Samira Marti

Isabel Martínez

Florian Scheuer

Samuel Ammann

Enea Baselgia

Marius Brülhart

Joël Bühler

Vera Frei

James Hines

Katrine Jakobsen

Niels Johannesen

Marko Köthenbürger

Wojtek Kopczuk

Simone Leibundgut

Raphael Parchet

Yasmine Perrinja- Quet

Laura Salathé

Sebastian Siegloch

Joel Slemrod

Dario Tortarolo

Does A Progressive Wealth Tax Reduce Top Wealth Inequality? Evidence From Switzerland *

Like in many other countries, wealth inequality has increased in Switzerland over the last fifty years. By providing new evidence on cantonal top wealth shares for each of the 26 cantons since 1969, we show that the overall trend masks striking differences across cantons, both in levels and trends. Combining this with variation in cantonal wealth taxes, we then estimate an event study model to identify the dynamic effects of reforms to top wealth tax rates on the subsequent evolution of wealth concentration. Our results imply that a reduction in the top marginal wealth tax rate by 0.1 percentage points increases the top 1% (0.1%) wealth share by 0.9 (1.2) percentage points five years after the reform. This suggests that wealth tax cuts over the last 50 years explain roughly 18% (25%) of the increase in wealth concentration among the top 1% (0.1%).

Introduction

Many advanced countries have experienced rising income and wealth inequality over the past decades. These trends have spurred discussions about potential institutional responses. In particular, the tax treatment of capital gains has been at the front and center of the policy debate across the globe in the last few years. One reason is that they make up the largest component of income at the top of the distribution, including notably the payoffs to the founders of successful businesses (Scheuer and Slemrod 2020). At the same time, capital gains are treated favorably by most countries' tax systems at the moment, mostly because of lower rates, taxation only based on realization, and various tax exemptions at death. Taken together, these two facts have been recognized as a key reason for the erosion of tax progressivity, so that the average tax rate of millionaires and billionaires can be lower than that of individuals further down the distribution (Leiserson and Yagan 2021).

To address this, the introduction of wealth taxes has been prominently discussed in several countries including the United States (Saez and Zucman 2019). This would ensure that rich households bear some tax burden even when not realizing any capital gains, which may dampen the rise in wealth inequality over time. However, we currently lack systematic evidence on how wealth taxation (and its progressivity) affects the evolution of wealth concentration. Our aim in this paper is to shed light precisely on this effect.

To do so, we exploit the decentralized structure of the Swiss wealth tax as a laboratory setting. While twelve European countries levied an annual tax on net wealth in the 1990s, by now only three-Norway, Spain, and Switzerland-still levy such a tax, and only Switzerland raises a level of government revenue comparable to recent proposals such as those put forward by Senators Bernie Sanders and Elizabeth Warren in the US (Scheuer and Slemrod 2021). The Swiss example is therefore of particular interest for the policy debate elsewhere.

Our first contribution is to construct novel time series, based on data from cantonal archives, for top wealth concentration in each of the 26 Swiss cantons since 1969. In particular, we calculate the top 10%, 5%, 1%, 0.1% and 0.01% wealth shares using data on the number of taxpayers and their total wealth in various brackets and by estimating a local Pareto distribution at the top. We find that the overall increase in wealth concentration at the national level masks striking differences across cantons, in terms of both levels and trends in within-cantonal inequality. Whereas some cantons (such as Zurich) have seen a reduction in their top 1% wealth share over the last 50 years, others (such as Schwyz) have seen theirs almost double. Since the cantons have freedom in designing their wealth tax schedules, this raises the question to what degree these diverging trends are driven by policy heterogeneity, and in particular by differences in wealth tax rates in the top bracket.

We therefore complement our information on cantonal wealth distributions with the cor-responding panel data on top marginal wealth taxes, going back to 1964. Cantons have frequently changed their top tax rates with an overall downward trend but significant variation. For instance, the highest rate in our data is 1.34% in Glarus in 1970, and the lowest is 0.13% in Nidwalden in 2014.

Combining these data sets, we then explore the link between the two. Our event study design allows us to estimate the dynamic effect of wealth tax reforms on the subsequent evolution of top wealth shares. Focusing on large tax reforms and controlling for income and bequest taxes, we find that cuts to the top marginal wealth tax rate in a given canton increase wealth concentration in that canton over the course of the following decade, and that tax increases reduce it. The effect is strongest at the very top of the distribution. For the top 1% and 0.1%, for instance, a reduction in the top marginal wealth tax rate by 0.1 percentage points increases their wealth share by 0.9 and 1.2 percentage points, respectively, five years after the reform (compared to an average wealth share of 34% for the top 1%, and 16% for the top 0.1%). This implies that the overall reduction in the progressivity of the wealth tax in the Swiss cantons over the last decades explains roughly a fifth (a quarter) of the increase in concentration among the top 1% (0.1%) over this time horizon.

While this is a sizeable portion, it is also clear that other factors must have played a more prominent role in shaping wealth inequality in Switzerland. This is not surprising because, despite the variation across cantons, the wealth tax is not very progressive in any of them, with moderate top rates especially compared to recent proposals in the US, and relatively low exemption amounts, which imply that a large swath of the population is subject to it.

Whereas we are interested in the relationship between progressive wealth taxation and wealth inequality, the growing literature on the behavioral response of declared wealth to wealth taxes has focused on the absolute effect (see Scheuer and Slemrod 2021 for an overview).1 Closest to our study is Brülhart et al. (2022), who also take advantage of variations in the wealth tax rate across Swiss cantons and over time using a similar event study design. Apart from the fact that we consider the distributional effects of wealth taxation rather than its effect on the total amount of reported wealth, our main contribution is that our data covers a much longer time period. Since 2003, the federal tax administration in Switzerland has published yearly wealth statistics for all cantons and Brülhart et al.'s (2022) panel analysis is based on this data. Instead, by collecting data from the cantonal archives directly, our time series go back to 1969. During the decades before 2003, there was more variation in tax rates across cantons, including notably some significant tax hikes, which became much rarer later on. Moreover, the overall level of tax rates was significantly higher in the 1970s than since the 2000s, and the degree of wealth concentration has changed sub-stantially since this earlier period. Thus, looking at a longer historical evolution is crucial to tackle our research question.

Our paper is organized as follows. Section 2 gives an overview of the Swiss wealth tax system. Section 3 describes our data construction, notably how we compute cantonal top wealth shares based on the archive data and using Pareto interpolations. Section 4 discusses our novel findings on inter-cantonal differences in wealth inequality since 1970. Finally, Section 5 presents the results from our cross-cantonal event study and Section 6 concludes.

Wealth taxation in Switzerland

The Swiss tax system is generally structured in three layers: the federal, cantonal and municipal level. There are 26 cantons and about 2,300 municipalities. The Swiss constitution gives the cantons considerable autonomy over taxation and public spending decisions. In 2018, total tax revenues at the federal level amounted to $70 billion, while all cantons and municipalities together raised another $77 billion in fiscal revenues (corresponding to 10% and 11% of GDP, respectively).

The wealth tax has a long tradition in Switzerland and in fact predates the modern income tax. The cantons have been taxing wealth since the early 18th century and this was their main source of revenue until World War I. Between 1915 and 1959, there was also a wealth tax at the federal level. Since then, there has been no federal wealth tax but all cantons must levy a comprehensive wealth tax, over which they have significant freedom in designing. 2 In the 1990s, twelve European countries levied an annual tax on net wealth. By now, only three-Norway, Spain, and Switzerland-still levy such a tax, with Switzerland raising more than three times as much revenue as a fraction of total revenues (3.9%) as any of the other countries (Scheuer and Slemrod 2021). In 2018, 9.6% of the total tax revenues of all Swiss cantons and municipalities was raised by the wealth tax ($7.5 billion).

Tax base

The base of the Swiss wealth tax is broad: in principle, all assets, including those held abroad, are taxable. Only common household assets and foreign real estate are exempt from taxation; moreover, pension wealth such as occupational pensions (the so-called second pillar of the Swiss retirement system, which complements the national social security system) and the balances held on some voluntary retirement savings accounts (the so-called third pillar of the retirement system) are exempt until the date of the payout. 3 The tax liability is based on net wealth, so taxpayers can deduct mortgages and other debt. All residents aged 18 and over are legally bound to submit an annual tax filing (children's wealth must be included in the parents' tax returns). Net wealth is self-reported, which constrains tax enforcement. However, there is a 35% withholding tax on the return to domestic financial assets, which can only be claimed back when those assets are declared in the wealth tax base.

Tax schedules

Each canton designs its wealth tax schedule. Eight cantons impose flat rates (above some exemption level) and the other 18 feature progressive schedules with multiple tax rate brackets. Each municipality then chooses a multiplier that is applied proportionally to the cantonal tax rate schedule. Hence, an individual's overall tax liability depends on both the canton and municipality of residence.

Exemption levels differ by canton but are relatively low (Scheuer 2020). For instance, in 2018, it ranged from about $55,000 in the canton of Jura to $250,000 in the canton of Schwyz (for married couples).

Tax rates have declined over time. In 16 of the 26 cantonal capitals, the annual top wealth tax rate was below 0.5% in 2018. Hence, the Swiss wealth tax is targeted at a large share of the population and is only moderately progressive. Indeed, it is not intended to redistribute the stock of wealth but to be payable out of the resulting income (Schweizerische Steuerkonferenz 2021). In the next section, we provide more detailed information on the historical evolution of the wealth tax burden across cantons.

Data and methodology

Top wealth tax rates

Our first data set includes the cantonal wealth tax rates for all 26 Swiss cantons. Due to the highly decentralized tax system in Switzerland, it offers substantial variation in wealth tax rates since cantons have frequently changed their tax schedules over the decades. Our data includes the average and the marginal wealth tax rates for each canton and year going back to 1955 (including the municipal multiplier for the cantonal capital or main municipality). Since our main interest is the concentration of wealth at the top, we focus on the top marginal tax rates. Moreover, because our data on cantonal top wealth shares only begin later, we use information on top marginal wealth tax rates since 1964 for the present analysis.

In 2018, the combined cantonal and municipal marginal wealth tax rates in the top bracket ranged between 0.1% (canton of Nidwalden) and 1.1% (canton of Geneva). In 1969, three cantons imposed top marginal wealth tax rates above one percent: 1.34% in the canton of Glarus, 1.11% in Graubünden and 1.0% in Basel-Landschaft. Figure 1 shows the average of the cantonal top marginal wealth tax rates over time. While, in 1969, the average of all cantonal top marginal wealth tax rates was 0.73%, it had decreased to 0.49% in 2018.

Overall, the 26 Swiss cantons can be divided in roughly three groups according to the trends in their wealth tax rates. In the first group, top tax rates have hardly changed over the last 50 years (see Figure 2 for some examples). Cantons in the second group have gradually lowered their wealth tax rates over time (see Figure 3). Finally, the third group consists of cantons that have changed tax rates more extensively, including notably some significant tax cuts over a short period of time in the recent past (see Figure 4).

Top wealth holdings

Our second data set collects information on reported wealth holdings from 1969 to 2018. Since 2003, the federal tax administration (Eidgenössische Steuerverwaltung, ESTV) has published yearly net wealth statistics for all cantons. For the prior years, we have collected data directly from the cantonal archives, and combined it with irregularly reported tabulations by the ESTV. All statistics report the number of taxpayers in different wealth brackets and the corresponding sum of total wealth. The brackets mostly range from a bracket for zero net wealth to one for a net wealth of more than $1 million. Not all cantonal time series are recorded identically; notably, the bracket thresholds differ across cantons and change over time. Most importantly, some cantonal sources exclude taxpayers with (nearly) no wealth (less than $1000). Before 1969, only 3 out of the 26 cantonal archives provided information about this lowest wealth bracket. Therefore, the available data for this period is insufficient to plausibly approximate the missing wealth at the bottom of the distribution. As a result, we confine our analysis to the years since 1969.

Some cantonal statistics report taxable wealth instead of net wealth. Taxable wealth is defined as net wealth minus the tax exemption amount. The taxable wealth statistics indicate a considerably more pronounced wealth concentration than those based on net wealth. There are two reasons: first, the exemption level matters relatively less for top wealth holders; and second, the statistics based on taxable wealth include a larger share of taxpayers with no wealth at all. Both effects increase the top wealth shares and thus lead to an overestimate of wealth inequality. To correct this break in the series, we rescale taxable wealth to match net wealth where both are available. We then apply the same scaling factor to the years where only taxable wealth is available. 4 The statistics published by the ESTV since 2003 are based on net wealth, so this issue only concerns the earlier years.

While the wealth statistics based on tax returns are the best available data to study the long-run evolution of wealth inequality in Switzerland, it is important to keep in mind some caveats. First, because net wealth is self-reported, and despite the withholding tax, misreporting and tax evasion cannot be excluded, especially for the wealthiest households. Indeed, Brülhart et al. (2022) argue that misreporting is the most important component to the overall behavioral response of reported wealth to changes in taxes. A second issue concerns the number of taxpayers. All adult residents have to submit a tax return each year. However, married and officially registered same-sex couples (since 2007) are jointly tax liable and show up as only one tax unit in the tax statistics. To make sure our time series are consistent across time and cantons, we calculate the total number of tax units using register data (adult population minus one half of the married adult population).

Pareto interpolation of top wealth shares

Our data sources report net wealth as well as the number of taxpayers in absolute brackets, defined between thresholds in Swiss Francs. Instead, our object of interest are the wealth shares of the various top quantiles of the wealth distribution. To estimate the wealth of a given percentile, we make use of the fact that wealth holdings at the top of the distribution approximately follow a Pareto distribution. The cumulative distribution function is given by

F(x) = 1 - k x α , k > 0, α > 1, x ≥ k,
where the parameters α and k need to be estimated. The probability density function takes the form

f (x) = αk α /x α+1 . Since f (z|z ≥ x) = f (z)/(1 -F(x))
, the average wealth w(x) of tax units with wealth larger than or equal to x is given by w

(x) = ∞ x z f (z|z ≥ x)dz = αx α ∞ x z -α dz = α α -1 x.
Hence, mean wealth above a given threshold x is given by the constant factor β ≡ α/(α -1) independent of the threshold x. Conversely, we can estimate the local Pareto parameter at any given threshold x from the equation

β = w(x) x .
Using these estimates, we then calculate the respective wealth shares of each of the top percentiles of interest.

4 Wealth inequality in Switzerland Dell et al. (2007) provided first estimates of the wealth distribution in Switzerland during the last century. For the years 1915 to 1957, they were based on the federal wealth tax, which (as described in Section 2) was levied irregularly as a war tax and eliminated after 1959. For 1940, Dell et al. (2007) extrapolated data from the canton of Thurgau. For some other years (1913, 1919, 1969, 1981, 1991 and 1997), they relied on wealth statistics published by the federal administration. Föllmi and Martínez (2017) updated these national top wealth share series using the wealth tax statistics published annually by the ESTV since 2003. Based on our data collected from the cantonal archives, we are able to paint both a more fine-grained and comprehensive picture of the evolution of wealth inequality in Switzerland over the past five decades. More importantly, rather than constructing only the national wealth distribution, we provide new evidence on inequality trends at the cantonal level. This variation across cantons will be crucial for analyzing the effect of wealth taxes on inequality.

The end result of our data construction is a panel data set including the wealth shares of various top percentiles (10%, 5%, 1%, 0.1% and 0.01%) of the adult population in each canton between 1969 and 2018. When some cantonal archives are missing information for some years, we linearly interpolate the top wealth shares between the years for which we have information for the corresponding canton.5

Historical evolution at the national level

Figure 5 shows the (wealth-weighted) average of the cantonal top wealth shares for the top 1%, 0.1% and 0.1-1%. 6 It indicates increasing wealth concentration in Switzerland since the mid 1970s: the average wealth share of the top 1% across cantons has risen from 30% to 42% most recently. The increase has been even more pronounced for the top 0.1%: their average wealth share has more than doubled over the course of the same time period from 11% to 23%. This suggests that the rising upward trend in wealth inequality is mainly rooted in the steep wealth growth at the very top of the wealth distribution. Indeed, the average share of the top 0.01% more than tripled from less than 4% to 12.5%, whereas the average top 10% share has increased more moderately from 65% to 75%.

It is important to keep in mind that pension accounts are tax exempt and therefore not included in the wealth statistics in our analysis. The mandatory wage deduction for savings in occupational retirement accounts, as well as the caps for voluntary savings in private retirement accounts, have constantly grown over the last fifty years. Since our calculations ignore these trends, we tend to over-estimate the increase in wealth inequality. Föllmi and Martínez (2017) provide approximate adjustments taking into account pension wealth for the national wealth distribution. While these corrections decrease the wealth share of the top 10%, the difference is much smaller for the top 1% and negligible for the top 0.1%. 7 Another caveat is that some cantons offer foreigners who live but do not work in Switzerland an exemption from regular taxation, subjecting them instead to a flat-rate tax based on their living expenses. Hence, the wealth held by these foreign nationals is not included in our data either. The rules for this alternative tax regime have been tightened recently, and it currently affects fewer than 5,000 individuals (Scheuer and Slemrod 2021). For Switzerland as a whole, Baselgia and Martínez (2022) estimate, based on data from the Swiss rich lists published by the business magazine BILANZ, that the top 0.01% wealth share is 16% instead of 12%. They caution, however, that the wealth reported by BILANZ may be systematically too high. Moreover, the bias from ignoring the wealth of foreigners subject to expenditurebased taxation varies by canton. For instance, most of those individuals live in the canton of Vaud, whereas several other cantons (such as Zurich, Schaffhausen, Basel-Landschaft and Basel-Stadt) have completely abolished the flat-rate tax regime.

Cantonal differences

Our data uncovers remarkable differences across the Swiss cantons in top wealth inequality and its evolution over the last 50 years. For illustrative purposes, we focus on the top 1%. Figure 6 presents the corresponding wealth shares for the six cantons Zürich, Nidwalden, Solothurn, Basel-Landschaft, Aargau and Schwyz. We pick these cantons because their time series are relatively complete and because they provide a good overview of the different inequality trends over time. It is immediately apparent that the top 1% wealth share in Nidwalden is of a different magnitude than in the other cantons. After falling over the course of the 1970s and then remaining constant until the end of the 1990s at the (already high) level of nearly 50%, it increased sharply to almost 70% most recently. This is the highest value across all cantons and years. Recall that Nidwalden is also the canton with the lowest top wealth tax rate for most of our study period.

Although starting out from a much lower level, the canton of Schwyz has experienced, in relative terms, an even more striking rise in wealth concentration. While its top 1% wealth share was 32% in 1969 and remained between 30 and 40% until the late 1990s, it has almost doubled since then and has now reached 60%.

By contrast, in the canton of Zürich, the top 1% share fell between 1969 and 1975 and has since then remained strikingly constant over time, always at a level slightly below 40% including most recently. We find similarly flat overall trends for the cantons of Bern, Graubünden, St. Gallen and Uri, although they have gone through larger fluctuations in between.

The most complete data series is available for the canton of Basel-Landschaft, with information for each year since 1969. Its top 1% wealth share has followed a pronounced U-shaped pattern over time: from 50% in 1969 to less than 30% in the 1990s and back to 44% in 2018. The canton of Solothurn has followed a similar trajectory over time.

Finally, the canton of Aargau has featured relatively low levels of concentration (around It is also useful to put these numbers in international comparison, such as relative to measures of wealth inequality in the United States. Compared to the U.S. top 1% wealth share of 35% in 2018 (World Inequality Database), 19 of the 26 Swiss cantons feature higher degrees of concentration, with Nidwalden (69%), Schwyz (60%), Basel-Stadt (57%), Obwalden (56%), Geneva (55%), and Zug (51%) coming out on top of the list. In view of the fact that the U.S. exhibits a higher degree of wealth concentration than many other advanced countries, we conclude that, from an international perspective, wealth inequality is relatively high in most Swiss cantons.

Wealth taxation and top wealth inequality

In this section, we combine our panel data on top wealth tax rates and top wealth shares to shed light on the relationship between the two: Do changes in the progressivity of the wealth tax affect the concentration of wealth down the road? For this purpose, we exploit the variation in the timing of wealth tax reforms across cantons.

Cross-canton event study model

We estimate an event study model of the following form (Schmidheiny and Siegloch 2020):

W i,t = K ∑ k=1 12 ∑ j=-4 β k j D k i,t-j + 12 ∑ j=-4 γ j X i,t-j + θ t + ω i + µ i t + ε i,t
where W i,t is the top wealth share in canton i and year t, D k i,t an event indicator for a wealth tax reform of type k in canton i and year t, X i,t a set of controls, θ t a time fixed effect, ω i a canton fixed effect, and µ i a canton-specific time trend. Since we aim at isolating the effect of wealth tax reforms on wealth concentration, the controls X i,t include both the top marginal income and the average estate net-of-tax rate in canton i and year t. 8 We distinguish K = 4 types of reform events to allow for potentially heterogeneous effects: tax cuts and hikes, as well as small and large tax changes, defined as smaller or larger than a cutoff of 0.05 percentage points in absolute value. The estimators β k j capture the dynamic effects of the events of interest-namely, small and large wealth tax cuts and hikes-on top wealth shares j years after the reform. We set β k -1 = 0 for all k to express the dynamic effects relative to the year prior to the reform. Moreover, we specify the effect window up to twelve years after the event and considering pre-trends up to four years prior to the event. This also corresponds to the leads and lags of the included controls.

Figure 7 displays the coefficients β k j for the top 1% wealth share as a dependent variable and for large reforms. The results suggest that cantonal tax cuts increase the cantonal top 1% wealth share up to 7 years after the reform, whereas tax hikes reduce it. The point estimates indicate an effect of a change in the top 1% wealth share between one and two percentage points (in either direction) but the statistical significance is only marginal (the effect of small reforms is insignificant). This needs to be compared to an average top 1% wealth share of roughly 34% in our estimation sample.

Figure 8 shows the corresponding results for the top 0.1% wealth share. In this case, the effects of wealth tax reforms display the same sign but are slightly bigger, notably when put in relation to the average top 0.1% wealth share of 16% across time and cantons. This suggests that the effect of wealth taxation on wealth inequality is concentrated at the very -10 % -8 % -6 % -4 % -2 % 0 % 2 % 4 % 6 % 8 % 10 % Effect on top 1% wealth share (pp change) top of the distribution. Indeed, our estimated coefficients for the top 0.01% (not shown) are of a similar absolute magniture (and thus twice as large relative to the baseline wealth share of that group of 8%), whereas the event study model with the top 10% wealth share as the dependent variable produces insignificant results.9 Of course, these results cannot necessarily be given a causal interpretation since tax policy decisions at the cantonal level could in principle have been anticipatory in nature or part of broader tax reforms. However, the insignificant pre-trends up to 4 years prior to a wealth tax reform reduce such potential endogeneity concerns. Tables 2 and3 in the Appendix contain the regression coefficients and standard errors (for the event study models with the top 1% and top 0.1% wealth share as the dependent variable, respectively), including γ j corresponding to the control variables X i,t-j , namely the top marginal income and the average estate net-of-tax rates.

Interpreting the magnitudes

To put these findings into perspective, it is useful to relate them to the magnitude of the typical wealth tax reforms in our data. Figure 9 shows the histogram of tax rate changes (in percentage points) in our sample period. Overall, there were 714 tax reforms (with more tax -10 % -8 % -6 % -4 % -2 % 0 % 2 % 4 % 6 % 8 % 10 % Effect on top 0.1% wealth share (pp change) Recall that Figures 7 and8 display the coefficients for reforms associated with a change in the tax rate of at least 0.05 percentage points, which effectively isolates approximately the largest 10% of all reforms. 10 The mean tax cut (hike) among this subset of reforms is a reduction (increase) in the top rate by 0.17 (0.1) percentage points. Taken together with the point estimates from our event study model, the implied magnitude of the effect on top wealth shares is therefore quite sizeable. For instance, it predicts that a 0.1 percentage points reduction in the top wealth tax rate would increase the top 1% wealth share by 0.9 percentage points five years after the reform, and the top 0.1% wealth share by 1.2 percentage points.

Indeed, the average top marginal wealth tax rate decreased from 0.73% in the mid-1970s to 0.49% in 2018. At the same time, the wealth share of the top 1% increased on average from 30% to 42%, and the top 0.1% wealth share more than doubled on average from 11% to 23%. Hence, this back-of-the-envelope calculation suggests that historical wealth tax cuts explain roughly 18% of the increase in wealth concentration among the top 1% over this time horizon, and 25% of the increase in wealth concentration among the top 0.1%. While this is a substantial portion, especially in view of the limited progressivity of the wealth tax in Switzerland, it means that other factors must have been more important in shaping the evolution of wealth inequality over the past few decades.

A potential concern with this interpretation is that reforms to the top marginal wealth tax rate may be correlated with simultaneous reforms to features of the tax schedule further down the distribution, notably the exemption amounts below which no wealth tax is due. For example, if cantonal policy makers aim for largely revenue-neutral reforms, they might lower the exemption amounts whenever they cut the top tax rate. In this case, our results might be driven by both components of the reforms rather than only by changes to the marginal wealth tax: If households further down the distribution who become subject to the wealth tax as a result of the reform (due to the lowered exemption amount) respond by accumulating less wealth, this would further (indirectly) increase concentration at the top. To address this, Figure 10 shows a scatter plot of the change to the top wealth tax (on the horizontal axis) and the change to the exemption amount (on the vertical axis) associated with each of the cantonal tax reforms in our sample. It reveals no systematic relationship between the two dimensions. In fact, exemption amounts were mostly increased over time (in real terms) and hardly ever reduced, despite considerable reductions in the top wealth tax rates. Thus, our results are likely due to changes in the progressivity of the wealth tax at the top rather than at the bottom of the distribution.

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Conclusion

In this paper, we first present novel evidence on wealth inequality in the 26 Swiss cantons since 1969. We find that the overall increase in wealth concentration at the national level masks striking differences across cantons, in terms of both levels and trends of withincantonal inequality. Our second contribution is to explore the effect of wealth taxation on wealth inequality exploiting policy heterogeneity across cantons over the last fifty years. Our event study design, based on large tax reforms and controlling for income and bequest taxes, shows that cuts to the top marginal wealth tax in a given canton increase wealth concentration in that canton over the course of the following decade, and that tax increases reduce it.

The effect is strongest at the very top of the distribution. For the top 0.1%, for instance, a reduction in the top marginal wealth tax rate by 0.1 percentage points increases their wealth share by 1.2 percentage points seven years after the reform. This implies that the overall reduction in the progressivity of the wealth tax over the last decades explains roughly a quarter of the increase in concentration among the top 0.1% over this time horizon. Note that, in 2018, this group only included roughly 5000 tax units. Thus, it is an impressively small number of the wealthiest households who benefited most from reduced wealth tax rates at the top.

Since our analysis is based on aggregate data at the cantonal level, it remains silent on the economic mechanisms underlying this effect. Uncovering the anatomy of the response of wealth inequality to wealth taxation-in terms of its mechanical, real savings, inter-cantonal mobility, misreporting, and asset pricing components, for example-would require more detailed micro data and is therefore beyond the scope of this paper. In their analysis of how declared wealth responds to changes in wealth taxes, Brülhart et al. (2022) make use of precisely such data for the cantons of Bern and Luzern. They find that about a quarter of the total response is due to mobility, a fifth due to house price capitalization, and the rest likely due to misreporting. Whether a similar decomposition holds for the response of top wealth concentration, rather than absolute wealth levels, is an interesting question for future research.

If a sizeable fraction of the effect arises from taxpayer mobility, then this also raises a conceptual question: At which level of geographical aggregation should we measure inequality? Suppose, for instance, that canton A lowers its top wealth tax rate, thereby inducing some wealthy households in another canton B to move to canton A. This would increase the top wealth share in canton A and reduce it in canton B without any change in the national top wealth share. This example illustrates that aggregating our estimates based on comparisons across cantons is not straightforward. Indeed, in the presence of a significant mobility response, our estimates would need to be scaled down correspondingly when used to predict the effect of a simultaneous reduction in wealth taxation in all cantons on wealth concentration at the national level.

Relatedly, it also raises the question whether we should care at all about inequality at the cantonal level, or rather only at the federal level. In fact, one may argue that there is no reason to stop at the level of an individual country, and that, ultimately, world-wide inequality is all that matters from a normative standpoint. Still, there is significant interest in withincountry inequality. One reason is that many political decisions, not least about tax policy and redistribution, are taken at the country level, and that there may be a feedback loop between within-country inequality and political decision making. By the same argument, since the Swiss cantons have a large degree of political autonomy, studying the evolution and determinants of wealth inequality at the sub-national level is equally important. Our paper takes a first step in this direction.

Our results also make clear that changes to wealth taxation are not the most important driver of the recent rise in wealth inequality in Switzerland. Indeed, the Swiss wealth tax was never intended to achieve a major redistribution of wealth, but rather to generate stable revenues for the cantons and municipalities. This is evident in the moderate tax rates, which even at the top are likely smaller than the rates of return to the wealth of the very rich, and the fact that a large portion of the population is subject to the wealth tax.

Other changes to the Swiss tax system over the last 50 years could have played a role. In particular, most cantons have abolished bequest taxes for direct descendants (Brülhart and Parchet 2014) and there is no bequest tax at the federal level. At the same time, bequests account for a considerable part of the wealth of the superrich in Switzerland. For instance, Baselgia and Martínez (2022) show that, most recently, 75% of the 300 richest individuals in Switzerland have been heirs. This is extremely high compared to the Forbes 400, the corresponding list for the United States. In 2018, 69% of the wealthiest Americans were self-made founders of their own businesses (Scheuer and Slemrod 2020). Quantifying the degree to which the erosion of the cantonal bequest taxes has contributed to long-run wealth inequality in Switzerland would be an interesting topic for further investigation.

Figure 1 :

 1 Figure 1: Top marginal wealth tax rates 1964-2018, average of cantonal rates

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Top marginal wealth tax rates in Fribourg, Basel-Stadt, Vaud, Neuchâtel and Geneva

Figure 5 :

 5 Figure 5: Top 1%, 0.1% and 1-0.1% wealth shares and top marginal wealth tax rates 1969-2018 (wealth-weighted averages across cantons)

Figure 6 :

 6 Figure 6: Top 1% wealth shares of the cantons Zurich, Nidwalden, Solothurn, Basel-Landschaft, Aargau and Schwyz 1969-2018

  = 294, # small hikes = 175, # large cuts = 34, # large hikes = 6, N = 1020, cantons: 26, years: 1976 -2015. Model includes canton and time FE, canton-specific trends, lags and leads of log top net-of-inheritance-tax and top net-of-income-tax rates. 90% confidence intervals, SEs clustered at canton level. Dependent variable: top 1% wealth share; average in estimation sample: 33.9%.

Figure 7 :

 7 Figure 7: Cross-canton event study, top 1% wealth share

  cuts = 294, # small hikes = 175, # large cuts = 34, # large hikes = 6, N = 1020, cantons: 26, years: 1976 -2015. Model includes canton and time FE, canton-specific trends, lags and leads of log top net-of-inheritance-tax and top net-of-income-tax rates. 90% confidence intervals, SEs clustered at canton level. Dependent variable: top 0.1% wealth share; average in estimation sample: 15.9%.

Figure 8 :

 8 Figure 8: Cross-canton event study, top 0.1% wealth share

  in top marginal wealth tax rate, in pp

Figure 9 :

 9 Figure 9: Cantonal wealth tax changes 1964-2018

Figure 10 :

 10 Figure 10: Cantonal changes to the top wealth tax and the exemption amount 1969-2018

  Note: Model includes canton and time FE as well as canton-specific trends, SEs clustered at canton level. Dependent variable: top 1% wealth share.

	Year relative to wealth tax change	Small cut	Small hike	Large cut	Large hike	log top net-of-estate-tax rates	log top net-of-income-tax rates
	-4	0.0022976 (0.0020295) (0.0038144) (0.0050576) (0.0083876) (0.0891568) (0.103685) -0.0031858 -0.001189 0.00271 0.1889398 0.1096511
	-3	0.0027329 (0.0021233) (0.003756) -0.0016244	-0.0010082 (0.0039427) (0.009724) 0.0052672	0.053156 (0.0634622) (0.0637724) 0.0253212
	-2	0.0020839 (0.0020935) (0.0031495) (0.0037763) (0.0093627) (0.0617059) (0.0832014) -0.0017135 0.0044265 0.0047558 0.064948 -0.1142434
	0	0.0010645 (0.0023383) (0.0036396) (0.0053442) (0.0104915) (0.0506904) (0.0533242) -0.00262 0.0116087 -0.0027763 0.0154428 -0.0171117
	1	0.0020149 (0.0024915) (0.003564) -0.0041061	0.0158724 (0.0071165) (0.0074566) (0.0814433) (0.0630007) -0.0080446 -0.0166484 -0.019266
	2	0.003395 (0.002812)	-0.0045465 (0.0034218) (0.0070065) (0.006087) 0.0144571 -0.0177016	0.0263441 (0.0640897) (0.0375767) -0.0479522
	3	0.0045326 (0.0026801) (0.0040457) (0.0066075) (0.0052283) (0.062072) -0.0035743 0.0107281 -0.0116437 0.009336	-0.0199642 (0.0304443)
	4	0.0039425 (0.0026098) (0.0030684) (0.0065124) (0.0045683) (0.0549504) (0.0241641) -0.0026286 0.014968 -0.0054516 0.0390477 -0.0032574
	5	0.0022235 (0.0025769) (0.0029955) (0.0085046) (0.0050247) (0.0475782) (0.038835) -0.002887 0.0156372 -0.0061575 -0.0134754 -0.0038553
		0.0025558 (0.0024987) (0.0030709) (0.0114197) (0.0067028) (0.0720875) (0.0229728) -0.0023962 0.0142353 -0.0025123 0.0632166 0.0167414
	7	0.0016832 (0.002224)	-0.0029766 (0.0032538) (0.0132075) (0.007296) 0.01908 0.0040213	0.2333809 (0.0792453) (0.0350742) -0.0141396
	8	0.001469 (0.0025894) (0.0029395) (0.0099914) (0.0070839) (0.0723615) (0.0242661) -0.001831 0.0109595 0.0013128 0.1514018 0.0302078
	9	0.0008653 (0.0026548) (0.0030323) (0.0093716) (0.0067707) (0.0853943) (0.0151072) -0.0004304 0.0070992 0.0013345 0.1061328 -0.0044352
	10	0.0001582 (0.0023999) (0.0034623) (0.0070931) (0.0051138) (0.065598) -0.0023406 -0.005262 -0.0005021 0.0892894	0.0475557 (0.0409621)
	11	-0.0007746 (0.0025855) (0.0036668) (0.0065559) (0.007485) -0.0024195 -0.0047227 0.0037486	0.0295935 (0.0682379) (0.0164015) -0.0232601
	12	-0.0009869 (0.0023105) (0.0033076) (0.0064668) (0.0079657) (0.0645107) (0.0550869) -0.0006997 0.0015392 0.0021267 0.1090395 0.0408898
	Constant	0.992283 1.189494
	N Groups N Observations 1002 26	R 2 within R 2 between 0.4699 0.7727 R 2 overall 0.2653

Table 2 :

 2 Cross-canton event study model for the top 1% wealth share Note: Model includes canton and time FE as well as canton-specific trends, SEs clustered at canton level. Dependent variable: top 0.1% wealth share.

	Year relative to wealth tax change	Small cut	Small hike	Large cut	Large hike	log top net-of-estate-tax rates	log top net-of-income-tax rates
	-4	0.001881 (0.0019647) (0.0039929) (0.0050858) (0.0086046) (0.0904797) (0.093234) -0.0022456 -0.0022969 0.00483 0.1429525 0.160743
	-3	0.002222 (0.0021907) (0.0038691) (0.0037324) (0.0099277) (0.061513) 0.0001204 0.0009048 0.0058096 0.0501811	0.0216165 (0.0529901)
	-2	0.0019531 (0.0021851) (0.0032527) (0.0039572) (0.0082074) (0.0600775) (0.0652993) -0.0003361 0.0050203 0.0026091 0.0982664 -0.0893154
	0	0.0013246 (0.0021654) (0.0043348) (0.0054106) (0.013202) -0.0006556 0.0137974 -0.0143376	0.0491401 (0.0347417) (0.0512172) -0.0074544
	1	0.002462 (0.0022067) (0.0041318) (0.0075811) (0.0107536) (0.0839255) (0.070348) -0.0021194 0.0192765 -0.0194086 0.0207446 0.0161065
	2	0.0031735 (0.0024697) (0.0036029) (0.0075215) (0.0096587) (0.0686057) (0.0361945) -0.0031897 0.0174902 -0.031182 0.0487051 -0.0473359
	3	0.0043674 (0.002497)	-0.0024796 (0.004441)	0.0129753 (0.0068345) (0.0099439) (0.0505065) (0.0361399) -0.0232158 0.0188918 -0.0671123
	4	0.0046294 (0.0023955) (0.0030902) (0.0073241) (0.0080322) (0.0483108) (0.0243231) -0.0016716 0.017926 -0.0155879 0.013602 0.021173
	5	0.0026645 (0.002486)	-0.0021085 (0.0026502) (0.0090204) (0.008296) 0.0204186 -0.0160419	-0.0655603 (0.047687)	0.0046424 (0.0490397)
	6	0.0032979 (0.0027402) (0.0028592) (0.0124937) (0.0068713) (0.0621205) (0.0260203) -0.0010331 0.0211501 -0.0120412 0.0035395 0.0053703
	7	0.0023027 (0.0024359) (0.0034237) (0.0142148) (0.0074954) (0.0717843) (0.0371136) -0.0010761 0.026211 -0.004494 0.1527247 0.0035248
	8	0.0021861 (0.0027077) (0.0031605) (0.0098094) (0.0081067) (0.0696155) (0.0301853) -0.0005634 0.0147989 -0.0075425 0.1000737 0.0200387
	9	0.00224 (0.002756)	0.001157 (0.0032542) (0.0085551) (0.0081934) (0.0795746) (0.0154614) 0.009812 -0.0044303 0.143336 0.0011738
	10	0.001999 (0.0024002) (0.0038338) (0.0054889) (0.0061475) (0.066082) -0.0003187 -0.0015225 -0.0060394 0.1389383	0.0664139 (0.0502499)
	11	0.0013585 (0.0025442) (0.0041113) (0.005228) -0.0004508 -0.0011343	-0.0020627 (0.0071496) (0.0641367) (0.0170583) 0.0405962 -0.0213298
	12	0.0011842 (0.002033)	0.0007849 (0.0032469) (0.0057216) (0.0082058) (0.0727604) (0.0570599) 0.0045363 -0.0023967 0.0369275 0.0382706
	Constant	-2.223493 (1.322405)
	N Groups N Observations 1002 26	R 2 within R 2 between 0.4699 0.7659 R 2 overall 0.3158

Table 3 :

 3 Cross-canton event study model for the top 0.1% wealth share

Recent studies include[START_REF] Seim | Behavioral responses to wealth taxes: Evidence from Sweden[END_REF],[START_REF] Zoutman | Yearly 2003 -2018 Schweizerische Steuerkonferenz "Vermögenssteuer natürlicher Personen[END_REF], Londoño-Velez and Avila-Mahecha (2019),Durán- Cabré et al. (2019),[START_REF] Agrawal | Wealth tax mobility and tax coordination[END_REF],[START_REF] Jakobsen | Wealth taxation and wealth accumulation: Theory and evidence from Denmark[END_REF] and[START_REF] Brülhart | Behavioral responses to wealth taxes: evidence from Switzerland[END_REF].

Bequests are taxed at the cantonal level, typically in the form of inheritance taxes. In the 1990s and early 2000s, however, most cantons abolished inheritance taxes for direct descendants[START_REF] Brülhart | Alleged tax competition: the mysterious death of bequest taxes in Switzerland[END_REF]. So far, all attempts to introduce an inheritance tax at the federal level have been unsuccessful.

While foreign real estate is not subject to the Swiss wealth tax, it is included when determining the relevant tax bracket and thus the individual tax rate.

Alternatively, we could add the exemption levels to taxable wealth. However, this would introduce measurement error because exemption amounts depend on marital status and the number of dependents of each taxpayer, and we do not have information on the composition of these characteristics by wealth bracket.

This only affects some years prior to 2003. Since then, the ESTV provides yearly wealth distribution statistics for each canton. See Table1in the Appendix for a detailed list of sources from which we have collected cantonal wealth statistics.

 6 The population-weighted averages across cantons feature similar levels and trends.

Accounting for pension wealth in our cantonal wealth statistics is beyond the scope of this paper, but an important task for future research.

Apart from wealth taxes, both bequest and income taxes at the cantonal level are potential drivers of wealth accumulation, and the corresponding rates have changed multiple times during our sample period. For instance, the average top marginal income tax rate across cantons rates rose from 20.5% in 1970 to 25.7% in 1982 and then decreased to 21.2% in 2018. In turn, the average inheritance tax rate across cantons first fluctuated between 1.5% and 2.5% and, since 2000, has decreased to 0.2% in 2018.

We obtain similar results when using the local Pareto parameter of the wealth distribution instead of top wealth shares as the dependent variable.

When using smaller cutoffs (such as 0.03 or 0.01 percentage points) to define large reforms, the point estimates decrease in magnitude, and the standard errors decrease at the same time due to the increasing sample size. As a result, our estimates remain statistically significant under these alternative definitions of large reforms.

Appendix

Canton(s) Year(s) Source All cantons 1


Table of Contents

Note:

The information in this document is provided as is and no guarantee or warranty is given that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability.

Introduction

In the current virtual reality field, each of current display (audition, visual and touch) are often indepently studied. But a more interresting approach consist to blend them to create a multi-modal virtual reality. The fact is putting more than one modality together does not result in the sum of the modalities. The experience can be enriched by such cooperation, but user can also feel destructive interaction between the modalities. This creates new challenges for the virtual reality creator. For the most cases, the modalities are not to be handled in the same way, thus creating diffuclties for integration. In the following, we focus on such challenges.

In order to be able prototype multimodal integration schemes and models, the LSC developed a generic framework called I-Touch [POC04a][POC04b] that serves also the rigid bodies prototyping demonstrator of the deliverable D7.5 and D7.6. This part will describe the components of the I-Touch framework. The fact the framework is multimodal is of great importance, in order to be able to evaluate the importance of each feedback. We tried to give each of the modalities the same 'importance', from an integration point a view, precisely in order to be able, after that, to modulate the user experience.

The I-Touch framework

Design of the I-Touch framework

The I-Touch framework is designed to be modular from its core to its interfaces. Although it is still in the development process, it already allows plug-ins (static linking in program) of different behaviors models and different collision detection algorithms. The framework architecture is given in the Figure 1. As it can be noticed, haptic interfacing, among other rendering capabilities, takes an important place. This particular focus meets the Touch-HapSys objectives in better understand haptics and its relation with the other modalities. However, this modality has the same 'rights' and 'duties' as other modalities. Precisely, the haptic modality can be as easily put in / removed from the simulation as any other.

From a pure technical point-of-view, the framework is divided in three main modules; each of them is further subdivided in as many sub-modules as needed. The I-Touch framework is completely object-oriented. This allows easy part replacement and improvements. It is implemented in a pure standard C++, and apart from the driver libraries, does not use platform dependent code. It can be easily ported to Linux or MacOsX, however it is designed to be best suited to MS Windows OS. Of course, this object-orientation makes also easy the sub classing of some part of the framework, in order to create completely new applications. The fact we can devise new application "on demand" is of considerable importance. Namely, we are also envisaging its use for dedicated psychophysical investigations.

The core system

The core system is responsible for handling the operating system (the platform specific code is inserted here), the configuration, and the basic functionalities of a physically-based simulation. It provides a basic scene graph for managing the various objects that composes the virtual scene. This core system can accept many simulation algorithms along with different input methods. Classical mathematical methods [START_REF] Dorst | Geometric algebra: A computational framework for geometrical aplications[END_REF] [2] and structures are also provided, for the easy prototyping/evaluation of new/existing algorithms. The core system also provides a very flexible configuration in XML, which parametrizes any aspect of the simulation, thus allowing the creation of test-beds rather easily.

In addition to configuration tools, a file format for holding together "geometrical" object properties has been devised. This format is open and flexible; moreover, additional data can be included and be ignored if not necessary to the simulation, even if it is an unknown data. An importer and exporter have been written for 3DSMax, along with C++ and C# libraries for loading efficiently theses files. This exporter can, for now, export geometry, normal, binormal and texture information, however, it does not save the relationship between texture information and texture map (this link has to be recreated in configuration files, however, the result is of course exactly the same). This data format, exporter, importer and C++ libraries have been conjointly developed by A. Pocheville and M. Brunel. Details concerning the implementation can be found in [START_REF] Brunel | Rendu multimodal haute fidélité, rapport de fin d'étude[END_REF].

The input and output system

While the input system needs to be flexible and needs to manage many different inputs, the output system should ensure high fidelity rendering along with adequate refresh rates according to the addressed modality/output. Of course, the output system is tightly related to multi-modal integration. One of the most important facts here is that the output system should be very flexible and allow almost any information as its input, in order to be able to prototype new integration schemes. However, there are limits to flexibility, in the end, each modality still requires special handling.

The simulation system

The simulation is composed of the simulation manager, and a set of simulation virtual objects.

The simulation system uses the core system for standard interaction with the computer and the user, and input/output system for multi-modal interaction. Collision detection algorithms are part of the simulation system. The simulation system, like any other part of the framework, can be replaced or modified rather easily. The simulation system can be decomposed in two subsystems: the collision detection subsystem and the behavior model (the physic step). The fact is that collision detection step can be also benchmarked to certain extend: for example, a penalty or a constraint based physic resolution has known inputs, we can then select detection collision algorithms that provides such information. The following schema illustrates this interaction: Impusle based method(s)

Multimodal integration

One of the most challenging issues of I-Touch is multimodal integration. Visual, audition and haptic senses have different refresh rates: from as low as 30Hz for visual interaction, up to as high as 10kHz for the 3D sound one. Integrating each of these modalities is not a trivial task. Others tentatives tried through parallelization of the computation on different computers.

Here, we decided to push the limits on focusing in using one computer, but this unveils some problems as exposed later.

Simulation engine flexibility

The fact that the simulation engine is completely flexible and modular allows the integration of different behaviors models with the same multimodal rendering. This is done through abstraction of the output methods. For example, haptic rendering can be easily derived to create new rendering. These rendering can then be easily switched. Moreover, since the simulation engine use the simulation objects as placeholders, alsmost any information can be provided to the output. For example, simulation objects are holding the whole normal map used in visual bump mapping and haptic bump mapping. This information can then be accessed at will.

However, the simulation engine has to provide some specific information to the output routines. For example, for the real-time 3D sound rendering, contact information (and changes in contact through time) are required. The immediate benefit of the ability to switch between renderings is that we can benchmark how well does a simulation engine behaves with multimodal rendering. For example, bounce models have difficulties in rendering contact information with sound, while they provide excellent rendering of bounce sounds. It can be observed that

Collision detection integration

The I-Touch framework has a certain amount of requirement from the collision detection module. It should be noted that due to the architecture of the framework, inter penetration never occurs. For I-Touch, objects can only be touching. Objects are considered to be touching when they are less than a certain predefined distance away from each other. We will refer to this distance as the tolerance of the collision system. When a contact occurs, the contact area should be represented by a group of point pairs, each of which is on the surface of one object. The points on each object should represent the contact region of this object. No point pairs should be duplicated in the list since each pair is later used to calculate the reaction force.

The collision detection module should be robust to handle polygon soups where duplicated vertices can be found and thus neighborhood information between triangles might be erroneous. The collision results should be calculated fast enough to leave the required time for the dynamical model to calculate the reaction forces at interactive rates.

Our proposed system provides the required information. It is composed of three layers:

The first is an acceleration layer that determines the triangle pairs having a high probability of colliding, thus needing further processing. The second layer performs our topology determination test on each of the pairs obtained from the first layer. This test obtains for each pair, the list of point pairs that defines the contact between these triangles. The last layer is where all the contact point pairs from the second layer are gathered, while eliminating duplicates. In the following, we will explain each of the three layers of our system.

Layer1

The first layer is an acceleration phase, based on object oriented bounding box hierarchies. Each hierarchy is constructed top down, starting by the first box at the object level, and proceeding by dividing the triangles into two groups, and constructing a bounding box for each group. The leaves of the trees of the hierarchies contain each, one triangle. This construction produces 2n boxes for each object, where n is the number of triangles for that object. In order to account for the tolerance, we have to enlarge each box of the tree by half the value of the tolerance in the direction of the three axes of the box. This action is crucial since we consider a pair of triangles to be in contact when they are within the tolerance from each other. An example is shown is the following figure.

When traversing the tree, if the current nodes are leaves we add the pair of triangles that they contain to the list of triangle pairs that we will process in layer 2. If the nodes are not both leaves, then we process their child boxes together. The output of this layer is the list of triangle pair that need further processing.

Layer2

The input of the second layer is the output of the first layer. We need to determine the contact type for each triangle pair in the list, which can be a point, edge or surface contact and for each case, find the points that delimit the contact zone for each triangle. We define the "margin" as the variation in the distance within which we consider the two vertices as parallel to a plan. For example, having v11, v12 and v13 as vertices of the first triangle and p2 as a plan of the second triangle with the tolerance fixed to 10 and the margin fixed to 2, then if

d(v11, p2) = 11 d(v12, p2) = 5 d(v13, p2) = 6
We consider that we are potentially in a case of edge-plan contact because we consider that v12 and v13 are both lying in p2 within the tolerance and the margin.

Our algorithm starts by calculating the distances from each vertex to the plan of the second. The distance is not a signed number, and do not depend on the direction of the normal. This is done to account for the case where the vertices of the triangles are given in the reverse order.

This first part of the algorithm has the following pseudo code t1 and t2 are the first and second triangle respectively v11, v12, v13,, v21, v22, v23 are the vertices of t1 and t2 respectively p1 and p2 are the plans of t1 and t2 respectively

For each vertex of t1

Calculate the distance for this vertex to p2

End for

For each vertex of t2

Calculate the distance for this vertex to p1

End for

If at lease one vertex of t1 is in t2 or at least one vertex of t2 is in t1

Determine the type of potential contact of t1 using the calculated distances Determine the type of potential contact of t2 using the calculated distances Call PlanContact(HighestContactOrder(t1,t2))

Else

Call EdgeEdge()

End if

In order to determine the type of potential plan contact, we start by identifying the vertex with the smallest distance, and then verify if the other two vertices of his triangles are within the margin. If both are, then we are in a potential plan-plan contact. If only one is, it is a potential edge-plan contact for this triangle. If none is, then it is a potential vertex-plan contact. All vertices within the margin and the closest one are called active.

PlanContact()

A plan contact can be a vertex-plan, edge-plan, or a plan-plan. In order to determine all the point pairs that define the contact, this function starts by checking if the active vertices of the two triangles are inside all the plans defined by the edges and the normal of the second triangle. If a vertex is, then we are sure it is a contact vertex since we already verified that it is within the tolerance, and we can add it and his projection on the second triangle are added to the result vertex pair list.

Then we check the edges of the two triangles, having both end points active, against each other. If their closest points are less then the tolerance away from each other then we add the closest points of the two triangles to the result pair list. Some conditions have to be verified in order to assure that no duplicate pair are added to the list, but will be omitted for simplicity.

For each active vertex v of t1

If v inside the triangular prism of t2

EdgeEdge()

If no vertex of the two triangles is in the plan of the other, then we must verify that we don't have edge-edge contacts before concluding with a no contact situation.

This test checks all pairs of edges again each other, and adds the closest point on two edges to the result list if the distance between their corresponding edges is less than the tolerance. We also have to check certain conditions to assure that no duplicates are added to the list.

The following is an edge-edge case

End for

An important remark is that the case of an edge penetrating the face of the second triangle, but having a distance to all three edges of the second triangle highest then the tolerance, is not detected.

This should not be a problem for I-Touch since this should not be allowed to happen.

When the end point of this edge reaches the plan of the triangle, the engine of I-Touch, prevents penetration, and thus the above mentioned situation never occurs.

Layer3

The last layer of the system deletes redundant pairs from the result list. This can be done by the brute force algorithm of order O(n 2 ), where n is the number of result pairs. Knowing that the number of results pair is usually not very high, such complexity can be acceptable in most cases. However, we designed an optimized duplicate pair remover, which checks only the results of the neighboring triangles for duplicate pair. Such a strategy is possible, since duplicates can only exist among adjacent triangles.

Such an optimized version should only be used when the models are well constructed, and neighborhood information for the triangles is correct. If it is not the case, it is better to use the brute force method.

The result of this layer is a list of point pairs, defining the contact between the two objects, and having no duplicates.

Sound integration

3D positional audio enhances the immersion of the operator in the simulation. We have two methods for rendering 3D sound: real-time rendering, and semi-real-time rendering. The realtime rendering uses information directly provided by the simulation, such as changes in the friction map to produce sound. It also uses object properties such as resonance frequencies to computes contact sounds [START_REF] Van Den Doel | FOLEYAUTOMATIC : Physicallybased Sound Effects for Interactive Simulation and Animation[END_REF] . While this is the correct method for producing friction and bounce sound, it suffers from several drawbacks. First of all, it is very computational-time consuming, and, in a system composed by only one processor, it can become the bottleneck of the simulation (and take the place of the collision detection!). Maybe relocating the sound computations could solve this problem. The other fact is that the sounds generated are, for now, less "realistic" than the ones produced by the second approach.

The following is an example of analysis of modal parameters of a material.

The semi-real-time sound rendering approach uses off-line recorded sounds of different materials in contact. These different sounds are stored in a database according to some material properties. They are used by the simulation as they are and the only amplitude and/or frequency modulation (pitch, volume...) are processed. This method can be seen as same as vertex transform followed by a texture pass in visual rendering.

The following figure stress out the difference between the two methods:

Visual integration

Relatively to the sound and the haptic rendering, the visual one is the easiest. We can use the same geometry as the one used for physics calculations, or a higher level, smoother one for better rendering. Objects are linked to rendering information, such as geometry, material and alpha information, and pixel and vertex shaders. This allows almost any rendering of the objects, from standard Gouraud-shaded plastic look, to advanced Phong-shaded semireflecting materials with bump mapping. Dynamic lighting is also supported. The visual rendering is completely configuration controlled, so there is a great flexibility in the rendering process. The fact that the visual rendering is well understood and has numerous techniques is very interesting here, because we can take inspiration from these techniques in others renderings.

Haptic integration

Our approach differs from the previous ones. Indeed we are conceptually considering that the haptic devices (interfaces) interact with the simulation and not the reverse way i.e. the haptic device does not drive the simulation. The point here is that haptic device can be removed at will. Even more, more than one haptic device should not pose any problem to this type of simulation, since haptic rendering is totally decorelated from simulation loop. Obviously, haptic devices can induce a change in the course of the simulation but they cannot compromise its integrity. To be clearer, the simulation does not take as granted what is needed from the input device and, in extreme cases, these particular inputs are ignored. In fact, this enhances considerably the stability of the interaction. For example, when the operator actions are toward violating a given non-penetration constraints, they are not considered integrally (as is the case for classical computer haptics API's). This is better for engine stbility, and allows for many new algorithms to be used. However, haptic feedback has to be consired from another point of view. The basic principle behind this is exmplained in the following figure: The fact that haptic integration is not considered as a special rendering allows new synergies between rendering to be investigated. One example of this is the recently implemented haptic

Simulation Loop

Haptic Device 1

Haptic Device 2 ASK ASK ANSWER ANSWER bump. As for visual bump mapping, we can simulate rough haptic surfaces through haptic bumps. We tried two different approaches: height based forces, and normal based forces. The basic principle is the same: the force computed by the simulation engine is slightly modulated by a term, which depends either on the height or the normal. In our actual implementation, haptic bump does only work with one contact point, but we are working on extension to multiple points.

The following figure explain the principle behind the haptic bump:

As far as "bump sensation" is concerned, the normal based force give superior results. The bump map used for haptic bump is exactly the same as the one used in the visual bump, thus the two modalities match perfectly and the rendering is coherent.

Thermal integration

While there have been advances in thermal integration, there remains difficulty in some areas:  First of all, we have to be able to reproduce any material from its thermal properties.

For now, we can only approximate the law of the thermal interaction between a human finger and a material through a Peltier pump, but many progress are still to be made, in order to be able to deliver almost exact replication of thermal sensations in the real worl.  Secondly, while the integration of the thermal sense is not very difficult in the simulation (mainly from its poor refresh rate, the human has not a high refresh rate thermal sense), it poses great difficulty from a practical point of view, where we should combine thermal with kinaesthetic feedback. An efficient device is yet to be devised.

Putting them together

Many synchronizing algorithms do exist, but for now, we did not experimented them, because they are time consuming when added to the simulation engine. Computation of different contacts, and then contact forces, takes most of the CPU time. As theses computation are the main bottleneck of the simulation, the rendering is sufficently fast. Moreover, the fact of multithreading does not harm this process, since the refresh rates are too high for an human to notice little differences.

The data is given to render in this order (immediately one after the other): haptics, sound and vision. In the future it would interesting to have scalable algorithms that would be totally disconnected from the physical simulation, we will then need synchronization algorithms. This would prevent that the new bottleneck (let's say, sound rendering) to harm the other displays. However, it will also mean that one of the modalities will have degraded performance, we will need to investigate how far we go in degradation in one of the modalities without ruining interaction.

Evaluation tools

The testing of research projects is made easy with I-Touch, however such a testing requires to analyze data from the simulation. In I-Touch, every simulation variable can be "tagged" from recording; this allows after-run simulation analysis through tools. For example, FPS data, or time taken to compute a frame (much more speaking than FPS in regard to performance) evolution can be viewed easily. The 'tagging' is done by providing the pointer to the variable. Then, each frame (or at a given time), the variable is recorded. At the end of the simulation, a file that can be viewed in a special viewer is written. The actual viewer is written in .Net and is provided with the I-Touch framework.

Also, the debugging facilities and text functions in I-Touch make it easy to dump data. However, we want to go further, and new real-time tools are in development. Such tools will render in real time evolution of variables, in numerous manners (time graphs, bars, standard text, etc.). Specific analysis tools to simulation should also be included, such as automatic reporting of number of contacts, physics calculation time, time spent in rendering or in other tasks -this step is almost done, but we have to link it with the visual rendering. This will create a complete and easy evaluation tool, in order to accelerate further the development of test cases. This step is very important in the relationship that this project can have with psycho-physical and others experiments.

Conclusion and future work

We have shown that we have made a prototype of software framework that allows the creation of many test cases. They all have in common experiments with operation, in a multimodal way. The haptic rendering is not considered with special concerns, thus facilitating the evaluation of haptic impact in a multimodal context. Integration of all these modailities in one framework in not an easy task. We have chosen the modularity path, which allows testing of and imaging new scenarios. However, there is an attempt to provide each of the modularity a coherent information, in order to allow a real symbiose between the different senses.

1 . 1 . 1 1 . 2 1 . 3 1 . 4 6 1. 5 1 . 7 1 . 8 1 . 9

 11112131465171819 INTRODUCTION...................................................................................................... THE I-TOUCH FRAMEWORK..................................................................................... Design of the I-Touch framework ..................................................................................................... The core system .................................................................................................................................. The input and output system ............................................................................................................. The simulation system ........................................................................................................................ 2 MULTIMODAL INTEGRATION.................................................................................Simulation engine flexibility .............................................................................................................. 1.6 Collision detection integration .......................................................................................................... 1.6.1 Layer1 ........................................................................................................................................... 1.6.2 Layer2 ........................................................................................................................................... 1.6.3 PlanContact() ................................................................................................................................ 1.6.4 EdgeEdge() ................................................................................................................................... 1.6.5 Layer3 ......................................................................................................................................... Sound integration ............................................................................................................................. Visual integration ............................................................................................................................. Haptic integration ............................................................................................................................ 1.10 Thermal integration ....................................................................................................................... 1.11 Putting them together .................................................................................................................... 1.12 Evaluation tools ............................................................................................................................. CONCLUSION AND FUTURE WORK.......................................................................16 REFERENCES...........................................................................................................

Figure 1 :

 1 Figure 1: I-Touch framework architecture.

Figure 2 :

 2 Figure 2: I-Touch interaction between collision detection and response

For every edge e1 of t

  For every edge e2 of t2 If d(e1, e2)<tolerance Add the closest points on e1 and e2 to the result list End if End for

Figure :

 : Figure : Principle of force handling.

Figure :

 : Figure : Example of haptic bump (combined with visual bump). The surface is in fact a plane.

We have succeeded in creating sample applications that use the flexibility of the I-Touch framework. However, integration with real psycho-physical tests is yet to be done. Further work is oriented toward refining I-Touch through its multimodal component to serve also as a psychophysics evaluation tool. Progressively our aim is to evolve it to a complete piece of software that can serve haptic research. Future improvements will include more intimate collaboration of the different modalities, and new ways of defining what a modility is from a software point of view.