Pierre-Alexis Chaboche
email: p.chaboche@protonmail.com

Fabien Pointurier

Pierre Sabatier

Anthony Foucher

Tales Tiecher

Jean P G Minella

Marcos Tassano

Amélie Hubert

Sergio Morera

Stéphane Guédron

Christophe Ardois

Béatrice Boulet

Catherine Cossonnet

Pablo Cabral

Mirel Cabrera

Guillermo Chalar

Olivier Evrard

Affiliations

Pu signatures allow refining the chronology of radionuclide fallout in South America

Keywords: Plutonium isotopes, Nuclear weapon tests, Geochronology, Soil erosion

Introduction

Anthropogenic radionuclides emitted by atmospheric nuclear weapon tests (NWT) conducted after 1945 have been widely used as powerful markers to date environmental archives and conduct climatic and environmental reconstructions [START_REF] Alewell | 239+240 Pu from "contaminant" to soil erosion tracer: Where do we stand?[END_REF][START_REF] Foucher | A worldwide meta-analysis (1977-2020) of sediment core dating using fallout radionuclides including 137 Cs and 210 Pbxs[END_REF][START_REF] Hancock | Plutonium as a chronomarker in Australian and New Zealand sediments: a comparison with 137Cs[END_REF][START_REF] Ritchie | Bibliography of publications of 137Cesium studies related to erosion and sediment deposition[END_REF]. This information is also used both to reconstruct soil erosion rates during the Great Acceleration [START_REF] Steffen | The trajectory of the Anthropocene: the great acceleration[END_REF] the period that started in 1950 and that is characterised by a general increase of the human imprint on global biogeochemical cyclesand for sediment source fingerprinting research (Evrard et al., 2020;[START_REF] Meusburger | A multi-radionuclide approach to evaluate the suitability of 239+240 Pu as soil erosion tracer[END_REF]. Over 90% of the total energy yield (440 Mt) associated with NWT was detonated in the Northern hemisphere, during tests that were mainly carried out by the United States (US) until 1958 in the Marshall Islands and then by the Former Soviet Union (FSU) between 1961 and 1962 at the Novaya Zemlya and Semipalatinsk sites [START_REF] Unscear | Sources and effects of ionizing radiation: United Nations Scientific Committee on the Effects of Atomic Radiation: UNSCEAR 2008 report to the General Assembly, with scientific annexes[END_REF]. The opening to the signature of the Partial Test-Ban Treaty on August 5, 1963 has significantly reduced global emissions of artificial radionuclides into the environment, with the end of the atmospheric tests conducted by the USA, FSU and the United Kingdom (UK). After initial tests in the Saharan Desert (1960)(1961)(1962)(1963)(1964)(1965)(1966), France conducted most of its nuclear experiments at the Centre d'Expérimation du Pacifique (C.E.P) in Polynesia, with 41 atmospheric tests conducted in the vicinity of Moruroa andFangataufa atolls between 1966 and1974. In the Northern Hemisphere, China carried out atmospheric tests in Lop Nor until 1980 (Fig. 1).

<Figure 1 here>

Depending on the type of test, its yield, firing altitude and latitude, radioactive debris attached to ambient aerosol particles were distributed between the troposphere and the stratosphere.

Tropospheric debris were dispersed within the latitude band of the initial injection and followed wind-driven trajectories, before being deposited onto the soil or ocean surfaces with residence times comprised between 30-70 days [START_REF] Bennett | Worldwide dispersion and deposition of radionuclides produced in atmospheric tests : Fallout from atmospheric nuclear tests : Impact on science and society[END_REF][START_REF] Holloway | Mean residence time of plutonium in the troposphere[END_REF]. Stratospheric deposition across the world accounts for the majority of global long-lived fission product residues, with atmospheric residence times comprised between 3 and 24 months depending on whether the debris were injected in the polar or the equatorial regions (UNSCEAR, 2000). In addition to the ubiquitous 137 Cs (T 1/2 = 30 years) found across the globe, the analysis of plutonium (Pu) isotopes including 239 Pu (T1/2 = 24 110 years) and 240 Pu (T1/2 = 6 561 years) offer the opportunity to discriminate between contrasted sources of radionuclides in general, and Pu in particular. Based on the available worldwide compilations, it is currently accepted that the deposition of 137 Cs and 239+240 Pu reached a peak in 1963 and 1964-1965 in the Northern and Southern hemispheres, respectively [START_REF] Cambray | Radioactive fallout in air and rain: results to the end of 1988[END_REF][START_REF] Turney | Global peak in atmospheric radiocarbon provides a potential definition for the onset of the anthropocene epoch in 1965[END_REF]. The delay observed between fallout peaks is attributed to the time required for allowing exchanges of nuclear fallout products between both hemispheres.

Although the fallout signatures associated with the tests conducted in the Northern Hemisphere have received a significant interest, much less is known regarding the deposition that took place in South America, e.g. in response to the nuclear tests conducted in the Southern hemisphere. Although the devices tested were less powerful than those developed by the USA and the FSU, the environmental impacts of the French nuclear experiments may have led to a distinct labelling of soils and sediments across this subcontinent. Sources of radionuclides in this part of the world are more limited, as they only include part of the emissions of the tests conducted by the i) USA and the UK in equatorial regions (Pacific Proving Grounds) until 1962, ii) FSU in the Artic, and iii) those conducted by France in Polynesia in the late 1960s and the early 1970s. The situation is also characterised by the lack of significant fallout in the Southern hemisphere following the Chernobyl (1986) and Fukushima (2011) nuclear power plant accidents [START_REF] Steinhauser | Comparison of the Chernobyl and Fukushima nuclear accidents: a review of the environmental impacts[END_REF].

The development of mass spectrometric techniques such as Inductively-Coupled Plasma Mass Spectrometry (ICP-MS) and Accelerator Mass Spectrometry (AMS) offered the opportunity to quantify ultra-trace levels of plutonium isotopes (fg to ag g -1 ) in a variety of contrasted environmental matrices (e.g. ice, soil, sediment, vegetation) after thorough chemical purifications.

Importantly, ICP-MS and AMS provided powerful techniques to quantify individually 240 Pu and 239 Pu, which was not possible in previous studies relying on alpha spectrometry, providing only the sum of both isotopes ( 239+240 Pu).

Early records of radioactive fallout in the Southern hemisphere were derived from the analysis of ice cores collected in Antarctica [START_REF] Koide | The 240 Pu/ 239 Pu ratio, a potential geochronometer[END_REF]. Although they were collected at ~75° latitude south (°S), i.e. in latitudinal bands far from those where France (22°S) or the USA (near the Equator) conducted detonations, their analysis demonstrated a change in the 240 Pu/ 239 Pu atom ratios with time. For example, the ratios found in ice deposited in the late 1950s dominated by the US tests in the Pacific Proving Ground (0.19-0.34) were higher than those accumulated in the early 1960s and dominated by the FSU tests (0.16-0.19), which indicates that Pu atom ratios provide relevant indicators to monitor potential changes in sources of nuclear fallout.

To address controversies [START_REF] Arnaud | Radionuclide dating ( 210 Pb, 137 Cs, 241 Am) of recent lake sediments in a highly active geodynamic setting (Lakes Puyehue and Icalma-Chilean Lake District)[END_REF][START_REF] Guevara | Comment on the article 'Radionuclide dating (Pb-210, Cs-137, Am-241) of recent lake sediments in a highly active geodynamic setting (Lakes Puyehue and Icalma-Chilean Lake District)[END_REF][START_REF] Magand | Response on the comment from Ribeiro Guevara and Arribere on the article[END_REF] regarding the sources and the timing of radionuclide fallout at the scale of South America, our study provides a novel compilation of already published and new original 240 Pu/ 239 Pu atom ratio measurements conducted in soils and sediments collected across the subcontinent. As previously hypothesized [START_REF] Chamizo | Presence of plutonium isotopes, 239 Pu and 240 Pu, in soils from Chile[END_REF][START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF], South America was, in addition to global fallout following nuclear tests carried out in the Northern hemisphere, also partially exposed to the fallout associated with nuclear tests performed in the Southern hemisphere, notably the French nuclear tests conducted in Polynesia. To the best of our knowledge, this contribution had never been calculated so far and its consequences for various research fields in Earth Science (soil redistribution assessment, climatic and environmental reconstructions) would deserve the discussion that is provided in the current research, given the potential of these analyses to refine spatial and temporal interpretations derived from routine radionuclide measurements.

Materials and Methods

Sample collection

Detailed information regarding study site and sample collection are provided in the original publications listed in Table 1.

Sediment core was collected in 2020 in the Rincón del Bonete (RDB-01, 31 cm long) reservoir using a 63 mm Uwittec gravity corer equipped with a 1 meter PVC liner tube available at the Universidad de la República (Montevideo, Uruguay). Coring site was selected on the vicinity of the dams in order to ensure the maximum accumulation of fine sediment.

Another 83 cm long sediment core (CG02, S 18°15.110', W 69°09.784') was collected in the western basin of Lake Chungará (Chile) in October 2014 at a water depth of 29.4 m, using a UWITEC gravity corer. Core opening and slicing were performed on the lake shore right after core collection to avoid transport-related disturbance. Detailed information for core sampling and processing are provided elsewhere [START_REF] Guédron | Late Holocene volcanic and anthropogenic mercury deposition in the western Central Andes (Lake Chungará, Chile)[END_REF].

Gamma spectrometry

Prior to gamma spectrometry measurements, samples were dried at 40°C for at least 24 h, ground and passed through a 2 mm sieve, then placed in plastic Marinelli beakers. Gamma-ray measurements of 137 Cs (662 keV) in all samples were obtained using very low-background and high purity germanium detectors (HPGe) (Canberra) at Laboratoire des Sciences du Climat et de l'Environnement (LSCE, Gif-sur-Yvette, France) and the Laboratoire Souterrain de Modane (LSM, Modane, France). After quantification, 137 Cs activities (Bq kg -1 ) were decay-corrected to 2020 assuming a decay constant of 137 Cs λ = Ln2/30.2 y.

The 210 Pbxs activities (T1/2 = 22.3 years) were calculated by subtracting the supported activity (determined by using two 226 Ra daughters), 214 Pb activities (average count number at 295.2 and 351.9 keV) and 214 Bi activity (609.3 keV) from the total 210 Pb activity measured at 46.5 keV.

Sample preparation and purification procedure for plutonium analysis

The sample preparation and isotope composition measurements for samples collected in Lake Chungará and Rincón del Bonete reservoir were carried in the analytical expertise laboratory (Arpajon, France) of the Military Application Division (DAM) of the French Atomic Energy Commission (CEA). All acids used were of analytical grade and one or two process blanks were prepared under the same conditions as the samples. Dissolution, separation and purification of Pu followed a procedure almost similar than described in previous publications [START_REF] Evrard | Novel Insights into Fukushima Nuclear Accident from Isotopic Evidence of Plutonium Spread along Coastal Rivers[END_REF][START_REF] Jaegler | Plutonium isotopic signatures in soils and their variation (2011-2014) in sediment transiting a coastal river in the Fukushima Prefecture, Japan[END_REF]. Quantities of approximately 5 g of each sample were transferred to Pyrex® beakers, covered with watch glasses to prevent cross contamination and dry-ashed at 450 °C for ~ 15 h in an electric furnace to decompose organic matter. After cooling to room temperature, limited amount of 244 Pu (1 pg) was added as an isotopic dilution tracer for plutonium quantitative analysis.

Digestion of the samples was carried out through five successive acid leaching steps: one leaching with concentrated HNO3 (30 mL), one leaching with fuming HNO3 and H2O2 (1 mL during the second leaching), one leaching with concentrated HCl (30 mL) and a last one with 4 mol L -1 HCl (50 mL). It should be noted that between each leaching step, the solutions were back flow heated at 120-150°C for several hours before being evaporated to dryness using hot plates. Samples were then filtrated with a disposable 0.45 µm Nalgene filtering unit (Thermo Scientific, Rochester, NY, USA). Dissolved fractions were evaporated to dryness before being recovered with 8 mol L -1 HNO3 (30 mL). Insoluble fractions were treated separately and transferred in clean Savillex® PFA Teflon® beakers, where a first digestion for 1 h with HF (20 mL) was done. Then, concentrated HNO3 (5 mL) was added, followed by evaporation to dryness. Samples were recovered with HF (5 mL) and concentrated HNO3 (20 mL) and evaporated to dryness. Samples were finally recovered with concentrated HNO3 (25 mL) and H2O2 (1 mL), evaporated to dryness and recovered with 10 mL of 8 mol L -1 HNO3, before being added to the previous dissolved fractions and evaporated to dryness. Samples were recovered with 50 mL of 8 mol L -1 HNO3, and a few mg of NH2OH.HCl and NaNO2 were added to stabilize the (+IV) oxidation state of Pu and evaporated to dryness.

Samples were finally recovered with 50 mL of 8 mol L -1 HNO3 before Pu separation on chromatographic columns.

Plutonium purification was performed with a 20 mL-column filled with Dowex AG1x8 anionexchange resin (50-100 mesh AG1X8 and 100-200 mesh AG1X8 resins, 10 mL each), rinsed twice and conditioned with 60 mL of 8 mol L -1 HNO3. A second purification was carried out with a 2 mL-column filled with a Dowex AG1X4 anion-exchange resin (100-200 mesh resin). Samples were loaded in both columns, washed with 8 mol L -1 HNO3, 10 mol L -1 HCl and concentrated HCl before Pu fraction elution with a NH4I (1.5%)-12 mol L -1 HCl solution. A succession of evaporation to dryness and recovering was performed using concentrated HNO3. Finally, samples were recovered with 3 mL of 2% HNO3 for ICP-MS measurements.

Sample preparation and isotope composition measurements for soils located in the Conceição River catchment (Brasil), in Antarctica and Peru were carried in the Laboratoire de Métrologie de la Radioactivité dans l'Environnement (LMRE) of the Institut de Radioprotection et de Sûreté Nucléaire (IRSN). Sample preparation was described in detail in previous papers and includes a two-step procedure allowing quantification of 238 Pu and 239+240 Pu by alpha spectrometry, then separately 239 Pu and 240 Pu by SF-ICP-MS [START_REF] Bouisset | Contribution of close-in fallout from the French atmospheric tests in inventories of Cs-137, Am-241 and plutonium (238,239,240) in Gambier Islands (French Polynesia) -Signatures of stratospheric fallout in the Southern Hemisphere[END_REF]. Acids used were analytical grade for alpha spectrometry measurements, and superior quality for SF-ICP-MS. In summary, approximately 50g of samples were ashed and spiked with 242 Pu (NIST SRM 4334). Digestion of the samples was carried out during 4 days using HNO3, HCl and H2O2, followed by three successive coprecipitations of actinides. Plutonium separation was then performed using chromatography column with anion exchange resin AG1X8 (Bio-rad), conditioned in 8 mol L -1 HNO3, and eluted with 12 mol L -1 HCl/0.1 mol L -1 NH4I. The purified fraction containing Pu was evaporated, before being dissolved in concentrated HNO3 and prepared by electrodeposition onto stainless steel discs (cathode) and Pt (anode). Pu isotopes were measured during 14 days with a low level background alpha spectrometer (alpha-analyst, Canberra) with 450 mm 2 PIPS detectors and using Genie 2000 software (Canberra). Then, Pu electroplated samples were dissolved with concentrated HNO3 and HCl. The solution was evaporated and recovered with 8 mol L -1 HNO3. NaNO2 was added for valence adjustment of Pu to Pu(IV). Plutonium purification was performed using a column filled with a Dowex AG1X8 anion-exchange resin and 8 mol L -1 HNO3 before being eluted with HCl/HI. Finally, samples were evaporated to dryness and recovered with 0.5 mol L -1 HNO3 for ICP-MS measurements.

Isotope composition measurements by multiple-collector ICP-MS

Pu isotopic composition for samples collected in Lake Chungará and Rincón del Bonete reservoir were measured with a Multi-Collector ICP-MS ("Neptune Plus", Thermo Fisher Scientific Inc., Bremen, Germany) equipped with eight Faraday cups and five ion counters. Ions counters were used for simultaneous measurements of plutonium isotopes ( 239 Pu, 240 Pu, 241 Pu, 242 Pu, 244 Pu). Faraday cups were used to monitor uranium content ( 238 U, to correct for 238 U hydrides and peak tailing) and lead content (to correct for PbO2 interferences). An Aridus II (Cetac) as an introduction system, high efficiency sampler and skimmer cones, and a reinforced pumping in the interface were implemented to enhance the sensitivity (by approximately a factor of 10 with respect to the standard configuration). Two certified isotopic reference materials were used in ICP-MS measurements (SRM-947, SRM-948) for mass bias correction and plutonium isotopic verification.

For samples collected in Antarctica, Conceição (Brasil) and Peru, measurements were made with a sector field ICP-MS (SF-ICP-MS, Element 2, Thermo Fisher Scientific Inc., Bremen, Germany). The introduction system was made up of a PFA MicroFlow nebulizer (200 μL/min) connected to a high sensitivity desolvating system (apex Ω, ESI) and a cyclonic spray chamber (PC3 -ESI). A high capacity dry interface pump and a set of cones (Jet Interface) were used to enhance the sensitivity. Two reference material were used in alpha spectrometry and ICP-MS measurements (IAEA-410 -Bikini atoll sediment and IAEA-412 -Pacific Ocean sediment)

Two-sources un-mixing model

The fraction of plutonium originating from a source with a much lower 240 Pu/ 239 Pu atom ratio likely associated with French nuclear weapon tests (FNT) in the analysed samples was estimated as the first source in an empirical two-sources un-mixing. The second source being the global fallout (GF) associated with the pre-moratorium NWT. This model had been previously used to evaluate the contribution of fallout from two sources, where both GF and local/regional fallout have occurred on a given site [START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF][START_REF] Pittauer | Continuous transport of Pacific-derived anthropogenic radionuclides towards the Indian Ocean[END_REF][START_REF] Tighe | Local and global trace plutonium contributions in fast breeder legacy soils[END_REF].

The compilation of 240 Pu/ 239 Pu atom ratios published in the literature and the use of this twosources un-mixing model was used to identify those latitudinal bands with atom ratios differing from the average global fallout signature.

Assuming that the 240 Pu/ 239 Pu atom ratio determined in one sample (RS) is a mixture of both fallout sources, the first with a low 240 Pu/ 239 Pu atom ratio (RFNT) and the second with a ratio corresponding to that of the global fallout (RGF):

𝑅 𝑠 = 𝜒 𝐹𝑁𝑇 × 𝑅 𝐹𝑁𝑇 + (1 -𝜒 𝐹𝑁𝑇 ) × 𝑅 𝐺𝐹 (Eq. 1)
where χFNT and 1-χFNT are respective contributions of FNT and GF. Consequently, the proportion of plutonium originating from this source with a much lower 240 Pu/ 239 Pu atom ratio, likely associated with FNT, is calculated according to Eq. 2:

𝜒 𝐹𝑁𝑇 = 𝑅 𝑠 -𝑅 𝐺𝐹 𝑅 𝐹𝑁𝑇 -𝑅 𝐺𝐹 (Eq. 2)
The standard-uncertainty of the contribution u(χFNT) is determined by the combination of uncertainties associated with each variable in Eq. 2 and is expressed as follows:

𝑢(𝜒 𝐹𝑁𝑇 ) = 𝜒 𝐹𝑁𝑇 × √ ( (𝑅 𝑆 -𝑅 𝐺𝐹 )×(𝑅 𝐹𝑁𝑇 -𝑅 𝐺𝐹 ) ) 2 (Eq. 3)

The RGF signatures were previously determined from a worldwide survey in terrestrial soils conducted between 1970-1971 by the Environmental Measurements Laboratory (EML) [START_REF] Hardy | Global Inventory and Distribution of Fallout Plutonium[END_REF][START_REF] Krey | Mass isotopic composition of global fall-out plutonium in soil[END_REF] and analysed by TIMS (Thermal ionization mass spectrometry) for Pu atom ratio in the late 2000s (Fig. 2) [START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF]. Of note, radioactive fallout derived from the Chinese and French nuclear tests conducted after the sampling period could therefore not be recorded in these soil samples. In particular, approximately half of the tropospheric and stratospheric fallout associated with the FNT occurred after the sampling period. According to [START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF], RGF is defined as 0.178 ± 0.019, 0.173 ± 0.027 and 0.185 ± 0.047 for the 0-10°N, 0-30°S and 30-53°S regions, respectively. Owing to the fact that uncertainties associated with these values take into account FNT (in particular the 30-53°S regions), we assume that a RGF signature of 0.179 ± 0.010 is representative of the radionuclide fallout associated with the USA, FSU and UK tests at the scale of South America.

Most information published regarding RFNT in tropospheric fallout is based on the analyses conducted on samples collected in 1996 by the International Atomic Energy Agency (IAEA), where a 240 Pu/ 239 Pu atom ratio of 0.04 ± 0.01 can be estimated for samples collected in Fangataufa atoll where part of the French tests was conducted (see the Discussion section). For RS values greater than RGF, it is assumed that the Pu is fully derived from GF with corresponding contributions equal to 100%.

Results

Spatial extent of plutonium signatures in soils

Uncertainties and/or standard deviations associated with Pu ratios presented in this section are detailed in Table 1.

The 240 Pu/ 239 Pu atom ratios published in the literature and determined in the current research for four additional surface soil samples (0-5 cm) and two undisturbed soil profiles are displayed in Table 1 and in Fig. 2. Overall, 240 Pu/ 239 Pu atom ratios remain rather heterogeneous across South America, ranging from 0.041 ± 0.003 to 0.2045 ± 0.0046 with three distinct latitudinal patterns. For the equatorial latitudes (10°N-7°S), 240 Pu/ 239 Pu atom ratios in surface soils and undisturbed soil profiles are characteristic of the average global fallout signature (~0.18, Fig. 2).

Although information is scarce for the Southern part of the continent (latitude below 43°S), a similar pattern is observed in these regions although Pu atom ratios were found to be slightly higher in surface soils of Antarctica (0.19 ± 0.01), with a maximum 240 Pu/ 239 Pu atom ratio of 0.2045 ± 0.0046 observed in a soil profile from Punta Arenas, Chile (53°S, Fig. 2).

<Figure 2 here>

In contrast, the 240 Pu/ 239 Pu atom ratios found in soils located between 20 and 42°S on both sides of the Andean Cordillera significantly deviate from the average global fallout signature (Fig. 2), with the lowest 240 Pu/ 239 Pu atom ratio (0.0041± 0.003) found at 33°S in a surface soil sample of La Parva (Chile) [START_REF] Chamizo | Presence of plutonium isotopes, 239 Pu and 240 Pu, in soils from Chile[END_REF]. From this location, 240 Pu/ 239 Pu atom ratios found in surface soils gradually increased southwards and northwards to reach the average global signature in Southern and Northern Chile. Compared to the signatures found in other soil profiles of South America, those obtained in Santiago, Chile, and Buenos Aires, Argentina, differ from the integrated fallout signature for the 30-53°S regions (0.185 ± 0.047) [START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF], with 240 Pu/ 239 Pu atom ratios of 0.116 ± 0.02 and 0.108 ± 0.001, respectively (Table 1). In addition to the Buenos Aires sample collected on the Atlantic coast, the undisturbed soil profiles collected from the Conceição catchment (28°S, Brazil) analysed in the current research showed 240 Pu/ 239 Pu atom ratios remaining homogeneous with depth (0.135 ± 0.006), while being slightly lower than those found in the two soil profiles analysed by [START_REF] Kelley | Global distribution of Pu isotopes and 237 Np[END_REF] in Rio de Janeiro (0.168 ± 0.003)

and Angra dos Reis (0.178 ± 0.002) located further North in Brazil (23°S).

<Table 1 here>

3.2.Respective contributions of global vs. French fallout (FF) in soils

The proportion of plutonium originating from the FNT, calculated from the two-source unmixing model (see section 2.5.), is provided in Table 1 and displayed in Fig. 3. Accordingly, the highest contributions of FNT were calculated for surface soil samples collected on the western continental margin of South America, in La Parva (99.3 ± 7.5%, 33°S) and Chillan (48.8 ± 7.7%, 36°S) both located in Chile (Fig. 3). In undisturbed soil profiles, the proportion increased between 23°S to 35°S from 8.0 ± 6.8% in Rio de Janeiro, Brazil, to 51.3 ± 5.1% in Buenos Aires, Argentina.

On the contrary, the contribution of GF remained unambiguously the main source of Pu found in soils located in the Northern and Southern parts of the subcontinent, with proportions generally exceeding 80% (Fig. 3).

<Figure 3 here> 3.3. Temporal evolution of fallout signature in sediment core profiles

The 210 Pbxs dating method was used to provide the chronology of two sedimentary sequences collected in Lake Chungará (Chile, 18°S -69°W) and in the Rincón del Bonete

Reservoir (Uruguay, 32°S -56°W). The Constant Flux Constant Sedimentation (CF:CS) was applied to determine 210 Pbxs age models for these sediment cores [START_REF] Bruel | serac: AR package for ShortlivEd RAdionuclide Chronology of recent sediment cores[END_REF]Goldberg, 1963;[START_REF] Krishnaswamy | Geochronology of lake sediments[END_REF]) (Table 2).

<Table 2 here>

For Lake Chungará, a mean sedimentation rate of 0.95 ± 0.04 mm yr -1 (1σ) over the top 10 cm was previously determined [START_REF] Guédron | Late Holocene volcanic and anthropogenic mercury deposition in the western Central Andes (Lake Chungará, Chile)[END_REF]. Samples from successive layers had to be regrouped as there was not sufficient material to perform high temporal-resolution analyses.

Plutonium isotopes were measured for three samples covering the periods 1908-1955 (5.75 -9.75 cm), 1955-1987 (2.75-5.25 cm) and 1987-2014 (2.75-0 cm) according to the CF:CS model (see Supplementary Fig. S1). The 240 Pu/ 239 Pu atom ratio gradually decreased with time from 0.252 ± 0.015 to 0.127 ± 0.013 between 1987-2014 (proportion from FNT, 37.3 ± 10.7%).

Sample CG6-11 corresponding to sediment deposited between 1955-1987, where the maximum activity peaks of 137 Cs were detected, showed a 240 Pu/ 239 Pu atom ratio of 0.156 ± 0.009 (FNT: 16.3 ± 8.9%) (Fig. 4).

<Figure 4 here>

Higher down-core resolution with depth was available for the sediment core collected in the Rincón del Bonete Reservoir, in Uruguay (Fig. 5). Importantly, sampling reached the bedrock material and it was therefore assumed to cover the entire sedimentation period since the reservoir has become operational early in the 1950s. In contrast, the CF:CS model applied to the 210 Pbxs activities calculated a mean accumulation rate of 11.03 ± 1.31 mm yr -1 (1σ) over the uppermost 30 cm, which indicated that this sediment was deposited since 1991 ± 4 years onwards (see Fig. S2).

The 240 Pu/ 239 Pu atom ratios showed the occurrence of uniform values with depth, ranging from 0.082 to 0.107 with an average signature of 0.092 ± 0.015, indicating a mean contribution of 62.2 ± 12.2% from FNT in this sediment core.

<Figure 5 here>

Discussion

Signatures and spatial extent of French radioactive debris

The complete cessation of nuclear weapon testing by France took place in January 1996 following a final series of six underground tests conducted in Polynesia. The IAEA was mandated on the same year to assess the radiological situation at the C.E.P (IAEA, 1998). Approximately 300 samples were collected in contrasted matrices (e.g. corals, soil profiles, aerosols, etc.) and analysed for their content in 238 Pu, 239+240 Pu and 241 Am. Based on this survey, it was concluded that most of the residual radioactivity in both atolls was derived from safety trials and barge tests, which have led to significant local fallout. Then, in 2005, part of these samples was analysed by means of AMS, and 240 Pu/ 239 Pu atom ratios characteristic of low yield detonations were observed [START_REF] Hrnecek | Determination of plutonium in environmental samples by AMS and alpha spectrometry[END_REF] (see Table S1). In safety trial sites of Moruroa, a 240 Pu/ 239 Pu atom ratio of 0.019 was determined, in close agreement with the signatures lower than 0.03 measured by [START_REF] Chiappini | 240 Pu/ 239 Pu isotopic ratios and 239+ 240 Pu total measurements in surface and deep waters around Mururoa and Fangataufa atolls compared with Rangiroa atoll (French Polynesia)[END_REF] in lagoon sediments. In other areas of Moruroa atoll affected by detonations conducted on barges, 240 Pu/ 239 Pu atoms ratios were found to be slightly higher and close to 0.05 in topsoils and loose coral rocks [START_REF] Chiappini | 240 Pu/ 239 Pu isotopic ratios and 239+ 240 Pu total measurements in surface and deep waters around Mururoa and Fangataufa atolls compared with Rangiroa atoll (French Polynesia)[END_REF][START_REF] Hrnecek | Determination of plutonium in environmental samples by AMS and alpha spectrometry[END_REF]. For samples collected at Fangataufa atoll, similar low 240 Pu/ 239 Pu atom ratio signatures of ~0.05 associated with the Rigel barge tests (125 kT) conducted in 1966 were outlined in these studies. Recently, the contribution of close-in fallout to the Pu deposition was investigated in Gambier Islands located 400 km south-east from the C.E.P. The authors found an average 240 Pu/ 239 Pu atom ratios of 0.0394 ± 0.0062 in seven undisturbed soil profiles [START_REF] Bouisset | Contribution of close-in fallout from the French atmospheric tests in inventories of Cs-137, Am-241 and plutonium (238,239,240) in Gambier Islands (French Polynesia) -Signatures of stratospheric fallout in the Southern Hemisphere[END_REF].

With the exception of four devices detonated on barges and five safety trials conducted at the soil surface, 34 atmospheric tests were carried out from balloons tethered at an altitude of several hundred meters above the atolls. Of note, this type of device has been used to limit both local and regional fallout in Polynesian regions, thereby injecting large proportions of fission products into the upper troposphere or stratosphere depending on the yield. Despite the lack of public information regarding individual signatures for these tests conducted in altitude, the available data suggests that the 240 Pu/ 239 Pu atom ratios associated with the original low-yield devices remained in the same range (0.03 -0.07) as other weapon grade devices, such as those detonated in Australia, Nevada Test Site and in the Pacific Proving Grounds by the UK and the U.S [START_REF] Froehlich | Time-resolved record of 236 U and 239,240 Pu isotopes from a coral growing during the nuclear testing program at Enewetak Atoll (Marshall Islands)[END_REF][START_REF] Hamilton | Frequency distribution, isotopic composition and physical characterization of plutonium-bearing particles from the Fig-Quince zone on Runit Island, Enewetak Atoll[END_REF][START_REF] Hicks | Nevada test site fallout atom ratios[END_REF][START_REF] Muramatsu | Measurement of 240 Pu/ 239 Pu isotopic ratios in soils from the Marshall Islands using ICP-MS[END_REF][START_REF] Tims | Plutonium isotope measurements from across continental Australia[END_REF]. In the light of the publicly available information, the RFNT of 0.04 ± 0.01 used in the un-mixing model previously described appears to be the most relevant to date.

Owing that a large proportion of FNT were conducted during the southern winter when a dominant general circulation of air masses at mid-latitudes is characterised by West to East directions [START_REF] Cai | Climate impacts of the El Niño-southern oscillation on South America[END_REF], most of the radioactive particles injected into the troposphere following kiloton explosions were likely first deposited in South America during the subsequent days or weeks. This hypothesis is further supported by a set of declassified documents and maps published by the French Ministry of Defence in 2013 regarding the spatial extent and trajectories of radioactive clouds released from nuclear tests conducted by France in 1967 and 1968. In addition, the low 240 Pu/ 239 Pu atom ratios observed in surface soils from La Parva (Chile) in particular (ranging from 0.024 to 0.067, average 0.041 ± 0.003) [START_REF] Chamizo | Presence of plutonium isotopes, 239 Pu and 240 Pu, in soils from Chile[END_REF], and in soils located at similar latitudes as the C.E.P in general, strongly support the occurrence of direct tropospheric fallout of FNT debris in these regions of South America.

In contrast, the contribution of stratospheric fallout from high-yield FNT to the Pu atom ratios in soils cannot be estimated based on the current information available in the literature.

Information regarding the 240 Pu/ 239 Pu atom ratios signatures from stratospheric injections is more limited and based on the analysis of air filters collected by the Health and Safety Laboratory (HASL) from 1963 to 1970 at 40°S, 70°W between 10 and 16 km of altitude (Fig. 6). Similarly to what can be observed in sediment cores analysed in the current research, the 240 Pu/ 239 Pu atom ratios decreased from a GF signature (0.181 ± 0.005, 1σ) between 1963 -1966 to a lower signature (0.11 ± 0.02, 1σ) in 1969-1970, which is in good agreement with those values reported for soils and sediment cores collected at similar latitudes.

Impact of plutonium signatures to improve sediment dating.

Although not assessed in the current study, potential 137 Cs and plutonium mobility in sediment cores has to be considered through additional analyses, such as grain size, mineralogy and organic matter properties.

A plutonium isotopic signature characteristic of the high yield atmospheric tests conducted in the 1950s by the US in the Pacific Proving Ground (Fig. 1) is observed in the lower part of the Chungará sediment core with 240 Pu/ 239 Pu atom ratio of 0.252 ± 0.015 in sample CG12-20 [START_REF] Muramatsu | Measurement of 240 Pu/ 239 Pu isotopic ratios in soils from the Marshall Islands using ICP-MS[END_REF] confirmed that the sampled sequence covered the most recent period (post-1990) when land use change has dramatically modified sediment dynamics in this river catchment, with a brutal transition from "pristine" natural grassland conditions to their conversion into cropland.

Considering the absence of a Pu signature characteristic of the 1964-1965 fallout peak, we demonstrate the added value of analysing 240 Pu/ 239 Pu atom ratios in addition to 137 Cs and/or 210 Pbxs in sediment profiles collected in this region of South America. This will avoid any age model misinterpretation that may arise from an incorrect sediment dating from short-lived radionuclides (see supplementary Fig. 2) or the lack of other local knowledge (e.g., regional land use changes).

Wider implications

South America has been exposed to significant land use change since the 1960sa period that coincides with that of the main radioactive fallout [START_REF] Winkler | Global land use changes are four times greater than previously estimated[END_REF]. The conversion of natural forests and grassland into cropland may result in the degradation of terrestrial ecosystems and to an excessive sediment supply to lacustrine environments and river systems, as observed in Models converting 137 Cs measurements [START_REF] Mabit | Assessment of soil erosion and sedimentation: the role of fallout radionuclides[END_REF] into quantitative estimates of erosional and depositional rates over the past decades are based on the assumption that radionuclide fallout peaked in 1963. Based on the undisturbed soil profiles of Conceição, Brazil, we showed that 30% of Pu deposition could be attributed to FNT at this site. Recalibrating these soil erosion models for applications in mid-latitudes of South America (by shifting the fallout peak to the period 1965-1970 instead of 1963 as it is currently done) would better constrain soil redistribution rates calculated in these regions.

In addition to the analysis of 240 Pu/ 239 Pu atom ratios, 238 Pu (T1/2 = 88 years) fallout peak in 1968, resulting from the atmospheric disintegration of a nuclear auxiliary power generator in 1964 (SNAP9-A), may provide an additional time marker [START_REF] Hancock | Plutonium as a chronomarker in Australian and New Zealand sediments: a comparison with 137Cs[END_REF] to confirm the occurrence of a radioactive fallout peak in 1970 in some regions of South America as suggested in our study.

Accordingly, the current research encourages the scientific community to continue analysing Pu in general, and 240 Pu/ 239 Pu atom ratios in particular, to improve our understanding of the spatial distribution of Pu fallout and its chronology in the Southern Hemisphere. This improved knowledge will undoubtedly provide a powerful anthropogenic marker that will contribute refining the dating of sediment cores, with a focus on those processes that took place during the 1960s and the 1970s when the conversion of natural areas into agricultural land increased around the world in general, and in Southern America in particular.

Conclusions

A novel compilation of already published and new original plutonium isotopes ( 239,240 Pu) measurements conducted in lake sediments and soils allowed us to update our knowledge of the spatial and temporal distribution of artificial radioactive fallout in South America. Significantly lower 240 Pu/ 239 Pu atom ratios differing from the average global fallout signature were found in the 20-45° latitudinal band and attributed to the higher contribution of the French atmospheric nuclear weapon tests (1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974) to the deposition of plutonium in these soils and sediments. These results indicate that fallout radionuclides from French atmospheric tests should be considered to avoid any

Fig. 1 .

 1 Figures and Tables

Fig. 2 .

 2 Fig. 2. Latitudinal distribution of 240 Pu/ 239 Pu atom ratios in undisturbed soil profiles and surface soils across continental South America and Antarctica published in the literature (circle) and analysed in the current research (square). The red rectangles indicate the integrated fallout signatures in different latitudinal ranges as determined from soils collected in 1970-1971 (19), while the blue rectangle (RFNT) indicates the French fallout signature reported in previous studies. The average global fallout signature in soils (~0.18) is displayed as a red dashed line.

Fig. 3 .

 3 Fig. 3. Respective contributions of global and French fallout in South America surface soil samples and undisturbed soil profiles published in the literature and analysed in the current research. Contribution uncertainties for each sample can be found in Table 1. Soil samples from Antarctica are not displayed for clarity. The location of the Centre pour l'Expérimentation du Pacifique (C.E.P.) is shown in the inset world map.

Fig. 4 .

 4 Fig. 4. Temporal evolution of 240 Pu/ 239 Pu atom ratios (red squares) and 137 Cs activities (black dots) in sediment core collected in Lake Chungara (Chile). The French (RFNT, in blue) and global (RGF, in red) 240 Pu/ 239 Pu atom ratio fallout signatures used in the current research are provided as an indication.

Fig. 5 :

 5 Fig. 5: Temporal evolution of 240 Pu/ 239 Pu atom ratios (red squares) and 137 Cs activities (black dots) in sediment core collected in Rincón del Bonete reservoir (Uruguay). The French (RFNT, in blue) and global (RGF, in red) 240 Pu/ 239 Pu atom ratio fallout signatures used in the current research are provided as an indication. Please note the scale difference for reporting the 240 Pu/ 239 Pu atom ratios in the sediment core.

Fig. 6 .

 6 Fig. 6. 240 Pu/ 239 Pu temporal evolution in air filters collected by HASL between 1965 and 1971 at 40°S latitude (black dots, no uncertainties for these samples were provided in the initial database), plotted along with French nuclear test yields over the same period. The RFNT (in blue) and RGF (in red) fallout signatures used in the current research are provided as an indication

  . According to the 210 Pbxs age model, sample CG6-11 corresponds to sediment deposited between 1955 and 1987, thereby reflecting the main radioactive fallout period and containing fallout originating from tests conducted in both the Northern and Southern hemispheres. The 240 Pu/ 239 Pu atom ratio of 0.156 ± 0.009 determined in this sample is in perfect agreement with the signature proposed by[START_REF] Bouisset | Contribution of close-in fallout from the French atmospheric tests in inventories of Cs-137, Am-241 and plutonium (238,239,240) in Gambier Islands (French Polynesia) -Signatures of stratospheric fallout in the Southern Hemisphere[END_REF] for the Southern hemisphere (0.157 ± 0.011), including fallout from all the Nuclear Weapon States. A similar value was observed in an undisturbed soil profile collected by the EML in Lima, Peru (0.1565, 12°S), suggesting that FNT debris, although they are less abundant, can also be observed in the Northern part of South America (Fig.3). The upward circulation that occurs in the equatorial regions (known as Hardley circulation) has likely limited the deposition of FNT debris in these zones, as suggested by the 240 Pu/ 239 Pu signature of ~0.18 determined in French Guiana and soil surface samples collected in Peru in the framework of the current research. In contrast, the decreasing atom ratio observed in

	sample CG1-5 (0.127 ± 0.013) for Lake Chungará sediments deposited between 1987-2014 is very
	likely related to the higher proportion of FNT debris.
	The 240 Pu/ 239 Pu atom ratios in sediment deposited in the Rincón del Bonete Reservoir,
	although they are significantly lower than those measured in Lake Chungará, remained in
	agreement with the plutonium signatures reported for undisturbed soil profiles collected at similar
	latitudes by Kelley et al. (1999) (0.1156 ± 0.0017 and 0.1075 ± 0.0007 in Santiago and Buenos
	Aires, respectively). The CF:CS 210 Pbxs age model for this sediment core indicates sediment
	accumulation since 1990. The analysis of 240 Pu/ 239 Pu atom ratios in this lacustrine archive

  the Rincón del Bonete Reservoir. Accordingly, improving the chronology of this reconstruction is of utmost interest in this part of the world. The current research demonstrates that the analysis of240 Pu/ 239 Pu ratios in addition to that of 137 Cs and 210 Pbxs in lacustrine archives may contribute to this ultimate objective given the strong spatial and temporal variations of this ratio compiled across the continent and demonstrated in this study.Low levels of 137 Cs analysed in soils and sediment of the Southern hemisphere outline the need to find alternative chronomarkers to reconstruct soil redistribution rates and/or to cross-check and validate the dating of sediment archives. Pu is a powerful candidate to achieve this goal as it may allow further distinguishing between different time periods corresponding to contrasted nuclear testing stages or fallout originating from nuclear experimentations carried out by different countries based on the reconstruction of 240 Pu/ 239 Pu temporal evolution in sediment cores. The 240 Pu/ 239 Pu atom ratio should therefore be analysed in sediment core layers showing a single or multiple137 Cs fallout peaks to avoid any misinterpretation in the core dating and, consequently, on the chronology of the reconstruction of environmental processes that have occurred in this subcontinent in general, and in the central part of South America in particular.

Table 1 .

 1 240 Pu/ 239 Pu atom ratios in soils of South America published in the literature and analysed in the current research. a Sampling carried out in other studies and analysed for Pu isotopes by the authors; b 240 Pu/ 239 Pu atom ratios as reported in Table2fromKelley et al., 1999; c "Weighted mean of duplicate aliquots from the same sample collection" (19); * Samples analysed at the IRSN; ** Samples analysed at the CEA/DAM.

	Country	Latitude	Longitude	Study Area / Sample label	Elevation (m)	Sampling year	240 Pu/ 239 Pu	Proportion of Plutonium from French fallout	Measurement technique

Acknowledgments

Pierre-Alexis Chaboche received a PhD fellowship from the University of Versailles-Saint-Quentin-en-Yvelines (a founding member of University Paris-Saclay). Collaboration between France and Uruguay for sediment sampling is supported by an applied research project from the Fondo Maria Viñas (FMV_1_2019_1_156244) funded by the National Agency of Research and Innovation (ANII, Uruguay). The Antarctic samples were obtained thanks to a project of the Instituto Antartico Uruguayo (IAU) (2015)(2016). The authors thank the Laboratoire Souterrain de Modane (LSM) facilities for the gamma spectrometry measurements.

Data and materials availability: All data are available in the main text and/or the Supplementary Materials.

misinterpretation in sediment core dating and to refine the chronology of the reconstruction of environmental processes in this region of the world.

Competing interests:

The authors declare that they have no competing interests.


Christophe Hachon

T Seegmuller

A D'autume

P Belan

H Cremer

A Dynamic Explanation of the Crisis of the Welfare State *

Keywords: Welfare State, Indeterminacy, Education, Redistribution. JEL Codes: D72, H55, I2

Although the crisis of the Welfare State has been evoked for quite a long time, gures show that such a phenomenon has arisen only recently. Furthermore, it is not a common feature in all developed countries. This paper aims at explaining these two empirical facts. We use an overlapping generations model in which agents decide to educate themselves or not endogenously. Furthermore, at each date, the working population vote on the size of a redistributive policy. Firstly, we show that the share of the educated population can be the engine of the crisis of the Welfare State. Moreover, our paper emphasizes that the expectations of agents about the size of redistributive policies, can explain the timing dierential in the crisis of the Welfare State between developed countries.

Introduction

Since the end of the second World War, the role of governments has dramatically expended. Indeed, they set up an objective solidarity which mainly aims at insuring agents against life risks, and at reducing wealth inequalities [START_REF] Merrien | L'Etat-Providence[END_REF], Barr 2002). However, for a few years, a growing literature consider that the Welfare State is in "crisis" or that it will be soon. The Welfare State would suer from a questionable eciency, from a legitimacy crisis, and from nancial diculties [START_REF] Rosanvallon | La crise de l'Etat-providence[END_REF]. Concerning the rst point, the literature argues that the private sector provides a better alternative to public expenditures because of the improvement of its eciency [START_REF] Bergh | Explaining the Survival of the Swedish Welfare State: Maintaining Political Support Through Incremental Change[END_REF]. Moreover, the size of transfers has a signicant distorsive impact on economic decisions. Secondly, the opacity of the management of public funds would imply that citizens do not necessarily consider any further expenditure as useful. Finally, the aging of the population, the increase in the unemployment (notably in Europe), or the improvement of the quality of health services have increased the burden of taxation. Although the "crisis" of the Welfare State is detailed in a fair number of studies, empirical data do not conrm their assertions. Indeed, the changes in the size and in the structure of the Welfare State seem to be incremental [START_REF] Bergh | Explaining the Survival of the Swedish Welfare State: Maintaining Political Support Through Incremental Change[END_REF]). However, it is possible to distinguish some trends about the size of the public sector in developed countries1 (see gure 1). More specically, countries can be classied in two groups. In the rst one, the scal burden has increased or is remained stable. France, Germany, the United-Kingdom and the United-States among others, belong to this group. In the second one, the scal burden has decreased for a few years. Canada and the Netherlands describe such a trend. Consequently, even if the term "crisis" does not seem to be the most appropriate to describe the situation, it seems that some countries have questioned the size of their Welfare State. Let us now consider the case of the European Union. In gure 2, it is shown that the size of the public sector has begun to decrease for a few years. Consequently, in average, countries have begun to question the size of their government.

In this paper we propose a model which is compatible with the previous facts. We assume that tax revenue is only used to redistribute resources between socio-economic groups. Our main argument is that the evolution of the political support for redistribution can explain the dynamics of the Welfare State. Moreover, in our model, the period from which the size of public spending decreases depends on the expectations of agents. This Some papers have previously studied the endogenous determination of the size of public spending (identied with the expenditures of redistribution) using political equilibrium models [START_REF] Meltzer | A Rational Theory of the Size of Government[END_REF]Richard 1981, Persson and[START_REF] Persson | Political Economics: Explaining Economic Theory[END_REF]. In these static models, agents supply their labor endogenously, given their productivity endowment and given the size of public transfers. Moreover, it is assumed that agents vote on the size of public spending knowing their distorsive impact on labor supply. Public transfers are nanced through a tax rate on wages, and are used to nance a at benet. The equilibrium tax rate is determined by the agent endowed with the median wage. Belletini and Ceroni (2007) use the same kind of model, but they assume (i) an exogenous labor supply, and (ii) that agents can face a liquidity constraint. In that case, the median voter is not necessarily the agent endowed with the median wage. Compared to these models, we add a dynamic environment. Furthermore, as in Belletini and Ceroni (2007), we do not consider the case of an endogenous labor supply to keep the model tractable.

Some other papers have studied the linkages between growth and redistribution [START_REF] Alesina | Distributive Politics and Economic Growth[END_REF]Rodrik 1994, Persson and[START_REF] Persson | Is Inequality Harmful for Growth ?[END_REF]. In these second kind of papers, the tax rate bears on the return of savings. Consequently, the scal system has a distorsive impact on capital accumulation. These transfers are used to nance at benets for [START_REF] Persson | Is Inequality Harmful for Growth ?[END_REF], and to nance Government spending on productive activities for [START_REF] Alesina | Distributive Politics and Economic Growth[END_REF]. The common feature of these two papers is to show that more inequalities lead to a higher demand for redistribution, which increases the tax rate on capital accumulation. It implies that the growth rate of the economy decreases. These models can be considered as the rst models which include both a dynamic environment and a demand for redistribution. However, in their models, the structure of the population is constant over time and they only consider the dynamics of the income distribution. [START_REF] Perotti | Political Equilibrium, Income Distribution and Growth[END_REF] incorporates an endogenous determination of the structure of the population à la [START_REF] Galor | Income Distribution and Macroeconomics[END_REF], into a model in which the size of the social security system is endogenously determined. However, he does not study the dynamics of the structure of the population2 .

Our paper completes this literature on two signicant points. (1) These papers use static models. It implies that all changes in the size of the Welfare State come from modications of some relevant parameters such as wage inequalities. Consequently, their models cannot provide an explanation of the dynamics of the size of Government. (2) Because the structure of the population is exogenous, these models do not study the impact of the increase in the share of the educated population on the demand for redistribution. However, the dynamics of modern economies also has an impact on the structure of the population, which changes the political support for intra-generational transfers. [START_REF] Acemoglu | Technical Change, Inequality, and the Labor Market[END_REF] reported two stylized facts concerning the evolution of the structure of the population. (1) There has been a large increase in the supply of skills in the US economy over the past sixty years. (2) There has been an increase in the relative return (the wage dierential) of education. Firstly, an increase in the skill premium has a positive impact on the share of the educated population. Moreover, following an increase in the share of the educated population, the stock of knowledge in the economy raises, which increases the marginal productivity of skilled agents more than that of unskilled agents. We explore these two causalities.

In this paper, we assume an overlapping generations model in which agents live for two periods. In the rst period of their life, agents decide to educate themselves or not, while in the second one agents work and supply one unit of labor inelastically. At each period t, the working population vote on the size of the Welfare State which can take two values to simplify the model: a high and a low tax rate. Tax revenue is redistributed through at benets to workers. It implies that educated agents (who receive high wages) vote for the low tax rate, while uneducated agents (who receive low wages) vote for the high tax rate. As long as uneducated (educated) agents have the majority, the high (low) tax rate is chosen. Consequently, the structure of the population determines the size of the Welfare State in our model. Our accumulation variable is knowledge capital. The share of the educated population has a positive impact on the stock of knowledge. Ceteris paribus, a higher knowledge capital level has a positive impact on the share of the educated population because of the increase in the wage dierential between the educated and the uneducated population. Conversely, the higher the tax rate (redistribution) is, the less agents decide to educate themselves. Finally, as the tax rate of period t has an impact on the educational choices of agents born at period t-1, there can be indeterminacy. Indeed, for a reasonably high value of the wage dierential3 , if agents expect a low tax rate, then only a few of them decide to remain uneducated, and the low tax rate will actually be chosen. Conversely, if agents expect that the high tax rate will be chosen, then only a few of them decide to educate themselves, and the high tax rate is actually chosen.

This theoretical structure enables us to distinguish four relevant scenarii about the dynamics of the Welfare State. The rst one is the following. Assuming that the initial wage dierential is low, then at the rst period, only a few agents decide to educate themselves and the majority of people (uneducated agents) vote for high vertical transfers. Then, because of the knowledge accumulation, the wage dierential increases [START_REF] Acemoglu | Technical Change, Inequality, and the Labor Market[END_REF] and more agents decide to educate themselves. However, the educational decision also depends on the size of vertical transfers. Consequently, for a reasonably high value of the wage dierential, if agents expect high vertical transfers, most of them remain uneducated, and actually, uneducated agents keep the majority. But if agents expect small vertical transfers, then most of them decide to educate themselves and educated agents actually get the majority, and they vote for a small Welfare State. Finally, once the wage dierential is very high, whatever the size of vertical transfers, the share of the educated population is so high that agents know that a small Welfare State will be chosen in the next period. This scenario could describe the empirical evidence which we mentioned above. Indeed, in a rst time the size of the Welfare Sate is signicant because of a political support in favor of vertical redistribution. Then, the increase in the share of the educated population decreases the demand for vertical redistribution. Consequently, agents vote for smaller public spending. In the second scenario, the cost of education is so high that the economy reaches the low steady state before educated agents get the majority. Then, in the long run, the high tax rate is chosen, and no crisis of the Welfare State takes place. In the third scenario, we show that an economy uses a high tax rate as long as agents expect that the same tax rate will be chosen at the next period. However, once agents expect that the Welfare State will be less generous, then a larger share of population decides to educate itself and the low tax rate is chosen at steady state. Finally, in the fourth scenario, we show that there can appear cycles in the size of the Welfare State. It means that an economy can switch from high public spending to smaller one, according to the beliefs of agents. This model allows us to study the dynamics of the Welfare State and to emphasize some properties related to indeterminacy. Moreover, our model suggests that the evolution of the Welfare State, and notably the "crisis of the Welfare State", is the result of new political pressures because of the endogenous dynamics of the structure of the population. To our knowledge, the literature on the welfare state has still not considered this problem previously.

This paper is organized as follows. Section 2 presents the basic structure of the model. In section 3, we study the equilibrium and the dynamics of the economy. Section 4 provides some concluding remarks.

The Model

We assume an overlapping generations model in which agents live for two periods4 . During their rst period of life, agents decide to educate themselves or not. It is a binary choice. If an agent decides not to educate himself, then he spends the time of his rst period of life on leisure5 . The utility level of leisure is normalized to 0. Conversely, if an agent decides to educate himself, then he has to bear a psychological cost related to the learning process. In this paper, we assume that it represents the time which cannot be spent for leisure. When an agent is born, he is randomly endowed with an educational cost (θ). This cost reduces the utility level of agents who decide to educate themselves (-v(θ)).

θ necessarily belongs to the interval Ω θ = [θ, θ]. f (θ) and F (θ) denote the density function and the cumulative distribution function of θ respectively. They are such that f (θ) > 0, ∀θ ∈ Ω θ , F (θ) = 0 and F ( θ) = 1. f (θ) also denotes the fraction of the population endowed with a cost θ.

We make the following assumption concerning the function v(θ):

Assumption 1: v( θ) > 0, v(θ) is of class C 1 and v (θ) > 0, ∀θ ∈ Ω θ .
The rst part of this assumption only means that there exists a fraction of the population for whom the time spent on education is a cost. Otherwise, every agent would choose to be educated. It also implies that our model does not exclude the case in which education provides more utility than leisure for a part of the population. For these people, education is a way to blossom out. It is the case i6 v(θ) < 0.

The last part of this assumption implies that a longer time spent on the learning process reduces leisure time, and thus the utility level.

During their second period of life, every agent works and supplies one unit of labor inelastically. At period t, the wage level of an educated agent is denoted by w e t , while that of an uneducated agent is denoted by w u t . At each period t, each worker pays a proportional tax (τ t ) on his wage to nance the Welfare State. The Government uses this scal revenue to nance at transfers (b t ). All agents benet from these transfers.

Moreover, we assume that there only exists one good in the economy which is used as a numeraire.

Consumers

At period t, the budget constraint of an agent born at period t -1 can be written:

W i t = w i t (1 -τ t ) + b t (1)
with i ∈ {e, u}. At period t -1, an agent expects the wealth which he will receive when he works. The expectation bears on the wage level (w i,a t ) and on the size of the welfare state (τ a t , b a t ). Indeed, our model includes a dynamics of wages, and at each period, the size of the welfare state is endogenously chosen by the population.

Then, at period t -1 we have:

W i,a t = w i,a t (1 -τ a t ) + b a t (2)
Each agent is assumed to consume his wealth when he works. The utility level of an agent born at period t -1 can be written 7 :

U i t = -(1 -I)v(θ) + ln W i t ( 3
)
with I = 0 if an agent decides to educate himself, and I = 1 if he decides to remain uneducated. At period t -1 an agent decides to educate himself i:

U u,a t < U e,a
t . This is equivalent to:

ln W e,a t W u,a t > v(θ) (4)
The use of the log-utility function implies that educational choices depend on the wealth inequality ratio. An increase in this ratio corresponds to an increase in the opportunity cost of remaining uneducated.

Equation (4) will enable us to dene a threshold value θt such that, given the expectation of τ a t , agents for whom θ is smaller than θt decide to educate themselves, while agents endowed with a θ higher than θt decide to remain uneducated. Consequently, F ( θt ) denotes the share of the educated population.

Knowledge Capital

In our economy, the accumulation variable is knowledge capital. h t denotes the knowledge capital, i.e. the available stock of knowledge at period t for each agent. In doing so, we assume that agents (educated and uneducated) only dier by their ability to transform knowledge capital into productivity, i.e. educated and uneducated agents dier by their ability to transfrom the stock of knowledge into a useful information for the production process. The dynamics of the knowledge capital is assumed to have the following form:

h t+1 = Ψ(h t , θt )
(5) 7 We use this utility function because of the tractability of our model.

with8 Ψ 1 () > 0 and Ψ 2 () > 0. It implies that the current stock of knowledge has a positive impact on the future stock of knowledge9 . Moreover, the share of the educated population at period t has a positive impact on the knowledge capital of period t + 1 because these people do research.

The timing of the model is the following. Agents decide at period t -1 to educate themselves or not. Education consists in the acquisition of methods to transform and to improve the stock of knowledge into useful information for the production process. As for uneducated agents, their ability level is not sucient to use the stock of knowledge eciently. At period t, agents use the methods which they have acquired when they were young, given the current knowledge capital level. Moreover, educated agents develop new knowledge by research activities, which will only be available at period t + 1. It implies that there need time before this new knowledge can be used.

The previous assumptions imply that when an agent is born at period t -1, he observes the current knowledge capital h t-1 and the share of the educated population θt-1 . Consequently, he rightly expects the value h t . His only problem concerns the expectation of the size of the Welfare State of period t.

Firms

We assume that the technology of rms has the following form:

Y t = f (h t , L e t , L u t ) (6)
with f i () > 0, ∀i ∈ {1, 2, 3}. It means that the stock of knowledge and both kinds of labor have a positive impact on the production level. More specically, rms take the available stock of knowledge of the current period (h t ) as given. h t is a positive externality for each rm. However, rms choose the quantity of labor which they decide to use in the production process. We make two further assumptions about the properties of the production function: f ii () < 0, and f ij () > 0 if i = j. These assumptions notably imply a decreasing marginal productivity of both kinds of labor. Assuming a perfect competition on the nal good market and on the inputs markets, we have:

w e t = f 2 (h t , L e t , L u t ) > 0 (7) w u t = f 3 (h t , L e t , L u t ) > 0 (8)
Finally, we make the following assumption about the wages of agents:

Assumption 2: w e t > w u t and

f 21 f 2 h > f 31 f 3 h.
The rst part of this assumption implies that the wage level of educated agents is higher than the wage level of uneducated agents. Thus, it is implicitly assumed that the marginal productivity of unskilled workers is always nite, even if the quantity of unskilled workers is small. Formally, it means that f 2 (h, N, 0) > f 3 (h, N, 0), ∀h, i.e. even if all agents decide to educate themselves and if rms employ them, then the scarcity of unskilled workers is not sucient for their wage level to become higher than that of skilled workers.

The second part of this assumption implies that the elasticity of the wage of skilled agents with respect to knowledge capital is higher than the elasticity of the wage of unskilled agents with respect to knowledge capital. Ceteris paribus, it means that the increase in the stock of knowledge increases the wage inequality ratio w e /w u [START_REF] Acemoglu | Technical Change, Inequality, and the Labor Market[END_REF]. In the Acemoglu's paper (2002), the stock of knowledge is represented by a technological bias in favor of skilled workers.

Government

The Government budget constraint can be written:

N τ t wt = N b t (9)
knowing that τ t is endogenously determined at each period by a voting procedure. The political equilibrium is detailed below.

The Dynamic Equilibrium

The dynamic equilibrium of this economy is the sequence {θ t , τ t , h t } t=0,..,+∞ which satises the rst order conditions of rms (( 7) and ( 8)), the educational choices of agents (4), and the government budget constraint (9). This sequence is such that the input markets and the output one are in equilibrium at each period. Thus, at each period we have:

L e t = N F ( θt ) (10)
L u t = N (1 -F ( θt )) (11)
Before studying the dynamics of the economy, we have to specify some properties of wages in equilibrium.

Some Properties of Wages in Equilibrium

Given equations ( 10) and ( 11) the wages of educated and uneducated agents can be written:

w e (h t , θt ) = f 2 (h t , N F ( θt ), N (1 -F ( θt ))) (12)
w u (h t , θt ) = f 3 (h t , N F ( θt ), N (1 -F ( θt ))) (13)
Our assumptions on the production function imply that that the knowledge capital (h t ) has a positive impact on both w e and w u (w e 1 () > 0 and w u 1 () > 010 ). However, it can also be shown that the share of the educated population (F ( θt )) has a negative impact on the wage level of educated agents (w e 2 () < 0), and a positive one on the wages of uneducated agents (w u 2 () > 0). This result comes from the decreasing marginal productivity of labor.

The average wage of the economy can be written:

wt = F ( θt )w e (h t , θt ) + (1 -F ( θt ))w u (h t , θt ) ≡ w(h t , θt ) (14)
The function w(h t , θt ) is such that w1 () > 0. An increase in the knowledge capital level has a positive impact on the average wage of the economy because of the increase in the wages of educated and uneducated agents (with θt constant). However, the share of the educated population has an ambiguous impact on the average wage of the economy. Dierentiating equation ( 14) with respect to θt we get:

∂ w(h t , θt ) ∂ θt = f ( θt )(w e -w u ) A>0 + F ( θt ) ∂w e (h t , θt ) ∂ θt B<0 + (1 -F ( θt )) ∂w u (h t , θt ) ∂ θt C>0 (15)
Elements A and C are both positive. Element A denotes the increase in the average wage because of the increase in the share of the population having the higher wage level. Element C denotes the increase in the wage level of uneducated agents because of the bigger scarcity of this input. Finally, element B is negative because of the smaller scarcity of the educated population.

Using this preliminary analysis and equation ( 5), we can determine the share of the educated population at each period. However, this choice depends on the result of the voting procedure. Consequently, we rst study the determination of the share of the educated population and the size of transfers from the Government. Finally, we consider the dynamics of our economy, i.e. the sequence {θ t , τ t , h t } t=0,..,+∞ .

3.2

The Share of the Educated Population At period t -1, every agent for whom the following inequality is (not) satised, decides to educate himself (to remain uneducated):

ln w e t (1 -τ a t ) + τ a t wt w u t (1 -τ a t ) + τ a t wt > v(θ) (16)
If it exists, the threshold value θt is dened by:

LHS(h t , θt , τ a t ) ≡ ln w e (h t , θt )(1 -τ a t ) + τ a t w(h t , θt ) w u (h t , θt )(1 -τ a t ) + τ a t w(h t , θt ) = v( θt ) (17)
Lemma 1 (i) There exists a unique threshold value θt such that θ < θt < c i LHS(h t , θ, τ a t ) > v(θ) and LHS(h t , θ, τ a t ) < v( θ). (ii) This threshold is such that ∂ θt /∂τ a t < 0 and ∂ θt /∂h t > 0.

Proof : See the appendix of this paper.2

The rst element of the second part of lemma 2.1 shows that the size of the welfare state has a negative impact on the share of the educated population. Indeed, because the Government uses at transfers, they redistribute resources in favor of unskilled agents. Consequently, an increase in the size of the Welfare State increases the opportunity cost to educate oneself.

The second element of the part (ii) of the lemma shows that, for a given tax rate, more agents decide to educate themselves if the knowledge capital level grows. Indeed, as the knowledge capital raises, the wage inequality ratio (w e /w u ) increases because of the larger marginal impact of the knowledge capital on the wages of educated agents than on the wages of uneducated agents. Thus, the stock of knowledge increases the cost of remaining uneducated, and more agents decide to educate themselves.

Thanks to lemma 2.1, we can dene a function θ() such that:

θt = θ(h t , τ a t ) (18)
with θ1 () > 0 and θ2 () < 0.

The Political Equilibrium

In our economy, the government is only an institution which applies the decision taken by the majority of the population. Among the working population, each group of agents (educated and uneducated agents) is homogenous, thus all educated agents vote for the same tax rate. As for uneducated agents, they vote for the same tax rate which is dierent from that of educated agents.

We assume that agents have to choose between two tax rates, τ and τ , such that:

1 > τ > τ > 0.
Agents choose the tax rate which maximizes their wealth level. The wealth level of an agent of type i can be written:

W i t = w i t (1 -τ t ) + τ t wt (19)
The wealth of educated agents is a decreasing function of τ t (assumption 2), while the wealth of uneducated agents is an increasing function of τ t . It implies that educated agents vote for the lower tax rate τ , while uneducated agents vote for the higher tax rate τ .

The political equilibrium is:

τ t = τ if F ( θt ) > 1/2 τ if F ( θt ) < 1/2
We conclude from this analysis that there exists a threshold θt such that:

F ( θt ) = 1/2 (20)
As long as θ(h t , τ a t ) < θt , uneducated agents have the majority and the higher tax rate (τ) is chosen. Conversely, once θ(h t , τ a t ) > θt , educated agents have the majority and the lower tax rate τ is chosen. It implies that the expectations of agents concerning the tax rate has an inuence on the result of the vote. We dene two threshold values for the knowledge capital level: h a and h b , such that:

F ( θ(h a , τ )) = 1/2 (21) and F ( θ(h b , τ )) = 1/2 (22)
Given that θ1 () > 0 and that θ2 () < 0, it is straightforward to show that h b > h a . More specically, dierentiating equation ( 20) with respect to h t and τ a t we obtain:

dh dτ a = θ2 (h, τ a ) θ1 (h, τ a )
This equation means that the gap between h a and h b is all the more small as the ratio between the marginal impact of the tax rate and the marginal impact of the stock of knowledge is low. It means that the dierential between the elasticity of the wages of educated agents and the elasticity of the wages of uneducated agents with respect to the stock of knowledge plays a signicant role. If this dierential is high, then a small increase in h provides a strong incentive to educate oneself. Consequently, if τ highly increases, then it could be sucient that h slightly increases for educated agents to get the majority.

Moreover, if the distance between h a and h b is small, then h a and h b take low (high) values if a large fraction of the population has small (big) educational costs. It could represent some other specicities of education which are not explicitly introduced in this paper such as low monetary or cultural costs of education. It means that the educated population gets the majority for low (high) values of the stock of knowledge.

Lemma 2 sums up the result of the vote for dierent values of h t .

Lemma 2 If h t < h a , then no agent born at period t -1 expects that τ can be adopted and τ t = τ . For h t > h b , no agent expects the tax rate τ can be adopted and thus τ t = τ . For h a < h t < h b , then we have self-fullling prophecies.

For h t < h a , the knowledge capital level is so small that agents know at period t -1 that uneducated agents will have the majority at period t. Conversely, for h t > h b , the knowledge capital level is so high that agents know at period t-1 that educated agents will have the majority at period t. For the intermediate case (h a < h t < h b ), if agents expect that the low (high) tax rate will be chosen, then a large (small) share of the population will educate itself and educated agents will (not) actually have the majority.

3.4

The Dynamics

The stock of knowledge of period t only depends on the share of the educated population of period t -1 and on the knowledge capital level h t-1 (equation ( 5)). Assuming that both can be observed without cost, agents know at period t -1 the value of the knowledge capital of period t (h t ). However, agents born at period t -1 expect the result of the vote of period t. This expectation has an impact on the share of the educated population at period t, and thus, it has an impact on the knowledge capital of period t + 1.

With equation ( 18), equation ( 5) can be rewritten:

h t+1 = Ψ(h t , θ(h t , τ a t )) ≡ Φ(h t , τ a t ) (23)
Using the properties of the function Ψ(), we have Φ 1 () > 0, i.e. a higher knowledge capital level at period t has a direct positive impact on h t+1 , and an indirect impact through the share of the educated population of period t. Given Lemma 2.2, the dynamics of our economy is described by:

h t+1 = Φ(h t , τ ) if h t < h a , or if h a < h t < h b and τ a t = τ Φ(h t , τ ) if h t > h b , or if h a < h t < h b and τ a t = τ
h t+1 is directly inuenced by choices made at period t-1. Indeed, the share of the educated population of period t is determined at period t -1. However, this decision depends on the expected tax rate of period t. Consequently, for low (high) values of knowledge capital, agents know that the high (low) tax rate will be chosen at period t, and only a few (most) people decide to educate themselves, which has a negative (positive) impact on h t+1 . For h a < h t < h b , if agents expect the tax rate τ to be chosen at period t, then only a few people decide to educate themselves, which implies that uneducated agents will actually have the majority. At period t -1, the expectation of a high tax rate at period t has a negative impact on knowledge capital of period t + 1.

Assumption 3: Φ 11 (h t , τ a t ) < 0 and Φ(0, τ a t ) ≥ 0.
The rst part of this assumption means that the function Φ() is a strictly concave function of h t , which is a sucient condition for the stability of the steady state. The second part of this assumption implies that there exists a unique non-trivial steady state for knowledge capital. Let us call a potential steady state, a steady state which should be obtained if an economy kept the same tax rate τ at every period. h and h denote the steady state knowledge capital levels for the tax rates τ and τ respectively. Given assumption 4 and the properties of the functions Φ() and θ(), it is straightforward to show

6 - h t+1 h t h h Φ(h t , τ ) Φ(h t , τ )
Figure 3: Potential steady states that h and h are stable, and that h > h. This result comes from the negative impact of redistributive policies on the share of the educated population (see gure 3).

The dynamics of our economy is not trivial and we have to consider some cases according to the respective positions of h, h, h a and h b11 . We assume that the initial value of knowledge capital h 0 is given. The rst generation of agents born at period -1, decide to educate themselves or not, expecting the values taken by h 0 and by τ a 0 .

Case 1: h > h b and h > h b

Figure 4 illustrates that case, which occurs as long as the gap between h a and h b is not too high and if a large share of the population has small educational costs. The gap between h a and h b is not too high if the knowledge capital has a dramatic impact on wage inequalities. Moreover, educational costs are small if there are no monetary or cultural barriers too signicant concerning the access to education.

In this rst case, the economy cannot converge towards the low steady state h, because at that point, educated agents have the majority and the tax rate τ is chosen. A large

6 - h t+1 h t h h a h b h 0 Figure 4: Illustration of case 1
Welfare State is only a temporary equilibrium. Let us consider that the initial value of knowledge capital is small (h 0 < h a ). Figure 4 gives two examples of trajectories. h 0 is so small that agents born at period -1 know that the high tax rate will be chosen at period 0. However, given the value h 1 , agents born at period 0 can have two kinds of expectations. They can either expect that τ 1 will be high and in that case the knowledge capital follows the low trajectory; or they can expect that τ 1 will be low and the knowledge capital follows the high trajectory. In the long-run, whatever the initial value of h, the economy will have a small Welfare State τ and knowledge capital will converge towards h. Nevertheless, self-fullling prophecies can change the dynamics of the economy and the date from which a small Welfare State is adopted12 . This case emphasizes that the size of the Government of near countries can follow dierent trajectories according to the beliefs of agents. Then, the "crisis of the Welfare State" can appear at dierent dates for countries having near initial characteristics.

Case 2: h a > h and h a > h The low access to education implies that the wage dierential has to be very high for a large share of agents decide to educate themselves. Consequently, the economy reaches the low steady state h before the educated population gets the majority. A wide Welfare State is chosen at all periods. This case could explain why some countries do not seem to question the size of their Welfare State, and go on to use a high tax level (see gure 1).

In the two previous cases, the technology of the production process is such that the knowledge capital has a signicant impact on the wage gap, and then on educational choices. Let us remove this assumption. It implies that the redistributive policy has henceforth a dramatic impact on educational choices. More formally, the distance between the thresholds h a and h b becomes high. signicant but not too high. Moreover, the value h b above which agents know that the low tax rate will be chosen at the next period, is smaller than the potential steady state of the knowledge capital. It means that the technology of the knowledge accumulation is very ecient.

In gure 6 we have represented the dynamics of an economy whose initial knowledge capital (h 0 ) is very low. In the long-run, the economy can converge towards the low steady state h such that a majority of the population votes for a high Welfare State. The economy can remain in this situation as long as expectations are stable. However, if a generation expects that a low tax rate will be chosen at the next period, then only a few agents decide not educate themselves, and the educated population actually gets the majority. Consequently, the crisis of the Welfare State occurs only if agents expect that it will do. Using this analysis, countries of gure 1 can dier in the long-run only because of dierent expectations. Compared to the rst case, here the expectations aect the steady state of the economy. policies than on the elasticity of wages with respect to the stock of knowledge. Moreover, the technology of the accumulation of the knowledge capital is not very ecient because the highest steady state (h) is smaller than h b . In this specic context the economy can switch from one steady state to the other according to the beliefs of agents about the size of the Welfare State. For example, let us assume that h 0 is suciently small for agents to expect a large Welfare State to be adopted at period 0. If this belief endures for every following generation, knowledge capital will converge towards h which is associated with a large social security system. But, once at steady state, if the beliefs of agents change and if young agents expect that the Welfare State will be less generous, then a large part of the population decides to educate itself and τ will actually be chosen. If this belief endures, the economy will converge towards h with a less redistributive Welfare State.

In the same way, once the economy reaches this new steady state, if some generations of agents believe that a large welfare state will be chosen, the economy will come back to the steady state h.

For the moment, the existence of the Welfare State is too recent to invalidate or to conrm the plausibility of such a case. So, it might be read as a theoretical result about the cycles in a political economy model with indeterminacy.

Conclusion and Discussion

In this paper we have tried to provide some explanations about the dierences observed in the dynamics of the size of the Welfare States of developed countries (gure 1)14 . We notably explain why some countries keep a high level of public spending, whereas others have begun to reduce the scal burden of their Government. Our main argument is that the demand for redistribution can be engine of the dynamics of the Welfare Sate. It implies that the structure of the population, which inuences the demand for redistribution, plays a main role to study the previous empirical facts. We have shown that some scenarii can be considered. In case 1, the beliefs of agents can change the period from which the size of public spending decreases. In case 2, the steady state value of the knowledge capital is too small for educated agents get the majority. Then, a wide Welfare State is chosen at all periods. In case 3, the beliefs of agents can change both the dynamics and the steady state of the economy. Indeed, an economy keeps a wide tax rate as long as agents expect that such a situation will remain the same at the next period. However, once a generation expect that a change will occur, then the economy adopts a low Welfare State. In case 4, we show that that there can appear cycles in the size of public spending.

This paper emphasizes that the dynamics of the size of a Government does not only depend on objective factors such as the wage inequality ratio,... but that it also depends on subjective factors such as the beliefs of agents. We have shown that it can have a signicant impact on the result of the voting procedure. It also proves the importance of the public debate and of the diusion of a belief. Some agents, whose educational costs are low, have an incentive to convince every others that the Welfare State will be less generous at the next period, for it actually to be the case. This role played by subjective factors has already been studied in a few papers. In Alesina and Angeletos (2005), the beliefs of agents about the fairness of the economic system determines the demand for redistribution. Bisin andVerdier (2000, 2002) also use the indeterminacy property in a cultural transmission model in which agents vote on the size of the Welfare State. In this model, there are two kinds of agents: those who like a wide Welfare State, and those who dislike it. Parents prefer that their children have the same preferences as they, and parents have to make an eort for this is the case. Parents also take into account the result of the vote which will occur when their children will be in the working population. Bisin andVerdier (2000, 2002) show that there can be indeterminacy in the initial periods, but that, in the long-run, an economy converges either towards a low Welfare State, or towards a high Welfare State.

Finally, this paper has only considered one aspect of the Welfare State: the vertical redistribution. Nevertheless, it also provides insurance against life risks, public spending in infrastructure,.... Our story is only one way to consider the role of the Welfare State and its dynamics. These elements should be considered to tell the complete story of the dynamics of Welfare States.

APPENDIX

This appendix provides the proof of lemma 2.1.

(i) The RHS is a strictly increasing function of θt by assumption (see assumption 1). There only remains to proove that the LHS is a decreasing function of θt .

Dierentiating the ratio of the LHS of equation ( 17), we obtain an expression of the following form: Dierentiating the ratio of the LHS of equation ( 17) with respect to h t , we obtain an expression of the following form: Using equations ( 14) we obtain:

(1 -τ a ) ∂w e ∂h w u -∂w u ∂h w e w u (h, θ)(1 -τ a ) + τ a w(h, θ)

2 > 0
The numerator is positive under assumption 2 and if τ a < 1. As we assume that τ cannot be equal to 1 then this condition is satised.

Figure 1 :

 1 Figure 1: Tax Revenue as percentage of GDP for some developed countries. Source: OECD.

Figure 2 :

 2 Figure 2: Tax Revenue as percentage of GDP in the European Union (19 countries) between 1990 and 2007. Source: OECD. The line is get using the command Smoothing-Spline plot in S-Plus.

Figure 5 Figure 5 :

 55 Figure5illustrates that case, which occurs as long as the gap between h a and h b is not

Figure 6 :

 6 Figure 6 illustrates that case. It occurs as long as the distance between h a and h b is

Case 4 :

 4 Figure7illustrates that case, which occurs as long as the gap between h a and h b is high, i.e. if the share of the educated population depends more on the redistributive

Figure 7 :

 7 Figure 7: Illustration of case 4

  ∂w e ∂ θ (1 -τ a )(w u (1 -τ a ) + τ a w) -∂w u ∂ θ (1 -τ a )(w e (1 -τ a ) + τ a w) + τ a (1 -τ a )(w u -w e ) ∂ w ∂ θ w u (h, θ)(1 -τ a ) + τ a w(h, θ) 2Using equations (14) and (15) this expression becomes:(1 -τ a )w u ∂w e ∂ θ -∂wu ∂ θ (1 -τ a )(w e (1 -τ a ) + τ a w) w u (h, θ)(1 -τ a ) + τ a w(h, θ) 2 <0 +τ a (1 -τ a )(w u -w e )[f ( θ)(w e -w u ) + (1 -F ( θ)) ∂w u ∂ θ ] w u (h, θ)(1 -τ a ) + τ a w(h, θ) 2 <0 (ii)The LHS is a strictly decreasing function of τ a t .

  ∂w e ∂h (1 -τ a )(w u (1 -τ a ) + τ a w) -∂w u ∂h (1 -τ a )(w e (1 -τ a ) + τ a w) + ∂ w ∂h τ a (1 -τ a )(w u -w e ) w u (h, θ)(1 -τ a ) + τ a w(h, θ) 2

In this paper the size of the public sector is represented by the share of the tax revenue in the GDP.

Only Bisin and Verdier (2000, 2002) consider a vote on public spending and an endogenous structure of the population in a cultural transmission model.

We dene more precisely in the model what the term "reasonnably high" means.

The size of generations does not matter in our model because there are no inter-generational transfers. Consequently, the size of each generation is assumed to be constant over time, and is denoted by N .

The length of each period is normalized to 1. We assume that education only has a leisure cost.

In this paper "i" means "if and only if".

g i denotes the derivative of g() with respect to its ith argument. g ij denotes the derivative of g i with respect to its jth argument.

This is a standard assumption in the human capital literature.

Some simple calculations imply the result.

We do not study all cases but only the ones which we consider to be the most relevant.

As inBisin and Verdier (2000, 2002), the beliefs of agents matter to study the dynamics of our economy. However, we assume that there is no coordination problem, i.e. agents share the same beliefs about the future policy.

Let us recall that this is the case as long as the knowledge capital has a signicant impact on the wage gap between educated and uneducated agents.

Our main objective was not to explain the initial dierences in the level of taxes which has already been widely studied in the literature.