Guillaume Bérard
email: guillaume.berard@univ-amu.fr

Alain Trannoy
email: alain.trannoy@univ-amu.fr

Stephen Bazen

Pierre Cahuc

Habiba Djebbari

Emmanuel Flachaire

Xavier Joutard

Barbara Petrongolo

Marc Sangnier

The impact of the 2014 increase in the real estate transfer taxes on the French housing market

Keywords: JEL Classification: H71, R21, R31, R51 local government, real estate market, transfer taxes, natural experiment

This paper estimates the effects of an increase in the share of the real estate transfer taxes (RETT) rates going to the French départements from 3.80% to 4.50%. Not all the départements voted the RETT increase on the same date, which is the starting point of a natural experiment. Using a difference-in-differences design, we estimate two main effects. (1) An anticipation effect, one month before the implementation of the reform, in order to avoid the RETT increase.

(2) A retention effect in the post-reform period. In the end, the net effect (retention minus anticipation) corresponds to an average drop in transactions of around 6% over the first three months after the reform, that is, approximately 15,000 transactions lost at national level. If we find a short term effect of the reform, we do not find evidence of a medium-or long-term effect.

T he 2014 reform of the French real estate transfer taxes (RETT), allowing an increase in départements' 1 tax rate of 0.7 percentage point (from 3.80% to 4.50% of the tax base), aimed to raise their tax revenue, in a context of state grants reductions and increasing social spending. Not all départements implemented the RETT increase, and not at the same time, which is the starting point for a natural experiment. Even though it is not a purely random experiment, we show in the course of the paper that there was no departmental selection bias in choosing the tax increase: this choice was not correlated with the local housing market or political features.

The RETT, also called stamp duties land taxes, or droits de mutations in French 2 , are taxes levied on all ownership transfers of real estate or land (Box 1). The RETT are an important source of revenue for the French départements: they represent around €10 billion per year. However, while the RETT as the other transaction costs (notary and experts' fees) cannot be financed through mortgages and must be paid first by the buyer and in addition to the downpayment, the possible negative impact on the housing market was not evaluated or even discussed when the reform was implemented. In this study, we use open access data on the monthly number of transactions and real estate tax bases by département and implement a difference-in-differences framework using a quasi-myopic model 3 as developed by [START_REF] Malani | Interpreting Pre-Trends as Anticipation: Impact on Estimated Treatment Effects from Tort Reform[END_REF]. We assume that there were two main effects due to this reform, (1) an anticipation effect from the buyers and sellers to avoid the tax increase (timing response), and (2) a retention effect in post-reform period: a classic depressing effect of a tax on the equilibrium quantity. Note that in the very short-run, this effect is composed of re-timing due to the anticipated transactions (intertemporal substitution by those who would have purchased a real estate property anyway), and of extensive margin responses (those who would have purchased a property in the absence of the reform).What we are looking for is the behavioral response in terms of timing and extensive margin of the agents (i.e. buyers and sellers). Finally, we evaluate the tax elasticity of the tax bases to the RETT.

We estimate that there was an anticipation effect of 26% on the volume of transactions, meaning that buyers and sellers reacted to the RETT increase, the month just before the implementation of the tax increase, by bringing forward their sale date. We also estimate the average monthly retention effect for the three months following the tax increase at around 14% of the volume of transactions (assuming no effect on the sale price) -with 49% of this loss due to a pure extensive margin effect (and not to re-timing) -meaning that the tax increase had a negative impact on the housing market. All in all, the average monthly net effect corresponds to a drop in transactions of around 6% over a period of three months following the implementation date. None of the estimates after these three months are significantly different from zero. The corresponding rough estimate of the lost transactions is around 15,000. Therefore, there is some compelling evidence of a strong short-term effect, but no mediumor long-term effect. Furthermore, we estimate that the elasticity of the tax bases to the tax increase is about -0.42, meaning that tax bases decreased by 0.42% for a 1% increase in the RETT rate (i.e. there is a loss in the tax bases which reduces the gains of tax revenues for the local budgets). Computing the Laffer rate, we conclude that départements' tax revenues are still on the increasing side of the Laffer curve. Note that our results are valid in a partial equilibrium framework: we do not estimate the possible general equilibrium resulting from the distortion of the housing market. Finally, we run a series of robustness checks such as a placebo test, a self-selection test and control for possible changes in local economic conditions, confirming that our results are unbiased and robust. 1 2 3

Literature review

Previous literature on the impact of the increase in the rate of the RETT is relatively recent, as the first empirical estimation was published in 1993, and other main theoretical and empirical evaluations were mostly done over the past four years. As this stream of papers follows a natural development, we present them in chronological order for a better understanding.

The first in-depth research on RETT was done by [START_REF] Benjamin | Real Estate Transfer Tax and Property Values: The Philadelphia Story[END_REF], who analyze the effect of an increase of 45% in Philadelphia's transfer taxes in 1988. The specificity of the transfer taxes of this city compared to French ones, is that the payment of the tax is shared equally between the seller and the buyer. The authors focus on the effect of the RETT increase on the sale price of residential property, using a hedonic model and micro data (around 350 transactions). Unfortunately, they could not estimate the impact on the volume of transactions. Nevertheless, they find a decrease in postreform prices equal to the tax increase, meaning that the burden of the tax increase rests on the seller, at least in the short-run.

The next paper, [START_REF] Ioannides | Structural Estimation of Residential Mobility and Housing Tenure Choice[END_REF], is not directly related to the RETT's impact, but more generally to residential mobility, and to the decision of moving, and whether to rent or to own. This article develops a theoretical model of housing tenure choice and residential mobility which is used as a basis in many following papers. The authors find that home-owners are responsive to housing market conditions by adjusting their stock. Their empirical estimates suggest that proportional monetary transaction costs are not worse than lump-sum transaction costs in households' mobility decisions, and that housing price increases seem to discourage renters from moving and from owning after moving. Then, housing price appreciation is likely to have Box 1 -The real-estate transfer taxes system in France

The RETT are levied on all transfers of ownership of real estate and land. The French law distinguishes between two types of transfers, (1) the droits de mutation à titre onéreux (DMTO), which are based on transfers of ownership further to a sale, and (2) the droits de mutation à titre gratuit, which are based on transfers of ownership further to a donation or inheritance. Unlike in the United Kingdom or some counties in the United States, the RETT in France are proportional and not progressive. However, different rates exist; they depend on the characteristics of the real estate, the denomination of the buyer and seller (i.e. individual or professional) and the type of transfer. The RETT are calculated on the tax base after abatements, which are very scarce and small; therefore, the tax base reflects the real estates' sale price in almost all cases.

Three tax regimes of RETT exist in France, with different applicable rates:

(1) Régime de droit commun. It applies to the DMTO on real estate exempted from Value Added Tax (VAT): established properties (more than 5 years old), new constructions (less than 5 years old) sold between individuals (except if the seller has bought it in off-plan (VEFA)), and the buildable lands sold between individuals. The applicable rate for this tax system is decomposed as follows: 3.80% goes to the départements (rate before the 2014 reform which we are interested in), 1.20% goes to the municipalities, and finally 2.37% applied to the départements' tax rate goes to the central government (for tax base and collection fees) (i.e. 0.09% of the tax base). Thus, the total rate for this regime was 5.09% before the reform, and is at 5.81% now for the départements which have implemented the reform.

(2) Régime dérogatoire. First, it applies to the DMTO subject to VAT: the new constructions and lands sold by a professional or the new constructions bought in off-plan and resold between individuals, at the rate of 0.715%. Secondly, it applies to all the droits de mutation à titre gratuit, at the rate of 0.60%. Within this tax regime, the part of transactions of droits de mutations à titre gratuit is roughly 40%.

(3) Exonération des droits de mutation. This tax system applies only to the acquisition done by the State or local authorities, so there is a total tax exemption.

The average period between the signature of the preliminary sale agreement (between the seller and the buyer), and the bill of sale is 3 months. The minimum is 1 month due to the legal period of withdrawal.

The transaction costs (i.e. the RETT, the notary and experts' fees) of a house or land sale are paid by the buyer, and must be paid in full when the bill of sale is signed. Before the reform, the average rate of the transaction costs for real estate subject to the Régime de droit commun was around 7%. These transaction costs are collected by the notary on behalf of the Treasury Department (Direction Générale des Finances Publiques -DGFiP).

Following this, all the transfers of ownership and their details (e.g. number of transactions, sale price, tax revenue, locality, owners' identities) are registered by the Service de publicité foncière, which depends on the Treasury Department, except for the Alsace-Moselle territory. Composed of the départements of Moselle (57), Bas-Rhin (67) and , this territory has its own registration utility: the Livre foncier. This situation is due to the particular legal status of this territory, inherited from the German annexation of 1870.

It is important to note that, in almost any case, the transaction costs cannot be financed through mortgages. In other words, the transaction costs must be paid first and in addition to the downpayment. Therefore, even a slight increase of the RETT could have a large impact on the behavior of the buyers, because it increases out-of-pocket contributions, and thus may have large impacts on the housing market. strong effects on the housing rental market. Using the same theoretical framework, [START_REF] Van Ommeren | New Evidence of the Effect of Transaction Tax Costs on Residential Mobility[END_REF] assess the impact of an increase in transaction costs in the Netherlands that are close to the transfer taxes in France, as they are ad valorem transaction costs, mostly paid by the buyer. Using duration models, they find that a 1% increase in the transaction costs decreases mobility by 8%, which is quite significant, but they did not conclude about the time length of this effect. They deduce that transaction costs could imply lock-in effects, leading to a negative impact on the housing market and the labor market. Their conclusion is that a decrease or an abolition of the buyer's transaction costs would improve home-owners' mobility.

A more recent study by [START_REF] Dachis | The Effects of Land Transfer Taxes on Real Estates Market: Evidence from a Natural Experiment in Toronto[END_REF] estimates the effect of the implementation of a progressive transfer taxes in Toronto (1.1% on average), paid by the buyer. Unfortunately, they could not consider a potential anticipation effect. Combining difference-in-differences and regression discontinuity designs on a large sample, they estimate that the new tax decreased the volume of transactions by 14%, and the sale price by a proportional amount to the transfer taxes. Their theoretical model predicts a welfare loss of about $1 for every $8 in tax revenue raised. They conclude that the RETT should be removed in favour of the property tax. [START_REF] Davidoff | How Do Stamp Duties Affect the Housing Market?[END_REF] assess the Australian's progressive RETT reform. Instrumenting the endogenous RETT variable, they obtain similar results to [START_REF] Benjamin | Real Estate Transfer Tax and Property Values: The Philadelphia Story[END_REF]: increases in transfer taxes lower housing prices, suggesting that economic incidence falls on sellers. They also evaluate that such increases have a negative impact on owners' mobility, and that this effect increases over time. [START_REF] Besley | The Incidence of Transaction Taxes: Evidence from a Stamp Duty Holiday[END_REF] evaluate the impact on the housing market of a RETT holiday in the United Kingdom, using data on sale price and number of transactions. They develop detailed and convincing empirical evaluations as well as a bargaining model. They find a significant increase of around 8% in the volume of transactions following the tax holiday, but only in the short-run. Their theoretical model allows them to estimate that 60% of the tax decrease accrues to the buyer. [START_REF] Kopczuk | Mansion Tax: The Effect of Transfer Taxes on the Residential Real Estate Market[END_REF] estimate the effect of a specific RETT on high value housing in New-York, called the mansion tax. They assess that this tax creates a notch (i.e. a discontinuity in the tax liability), with a surplus of selling below, and a large gap above the threshold. They estimate that the volume of missing transactions above the threshold is greater than the volume of transactions below. They conclude that this is due to the bargaining of buyers and sellers, and that this particular tax impacts negatively the search market around the notch, and is inefficient in terms of tax revenue. [START_REF] Slemrod | The Behavioral Response to Housing Transfer Taxes: Evidence from a Notched Change in D.C. Policy[END_REF] estimate the behavioral responses to a change in the RETT's notch in Washington D.C. They provide a useful model of bargaining between sellers and buyers, with progressive tax. Using a difference-indifferences design, they find no evidence of a lock-in effect, but they estimate a slight timing effect, which corresponds to an anticipation effect to avoid the tax increase. Furthermore, they conclude that buyers and sellers are more able to adjust the sale price in response to the tax increase than to modify the sale date (which works only with progressive RETT). Finally, [START_REF] Best | Housing Market Responses to Transactions Taxes: Evidence from Notches and Stimulus in the UK[END_REF] also analyze some notches in the housing market in the United Kingdom, due to the progressive RETT. Their findings are similar to those of the previous article: there is some distortion of the housing market across marginal tax rates. Analyzing the same tax holiday as [START_REF] Besley | The Incidence of Transaction Taxes: Evidence from a Stamp Duty Holiday[END_REF], they find similar results regarding the volume of transactions: the elimination of 1% transfer taxes increased housing market activity by 20%. Therefore, there are large timing and extensive margin responses from buyers and sellers due to the RETT modification.

To summarize, transaction costs have a negative impact on residential mobility. They lead to suboptimal equilibrium on the housing market: they distort owners and renters' choices between staying or moving and renting or owning. Empirical literature on RETT mainly resorts on difference-in-differences and regression discontinuity designs using the features of quasi-natural experiments. They proved that transfer taxes are highly distortionary in the short-run, in terms of volume, price and timing of transactions. Medium-and long-run effects are more ambiguous, and depend on the specificity of each legal system and local conditions. The specificity of the French case is that the RETT are proportional, they accrue to the buyer only, and it is a potential widespread reform that impacted the whole country.

Context of the reform in France

As explained by all the official documents and newspapers, there are two main reasons why the government and the départements wanted to increase the RETT. (1) For several decades, a process of decentralization and fiscal autonomy of local authorities has been engaged. As a result, State grants decreased drastically. Moreover, in 2010, the State abolished the business tax, one of the main sources of tax revenue for local authorities. (2) The growth in the real estate market between 2000 and 2007 enabled départements to follow the pace of growth of local public expenditures until the financial crisis of 2007. Then, with the economic downturn, the revenues generated by the transfer tax dropped. Simultaneously, the amount of social spending of the départements (RSA, APA and PCH 4 especially) increased sharply, and both factors resulted in a financial stranglehold. Thus, in the framework of the Pacte de confiance et de responsabilité entre l'État et les collectivités territoriales, the Prime Minister and the local councilors discussed the possibility of an RETT's increase, to help the départements which were struggling with their finances.

Therefore, we can argue that the implementation or the non-implementation of the treatment was not due to a willingness to stimulate the housing market, or to help buyers and sellers through fiscal policy. This policy change was mainly driven by reasons entrenched in the financial turmoil of the départements, then it is as random. The draft Finance Act for 2014 was publicly announced on September 25, 2013, and relayed the information of a first agreement between the départements and the French Government about an increase in the RETT's Régime de droit commun (cf. Box 1). Most of the départements announced whether they would increase the RETT and when during the first semester of 2014.

At this stage, we can argue that both buyers and sellers of property were aware of the reform and its date of implementation, and whether the département where they intended to buy or sell would increase the tax 5 . They then could anticipate the reform by bringing forward the sale date, in order to avoid the tax increase.

The RETT reform was enacted on December 29, 2013, by the article 77 of the Finance Act for 2014, allowing the départements that are willing to do so (i.e. the implementation of a tax rise remains optional), to increase their part of RETT's Régime de droit commun by a maximum of 0.7 percentage point. It means that the rate of the RETT going to the départements can rise from 3.80% to 4.50% (i.e. an increase of 18.42% of the RETT departmental's part). Furthermore, at this time, the reform was enacted as temporary and should have been implemented only on the agreements finalized between March 2014 and February 2016; afterwards, the RETT should have gone back to 3.80% maximum. However, on December 29, 2014, the article 116 of the Finance Act for 2015 made permanent the possibility for the départements to rise their part of the RETT up to a rate of 4.50%. The choice to increase the tax or not and the level falls to the local councilors. The 4.50% rate is an upper limit, and the départements can set whatever rate suits them between 1.20% and 4.50%. However, in practice, every département that chose to raise the RETT has increased them by the maximum amount (i.e. up to 4.50%) 4 5 6 .

A first group of 61 départements implemented the reform on March 1, 2014, a second group of 20 départements on April 1, 2014, a third group of 2 départements on May 1, 2014, a fourth group of 7 départements on June 1, 2014, a fifth group of 4 départements on January 1, 2015 and finally a group composed of 2 départements on January 1, 2016. However, this last group is not considered as treated in our estimates, as we stop the period of estimation in October 2015. Finally, 5 départements 7 are still currently 8 at 3.80%.

From the full sample of départements (i.e. 101), we remove 9 départements, because of lack of data, or because we strongly suspect them to have heterogeneous housing market and/or unobservables that affect their housing market differently over time. Those départements are the 3 départements of Alsace-Moselle territory for the reasons already defined above (cf. also Box 1), the 5 overseas départements, and finally the département of Paris (75). Figure I 4. Revenu de solidarité active (RSA), Allocation personnalisée d'autonomie and Prestation de compensation du handicap.

5. An additional argument to demonstrate that they were aware of the reform, is that buyers and sellers usually mandate a real estate agent and/ or a notary for the matching process and afterwards some counsels about the documents (e.g. expert diagnostics, property tax notice), needed for the preliminary agreement. These brokers and experts are well-informed about the changes of the real estate legal context. 6. Except Côte d'Or ( 21) which increased them until 4.45%; thus, we consider it as being at 4.50% in the estimates. 7. Indre (36), Isère (38), Morbihan (56), and two overseas départements Martinique (972) and Mayotte (976). 8. May 2017.

shows a map of the implementation schedule of the reform.

Purpose of the evaluation

In the following evaluation, we focus on two main potential effects, although three effects may be distinguished:

(1) Anticipation effect. As the reform was publicly announced far ahead, we assume that the buyers and sellers were not caught off guard, and thus many of them may have chosen to bring forward the sale date in order to avoid the tax increase in their départements. This behavior can be referred to as a dynamic optimisation effect, or timing response. This period of anticipation should precede the implementation month. This assumption seems plausible when observing the trends in the number of transactions and the total tax base of the Régime de droit commun from January 2012 to October 2015 (Figure II). Indeed, some distinct peaks appear just before the date of implementation. We expect no effect on prices during the anticipation period. Two reasons can vindicate this guess. In the first place, as said in the previous sections, the sale price is set during the preliminary agreement, which is signed around 3 months before the sale date, and thus people who anticipated could have changed only the sale date and not the sale price set by agreement. Furthermore, as both the seller and the buyer are interested in avoiding the tax increase, no bargaining on price should have occurred.

(2) Retention effect. We expect the housing market to be impacted durably by the increase in the RETT, preventing some buyers from moving and accessing to ownership. Thus, a decrease in the volume of transactions should be observed. The retention effect should begin at the implementation date of the reform, and could have either persisted or diminished over time ("resilience of the market"). Note that in the very short-run, this effect is composed of re-timing due to the anticipated transactions (intertemporal substitution by those who would have purchased a real estate anyway), and of extensive margin responses (those who would have purchased a real estate in the absence of the reform). If the extensive margin effect is dominating the timing effect, we may observe a lock-in effect (e.g. buyers could have chosen to renounce to buy, to postpone their purchase, or to rent rather than to become owners).

(3) Price effect. Theory also suggests a slight effect on sales prices: due to the extensive margin response, the demand must have decreased while the supply must have remained the same; therefore, the bargaining power of the buyers must be higher, the competition between sellers must increase, and some would be willing to decrease their selling price. However, this is a strong assumption knowing that the French housing market is price-sticky. In France, RETT must be paid by the buyers, and knowing that the housing market is rigid, sellers have a greater bargaining power. Then, unlike Philadelphia's RETT reform -where RETT's payment is divided in half between buyer and seller -studied by [START_REF] Benjamin | Real Estate Transfer Tax and Property Values: The Philadelphia Story[END_REF] who estimated that housing prices decreased, it is less likely to observe the same phenomenon in France. Moreover, the RETT in France are proportional and not progressive. Hence, the agents have less interest in changing the sale price -compared to progressive RETT -and more in changing the sale date [START_REF] Davidoff | How Do Stamp Duties Affect the Housing Market?[END_REF][START_REF] Slemrod | The Behavioral Response to Housing Transfer Taxes: Evidence from a Notched Change in D.C. Policy[END_REF]. Furthermore, the data we use are not very suited to test this price effect. We hence focus on the first two effects.

Data

To undertake this evaluation, we use two main variables, which are the number of transactions and the tax bases, both by département Following this, we applied some correction to these raw datasets, in order to make them match to the months when the bill of sale is signed (and not to the months of tax revenue collection).

Our control variables include the unemployment rates, the number of new residential construction, population, property tax rates, and three local variables on the départements' finances. Data on the unemployment rates come from Insee 11 and are quarterly data by département for metropolitan France, and yearly data for the overseas départements, both seasonally adjusted. In order to estimate monthly data, we made linear interpolation. Data on the new residential construction (monthly building permits by département) come from the database Sit@del2, and are compiled by Insee. The estimated population on January 1 of each year in each département, using the Insee annual census. The property tax rates voted each year by the départements, from the 9 1 0 1 1 DGFiP 12 . Three local variables from the DGFiP-DGCL 13 in order to "compute an index of good administration" of the local governments. These variables are the salary cost, the operating revenue (which the total local taxes revenues) and the social spending, all per capita, by département.

Finally, we also use other variables in order to check for possible unobservables that could affect the sample groups differently over time, not included as covariates in the estimates because they do not fit to our panel data. Indeed, these variables do not vary across the regressed period, then their effect ought to be captured by the département and month fixed effects. These data are two local variables from Insee, in order to make a comparison of the treated and control groups from their inherent housing market, which are: the share of social housing and secondary residence, within the total number of housings, by département.

Empirical strategy

In order to estimate the effects of the RETT increase, we use a difference-in-differences framework [START_REF] Donald | Inference with Difference-in-Differences and Other Panel Data[END_REF]. To undertake our difference-in-differences design (Box 2), we divide our sample in two groups:

(1) the treatment group, composed of the départements that implemented the reform during the period from March 2014 to January 2015; (2) the control group, composed of the départements which had not yet implemented the reform at the estimated month (i.e. these départements were still on their pre-reform period) and of the 4 départements which remained at a RETT rate of 3.80% during our regressed time period: the (final) control group.

Number of Transactions

Number of Transactions

dt � � � � � � = dt Fidji ( ) × ×� � � MEDOC � � � . Tax

Box 2 -Validity of the difference-in-differences design

The most important hypothesis in the difference-indifferences framework is the common trend assumption, which assumes the evolution of the variable of interests would have been the same for the treatment and the control groups, without the reform. This assumption could be violated if there are some exogenous shocks or unobservables, which affect differently the groups over time. However, the trends of the outcome variables over the full sample period and the estimated period, show that they followed exactly the same trend and level until the reform, except for the département of Paris (75) and overseas départements (DOM). There is sometimes a slight difference in the trend of the May 2014 group (cf. Figures II andA). Those observations deserve further enquiry, which we perform below, and later in the robustness checks section.

Test on possible self-selection: logit

The binary logit is used to test whether there is a selection bias in the départements which implemented the tax increase, compared to the départements which did not (i.e. (final) control group). We use a binary logit over the period from January 2008 to December 2013.

Y X dt x x dt = + = ∑ β ε 1 9 (1)
In this equation, Y dt is equal to 1 if the département implemented the tax increase, 0 otherwise; X dt corresponds to one of the variables of interest or control, in a département d, in period t.

Estimates are reported in the online add-on C5. Estimates of the Table C5-1 show that the coefficients are close to zero, even if they are statistically significant. The choice to increase the tax is almost not correlated with these variables. It means that there is no selection bias of the treated départements. They did not do it because of a possible difference in the explaining variables, compared to the départements which chose to remain at 3.80%.

Placebo test

The placebo test is used to check empirically the validity of the common trend assumption, by regressing our variable of interests in a pre-reform period, and prior the period used in the standard regressions (i.e. January 2012 to October 2015). To implement this test, we use the period from January 2008 to October 2011, and regress the month-based model (see below) on the outcome variables, using the same groups. We define our dummies for anticipation and retention as being the same than equation (2) below, but the periods are moved back of four years.

For instance, for the départements which implemented the reform on March 1, 2014, the dummy for the anticipation effect (T d -1) is equal to 1 in February 2010, 0 otherwise.

Estimates are reported in the Online complement C5. Table C5-2 shows no coefficients significantly different from zero at the 10% level, in all the variables of interest; meaning that the trends of the treatment and control groups are the same before the implementation of the reform. Thus, the common trend assumption appears valid, and our difference-in-differences design can be implemented. The specificity of our difference-in-differences framework is that there is an attrition of the control group over the regressed period, and an increase of the treatment group ( 2015), in order to estimate whether there have been some different effects and heterogeneous shocks between all groups and subgroups.

We limit our estimations to the period from January 2012 to October 2015, for two reasons. First, we choose to start from January 2012 to avoid a possible noise from the reduction of the period of transmission of the bill of sale by the notaries from 2 months to 1 month (which occurred in 2011). Second, we stop the study in October 2015 because on January 1, 2016, the Mayenne (53), one of the département of the (final) control group, implemented the tax increase. Consequently, its anticipation period should begin in November 2015 (date of its public announcement) (Box 3).

We estimate two models, respectively termed month-based model and parsimonious model.

Month-Based Model

The aim of this model is to see the dynamics of the anticipation and the retention effects in the pre-treatment and post-treatment periods. We attempt to estimate how quickly the outcome variables react to the reform of the RETT, and how they evolve over time (e.g. how long the retention effect lasts). In order to perform these estimations, we use monthly leads and lags: 6 month leads for the anticipation effect, and 20 month lags for the maximum retention effect (keeping in mind that 20 months is the full period of retention: March 2014 to October 2015). This model can be seen as a sensitivity check against the parsimonious model. The model with monthly regressors is shown in the following equation:

logY A nticipation dt d t j Aj d,t T j k Oct d = + + + = = - = ∑ ∑ α λ β � � � .� 1 6 0 2015 β β ρ ε Rk d,t T k d t d t Retention X d � � = + + + (2)
where T d is equal to the implementation month of the reform in a département d.

Box 3 -Standard event study model with anticipation

To estimate properly the anticipation and retention effects, we use a standard event study model (see for instance [START_REF] Jacobson | Earnings Losses of Displaced Workers[END_REF] as proposed in [START_REF] Malani | Interpreting Pre-Trends as Anticipation: Impact on Estimated Treatment Effects from Tort Reform[END_REF] that allows to estimate properly effects of a treatment, when there are expectations and anticipations from the treated population, as it is the case in our evaluation. Indeed, as explained in the previous cited papers, when there are anticipations, the full treatment effect depends on both the ex-ante and the ex-post effects. Therefore, they must be estimated simultaneously to avoid a bias in the estimations. They propose two models: (1) the quasi-myopic model which is based on a standard event study specification, and

(2) the exponential discounting model. In this paper, our preferred model is the quasi-myopic model for two main reasons. First, the quasi-myopic model is easier to implement than the exponential discounting model and provides equal or better estimates when there is a finite and known period of anticipation, as in this natural experiment. Secondly, the exponential discounting model requires a structure on the error term, and in addition it assumes that people discount the future exponentially and have rational expectations, which is a strong assumption. In addition, the models include X dt , a vector of 7 time-variant control variables that could affect the outcome variable Y dt , α d , which controls for département time-invariant characteristics (département fixed effects), and λ t , which controls for differences across months shared by the sample groups (month-fixed effects).

Finally, the error term ε dt , clustered by département, and captures the département × month shocks to the variable Y dt [START_REF] Wooldridge | Fixed Effects and Related Estimators for Correlated Random-Coefficient and Treatment-Effect Panel Data Models[END_REF]. This error term is assumed to be uncorrelated with the regressors, and problems could occur using a within estimator in a difference-indifferences framework, especially in the case of time-variant omitted variables that affect differently the sample groups.

Parsimonious model

The following model is similar to the regression developed by [START_REF] Best | Housing Market Responses to Transactions Taxes: Evidence from Notches and Stimulus in the UK[END_REF].

It is our benchmark because it is parsimonious. Indeed, as shown in the estimates of the month-based model, the anticipation effect only occurs the month before the implementation, while the retention effect seems to last only in a short-term of 3 months after reform. The anticipation effect in T d -1 is estimated by β  A1 (positive timing effect), the mean retention effect is estimated by β  2 (negative effect due to re-timing + extensive margin response) and the mean effect post retention is estimated by β  3 . To interpret the raw coefficients, see Box 4. The estimates show an increase of around 25% the month just before the implementation of the reform (i.e. T d -1), significant at the 1% level. None of the other anticipation-period coefficients are significantly different from zero (except the coefficient for T d -5, equal to 5.6% 14 ), meaning the anticipation effect is concentrated over the month just before the date of implementation. Both specifications suggest that buyers and sellers really agreed to escape the tax increase, and consequently, they brought forward the sale date of one month.

Results

Month-based model

The estimates with the month-based model show a large decrease in the tax bases the first month of the RETT increase (i.e. T d ), of around 22%, 9.5% the second month after reform (i.e. T d + 1), and 4.6% the third month (i.e. T d + 2), all significant at the 1% and 5% level (Table 2-A). 1 4 None of the other 14. This coefficient is quite puzzling and we speculate that it might correspond to a possible first anticipation during the last quarter of 2013, following the draft Finance Act.

Box 4 -Interpreting results from a log-level model

As all the models are estimated in log-level, and as our independent variables displayed in the tables of results are dummies, exp β ( )-

(

)×

1 100 can be interpreted as: by how many percent the dependent variable Y has evolved in the situation where D = 1, compared to D = 0 (D represents the dummy variable of the treatment).

An admissible approximation is β × 100% when the coefficient is lower than 0.10. Note that all the results displayed in the tables are the raw estimated coefficients. Notes: Month 0 (solid vertical line) is the month of implementation of the reform in a given département; vertical dotted lines indicate statistically significant months As the month-based model is log-level, the «gross» coefficients should be calculated using the following method in order to be interpreted as a percentage, as shown in the graph above: (exp (β) -1) × 100. These effects are estimated from monthly total tax bases by département, thus represent the change in the volume of transactions assuming that prices was unchanged. Reading note: One month after the implementation of the reform, the volume of transactions decreased by around 10% in the départements which increased the RETT. Sources: CGEDD from DGFiP (MEDOC), Assiettes des droits de mutation immobiliers par département, Insee, Construction de logements (Sit@del2), Taux de chômage localisés, Estimation de population au 1 er janvier, DGFiP, Taux de fiscalité directe locale (TFPB), DGFiP-DGCL, Les budgets primitifs des départements, from 2012 to 2015; authors' computation.

coefficients are significantly different from zero. It proves that most of the retention effect took place the first three months after the reform, and the effect vanished later on as the plot of estimated coefficients of the monthly effects shows (cf. Figure IV). The cumulated decrease in the months following the reform is higher than the increase of 25% in T d -1 (see Figure V). This proves that the estimated fall in the number of transactions is not only due to re-timing (i.e. anticipated transactions that already occurred in T d -1).

Parsimonious model

Table 2-B shows the estimates of the parsimonious model for different specifications, introducing one by one the monthly-fixed effect, the département fixed effect and the control variables. For the anticipation effect, once we introduce monthly fixed effects, that is, we are really adopting the difference-in-differences estimation strategy, neither the coefficients nor the standard errors really change with or without covariates (columns (4) to ( 6)). We find that there was an anticipation in T d -1, of around 26%, significant at the 1% level. The average monthly retention effect during the three months following the implementation is around -14%, and significant at the 1% level (columns (4) to ( 6)), while we see no effect significantly different from zero in the period post retention when introducing the monthly fixed effects (columns (4) to ( 6)).

Net effect

One may want to compute the net retention effect (Mian & Sufi 15 , 2012; Best & Kleven, 2018). Indeed, the retention effect which has been evaluated so far is magnified by the strong anticipation effect in T d -1 which creates a "loss" of transactions the following month 15. We cannot implement the very same method proposed in this paper, because of the different waves of the implementation process. (i.e. re-timing). The upshot is an increase of the estimated negative effect.

Table 2-A Estimates for the month-based model

Retention effect (T d + 2) (β  3 R2 ) -0.047** (0.023) Retention effect (T d + 3) (β  3 R3 ) 0.00085 (0.029) Retention effect (T d + 4) (β  3 R4 ) 0.
The coefficient β  A1 = 0.23 (0.021) from equation (3) implies that the anticipation of the reform increase the volume of transactions by 26% the month just before the implementation, and the coefficient β  2 = -0.15 (0.021) implies that the average monthly activity was 14% lower in the treated départements the 3 months following the implementation. These estimates together imply that -β  A1 /(3β  2 ) = 51% of the retention effect was a re-timing effect due to the anticipated transactions (intertemporal substitution by those who would have purchased a real estate anyway), and the remaining 49% was an extensive margin effect (those who would have purchased a real estate in the absence of the reform).

A new piece of evidence is brought by Figure V, which plots the cumulative sum of the coefficients from the month-based model starting from one month before the implementation month (i.e. T d -1). It shows that the magnitude of the retention effect is higher than the one of the anticipation effect, and that in the months following the implementation date, the cumulative sum is always negative. Performing a Wald test on the sum of the coefficients from T d -1 to T d + 2, we can reject at the 5% level the hypothesis that this add-up to zero (H 0 ). In fact, it is even true for a period of 5 months from T d -1 to T d + 3 (Figure V and the double arrow), except for the month of implementation of the reform (i.e. T d ), which implies that the re-timing is almost completely absorbed in the first month of implementation. Beyond four months following the implementation date, we cannot reject that the evolution of the treatment and the control groups are similar.

The cumulative sum of the coefficients up to T d + 2 is equal to -0.18. Dividing it by 3 (i.e. number of months of the retention effect), we find an average monthly net effect of -5.8% over three months. The same computation up to T d + 3 (-0.17) gives an average monthly net effect of -5.5% during four months. We have then strong evidence of a short-term effect.

Using the coefficients from the parsimonious model (cf. Table 2-B) and applying a similar computation 16 , we find an average monthly net proportional change of -7% during the three initial months following the implementation date.

Taking advantage of both estimations and giving more weight to the monthly estimation, we then conclude to a short-term drop between 5.5% and 7% per month during three months after the implementation date (i.e. approximatively 6%), and no medium-or long-term response afterwards. This net monthly effect helps to provide a rough estimate of the number of missed transactions in 2014 due to the rise of the RETT. As shown in the following graph (Figure VI), the yearly number of transactions in the Régime de droit commun at the national level were around 1,050,000 on the verge of 2014. We could offer a rough estimate of the drop that would have occurred, were the implementation nationwide. We should observe a drop of the number of transactions of around 18% at the end of the three month period, that is 16,142 � , � � % 16 142 18 In fact, only 93% 17 of the départements implemented the measure over the regressed period. Then, the true effect is closer to 15,000, which is approximatively 1/4 of the drop that we can detect on Figure VI (see the circle, which brings about the total number of real estate transactions 18 ). This computation surely underestimates the true effect since we ignore any interdependence between local markets. It should be considered as a lower bound of the true effect. 1 7 1 8

Unfortunately, we cannot conclude on the issue whether this loss in transaction of the three (four) initial months is reversed (i.e. the entire response to the reform is a timing response). On the one hand, a piece of evidence in favor of no recovering is the fact that the monthly coefficients are not significant after T d + 2 in the monthly regression (cf. Table 2-A), contrary to [START_REF] Mian | The Effects of Fiscal Stimulus: Evidence from the 2009 'Cash for Clunkers' Program[END_REF] who got statistically significant reversal coefficients. On the other hand, a piece of evidence in the other direction is the fact that looking at the Wald test of the nullity of the sum of the coefficients, we cannot reject the hypothesis that these coefficients add up to zero beyond five months (i.e. after T d + 3). However, we should not forget that the design of the implementation introduces a noise beyond T d + 3 due to the attrition of the control group (cf. Table 1), in addition to the decrease in magnitude of the effect, which reduces the statistical power. Furthermore, the statistical power of the cumulative effect weakens also mechanically as we extend the horizon by adding an extra noise for each month added, as shown by the pattern of the Wald test's p-value. So, we choose to let this issue unsettled at this stage, and we conclude that this point needs further investigations.

Tax elasticity and the Laffer curve

In this section we are interested in quantifying the response of the total tax bases to a onepercent increase in the tax. Since we found that the net effect is estimated to be around -6% whereas the increase in tax rates is about 14.15% 19 the elasticity is:

ε τ τ τ τ TB Y Y Y = ∂ ∂ ≅ ≅ = log log � � . . . ∆ ∆ - - 0 06 0 1415 0 42 (4)
This means that tax bases decreased by 0.42% for a 1% increase in the RETT's rate (i.e. there is a loss in the tax bases which reduces the gains of tax revenues for the local budgets).

Following this, we want to compute � � τ τ > the rate from which the tax revenues of the départements would be maximum, then, begin to decline for each � � τ τ > (i.e. the maximum of the Laffer curve).

A small change in the tax rate changes Y by:

∂ ∂ = + ∂ ∂ + ( ) = -+       τ τ τ τ τ τ ε . � � � � � � � � � � � � � � � � � � � � � Y Y Y Y 1 1 1 (5) where ε ε τ τ � log log � � ≡ =- ∂ ∂ + ( ) + TB Y 1 1
is the elasticity of the tax bases with respect to tax-inclusive prices. The Laffer rate sets the above expression to zero:

τ ε = - � � 1 1 (6)
To compute this rate, we use the following expression:

� � � � � � . . . � � � . � ε ε τ τ τ τ TB TB 1 1 0 42 1 0 0509 0 0509 8 7 + = + ≅ + ≅ (7)
Then, replacing ( 7) in ( 6) implies that � � τ τ > ≅ 13%, and that the départements' tax revenues are still on the increasing part of the Laffer curve 20 .

Discussion

The main result -only a short-term effect of the reform -raises interesting issues. At first glance, when increasing tax, we should expect the market to be negatively impacted durably.

Nevertheless, in our case, the extensive margin effect is estimated to last only 3 months, and we see no difference between treated and controlled départements beyond. We provide three possible explanations for this result. (1) It is possible to build theoretical model of housing investment where the long-term effect is ambiguous.

(2) This short decrease could correspond to the shifting time of demand (people should buy anyway). Indeed, the average time for a housing contract in France is 3 months. So, perhaps that these three months of decrease correspond to the time spell for sellers to find new buyers, after that the first buyers gave up on buying when being informed of the tax increase. Moreover, those buyers could have decided to buy at lower price. They renounced to real estates that they were looking for, in order to buy some with lower characteristics and amenities a few months later. (3) It could be related to a cognitive bias from the agent. As developed by the Nobel Prize Richard Thaler, people do not feel price differences "equally" when prices are big. For instance, people are ready to pay a relatively important "cost" to save €10 for a small purchase (e.g. at a restaurant); at the same time, they think that a €200,000 and a €205,000 housing are almost of the same values, except the deviation is €5,000! Nevertheless, our study faces two main limitations. One is a possible spillover effect, due the fact that some buyers could have "voted with their feet" may introduce a bias. More precisely, some buyers who were willing to buy real estate in a treated département neighboring a controlled département, in an area close to the border, could have chosen to buy in the controlled département because of the reform. In further studies using micro data, this spillover effect could be estimated with a regression discontinuity design [START_REF] Hahn | Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design[END_REF][START_REF] Imbens | Regression Discontinuity Designs: A Guide to Practice[END_REF], by clustering the neighboring treated and controlled départements. Defining a band of a few kilometers around the border to make the difference between treated and controlled, and between the housing markets in the center of the treated départements, compared to their housing market at this border.

Nonetheless, we guess that this effect is small in magnitude, as real estate are heterogeneous goods, including their localization. The other is a possible lack of control variables, because we could not get all the desired data (monthly and by départements), especially the rent by département. Nevertheless, we attempt below to check for possible unobservables or heterogeneity between départements, and we assume that most of the possibly omitted covariates are time-invariant, thus captured by the fixedeffects estimator. 2 0

One could also argue that there is a selection bias, because the départements that did not implement the RETT increase, are different in some points to the others. That does not seem likely when looking at the trends of the outcome variables (cf. Figures II andA). Furthermore, when looking at the distributions and trends of the other local variables between groups (see Online complement C1), there is no marked difference between the treated and control groups. Population, property tax rates, index of "good administration" and their inherent housing market show no differences between groups, and between them and national statistics. Ultimately, what we are interested in here is the elasticity of buyers' and sellers' supply and demand in real estate, while the choice of the reform implementation falls to the local councilors. Those decision makers are elected, and thus one could think that there is a correlation between them and the population (composed of the buyers and sellers). However, the point in case is to know whether those elasticities are correlated with the choice to implement the reform or not. Such independence assumption is difficult to 20. Notice that the elasticity estimate would be higher using the gross estimates (rather than net estimates).

check. Nonetheless, we attempt below to test for a possible bias from the political color of the local governments.

The main selection problem, in natural experiments including a local fiscal policy reform, is the political color of the local councilors that decided to implement (or not) the tax increase. Indeed, in our study one could argue that leftwing or right-wing départements might have implemented the reform differently. However, the proportion of left-wing and right-wing départements which implemented the tax increase (or not), is exactly the same as the distribution of left-wing and right-wing départements among the whole country (Table 3). Furthermore, in the 2015 departmental elections, 28 départements switched from the left-wing to the right-wing, and only one switched from the right-wing to the left-wing. The new political distribution of the local councils is: 34 for the left-wing and 67 for the right-wing. Therefore, the distribution has shifted between political wings, but no département has decided to decrease the RETT, while they have had the possibility to do so.

More elements are discussed in the Online complement C4.

Robustness checks

As suggested in [START_REF] Meyer | Natural and Quasi-experiments in Economics[END_REF], we multiply the tests of robustness, in order to check the validity of our results. Developments and estimates are reported in the Online complement C5.

(1) Alternative dependent variable. Alternative dependent variable (total tax bases of the Régime dérogatoire) is used to test whether the results are biased because there was an exogenous shock affecting the housing markets of the two groups differently. Results of Table C5-3 show no coefficient significantly different from zero at the 10% level, for the substitute outcomes. Then, it appears that our results are not biased: there was no shock affecting differently the housing markets of the two groups during the regressed period.

(2) Estimations using different period. We check the validity of our results to the choice of the period and sample groups. Table C5-4 shows estimates close to the ones found in the main estimations. The main effect in which we are interest in being similar to our first estimates, they appear robust to the choice of the estimation period.

(3) Changes in local economic conditions. As the results that we obtain could be impacted by an exogenous economic shock, affecting the sample groups differently, we test for this kind of changes in the local economic conditions. To implement this test, we use the same method as in [START_REF] Benzarti | Who Really Benefits from Consumption Tax Cuts? Evidence from a Large VAT Reform in France[END_REF]. Results for both models presented in Tables C5-5 and C5-6 show only slight differences between the estimates and our main results. We can therefore conclude that our estimates are robust, and that no exogenous local economic shocks affected differently our groups.

(4) Regressing by treatment subgroups. We re-estimate the parsimonious model where we allow for a possible heterogeneity for the different subsets of treated groups. Results of this regression are displayed in Table C5-7. The anticipation effect is non-significant for the January subgroup and for the other subgroups spans a large range between 16% (May) and 45% (April). The retention effect is also non-significant for the January subgroup and is comprised between -10% (March) and -17% (May) for the other subgroups. None of the coefficients of the post retention period are significantly different from zero. It is not very surprising that there is some heterogeneity in the local-market responses.

(5) Removing possibly heterogeneous groups. We may suspect a possible heterogeneity or unobservables that affect differently May 2014 and January 2015 groups over time. In order to test this hypothesis, we estimate our coefficients removing either January 2015 or May 2014 group or both, from the estimated sample. Tables C5-8 shows only slight differences between the estimates and our main results. We can conclude that our findings are robust to the choice of the sample, and to a possible bias from heterogeneous départements.

* * *

To conclude, we find evidence that the RETT increase had an impact on the housing market in line with the economic literature. We bring empirical evidence that two behavioral responses took place. We show extremely compelling estimates of a short-term timing response to an anticipated tax increase. People brought forward transactions to the month before the tax increase. The number of transactions rocketed by 26% the month preceding the implementation of the reform. Second, the volume of transactions fell by around 14% on average per month during the three months following the rate change, whose 51% of this loss is due to re-timing. The two effects do not cancel out. All in all, the average monthly net effect corresponds to a transaction drop of around 6% over the three months following the implementation date, assuming no sale-price changes. Such assumption appears realistic, as the RETT system in France is proportional and the RETT's payment accrues to the buyer. Buyers and sellers can more easily agree in changing the sale date rather than the sale price [START_REF] Benjamin | Real Estate Transfer Tax and Property Values: The Philadelphia Story[END_REF][START_REF] Davidoff | How Do Stamp Duties Affect the Housing Market?[END_REF][START_REF] Slemrod | The Behavioral Response to Housing Transfer Taxes: Evidence from a Notched Change in D.C. Policy[END_REF], a behavior supported by the large anticipation effect. Nonetheless, it is difficult without noncumulative monthly data on the number of transactions and hedonic estimation, to disentangle the effect on the volume of transactions from the price effect. We find compelling evidence of a sizable short-term effect -but no medium or long-run effectmeaning that there is a strong "resilience" from the housing market (people should buy anyway). Moreover, we estimate that the short-term elasticity of the tax base to the tax rate is around -0.42, meaning that there is a loss of 42% in the tax revenues with respect to a situation of no behavioral response, the first quarter after the reform. Computing the Laffer tax rate, we conclude that départements' tax revenues are still on the increasing side of the Laffer curve. Note that our results are valid for partial equilibrium. We do not estimate the possible other general equilibrium aspects resulting from the distortion of the housing market, such as changes in investment from the local governments or impact in the labor market. Applied to national transactions data, our estimate means that around 15,000 transactions were missing because of the transfer tax increase. This estimate is likely a lower bound of the true impact.

This evaluation can be extended in three ways: using the non-cumulative monthly data on the number of transactions; doing a precise estimation of the price effect through hedonic model (using the notarial databases BIEN -Base d'informations économiques notariales -and Perval); implementing a regression discontinuity design to estimate the possibility that buyers could have "voted with their feet" (i.e. spillover effect).

Finally, our results might be used to discuss the impact of future RETT reforms, and anticipate the effect on the housing market, in particular on buyers and sellers behavior. Even if the RETT rise was a "good deal" for the départements in terms of tax revenue, the distorting effect of the tax reform was assessed: some people who could have become owners or moved from a place to another, did not because of the reform (i.e. lock-in effect). Consequently, in line with the findings of [START_REF] Van Ommeren | New Evidence of the Effect of Transaction Tax Costs on Residential Mobility[END_REF], we conclude that the RETT increase has a negative sizable (short-term) impact on mobility and well-being.

(

  Sources: DGFiP and Légifrance, Bulletin officiel des Finances publiques -Impôts 2017)

  Figure I Map of the RETT-increase implementation by Département

  Figure II Monthly (12-month cumulative) number of transactions from January 2012 to October 2015, by sample and implementation groups

  Figure A Monthly (12-month cumulative) tax base of the Régime de droit commun trends from October 2000 to February 2016, by sample groups

  Figure III Maps of the treatment and control départements A -March 2014 B -April 2014

  Table 2-A shows the estimates of the monthbased model, where the dependent variable is the total tax base of the Régime de droit commun, and are illustrated by Figure IV which shows a plot of the coefficient and confidence intervals (Table C-2 in the Online complement shows the detailed coefficients).

  Figure IVEffect of the reform on the volume of transactions, month by month before and after the implementation date

  Total tax bases of the Régime de droit communAnticipation effect (T d -5) (β 

  16. (0.23 + 3 x (-0.15)) / 3 = -0.073.

  Number of Transactions is equal to the previous two years' average monthly number of transactions (i.e. 89,681).

  Figure VCumulative effect of the reform on the volume of transactions, month by month before and after the implementation date

  Figure VI 12 month-moving-average total number of real-estate transactions

  The number of département used to compute the percentages. Notes: The party of the local government corresponds to the political color when the RETT increase was voted. Then, it corresponds either to the 2011 or 2015 departmental elections. Coverage: Whole France. This computation was made among all the départements (i.e. 101). Sources: Ministère de l'Intérieur, résultats des élections cantonales 2011 et départementales 2015.

  and month. The source of these variables is the Conseil général de l'environnement et du développement durable (CGEDD). The raw data on the RETT come from the Service de publicité foncière (datasets MEDOC and Fidji) and are compiled and modified by the CGEDD, before being made available at the departmental level in open access.

	MEDOC provides the tax revenue by dépar-
	tement and month, and is exhaustive. Fidji
	provides the tax revenue and the number of
	transactions also by département and month,
	but it presents the inconvenience to be not com-
	pletely exhaustive (1% of the transactions are
	missing). To solve this problem, the CGEDD
	uses both databases, and applies a correction
	coefficient 9 , in order to recover an estimation
	of the total number of transactions by départe-
	ment 10 . Next, the CGEDD computes the monthly
	total tax bases of each département, by dividing
	the tax revenue by the corresponding RETT rate,
	and publishes in open access two datasets.

The first one provides monthly data on the number of transactions of the Régime de Droit Commun by département, for the period from April 2004 up to now (from MEDOC + Fidji), but those data are computed on a 12-month cumulative basis. The second one provides monthly total tax bases (raw and 12-month cumulative) of the Régime de droit commun and the Régime dérogatoire separately and by département, for the period from January 2000 up to now (from MEDOC). The data on the Régime de droit commun (whereby the reform is implemented) are composed approximately of 95% of established properties (whose 15% of non-residential premises) and around 5% of lands. We only use this second dataset because it is impossible at this stage to recover the simple monthly data for the first dataset.

  Table 1 and Figure III). We subdivide the treatment group in five subgroups (Table A1 in Appendix), where the treated départements are clustered by date of implementation (i.e. March 2014, April 2014, May 2014, June 2014 and January

Table 1 Size of the treatment and control groups over the estimated period, by date of implementation

 1

	Period (from)		Treatment	Group	Control	Total
		March 2014	58		34	92
		April 2014	76		16	92
	Treatment	May 2014	78		14	92
		June 2014	85		7	92
		January 2015	88		4	92
	Notes: Numbers correspond to the number of départements. Dates correspond to the month of implementation, and are different from the treat-
	ment subgroups.

  Post.Retention d,t ∈ [T d + 3, Oct. 2015]

	+ ρX dt + ε dt	(3)
	where T d is equal to the implementation month of the reform in a département d.
	Anticipation d,t = T d -1 is a dummy variable
	equal to 1 only the month preceding the imple-
	mentation month (i.e. T d ) in a département d, 0 otherwise. For instance, in the départe-
	ments that increased the RETT in March 2014,
	Anticipation d,t = T d -1 = 1 in February 2014; in
	the départements that implemented the reform
	in April 2014, Anticipation d,t = T d -1 = 1 in March 2014.

After this short-term effect, the coefficients are non-different from zero, meaning that the housing market should have reached a new steady state. Then, we developed the following model in order to estimate the average effects of these three periods.

logY dt = α d + λ t + β A1 Anticipation d,t = T d -1 + β 2 Retention d,t ∈ [T d , T d + 1, T d + 2] + β 3 Retention d,t ∈ [T d , T d + 1, T d + 2] is equal to 1 if a

RETT increase is implemented in a département d, and the month t belongs to its 3 first months following the implementation date, 0 otherwise.

Post.Retention d,t ∈ [T d +

3, Oct. 2015]

is equal to 1 if a RETT increase is implemented in a département d, and the month t belongs to the period after the 3 first months following the implementation date, 0 otherwise.

Table 2 -B Estimates for the parsimonious model

 2 This table reports estimates of equation 3, using within estimator. Outcome variable is in log in the estimations. In this table T d corresponds to the month of implementation of the reform in a département d. Standard errors, given in brackets, are clustered by département. FE indicates fixed effects. Stars indicate significance level: * p<0.1, ** p<0.05 and *** p<0.01. Sources: cf. Table 2-A.

			Total tax bases of the Régime de droit commun
		(1)	(2)	(3)	(4)	(5)	(6)
	Anticipation effect (T d -1) (β  A1 ) 3	0.19*** (0.014)	0.19*** (0.016)	0.18*** (0.016)	0.23*** (0.021)	0.23*** (0.021)	0.23*** (0.021)
	Mean retention effect (β  2 ) 3	-0.16*** (0.011)	-0.15*** (0.014)	-0.17*** (0.013)	-0.14*** (0.022)	-0.15*** (0.021)	-0.15*** (0.021)
	Mean effect post retention (β  3 ) 3	0.031*** (0.0050)	0.049*** (0.010)	0.036*** (0.0099)	-0.0099 (0.026)	-0.018 (0.025)	-0.016 (0.024)
	Adjusted R²	0.055	0.067	0.070	0.64	0.64	0.65
	Observations	4,232	4,232	4,232	4,232	4,232	4,232
	Monthly FE	No	No	No	Yes	Yes	Yes
	Département FE	No	No	Yes	No	Yes	Yes
	Control Variables	No	Yes	Yes	Yes	No	Yes
	Notes:

Table 3 Distribution of the départements' political color, by implementation or non-implementation of the RETT increase

 3

	Left-Wing	Party	Right-Wing	Total

APPENDIX ____________________________________________________________________________________________

Table A1

Sample groups with subdivision of the treatment group (a) in subgroups by date of implementation


Henri Gouin
email: henri.gouin@meca.u-3mrs.fr

The wetting problem of fluids on solid surfaces: Dynamics of lines and contact angle hysteresis

In 1805, Young was the first who introduced an expression for contact angle in static, but today, the motion of the contact-line formed at the intersection of two immiscible fluids and a solid is still subject to dispute. By means of the new physical concept of line viscosity, the equations of motions and boundary conditions for fluids in contact on a solid surface together with interface and contact-line are revisited. A new Young-Dupré equation for the dynamic contact angle is deduced. The interfacial energies between fluids and solid take into account the chemical heterogeneities and the solid surface roughness. A scaling analysis of the microscopic law associated with the Young-Dupré dynamic equation allows us to obtain a new macroscopic equation for the motion of the contact-line.

Here we show that our theoretical predictions fit perfectly together with the contact angle hysteresis phenomenon and the experimentally well-known results expressing the dependence of the dynamic contact angle on the celerity of the contact-line. We additively get a quantitative explanation for the maximum speed of wetting (and dewetting).

INTRODUCTION

We propose a dynamic model of motion of the contact system between two fluids and a solid surface [1][2][3]. Comparisons between our results and recent experiments or behaviours from statistical physics show an excellent fit on the qualitative level as well as the quantitative one [4][5][6]. The equations of fluid motions are those of Navier-Stokes [7,8]. The boundary conditions are expressed at the same time on different surfaces by dynamic Laplace formulae, but also on the contact-line by a differential equation using the new term of line viscosity. This differential equation, namely the expression of the Young-Dupré condition of the dynamic contact angle, takes into account the line viscosity but also the inhomogeneousness of all types on the solid surface [9][10][11][12]. It gives the microscopic behaviour of the Young contact angle. The inhomogeneousness is distributed at distances going from a few tens to a hundred Angströms. This distance is that of the operating ranges of intermolecular forces which command the surface energies. With the lower part of this scale, energies are homogenized and distances are no longer significant on a macroscopic scale [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF][START_REF] Gouin | [END_REF]. The hysteresis behaviour depends solely on the physico-chemical properties of the solid surface and on the celerity of the contact-line. Its universal form makes it possible to discuss the general case independently of any particular apparatus. All the efforts on the contact-line do not have the same effects. In our scaling, some of them produce a work that tends to zero and they do not appear in the macroscopic expression of the contact system motion. It is noteworthy that the weak differences between potentials have huge implications. This may seem surprising at first sight. It is due to the fact that the contact-line is massless and consequently its motion equation is not in the same form as for material systems [15]. We can note finally that the often proposed concepts of Young angles (apparent, measured, real etc.) [16,17] no longer appear in our model; only the expression of the angle associated with the macroscopic contact-line behaviour is meaningful in the framework of continuum mechanics. Jumps on the inhomogeneousness are also considered; a shift factor is proposed by Hoffman correcting the relation between the line celerity and the Young angle [7]. They appear unnecessary in our model. Lastly, other models using a notion of dynamic surface energy are also considered in the literature [18]. They are out from the framework of our study.

KINEMATICS OF A FLUID AT A CONTACT LINE AND DISSIPATIVE FUNCTION

To solve a boundary-value problem associated with the governing equations, it is necessary to detail what conditions should be imposed near a moving contact-line. The difficulty arises with the no-slip condition usually considered in viscous fluid problems. In fact two main problems crop up in the physics of contactline. The first one comes from the kinematics of the contact-line itself. It is thought there is a contradiction in assuming the no-slip condition at a solid wall and expecting the fluid to displace another one (for example a liquid displaces air). The classical concept of fluid kinematics does not satisfactorily describe the solidfluid-fluid intersection region. In a larger fluid motion axiomatic, a no-slip boundary condition and a moving contact-line are kinematically compatible concepts. One fluid undergoes a rolling-sticking motion while the motion of the other fluid is more complex. No constitutive equation for the bulk fluids is imposed. The derived properties of the motion are characteristic of Newtonian as well as non-Newtonian fluids. The wall shape roughness is not excluded. The key idea is that a fluid which does not slip on a solid surface does not preclude the possibility that at some instant a fluid material point may leave the surface: the velocity of the fluid must equal the solid velocity at the surface. The representation is a consequence of experimental, theoretical and numerical data obtained in the literature. When the fluid advances, the particles of the fluid interface come and stick on the solid wall according to a movement of adhesion similar to an adhesive tape that one would stick on the wall. These various observations are schematized in a main case represented in the plane section in Fig. 1. The contactline acts in a similar way as a "shock wave" and is not a material line: along the contact-line, the velocity of the fluid is discontinuous and we emphasize that the multi-valued velocity of the fluid at the contact-line comes from the fact that the two fluids stick a solid surface [4].

D t (L ) 1 D t (L ) 2 " D t (L ) 2 ' S Σ t 1 Σ t 1 ' Σ t 3 Σ t 2 t Γ (a)
Motion of the contact line Figure 1. Typical motion of fluids in contact on a solid surface: the contact-line moves along the edge ∑ t1 of S towards ∑' t1 . The fluid L 1 is presented solely in one field D t (L 1 ) and moves along material surface ∑ t2 which has a common interface with the fluid L 2 . The velocity of the fluid L 1 is equal to that of the fluid L 2 along the interface ∑ t2 .The particles of the fluid L 1 belonging to ∑ t2 come into contact with S along the line Γ t and then adhere to ∑ t1 . The fluid L 2 is made up of two independent fields D' t (L 2 ) and D" t (L 2 ). A common material surface ∑ t3 separates the two parts of the fluid L 2 . The particles from L 2 belonging to ∑ t2 come into contact with S along Γ t then fork on ∑ t3 and move away from Γ t . The particles of L 2 in contact with ∑' t1 are reached by Γ t and subsequently move away from Γ t along ∑ t3 . We obtain three other similar cases in exchanging fluids L 1 and L 2 and the direction of the motion.

The second problem arises from the dissipative function of a Newtonian fluid at the contact-line between two immiscible fluids and a solid boundary [19]. The condition of adherence generates a rate of energy dissipation which becomes infinite along the contact-line. We will adopt the following framework: At the distance e from the contact-line the viscosity stresses are those of Navier-Stokes and the liquid adheres to the wall. Near the contact-line (distance less e), the liquid slips on the wall and the flow is no longer Newtonian. On the solid surface S, the variations of the fluid velocities are along S and for a non isotropic case of a linear viscous stress tensor, the principal part of the viscous stress vector is in the form:

K v = -ν ∂v 1 ∂x n
where n is the unit normal vector at the contact-line along S, x the curvilinear coordinate orthogonal to the contact-line and v 1 the fluid velocity component along the x-coordinate.

Assuming that ν is a constant near the contact-line, we integrate K v between point P e of curvilinear coordinate -e and point P on the contact-line. Then T v = -∫ -e 0 K v dx and the form of the line viscosity vector T v for the linear model is

T v = τ n with τ = -ν u (1)
where u is the contact-line celerity. This line viscosity vector T v has the physical dimension of a force per unit of length. Modern computers calculate the motion of a few thousands of molecules of liquid in the vicinity of the contact system. The slip condition is fully justified but only at a few Angströms distance e from the contactline (one or two molecular distances). Beyond this zone, the liquid adheres to the wall and the Navier-Stokes equations seem to be a quite adapted model. In the e-zone of slipping, the tensor of viscosity is expressed by the single term T v regrouping the non-Newtonian stresses of the contact system. Far from the critical conditions, the contact-line has a thickness of a few molecules and merges with the e-zone of slipping: on the scale of a continuous medium, the slip zone appears with a null thickness and merges into the contact-line. The well-known paradox of the dissipation rate becoming infinite in the vicinity of the contact-line becomes a purely mathematical paradox. Indeed, the singularity of the dissipative function comes from a term in the form log(R/e) where R is a macroscopic length scale. This term becomes infinite when e tends to zero. Due to the fact that e is of an order of two molecular distances, log(R/e) remains bounded (for example with R/e = 10 10 , log(R/e) is lower than 30) and the singularity does not have a physical existence. For slow motions, the dissipative terms coming from the Newtonian dissipative rate and from the line viscosity ν are small enough for the flow to be considered as isothermal.

THE DYNAMIC YOUNG-DUPRE EQUATION

The Lagrange-d'Alembert principle of virtual works can be handled by the usual techniques of variation calculus. The equations of fluid motions are those of Navier-Stokes including the condition of adherence to the solid wall. The boundary conditions are expressed at the same time on different surfaces by dynamic Laplace formulae, but also on the contact-line by a new dynamic Young-Dupré equation using the new term ν of line viscosity:

σ 2 cos θ + σ 1 + ν u = 0 (2)
where σ 1 is the difference between the surface energies of the two fluids and the solid, σ 2 is positive and constant along the fluid-fluid interface, θ represents the Young angle associated with the fluid-fluid interface and the solid wall. To be in accordance with the second law of thermodynamics ν must be positive. It is immediately noticed that with the advance of the contact-line, u is positive and the dynamic angle is higher than the equilibrium angle. The result is reversed when u is negative.

A CLASSICAL EXAMPLE OF EXPERIMENT AND THE SCALING BEHAVIOUR OF THE CONTACT-LINE

The apparatus is a cylindar tube of radius "a" with a vertical symmetry axis [20]. A liquid fills the cylinder up to a position determined by a piston (Fig. 2a). The liquid is in contact with air. The air-liquid constant surface energy is σ 2 . The wall is inhomogeneous with a surface energy σ 1 = σ sl -σ sv (difference between the superficial energies of solid-liquid and solid-air) depending on the geometrical position of the meniscus. The value σ 1 is invariant around the vertical axis. The surface energy on the interval AP of the wall is such that:

σ 1 = σ 10 -K sin ( z εa ) (3)
where σ 10 is a constant surface energy, ε is a small dimensionless parameter and the scalar K is positive and smaller than σ 10 . (This expression is only an explicit case. For any periodic smooth function, calculations are unaltered). The methods of similarity make it possible to consider slow motions so that the dynamic Laplace equations have the simple form of the static case. Because of the symmetry of revolution, the meniscus may be considered as a spherical cap (such is the case for speeds lower than a few cm.s -1 in capillary tube of diameter less than a few millimeters and for a liquid as the glycerol). We trace the various potentials when the piston is at rest (z = l ). In suitable units, we take the values a = 1, K = 0.005, σ 10 = -0.5, σ 2 = 1. The curves are represented in the vicinity of the z-coordinate z m associated with θ m = Arcos 0.5 (coordinate z is defined except for an additive constant). The potentials W o , W e and W s have been shifted up or down (since potentials are defined except for an additive constant). Note that W o is the limit of W e when ε tends to 0, but it is not the same for W s which is basically different: tangent lines at the limiting points zand z + are parallel to the z-axis and the function g(z) = -∂W s /∂z is of square root type at these points.

ν dz dt = F(z,t) + K sin ( z εa ) (4)
with F(z,t) = 2λ (1+λ 2 ) σ 2 -σ 10 and λ = h a .

We focus on the most characteristic case for which |

σ 10 σ 2 | < 1.
We consider two significant cases encountered in experiments: piston at rest and piston in uniform motion (such as L(t) = v o t). Consequently,

ν dl dt = f(l ) + K sin ( l + L(t) εa ) -ν v o (5)
with l = z-L(t) and f(l ) = F(z,t). Differential equation ( 5) yields a single solution l ε (t) when l ε (0) = l o . We obtain the fundamental result that gives the behaviour of l ε (t) when ε tends to zero [15].

Asymptotic behaviour of the contact-line

When ε tends to 0, l ε (t) converges uniformly to the solution l (t) of the differential equation:

ν dl dt = g(l ) (6)
(f 2 (l ) -K 2 ) 0.5 -ν v o if l < l - with l (0) = l o and g(l ) = -ν v o if l -< l < l + -(f 2 (l ) -K 2 ) 0.5 -ν v o
if l + < l where land l + are two constant abscissa verifying the relations f(l -) = K and f(l + ) = -K.

When the piston is fixed Eqs (5-6) become Potentials W o and W s are convex functions.

The piston is fixed

(v o = 0).
The differential equation corresponding to the asymptotic behaviour of Eq. ( 5) when ε tends to zero is associated with potential W s . The driving force g(l ) has singularities of the square root type.

During the scaling -i.e. when ε tends to zero -W s cannot be regarded as the limit of the sum of separate energies associated with the mean energy W o and the energy of perturbation (W e -W o ). This shows that by changing the scale we lose the property of additivity of energy for the solution of the limit differential equation: W s is not the potential energy limit (Fig. 2b).

A physical interpretation of the previous limit behaviour can be given: when ε tends to zero, the potential W e admits on the range [l -, l + ] a large number of local minima whose respective distances converge toward zero with ε. For any initial position l o in [l -, l + ] the closest local minimum is reached in an infinite time. When ε tends to zero, any initial position l o of the interval [l -, l + ] is located between two local minima whose gap tends to zero with ε. On a macroscopic scale the local minima confound themselves with the initial value which becomes a "position of equilibrium".

Note that an angle θ A (0) = Arcos ( -σ 10 -K σ 2

) corresponds to the l --position. In the same way, an angle θ R (0) = Arcos ( -σ 10 + K σ 2

) corresponds to the l + -position.

Then, θ R (0) ≤ θ m ≤ θ A (0) with θ m = Arcos ( -σ 1 0 σ 2

).

This "static case" indicates that the Young angle θ is included in the interval [θ R (0), θ A (0)]. The final value of θ denoted θ f depends on the initial position l o of l and consequently on the initial value θ o of θ. We obtain the asymptotic behaviour.

θ o ∈ ] θ R (0), θ A (0) [ ⇒ θ f = θ o θ o ≤ θ R (0) ⇒ θ f = θ R (0) θ o ≥ θ A (0) ⇒ θ f = θ A (0) 4.1.2. Piston in uniform motion (v o ≠ 0).
It is easy to prove that for any initial condition l o , a constant solution to Eq. ( 6) is reached in a finite time. We get v o = u. When the piston advances, v o = u > 0 and when the piston recedes, v o = u < 0. The constant solutions of Eq. ( 6) verify f 2 (l ) -K 2 -ν 2 u 2 = 0 The value of the advancing angle θ A (u) is

θ A (u) = Arcos ( -σ 10 -( K 2 + ν 2 u 2 ) 0.5 σ 2 ) (7)
The value of the receding angle θ R (u) is

θ R (u) = Arcos ( -σ 10 + ( K 2 + ν 2 u 2 ) 0.5 σ 2 ) (8)
We deduce the inequalities

θ R (u) ≤ θ R (0) ≤ θ m ≤ θ A (0) ≤ θ A (u)

COMPARISON BETWEEN RESULTS, EXPERIMENTAL DATA AND OTHER MO-DELS OBTAINED IN THE LITERATURE

The dynamic line tension behaviour

Following the expression of the solid surface energy, the dynamic line tension admits a macroscopic behaviour associated with the differential equation ( 6). In the case of a piston at rest, the contact-line motion is governed by the equation

ν dz dt = ± [(σ 2 cos θ + σ 10 ) 2 -K 2 ] 0.5 (9)
where sign + or -depends on the direction of the line motion and K = max | σ 1 -σ 10 | is the maximum of fluctuations in the fluid-solid energy with respect to its average value. The average value of the line tension due to surface energies is τ = σ 2 cos θ + σ 10 . The line tension τ has the dimension of a force by unit of length as |T v | does. Then τ m = -K represents the value of τ at θ = θ A (0).

If angle θ is close to θ A (0), we obtain

ν dz dt ≈ (2 τ m ) 0.5 (τ -τ m ) 0.5 (10)
If the contact angles are small while expanding θ to order 2, we obtain

θ 2 -θ A 2 (0) = β (τ -τ m )
where β is a suitable constant.

The results are extendable to the case of a piston in advancing motion. Eq. ( 10) is unchanged but the dynamic angle of contact is such that

(σ 2 cos θ + σ 10 ) 2 -K 2 -ν 2 u 2 = 0
If we note τ u = -((σ 2 cos θ + σ 10 ) 2 -ν 2 u 2 ) 0.5 , the previous results are unchanged but | τ u | < | τ m |. In the case where the contact-line recedes it is easy to present calculations in the same way and to obtain analogous results. These results agree with those obtained in the literature by molecular statistics or single defects [5,21].

Limit velocities of the contact-line

Limit velocities for advancing and receding contact-lines are revealed in experiments [22]. Indeed, the advancing dynamic Young angle must be lower than π. The associated celerity of the contact-line is u π and ν u π = [(σ 2 -σ 10 ) 2 -K 2 ] 0.5 (11) In the same way the receding dynamic Young angle must be higher than 0. The associated celerity of the contact-line is u o and

ν u o = -[(σ 2 + σ 10 ) 2 -K 2 ] 0.5 (12)
The knowledge of σ 2 , σ 10 , u o and u π allows us to determine K and ν: the previous results make it possible to determine surface quality and line viscosity by simple measurements.

Connection between the dynamic contact angle, the line celerity and the line viscosity

The purpose is not to take the whole of the results obtained in the literature but to show by comparison with simple experiments that our model leads to qualitative and quantitative behaviour of the contact system. [START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF] from [6]). Plots from figures (c) and (d) are associated with the data of G.M. Femigier and P. Jenffer (ref. [12] from [6]). Solid curves are ajusted with our results computed by Eq. (7). The solid surfaces are assumed to be smooth enough for the hysteresis to be small. It is easy to adjust the values of σ 10 , σ 2 and ν to fit with experimental data denoted by points.

We use the experimental results for liquid flows in contact with air in capillary tubes. In the model suggested by relation (3), the solid surface inhomogeneousness is represented by the factor K. In fact, K << σ 10 and in relations ( 11) and ( 12) the term K can be neglected with regard to the terms σ 2 -σ 10 and σ 2 + σ 10 . In order to be in experimental conditions, we suppose the diameter of the tube to be about 1 mm. Values of the viscosity and those of the liquid-air surface energy are given in textbooks. We deduce the numerical relations between the capillary number Ca = μu / σ 2 and u. These expressions are given in c.g.s. units. For water we obtain u = 7200 Ca and for glycerol u = 4.2 Ca. The experimental curves yield limit velocities of the contact-line. They are associated in various measurements with the values of Ca ranging between 10 -2 and 10 -3 . In the case of an air-glycerol interface, these values correspond to limit celerity u π included between 0.042 cm.s -1 and 0.42 cm.s -1 and in the case of an air-water interface to limit celerity u π included between 72 cm.s -1 and 720 cm.s -1 .

For glycerol and a limit celerity of 0.42 cm.s -1 and for a wetting static angle θ A (0) of about 50° we obtain a line viscosity of about 240 c.g.s. In the case of water and for a limit celerity of 100 cm.s -1 , the line viscosity is about one c.g.s. unit. Due to the condition associated with the dimensionless Weber number, in order to obtain a spherical cap for the liquid-air interface, the case of water is obviously less realistic than the case of glycerol.

CONCLUSION

It is the first time a complete method of the wetting problem of the contact-line is presented in dynamics.

The results are applicable in different fields as well as physics, physical-chemistry, biology and engineering.

The model proposed accounts for the phenomenon of hysteresis of the Young angle and expresses the movement of the line. Fluid motions are continuous along the contact system but velocities are discontinuous on the contact-line.

The microscopic motion of the contact system is governed by a simple differential equation. When the irregularities of liquid-wall surface energy vary over length that tends to zero relative to the size of the capillary tube, the solution of the microscopic motion tends to a limit which is the solution of a new differential equation. This differential equation of the macroscopic or homogenized motion is not the limit of the microscopic one.

The paradox of an infinite rate of dissipation in the contact system is solved: near the contact-line, the viscous stress tensor is replaced by a viscous tension of line. The results of the example are independent of the tube radius. Indeed, the hysteresis behaviour depends solely on the physico-chemical properties of the solid surface and on the celerity of the contact-line. Its universal form makes it possible to discuss the general case independently of any particular apparatus. Relations ( 11) and ( 12) can be written without difficulty utilizing the inhomogeneousness of the solid surface in a different form from Eq. ( 3). The most significant innovation is the introduction of the new concept of line viscosity. This term is essential to the construction of our model. Its value is obtained by using experimental measurements. It allows us to obtain the microscopic behaviour of the contact-line. The scaling of this behaviour to a macroscopic level yields the hysteresis phenomenon of the Young contact angle and the main experimental well-known results.

A complete study of our model in the wetting problem and the analysis of fluid motion near a contact-line are presented in [START_REF] Gouin | Continuum Mechanics and Thermodynamics[END_REF].

  Figures 2.Figurea: "The apparatus". In the axisymmetric orthonormal axes O i k, z denotes the position of the contactline, L(t) determines the position of the piston (z = l + L); the Young angle between the meniscus and the wall is noted θ; h is the algebraic height of the meniscus. Figureb: "Potentials associated with the motion equation of the contact-line". We trace the various potentials when the piston is at rest (z = l ). In suitable units, we take the values a = 1, K = 0.005, σ 10 = -0.5, σ 2 = 1. The curves are represented in the vicinity of the z-coordinate z m associated with θ m = Arcos 0.5 (coordinate z is defined except for an additive constant). The potentials W o , W e and W s have been shifted up or down (since potentials are defined except for an additive constant). Note that W o is the limit of W e when ε tends to 0, but it is not the same for W s which is basically different: tangent lines at the limiting points zand z + are parallel to the z-axis and the function g(z) = -∂W s /∂z is of square root type at these points.

Figure

  Figures 2.Figurea: "The apparatus". In the axisymmetric orthonormal axes O i k, z denotes the position of the contactline, L(t) determines the position of the piston (z = l + L); the Young angle between the meniscus and the wall is noted θ; h is the algebraic height of the meniscus. Figureb: "Potentials associated with the motion equation of the contact-line". We trace the various potentials when the piston is at rest (z = l ). In suitable units, we take the values a = 1, K = 0.005, σ 10 = -0.5, σ 2 = 1. The curves are represented in the vicinity of the z-coordinate z m associated with θ m = Arcos 0.5 (coordinate z is defined except for an additive constant). The potentials W o , W e and W s have been shifted up or down (since potentials are defined except for an additive constant). Note that W o is the limit of W e when ε tends to 0, but it is not the same for W s which is basically different: tangent lines at the limiting points zand z + are parallel to the z-axis and the function g(z) = -∂W s /∂z is of square root type at these points.

  Figures 2.Figurea: "The apparatus". In the axisymmetric orthonormal axes O i k, z denotes the position of the contactline, L(t) determines the position of the piston (z = l + L); the Young angle between the meniscus and the wall is noted θ; h is the algebraic height of the meniscus. Figureb: "Potentials associated with the motion equation of the contact-line". We trace the various potentials when the piston is at rest (z = l ). In suitable units, we take the values a = 1, K = 0.005, σ 10 = -0.5, σ 2 = 1. The curves are represented in the vicinity of the z-coordinate z m associated with θ m = Arcos 0.5 (coordinate z is defined except for an additive constant). The potentials W o , W e and W s have been shifted up or down (since potentials are defined except for an additive constant). Note that W o is the limit of W e when ε tends to 0, but it is not the same for W s which is basically different: tangent lines at the limiting points zand z + are parallel to the z-axis and the function g(z) = -∂W s /∂z is of square root type at these points.

Figures 3 .

 3 Figures 3. "Comparison between theoretical and experimental results". Due to the universality of our model, the macroscopic contact angle is plotted as a function of Ca = μu / σ 2 . The Ca-axis is on a logarithmic scale. Plots from figures (a) and (b) are associated with the data of J.P. Stokes et al. (ref.[START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF] from[6]). Plots from figures (c) and (d) are associated with the data of G.M. Femigier and P. Jenffer (ref.[12] from[6]). Solid curves are ajusted with our results computed by Eq.(7). The solid surfaces are assumed to be smooth enough for the hysteresis to be small. It is easy to adjust the values of σ 10 , σ 2 and ν to fit with experimental data denoted by points.

The simplest way is to consider some surface inhomogeneousness given by a model represented with relation (3). The values of σ 2 , σ 10 and K allow us to draw the graphs of applications given by ( 7) and (8). Only the ratio σ 10 / σ 2 and the value of K are significant. We present in Figures 3 experimental layouts drawn in the literature for an advancing motion of the contactline. The similarity of the theoretical graph and experimental data is striking. It makes it possible to obtain the line viscosity value ν and the limit velocities u o and u π .