Thibault Darcillon
email: thibault.darcillon@univ-paris1.fr.

What Causes Labor-Market Volatility? The Role of Finance and Welfare State Institutions

Keywords: Labor-market volatility, financial development, social security expenditure, compensation hypothesis Volatilité du marché du travail, développement financier, dépenses de sécurité sociale, hypothèse de compensation JEL Classification: F41, I3, P16

Using fixed effects panel data models on a sample of 15 OECD countries over the period 1970-2007, this article explores the linkages between labor-market volatility, financial development and welfare state institutions. We analyze the interacted impact of financial development on the one hand and welfare state institutions (i.e., overall social spending) on the other hand on volatility of hours worked and volatility of wages. Our results indicate that financial development is associated with higher volatility on labor-markets. Estimates of the marginal effects show that overall social spending increasingly reduces labormarket volatility with the degree of financial development, and more specifically for low-skilled workers through compensation mechanisms. Finally, we control for potential reversed causality by running IV-GMM estimations suggesting that increasing financial development has not threatened the governments' ability to play an active role in cushioning fluctuations on labor markets.

Résumé

VOLATILITÉ SUR LE MARCHÉ DU TRAVAIL, DÉVELOPPEMENT FINANCIER ET DÉPENSES DE SÉCURITÉ SOCIALE

A l'aide de modèles à effets fixes sur données de panel sur un échantillon comprenant 15 pays de l'OCDE de 1970 à 2007, cet article s'intéresse aux interactions entre le niveau de développement financier et les institutions de protection sociale (mesurée à l'aide du poids des dépenses de sécurité sociale dans le PIB) sur la volatilité sur le marché du travail. Plus particulièrement, nous cherchons à expliquer les déterminants de la volatilité du nombre d'heures travaillées et

Introduction

The recent financial crisis and the subsequent economic recession have caused an unprecedented increase in the unemployment rate in most OECD countries: the unemployment rate rose from an average of 5.96% in 2008 to 8.13% in 2009. In the wake of the subprime crash, a majority of political leaders -on both the left and on the right -sharply damned financial capitalism. 1 Many political leaders and scholars have stressed the consequences of increasing instability on financial markets on the whole economy at the international and domestic levels.

The main issue that we address in this paper deals with the impact of financial markets at the international and domestic levels on the labor market instability. According to the standard efficient market hypothesis, more efficient financial markets should contribute to reducing significantly risks in the economy, including the risks for workers. We argue that the instability of financial markets can impact the firms' financial strategy which can affect the fluctuations on labor markets. Data on labor market volatility reveal that the volatility of wages growth has increased since several decades, and this more specifically for low-skilled workers. Our main hypothesis is that financial markets are likely to affect the workers' welfare, and more specifically for low-skilled workers. This arguments is directly derived from the [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF] work who argues that trade development intensified competition from low-wage countries and may be harmful particularly for low-skilled workers. Trade openness has increased indeed the elasticity of labor demand for this category of workers. We assume in this paper that financial development -at the international level as well as at the domestic level -may have amplified this effect. This paper explores the role of increasing financial development and its interactions with welfare state institutions on labor-market volatility in 15 OECD countries. First, recent literature in economics has focused on the role of financial development on macroeconomic volatility (i.e., volatility in output or consumption growth). Few papers have analyzed the relationship between financial development and labormarket volatility. We argue that financial development might also affect the fluctuations and the distribution of risks on labor markets. We distinguish two different channels: in a first channel, financial openness through an increase in foreign direct investment (FDI) has contributed to the increase in labor-market volatility; in a second channel, financial development has led financial participants to use more sophisticated instruments and to take excessive risks which has created instability on labor markets.

In his seminal work, [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF] argues that globalization is associated with an increased dispersion of earnings among low-skilled workers and higher volatility in wages and hours worked. Trade openness has therefore asymmetric effects between owners of capital and high-skilled workers on the one hand and semiskilled and unskilled workers on the other hand. The first category is composed on very mobile actors whereas globalization has increased the elasticity of the demand for the services provided by workers from the second category of workers. One of the other consequences associated with increased trade openness is that wages for unskilled workers have been eroding and then labor-market instability and insecurity have been increasing. These two phenomena have resulted in an inward shift in the demand curve and a flattening of the demand curve for low-skilled labor. Consequently, according to [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF], these changes would be associated with increased volatility in wages and hours worked. OECD (2007) provides empirical evidence suggesting that labor demand has become more responsive to shocks. Empirical findings also indicate that the elasticity of labor demand is considerably higher for low-skilled workers than for high-skilled workers. One of the potential explanation of this would be that easier offshoring of production may have significantly raised the volatility of employment and wages. We suppose in this paper that financial development at the international and domestic levels has increased short-term fluctuations in labor demand: first, financial development at the domestic and international levels reduces the bargaining position of labor [START_REF] Jayadev | Capital account openness and the labour share of income[END_REF]. Accordingly, it was very frequently shown that financial development and more specifically financial openness (through foreign direct investment) has contributed to making employers more sensible to labor costs (International Institute for Labour Studies, 2008). We argue that this effect should be larger for low-skilled workers than for high-skilled workers according to the 'capital-skill complementarity' hypothesis [START_REF] Griliches | Capital-skill complementarity[END_REF]. In reaction to this more insecure context, Rodrik argues that gov-ernments will increase welfare state spending to compensate the negative impact of globalization on the welfare of low-skilled workers. Beyond trade globalization, we extend Rodrik's argument to financial development including an international dimension with financial openness and a domestic dimension with financial deregulation.

We use to measure labor market instability two different proxies for labor market volatility. Using the EU KLEMS database that provides data on averages wages and hours worked by skill groups from 1970 to 2007, we compute two different measures of labor-market volatility as the rolling standard deviation for growth rates of average wages and of hours worked over a five-year-window at the aggregate level to obtain a measure of volatility. To capture financial development, we use different measures of financial integration at the international and domestic levels. We use social security transfers as a percentage of GDP as a measure of welfare state institutions. As robustness checks we use an alternative measure of welfare state institutions: government employment.

Using panel data on 15 OECD countries from 1970 to 2005, we first run PCSE/OLS regressions. Then, to consider the argument that globalization would reduce the governments' ability to compensate labor-market volatility but also to address the endogeneity issue of welfare state institutions, we run IV-GMM estimations. Our results indicate strong evidence of a positive effect of financial integration at the international and domestic levels on volatility of wages and of hours worked. We find that financial development increases the volatility of hours particularly for low-skilled workers and the volatility of wages particularly for high-skilled workers. In addition, our main argument of complementarity between financial development and welfare state institutions is empirically verified: social security transfers contribute to the reduction in labor market volatility associated with financial development.

The paper is organized as follows. Section 2 presents our main argument on the linkages between financial development, risk diversification and labor market instability. In Section 3, we present our data. we conduct in Section 4 a regression analysis. Section 5 concludes.

Labor-market volatility, financial development and welfare state institutions

In The General Theory of Employment, Interest, and Money, [START_REF] Keynes | The General Theory of Employment Interest and Money[END_REF] underlined the dual role of financial markets. On the one hand, stock markets facilitate productive investments by transferring risk from firm owners to capital holders. In this line, [START_REF] Keynes | The General Theory of Employment Interest and Money[END_REF] has stressed that capital markets by increasing financial liquidity play a positive role: higher liquidity on the financial markets that allow to convert a security into currency immediately and freely has the main advantage of increasing the access to abundant capital inflows. On the other hand, the expansion of security markets contributes to making more complex investment decisions. Henceforth, lenders are able to pledge funds for a whole lifespan of capital goods but also have the possibility of reviewing their commitments at any time by selling their assets. In this sense, "with the separation between ownership and management which prevails to-day and with the development of organised investment markets, a new factor of great importance has entered in, which sometimes facilitates investment but sometimes adds greatly to the instability of the system" (Keynes,emphasis added).

Alternatively, according to the efficiency market hypothesis, the risks should be diversified to be reduced. The diffusion of financial innovations -in the first place the operations of 'securitization' -that took place in the 1980s in the wake of financial liberalization process played a considerable role in the process of the 'Great Moderation' that refers to a reduction in the volatility of business cycle fluctuations starting in the mid-1980s. In other words, the advanced economies would have been less volatile and safer: the Great Moderation thus denotes a new era of low risks and high returns. In that sense, securitization played a central role in the Great Moderation: the aim of securization is to promote liquidity on financial markets by improving the possibilities of risk diversification [START_REF] Orléan | De l'euphorie à la panique : penser la crise financière, Cepremap[END_REF]. As we review in the related literature, financial development may have contributed to a significant reduction in output volatility in the 1980s-1990s, but at the same time labor-market volatility has sharply increased, and this more particularly for low-skilled workers (See Table 1). In this vein, a report from the International Institute for Labour Studies (2008) finds mixed evidence on a positive effect of financial market development on employment growth, partly because of increased turbulence on the labor market.

In addition, the recent financial crisis has brought back the thesis of 'financial fragility' developed by [START_REF] Minsky | OECD Workers in the Global Economy: Increasingly Vulnerable?[END_REF]. The argument of the thesis of financial fragility is that, in times of prosperity, the state of confidence increases as the participants on financial markets underestimates the risk of bubble bursting. Financial innovationssuch as derivatives -have contributed to the efficiency of financial markets by reducing transaction costs, increasing financial liquidity and by enhancing the diffusion of the information. At the same time, these innovations can lead the agents to underestimate or misperceive the risks on the market, and thus to take excessive risks [START_REF] Capelle-Blancard | Les marchés dérivés sont-ils dangereux?[END_REF]. In line with Minsky's analysis, Aglietta and Rebérioux (2005) share the main idea that financial markets are inherently unstable: moreover, beyond inefficiencies in asset pricing method, shareholder value maximization strategy is incongruent with chronic financial instability. Following this viewpoint, we argue that instability on financial markets may directly impact fluctuations on labor markets. 2The aim of this paper is to analyze the relationship between financial development and labor-market volatility. The impact of increasing financial development on macroeconomic volatility (and more specifically on output volatility) has remained largely theoretically and empirically explored. [START_REF] Bekaert | Growth volatility and financial liberalization[END_REF] and [START_REF] Eozenou | Financial Integration and Macroeconomic Volatility: Does Financial Development Matter?[END_REF] suggest that countries with high levels of financial integration experience a greater reduction in consumption growth volatility. More specifically, [START_REF] Eozenou | Financial Integration and Macroeconomic Volatility: Does Financial Development Matter?[END_REF] results indicate that consumption growth volatility is found to increase with the degree of financial integration in countries with low level of financial development and to decrease in countries with high level of financial development. More recently, [START_REF] Popov | Financial Liberalization, Growth, and Risk[END_REF] finds that the effect of financial liberalization is positive on growth but at the same time financial liberalization is correlated with higher tail risk (i.e., an increase in the left skewness). [START_REF] Stiglitz | Capital Market Liberalization, Economic Growth, and Instability[END_REF] argues that capital market liberalization has not enhanced growth but rather produced higher instability on growth. A considerable amount of research effort has dealt with the implications of increasing financial development on growth or unemployment dynamics. In this article, we want to test empirically the relationship between financial development and the volatility of the employment and of wages of workers with different skills.

Financial development and labor-market volatility

A majority of theoretical and empirical papers interesting in evaluating the impact of financial globalization on labor-market finds contrasting effects. It is frequently argued by financial economists that financial development increases the risk-diversification possibilities and thus would be associated with reduced volatility. This thesis is theoretically supported by [START_REF] Erauskin | Financial openness, volatility, and the size of productive government[END_REF] who proposes a model for a stochastically growing small open economy in continuous time where government spending is productive. In the same vein, [START_REF] Bertola | Finance and Welfare States in Globalising Markets[END_REF] finds evidence of substitution, rather than complementarity, between the trade openness and financial integration in regard with government size. This finding suggests that economic globalization would decrease demand for redistribution in countries where financial markets are well developed and thus are a good substitute for government policies. In other words, better financial markets reduce the positive effect of trade openness on the size of government. In this case, financial markets allowing a better riskdiversification are not responsible for generating new risks but, on the contrary, are considered as efficient instruments to cover risks related to international competition. By contrast, [START_REF] Thesmar | Financial Market Developments and The Rise in Firm Level Uncertainty[END_REF] demonstrate that financial development, by broadening of the pool of external investors (both at the domestic and international levels), improves risk sharing but also encourages firms to adopt more profitable and riskier strategies including for non-listed firms. Financial development has a general equilibrium effect on wages and price and also affects the strategies of the non-listed firms: this will result in an overall increase in the uncertainty of sales, employment and profits in all firms. The model also predicts that the effects of financial development are stronger when the competition on the product market is tougher. In this sense, we argue that financial development should lead to higher volatility in employment and in wages.

Another argument relates to the thesis of financial fragility: it has been often argued that financial development has the potential to increase the likelihood of major economic crises [START_REF] Saillard | The Role of Complementarity and the Finance Liberalization in the Financial Crisis[END_REF]. In advanced as well as in emerging countries, the frequency of banking and/or financial crises and the degree of labor-market volatility have simultaneously increased since the early of the 1980s (International Institute for Labour Studies, 2008). By generating negative externalities in the whole economy, specifically in periods of financial crisis, financial development thereby may have an increasing-effect on labor-market volatility. [START_REF] Pagano | Finance and employment[END_REF] find strong support for the thesis of 'dark side' of financial development claiming that a developed financial market amplifies the repercussions of financial crises on employment. Results seem to show that, during banking crises negative shocks disproportionately impact employment growth in financially dependent sectors in countries with more highly developed financial systems. Consequently, financial development is considered as a risk per se as underlined by some scholars (e.g. [START_REF] Stiglitz | Capital Market Liberalization, Economic Growth, and Instability[END_REF]: as noted by [START_REF] Pagano | Finance and employment[END_REF], "the recent events have administered a sharp reminder that the financial markets may themselves be a source of risk, rather than a mechanism to price and share it; and financial sophistication may itself be a source of instability, if it encourages excess risk-taking".

Hypothesis 1 Financial development at the international and domestic levels is associated with an increase in labor-market volatility.

The improved risk-diversification possibilities that offers financial development are, however, not equal among individuals. Buch and Pierdzoch (2013) find strong evidence of a positive impact of higher degree of financial integration on the volatility of hours worked, and this particularly for low-skilled workers. They propose a dynamic stochastic general equilibrium model of a small open economy: financial globalization is captured by portfolio adjustment costs, and the costs of adjusting hours worked are different between high-and low-skilled workers. The two authors show that financial globalization gives rise to an increase in the volatility of hours worked: financial globalization is associated with a rise in output volatility that may trigger more additional hires and fires of low-skilled workers than of high-skilled workers because the adjustment costs per unit of high-skilled workers are lower for high-skilled workers than for low-skilled workers. Alternatively, their results also indicate that financial globalization leads to more wage volatility with similar effects across skill groups.

In a recent paper, [START_REF] Pagano | Finance and employment[END_REF] find that financial development at the domestic level increases the volatility of employment, i.e. when the cross-industry dispersion in stock returns is high (reflecting profit shocks). In other words, financial development favors labor reallocation across industries from the 'weaker' to the 'stronger' industries. As a result, if financial development is prone to increase the expected labor income for workers, it also raises its variance. Workers with high degree of risk aversion will consider that the increase in their labor income risk outweighs the benefits from an increase in expected income and will probably be more resistant against more volatility in financial and labor markets. Because of higher risks of becoming unemployed or jobless, lower income groups are more exposed to risk than higher income groups and have thus a higher degree of risk aversion.

In addition, it is frequently claimed in the literature on earnings volatility that the access to financial assets allow households to smooth consumption: following this literature, because low-skilled and low-paid workers have poorer access to financial markets it becomes more difficult for these households to smooth consumption if they are hit by negative income shocks (OECD, 2011). However, recent analyses (Rajan, 2010) have shown that political authorities in many countries (in the first place the United States) have made easier the access to financial markets for the poorer households. In that vein, following the thesis of financial fragility, the potential negative externalities on the whole economy would be larger in times of financial crisis in highly financially developed countries. And this because financial deregulation has made easier the access to credit for low-income households and has thus increased the proportion of households owning shares and financial assets. Furthermore, as witnessed the recent financial crisis of 2007-2008, the bursting of assets bubbles may have direct impact on the economy's real sectors: the negative shock on the economy at large -and more specifically the reduction in credit availability (due to deterioration of the financial industry's intermediary function) -are more likely to affect lower wage-earners [START_REF] Boyer | Les financiers détruiront-ils le capitalisme ?[END_REF].

To sum up, we have shown that labor-market risks are more concentrated on low-skilled workers whereas financial development should theoretically mitigate labor-market volatility for high-skilled workers because of improved possibilities of risk-sharing and of reduced consumption volatility. As a consequence, the risks on labor market being unevenly distributed across skill groups, we assume that financial development should have strong distributional effects and should more impact lowskilled workers' welfare. We propose to analyze the political-economic implications of the consequences of increased labor-market volatility.

Hypothesis 2 Financial development at the international and domestic levels is associated with an increase in labor-market volatility, more particularly for low-skilled workers.

Interactions between financial development and welfare state institutions

We investigate in this section the relationship between welfare state institutions and labor-market volatility. In a political-economy perspective, we argue that governments will implement specific welfare programs to compensate the negative impact of financial development on workers' welfare. Our argument is based on the 'compensation hypothesis' that stresses that increasing globalization generates new risks and thus will result in an increase in welfare state spending.

Political scientists (e.g. [START_REF] Cameron | The Expansion of the Public Economy[END_REF]Katzenstein, 1984[START_REF] Garrett | Partisan Politics in the Global Economy[END_REF] and economists (e.g., [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF]1998) analyzed the role of government faced with an increasing instability in a more globalized world. According to the proponents of the 'compensation hypothesis', governments will continue to provide social compensation to workers who are affected by the new risks engendered by globalization. 3 The so-called 'compensation hypothesis' is based on two different mechanisms:

1. First, in reaction to this new insecure context, voters will increasingly support social protection and social insurance (political demand);

2. Then, in response to this political demand, governments will supply more generous welfare state programs (political supply);

In other words, more generous welfare programs will compensate globalization 'losers' for the risks associated with increased international competition and volatility. This literature assumes that some categories of workers would be exposed to higher risks. Two different approaches are prevalent in identifying globalization winners and globalization losers [START_REF] Gourevitch | Political Power and Corporate Control: The New Global Politics of Corporate Governance[END_REF][START_REF] Walter | Globalization and the Welfare State: Testing the Microfoundations of the Compensation Hypothesis[END_REF]. First, the sectoral Ricardo-Viner model that stresses the sectors' degree of exposure to global competition: workers in exposed sectors are more likely to suffer from globalization than workers in sheltered sectors. Second, the Stolper-Samuelson model that stresses factor endowments as central determinant: capital holders (including high-skilled workers) have comparative advantages in developed economies. In this paper, we adopt an unidimensional approach by focusing only on skill levels since data provides only information on the 'skill dimension'.

Empirically, [START_REF] Cameron | The Expansion of the Public Economy[END_REF], Katzenstein (1984), [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF]1998), [START_REF] Garrett | Partisan Politics in the Global Economy[END_REF] and [START_REF] Swank | Global Capital, Political Institutions, and Policy Change in Developed Welfare States[END_REF] find all a positive relationship between trade openness and the size of government, and more specifically the size of the welfare state. In their seminal works, [START_REF] Cameron | The Expansion of the Public Economy[END_REF] and Katzenstein (1984) find that globalization leads to welfare state expansion particularly in small countries. In this line, [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF] finds a positive relationship between lagged exposure to trade as a share of GDP from 1980 to 1989 and government expenditures as a percentage of GDP from 1990 to 1992. Using Swiss survey data, [START_REF] Walter | Globalization and the Welfare State: Testing the Microfoundations of the Compensation Hypothesis[END_REF] proposes to test the microfoundations of the 'compensation hypothesis' and finds out that low-skilled workers (i) are more likely to feel insecure, (ii) are more in favor of higher state involvement and then (iii) are more likely to support left-wing parties. 4 Conservely, Meinhard and Potrakfe (2012), in a recent paper, challenge the traditional view of the 'compensation hypothesis': they find no empirical evidence indicating that economic globalization increases the demand for social insurance.

More interestingly for our purpose, [START_REF] Fatás | Government size and automatic stabilizers: international and intranational evidence[END_REF] find that government expenditure, through the automatic stabilizers, has a reducing-effect on output growth volatility. The automatic stabilizers can absorb the shocks associated with higher volatility and instability better [START_REF] Stiglitz | Capital Market Liberalization, Economic Growth, and Instability[END_REF]. In the same vein, OECD (2011) points out the importance of taxes (such as unemployment benefits and social assistance) in reducing the impact of earnings fluctuations. Hypothesis 3 Welfare state institutions reduce labor-market volatility when the influence of financial markets at the international and domestic levels becomes larger.

By contrast, as underlined above, [START_REF] Erauskin | Financial openness, volatility, and the size of productive government[END_REF] argues that the role of government in the mitigation of volatility is lower in an economy without restrictions on capital account because financial development is associated with a better riskdiversification. [START_REF] Carmignani | Macroeconomic risk and the (de)stabilising role of government size[END_REF] find that higher volatility is associated with larger government size but with a destabilizing effect. Overall, all these results suggest that financial development cannot be considered as a risk factor but seems rather to discipline government size. From this perspective, and in line with [START_REF] Bertola | Finance and Welfare States in Globalising Markets[END_REF] argument, welfare state institutions would have no impact on labor-market volatility when the influence of the financial markets becomes larger.

Data and trends

Measuring labor-market volatility

More volatile labor markets should reflect more dynamic and innovative markets that encourage the reallocation of labor [START_REF] Pagano | Finance and employment[END_REF]. In that sense, labor market volatility reflects a large dispersion of productivity/profitability across industries. On the other hand, an increase in the volatility on labor markets is also closely related to an increase in the dispersion of the level of earnings across workers. Following the methodology used by [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF]1998), we measure the degree of instability on labor markets by computing the volatility of hours worked (as a proxy for instability in employment) and the volatility of wages (reflecting the instability in earnings).

To obtain a measure of labor-market volatility, we use the EU KLEMS Database from the OECD that provides data on total hours workers by persons engaged and total labor compensation by skill groups (HS for high-skilled workers and LS for lowskilled workers) from 1970 to 2005. 5 We use two measures of labor-market volatility (See Appendix A for the construction of these two measures).

Figures 1 and2 plot labor-market volatility, distinguishing labor-market volatility by skill groups. Table 1 also provides summary statistics on the level of labor-market volatility by decade and by skill group.

[INSERT TABLE 1 ABOUT HERE]

Figure 1 as well as Table 1 reveal some differences in the level of volatility of hours worked by skill groups. One can see important cross-national variations in levels and in trends: Continental European countries (such as France, Germany, Netherlands or Belgium) have experienced low and stable volatility of hours worked. However, some Anglo-Saxon countries (such as Australia, Ireland, the United Kingdom and to a lesser extent the United States) have higher variations in volatility. Interestingly, Northern European countries (such as Finland and Sweden) as well as Spain are closer to the Anglo-Saxon countries than to the first category of countries. It is striking that in some countries (i.e., Italy, the Netherlands, the United Kingdom or the United States) low-skilled workers have been particularly confronting to increasing volatility than other categories of workers. This trend is compatible with our argument that financial development has mainly affected low-skilled workers.

As regards the volatility of wages, most of the countries in our sample have experienced greater volatility of wages than the volatility of hours worked (Table 1). Volatility of wages reached a peak during the 1980s and is decreasing since then but with relatively high levels of volatility. About differences across skill groups, one finds that the volatility of wages for low-skilled workers is slightly higher than high-skilled workers (Figure 2). There is much less cross-national variations in the volatility of wages: as indicated in Table 1, most of the OECD countries (excluding the United States that presents very stable levels of volatility) experienced an increase in the volatility of wages during the 1980s and show similar patterns.

[INSERT FIGURES 1 AND 2 ABOUT HERE]

Data on financial development and welfare state institutions

Our main explanatory variables are proxies for the degree of international integration of financial markets and the degree of development of the domestic financial system. First, to capture the international dimension of financial development, we use a common measure of international financial integration provided by [START_REF] Lane | The external wealth of nations mark II: Revised and extended estimates of foreign assets and liabilities, 1970-2004[END_REF]: total assets and liabilities over GDP. 6 External assets and liabilities are including (i) portfolio investment (subdivided into equity securities and debt securities), (ii) foreign direct investment that refers to equity participations above 10%, (iii) financial derivatives and (iv) reserve assets. Second, to capture the domestic dimension of financial development, we use one measure of domestic financial development: the stock market capitalization ratio to GDP. The stock market capitalization ratio gives a measure of stock market activity, i.e. to what extent the stock market can efficiently allocate capital to investment projects. It is assumed that a more developed financial market also increases the investors' opportunities for risk diversification.

Consequently, this indicator reflects the capacity of stock markets to provide external financing.

To account for welfare state institutions, we use the overall social security expenditure as a percentage of GDP, taken from [START_REF] Armingeon | Comparative Political Data Set I 1960-2010[END_REF], which is a very standard indicator for a country's effort in social protection. This variable is including social assistance grants and welfare benefits paid by general government (benefits for sickness, old-age, family allowances, . . . ). Database on public social expenditure and revenue data is freely downloaded from OECD National Accounts Statistics.

Control variables

Our choice of control variables is in line with the existing literature. To consider [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF]1998) argument that trade openness increases wage elasticity of firms, and thus increases labor-market volatility, particularly for low-skilled workers, we compute the sum of imports and exports as a percentage of GDP (Trade openness). Because low-skilled workers have higher risks of becoming unemployed or jobless and with more persistent periods of unemployment, we control for unemployment rates as a determinant of labor-market volatility. We use the OECD time series on standardized unemployment rates (Unemployment rate). Data on trade openness and unemployment rate is provided by [START_REF] Armingeon | Comparative Political Data Set I 1960-2010[END_REF]. Finally, we control for macroeconomic volatility that is expected to impact labor markets and then to increase their volatility. We calculate the volatility of output growth by computing the rolling standard deviation for growth rates of real GDP over a five-year-window (Volatility of output growth).

Our sample is composed of 15 OECD countries (Australia, Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, Spain, Sweden, United Kingdom and United States). Table A.1 in Appendix displays descriptive statistics for the variables used in the regressions.

The aim of this paper is to analyze the impact of financial development on labormarket volatility. More specifically, we want to test the argument that the impact of financial development on the labor-market volatility is conditional to specific welfare state institutions. We introduced one interactive term between financial development and welfare state institutions. We will estimate the following relationship:

σ(y it ) = β 0 + β 1 • FIN it + β 2 • SPEND it + β 3 • FIN it • SPEND it + β k • Σ k X k,it + µ t + µ i + ǫ it (1)
where σ(y it ) denotes rolling standard deviation for growth rates of hours worked (or of wages) over a five-year-window, FIN it the variable capturing financial development, SPEND it the variable capturing the social security transfers as a percentage of GDP, µ t country fixed effects, µ i time fixed effects, k X k,it a set of control variables, and ǫ it an error term.

In this article, we want to explore whether governments will react to fluctuations on labor-market by increasing social security expenditure. Our principal argument is based on the idea that the more financially developed a country is, the more governments will play an active role in compensating negative externalities associated with increasing financial development. In other words, the effect of welfare state institutions on labor-market volatility is conditional on specific levels of financial development, as expressed as follows:

∂σ(y it ) ∂SPEND it = β 2 + β 3 • FIN it (2)
In Equation ( 1), the coefficient β 1 can be interpreted as the effect of the social security transfers on labor-market volatility but when FIN it equals zero. 7Accordingly, the t values obtained from an interactive model indicate the effect of an independent variable on the dependent variable but depending on particular levels of another independent variable: hence, it is not surprising that insignificant variables can produce significant marginal effects [START_REF] Friedrich | In Defense of Multiplicative Terms in Multiple Regression Equations[END_REF].

Thanks to a specific STATA procedure, we report marginal effects of the social security transfers at different sample values of financial variables (minimum, mean minus one standard deviation, mean, mean plus one standard deviation, maximum).

We begin by using a simple fixed effect panel data model with a panel corrected standard errors (PCSE) estimator provided by [START_REF] Beck | What To Do (and Not To Do) with Times-Series-Cross-Section Data in Comparative Politics?[END_REF]. Before running regressions, we run unit root tests to check whether our variables are stationary or not.We find that our variables in levels are stationary, sometimes with a drift. Then, we are able to use the ordinary least squares (OLS) with [START_REF] Beck | What To Do (and Not To Do) with Times-Series-Cross-Section Data in Comparative Politics?[END_REF] panel corrected standard errors (PCSE) as homoskedasticity and indepedence of errors are not verified. Finally, we find that the data does not have first-order autocorrelation. Then, we address the issue of reverse causality by running IV-GMM estimations: welfare state institutions can be endogenous to economic conditions: as noted by [START_REF] Rodrik | Why Do More Open Economies Have Bigger Governments?[END_REF] and [START_REF] Fatás | Government size and automatic stabilizers: international and intranational evidence[END_REF], governments should be capable of stabilizing labor-market fluctuations if such economies are more volatile. In order to take into account the effect of labor-market volatility on social security expenditure, and hence to address the endogeneity problem of the welfare state variable, we run instrumental-variable (IV) regressions. Here are considered as endogenous variables the welfare state variable as well as the interactive term between welfare state variables and our financial variables.

But finding good instruments is always a difficult task. 8 On the basis on theoretical considerations and empirical analysis, we choose several instruments. First, we want to test the argument that economic and financial globalization may have gradually contributed to the reduction in the governments' ability to finance generous welfare programs in accordance with the 'efficiency hypothesis'. Accordingly, globalization has created new constraints on public budgets what would be associated with a 'welfare retrenchment' or what [START_REF] Pierson | Coping with Permanent Austerity: Welfare State Restructuring in Affluent Democracies[END_REF] called the 'era of austerity'. So, we include as first instruments several variables related to the budgetary situation in the economy all provided by Armingeon et al. (2010): adjusted annual deficit, total receipts of general government/GDP, long-term interest rate on governments bonds, and gross government debt/GDP. Then, we seek to investigate the specific determinants of social security expenditure: we include in our regressions cyclical economic variables (i.e., real GDP growth, and current account balance/GDP), structural economic variable (i.e., total population), institutional variables (i.e., net unemployment replacement rates, trade union density rate and financial openness index from Chinn and Ito, 2008) and finally political economy variables (government ideological orientation, Rae's index of electoral fractionalization of the party-system, political voting system, political regime, and Herfindahl index of political competition).9

Finally, we submit those instruments to a series of tests to confirm its validity: for the first test, we never reject the null hypothesis that our overidentifying restrictions are correct. For the other tests, we always reject the null hypothesis of non-significance not suggesting the existence of any problems with our instrumentation.

Estimation results

The regression results when using volatility of hours worked growth as dependent variable are reported in Tables 2. Each financial variable is regressed against our two dependent variables for high-skilled [HS] workers and for low-skilled [LS] workers. Focusing on the impact of financial variables, our results indicate a positive and significant relationship between our dependent variable and our two financial variables: financial development at the international and domestic levels is associated with higher volatility of hours workers. We find as expected that the impact is larger and statistically significant for low-skilled workers. Very surprisingly, we do not find a differentiated effect of financial development on the dependent variable across skill groups. Financial development is also associated with an increase in the volatil-ity of hours workers for high-skilled workers but the coefficient is not statistically significant.

Then, focusing on the welfare state variable, we find a strong reducing effect of government size on the volatility of hours worked. More interestingly, to account for the complementarity or substituability effects between welfare state institutions and financial development, we compute marginal effects for specific levels of each financial variable. One can see that the marginal effect is decreasing with the level of financial variables. These results seem to support the 'compensation hypothesis': social security expenditure significantly contributes to the reduction in the volatility of hours worked as financial development is increasing at the domestic and international levels. Using the PCSE/OLS estimator, we find that the compensation effect is larger for high-skilled workers. By contrast, as expected, we do find, when using the IV-GMM estimator, a larger and statistically significant effect for low-skilled workers who are more affected by the turbulences on the labor market. Controlling for endogeneity in the IV-GMM regressions allows us to deal with the simultaneity bias between our dependent variable and the welfare state variable. In addition, one should remark that the size of the coefficients associated with our several financial variables and with the marginal effects are slightly higher than the coefficients found in the first regressions using PCSE/OLS method in models (5) to ( 8). This finding might suggest that, contrary to the 'efficiency thesis', financial integration which may be associated with higher budgetary discipline will not lead to a reduction in the governments' ability of cushioning labor-market fluctuations.10 Among control variables, we find that unemployment and volatility of output growth are both correlated with an increase in the standard deviation of hours worked growth. However, results associated with trade openness are not statistically robust.11

[INSERT TABLE 2 ABOUT HERE]

We run similar regressions using the volatility of wages growth as dependent variable (Table 3). As expected, we find that the volatility of wages and our different measures of financial development are all positively correlated. But, surprisingly, we find that high-skilled workers are sightly more affected by an increase in volatility of wages than low-skilled workers except in models (3) and (4) for the international dimension of financial development. This result contrasts with Buch and Pierdzioch (2013) who find that the effect of financial globalization on the volatility of wages is roughly the same for high-and low-skilled workers.

How should we explain that high-skilled workers are more affected by the financial development at the domestic dimension (models ( 7) and ( 8))? The more frequent use of performance-related pay (PRP) and other incentive devices by top managers and high-skilled workers may result in the increase in the earnings volatility (OECD, 2011). Moreover, [START_REF] Thesmar | Financial Market Developments and The Rise in Firm Level Uncertainty[END_REF] show that the development of financial markets increases the uncertainty on employment and profits. These specific earnings schemes, such as PRP and other incentive devices, are indexed on the firms' profits (Boyer, 2005) making the wages of high-skilled workers more volatile.

Very surprisingly, we find that higher social security expenditure is associated with an increase in the volatility of wages. This result is in line with the argument of [START_REF] Carmignani | Macroeconomic risk and the (de)stabilising role of government size[END_REF] claiming that government would have a destabilizing effect on macroeconomic volatility. However, if we look at the marginal effects, one can see that the marginal effect of social security expenditure on the volatility of wages is decreasing with the level of financial integration at the international and domestic levels. As expected, we find a larger compensation effect for low-skilled workers than high-skilled workers in the face of financial globalization (models (3) and ( 4)). However, we find, including in the IV-GMM regressions, a larger compensation effect of welfare state institutions for high-skilled workers than low-skilled workers in the face of the development of the financial markets at the domestic level.

Among control variables, we find that only unemployment rate has a positive and significant influence on our dependent variable. Trade openness and macroeconomic volatility give insignificant coefficients.

[INSERT TABLE 3 ABOUT HERE]

Conclusion

The aim of this paper was to assess the consequences of increasing financial development on labor-market instability and to explore the role of welfare state institutions in the face of a more unstable environment. Based on the theory of financial fragility [START_REF] Keynes | The General Theory of Employment Interest and Money[END_REF]Minksy, 1986;Aglietta and Rebérioux, 2005), we provide in this paper evidence of a positive relationship between the expansion of financial markets and the instability on labor markets. Empirical work has shown that macroeconomic volatility decreased during the 1980s and the 1990s suggesting a new era of low risks and high returns ('Great Moderation'). Financial development, by contrast, has not contributed to a risk reduction on labor markets, and this despite improved possibilities of risk-diversification. Thus, our analysis on panel data shows that financial development is quite associated with higher labor-market instability measured by the volatility of hours worked and the volatility of wages. Controlling for potential reverse causality between our dependent variables and welfare state institutions, we find that this impact is higher for low-skilled workers than for high-skilled workers. More specifically, results clearly indicate that low-skilled workers are more affected by higher volatility of hours worked. Results are, however, less robust on the impact on the volatility of wages when controlling or not for potential reverse causality.

One major contribution of this paper is to analyze the interdependency across financial development and welfare state institutions: we find that welfare state institutions measured by social security transfers as a percentage of GDP contribute to the reduction in labor-market volatility. Furthermore, our results suggest that this impact is conditional to the degree of financial exposure: the more financially developed an economy, the more social security expenditure has a reducing-effect on labor-market volatility. This result seems support the 'compensation hypothesis': higher financial development may increase the insecurity feeling, more particularly for low-skilled workers who will support more generous welfare programs. Governments therefore continue to provide ample compensations to their constituency, and this although growing globalization and tax competition would have made more difficult for governments to finance generous welfare programs. These findings challenge the 'efficiency hypothesis' considering that welfare state institutions, because they are financially unsustainable, need to be reformed. [START_REF] Saint-Paul | The Political Economy of Labour Market Institutions[END_REF] argues that 'suboptimal' institutions can be resilient if they rely on a solid political base.

This paper provides a political-economy explanation of complementarities between financial development on the one hand and welfare state institutions on the other hand. The evidence of the 'compensation hypothesis' indicates that financial development has not reduced but increased government involvement in providing specific compensation mechanisms. Beyond increased volatility of hours worked and of wages, [START_REF] Rodrik | Has Globalization Gone Too Far[END_REF] predicts that globalization may also have caused an increase in the dispersion of earnings among low-skilled workers. [START_REF] Darcillon | Do Interactions between Finance and Labor Market Institutions Affect Wage Distribution?[END_REF] proposes to analyze the relationship between financial development at the domestic level and wage inequality. More specifically, we show that financial development is associated with higher wage inequality particularly in countries with weak labor labor market institutions. In line with our present research, [START_REF] Breen | Income Inequality and Macroeconomic Volatility: An Empirical Investigation[END_REF] find a strong correlation between macroeconomic volatility and income inequality measured by the Gini coefficient. In other words, countries where output is very volatile are also more unequal.

Bibliography

Aglietta, M., and Rebérioux, A. 2005. Corporate governance adrift. A critique of shareholder value, Cheltenham, Edward Elgar Amable, B., Gatti, D., and Schumacher, J. 2006. Welfare-State Retrenchment: The Partisan Effect Revisited, Oxford Review of Economic Policy, vol. 22, no. 3, 426-44. the Kleibergen-Paap (or Cragg-Donald) test of the weak-instruments problem that arises when the correlations between the endogenous regressors and the excluded instruments are nonzero but small. The null hypothesis is that the estimator is weakly identified in the sense that it is subject to bias that the investigator finds unacceptably large. To be able to reject the null that the size of the test is unacceptably large the Cragg-Donald F-statistic must exceed the tabulated critical value. See [START_REF] Baum | Enhanced routines for instrumental variables/GMM estimation and testing[END_REF] for more details on testing the relevance and validity of instruments. . 15 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 1970 1980 1990 2 .3 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 1970 1980 1990

Figure 2 :

 2 Figure 2: Volatiltiy of wages (by skill groups)

Table 1 :

 1 Labor-market volatility (by decade and skill group)

	Variables	1970-1979	1980-1989	1990-1990	2000-2007
		Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
	Hours worked volatility abcHigh skill abcLow skill	0.013 0.017 0.020	0.008 0.001 0.009	0.011 0.022 0.016	0.0066 0.016 0.016	0.013 0.025 0.023	0.007 0.017 0.019	0.008 0.022 0.027	0.003 0.022 0.030
	Wages volatility abcHigh skill abcLow skill	0.077 0.077 0.080	0.031 0.034 0.030	0.123 0.134 0.136	0.058 0.065 0.062	0.097 0.098 0.100	0.04 0.039 0.039	0.083 0.082 0.089	0.033 0.032 0.039

Table 2 :

 2 Impact of financial development on standard deviation of hours worked growthNote: (Panel corrected) standard errors in parentheses; * p < 0.10, ** p < 0.05, *** p < 0.01; HS = high-skilled workers and LS = low-skilled workers Endogenous variables: Social expenditures/GDP, Total assets × Social exp. Instruments: Social expenditures/GDP (lag), adjusted annual deficit/GDP, total receipts of general government/GDP, long-term interest rate on government bonds, gross government debt/GDP, real GDP growth, current account balance/GDP, total population, net unemployment replacement rate, trade union density rate, financial openness index, government ideological orientation, Rae's index of electoral fractionalization of the party-system, political voting system, political regime, Herfindahl index of political competition STATA provides different tests: (1) Overidentifying restrictions (Sargan-Hansen-J statistic): the null hypothesis of this first test is that the instruments are valid instruments, i.e., uncorrelated with the error term, and that the excluded instruments are correctly excluded from the estimated equation. If the null is not rejected, the overidentification restrictions are valid; (2) Underidentification test (Kleibergen-Paap LM statistic): the underidentification test is relevant to verify whether the equation is quite identified: in this case, instruments are relevant in the sense that they are correlated with assumed endogenous regressions. The null hypothesis cannot be rejected; (3) Endogenous regressors tests (Anderson-Rubin χ 2 test and Anderson-Rubin F-test): the Anderson-Rubin tests are tests of the significance of endogenous regressors in the structural estimated equation: the null hypothesis tested is that the coefficients of the endogenous regressors in the structural equation are jointly equal to zero; (4) Weak-identification test (Kleibergen-Paap F-statistic or Cragg-Donald):

		HS	LS	HS	LS	HS	LS	HS	LS
		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
	Total assets plus liabilities/GDP Stock Market Capitalization/GDP Social expenditures/GDP Total assets × Social exp. Stock market cap. × Social exp. Trade openness Unemployment rate Volatility of output growth Constant	0.0025 (0.0016) -0.0031*** (0.0006) -0.0003** (0.0001) 0.0002* (0.0001) 0.0016*** (0.0005) 0.0031*** (0.0010) 0.0267* (0.0138)	0.0032** (0.0015) -0.0033*** (0.0010) -0.0002 (0.0002) -0.0001 (0.0002) 0.0020*** (0.0006) 0.0030** (0.0015) 0.0496** (0.0214)	0.0058 (0.0039) -0.0026*** (0.0009) -0.0007** (0.0004) 0.0002 (0.0002) 0.0010*** (0.0004) 0.0057*** (0.0015)	0.0304*** (0.0069) 0.0010 (0.0014) -0.0026*** (0.0006) 0.0011*** (0.0003) 0.0022*** (0.0006) 0.0050** (0.0024)	0.0101 (0.0094) -0.0033*** (0.0009) -0.0003 (0.0007) -0.0003 (0.0007) -0.0001 (0.0002) 0.0013*** (0.0005) 0.0061*** (0.0014) 0.0437** (0.0204)	0.0351** (0.0148) -0.0039*** (0.0015) -0.0030*** (0.0009) -0.0030*** (0.0009) -0.0003 (0.0002) 0.0023*** (0.0008) 0.0058*** (0.0018) 0.0893*** (0.0323)	0.0340 (0.0328) -0.0035*** (0.0011) -0.0022 (0.0024) -0.0022 (0.0024) -0.0002** (0.0001) 0.0017*** (0.0005) 0.0062*** (0.0018)	0.1601*** (0.0525) -0.0003 (0.0018) -0.0116*** (0.0039) -0.0116*** (0.0039) 0.0003 (0.0002) 0.0033*** (0.0008) 0.0025 (0.0030)
	Marginal effects of social security expenditures conditional to financial level
	Min Mean_less_sd Mean Mean_plus_sd Max	-0.0032*** (0.0006) -0.0029*** (0.0006) -0.0038*** (0.0007) -0.0046*** (0.0010) -0.0122*** (0.0039)	-0.0033*** (0.0010) -0.0032*** (0.0010) -0.0037*** (0.0011) -0.0043*** (0.0013) -0.0085** (0.0043)	-0.0023*** (0.0008) -0.0016* (0.0010) -0.0042*** (0.0006) -0.0068*** (0.0013) -0.0268*** (0.0084)	-0.0011 (0.0015) 0.0004 (0.0018) -0.0052*** (0.0018) -0.0109*** (0.0022) -0.0547*** (0.0151)	-0.0033*** (0.0009) -0.0033*** (0.0009) -0.0035*** (0.0011) -0.0036*** (0.0013) -0.0041* (0.0022)	-0.0040*** (0.0015) -0.0047*** (0.0015) -0.0060*** (0.0017) -0.0073*** (0.0018) -0.0189*** (0.0029)	-0.0036*** (0.0009) -0.0041*** (0.0007) -0.0050*** (0.0011) -0.0059*** (0.0020) -0.0092* (0.0056)	-0.0009 (0.0016) -0.0033*** (0.0012) -0.0083*** (0.0017) -0.0133*** (0.0032) -0.0311*** (0.0091)
	Observations R-squared Fixed effects Estimator Hansen J-statistic (p-value) Kleibergen-Paap LM χ 2 -statistic (p-value) Anderson-Rubin χ 2 test (p-value) Anderson-Rubin F-test (p-value) Kleibergen-Paap LM F-statistic	437 0.4610 Yes PCSE/OLS -----	437 0.4108 Yes PCSE/OLS -----	278 -Yes IV-GMM 0.1108 0.0000 0.0000 0.0050 2.300	278 -Yes IV-GMM 0.2981 0.0000 0.0000 0.0050 2.300	278 0.5494 Yes PCSE/OLS -----	278 0.4777 Yes PCSE/OLS -----	278 -Yes IV-GMM 0.4951 0.0000 0.0000 0.0009 5.277	278 -Yes IV-GMM 0.9078 0.0001 0.0000 0.0009 5.277

Table 3 :

 3 Impact of financial development on standard deviation of wages growth Note: (Panel corrected) standard errors in parentheses; * p < 0.10, ** p < 0.05, *** p < 0.01; HS = high-skilled workers and LS = low-skilled workers Endogenous variables: Social expenditures/GDP, Total assets × Social exp.Instruments: Social expenditures/GDP (lag), adjusted annual deficit/GDP, total receipts of general government/GDP, long-term interest rate on government bonds, gross government debt/GDP, real GDP growth, current account balance/GDP, total population, net unemployment replacement rate, trade union density rate, financial openness index, government ideological orientation, Rae's index of electoral fractionalization of the party-system, political voting system, political regime, Herfindahl index of political competition

		HS	LS	HS	LS	HS	LS	HS	LS
		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
	Total assets plus liabilities/GDP Stock Market Capitalization/GDP Social expenditures/GDP Total assets × Social exp. Stock market cap. × social exp. Trade openness Unemployment rate Volatility of output growth Constant	0.0120*** (0.0021) 0.0045*** (0.0010) -0.0014*** (0.0003) 0.0005* (0.0003) 0.0036*** (0.0009) 0.0007 (0.0023) -0.0138 (0.0280)	0.0090*** (0.0024) 0.0043*** (0.0010) -0.0012*** (0.0003) 0.0004 (0.0003) 0.0032*** (0.0010) 0.0040 (0.0027) 0.0152 (0.0258)	0.0272*** (0.0072) 0.0045** (0.0022) -0.0027*** (0.0008) -0.0011** (0.0005) 0.0045*** (0.0012) -0.0105*** (0.0029) --	0.0355*** (0.0120) 0.0076*** (0.0027) -0.0035*** (0.0013) -0.0007 (0.0007) 0.0044*** (0.0015) -0.0094*** (0.0031) --	0.1149*** (0.0231) 0.0061*** (0.0011) -0.0070*** (0.0015) 0.0003 (0.0004) 0.0033*** (0.0011) 0.0025 (0.0029) -0.0602* (0.0352)	0.0977*** (0.0201) 0.0064*** (0.0012) -0.0055*** (0.0014) 0.0003 (0.0003) 0.0022** (0.0010) 0.0086** (0.0035) -0.0557 (0.0356)	0.4186*** (0.1099) 0.0138*** (0.0040) -0.0314*** (0.0079) -0.0008* (0.0004) 0.0053*** (0.0017) -0.0102* (0.0054)	0.3640*** (0.1131) 0.0143*** (0.0041) -0.0273*** (0.0080) -0.0004 (0.0004) 0.0026 (0.0018) -0.0024 (0.0055)
	Marginal effects of social security expenditures conditional to financial level
	Min Mean_less_sd Mean Mean_plus_sd Max	0.0042*** (0.0010) 0.0053*** (0.0010) 0.0015 (0.0009) -0.0023* (0.0013) -0.0316*** (0.0069)	0.0041*** (0.0010) 0.0050*** (0.0010) 0.0017* (0.0009) -0.0015 (0.0015) -0.0267*** (0.0084)	0.0040* (0.0021) 0.0061** (0.0025) -0.0012 (0.0017) -0.0085*** (0.0029) -0.0646*** (0.0187)	0.0070*** (0.0026) 0.0096*** (0.0033) 0.0002 (0.0019) -0.0093** (0.0045) -0.0821*** (0.0309)	0.0057** (0.0010) 0.0042*** (0.0009) 0.0012 (0.0011) -0.0018 (0.0015) -0.0126*** (0.0037)	0.0060*** (0.0011) 0.0049*** (0.0010) 0.0025*** (0.0009) 0.0002 (0.0012) -0.0083*** (0.0035)	0.0112*** (0.0036) 0.0054** (0.0024) -0.0081*** (0.0029) -0.0217*** (0.0058) -0.0698*** (0.0176)	0.0127*** (0.0037) 0.0070*** (0.0026) -0.0047 (0.0031) -0.0165*** (0.0061) -0.0582*** (0.0180)
	Observations R-squared Fixed effects Estimator Hansen J-statistic (p-value) Kleibergen-Paap LM χ 2 -statistic (p-value) Anderson-Rubin χ 2 test (p-value) Anderson-Rubin F-test (p-value) Kleibergen-Paap LM F-statistic	362 0.5494 Yes PCSE/OLS -----	362 0.4777 Yes PCSE/OLS -----	332 -Yes IV-GMM 0.1192 0.0000 0.0012 0.0020 8.851	332 -0.5494 Yes IV-GMM 0.1231 0.0009 0.0016 0.0025 5.352	248 0.4777 Yes PCSE/OLS -----	248 -Yes PCSE/OLS -----	230 -Yes IV-GMM 0.5475 0.0002 0.0001 0.0005 3.894	230 Yes IV-GMM 0.2615 0.0004 0.0000 0.0000 3.442

24 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

For instance,"We have over-mutualized risk and so diluted accountability. If the risks are shared ad infinitum, there's no longer anyone accountable. [. . . ] Purely financial capitalism is a perversion that could threaten the global economy" (French President N. Sarkozy, speech in Davos, January

27, 2010).2 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

Numerous studies in human resource management have analyzed the impact of increasing diffusion of the logic based on shareholder value maximization on labor relations and more specifically on labor flexibility. Thus, the implementation of shareholder primacy encourages the firms of pursuing a cost-reduction strategy of human resource management[START_REF] Deakin | Corporate governance, labour relations and human resource management in Britain and France: convergence or divergence?[END_REF]. 5 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

[START_REF] Iversen | The Causes of Welfare State Expansion[END_REF] challenge this view: they argue that growing insecurity does not result from increasing globalization but from the process of deindustrialization.

About the relationship between insecurity feeling and partisan preferences, Guillaud (2013) finds that occupation, income and risk aversion are all shaping the individual preferences for redistribution. For instance, service workers, craftsmen, machine operators and elementary workers are more in favor of redistribution than the reference category, represented by office clerks.

All data are available on the website: www.euklems.net.

Data are freely available on Philip R. Lane's website: http://www.philiplane.org/EWN.html 11 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

Determining the significance of the effect of SPEND it on σ(y it ) conditional on FIN it values, we compute the standard error of the sum (β 1 + β 3 • FL it ) as follows:se ( β 1 + β 3 •FIN it ) = var β 1 + FIN 2 it • var β 3 + 2FIN it • cov β 1 β 3 (3)13 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

See Angrist and Pischke (2009) for a full discussion.

Fiscal data, and total population, trade union density rate, Rae's index of electoral fractionalization of the party-system, political regime and political voting system are provided byArmingeon et al. (2010); current account balance/GDP is found in the Word Bank Indicators database; net unemployment replacement rates are provided by[START_REF] Caminada | Unemployment Replacement Rates Dataset among 34 Welfare States, 1971-2009. An update, extension and modification of the Scruggs' Welfare State Entitlements Data Set[END_REF]; government ideological orientation is an extended version of the index proposed byAmable et al. (2006) and the Herfindahl index of political competition is given from the Database of Political Institutions.

Results from first-stages (not shown) indicate a negative and significant coefficient between social security expenditure and fiscal cyclical variables.

Moreover, finding a highly significant coefficient in models including proxies for international financial integration raises the risk of multicollinearity between trade openness and total assets plus liabilities over GDP. Regressions by dropping the variable of trade openness give similar results.

All data are available on the website: www.euklems.net. 20 Documents de Travail du Centre d'Economie de la Sorbonne -2013.70

Appendix 1: Construction of data

To obtain a measure of labor-market volatility, we use the EU KLEMS Database from the OECD that provides data on total hours workers by persons engaged and total labor compensation by skill groups (HS for high-skilled workers and LS for low-skilled workers) from 1970 to 2005. 12 We use two measures of labor-market volatility:

• Hours worked: We calculate hours worked h f it for each skill level f (with f = {HS; LS}) in country i for the year t:

with H_EMP it denoting the total hours worked by persons engaged and H f it the share in total hours worked by the group of workers f (high-, or low-skilled workers);

• Average wages: We calculate average wages w f it for each skill level f (with f = {HS; LS}) in country i for the year t:

with LAB it denoting labor compensation converted into constant US dollar (by using the exchange rate series from the Penn World Tables and by deflating by the U.S. output price index) and LAB f it the share in total labor compensation to the group of workers f (high-, low-skilled workers);

By using these data, we obtain respectively the growth rate of hours worked and wages. We compute the rolling standard deviation for growth rates over a five-year-window:

where y it is the growth rates of hours workers and of wages and ȳt+k is the corresponding mean.

Appendix 2: Descriptive statistics


Bruno Colbois
email: bruno.colbois@unine.ch

Patrick Verovic
email: verovic@univ-savoie.fr

TWO PROPERTIES OF VOLUME GROWTH ENTROPY IN HILBERT GEOMETRY

The aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.

Introduction

A Hilbert domain in R m is a metric space (C, d C ), where C is an open bounded convex set in R m and d C is the distance function on C -called the Hilbert metric -defined as follows.

Given two distinct points p and q in C, let a and b be the intersection points of the straight line defined by p and q with ∂C so that p = (1 -s)a + sb and q = (1 -t)a + tb with 0 < s < t < 1. Then d C (p, q) := 1 2 ln[a, p, q, b] ,

where [a, p, q, b]

:= 1 -s s × t 1 -t > 1
is the cross ratio of the 4-tuple of ordered collinear points (a, p, q, b) (see Figure 1). We complete the definition by setting d C (p, q) := 0 for p = q. The metric space (C, d C ) thus obtained is a complete non-compact geodesic metric space whose topology is the one induced by the canonical topology of R m and in which the affine open segments joining two points of the boundary ∂C are geodesic lines. It is to be mentioned here that in general the affine segment between two points in C may not be the unique geodesic joining these points (for example, if C is a square). Nevertheless, this uniqueness holds whenever C is strictly convex.

Moreover, the distance function d C is associated with the Finsler metric F C on C given, for any p ∈ C and any v ∈ T p C ≡ R m (the tangent vector space to C at p), by 2). Then, given any arbitrary norm • on R m , we can write Thanks to this measure, we can make use of a concept of fundamental importance, the volume growth entropy, which is attached to any metric space. Very often, this notion is introduced for cocompact metric spaces and is defined as follows in Hibert geometry.

F C (p, v) := 1 2 1 t -+ 1 t + for v = 0 , where t -= t - C (p, v) and t + = t + C (p, v) are the unique positive numbers satisfying p -t -v ∈ ∂C and p + t + v ∈ ∂C, and F C (p, v) := 0 for v = 0. Remark. For p ∈ C and v ∈ T p C ≡ R m with v = 0, we will define p -= p - C (p, v) := p -t - C (p, v)v and p + = p + C (p, v) := p + t + C (p, v)v (see Figure
F C (p, v) = 1 2 v 1 p -p -+ 1 p -p + . p p + p - v v ∂C
Let (C, d C ) be a Hilbert domain in R m admitting a cocompact group of isometries, for which we may assume 0 ∈ C since translations in R m preserve the cross ratio.

h(C) := lim R→+∞ 1 R ln[µ C (B C (0, R))] .
Now, when we drop cocompactness, this limit still exists in the case when the boundary ∂C of C is strongly convex (see [START_REF] Colbois | Hilbert geometry for strictly convex domains[END_REF]) or in the case when C is a polytope (see [START_REF] Vernicos | Spectral radius and amenability in Hilbert geometry[END_REF]), but it is not known whether this is true in general as stated in [10, question raised in section 2.5].

Therefore, the main goal of this paper is to answer to this question and to show that the answer is negative.

Main Theorem. There exists a Hilbert domain

(C, d C ) in R 2 with 0 ∈ C that satisfies lim sup R→+∞ 1 R ln[µ C (B C (0, R))] = 1 , and
lim inf R→+∞ 1 R ln[µ C (B C (0, R))] = 0 .
The proof of this theorem will be given in the third section by constructing an explicit example which is a convex 'polygon' with infinitely many vertices having an accumulation point around which the boundary of the 'polygon' strongly looks like a circle.

The intuitive idea behind this construction is that, depending on where we are located in the 'polygon', its boundary may look like the one of a usual polygon -and hence the volume growth entropy behaves as if it were vanishing -(this corresponds to the lim inf part in the theorem) or like a small portion of a circle (around the accumulation point) -and hence the volume growth entropy behaves as if it were positive -(this corresponds to the lim sup part in the theorem).

On the other hand, using such 'polygons' with infinitely many vertices and considering the same techniques as in the proof of the main theorem, we show that there are other Hilbert domains in the plane than polygonal ones whose volume growth entropy is zero. This is stated in Theorem 4.2.

For further information about Hilbert geometry, we refer to [START_REF] Busemann | The geometry of geodesics[END_REF][START_REF] Busemann | Projective geometry and projective metrics[END_REF][START_REF] Egloff | Uniform Finsler Hadamard manifolds[END_REF][START_REF] Goldman | Projective geometry on manifolds[END_REF][START_REF] Vernicos | Introduction aux géométries de Hilbert[END_REF] and the excellent introduction [START_REF] Socié-Méthou | Comportements asymptotiques et rigidités des géométries de Hilbert[END_REF] by Socié-Méthou.

About the importance of volume growth and topological entropies in Hilbert geometry, we may have a look at the interesting work [START_REF] Crampon | Entropies of compact strictly convex projective manifolds[END_REF] by Crampon and the references therein.

Preliminaries

This section is devoted to listing the key ingredients we will need in the present work.

Notations. From now on, the canonical Euclidean norm on R 2 will be denoted by • .

On the other hand, for any three distinct points a, b and c in R 2 , we will denoted by abc their open convex hull (open triangle), and by ∢(bac) the sector defined as the convex hull of the union of the half-lines a + R + -→ ab and a + R + -→ ac.

The first ingredient, whose proof can be found for example in [16, page 69], is classic and concerns the hyperbolic plane given here by its Klein model (B 2 , d B 2).

Proposition 2.1. We have

(1) d B 2(0, p) = atanh( p ) for any p ∈ B 2 , and (2)
µ B 2(B B 2(0, R)) = π 2 sinh 2 (R) for any R > 0.
The (1) Given any two distinct points p, q ∈ C, we have d C (p, q) d D (p, q) with equality if and only if

(p + R + -→ pq) ∩ ∂C = (p + R + -→ pq) ∩ ∂D and (p + R - -→ pq) ∩ ∂C = (p + R - -→ pq) ∩ ∂D hold.
(2) For any p ∈ C, we have vol(B C (p)) vol(B D (p)).

(3) For any Borel set A ⊆ C, we have µ C (A) µ D (A).

Proposition 2.3. If Q := (-1, 1)×(-1, 1) ⊆ R 2
denotes the standard open square, then for any p = (x, y) ∈ Q we have

2(1 -x 2 )(1 -y 2 ) vol(B Q (p)) 4(1 -x 2 )(1 -y 2 ) .
The last ingredient can be found in for all R > 0 and p ∈ B C (0, R).

We now give two technical lemmas which will be used for proving both Theorem 3.1 and Theorem 4.2. (2)

µ C B C (0, R) ∩ P 0Q 2πR 2 . Q P -P -Q 0 B T (0, R) B C (0, R) ∂T ∂C Figure 3. Comparing B C (0, R) ∩ P 0Q and B T (0, R) ∩ P 0Q
Proof (see Figure 3).

• Point 1. The equality case in Point 1 of Proposition 2.2 proves that any point p ∈ P 0Q satisfies d C (0, p) = d T (0, p), and hence we get S C (0, R)

∩ P 0Q = S T (0, R) ∩ P 0Q. Then, writing B C (0, R) = r∈[0,R) S C (0, r), we have B C (0, R) ∩ P 0Q = B T (0, R) ∩ P 0Q.
• Point 2. The previous point implies

B C (0, R) ∩ P 0Q ⊆ B T (0, R) ⊆ T ⊆ C , and hence (2.1) µ C B C (0, R) ∩ P 0Q µ T (B T (0, R)) .
Now, if f denotes the unique linear transformation of R 2 such that f (P ) = (1, -1) and f (Q) = (1, 1), we have

f (T ) = Q := (-1, 1)×(-1, 1) ⊆ R 2 (standard open square) .
The cross ratio being preserved by the linear group GL(R 2 ), the map f induces an isometry between the metric spaces (T , d T ) and (Q, d Q ) with f (0) = 0, and thus we obtain

µ T (B T (0, R)) = µ Q (B Q (0, R)).
But Proposition 2.3 yields

µ Q (B Q (0, R)) = 8 tanh(R) 0 x 0 π vol(B Q (x, y)) dy dx 4 tanh(R) 0 x 0 π (1 -x 2 )(1 -y 2 )
dy dx = 2πR Then for any R > 0 we have

µ C B C (0, R) ∩ ∢(A0B) πe 8R vol(B C (0, 1)) × A0B/2 ,
where A0B is the spherical distance between the vectors -→ 0A and -→ 0B ( i. e. , the unique number

θ in [0, π] defined by cos θ = -→ 0A/ -→ 0A , -→ 0B/ -→ 0B ∈ [-1, 1]
, where • , • stands for the canonical Euclidean scalar product on R 2 ).

Proof.

Since we have 1/vol(B C (p)) e 8R /vol(B C (0, R)) for every p ∈ B C (0, R) by Proposition 2.4, one can write

µ C B C (0, R) ∩ ∢(A0B) πe 8R vol(B C (0, R)) vol B C (0, R) ∩ ∢(A0B) πe 8R vol(B C (0, 1)) vol B 2 ∩ ∢(A0B) (noticing B C (0, 1) ⊆ B C (0, R) ⊆ B 2 ) = πe 8R vol(B C (0, 1)) ×π× A0B/(2π) = πe 8R vol(B C (0, 1)) × A0B/2 ,
where we used vol(B 2 ) = π.

Entropy may not be a limit

We prove in this section the main result of this paper which states that the volume growth entropy for a Hilbert domain may not be a limit. To this end, we will approximate a disc in the plane by an inscribed 'polygonal' domain with infinitely many vertices that have two accumulation points around which the boundary of the 'polygonal' domain looks very strongly like the boundary of the disc.

Let (n k ) k 0 be the sequence of positive integers defined by

n 0 := 3 and ∀ k 0, n k+1 = 3 n 2 k .
It is increasing and satisfies n k -→ +∞ as k -→ +∞.

Next, define the sequences (α k ) k 0 and (θ k ) k 0 in R by α k := 2π/n k together with

θ 0 := 0 and ∀ k 1, θ k := k-1 ℓ=0 α ℓ = 2π k-1 ℓ=0 1 n ℓ .
Finally, consider the sequence (M k ) k 0 and the family (P k (j)) (k,j)∈{(k,j)∈Z 2 | k 0 and 0 j n k } of points in S 1 defined by

M k := (cos(θ k ) , sin(θ k )) and P k (j) := (cos(θ k + α k j/n k ) , sin(θ k + α k j/n k )) ,
and denote by C the open convex hull in R 2 of the set {P k (j), -P k (j) | k 0 and 0 j n k } .

Then we get the following (see Figure 4):

Theorem 3.1. We have (1) h(C) := lim sup R→+∞ 1 R ln[µ C (B C (0, R))] = 1, and (
2) lim inf R→+∞ 1 R ln[µ C (B C (0, R))] = 0. M 0 M 1 M ∞ M 2 0 S 1 P 0 (1) P 0 (2) ∂C α 0 /n 0 α 0 /n 0 α 0 /n 0 θ 1 Figure 4.
A Hilbert domain in the plane whose entropy is not a limit Remarks. 1) For all k 0, one has

P k (0) = M k and P k (n k ) = M k+1 .
2) For all ℓ 0, we have n ℓ 3 ℓ+1 (by induction and using 9 m m for all integer m 0), and hence the increasing sequence (θ k ) k 0 converges to some real number θ ∞ which satisfies 0 < θ ∞ < π (since we have

+∞ ℓ=0 1/3 ℓ+1 = (1/3) +∞ ℓ=0 (1/3) ℓ = 1/2 and (n ℓ ) ℓ 0 = 3 ℓ+1 ℓ 0 ).
In order to prove this theorem, we shall use two different sequences of balls about the origin. The first one corresponds to the sequence of radii r k := ln(n k ) for k 0 that makes look the balls like those in the Klein model (B 2 , d B 2) as k -→ +∞, from which we get Point 1. The second one corresponds to the sequence of radii R i := n i for i 0 that makes look the balls like those in a polygonal domain as i -→ +∞, leading to Point 2.

Proof of Theorem 3.1.

• Point 1. Since we already have h(C) 1 by [1, Theorem 3.3], let us prove h(C) 1.
Consider the sequence of positive numbers (r k ) k 0 defined by r k := ln(n k ).

Fix k 0, and let (p k (j)) 0 j n k -1 be the sequence of points in R 2 defined by

p k (j) ∈ [0, P k (j)] and d C (0, p k (j)) = r k .
Then fix j ∈ {0, . . . , n k -1}, and let P := P k (j), Q := P k (j + 1), p := p k (j) and q := q k (j + 1) (see Figure 5).

0 P = P k (j) Q = P k (j + 1) ∂C S 1 p = p k (j) q = p k (j + 1) B C (0, r k ) α k /n k
r k = d C (0, p) = d B 2(0, p) and r k = d C (0, q) = d B 2(0, q) ,
which implies p = q = tanh(r k )

by Point 1 of Proposition 2.1.

Therefore, if a denotes the midle point of p and q, we get

a = tanh(r k ) cos[α k /(2n k )] = tanh(r k ) cos(π/n 2 k ) .
Defining ρ k := d B 2(0, a) and using again Point 1 of Proposition 2.2 together with the formula tanh(ln x) = x 2 -1 x 2 + 1 which holds for any x > 0, one can write

1 -tanh(ρ k ) = 1 -tanh(r k ) cos(π/n 2 k ) = 1 - n 2 k -1 n 2 k + 1 cos(π/n 2 k ) = 2/n 2 k + •(1/n 2 k ) ∼ 2/n 2 k as k -→ +∞ . (3.1)
On the other hand, the inclusions

B B 2(0, ρ k ) ∩ P 0Q ⊆ p0q ⊆ C ⊆ B 2 yield µ C (p0q) µ B 2(p0q) µ B 2(B B 2(0, ρ k ) ∩ P 0Q) = α k /n k 2π ×µ B 2(B B 2(0, ρ k ))
by Point

0 such that µ C (p k (j)0p k (j + 1)) = µ C (p0q) 1/3
holds for every k k 0 .

We then get

µ C (B C (0, r k )) n k -1 j=0 µ C (p k (j)0p k (j + 1)) (1/3)n k
for each k k 0 since we have p k (j)0p k (j + 1) ⊆ B C (0, r k ) for every j ∈ {0, . . . , n k -1} (indeed, balls of a Hilbert domain are convex), and hence

ln[µ C (B C (0, r k ))] r k ln[(1/3)n k ] ln(n k ) , which yields ln[µ C (B C (0, r k ))] r k -→ 1 as k -→ +∞.
This gives the first point of Theorem 3.1.

• Point 2. Consider the sequence of positive numbers (R i ) i 0 defined by R i := n i .

Then we have (see Figure 6)

Theorem 4.2. The volume growth entropy of d C satisfies

h(C) = lim R→+∞ 1 R ln[µ C (B C (0, R))] = 0 .
Remark. More precisely, we will show in the proof of this result that the volume µ C (B C (0, R)) of the ball B C (0, R) actually has at most the same growth as R 3 when R goes to infinity. This proves Theorem 4.2.

P ∞ P 0 P 1 P 2 S 1 ∂C 0 -P ∞ -P 0 -P 1 -P 2

Figure 1 .

 1 Figure 1. The Hilbert metric d C

Date:

  November 8, 2013. 2000 Mathematics Subject Classification. Primary: global Finsler geometry, Secondary: convexity.

Figure 2 .

 2 Figure 2. The Finsler metric F C

  If for any R > 0 we denote by B C (0, R) := {p ∈ C | d C (0, p) < R} the open ball of radius R about 0 in (C, d C ), then the volume growth entropy of d C writes

  two following results have been established in [7, Proposition 5 and Proposition 6]. Proposition 2.2. If (C, d C ) and (D, d D ) are Hilbert domains in R m satisfying C ⊆ D, then the following properties are true:

[ 6 ,Proposition 2 . 4 .

 624 Proof of Theorem 12]. Given any Hilbert domain (C, d C ) in R m satifying 0 ∈ C, we have vol(B C (0, R)) e 8R ×vol(B C (p))

Lemma 2 . 1 .

 21 Let (C, d C ) be a Hilbert domain in R 2 with 0 ∈ C, and let P , Q be distinct points in R 2 such that the affine segments [P, Q] and [-P , -Q] are contained in the boundary ∂C. If T is the open quadrilateral in R 2 defined as the open convex hull of P , Q, -P and -Q, then for any R > 0 we have (1) B C (0, R) ∩ P 0Q = B T (0, R) ∩ P 0Q, and

Figure 5 .

 5 Figure 5. Showing lim sup k→+∞ ln[µ C (B C (0, r k ))] r k > 0 with r k := ln(n k )

Figure 6 .• 2 .••

 62 Figure 6. A non-polygonal Hilbert domain in the plane with zero entropy

  Lemma 2.2. Let (C, d C ) be a Hilbert domain in R 2 which satisfies 0 ∈ C ⊆ B 2 , and let A, B be two distinct points in S 1 .

		2 ,
	which gives µ C B C (0, R) ∩ P 0Q	2πR 2 from Equation 2.1.

  3 of Proposition 2.2 and since Euclidean rotations induce isometries of (B 2 , d B 2), from

	which one obtains
	(3.2)	µ C (p0q)	π 2	sinh 2 (ρ k )/n 2 k
	by Point 2 in Proposition 2.1.
	Now, since we have sinh 2 (x) = tanh 2 (x)/(1 -tanh 2 (x)) for all x ∈ R, Equation 3.1 implies
	π 2	sinh 2 (ρ k )/n 2

k ∼ π/8 as k -→ +∞ , from which Equation 3.2 insures the existence of an integer k 0

Fixing an integer i 0, we can write the decomposition

with M ∞ := (cos(θ ∞ ) , sin(θ ∞ )) ∈ ∂C (recall that θ ∞ is the limit of the sequence (θ k ) k∈N : see the second remark following Theorem 3.1). * First step. Here, we deal with the two first terms in Equation 3.3.

For each k 0 and j ∈ {0, . . . , n k -1}, let T k (j) be the open rectangle that is equal to the open convex hull in R 2 of P k (j), -P k (j), P k (j + 1) and -P k (j + 1).

Then, by Lemma 2.1, we have

Next, we focus on the third term in Equation 3.3.

Lemma 2.2 with

where τ := π 2 /vol(B C (0, 1)) is a positive constant.

But for any ℓ i we have

with n i /n 2 ℓ 1/n i from the monotone increasing of the sequence (n ℓ ) ℓ 0 .

Hence, since 1/n i -→ 0 as i -→ +∞, there exists an integer i 0 0 such that for all ℓ i one has e 8R i /n ℓ+1 3 -n 2 ℓ /2 whenever i i 0 . Equation 3.6 then implies

for all i i 0 (notice that we have n 2 ℓ /2 9 ℓ ℓ for any ℓ 0: see the second remark following Theorem 3.1). Now we have 3 -i -→ 0 as i -→ +∞, and thus there exists an integer i 1 i 0 such that for all i i 1 one has

1 . * Third step. Combining Equations 3.3, 3.4, 3.5 and 3.7, we eventually get

i for all i i 1 (since we have R ℓ := n ℓ 3 ℓ+1 ℓ + 1 1 for every ℓ 0), and hence

This proves the second point of Theorem 3.1.

Remark. Considering the proof of Point 1 in Theorem 3.1, we can observe that the conclusion h(C) > 0 we obtained is actually true for any sequence of positive integers (n k ) k 0 provided the sequence (θ k ) k∈N converges to some real number θ ∞ which satisfies 0 < θ ∞ < π.

Non-polygonal domains may have zero entropy

In this section, we construct a Hilbert domain in the plane which is a 'polygon' having infinitely many vertices and whose volume growth entropy is a limit that is equal to zero. This 'polygon' is inscribed in a circle and its vertices have one accumulation point.

Before giving our example, let us first recall the following result proved in [START_REF] Vernicos | Spectral radius and amenability in Hilbert geometry[END_REF]:

Given any open convex polytope P in R m that contains the origin 0, the volume growth entropy of d P satisfies

Remark. Another -but less direct -proof of this theorem consists in saying that (P, d P ) is Lipschitz equivalent to Euclidean plane as shown in [START_REF] Bernig | Hilbert geometry of polytopes[END_REF] (and in [START_REF] Colbois | Hilbert geometry for convex polygonal domains[END_REF] for the particular case when n := 2), and hence h(P) = 0 since the volume growth entropy of any finite-dimensional normed vector space is equal to zero. Now, let us show that having zero volume growth entropy for a Hilbert domain in R 2 does not mean being polygonal, that is, that the converse of Theorem 4.1 is false.

Let (P n ) n∈N be the sequence of points in S 1 defined by