Emilie Alberola
email: emilie.alberola@caissedesdepots.fr

Julien Chevallier

European Carbon Prices and Banking Restrictions : Evidence from Phase I (2005-2007)

Keywords: Q28, Q52, Q58 Carbon emissions trading, EU ETS, Banking, Borrowing, Hotelling rule, Ellerman-Parsons ratio

The price of European Union Allowances (EUAs) has been declining at far lower levels than expected during Phase I (2005-2007). Previous literature identifies among its main explanations over-allocation concerns, early abatement efforts in 2005 and possibly decreasing abatement costs in 2006. We advocate low allowance prices may also be explained by banking restrictions between 2007 and 2008 which undermine the ability of the EU ETS to provide an efficient price signal. Based on a Hotelling-type analysis, our results suggest EUA prices do not reflect adequately abatement costs. We also give evidence that the French ban on banking and the expected allowance scarcity at the end of Phase I computed by the Ellerman-Parsons ratio contribute to the explanation of low EUA prices. This situation may be interpreted as a sacrifice of the temporal flexibility offered to industrials in Phase I to give a chance to correct design inefficiencies and achieve an efficient price pattern leading to effective abatement efforts in Phase II.

Each year, when abatement together with endowment of allowances are above emissions levels, regulated agents may bank surplus allowances for potential later use. Conversely, if regulated agents do not abate enough to cover their emissions level with their actual endowment, they may borrow allowances from future allocations. A fundamental statement on emissions trading is that allowance trading is efficient over time only if banking and borrowing are authorized [START_REF] Rubin | A model of Intertemporal Emission Trading, Banking, and Borrowing[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]). With such provisions, market prices reflect opportunity costs leading to an efficient choice of abatement measures [START_REF] Schleich | Banning Banking in EU Emissions Trading?[END_REF]). The most prominent example of the key role of such provisions is the US Acid Rain Program where banking has been a major feature in the success 1 of this emissions trading scheme [START_REF] Ellerman | Markets for Clean Air: The US Acid Rain Program[END_REF], [START_REF] Ellerman | The Efficiency and Robustness of Allowance Banking in the U.S. Acid Rain Program[END_REF]).

None empirical analysis on the impact of the intra-period banking coupled with a ban on inter-period banking on EUA prices has yet been realised. In a game simulation, [START_REF] Ehrhart | The Role of Auctions and Forward Markets in the EU ETS: Counterbalancing the Cost-Inefficiencies of Combining Generous Allocation with a Ban on Banking[END_REF] found that a ban on banking leads to an inefficient adjustment with first an under-investment in abatement technologies and a low allowance price during 2005-2007; and second a more stringent cap with a price peak and over investment in emission reduction during 2008-2009. Among the main explanations of low allowance prices towards the end of Phase I, the previous literature identifies over-allocation concerns, early abatement efforts in 2005 due to high allowance prices, and possibly decreasing abatement costs in 2006 due to abnormal temperatures and switching from coal-to gas-fired electricity in a context of falling natural gas prices com-1 Introduction 3 pared to coal [START_REF] Buchner | The European Union Emissions Trading Scheme: Origins, Allocation, and Early Results[END_REF], [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF]). Therefore, a thorough analysis of banking and borrowing provisions is missing and it appears necessary to disentangle those effects on allowance prices which develop differently in the following two cases. If inter-period banking is allowed, it is reasonable to expect that allowance price changes do not exceed Hotelling's rule, rising at the market rate of interest 2 . If inter-period banking is restricted, lower Phase I prices and higher Phase II prices are expected. The former result is due to the validity of allowances which is shorter than the time horizon required by investors. The latter is due to increased allowance scarcity compared to full inter-period banking.

Within 2005-2007, participants are expected to use unrestricted banking and borrowing. Based on this assumption, we test in an empirical approach whether the allowance price pattern is consistent with a competitive equilibrium in the intertemporal market when banking and borrowing provisions are allowed. [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF]'s theoretical model of the intertemporal allowance market applied by [START_REF] Helfand | Testing for dynamic efficiency of the Sulfur Dioxide Allowance Market[END_REF] to the US SO 2 market guides our Hotelling-type analysis. The impact of the late restriction on inter-period banking is evaluated through dummy variables representing official communications between France, Poland and the European Commission (EC). The probability of allowance shortage at the end of Phase I computed by [START_REF] Ellerman | Shortage, Inter-period Pricing, and Banking[END_REF] ratio (EPR) 3 is used to characterize how the allowance price reacts to the environmental policy constraint during Phase I.

Compared to the literature on efficient banking in the US SO 2 Program, our results show the inter-period banking restriction undermines the ability of the EU ETS to provide an efficient price signal. Phase I allowances do not appear to reflect adequately abatement costs. Following the disclosure 2 With complete information and emission targets being known, [START_REF] Hotelling | The Economics of Exhaustible Resources[END_REF] shows the percentage change in net-price per unit of time of an exhaustible resource should be equal to the discount rate to maximize the present value of the resource capital over the extraction period. Otherwise, arbitrage between periods is possible and the allowance price path will lead to non socially desirable outcomes such as a concentration of emissions on early periods [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF]). If borrowing is allowed, then the allowance price follows the Hotelling rule for exhaustible resources. If borrowing is forbidden, then the allowance price rises at a rate inferior to the interest rate.

3 Briefly defined as the December, 2007 delivery allowance price over the December, 2008 delivery allowance price plus a 40 = C penalty for non compliance.

of 2005 verified emissions on April, 2006 we provide a sharper explanation of low EUA prices with both the EPR ratio and French official communications related to the ban on banking being significant. These results are robust to the introduction of energy market shocks and weather conditions previously identified as being the main determinants of EUA prices.

The paper is organized as follows. Section 2 provides an assessment of the banking and borrowing provisions in the EU ETS. Section 3 tests the effects of banking restrictions on allowance prices and presents the results.

Background

This section reviews the environmental and economic effects of banking and borrowing, the motives that led Member States to ban banking and borrowing between 2007 and 2008, and the allowance price development along with its charateristic structural break on April, 2006.

Environmental and economic effects of banking

In terms of environmental effects, allowance banking and borrowing may change the temporal path of emissions and aggregate emissions. While banking reduces social damages in presence of a convex damage function coming from emissions and stricter future standards [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF]), unrestricted borrowing may have negative consequences with a concentration of emissions on early periods by delaying abatement decisions 4 . To correct these unwanted allowance paths, the regulator may introduce a non unitary Intertemporal Trading Ratio (ITR) 5 including interests on banking and dis-couraging borrowing : if firms borrow a lot of allowances in early periods, they will reimburse more allowances than actually used in the next period. Furthermore, banking provisions could affect the rate of non-compliance and the resulting excess emissions6 .

In terms of economic effects, the theoretical literature suggests allowance banking and borrowing may improve economic efficiency under specific assumptions7 . First, banking links future allowance prices to the spot price as stated by [START_REF] Maeda | Impact of Banking and Forward Contracts on Tradable Permit Markets[END_REF]. Second, banking and borrowing improve price stability [START_REF] Ellerman | The Temporal Efficiency of SO2 Emissions Trading[END_REF]). If inter-period banking is not allowed, allowance prices are likely to be unstable at the end of each compliance period. In case of surplus, allowances are worthless and their price should fall to zero. In case of excess emissions, the allowance price should rise sharply at the end of the period. Third, banking and borrowing improve liquidity in the allowance market by increasing the quantity of allowances available to the market8 and the volume of allowances traded [START_REF] Godby | Emissions Trading with Shares and Coupons when Control of Discharges is Uncertain[END_REF]).

Following this review of the various effects of banking and borrowing that need to be accounted for when tailoring environmental regulation, we expose in the next section the motivation of MS leading to the inter-period ban on banking.

Reasons to ban banking between 2007 and 2008

MS decided to prohibit the transfer of unused allowances from the EU ETS Phase I (2005-2007) into the Kyoto Protocol (KP) first commitment period (2008)(2009)(2010)(2011)(2012). Therefore, installations covered by the EU ETS may not use atmosphere", increased by the difference between the discount rates of firms and the social planner.

banked allowances during Phase I to meet their obligations during Phase II (2008)(2009)(2010)(2011)(2012) except through advance agreements for purchase of Certified Emissions Reductions (CERs)9 . Neither may installations use borrowed allowances between the two periods10 .

Discussions on the banking feature were characterized by sudden changes. At the beginning of the EU ETS in 2005, all MS decided, with the exception of Poland and France, against the inter-period transfer of allowances: all allowances not surrendered by the end of 2007 will be cancelled and not transferred to the subsequent five-years implementation period. Since then, Poland and France decided to ban the transfer of allowances to Phase II.

Two main reasons may explain this inter-period ban on banking by MS vis-à-vis their Kyoto commitment [START_REF] Ehrhart | The Role of Auctions and Forward Markets in the EU ETS: Counterbalancing the Cost-Inefficiencies of Combining Generous Allocation with a Ban on Banking[END_REF], [START_REF] Schleich | Banning Banking in EU Emissions Trading?[END_REF]). First, the transfer of banked allowances from 2007 to 2008 may weaken the ability of a MS to meet its Burden-Sharing Agreement target starting in 2008 and second, large quantities of banked allowances may trigger the Commitment Period Reserve (CPR) rule which postulates to keep a reserve of Assigned Amount Units (AAUs)11 above 90%. Second, the interperiod ban on banking avoids negative side effects at the EU ETS level. It might have been problematic for MS to forecast in 2006 the amount of banked allowances when drawing up their National Allocation Plans (NAPs) for 2008-2012. In presence of unexpected large amounts of banked allowances, sectors non-covered by the EU ETS need to make additional abatement efforts12 . As displayed in Figure 1, the dataset is divided into two sub-periods due to the presence of one structural break following the simultaneous releases of 2005 verified emissions by the Walloon Region of Belgium, France and Spain which serve as a proxy for the adjustment of agents' expectations. Using the method developed by [START_REF] Lee | Testing the Null of Stationarity in the Presence of a Structural Break[END_REF] and [START_REF] Lee | Minimum LM Unit Root Test with Two Structural Breaks[END_REF] 14 that endogenously looks for structural breaks while testing for the existence of a unit root, we identify April 20, 2006 as a breakpoint in our dataset and exclude extreme price changes from our regressions. Two subperiods need to be considered : the "before the compliance break" period from 2005/07/01 to 2006/04/20 and the "after the compliance break" period from 2006/06/22 to 2007/05/31. This endogenous structural break may be associated to institutional features of the EU ETS during Phase I. As 54% of the EUA spot price adjustment was made within four days 15 starting on April 24, 2006 this break eliminates prior uncertain information and reveals agents' net short/long positions.

As expressed before, the main explanations of low EUA prices toward zero the end of Phase I include over-allocation concerns, the possibility early abatement and the influence of climatic and energy variables [START_REF] Buchner | The European Union Emissions Trading Scheme: Origins, Allocation, and Early Results[END_REF], Mansanet-Bataller et al. ( 2007)) The model estimated in the next section evaluates the specific role played by banking and borrowing provisions that may contribute to a sharper explanation of low allowance price levels.

Analysis

In this section, we describe first the theoretical framework from which we derive our estimation strategy. Second, we explicit our econometric specification. Third, we present the data sources. Fourth, we interpret the results.

Rationale behind a Hotelling-type analysis

The model we estimate is based on two strands of literature developped by [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] and [START_REF] Slade | Hotelling Confronts CAPM: A Test of the Theory of Exhaustible Resources[END_REF] which were first applied to the US SO 2 market by [START_REF] Helfand | Testing for dynamic efficiency of the Sulfur Dioxide Allowance Market[END_REF].

First, [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF] studies the banking behavior of regulated industrials in the US Acid Rain Program and implicitly the behavior of spot prices in a stochastic, continous-time, infinite horizon model for allowance allocation, use and storage. Under certainty, the model predicts that the price path would increase smoothly at the rate of interest according to the Hotelling rule. Under uncertainty, the model features risk-neutral agents minimizing their abatement costs under a banking constraint and with a discount rate specific to risky assets in the spirit of the capital asset pricing model (CAPM). The solution to this problem is a continuous time version of [START_REF] Pindyck | The Present Value Model of Rational Commodity Pricing[END_REF]'s model of rational commodity pricing:

E t [p t+1 ] = (1 + µ)p t -ψ t (1)
with E t the expectation value given the information available at time t, p t+1 the commodity price at time t + 1, r the risk free interest rate, ρ the risk premium specific to holding allowances as an asset in a diversified portfolio of investments, µ = r + ρ , and ψ t a convenience yield16 . Eq. ( 1) therefore represents the basic relationship we want to test. Second, assuming an allowance may be considered as an exhaustible resource17 , [START_REF] Slade | Hotelling Confronts CAPM: A Test of the Theory of Exhaustible Resources[END_REF] provide an analogous theoretical framework by maximizing the expected present value of a risk adjusted profit with re-spect to the state of the resource bank 18 and a random productivity shock. At the equilibrium, the evolution of the allowance price p t follows a Hotelling-CAPM relationship between the risk-free interest rate, the investment rate of return in a diversified portfolio and the risk premium specific to the asset similar to eq. ( 1).

As developed in [START_REF] Helfand | Testing for dynamic efficiency of the Sulfur Dioxide Allowance Market[END_REF], we rearrange eq. ( 1) to isolate first-difference prices on the left-hand side:

E t p t+1 -p t = r f t p t + ρ t p t -ψ t (2) Rewriting ρ t = σ am σ mm (r m t -r f t )
, which is standard practice for CAPM, yields:

E t p t+1 -p t = r f t p t + σ am σ mm (r m t -r f t ) p t -ψ t (3)
where r f t is the risk-free rate, r m t is the rate of return on the market portfolio, σ am is the covariance between the rate of return of EUA prices and r m t , and σ mm is the variance of r m t . The first term r f t p t represents the Hotelling rule for cost-minimizing intertemporal arbitrage in the EU ETS. The second term σam σmm is the risk premium for holding allowances as part of a diversified portfolio. The expression (r m t -r f t ) is the excess return on the market portfolio at time t.

Since the expected value of p t+1 is known only with errors at time t, we substitute E t p t+1 by p t+1 + t+1 :

p t+1 -p t = r f t p t + σ am σ mm (r m t -r f t ) p t -ψ t + t+1 (4)
where the dependent variable is the first log-differenced EUA price series and the error term. Next, assuming the convenience yield is constant (ψ t = ψ) and adding dummy variables to get rid of extreme price changes, we get:

p t+1 -p t = α + β 1 r f t p t + β 2 (r m t -r f t ) p t + β 3 p t min + β 4 p t max + t+1 (5)
where α = -ψ, β 2 = σam σmm , p t min and p t max the dummy variables for min-imum and maximum price changes. In eq. ( 5), the null hypothesis β 1 = 1 tests the Hotelling rule and β 2 provides information on the CAPM risk premium for CO 2 allowances which is the difference between the expected return on allowances and the return of the risk-free asset.

Environmental policy constraint

The EC defined the environmental policy constraint by validating each NAP before the launch of the EU ETS on January 1, 2005 even if some NAPs were validated after the beginning of the scheme. During 2006-2007, MS are currently negociating their NAP for Phase II.

As noted previously, France and Poland initially decided to authorize industrials to transfer allowances to Phase II. Governments changed this option during NAPs II negotiations until the EC final decision on March 26, 2007 that validated the ban on banking. To capture these effects, we introduce two dummy variables, banf r for France and banpl for Poland, that correspond to official communications between MS and the EC leading to the restriction on banking as summarized in Table 1.

Without banking and borrowing provisions between Phase I and II, Ellerman & Parsons (2006) stated "it is virtually certain that the EU ETS will then be either long or short; the likelihood of a perfect match between first period EUAs and emissions are extremely small. This binary outcome places a limit on first period prices that, when coupled with the constraint on interperiod banking, allows a probability of shortage to be calculated taking into account all the uncertainties weather, economic growth, energy prices, and the abatement response to carbon prices". In this perspective, the probability of scarcity expected by market participants at any point in time is defined as the ratio between Phase I future price series of delivery December, 2007 and Phase II future price series of delivery December, 2008 plus 40 = C which represents the penalty19 : Therefore, the higher the expected allowance scarcity, the higher EUA prices. As shown in Figure 2, the expected allowance scarcity is largely reflected in allowance price changes.

Introducing the banf r, banpl and epr variables relative to the environmental policy constraint, eq. ( 5) becomes:

p t+1 -p t =α + β 1 r f t p t + β 2 (r m t -r f t ) p t + β 3 p t min + β 4 p t max + β 5 epr + β 4 banf r + β 6 banpl + t+1 (6)
with epr the probability of EUA allowance shortage at the end of Phase I, banf r and banpl the dummy variables related to the ban of banking by France and Poland. Eq. ( 6) constitues the general form of the model we estimate on the full period and corresponding two sub-periods.

Energy market shocks

To avoid model misspecification, we introduce Brent prices, Natural gas prices and weather conditions that were identified as being the main determinants of EUA prices by [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF]. This final step also serves as robustness check of the results obtained in eq. ( 6) which becomes:

p t+1 -p t =α + β 1 r f t p t + β 2 (r m t -r f t ) p t + β 3 p t min + β 4 p t max + β 5 epr + β 6 banf r + β 7 banpl + β 8 brent + β 9 ngas + β 10 temp + t+1 (7)
with brent the Brent price series, ngas the Natural gas price series20 and temp an European temperature index defined below.

4 Data 14 4 Data

The carbon price

The EUA price is determined on several markets: the over-the-counter (OTC), spot and futures markets. The most liquid market is the OTC market, where transactions are usually operated through industrials or brokers and consequently price data is confidential, or available through commercial energy consultancies 21

CAPM rates of return

The risk-free rate of return (r f t ) is the 3-months Euribor presented as annual percentages with daily data frequency. The rate of return on the market portfolio of risky assets (r m t ) is the Dow Jones EURO STOXX 50 Index annual return with a daily data frequency. We convert each daily observation to a daily interest rate. Thus, r f t and r m t are expressed in percentage points at daily rates (see Figure 3 in the Appendix).

Energy prices

On energy markets, the Brent price (brent in US $ per baril) is the daily brent crude futures price negotiated on the Intercontinental Exchange (ICE), the European leader on futures energy exchange. The Natural gas price (ngas in = C per Mwh) is the daily natural gas price negotiated on Zeebrugge Hub. The euro-dollar exchange rate provided by the European Central Bank is used to ensure all variables are transformed to the same currency. Descriptive statistics for energy variables are presented in Table 5 (see the Appendix).

Temperatures

Concerning climate conditions, the daily data of the European temperature index published by Tendances Carbone 22 is used. It is equal to the average of national temperature Powernext weather indices (in • C) weighted by the share of each NAP in the total of four countries. These national indices are the mean temperature for Spain, France, Germany and United Kingdom, calculated as the average of the temperatures at the representative regional weather station weighted by the regional population.

Stationarity tests

Because econometric results may be unreliable if the dependent variable is non-stationary, we first need to test the stationarity of all price series and their first-difference. One possible complication of unit root tests for stationarity is that the presence of structural changes during the time series may make rejection of a unit root more difficult [START_REF] Perron | The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis[END_REF]). We performed usual unit root tests (ADF, PP, KPSS) and found that all price series are characterized by a unit root. When tests are applied on series in first differences, they are found to be stationary. In other words, all prices series are integrated of order 1 (I(1)) 23 .

Results and discussion

Results of equations ( 6) and ( 7) for the full period and the two sub-periods are presented in Table 2, 3 and 4. Estimations are computed using OLS and the Newey-West procedure to correct for serial correlation and generate 22 Tendances Carbone is the monthly bulletin of the European Carbon Market published by the Caisse des Dépôts and Powernext Carbon.

23 Detailed results of the unit root tests are available upon request to the authors.

robust standard errors (NW-OLS). The dependent variable in this analysis is the first log-differenced EUA price series 24 . Based on its correlogram, the true data generating process is characterized as an ARMA(p,q) of order 1. This is confirmed by autoregressive and moving average coefficients being statistically different from 0. The quality of the regressions is verified through the following diagnostic tests: the simple R-squared, the adjusted R-squared, the p-value of the F-test statistic (F -Stat), the Durbin-Watson statistic (D.W.), the p-value of the the Breusch-Godfrey Serial Correlation Lagrange Multiplier test (LM ), the p-value of the White heteroskedasticity test (W hite test), the Akaike Information Criterion (AIC) and the Schwartz Criterion (SC). For each regression, the Lagrange-Multiplier test indicates residuals are not autocorrelated. Robustness checks concerning the choice of the rate of return on the market portfolio of risky assets, the determination of the structural break and the presence of heteroskedasticity in coefficient estimates are detailed in the Appendix.

Full sample

Regression results for the full sample are given in Table 2. The significance of r f t p t in row (1) is not a primary concern, since it tests the restriction β 1 = 0. For the validation of the Hotelling rule, we are rather interested in the null hypothesis β 1 = 1. Thus, we calculate the confidence interval where the true value of the β parameter has a 95% probability to be according to the formula: CI = [ β ± 2.11 * Std.error]. These calculations 25 lead us to reject the Hotelling rule in full period. The interpretation is that with the inter-period banking restriction the EU ETS does not achieve a competitive equilibrium that would minimize compliance costs through the use of temporal flexibility mechanisms. Several other inferences may be made from the Hotelling-CAPM model. The lack of significancy of the β 2 coefficient suggests CO 2 allowances do not bear a risk-premium as part of a diversified commodities portfolio. Both ptmin and ptmax dummy variables are signifi-24 Thus, we are interested in the growth rate of the dependent variable. 25 Not shown but available from the authors by request. cant respectively at 1% and 5% in row (1) which improves the quality of this regression.

What concerns environmental policy indicators, the banf r variable is significant at 5% in row (1). Its positive sign suggests the French restriction on banking between Phase I and II contributes to the decline of EUA price changes toward zero until the end of Phase I. The banpl variable is not significant. The EPR ratio is positive and significant in row (1) at 1%. Allowance price changes therefore react to the EPR ratio with the expected sign, i.e. the higher (lower) the expected allowance scarcity the higher (lower) the allowance price.

Coefficient estimates are robust to the introduction of energy market shocks. Results for banf r and epr in row (2) are consistent with those in row (1). ngas and ngas(-1) affect positively EUA price changes at 5%, which is in line with previous literature on EUA price determinants [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF]). The fact that temp is not significant suggests abnormal climatic events with respect to seasonal average should be used instead of the European temperature index here.

"Before the compliance break"

Regression results for the sub-period "before the compliance break" are given in Table 3. Similarly to the full period, calculations of the confidence intervals lead to the rejection of the Hotelling rule. At this point in time, the only environmental policy indicator is the epr variable, which is not significant in row (3). This result suggests agents were not trading allowances based on the expected allowance scarcity. Despite the introduction of ngas and ngas(-1) at 1% significance with the expected sign, the epr variable remains non significant in row (4).

While allowance price changes seemed to react to other energy market shocks, it appears more difficult to ascertain the effects of the banking restriction before the compliance break for two main reasons. First, discussions on the banking provisions were still under way between France, Poland and the EC. Second, market participants had incomplete information about the

"After the compliance break"

Regression results for the sub-period "after the compliance break" are given in Table 4. Calculations of the confidence intervals also yield to the rejection of the Hotelling rule. The banf r variable is positive and significant at 1% in row (5) which, similarly to the full sample, suggests the French ban on banking contributes to the explanation of low EUA prices until the end of Phase I. The EPR ratio becomes significant again at 5% and positive in row (5) suggesting market participants pay more attention to the expected allowance scarcity revised after the 2005 compliance. Those results are stable in row ( 6). The introduction of the brent variable at 5% significance is similar in sign to previous literature [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF]).

Discussion

On the failure of the Hotelling rule

The failure of the Hotelling rule during Phase I of the EU ETS is not worrying in itself. Recent empirical applications state even if it fails such analysis still brings a better understanding of the intertemporal scheduling of the resource use overtime [START_REF] Heal | A Celebration of Environmental and Resource Economics[END_REF]). In the context of the EU ETS, the rejection of the Hotelling rule implies EUA prices do not adequately reflect abatement costs at the installation level during 2005-2007. Indeed, [START_REF] Helfand | Testing for dynamic efficiency of the Sulfur Dioxide Allowance Market[END_REF] who also reject the Hotelling rule for the SO 2 allowance price pattern state "under the first fundamental theorem of welfare economics, evidence of competitive equilibrium would imply dynamic efficiency. In this case, dynamic efficiency involves minimizing present-value cost of compliance with the intertemporal emission regulation". As noted previously, the efficiency of allowance trading is linked to the authorization of flexibility instruments where agents may trade allowances not only spatially but also through time, such as full banking and restricted borrowing [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF], [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF]). With- out such provisions, market prices do not reflect opportunity costs leading to an efficient choice of abatement measures [START_REF] Schleich | Banning Banking in EU Emissions Trading?[END_REF]). [START_REF] Kronenberg | Should we worry about the failure of the Hotelling rule?[END_REF] provides other reasons to justify this failure, for instance by paying attention to the fact that allowances are characterized by a costless extraction or by focusing on strategic interactions between firms26 .

On institutional learning

Our results suggest an increasing influence of the environmental policy constraint on EUA price changes. This evidence may be approximated first by the significance of the EPR ratio. Before the price adjustment on April, 2006 allowance trading may be characterized as uncertain with heterogeneous expectations related to the EUA price pattern and only the release of 2005 verified emissions gave industrials a hint about their net short/long positions. After the compliance break, market participants form their anticipations more accurately in a context of a low environmental policy constraint coupled to a ban on inter-period banking which explains why the EPR ratio becomes significant again. Second, we may deduce from official communications between France and the EC that the French ban on banking significantly affected allowance price changes and contribute to a sharper explanation of low allowance prices. As EUA price changes are statistically and significantly affected by institutional events such as the April, 2006 structural break, the French ban on banking and the EPR indicator of a low allowance scarcity, a main finding of our tests lies in the evidence of institutional learning within Phase I.

On the influence of energy markets

As stated by [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF], EUA prices are affected by Brent and Natural gas market shocks. However, the temp variable is not significant. Our point is to show the influence of environmental policy indicators is robust to the introduction of energy market shocks: signs and coefficient estimates for the banf r dummy and the epr ratio remain significant in full period (row 3) and after the break (row 6).

Conclusion

This analysis estimates the impact of banking restrictions on EUA prices.

Beginning at 8 = C on January 1, 2005 allowance prices initially rose to 20-30 = C, and have been declining steadily since the disclosure of 2005 verified emissions on April, 2006. Previous literature identified among the main explanations of this price pattern over-allocation concerns, early abatement efforts in 2005 and possibly decreasing abatement costs in 2006 due to the interaction with energy markets and climatic events [START_REF] Buchner | The European Union Emissions Trading Scheme: Origins, Allocation, and Early Results[END_REF], [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF]). Our work demonstrates banking and borrowing provisions adopted during Phase I and II may also contribute to the explanation of low EUA prices.

First, with unrestricted banking within Phase I, we give evidence that allowance prices do not reflect adequately abatement costs leading to abatement efforts based on a Hotelling-type analysis [START_REF] Schennach | The Economics of Pollution Permit Banking in the Context of Title IV of the 1990 Clean Air Act Amendments[END_REF], [START_REF] Slade | Hotelling Confronts CAPM: A Test of the Theory of Exhaustible Resources[END_REF], [START_REF] Helfand | Testing for dynamic efficiency of the Sulfur Dioxide Allowance Market[END_REF]). With regard to high allowance prices until the compliance break, early abatement occurred in the EU ETS to meet the cap requirements. After the 2005 compliance and the correction of the expectation error concerning the amount of abatement required to comply with the cap, the restriction on inter-period banking limited additional abatement27 .

Second, with restricted banking between Phase I and II, we estimate the effects of two environmental policy indicators related to banking provisions on EUA prices. The Ellerman & Parsons ratio is used to measure the probability of allowance shortage at the end of Phase I expected by market participants. French official communications to the European Commission concerning banning banking between 2007 and 2008 in its NAP II are used as a dummy variable. Before the compliance break, EUA prices are not affected by the expected allowance scarcity at the end of 2007. After the compliance break, market participants form their anticipations more accurately in a context of a low environmental policy constraint and the impossibility to transfer allowances from Phase I to Phase II. Both indicators become significant in explaining the low EUA price pattern, as for the full period sample. These results are robust to the introduction of energy market shocks.

Considering Phase I as a learning period, these results give insight into the possible sacrifice of the banking instrument to limit the transfer of design inefficiencies in Phase II. Finally, for Member States to take advantage of the banking provision in longer-term mitigation plans, the publication of the net amount of allowances either banked or borrowed at the end of each compliance period by the European Commission may be of precious use.

Robustness checks

As noted previously, adding energy variables to the model serves as a first robustness check for coefficient estimates. Concerning the choice of the rate of return of a diversified portfolio, the inclusion of the Euronext 100 Index instead of the Dow Jones EURO STOXX 50 does not change the results since financial market places are strongly correlated. The Hotelling rule is still rejected in all models. The determination of the structural break is further proofed by a Chow breakpoint, which rejects the null hypothesis that the sample does not contain a structural break at a 5%. When the White test shows evidence of heteroskedasticity, a GARCH(p,q) model of order 1 has been implemented using Bollerslev Wooldrige robust standard errors and covariance for each period. GARCH coefficients are stable with significant estimates in the mean and variance equations. Since both estimation techniques yield similar results, we present only NW-OLS coefficients to simplify the exposition. A journal of those results may be obtained upon request to the authors.

  Union Emissions Trading Scheme (EU ETS) launched in 2005 to help Member States (MS) achieve their Kyoto target to reduce 1990 emissions by 8% during 2008-2012, covered installations are only allowed for banking and borrowing allowances within 2005-2007 and within 2008-2012.

Figure 1 :

 1 Figure 1: EUA Spot anf Futures Prices from 2005/07/01 to 2007/05/31 Source: Powernext Carbon and ECX

Figure 2 :

 2 Figure 2: Probability of EUA shortage à la Ellerman & Parsons (2006) Source: ECX

Table 1 :

 1 Official communications between France, Poland and the European Commission regarding NAPs II

	Source: European Commission Environment DG
	Steps	Action		France	Poland
	NAP II submission	Letter dated	2006/09/28 2006/06/30
		EC registration 2006/09/28 2006/07/06
	Additional information Letter dated	2006/10/27 2006/12/29
				2006/12/29 2007/01/09
				2007/01/17
				2007/03/14
				2007/03/15
		EC registration 2006/11/08 2007/01/08
				2007/01/05 2007/01/23
				2007/01/23
		Withdrawing	2006/11/28
	EC decision			2007/03/26 2007/03/26
	P r(scarcity) =	EU A Dec.2007 40 = C + EU A Dec.2008

  . The most liquid futures market is the European Climate Exchange (ECX) and the most liquid spot market is Powernext Carbon launched on June, 2005. We use the daily EUA spot price (p t in = C per ton of CO 2 ) negotiated on Powernext carbon. The data range goes from 2005/07/01 to 2007/05/31, i.e. from the start of the Powernext Carbon trading platform until the period where Phase I allowance prices gear toward zero.

Table 2 :

 2

		Hotelling-CAPM Model: Full sample results
		(1)	(2)
	p t (-1)	-0.5131***	-0.5145***
		(0.1239)	(0.1239)
	ma(1)	0.8191***	0.8189***
		(0.0830)	(0.0830)
	Constant	-0.0345***	-0.0344***
		(0.0115)	(0.0115)
	r f t p t	-0.0026	-0.0031
		(0.0149)	(0.0140)
	(r m t -r f t ) p t	-0.0001	-0.0009
		(0.0090)	(0.0089)
	p t min	-0.1281***	-0.1262***
		(0.0218)	(0.0211)
	p t max	0.0387**	0.0386**
		(0.0199)	(0.01995)
	epr	0.0908***	0.0904***
		(0.0315)	(0.0316)
	banfr	0.0219**	0.0216**
		(0.0106)	(0.0107)
	ngas		0.0016**
			(0.0009)
	ngas(-1)		0.0016**
			(0.0008)
	brent
	banpl
	temp
	R-squ.	0.1906	0.1929
	Adj. R-squ.	0.1747	0.1737
	F-Stat	0.0000	0.0000
	D.W.	1.9379	1.9383
	LM test	0.7237	0.7110
	White test	0.0000	0.0000
	AIC	-3.0428	-3.0737
	SC	-2.9727	-2.9685
	* * * indicates 1% significance, * * 5%
	significance and

* 10% significance.

Table 3 :

 3

	Hotelling-CAPM Model: "Before the compliance break" results
		(3)	(4)
	p t (-1)	-0.7943***	-0.7409***
		(0.0828)	(0.0921)
	ma(1)	0.9853***	0.9822***
		(0.0206)	(0.0207)
	Constant	-0.0141	-0.0099
		(0.0137)	(0.0123)
	r f t p t	-0.0011	-0.0049
		(0.0053)	(0.0059)
	(r m t -r f t ) p t	-0.0001	-0.0009
		(0.0033)	(0.0039)
	p t min	-0.0246**	-0.0385***
		(0.0114)	(0.0103)
	epr	0.0393	0.0317
		(0.0323)	(0.0282)
	ngas		0.0021***
			(0.0007)
	ngas(-1)		0.0022***
			(0.0007)
	p t max
	brent
	temp
	R-squ.	0.1245	0.2448
	Adj. R-squ.	0.0889	0.2051
	F-Stat	0.0008	0.0000
	D.W.	1,9744	1.8448
	LM test	0.8199	0.2698
	White test	0.0053	0.5200
	AIC	-4.6297	-4.7535
	SC	-4.4843	-4.5727
	* * * indicates 1% significance, * * 5%
	significance and * 10% significance.

Table 4 :

 4

	Hotelling-CAPM Model: "After the compliance break" results
		(5)	(6)
	p t (-1)	-0.3135**	-0.2443*
		(0.1584)	(0.1409)
	ma(1)	0.7123***	0.6434***
		(0.1440)	(0.1274)
	Constant	-0.0366***	-0.0351***
		(0.0142)	(0.0135)
	r f t p t	-0.1904	-0.1994
		(0.1507)	(0.1454)
	(r m t -r f t ) p t	0.0054	0.0129
		(0.0169)	(0.0169)
	p t min	-0.1306***	-0.1211***
		(0.0245)	(0.0293)
	epr	0.1343**	0.1312**
		(0.0586)	(0.0559)
	banfr	0.0310***	0.0359***
		(0.0110)	(0.0106)
	brent		0.0062**
			(0.0028)
	p t max
	banpl
	ngas
	temp
	R-squ.	0.2570	0.2754
	Adj. R-squ.	0.2300	0.2456
	F-Stat	0.0000	0.0000
	D.W.	1.9417	1.9670
	LM test	0.5502	0.7227
	White test	0.0002	0.0002
	AIC	-3.0390	-3.0554
	SC	-2.9041	-2.9054
	* * * indicates 1% significance, * * 5%
	significance and * 10% significance.

Table 5 :

 5 Descriptive Statistics* *for the EUA first log-differenced price series, the brent and natural gas price series computed as forecast errors.

		Mean Median Max.	Min.	SE	Skew. Kurt. N
	Full period
	p t	-0.009 0.000	0.297 -0.437 0.056 -1.332 14.708 492
	brent 0.001 0.008	3.757 -2.973 1.068 -0.018 2.797 492
	ngas -0.001 -0.145	11.542 -10.570 1.672 0.997 14.809 492
	Before the compliance break
	p t	0.001 0.001	0.085 -0.134 0.025 -1.044 8.775 208
	brent 0.015 -0.006	3.757 -2.827 1.066 0.165 3.024 208
	ngas -0.007 -0.298	11.542 -10.570 1.950 1.056 14.120 208
	After the compliance break
	p t	-0.017 -0.008	0.169 -0.288 0.064 -0.929 6.122 241
	brent -0.017 0.013	2.151 -2.052 0.822 -0.080 2.602 241
	ngas 0.016 -0.010	7.206 -6.844 1.484 0.736 11.159 241

There may also be a risk of building a bank of borrowed allowances too large by the end of the period. This hypothesis, suggested by[START_REF] Grubb | Allocation and Competitiveness in the EU Emissions Trading Scheme: Policy Overview[END_REF], would imply an additional constraint during the negotiation phase of the post-2012 period of the EU ETS if the amount of borrowed allowances accumulated by industrials, and that could legally be transferred to the next period, appears unsustainably large.

[START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF] suggest the implementation of a discount rate equal to the industry average rate of interest used to finance medium term capital expenditures. For greenhouse gases, the optimal rate of intertemporal substitution has been suggested by[START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF] as being "the ratio of current marginal stock damages to the discounted future value of marginal stock damages less the decay rate of emissions in the

[START_REF] Cason | Emissions Variability in Tradable Permit Markets with Imperfect Enforcement and Banking[END_REF] find that banking increases non-compliance and total emissions in experiments with weak enforcement. Borrowing provisions could increase the rate of compliance in current period, but that would only shift excess emissions to the next period.

i.e. the abatement cost function does not change over time and there is complete information on the marginal damage function and emissions of sources[START_REF] Rubin | A model of Intertemporal Emission Trading, Banking, and Borrowing[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]).

This positive effect of banking may be distorted, for instance by initial allocation, leading to the Hot Air situation in the Kyoto Protocol.

i.e. allowances from Clean Development Mechanism (CDM) projects.

In practice, the link between the European and international transactions registries, respectively the Community Transaction Log and the International Transaction Log will be effective before the end of 2007. This communication will allow deliveries of credits from CDM projects on European registries.

i.e. allowances from the KP.

The relationship between sectors covered / not covered by the EU ETS adds another layer of complexity to the scheme, given that mandatory national abatement targets were signed at the national level under the Kyoto Protocol.

1.45% according to the EU Press Release IP/06/612 on May 15, 2006 available at http://europa.eu.

In this method, data themselves suggest the possible timing of structural breaks. Their GAUSS code may be found at http://www.cba.ua.edu/ jlee/gauss/ls.txt.

According to[START_REF] Ellerman | Markets for Clean Air: The US Acid Rain Program[END_REF], agents may benefit from holding a stock of allowances on hand to buffer itself against unexpected changes in emissions, which is called a convenience yield.

According to[START_REF] Liski | On Pollution Permit Banking and Market Power[END_REF], the following differences may be highlighted. First, on 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 extracted. 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 penalty will be 100 = C during Phase II and industrials also need to surrender a compensating amount of allowances.

Following Helfand et al. (2006), we compute forecast errors using the "one-step ahead" forecast method for energy variables.

For instance, the London Energy Brokers Association (LEBA) produces each trading day an index price using the volume weighted average of EUA trades since December 2006.

For instance, lobbying activites may take place within each allocation plan to extract more allowances from the regulator based on past emissions as a monopoly rent.

Indeed, as stated by[START_REF] Ellerman | The Efficiency and Robustness of Allowance Banking in the U.S. Acid Rain Program[END_REF] on the evaluation of the US SO 2 program, "when allowed in phased-in cap-and-trade programs, banking can be expected to produce more abatement and higher allowance prices in the early phases of the program".

Acknowledgments: Emilie Alberola is grateful for financial support to the ADEME (French Government Agency for Environmental and Energy Management) and the Caisse des Dépôts. The authors would like to thank Pr. Gunther Capelle-Blancard (University Paris 1), Dr. Karsten Neuhoff (University of Cambridge) and seminar participants at the Envecon: Applied Environmental Economics Conference (London, UK, 2007) and the XII. Spring Meeting for Young Economists (Hambourg, Germany, 2007) for helpful comments and suggestions. The authors are also indebted to Dr. Stéphanie Monjon (ADEME) and Dr. Christian de Perthuis (Head of the Caisse des Dépôts Climate Task Force) for their precious reading on earlier drafts. The usual disclaimer applies.

Appendix

A graph of the rates of return for 3 Months-Euribor and Dow Jones EURO STOXX 50 is given in Figure 3. Table 5 shows descriptive statistics for the energy price series under consideration with SE the standard errors, Skew. the skewness, Kurt. the kurtosis and N the number of observations.


Pierre-Andr Jouvet

Philippe Michel

Pierre Pestieau

Public and private environmental spending. A political economy approach *

This paper studies the determination of public investment in environmental quality when there are private alternatives. Public investment is chosen by majority voting. When consumption and environmental quality are complementary one may observe a solution of the type "ends against the middle.

Introduction

There are various ways of dealing with environmental quality and with policies, public or private, aimed at maintaining it. In this paper we look at a problem such as the quality of water that deteriorates at a constant rate and can be preserved collectively or privately. We adopt a dynamic model of successive generations. Bequest motive is the source of savings. At each generation, the quality of environment can be enhanced by a public investment which is particularly efficient because of the public good characteristic of the environment. This public investment is financed by a flat rate income tax. Individuals differ in their labor productivity and in their initial endowment.

Hence, the public environmental policy can be viewed as progressive. Each individual can also invest privately in environmental quality; this investment does not have any externality. To pursue with the example of water quality, one can think of collective purification plant and of individual water filters.

Given the progressivity of the environmental policy, well-to-do households would prefer individual over collective techniques even though the latter is much more efficient than the former. Henceforth, if the tax rates and thus the level of public investment are chosen by votes, one can expect that the most preferred tax rate decreases with income and indeed reaches zero for some above average income. This conjecture is not necessarily verified for low incomes. We assume that an individual's utility depends on consumption, environmental quality and the utility of his children. If consumption and environment are strong complements, a poor individual can vote against a too high tax rate that would lead him to a too low consumption level. Indeed we show that in case of complementarity between consumption and environmental quality, we can have a coalition of low and high income individuals opposing middle income individuals in the determination of the tax rate. This is what Epple and Romano (1996) call "ends against the middle" and it is a case where the median voter theorem does not apply. To get this particular result, one needs complementarity between consumption and the publicly provided private good and the possibility of supplementing public provision by private purchases.

We have thus two key features of our paper, the possibility of private contribution and the complementary between consumption and environment.

There is little work on the issue of voting for environmental quality. Note a recent paper by [START_REF] Kempf | Is inequality harmful for the environment?[END_REF] who show that public spending tends to be larger in societies with less inequality.

There is also a paper by [START_REF] Aidt | Political internalization of economic externalities and environmental policy[END_REF]

The Model

At each period of time N altruistic agents live and work for one period. There are I type of individuals. Each agent of type i is characterized by her/his labor productivity α i where i = 1, ..., I. In the economy there is a proportion p i of agents of type i and we assume:

I i=1 p i α i = 1 (1)

Consumers

In period t, each agent of type i can improve her/his own environmental quality in a private way by an environmental expense e it . The environmental quality for an agent is given by :

q it = e it + Ψ t ( 2
)
where Ψ t represents the contribution of public investment to the individual's environmental quality. We assume additivity of e and Ψ for the sake of simplicity.

Agent i 's budget constraint is given by:

c it = (1 -τ t )α i w t + R t x it-1 -e it -x it (3)
where c it is consumption, w t is wage per unit of efficient labor, R t is the return to capital investment. This agent receives a bequest x it-1 and gives

x it to her/his child (constant population implies that each parent has only on child). τ t is the environmental tax. Bequest are constraint to be non negative, x it ≥ 0.

Any agent of type i born in period t derive utility from consumption c it environmental quality q it and his/her child's utility. Agent's preferences for c and q are supposed to be homothetic and represented by a utility function U (c it , q it ). U (.) is an increasing, strictly concave, twice differentiable function and verifies Inada conditions. Then, the marginal rate of substitution is defined by:

U c (c, q) U q (c, q) = h( q c ) (4)
where h(.) is an increasing bijection on ++ .

At period t, the welfare of an agent of type i, V it , is defined by:

V it = U (c it , q it ) + γV it+1 (5)
with γ the degree of altruism, γ ∈ (0, 1). Agent i solves the following problem:

                     max c it ,q it ,x it ,e it ∞ t=0 γ t U (c it , q it ) s.c. c it = (1 -τ t )α i w t + R t x it-1 -e it -x it , q it = e it + Ψ t ,
x it ≥ 0, e it ≥ 0.

Government

The public side of environmental quality Ψ t depends on total public spending,

E t : Ψ t = Ψ(E t ).
Function Ψ is a concave and twice differentiable: Ψ > 0, Ψ < 0 with Ψ(0) = 0, Ψ (0) = +∞ and Ψ (+∞) = 0. The specification of Ψ is crucial. The return to private spending is constant and equal to one.

Whatever the number of agents, the marginal return of public spending decreases from +∞ to 0. Low (or not to large) E is more efficient than private spending, but for enough large E, it is the opposite. These properties imply that on pure efficiency grounds (identical individuals) public investment should prevail up to the point where its marginal return (Ψ ) equals unity (the return of the private technology).

Because of its public good nature public investment offers another advantage:

it dominates private investment up to the point where its marginal return times N (N Ψ ) equals 1.

As an illustration take Ψ(E) = AE ε where ε < 1 and A > 0 is a scale factor.

In the identical individuals case, environmental quality is given by

q = e + AE ε
subject to e + E/N being a constant. It is thus clear that e = 0 as long as

E ≤ (εN A) 1 1-ε
In the following we assume that in the first-best case wherein individuals are made identical the optimal E denoted E * is below that upper bound. In the second best with heterogenous individuals this condition does not hold anymore.

The environmental tax, τ t , is used to finance E, public environmental spending. The government's revenue constraint is given by

E t = I i=1 p i N α i τ t w t = τ t N w t (6)

Firms behavior

At each period t, competitive firms produce an homogeneous good Y t with capital K t and labor L t . We assume a well-behaved production function (increasing, concave and homogeneous of degree one),

Y t = F (K t , L t )
Assuming a total depreciation of capital after one period1 , a representative firm, in period t, maximizes its profits π t ,

π t = F (K t , L t ) -w t L t -R t K t
With perfect competition, price w t and R t are given and factors prices are equal to there marginal productivities,

F L (K t , L t ) = w t (7) F K (K t , L t ) = R t (8)

Equilibrium for a given policy

The equilibrium conditions for an agent i are given by the following first order conditions:

-U c (c it , q it ) + U q (c it , q it ) ≤ 0; = 0 if e it > 0 (9)
and

-U c (c it , q it ) + γR t+1 U c (c it+1 , q it+1 ) ≤ 0; = 0 if x it > 0. ( 10
)
The intertemporal equilibrium is defined, for a given sequence of government decisions τ t , by a sequence of prices w t and R t , and individual variables satisfying all the equilibrium conditions. The government decisions satisfies its budget constraint (6. Consumers decisions maximize their utility 5 which yields ( 9) and (10). Firms decision implies [START_REF] Mcausland | Voting for pollution policy: the importance of income inequality and openness to trade[END_REF] and (8).

The capital stock is equal to the sum of bequests,

K t+1 = i p i N x it = N x t (11)
where x t is the mean of bequests. The return to bequest is defined by the marginal productivity of capital (8). The markets of labor and good are clear, respectively,

L t = i p i α i N = N (12)
and

Y t = N c t + N e t + E t + N x t ( 13
)
where

c t = i p i c it , e t = i p i e it .
The initial conditions x i,-1 ≥ 0 for all i = 1, ..., I are given with x i,-1

such that i p i N x i,-1 = K 0 > 0
3 Equilibrium and optimum in the steadystate With a constant tax rate τ t = τ , the steady state satisfies E = τ N w, L t = N ,

K t+1 = K, k = K/N = i p i x i = x, e it = e i , x it = x i , c it = c i = (1-τ )α i w+ (R -1)
x i -e i and q it = q i = e i + Ψ(E). A positive stock of capital implies that at least one bequest x i is strictly positive and therefore with relation (10), we obtain the modified golden rule,

F K (k, 1) = 1 γ ( 14
)
with k = k being the stationary stock of capital. Then, at the steady-state equilibrium R = R = 1/γ, and w = w = F L ( k, 1) and type's i's individuals have a life-cycle income:

ω i = (1 -τ )α i w + ( R -1)x i (15)
In the long run wealth distribution depends on x i,-1 and on the dynamics.

When there is no constraint on bequest each altruistic agent has the same behavior as an infinite lived agent faced to the following intertemporal budget constraint,

∞ t=0 ρ t (c it + e it ) = x i,-1 + ∞ t=0 α i ρ t w t (1 -τ ) ≡ Ω i ( 16
)
where ρ t = ρ t-1 /R t , with ρ 0 = 1, are the discount factors. Then, the long run net wealth distribution depends on the distribution of Ω i and on the x i,-1 .

When the ranking of x i,-1 is the same as α i ranking, the distribution of net wealth is the same as the distribution of labor productivities α i . In order to simplify our study, we assume that bequests, at the stationary equilibrium, proportional to the labor productivities, i.e. x i = α i x = α i k.

3.1 Laissez-faire (E = 0, τ = 0)

Each agent i maximizes his/her utility subject to his/her budget constraint with τ = 0, i.e. maxU (c i , e i ) subject to

c i + e i = ω i = α i w + ( R -1)α i k.
Then the relation (9) implies

U c (c i , e i ) = U q (c i , e i ) ( 17
)
which is equivalent to e i /c i = h -1 (1) ≡ µ. Therefore, with the assumption of homothetic preferences we obtain that private environmental spending is proportional to consumption, e i = µc i and to the net income, ω i , (and also to α i ).

Social Optimum and the decentralization

In a centralized economy after redistribution all agent have the same consumption, c i = c and e i = e. The environmental quality is defined by The solution with e * = 0 satisfies

q = e + ψ(
U c (c * , ψ(E * )) = N ψ (E * )U q (c * , ψ(E * )) (18)
The condition for e * = 0 is equivalent to ψ (E * ) ≥ 1/N , at steady state the productivity of public spending is large than one of private.

Contrasting (17) and ( 18) is interesting. In a laissez-faire setting, given our assumption on preferences, each individual contributes to the quality of his/her environment. In the first-best optimum, we expect that E * > 0 and e * = 0. This mainly depends on Ψ (E) and on N . But this is just an assumption. Nothing precludes Ψ to be so inefficient and N to be so low that even in the first-best, private contribution would be the best device to maintain environmental quality.

We now move to a positive setting and try to see what would be the private contribution for a given value of τ and thus of E = τ N w.

Private environmental contribution for a given public policy

-Case e i > 0.

If e i > 0 then the relation (9) implies

U c (c i , q i ) = U q (c i , q i ) (19)
which is equivalent to q i /c i = h -1 (1) ≡ µ. The budget constraint is

c i + e i = (1 -τ )α i w + ( R -1)α i k ≡ ω i (20)
and with µc i = q i = e i + Ψ(E) we obtain

e i (1 + 1 µ ) = ω i - Ψ(E) µ ( 21
)
Thus e i > 0 if and only if ω i > Ψ(E)/µ. This condition is equivalent to

h(Ψ(E)/ω i ) < h(µ) = 1 and thus e i > 0 if and only if U c (ω i , Ψ(E)) < U q (ω i , Ψ(E)
). An agent chooses a positive environmental contribution, e i > 0, if the consumption of her/his net income ω i induce a lower consumption marginal utility than the environmental quality financed by the government.

-Case e i = 0

If e i = 0, then U c (c i , q i ) ≥ U q (c i , q i ) with q i = Ψ(E) and

c i = ω i . The condition for e i = 0 is U c (ω i , Ψ(E)) ≥ U q (ω i , Ψ(E)).
The two cases can be resumed in the following way,

e i = max{0, µc i -Ψ(E)} (22)
and the corresponding stationary welfare V i (E) satisfies

V i (E) = ∞ 0 γ t U (c i , q i ) = 1 1 -γ U (c i , q i ) (23)
We now turn to the central section of this paper: the determination of τ or E through majority voting. We first have to characterize the individual's indirect utility with E or τ as argument.

Voting Equilibrium

We show that the welfare function of an agent of type i, V i (E) is single peaked and we study the variations of his/her preferred public spending level with respect to his/her productivity parameter α i . After that we analyze the result of the vote.

Study of the welfare function

We introduce two life-cycle utility functions for the constrained case and for the unconstrained one.

Given by relation ( 15) and τ w = E/N , we write the income net of bequest of agent of type i as the following function of E,

ω i = (1 -τ )α i w + ( R -1)α i k = α i ω -α i E/N ≡ ω i (E) ( 24
)
where ω = w + ( R -1) k. We denote

U 0 i (E) = U (ω i (E), Ψ(E)) ( 25
)
as the life-cycle utility when consumption is equal to ω i (E) and thus his/her environmental quality is Ψ(E).

We show in the appendix that the strictly concave function U 0 i (E) reaches its maximum at some point E 0 i in(0, N ω).

The "unconstrained" life-cycle utility is defined by choosing e i , positive or negative, which maximizes

U (ω i (E) -e i , Ψ(E) + e i ) ( 26
)
This maximum is reached when the partial derivatives U c and U q are equal and this is equivalent to

q i = Ψ(E) + e i = µc i = µ(ω i (E) -e i ).
Thus the maximum of ( 26) is

U 1 i (E) = U (c 1 i (E), µc 1 i (E)) ( 27
)
where

c 1 i (E) = 1 1+µ (ω i (E) + Ψ(E)). The strictly concave function U 1 i (E) reaches its maximum at E 1 i , the solution of Ψ (E 1 i ) = α i /N .
¿From relation (21), the constraint e i ≥ 0 is binding if and only if

E ≥ E i ,
where E i is the solution of Ψ(E i ) -µω i (E i ) = 0. Thus we have

V i (E) =    1 1-γ U 0 i (E) if E i ≥ E i 1 1-γ U 1 i (E) if E i ≤ E i
We show in the appendix

Proposition 1

The function V i (E) is single peaked and it reaches its max-

imum either at E 1 i if E 1 i ≤ E i or at E 0 i if E 1 i > E i . In the later case, E 0 i belongs the the interval (E i , E 1 i ).
It may help the intuition to represent the problem at hand graphically.

Figure 1a and 1b present the indirect utilities U 0 i and U 1 i for the two cases. For E i < Ēi , U 0 i prevails and for E i > Ēi , U 1 i prevails. The relevant indirect utility is given by the thick single-peaked curve.

Insert Figure 1a and Figure 1b ¿From these two figures we obtained the most preferred value of E for an individual of type α i .

Variations of the preferred public spending level

• For an agent of type i, the unconstrained preferred public spending

E 1 i = E 1 (α i ) is the solution of Ψ (E 1 i ) = α i /N . This level is feasible (with e i ≥ 0) if and only if E 1 (α i ) ≤ E i , where E i is the solution of Ψ(E i ) -µω i (E i ) = 0.
These conditions:

E 1 (α i ) ≤ E i and E i N + Ψ(E i ) µα i = ω are equivalent to g(α i ) = E 1 (α i ) N + Ψ(E 1 (α i )) µα i ≤ ω (28) Since E 1 (α) = Ψ -1 (α/N ) is decreasing function of α, g(α)
is decreasing and the agents of type i reach their unconstrained preferred level E 1 (α i ) if and

only if α i ≥ α, where α is the solution of g( α) = ω. For these agents, E 1 (α i )
is a decreasing function of the productivity parameter α i .

• If an agent of type i is constrained, i.e. α i < α, his or her preferred level of public spending is E 0 i = E 0 (α i ) which is the solution of equation (A2) in the appendix:

φ(E 0 i , α i ) ≡ Ψ (E 0 i ) - α i N h( Ψ(E 0 i ) α i ( ω -E 0 i /N ) ) = 0
The function φ(E 0 i , α i ) is decreasing with respect to E 0 i (since Ψ is decreasing and h is increasing) and its derivative with respect α i is equal to

∂φ(E 0 i , α i ) ∂α i = 1 N (z 0 i h (z 0 i ) -h(z 0 i )) (29)
where

z 0 i = Ψ(E 0 i ) α i (b ω-E 0 i /N
) is the ratio of environmental quality to consumption. We have shown the following:

The function h has the constant elasticity 1/σ. Then, if σ > 1, the preferred public spending level is decreasing with

α i . But E 0 (α i ) is increasing if σ < 1.
Therefore, if we now consider a vote on the environmental tax τ we distinguish the following possibilities:

• If σ > 1, the preferred public spending is a decreasing function of α i . Then, given the single peakedness of preferences, the median voter theorem applies. The individual of type α m (median productivity) is therefore decisive.

• If σ = 1, E 0
i is constant, the median voter theorem applies as well and under the assumption α m ≤ α there is a majority vote in favor of E 0 i .

• If σ < 1, E 0 i is an increasing function of α i . Then, the median voter theorem does not apply and we have to use the Epple-Romano approach.

That is, the voting equilibrium involves the worker with middle productivity individuals voting against a coalition of the lower productivity and the higher productivity individuals. We know that if α i > α, the desired level of environmental quality decreases with α i . For α i < α, any profile can be observed. Using a CES utility we know that it decreases also with α i if σ > 1 and then the Condorcet winner is the median productivity α m worker. If σ < 1, we have the "ends against the middle" solution. On Figure 2b there is a coalition of workers with α i < α 0 and α i > α 1 against workers with α 0 < α i < α 1 .

Insert Figure 2a and Figure 2b The intuition is very close to that of Epple-Romano (1996) or [START_REF] Casamatta | The political economy of social security[END_REF]. With complementarity low ability workers are not going to vote for a too high tax as their consumption is closely related to the net of tax wage: αw (1 -τ ). With substituability between c and q, they instead vote for high rate of taxation realizing that they get a lot Ψ (E) for paying little τ αw i .

In these figures for the sake of intuition, we assume that the median wage w m < w = 1 which is standard and that α > 1, which is less standard and implies that only the richer workers privately contribute to the quality of their environment. Comparing Figures 2a and2b we also see that the level of public investment tends to be lower in the case when "the end meet the middle".

We now look at the amount of private contribution that result from majority voting. We know that Ēi , the value of public spending that makes workers of type i indifferent between contributing and not contributing, increases with α i . Let us denote α the productivity for which Ēi and the majority choice of E are equal. As Figures 3a and3b indicate all individuals with productivity above α will contribute to environmental quality. We note that Ē intersect the curve with the most preferred E at α. Not surprisingly there will be more private contribution where "ends meet the middle" than when the median voter is decisive.

Insert Figure 3a and Figure 3b 5 Conclusion

In this paper we have considered the case where environmental quality can be maintained by either public investment or private contribution. Public investment is financed by a flat rate tax which implies that workers with income below average benefit from it. There is another reason why one could prefer public investment, namely the technology. Given the public good nature of environmental quality, public investment is more efficient than private one. This question is dealt within a growth model of successive generations where the motive for saving is parental altruism toward children. As a consequence, the modified golden rule is achieved in the long run and this drives some of the results. We are interested by the political economy choice of public investment by workers of different productivity. We show that preferences are single peaked in that in some case one readily applies the median voter theorem and in other cases one has to use the so-called "the ends meet the middle" approach.

One of the avenues of further research is to introduce the idea of bequeathing not only financial or human capital, but also environmental quality.

We would introduce a lag in the way public investment affect the environmental quality whereas there would not be any lag for private protection.

Taking the example of water, public infrastructure investment takes time whereas domestic purification devices have an instantaneous effect. These differential lags make the analytic more difficult. Note that we could also allow for a choice between private contribution to public investment and going through the political process. Suppose that parents have the choice of helping their children by contributing s i to a public investment or to invest time and money in a political process such as the one described here. There would be an interesting arbitrage between the efficiency loss linked to the "tragedy of the commons" and the loss associated with a redistributive political process.

Another extension we are thinking of is to link environmental deterioration to production which would take us away from the very convenient modified golden rule.

U 0 i (E)

  who analyzes environmental policy in a common agency model of politics. Competition between lobby groups keeps the economy away from the efficiency Pigouvian rule. Also McAusland (2003) looks at the issue of voting for pollution policy within an open economy setting. She shows that poorer voters may be the greener voters within the electorate for reasons close to those developed in our paper. Even though richer voters are in favor of higher provision of environmental quality they may be unwilling to pay proportionality more. Finally, Jouvet et al. (2000) discuss the issue of environmental quality in a dynamic setting with altruism but without any political economy feature. The remainder of this paper is organized as follows. Next section presents the model. Section 3 focuses on the steady-state solution; it gives the optimality conditions and the laissez-faire solution. Section 4 gives the voting equilibrium. A final section concludes.

  E). The central planer maximizes U (c, e + ψ(E)) with respect to c, e and E subject to c + e + E/N = f ( k) -k, c ≥ 0, e ≥ 0 and E ≥ 0.

22 6 - 6 -Figure 1b : 1 .

 661b1 Figure 1a: The case E 1i < E i (e i > 0)

  is decreasing for E ≥ E i . Thus the maximum of V i (E) is reached atE 1 i and V i (E) is single peaked. c) If E 1 i > E i , then U 1 i (E) is increasing for E ≤ E i , and for E ≥ E 1 i U 0 i (E) is decreasing. Thus the maximum of V i (E) is reached at E 0 i which belongs to the interval (E i , E 1 i) and V i (E) is single peaked.

Or equivalently that F (K, L) includes capital after depreciation

Proposition 2 If α i ≥ α, the preferred public spending level of the agents of type i is E 1 (α i ) which is a decreasing function of the productivity parameter α i . For α i ≤ α, the preferred public spending level of the agents of type i is E 0 (α i ) and the derivative of E 0 (α i ) has the same sign as the elasticity of h at z 0 i minus 1, where

The political equilibrium

Only if the preferred public spending level is a monotonic function of the productivity the median voter theorem applies. This is the case when the elasticity of the function h is smaller or equal to 1: the two functions E 0 (α)

and E 1 (α) are non-increasing.

Proposition 3 If the elasticity of h is smaller or equal to 1, then the preferred public spending level (and the corresponding tax rate) is non-increasing

with respect to the productivity parameter. Thus the political equilibrium is the level preferred by the median voter (see figure 2a).

When the elasticity of h is not smaller or equal to 1, the analysis of the voter is a more complex. In order to obtain explicit results we consider a constant elasticity of substitution (CES) utility function,

where σ is the elasticity of substitution and β the environmental preferences parameter, β > 0. For this function, we have

The function U 0 i (E) is defined and strictly concave on the interval (0, N ω), since U (c, q) is increasing and strictly concave, and

and Ψ(E) are concave. Its derivative

tends to +∞ (resp. -∞) when E tends to 0 (resp. to N ω). Thus U 0 i (E) reaches its maximum at E 0 i where its derivative is equal to 0. Using U c /U q = h(q/c) we obtain

Then the constraint e i ≥ 0 is binding and at c i = ω i (E) and q i = Ψ(E), we have U c ≥ U q and

The