terminal period imply µ(T ) = 0 and µ(t) = 0. From eq.( 9), it is possible to identify a price distortion condition analogous to the market power condition (5):

-C 1 [e 1 (t)] = -C i E -e 1 (t) N -1 1 + 1 N -1 C i C i [e 1 (t) -e 1 (t)] -C 1 [e 1 (t)] = P (t) 1 + 1 N -1 ε i [e 1 (t) -e 1 (t)]
On the left hand side of the equation, I have the large agent's MAC. On the right hand side, I recognize the price distortion as a function of fringe agents' elasticity and the large agent's permits endowment.

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 result is similar to the benchmark case where the large agent owns all the stock of permits.

Next, I examine the form of the solution for the second form of fringe agents' reaction function.

Second case

Replacing P t by eq.( 6), the large agent's optimization program becomes:

                     min e 1 T 0 e -rt C 1 [e 1 (t)] -{C i [e i (t)] + λ(t)} [e 1 (t) -e 1 (t)] dt Ḃ1 (t) = e 1 (t) -e 1 (t) E = T 0 e 1 (t)dt + T 0 N i=2 e i (t)dt B 1 (0) = 0 C 1 [e 1 (0)] = 0
The resolution method is the same as in the first case and the following comment arise:

CONCLUSION

22

• if B 1 (T ) = 0, then µ(T ) > 0, µ(t) > 0 and B i (T ) = 0: in case both agents clear their permits bank at the end of the period, the shadow values of a unit in the bank is positive. Both values of µ(t) and λ(t) are known. For the large agent, the shadow value of a unit of emission in the bank measures the marginal utility of the state at time t along the optimal trajectory. For fringe agents, λ(t) reflects the highest hypothetical price at which they would be willing to pay for an additional permit at time t.

Rearranging as above and setting {B i (T ) = 0, B 1 (T ) = 0, λ(t) > 0, µ(t) > 0} yields:

-

C 1 [e 1 (t)] + µ(t) = -C i E -e 1 (t) N -1 1 + 1 N -1 C i C i [e 1 (t) -e 1 (t)] -λ(t) -C 1 [e 1 (t)] + µ(t) = P (t) 1 + 1 N -1 ε i [e 1 (t) -e 1 (t)] -λ(t)
This condition means that 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.

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 permis efficiently (be it through output-based methods, benchmarking, minimum price auctioning, etc.).

Conclusion

The description of the institutional environment 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 it seems overall difficult 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.

23

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. Some similarities have been underlined between a benchmark case where the large agent owns all the stock and the model with grandfathered permits: 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 & 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.

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] 43 , 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.

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.

Figure 1 :

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

Figure 2 :

 2 Figure 2: Potential for Market Power by Country (in absolute terms) Source: CITL (2007) and CDC (2006)

Figure 3 :

 3 Figure 3: Potential for Market Power by Country (in percentage) Source: CITL (2007) and CDC (2006)

Table 2 :

 2 Table 1 is that under all scenarios Russia would meet A Survey of Projections for Russia's Surplus under the KP.

	11 See Kolstad (2005).
	12 p.207
	13 p.295
	14 p.2
	15 Based on the MIT-EPPA database that aggregates FSU countries.

a Scenario with Annex B Trading. b Scenario assuming trading without the US. c International Energy Agency. d Cambridge Economic Policy Associates. Scenario with a 2% energy intensity reduction. a In million tonnes of carbon equivalent (MtCe). b M tCO2 c scenario assuming only Russia exercises monopoly power d adapted from Korpoo et al. (2005) e adapted from Korpoo et al. (2005) f M tCO2

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].

This was mainly due to the fact that no major international agreement on greenhouse gases allows borrowing to a full extent to date (personal contacts with M. Liski).

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]

Namely the exhaustion and terminal conditions.

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],[START_REF] Holtsmark | Russian Behaviour in the Market for Permits under the Kyoto Protocol[END_REF].

Tendances Carbone is published by the French Caisse des Depots and is available at <http://www.caissedesdepots.fr/>, accessed on November, 2006.

available at <http://ec.europa.eu/environment/ets/>, accessed on June, 2007

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".

The iREP is monitored by the Minister of the Environment and displays public information at <http://www.pollutionsindustrielles.ecologie.gouv.fr/IREP/index.php>(accessed on November, 24th 2006

). 27 Workshop "Evaluation of the EU ETS", April 2007, Caisse des Dépôts,

Paris 28 Permits that were bought cannot be banked. Besides, only "green" firms that have effectively reduced emissions may bank allowances in Poland.

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 SO2 Emissions Trading[END_REF], but they reflect the specific requirements of the Acid Rain Program (USA).

 38 See also[START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF], p.

231.39 For instance, the penalty is

equal to 40= C and 100= C 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.40 Compared to a situation where profits are unconstrained, abatement costs appear in order to meet the emission cap ei.

The adoption of a discount rate penalizing borrowing may remove some of the perverse incentives whereby agents concentrate emissions on early periods, which is not socially optimal.


Marie-Laure Breuillé
email: breuille@wz-berlin.de

Francis Bloch

Alessandra Casella

Robert Gary-Bobo

Pierre-André Jouvet

Kai Konrad

Katheline Schubert

Tradable De…cit Permits: A Way to Ensure Sub-National Fiscal Discipline?

Keywords: JEL Classi…cation: E61, E62, G14, H6, H7 Policy Coordination, Tradable Permits Market, Soft budget constraint

This paper proposes a system of tradable de…cit permits for implementing budgetary austerity at the local level. We evaluate the e¢ ciency of the …scal retrenchment allocation in a dynamic setting with a commitment problem. The way rights are allocated and traded on the market turns out to be decisive for the cost-e¤ectiveness of the system. Indeed, the inability of the State to commit dynamically to a sharing rule of de…cit rights generates perverse incentives which a¤ect the local market. The market turns out to be ine¢ cient -with heterogeneous jurisdictions -unless the State allows local decision-makers to trade permits through time.

Introduction

The criteria for …scal convergence implemented by the Maastricht Treaty, and strengthened by the Stability and Growth Pact (SGP) signed in Amsterdam (1997) compel each EMU Member State to a national budgetary discipline: the public de…cit must remain below 3% of GNP and the public debt must not exceed 60% of GNP. To enforce budgetary rules, the SGP is based on both preventive and repressive mechanisms which may lead to …nancial sanctions imposed on the non-complying Member State's central government.

A national internal incentives problem stems from the fact that de…cit and debt ratios apply to the consolidated budget of all public administration bodies -including local jurisdictions -whereas only central authorities are held responsible for violations. Without a mechanism for explicitly sharing the e¤orts made to curb de…cits, local jurisdictions would conduct tax and spending policies without taking their impact on the consolidated de…cit into account. The central government can therefore be penalized, despite the fact that its budget is balanced, if local jurisdictions run excessive de…cits. To abide by the terms of European commitments, the central government should therefore design and implement coordination mechanisms, that have the power to internalize budgetary externalities and to ensure the overall compatibility of local policies with national objectives [START_REF] Batabyal | Consistency and Optimality in a Dynamic Game of Pollution Control I: Competition[END_REF] . In an "austerity" context, a crucial question is how the central government will share de…cit-reduction e¤orts with local authorities. In other words, how can we solve this "tragedy of the commons"? Most of the EMU Member States have adopted or strengthened …scal rules at the local level -either enforced by the central government (Finland, Italy, Portugal) or set up by cooperative institutions (Belgium, Denmark, Germany and Spain) -in order to improve …scal coordination. Among the wide variety of …scal rules, the most widespread are budget balance requirements, restrictions on borrowing and, to a lesser extent, limits on the ability to increase spending or the tax burden [START_REF] Baumol | The Theory of Environmental Policy[END_REF] . These traditional instruments for de…cit regulation are certainly simple and easily understandable, but they often lack ‡exibility -especially in bad times -and do not take the heterogeneity of the costs of enforcing the standard into account. Because local jurisdictions have di¤erent administrative rigidities, di¤erent electoral motivations, and di¤erent costs for public good provision, a uniform quantitative constraint imposes widely diverging costs which turns out to be ine¢ cient. With a large number of local jurisdictions, it becomes particularly costly, not to say impossible, to extract all the necessary information about the marginal cost of …scal retrenchment. When marginal costs vary according to time period, such a cost could not even be inferred from the response to the instrument. The implementation of Pigovian taxes thus also becomes ine¢ cient. An alternative mechanism would be the implementation of a local market for tradable de…cit permits. In Austria [START_REF] Böhringer | On the design of optimal grandfathering schemes for emission allowances[END_REF] , an internal stability pact introduced in 1999 -and revised in 2001 -implicitly includes the possibility of establishing a market for de…cit permits which are transferable between Land governments and municipalities within the Land4 .

A system of tradable de…cit permits for implementing the …scal constraints of the Maastricht treaty has already been studied by Casella (1999), but at the EMU level. Casella has based her approach on the environmental economic literature5 by considering de…cits as a form of pollution. According to this literature, independently of the initial distribution, the trading of permits on a competitive market ensures an e¢ cient decentralised allocation of pollution. Any polluter whose the marginal abatement cost is lower than the price at which permits are trading will have an incentive to abate more by selling permits, and vice versa [START_REF] Casella | Market Mechanisms for Policy Decisions. Tools for the European Union[END_REF] . At the market equilibrium, marginal abatement costs are equalized, and the target is achieved at minimum cost. The parallel to the regulation of the de…cit behaviour -at the EMU level or the national level -is immediate and... tempting! In a static setting, Casella (1999) shows that the implementation of a tradable de…cit permits market as a mechanism for sharing budgetary discipline among the EMU Member States is e¢ cient. The optimal allocation equalizes the marginal abatement cost of …scal retrenchment across all countries. However, …scal coordination is mainly a dynamic issue.

The de…cit is a form of stock pollutant: each unit of current de…cit adds permanently to the stock of the debt and to the level of the damage. In a dynamic setting, a soft budget constraint [START_REF] Coase | The Problem of Social Cost[END_REF] problem may arise if the behaviour of local jurisdictions in the market is in ‡uenced by the expectation of receiving additional de…cit rights from the State. Our paper aims at evaluating the e¢ ciency of a market for tradable de…cit permits at the local level, in a dynamic setting with a commitment problem [START_REF]Razionalizzazione e sempli…cazione dei documenti e delle procedure contabili[END_REF] . Is the market-based mechanism for allocating budgetary austerity at the local level still e¢ cient under a soft budget constraint policy? How does the ability of local jurisdictions to manipulate the distribution of rights a¤ect the market?

The originality of our paper comes from modelling an intertemporal budgetary game between the regulator -the State -and the regulated agents -the local jurisdictions -for the distribution of de…cit rights. The existence of the soft budget constraint is based on the local jurisdictions'interest in behaving strategically in order to extract extra rights from the State, and on the State's incentives to deviate ex post from the …scal discipline policy stated ex ante. In this sequential game, local jurisdictions act as Nash competitors with each other, but they are …rst-movers in a Stackelberg game with the State. The total amount of rights [START_REF] Crocker | The Structuring of Atmospheric Pollution Control Systems[END_REF] to be shared between the local and central levels is assumed to be exogenous in each period. The State -which only intervenes in the market via the distribution of rights -splits this initial stock between itself and the local jurisdictions so that it maximizes the welfare [START_REF] Coase | The Problem of Social Cost[END_REF] Following Rodden, Eskeland and Litvack (2003), a soft budget constraint can be de…ned as "the situation when an entity can manipulate its access to funds in undesirable ways". The inability of the rescuer to generate expectations of no bailout entails a soft budget constraint. See Kornai, Maskin and Roland (2003) for a survey. minus its own cost of being constrained. A local jurisdiction may buy permits from another jurisdiction with a rights surplus or it may sell permits but must own enough of them to cover its current de…cit in each period. The market price thus re ‡ects the current value of the budgetary austerity.

We subsequently allow de…cit permits to be traded over time. In such a setting, local jurisdictions may directly reduce their de…cit as well as buy, sell, bank and borrow de…cit permits in order to meet the applicable standards or to take advantage of any speculative opportunities. The intertemporal de…cit permits trading thus lowers the cost of compliance with the de…cit standards imposed by the State by allowing local jurisdictions to administer their budgets more ‡exibly over the two periods.

Our results can be summarized as follows. The sharing rule of de…cit rights designed by the State turns out to be easily manipulable by local decision makers. Due to its aim of equalizing the marginal welfare across all local jurisdictions, the State always …nds it optimal ex post to reallocate rights in the second period when a local jurisdiction increases its …rstperiod de…cit. We show that the soft budget constraint phenomenon results in ine¢ ciency of the local market for tradable de…cit permits, unless local preferences are identical. The opportunistic behaviour of local decision makers consists in reducing …rst-period budgetary e¤orts, and this exerts upwards pressure on the price. However, the intertemporal trading of de…cit permits restores the e¢ ciency, i.e. the marginal abatement costs equalization condition, due to the fact that local jurisdictions aim at smoothing their cost of being constrained over the two periods.

In the following, section 2 describes the model, the cost of being constrained and the functioning of the local market. Section 3 characterizes the rights allocation rule set by the State. Section 4 analyzes the impact of the soft budget constraint on the e¢ ciency of the tradable de…cit permits market. Section 5 introduces intertemporal ‡exibility in the permits trading and studies how this modi…es the behaviour of local jurisdictions. Concluding remarks are gathered in section 6.

Analytical framework

The framework is a simple two-period intertemporal model. The economy consists of n local jurisdictions with the same level in the hierarchy of public authorities and a State. An internal budgetary austerity policy -laid down by an international treaty or self-imposedcommits the State to keep the consolidated public de…cit, i.e. the sum of local jurisdictions' de…cit n P i=1 d it and its own de…cit10 d ct , below an exogenous amount11 d t :

n X i=1 d it + d ct d t ;
in each period t = 1; 2.

The budgetary austerity is costly both for the local jurisdictions and for the State.

Let C it ( it ) measure the cost to local jurisdiction i of reducing its primary de…cit to a level it in period t, that is, the di¤erence between the unconstrained utility, i.e. the local jurisdiction i runs a primary de…cit it , and the utility constrained by a budgetary rule, i.e. the local jurisdiction i runs it < it . This cost of being constrained, which di¤ers among local jurisdictions and through time [START_REF] Dales | Pollution, Property and Prices[END_REF] , may be interpreted as the value of foregone local public good provision, the value of the resources involved in improving e¢ ciency and (or) can be viewed as an electoral cost. Note that the cost does not depend on the de…cit d it but on the primary de…cit it , i.e. the real amount of new resources devoted to the objectives of the local jurisdiction. Similarly, we de…ne C ct ( ct ) as the cost to the State of being constrained to a primary de…cit ct in period t. These functions are assumed to be twice continuously di¤erentiable, decreasing

(C 0 ct ( ct ) 0, C 0 it ( it ) 0 8i) and convex (C 00 ct ( ct ) 0, C 00 it ( it ) 0 8i) in the amount of primary de…cit, with a unique minimum at zero ( C 0 ct ( ct ) = C 0 it ( it ) = 0 8i
) at the unconstrained levels ct and it . Hence additional costs from reducing one more unit of primary de…cit, i.e. C 0 ct ( ct ) and C 0 it ( it ), also called marginal abatement costs, are increasing. By simpli…cation, C 000 ct ( ct ) = 0 and C 000 it ( it ) = 0 8i.

Each local jurisdiction i provides a local public good in quantity G it …nanced by its primary de…cit it . Similarly [START_REF] Disegni Eshel | Optimal Allocation of Tradable Pollution Rights and Market Structures[END_REF] , the State provides a national public good in quantity G ct …nanced by ct . The citizens'utility generated from the public goods provision is represented by the welfare function W t (G 1t ; :::; G it ; :::; G nt ; G ct ), which is twice continuously di¤erentiable, increasing and concave in each argument, with W 000 t = 0 by simpli…cation. We rule out spillover e¤ects and substitutability between public goods, i.e. @ 2 Wt @G it @Gct = 0,

@ 2 Wt @Gct@G ct 1 = 0, @ 2 Wt @G it @G jt = 0 and @ 2 Wt @G it @G it 1 = 0 8i; j.
The State perfectly observes [START_REF] Ellerman | A Note on Tradable Permits[END_REF] the de…cit d it and debt D it levels but has imperfect information about the marginal abatement cost C 0 it ( it ) of each local jurisdiction i. In order to tackle this asymmetric information problem, it organizes a local market for tradable de…cit permits which is supposed to ensure optimal sharing of the budgetary austerity among the local jurisdictions. In each period t, the State allocates itself t d t = d ct and splits

(1 t ) d t = n P i=1 d it between local jurisdictions, where t 2 [0; 1]
. The local entity i may either reduce its de…cit to the initial stock of rights d it or be involved in the market, but it must have enough permits to cover the total de…cit d it at the end of the period t. Let p t denote the endogenous price of a permit in t.

The primary de…cit it of a local jurisdiction i in the period t is then de…ned as the total de…cit d it less the debt interest payment rD it , to which either the receipts from selling permits p t d it d it > 0 are added or the expenses from buying permits p t d it d it < 0 are deducted:

it = d it + p t d it d it rD it ;
where r is the debt interest, D i1 the exogenous level of the debt at the beginning of period 1 and D i2 = D i1 + d i1 the level of the debt at the beginning of period 2 [START_REF] Gersbach | Markets and Regulatory Hold-Up Problems[END_REF] . The State, being [START_REF] Disegni Eshel | Optimal Allocation of Tradable Pollution Rights and Market Structures[END_REF] There is no explicit taxation in our model but lump-sum taxes would not change the behaviour of the players. [START_REF] Ellerman | A Note on Tradable Permits[END_REF] In many countries, the State has many pieces of information relative to local jurisdictions' accounts, insofar as local budgetary documents are transmitted to central administrative services, or publicly disclosed, and because the Public Treasury acts as a tax collector on behalf of local jurisdictions. [START_REF] Gersbach | Markets and Regulatory Hold-Up Problems[END_REF] Assuming that the debt lives in…nitely, the debt reimbursement tends to zero.

not involved in the market, uses all the rights it owns (d ct = d ct 8t) under the assumption ct < ct so that its primary de…cit in the period t is

ct = d ct rD ct ;
where D c1 is the exogenous level of the …rst-period debt and

D c2 = D c1 + d c1 is the second- period debt.
Contrary to the environmental economic literature, it can be noted that the marginal abatement cost does not depend only on the use of the common-pool resource, i.e. the de…cit in our setting. The endowment of rights, the debt level , and the market price also a¤ect the cost of being constrained. We …rst point out an inverse relation between the endowment of rights and the marginal abatement cost (

@( C 0 it ) @d it = p t C 00 it < 0
). An increase in the initial stock of rights induces a downwards translation of the marginal abatement cost, and conversely, as shown below by the Figure 1.

@( C 0 it ) @d it = (1 p t ) C 00
it < 0) but it creates a loss for the future period (

@( C 0 it+1 ) @d it = rC 00
it+1 > 0) due to the increase in the debt interest payment.

In addition, the negative slope (

@( C 0 it ) @d it = (1 p t ) C 00
it < 0) of the marginal abatement cost increases with respect to the market price ( @ 2 ( C 0 it ) @d it @pt = C 00 it (1 p t ) C 000 it (d it d it ) > 0 under the assumption C 000 it = 0). As shown by the Figure 2, the straight line of the marginal abatement cost pivots around the point d it = d it following a rise in the market price spots" jurisdictions with a high concentration of de…cits and "cool spots" jurisdictions that run high surpluses.

( @( C 0 it ) @pt = C 00 it (d it d it )).
The sequence of decisions is as follows. Before the game, the State is assumed to decide on the …rst-period level of rights for each local jurisdiction i and for itself. This initial decision is exogenous to the game to be played. In each period t, local jurisdictions play as Nash competitors when choosing their level of de…cit (d it ) and trading permits on the local market. Conversely, they play as Stackelberg leaders towards the State, i.e. they expect the way it will share the rights in period 2 and take into account its reaction function in their choice of …rst-period local de…cit. We solve the model by backward induction.

The sharing rule of de…cit rights

The …rst-period sharing of de…cit rights is exogenous so we focus on the second-period problem. The State assumes that the local jurisdiction i will provide the public good in

quantity b G i2 = d i2 rD i2 , with D i2 = D i1 + d i1 .
:; b G n2 ; G c2 ) C c2 ( c2 ) s.t. n X i=1 d i2 + d c2 = d 2 with b G i2 = d i2 rD i2 8i
G c2 = c2 = d c2 rD c2
which yields the following …rst-order conditions:

W 2 G c2 C 0 c2 = W 2 b G i2 8i (1)
and

n X i=1 d i2 + d c2 = d 2 : (2)
The internal sharing of de…cit rights in the second period is such that the value from distributing one additional right to the State, in terms of welfare and abatement cost, is equal [START_REF] Hagem | The Design of a Dynamic Tradable Quota System under Market Imperfections[END_REF] Under the assumption @ 2 Wt @Gct@Gct 1 = @ 2 Wt @Git@Git 1 = 0, the …rst-period welfare function does not in ‡uence the second-period sharing rule.

to the welfare from distributing one extra right to the jurisdiction i, and that for all i. As a consequence, the sharing rule equalizes the marginal welfare across all local jurisdictions.

The design of the sharing rule is crucial because it conditions the behaviour of local jurisdictions. A key issue consists in determining the reaction function of the State, i.e. how it will modify the second-period sharing of rights following an increase in the …rst-period de…cit of a local jurisdiction i. If the State feels pressure to increase the jurisdiction i's de…cit rights in period 2, it will create a soft budget constraint. Conversely, a hard budget constraint policy will be implemented if it resists temptation to reallocate rights ex post.

From the di¤erentiation of the …rst-order conditions (1) and ( 2) with respect to d i1 , d i2 8i and d c2 , we derive the State's budgetary response [START_REF] Joumard | Fiscal Relations across Government Levels[END_REF] :

Proposition 1: The State cannot credibly commit on a sharing rule of de…cit rights: it always …nds it optimal to deviate ex post from the originally stated second-period distribution of rights following an increase in jurisdiction i's …rst-period de…cit. Formally,

d(d i2 ) d(d i1 ) = 1 r n 2 1 r P k6 =i X k ! det A > 0; d(d c2 ) d(d i1 ) = 1 r n 2 1 r det A < 0; d(d k2 ) d(d i1 ) = 1 r n 2 X k det A < 0;
with

X i = C 00 c2 2 W 2 G 2 c2 r 2 W 2 b G 2 i2 < 0 17 Note that n X k=1 " n X i=1 d(d i2 ) d(d k1 ) + d(d c2 ) d(d k1 ) # = 0: and det A = 1 r n 1 1 r n X i=1 X i ! > 0: Proof: Appendix 1
The State is then unable to commit dynamically. While it …nds it optimal ex ante to deny additional rights in order to impose budgetary discipline, it always …nds it optimal to reallocate rights ex post when a local jurisdiction increases its de…cit in the previous period. Additional de…cit rights distributed to local jurisdiction i are shifted onto the other jurisdictions and the State, constituting a negative externality. In the following section, we analyse the impact of the soft budget constraint on the local market for tradable de…cit permits.

4 The local market for tradable de…cit permits

Local jurisdictions'behaviour

The market is supposed to be perfectly competitive. Each local jurisdiction behaves as a price taker. The Stackelberg leader's position gives local decision makers a strategic advantage over the State: they choose their …rst-period de…cit expecting the second-period reallocation of rights. Each local jurisdiction i aims at minimizing its cost of being constrained and the expenditure in permits over the two periods [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF] , discounted at the rate e: min

d i1 ;d i2 2 X t=1 1 (1 + e) t 1 C it ( it ) + p t (d it d it ) s.t. i1 = (1 p 1 ) d i1 + p 1 d i1 rD i1 i2 = (1 p 2 ) d i2 + p 2 d i2 r (D i1 + d i1 ) d(d i2 ) d(d i1 ) ; d(d k2 ) d(d i1 ) 8k 6 = i The solution d i1 (p 1 ; p 2 ; d i1 ; d(d i2 ) d(d i1 )
) is implicitly de…ned by the …rst-order condition:

(1 p 1 ) C 0 i1 ( i1 ) + r 1 + e C 0 i2 ( i2 ) + p 2 1 + e ( C 0 i2 ( i2 ) + 1) d(d i2 ) d(d i1 ) = p 1 : (3)
Compared with the following condition derived by Casella (1999) in a static case

(1 p 1 ) C 0 i1 ( i1 ) = p 1 ;
two additional e¤ects are at work. The …rst ( r 1+e C 0 i2 ( i2 )) is due to the dynamic of the model. The increase in the …rst-period de…cit weighs the debt interest payment down which, in turn, raises the second period marginal abatement cost. The second e¤ect (

p 2 1+e ( C 0 i2 ( i2 )+1) d(d i2 ) d(d i1 )
) comes from the inability of the State to commit dynamically to an internal sharing of de…cit rights. The expectation of additional rights in the second period -which lower both the cost of being constrained and the expenditure in permits -encourages the local jurisdiction to run more de…cits. A direct implication of the soft budget constraint is that too much de…cit is undertaken relative to the e¢ cient level.

On the contrary, the condition which implicitly de…nes the second-period de…cit

d i2 (p 2 ; d i2 ; d i1 ): (1 p 2 ) C 0 i2 ( i2 ) = p 2 ; (4)
turns out to be the one derived in the static case with a hard budget constraint. This result is not surprising given the temporal horizon. In the second period, the decision maker does not take into account either the budgetary position he leaves its successor or the future sharing of rights. The second-period de…cit allocation thus equalizes the marginal cost of …scal retrenchment across all jurisdictions:

C 0 i2 ( i2 ) = p 2 (1 p 2 ) = C 0 j2 ( j2 ) 8i; j; (5)
which is a standard result in the environmental economic literature.

On the other hand, this condition is altered in the …rst period by the soft budget constraint, as stated by Proposition 2 and its corollary.

Proposition 2: The sharing of …scal retrenchment e¤orts among the local jurisdictions in the …rst period is conditioned by the expectation of the second-period rights reallocation.

The standard condition of marginal abatement costs equalization changes in a non-standard condition

C 0 i1 + p 2 (1 + e) (1 p 1 ) (1 p 2 ) d(d i2 ) d(d i1 ) = C 0 j1 + p 2 (1 + e) (1 p 1 ) (1 p 2 ) d(d j2 ) d(d j1 ) 8i; j: (6
d (d it ) dp t = C 0 it (1 p t ) C 00 it (d it d it ) 1 (1 p t ) 2 C 00 it : Proof: Given C 0 it 0 and C 00 it 0, d(d it ) dpt < 0 for d it > d it . On the contrary, d(d it ) dpt > 0 for (1 p t ) C 00 it (d it d it ) > C 0 it + 1.
Contrary to the standard result which states that the use of the common-pool resource decreases with the market price, the growing cost of buying permits may increase the de…cit of badly endowed jurisdictions. For the following, let us assume that the impact of the market price on the aggregated de…cit demand is negative:

Assumption 1: n X i=1 d (d it ) dp t = n P i=1 (C 0 it 1) (1 p t ) n P i=1 C 00 it (d it d it ) (1 p t ) 2 n P i=1 C 00 it < 0
Result 2: The …rst-period …scal retrenchment decreases with the softness of the budget constraint. Formally,

d (d i1 ) d d(d i2 ) d(d i1 ) = 1 (1+e) p 2 (1 p 2 ) (1 p 1 ) 2 C 00 i1 > 0:
The level of the …rst-period de…cit depends on the credibility of the second-period sharing rule. The expectation of additional rights results in an opportunistic behaviour which leads to lower budgetary e¤orts. These perverse incentives increase with the p 2 value of one additional right but decrease with the p 1 cost of one extra unit of de…cit in the …rst period.

Result 3: Under a hard budget constraint policy, an increase in p 2 always deters the local jurisdiction from running de…cit in the …rst period. On the contrary, the soft budget constraint phenomenon encourages the budgetary indiscipline of the local jurisdiction, following a rise in p 2 , for d(d i2 ) d(d i1 ) > r. Formally,

d (d i1 ) dp 2 = h d(d i2 ) d(d i1 ) r i (1 + e) (1 p 1 ) 2 (1 p 2 ) 2 C 00 i1 Proof: for d(d i2 ) d(d i1 ) = 0, d(d i1 ) dp 2 = r (1+e)(1 p 1 ) 2 (1 p 2 ) 2 C 00 i1 < 0 and for d(d i2 ) d(d i1 ) > r, d(d i1 ) dp 2 > 0. For d(d i2 ) d(d i1 ) > r (resp. d(d i2 ) d(d i1 ) < r
), the expected gain in terms of the valuation of additional rights is higher (resp. lower) than the increasing cost of the debt following a rise in p 2 which encourages (resp. discourages) the local policy-maker to borrow.

Finally, and obviously, the more rights the local jurisdiction owns, the lower its de…cit

d (d it ) d(d it ) = p t 1 p t < 0:
With a plethora of rights, the market price is equal to zero so that the distribution of rights does not a¤ect the budgetary behaviour.

The market price at the equilibrium

For a given initial sharing of rights, the …rst-period equilibrium price p m 1 on the tradable de…cit permits market is implicitly de…ned by the following market clearing condition:

n X i=1 d i1 p m 1 ; p 2 ; d i1 ; d(d i2 ) d(d i1 ) = n X i=1 d i1 (7)
and similarly, the second-period equilibrium price p m 2 is implicitly de…ned by:

n X i=1 d i2 p m 2 ; d i2 ; d i1 = n X i=1 d i2 ; (8)
under the assumption that the unconstrained demand is higher than the total rights:

n X i=1 d i1 > n X i=1 d i1 and n X i=1 d i2 > n X i=1 d i2 :
The case where the equilibrium price is zero is ignored as it is neither realistic nor interesting.

From the …rst-order conditions ((3) and ( 4)) and the market clearing conditions (( 7)

and ( 8)), we derive

p m 1 = p m 1 p 2 ; n X i=1 d i1 ; d(d 2 ) d(d 1 ) ! (9)
where

d(d 2 ) d(d 1 ) = d(d 12 ) d(d 11 ) ; :::; d(d n2 ) d(d n1 )
, and

p m 2 = p m 2 n X i=1 d i1 ; n X i=1 d i2 ! : (10)
Result 4: Under Assumption 1, the soft budget constraint exerts an upwards pressure on the …rst-period market price. Formally [START_REF] Kopits | Fiscal policy rules[END_REF] ,

dp m 1 d d(d i2 ) d(d i1 ) = @d i1 (:) @ d(d i2 ) d(d i1 ) n P i=1 @d i1 (:) @p 1 > 0:
Intuitively, the expectation of additional rights in the second period increases the …rstperiod demand for de…cit permits which, in turn, exerts pressure on the market resulting in a higher …rst-period price. However, the price increase will tend to counteract attempts to encourage de…cit, at least for net seller jurisdictions from Result 1.

Result 5: As expected, under Assumption 1, the market price is a decreasing function of the same period local de…cit rights ceiling [START_REF] Kornai | Resource-Constrained Versus Demand-Constrained Systems[END_REF] :

dp m t d n P i=1 d it = @ n P i=1 d it (:) @ n P i=1 d it + 1 n P i=1 @d it (:) @pt < 0
Result 6: Given Assumption 1, the …rst-period market price always decreases with the rise of the second-period market price under a hard budget constraint policy [START_REF] Kornai | The Soft Budget Constraint[END_REF] . The soft budget constraint reverses this negative intertemporal relation between the market prices for d(d i2 )

d(d i1 )
high enough 8i. Formally,

dp m 1 dp 2 = n P i=1 @d i1 (:) @p 2 n P i=1 @d i1 (:) @p 1
: [START_REF] Kopits | Fiscal policy rules[END_REF] The sign of this expression follows from Assumption 1 and Result 2.

Proof:

for d(d i2 ) d(d i1 ) = 0 8i, n P i=1 d(d i1 ) dp 2 = n P i=1 r (1+e)(1 p 1 ) 2 (1 p 2 ) 2 C 00 i1 < 0 which leads to dp m 1 dp 2 < 0 under Assumption 1. For n P i=1 d(d i2 ) d(d i1 ) r (1+e)(1 p 1 ) 2 (1 p 2 ) 2 C 00 i1 > 0, which is always the case for d(d i2 ) d(d i1 ) > r 8i, dp m 1 dp 2 > 0 under Assumption 1.
Under a hard budget constraint policy, the intertemporal relation between the market prices is always negative. The expected increase in the second-period cost of being constrained reduces the …rst-period demand for permits and thus the …rst-period market price. On the contrary, the increasing valuation of one extra right in the second period pushes the local jurisdiction to run more de…cits in the …rst period under a soft budget constraint policy which induces a positive relation between the market prices for d(d i2 ) d(d i1 ) high enough for all i.

Impact of the soft budget constraint at the equilibrium

@d m i1 (:) @ d(d i2 ) d(d i1 ) = @d i1 (:) @ d(d i2 ) d(d i1 ) + @d i1 (:) @p m 1 @p m 1 (:) @ d(d i2 ) d(d i1 ) + @d i1 (:) @p m 1 @p m 1 @p m 2 @p m 2 @ n P i=1 d i2 @ n P i=1 d i2 @ d(d i2 ) d(d i1 ) + @d i1 (:) @p m 2 @p m 2 @ n P i=1 d i2 @ n P i=1 d i2 @ d(d i2 ) d(d i1 ) @d m j1 (:) @ d(d i2 ) d(d i1 ) = @d j1 (:) @p m 1 @p m 1 (:) @ d(d i2 ) d(d i1 ) + @d j1 (:) @p m 1 @p m 1 @p m 2 @p m 2 @ n P i=1 d i2 @ n P i=1 d i2 @ d(d i2 ) d(d i1 ) + @d j1 (:) @p m 2 @p m 2 @ n P i=1 d i2 @ n P i=1 d i2 @ d(d i2 ) d(d i1 ) A change in d(d i2 ) d(d i1 ) directly in ‡uences d m i1 (:) -@d i1 (:) @ d(d i2 ) d(d i1 )
being always positive from Result 2 -while it a¤ects both d m i1 (:) and d m j1 (:) 8j 6 = i in an indirect manner through changes in equilibrium prices. The term @d k1 (:)

@p m 1 @p m 1 (:) @ d(d i2 ) d(d i1 )
for k = 1; :::; n represents the indirect e¤ect on jurisdiction k's level of unabated de…cit that occurs when p m 1 adjusts to the variation in demand, which is always negative for a net seller, i.e. d k1 > d k1 , but may be positive for a net buyer, i.e. d k1 < d k1 , from Results 1 and 4. The two other e¤ects go through the reduction in p m 2 following a higher amount of rights allocated at local level. From Results 1, 3 and 6 it turns out that the impact of d(d i2 ) d(d i1 ) via p m 2 on d m k1 (:) depends on the initial endowment as well as on the comparison between d(d i2 )

d(d i1 ) and r. We do not rule out the case where

@d m i1 (:) @ d(d i2 ) d(d i1 ) < 0 and @d m j1 (:) @ d(d i2 ) d(d i1 )
> 0. However, the following condition obtained by di¤erentiating (7) must be satis…ed:

@d m i1 (:) @ d(d i2 ) d(d i1 ) = X k6 =i @d m k1 (:) @ d(d i2 ) d(d i1 ) : If a rise in d(d i2 ) d(d i1
) causes an increase in local jurisdiction i's equilibrium level of unabated de…cit, the equilibrium levels of unabated de…cit for all other jurisdictions will decrease in equal magnitude, and conversely.

The ine¢ ciency of the tradable de…cit permits market

; d n2 2 X t=1 1 (1 + e) t 1 n P i=1 C it ( it ) + p t ( n P i=1 d it n P i=1 d it ) s.t. i1 = (1 p 1 ) d i1 + p 1 d i1 rD i1 8i i2 = (1 p 2 ) d i2 + p 2 d i2 r (D i1 + d i1 ) 8i d(d i2 ) d(d i1 ) ; d(d k2 ) d(d i1 ) 8k 6 = i; 8i:
The …rst-period and second-period budgetary e¤orts are implicitly de…ned by the following …rst-order conditions:

(1 p 1 ) C 0 i1 + r 1 + e C 0 i2 + p 2 1 + e ( C 0 i2 + 1) d(d i2 ) d(d i1 ) + X k6 =i p 2 1 + e ( C 0 k2 + 1) d(d k2 ) d(d i1 ) = p 1 8i (11)
(1 p 2 ) C 0 i2 p 2 = 0 8i: (12)
By comparing ( 11) and ( 3), it turns out that the optimal level of the …rst-period de…cit from the local planner's point of view is lower than the level of de…cit chosen by the self-interested local jurisdiction i 22 . Without any regulation on the market, the local jurisdiction i does not take into account the budgetary externality it generates for the other jurisdictions because the fact that the additional de…cit rights allocated to one jurisdiction are partially borne by others. The …rst-period market allocation is thus ine¢ cient from the local planner's point of view. On the contrary, the second-period market allocation is e¢ cient.

Suppose now that an omniscient and benevolent national planner, implementing a hard budget constraint policy, aims at minimizing the total costs of being constrained of the n heterogeneous local jurisdictions. It turns out that the optimal allocation equalizes the marginal abatement cost in each period. [START_REF] Kornai | Understanding the Soft Budget Constraint[END_REF] Indeed, the additional term

P k6 =i p2 1+e ( C 0 k2 + 1) d(d k2 ) d(di1) is negative from Proposition 1.

Intertemporal trading of de…cit permits

In addition to de…cit permits trading between local jurisdictions, we now allow de…cit permits trading through time. In such a setting, local jurisdictions may directly reduce de…cit as well as buy, sell, bank and borrow de…cit permits in order to meet applicable standards or to take advantage of any speculative opportunities. For a de…cit d i1 < d i1 , the local jurisdiction i generates a permits surplus which may be sold to other jurisdictions, or deposited in a bank account to be used by the local jurisdiction itself, or sold to other jurisdictions in period 2.

On the contrary, for a de…cit d i1 > d i1 , the local jurisdiction i may buy permits from other jurisdictions or borrow permits, but its budget must be balanced at the end of period 2. Let

x i1 > 0 (resp. < 0) denote the quantity of de…cit permits bought (resp. sold) and B i1 > 0 (resp. < 0) denote the stock (resp. the borrowing) of permits. The intertemporal de…cit permits trading thus lowers the cost of compliance with the de…cit standards imposed by the State by allowing local jurisdictions to administer their budget more ‡exibly over the two periods.

At …rst sight, the intertemporal trading of de…cit permits seems to be an additional tool at local jurisdiction's disposal for manipulating its …rst-period budgetary choice and obtaining more de…cit rights in the second period. By banking de…cit permits, it increases its …rst-period de…cit and thus receives additional rights in the second period.

The local jurisdiction i chooses the …rst-period de…cit d i1 , the second-period de…cit d i2 as well as the stock of permits B i1 minimizing its cost of being constrained and the expenditure in permits over the two periods:

min d i1 ;d i2 ;B i1 2 X t=1 1 (1 + e) t 1 [C it ( it ) + p t x it ] s.t. i1 = d i1 p 1 x i1 rD i1 i2 = d i2 p 2 x i2 r (D i1 + d i1 ) x i1 = d i1 d i1 + B i1 x i2 = d i2 d i2 B i1 : d(d i2 ) d(d i1 ) ; d(d k2 ) d(d i1 ) 8k 6 = i
This problem with intertemporal ‡exibility yields the following necessary conditions:

(1 p 1 ) C 0 i1 ( i1 ) + r 1 + e C 0 i2 ( i2 ) + p 2 1 + e ( C 0 i2 ( i2 ) + 1) d(d i2 ) d(d i1 ) = p 1 8i (13)
(1 p 2 ) C 0 i2 ( i2 ) = p 2 8i (14)
p 1 ( C 0 i1 + 1) = p 2 ( C 0 i2 + 1) 1 + e 8i:

The local jurisdiction i's stock of de…cit permits B i1 p 1 ; p 2 ; d i1 ; d i2 ; d i1 ; d i2 , given by the condition (15), equalizes the marginal costs of the two periods. To some extent, the local jurisdiction smoothes the cost of budgetary austerity over the two periods.

Proposition 3: The intertemporal trading of de…cit permits restores the e¢ ciency of the market for tradable de…cit permits Proof: directly from ( 14) and (15).

By allowing local jurisdictions to trade permits through time, the State thus restores the standard condition of marginal abatement costs equalization in the …rst period. The ine¢ ciency related to the soft budget constraint phenomenon vanishes.

Conclusion

The way rights are allocated and traded on the market is decisive for the cost-e¤ectiveness of the system. The inability of the State to commit dynamically to a sharing rule of de…cit rights generates perverse incentives. Local jurisdictions are encouraged to run more de…cits in the …rst period in order to receive more rights in the second period, which in turn exerts an upwards pressure on the …rst-period market price. The market turns out to be ine¢ cient -with heterogeneous jurisdictions -unless the State allows local decision makers to trade permits through time. Indeed, the intertemporal trading of de…cit permits restores the marginal abatement costs equalization condition due to the fact that local jurisdictions aim at smoothing their cost of being constrained over the two periods.

Appendix

7.A Proof of Proposition 1

The di¤erentiation of the …rst-order conditions (1) and ( 2) with respect to d i1 , d i2 8i, d c2 leads to: To simplify notation, in the following, we denote

d(d i1 ) = 1 r d(d i2 ) + C 00 c2 2 W 2 G 2 c2 r 2 W 2 b G 2 i2 d(d c2 ) 8i
X i = C 00 c2 2 W 2 G 2 c2 r 2 W 2 b G 2 i2
< 0.

7.B Proof of Proposition 2

By substituting C 0 i2 ( i2 ) by the expression given by the …rst-order condition (4) in the …rst-order condition (3), the n conditions (3) boil down to:

Figure 1 :

 1 Figure 1: E¤ects of the initial stock of rights

Figure 2 :

 2 Figure 2: E¤ects of an increase in the market price

Result 7 :

 7 The impact of local jurisdiction i's soft budget constraint on its cost minimizing level of unabated de…cit d m i1 p m 1 ; p m 2 ; d i1 ; d(d i2 ) d(d i1 ) as well as on local jurisdiction j's cost minimizing level of unabated de…cit d m j1 p m 1 ; p m 2 ; d j1 ; d(d j2 ) d(d j1 ) , when evaluated at the equilibrium prices, are obtained by di¤erentiating these two expressions d m i1 (:) and d m j1 (:) with respect to d(d i2 ) d(d i1 ) : Following an increase in d(d i2 ) d(d i1 ) , the respective demand for de…cit of local jurisdiction i and local jurisdiction j 8j 6 = i, change as follows:

  Suppose that an omniscient and benevolent planner towards the local jurisdictions acts instead of each decision maker on the local market over the two periods. In this joint-cost problem, the local regulator's problem is to minimize the total costs of being constrained of the n heterogeneous local jurisdictions, expecting the inability of the State to commit dynamically: min d 11 ; :::; d n1 d 12 ; :::

  i2 ) + d(d c2 ) = d(d 2 ) with d(d 2 ) = 0:

From

  compute the reaction function with the Cramer's rule. Q.E.D.

  1 ) C 0 j1 ( j1 ) + 1 1 + e p 2 (1 p 1 ) (1 p 2 ) d(d j2 ) d(d j1 )8i; j:Q.E.D.

  Due to the asymmetric informationthe local abatement cost being private information -the State is indeed unable to predict the market-based allocation of permits, the market price and, consequently, the local public good provision which justi…es the use of b G i2 8i. The State designs the sharing rule so that it maximizes the welfare function minus its cost of being constrained 16 :

max d 12 ;:::;d i2 ;:::;d n2 ;d c2 W 2 ( b G 12 ; :::; b G i2 ; ::

  To sum up, equilibrium market conditions are a¤ected by the inability of the State to commit dynamically. By di¤erentiating (3) and (4), we calculate respectively the e¤ects of changes in p 1 , p 2 , d i1 and d(d i2 ) d(d i1 ) on the local jurisdiction i's …rst-period cost-minimizing level of de…cit d i1 (p 1 ; p 2 ; d i1 ; d(d i2 ) d(d i1 ) ), as well as the impact of p 2 , d i2 and d i1 on d i2 (p 2 ; d i2 ; d i1 ). The main results are stated below. Result 1: A rise in the market price always induces a de…cit reduction for a net seller, i.e. a local jurisdiction with d it > d it , but may increase the de…cit of a net buyer, i.e. a local jurisdiction with d it < d it , in the same period. Formally,

	2 W 2 b G 2 i2	=	2 W 2 b G 2 j2	8i; j
	which results in
	d(d i2 ) d(d i1 )	=	d(d j2 ) d(d j1 )	8i; j:

) Proof: Appendix 2 Corollary of Proposition 2: Under a hard budget constraint policy, the condition of marginal abatement costs equalization is satis…ed again. Under a soft budget constraint policy, marginal abatement costs coincide for all local jurisdictions if, and only if, preferences for the local public good are identical

The need to secure …scal discipline is not a major concern only in EMU Member States. Each decentralised country aims at ensuring the sustainability of public …nances while exploiting the e¢ ciency gains of decentralisation

.[START_REF] Baumol | The Theory of Environmental Policy[END_REF] See[START_REF] Joumard | Fiscal Relations across Government Levels[END_REF] and[START_REF] Sutherland | Fiscal Rules for Sub-Central Governments: Design and Impact[END_REF].

This way has also been considered by Italia (Commissione Tecnica per la Spesa Pubblica, 1998).

In practice, it has not been implemented because the incentives for trading permits on a market have been blunted by a soft budget balance objective.

The theoretical foundation is based on the notion of property rights developed by[START_REF] Coase | The Problem of Social Cost[END_REF]. Dales (1968) pointed out its applicability for water and Crocker (1966) for air.[START_REF] Montgomery | Markets in licenses and e¢ cient pollution control programs[END_REF] formally solved the problem of proving the existence of cost-e¤ective permit market equilibrium. See Cropper and Oates (1992) for a survey.

A famous application of a market based system of tradable pollution permits follows the 1990 Clean Air Act Amendments.

The environmental economic literature has paid scant attention to the temporal inconsistency problem.See[START_REF] Batabyal | Consistency and Optimality in a Dynamic Game of Pollution Control I: Competition[END_REF],[START_REF] Gersbach | Markets and Regulatory Hold-Up Problems[END_REF],[START_REF] Marsiliani | Time inconsistency in environmental policy: tax earmarking as a commitment solution[END_REF],[START_REF] Petrakis | Location Decisions of a Polluting Firm and the Time Consistency of Environmental Policy[END_REF].

This amount implicitly ensures the long-term solvency of public administrations.

For a negative value of d it (resp. d ct ), the jurisdiction i (resp. the State) runs a surplus.

d t could correspond to 3% of GDP of a EMU Member State.

In a political economy model, this cost would be higher the year before the election.

The local decision-maker does not take the consequences of its budgetary decisions for the future periods (t > 2) into account. A variant would consist in adding a …nal-value term which captures the value of damages for all time periods after t = 2.

The sign of this expression follows from Assumption 1 and @dit(:) @dit < 0.

The sign of this expression follows from Assumption 1 and Result 3.