Viet-Anh Ha

George Volonakis

Hyungjun Lee

Marios Zacharias

Feliciano Giustino
email: fgiustino@oden.utexas.edu

Quasiparticle Band Structure and Phonon-induced Band Gap Renormalization of the Lead-free Halide Double Perovskite Cs 2 InAgCl 6

The lead-free halide double perovskite Cs 2 InAgCl 6 was recently designed in silico and subsequently synthesized in the lab. This perovskite is a wide-gap semiconductor with a direct band gap, and exhibits extraordinary photoluminescence in the visible range upon Na doping. The light emission properties of Cs 2 InAgCl 6 have successfully been exploited to fabricate stable single-emitter-based white light LEDs with near unity quantum efficiency. An intriguing puzzle in the photophysics of this compound is that the onset of optical absorption is around 3 eV, but the luminescence peak is found around 2 eV. As a first step toward elucidating this mismatch and clarifying the atomic scale mechanisms underpinning the observed luminescence, here we report a detailed investigation of the quasiparticle band structure of Cs 2 InAgCl 6 as well as the phonon-induced renormalization of the band structure. We perform calculations of bang gaps and effective masses using the GW method, and we calculate the phononinduced band structure renormalization using the special displacement method. We find that GW calculations are rather sensitive to the functional used in the density functional theory calculations, and that self-consistency on the eigenvalues is necessary to achieve quantitative agreement with experiments. Our most accurate band gap at room temperature is in the range 3.1-3.2 eV, and includes a phonon-induced gap renormalization of 0.2 eV. By computing the phonon-induced mass enhancement, we find that the electron carriers are in the weak polaronic coupling regime, while hole carriers are in the intermediate coupling regime as a result of the localized and directional nature of the Ag e g 4d states at the valence band top.

Introduction

The ABX 3 perovskite lattice is one of the most commonly occurring crystal structures. [START_REF] Filip | The geometric blueprint of perovskites[END_REF] Perovskites and perovskite-related materials find applications as ferroelectrics, 2,3 ferromagnets, [START_REF] Loudon | Charge-ordered ferromagnetic phase in La 0.5 Ca 0.5 MnO 3[END_REF] multiferroics, [START_REF] Tokunaga | Composite domain walls in a multiferroic perovskite ferrite[END_REF][START_REF] Mundy | Atomically engineered ferroic layers yield a room-temperature magnetoelectric multiferroic[END_REF] semiconductors, [START_REF] Xing | Long-Range Balanced Electron-and Hole-Transport Lengths in Organic-Inorganic CH 3 NH 3 PbI 3[END_REF][START_REF] Lin | Metal-Halide Perovskite Transistors for Printed Electronics: Challenges and Opportunities[END_REF][START_REF] Gedda | Ruddlesden-Popper-Phase Hybrid Halide Perovskite/Small-Molecule Organic Blend Memory Transistors[END_REF] superconductors, [START_REF] Bednorz | Possible high T c superconductivity in the Ba-La-Cu-O system[END_REF][START_REF] Murphy | New superconducting cuprate perovskites[END_REF] and solar absorbers. [START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites[END_REF][START_REF] Kim | Lead Iodide Perovskite Sensitized All-Solid-State Submicron Thin Film Mesoscopic Solar Cell with Efficiency Exceeding 9%[END_REF][START_REF] Green | The emergence of perovskite solar cells[END_REF][START_REF] Stranks | Metal-halide perovskites for photovoltaic and lightemitting devices[END_REF][START_REF] Eperon | Perovskite-perovskite tandem photovoltaics with optimized band gaps[END_REF][START_REF] Jung | Efficient, stable and scalable perovskite solar cells using poly(3hexylthiophene)[END_REF] Among the many families of perovskites, lead-based halide perovskites have attracted considerable interest owing to their extraordinary optoelectronic properties, leading to solar cells with energy conversion efficiencies as high as 25.5%. [START_REF] Jung | Efficient, stable and scalable perovskite solar cells using poly(3hexylthiophene)[END_REF][START_REF]Best Research-Cell Efficiencies[END_REF] Furthermore, the presence of a potentially toxic element in these compounds and their tendency to degrade in humid environments [START_REF] Wang | A Review of Perovskites Solar Cell Stability[END_REF][START_REF] Boyd | Understanding Degradation Mechanisms and Improving Stability of Perovskite Photovoltaics[END_REF] prompted the search for alternative, lead-free materials with the perovskite structure. [START_REF] Giustino | Toward Lead-Free Perovskite Solar Cells[END_REF][START_REF] De Angelis | The Prospect of Lead-Free Perovskite Photovoltaics[END_REF] One class of lead-free perovskites that has gained significant attention are the ordered halide double perovskites (HDP) or 'elpasolites'. [START_REF] Giustino | Toward Lead-Free Perovskite Solar Cells[END_REF] The first HDPs with band gaps in the visible range, Cs 2 BiAgCl 6 and Cs 2 BiAgBr 6 , have been designed and synthesized in early 2016. [START_REF] Slavney | A Bismuth-Halide Double Perovskite with Long Carrier Recombination Lifetime for Photovoltaic Applications[END_REF][START_REF] Mcclure | Cs 2 AgBiX 6 (X = Br, Cl) -New Visible Light Absorbing, Lead-Free Halide Perovskite Semiconductors[END_REF][START_REF] Volonakis | Lead-Free Halide Double Perovskites via Heterovalent Substitution of Noble Metals[END_REF] Subsequent work along this direction led to the discovery of additional new HDPs such as Cs 2 SbAgCl 6 , Cs 2 SbAgBr 6 , Cs 2 InAgCl 6 , Cs 2 AgTlCl 6 , Cs 2 AgTlBr 6 , and even the oxide Ba 2 AgIO 6 . [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF][START_REF] Volonakis | Oxide Analogs of Halide Perovskites and the New Semiconductor Ba 2 AgIO 6[END_REF][START_REF] Tran | Designing indirect-direct bandgap transitions in double perovskites[END_REF][START_REF] Slavney | Defect-Induced Band-Edge Reconstruction of a Bismuth-Halide Double Perovskite for Visible-Light Absorption[END_REF][START_REF] Slavney | Small-Band-Gap Halide Double Perovskites[END_REF][START_REF] Wei | Enhanced visible light absorption for lead-free double perovskite Cs 2 AgSbBr 6[END_REF] Many of these new compounds have since been employed in a range of applications, including solar cells, photocatalysts, light emitters, and photodetectors. [START_REF] Greul | Highly stable, phase pure Cs 2 AgBiBr 6 double perovskite thin films for optoelectronic applications[END_REF][START_REF] Luo | Cs 2 AgInCl 6 Double Perovskite Single Crystals: Parity Forbidden Transitions and Their Application For Sensitive and Fast UV Photodetectors[END_REF][START_REF] Wu | Highly Efficient and Stable Self-Powered Ultraviolet and Deep-Blue Photodetector Based on Cs 2 AgBiBr 6 /SnO 2 Heterojunction[END_REF][START_REF] Gao | High-Quality Cs 2 AgBiBr 6 Double Perovskite Film for Lead-Free Inverted Planar Heterojunction Solar Cells with 2.2% Efficiency[END_REF][START_REF] Igbari | Composition Stoichiometry of Cs 2 AgBiBr 6 Films for Highly Efficient Lead-Free Perovskite Solar Cells[END_REF][START_REF] Longo | Understanding the Performance-Limiting Factors of Cs 2 AgBiBr 6 Double-Perovskite Solar Cells[END_REF][START_REF] Yang | Recent Advances in Cs 2 AgBiBr 6 -Based Halide Double Perovskites as Lead-Free and Inorganic Light Absorbers for Perovskite Solar Cells[END_REF][START_REF] Kung | Lead-Free Double Perovskites for Perovskite Solar Cells[END_REF][START_REF] Pan | Cs 2 AgBiBr 6 single-crystal X-ray detectors with a low detection limit[END_REF][START_REF] Zhou | Synthesis and Photocatalytic Application of Stable Lead-Free Cs 2 AgBiBr 6 Perovskite Nanocrystals[END_REF][START_REF] Yang | Colloidal Synthesis and Charge-Carrier Dynamics of Cs 2 AgSb 1-y Bi y X 6 (X: Br, Cl; 0 <= y<=1) Double Perovskite Nanocrystals[END_REF][START_REF] Keshavarz | Tuning the Structural and Optoelectronic Properties of Cs 2 AgBiBr 6 Double-Perovskite Single Crystals through Alkali-Metal Substitution[END_REF][START_REF] Wu | The Dawn of Lead-Free Perovskite Solar Cell: Highly Stable Double Perovskite Cs 2 AgBiBr 6 Film[END_REF][START_REF] Yang | Stable, High-Sensitivity and Fast-Response Photodetectors Based on Lead-Free Cs 2 AgBiBr 6 Double Perovskite Films[END_REF] Of these HDPs, Cs 2 InAgCl 6 [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] turned out to be an exceptionally efficient emitter of warm white light. [START_REF] Luo | Efficient and stable emission of warm-white light from lead-free halide double perovskites[END_REF] At room temperature Cs 2 InAgCl 6 crystallizes in a face-centered cubic lattice with space group F m3m. The unit cell contains 10 atoms arranged in two corner-sharing octahedra centered on the Ag and In cations, respectively, with the Cl anions at the vertices and the Cs cation in the cuboctahedral cages, as shown in Fig. 1. The first optical measurements on this compound were performed on powder samples, and revealed an absorption onset at 380 nm (3.26 eV) and a broad photoluminescence (PL) peak centered at 608 nm (2.04 eV)

with full width at half maximum (FWHM) of 0.4 eV. [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] These observations were subsequently confirmed by UV-Vis spectra on single-crystal samples. [START_REF] Zhou | Composition design, optical gap and stability investigations of lead-free halide double perovskite Cs 2 AgInCl 6[END_REF] In Ref. 46 it was found that this PL signal can dramatically be enhanced by doping Cs 2 InAgCl 6 with Na. This discovery enabled the fabrication of single-emitter-based white-light LEDs with a record-high quantum yield of 86% and lifetimes up to 1000 hours. [START_REF] Luo | Efficient and stable emission of warm-white light from lead-free halide double perovskites[END_REF] Further work confirmed that Na and/or Bi doping lead to strong and broad PL, [START_REF] Luo | Efficient and stable emission of warm-white light from lead-free halide double perovskites[END_REF][START_REF] Li | Self-Trapped Excitons in All-Inorganic Halide Perovskites: Fundamentals, Status, and Potential Applications[END_REF][START_REF] Locardi | Emissive Bi-Doped Double Perovskite Cs 2 Ag 1-x Na x InCl 6 Nanocrystals[END_REF][START_REF] Liu | Design Optimization of Lead-Free Perovskite Cs 2 AgInCl 6 :Bi Nanocrystals with 11.4% Photoluminescence Quantum Yield[END_REF][START_REF] Manna | Tunable and Stable White Light Emission in Bi 3+ -Alloyed Cs 2 AgInCl 6 Double Perovskite Nanocrystals[END_REF][START_REF] Li | Ultrastable Lead-Free Double Perovskite Warm-White Light-Emitting Devices with a Lifetime above 1000 Hours[END_REF][START_REF] Schade | Crystallographic, Optical, and Electronic Properties of the Cs 2 AgBi 1-x In x Br 6 Double Perovskite: Understanding the Fundamental Photovoltaic Efficiency Challenges[END_REF] thus establishing Cs 2 InAgCl 6 as a new HDP with exceptional promise for LED applications.

Based on the large mismatch between the optical absorption onset (> 3 eV) and the PL peak (∼2 eV), and the observation of photo-chromism, Ref. 26 tentatively attributed the PL signal to the formation of photo-induced defects. On the other hand, Ref. 46 proposed that the PL originates from the formation of self-trapped excitons driven by strong electronphonon coupling in the form of Jahn-Teller distortions. A definitive assignment is further complicated by the fact that the magnitude of the band gap is not fully settled. For example, ab initio calculations using hybrid HSE or PBE0 functionals yield gaps ranging from 2.1 to 3.3 eV, [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] while GW calculations indicate gaps ranging from 3.1 to 3.3 eV. [START_REF] Luo | Efficient and stable emission of warm-white light from lead-free halide double perovskites[END_REF][START_REF] Leppert | Towards predictive band gaps for halide perovskites: Lessons from one-shot and eigenvalue self-consistent GW[END_REF] All these previous calculations do not include the effect of electron-phonon coupling, which could significantly renormalize the band gap. On the experimental front the optical gap is better understood, but quantitative differences remain. For example, UV-Vis spectra from different research groups yield optical gaps between 3.17 and 3.53 eV depending on sample morphology. [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF][START_REF] Tran | Designing indirect-direct bandgap transitions in double perovskites[END_REF][START_REF] Luo | Cs 2 AgInCl 6 Double Perovskite Single Crystals: Parity Forbidden Transitions and Their Application For Sensitive and Fast UV Photodetectors[END_REF][START_REF] Zhou | Composition design, optical gap and stability investigations of lead-free halide double perovskite Cs 2 AgInCl 6[END_REF][START_REF] Locardi | Emissive Bi-Doped Double Perovskite Cs 2 Ag 1-x Na x InCl 6 Nanocrystals[END_REF][START_REF] Manna | Tunable and Stable White Light Emission in Bi 3+ -Alloyed Cs 2 AgInCl 6 Double Perovskite Nanocrystals[END_REF][START_REF] Manna | Lattice Dynamics and Electron-Phonon Coupling in Lead-Free Cs 2 AgIn 1-x Bi x Cl 6 Double Perovskite Nanocrystals[END_REF][START_REF] Lee | Colloidal Synthesis of Lead-Free Silver-Indium Double-Perovskite Cs 2 AgInCl 6 Nanocrystals and Their Doping with Lanthanide Ions[END_REF][START_REF] Liu | Ag/In lead-free double perovskites[END_REF] In this work, as a first step toward elucidating the origin of the strong PL in Cs 2 InAgCl 6

and related materials, we focus on the accurate determination of the band structure of this compound by employing GW calculations and including the electron-phonon renormalization of the bands via the special displacement method. [START_REF] Zacharias | One-shot calculation of temperature-dependent optical spectra and phonon-induced band-gap renormalization[END_REF][START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF] In particular, we carry out a thorough sensitivity analysis of GW quasiparticle band gaps in relation to the density functional theory (DFT) starting point and the use of self-consistency on the eigenvalues. We determine accurate band structures and effective masses by means of Wannier interpolation, and we calculate the electron-phonon renormalization of the band gap and effective masses using the special displacement method.

The manuscript is organized as follows: In Sec. 3 we briefly review the theoretical and computational approach employed in this work, from the GW method to the special displacement method. The computational setup, including details on software and convergence tests, is described in Sec. 4. Section 5 presents our main results for the band gap, band structures, effective masses, and phonon-induced renormalization of the bands. In this section we also compare our results to experiments. In Sec. 6 we draw our conclusions and discuss the possible origin of the visible PL in Cs 2 InAgCl 6 .

3 Overview of methodology

GW calculations

The GW method [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF][START_REF] Aryasetiawan | The GW method[END_REF][START_REF] Golze | The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy[END_REF] constitutes the most reliable parameter-free approach for computing quasiparticle band structures in weakly-to moderately-correlated solids. In this method, quasiparticle eigenvalues including nonlocal exchange and dynamical correlation are obtained by replacing the DFT exchange and correlation potential V xc with the many-body self energy Σ = GW . [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF] In this symbolic expression, G is the electron Green's function, and is evaluated using DFT Kohn-Sham wavefunctions and eigenvalues, and W is the screened Coulomb interaction, which is evaluated within the random-phase approximation (RPA) using once again DFT wavefunctions and eigenvalues. The quasiparticle energy E nk is calculated perturbatively as

E nk = ε nk + Z nk ψ nk |Σ(ε nk ) -V xc |ψ nk
, where ε nk is the DFT eigenvalue corresponding to the electronic band n and wavevector k, and ψ nk is the associated wavefunction. Z nk is the quasiparticle renormalization, and the self-energy is evaluated at the unperturbed DFT energy, Σ(ε nk ). [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF][START_REF] Golze | The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy[END_REF] The above approach works best when the quasiparticle correction to DFT eigenvalues is relatively small. For large corrections, this method suffers from the sensitivity of the selfenergy to the DFT band structure and wavefunctions. [START_REF] Rinke | Combining GW calculations with exact-exchange density-functional theory: an analysis of valenceband photoemission for compound semiconductors[END_REF][START_REF] Fuchs | Quasiparticle band structure based on a generalized Kohn-Sham scheme[END_REF][START_REF] Rinke | Exciting prospects for solids: Exact-exchange based functionals meet quasiparticle energy calculations[END_REF][START_REF] Jiang | First-principles modeling of localized d states with the GW @LDA + U approach[END_REF][START_REF] Waroquiers | Band widths and gaps from the Tran-Blaha functional: Comparison with many-body perturbation theory[END_REF][START_REF] Van Setten | Automation methodologies and large-scale validation for GW : Towards high-throughput GW calculations[END_REF] In order to reduce this sensitivity, it is common to perform a few iterations where the eigenvalues used to calculate the selfenergy at each iteration are taken from the result of the previous iteration. [START_REF] Hybertsen | Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies[END_REF] This approach has been used for a variety of systems, ranging from molecules to semiconductors. [START_REF] Blase | First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications[END_REF][START_REF] Faber | First-principles GW calculations for DNA and RNA nucleobases[END_REF][START_REF] Blase | Charge-transfer excitations in molecular donor-acceptor complexes within the many-body Bethe-Salpeter approach[END_REF][START_REF] Marom | Benchmark of GW methods for azabenzenes[END_REF][START_REF] Wilhelm | GW in the Gaussian and Plane Waves Scheme with Application to Linear Acenes[END_REF][START_REF] Shishkin | Self-consistent GW calculations for semiconductors and insulators[END_REF] A simpler self-consistent procedure consists of updating a scissor correction instead of all individual eigenvalues. 100,101 The advantage of the self-consistent scissor is that the correction is well defined for all empty bands, and there is no need for extrapolations as in the case of eigenvalue-self-consistency. The scissor operator is simply defined as k ∆ k P c k , where P c k is the projector on the empty states with wavevector k, and ∆ k is a k-dependent scissor correction. In the presence of a scissor correction, the quasiparticle eigenvalues are evaluated as

E nk = ε nk + Z nk ψ nk |Σ(ε nk ) -V xc -∆ k P c k |ψ nk .
For completeness, in this work we also perform calculations using a k-independent scissor correction evaluated at the direct gap. Self-consistent-scissor GW calculations proceed in the same way as for eigenvalue-selfconsistency: the scissor correction determined at a given iteration is employed as a starting point for the evaluation of G and W in the next iteration, until convergence is achieved. For comparison, we also perform band structure calculations using the HSE functional, which has been very successful in calculations of extended solids. 102,103 GW and HSE calculations are notoriously expensive. In order to calculate band structures along high-symmetry lines and evaluate the derivatives needed for the effective masses, we employ Wannier interpolation. 104,105 In particular, we perform GW and HSE calculations on coarse Brillouin zone grids, and we interpolate the eigenvalues using maximally localized Wannier functions. 106 We evaluate effective masses by taking second order derivatives of the eigenvalues via finite differences. In particular, we evaluate 1/m αβ = h-2 ∂ 2 E nk /∂k α ∂k β , where h is Planck's constant and α, β indicate Cartesian directions. We evaluate the masses using second-order mixed finite difference formulas; the values reported below are the eigenvalues of the effective mass tensors.

Special displacement calculations

To investigate the phonon-induced renormalization of the band structure of Cs 2 InAgCl 6 , we employ the special displacement method. [START_REF] Zacharias | One-shot calculation of temperature-dependent optical spectra and phonon-induced band-gap renormalization[END_REF][START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF] This method consists of performing a band structure calculation for a supercell where the atoms have been displaced from their equilibrium sites. The displacements are chosen so as to reproduce the thermal mean square displacements including quantum zero-point fluctuations within a single supercell calculation. The special set of displacements is referred to as the ZG (Zacharias-Giustino) displacement, and it is understood as the mean-value point of the multi-dimensional Williams-Lax integral. [START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF] This method was successfully employed to calculate temperature-dependent band structure renormalization in numerous semiconductors and insulators. [107][108][109][110][111] The ZG displacement is given by:

∆τ κα (R) = q,ν S qν h 2N M κ ω qν (2n qν,T + 1) 1 2 2 Re e iq•R e κα,ν (q) , (1)
where ∆τ κα (R) indicates the displacement of the atom κ with mass M κ in the unit cell with lattice vector R, and N is the number of unit cells in the supercell. e κα,ν (q) is the polarization vector of the phonon (normalized within the unit cell) with wavevector q, branch index ν, frequency ω qν , and Bose-Einstein occupation n qν,T at the temperature T . The quantities S qν are signs (±1) determined so as to yield the correct quantum canonical average of the displacements in the thermodynamic limit of an infinitely-extended supercell. The summation in Eq. ( 1) is restricted to q-vectors that are neither time-reversal invariant nor time-reversal partner of another vector in the sum. The phonon frequencies and eigenvectors needed in Eq. ( 1) are obtained via density-functional perturbation theory. 112

To obtain the electron spectral function in the presence of electron-phonon interactions, we unfold the band structure calculated for the supercell with the ZG displacement. [START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF]113 The resulting bands include electron-phonon renormalization to second order in the atomic displacements as in the Allen-Heine theory, 114 as well as all higher orders of electron-phonon coupling and off-diagonal Debye-Waller corrections. [START_REF] Giustino | Electron-phonon interactions from first principles[END_REF]115 4 Computational setup and convergence tests

All DFT calculations were performed using planewaves and pseudopotentials, as implemented in the Quantum ESPRESSO suite. 116 We considered three exchange and correlation functionals: LDA, 117 PBE, 118 and PBEsol. 119 We employed optimized norm-conserving

Vanderbilt pseudo-potentials (ONCVPSP) 120 from the PseudoDojo library. 121 The valence configurations for Cs, Ag, and Cl are 5s 2 p 6 6s 1 , 4s 2 p 6 d 10 5s 1 , and 3s 2 p 5 , respectively. For In the library offers pseudopotentials with semicore states, 4s 2 p 6 d 10 5s 2 p 1 , and without semicore states, 4d 10 5s 2 p 1 . We tested both options to investigate the sensitivity of GW calculations to semicore electrons. Following the PseudoDojo recommendations we used a stringent kinetic energy cutoff of 100 Ry for all calculations. To confirm that this cutoff is sufficient for Cs 2 InAgCl 6 , we also tested 120, 140, and 160 Ry, and we found that 100 Ry yields lattice parameters and GW shifts within 10 -4 Å and 1 meV of the most accurate calculation at 160 Ry.

Structural optimization was carried out using very stringent convergent criteria for the total energy (10 -12 Ry), forces (10 -6 Ry/bohr), and pressure (10 -3 kbar). The optimized lattice constants are 10.19 Å, 10.66 Å, and 10.38 Å for the LDA, PBE, and PBEsol functional, respectively. As expected, the PBEsol functional produces the lattice constant closest to the experimental value of 10.47 Å. [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] Complete data about structural optimization is reported in Table 1.

All GW calculations were performed using the Yambo code 122,123 in conjunction with Quantum ESPRESSO. 116 We employed the Godby-Needs plasmon-pole model, 124 and set the plasmon energy to 21 eV based on calculations of the energy-loss function. In Fig. 2 we show our convergence tests for the kinetic energy cutoff of the dielectric matrix and the energy window used for the Green's function (or equivalently the number of empty states). We considered cutoffs between 4 and 12 Ry for the dielectric matrix, and between 15 and 90 eV for the Green's function (100 to 650 bands). In this test we employed an 8 × 8 × 8 Brillouin zone grid. Fig. 2 shows that using 10 Ry for the dielectric matrix and 600 bands is sufficient to obtain a band gap converged to within 10 meV. Using these converged parameters, we tested the sensitivity to the Brillouin zone sampling. To this end we repeated the calculations using uniform grids with 6 3 to 12 [START_REF] Paillard | Photovoltaics with Ferroelectrics: Current Status and Beyond[END_REF] points, and found that the gap changes by 7 meV at most. Therefore a 6 × 6 × 6 grid is already enough for our purposes. All subsequent calculations have been performed using 10 × 10 × 10 grids. For eigenvalue-self-consistent GW calculations, we calculated the quasiparticle shifts for 14 valence bands and 14 conduction bands, and interpolated the corrections for the other bands.

Below we report GW calculations performed using the three exchange and correlation functionals LDA, PBE, and PBEsol. The band structures, gaps, and effective masses reported below were obtained using the optimized lattice parameter for each functional, unless otherwise stated. We also tested the effect of spin-orbit coupling by using fully-relativistic ONCVPSP pseudopotentials, but we found that it does not affect band structures in any significant way, in agreement with previous work. [START_REF] Tran | Designing indirect-direct bandgap transitions in double perovskites[END_REF]125 Based on these results, we performed GW calculations without spin-orbit coupling.

HSE calculations were carried out starting from the PBE-optimized structure and a 10 × 10 × 10 Brillouin zone grid, using the adaptively compressed exchange operator implemented in Quantum ESPRESSO. 126 To facilitate Wannier interpolation, we excluded the first 27 bands which form a composite manifold below the valence bands for calculations in the presence of In semicore states (23 bands otherwise). This leaves 30 bands for Wannier interpolation. All calculations of Wannier functions were performed using the Wannier90 code. [127][128][129] Effective masses were evaluated by finite differences using a step size of 10 -3 Å-1 in reciprocal space, which guarantees an accuracy of 10 -2 m 0 , with m 0 being the free electron mass.

Phonon dispersion relations and eigenmodes were calculated using density-functional perturbation theory as implemented in the PHonon code of Quantum ESPRESSO. 116 For these calculations we used the PBE functional. The supercells with ZG displacements were obtained using the ZG.x code released with the EPW package. [START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF]130 We constructed special displacements in 3×3×3, 4×4×4, and 5×5×5 supercells, containing up to 1250 atoms. To lower the computational cost of these calculations, we used a reduced planewaves cutoff of 76 Ry, as suggested by the PseudoDojo library. 121 We checked that band structure computed To reduce the cost of these calculations, we analyze band structures for the 4×4×4 supercell, which yields a renormalization within 10% of our most converged value.

5 Results and discussion

Quasiparticle band gap and effective masses

The band structures of Cs 2 InAgCl 6 calculated using DFT, GW, and HSE are shown in Fig. 3.

This compound exhibits a direct gap at the Γ point. The conduction band minimum (CBM)

originates from In/Ag-5s and Cl-3p states, while the valence band maximum (VBM) results from the hybridization of Ag e g states (4d x 2 -y 2 and 4d z 2 ) with Cl-3p states. The flat band visible along the Γ-X line at the top of the valence bands originates from the hybridization of 4d x 2 -y 2 states of Ag with the Cl-3p x,y states, leading to a two-dimensional wavefunction confined within the xy plane. [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] An example of the iterative procedure for self-consistent GW calculations is given in In Table 2 we report a comprehensive dataset of band gaps computed using various self-consistency schemes for GW, starting from three DFT functionals, with or without In semicore states, and using the optimized or experimental lattice parameters. Key data is also reported in Fig. 5. As expected, semicore states play an important role in the calculated GW gaps, causing a blueshift of up to 0.4 eV as compared to calculation without semicore states [Table 2(a)]. This is understood in terms of the exchange self-energy being sensitive to the overlap between semicore states and band edge states. 131-133 Among the functionals employed, the DFT band gap is smallest for LDA and largest for PBE. This trend is reflected in the corresponding GW band gaps. The same trends are observed when performing all calculations using the experimental lattice constant, as shown in Table 2(b). The choice of using optimized or experimental lattice constant brings only a small change in the gap, in the range of 0.1 eV, as it can be seen in Table 2(c). Furthermore, the use of eigenvalue-selfconsistency or scissor-self-consistency is found to reduce the dependence of the GW band gap on the DFT starting point, approximately halving the differences between LDA, PBE, and PBEsol calculations. In comparison, HSE calculations yield too small a band gap, in the same range as the result of one-shot G 0 W 0 calculations (Table 2).

In Fig. 5 we compare our calculations with measured absorption onsets in UV-Vis spectra. Experimental data is from powder samples, 26 single crystalline samples, 33,47 films, [START_REF] Tran | Designing indirect-direct bandgap transitions in double perovskites[END_REF][START_REF] Liu | Ag/In lead-free double perovskites[END_REF] and nanocrystals. [START_REF] Locardi | Emissive Bi-Doped Double Perovskite Cs 2 Ag 1-x Na x InCl 6 Nanocrystals[END_REF][START_REF] Manna | Tunable and Stable White Light Emission in Bi 3+ -Alloyed Cs 2 AgInCl 6 Double Perovskite Nanocrystals[END_REF][START_REF] Manna | Lattice Dynamics and Electron-Phonon Coupling in Lead-Free Cs 2 AgIn 1-x Bi x Cl 6 Double Perovskite Nanocrystals[END_REF][START_REF] Lee | Colloidal Synthesis of Lead-Free Silver-Indium Double-Perovskite Cs 2 AgInCl 6 Nanocrystals and Their Doping with Lanthanide Ions[END_REF] Even after excluding nanocrystals which might be affected by quantum confinement, there is a significant scatter between reported gaps, from 3.2 eV 33 to 3.53 eV. [START_REF] Tran | Designing indirect-direct bandgap transitions in double perovskites[END_REF][START_REF] Liu | Ag/In lead-free double perovskites[END_REF] These values are shown in pink in Fig. 5, and the experimental range is indicated by a gray horizontal stripe.

A comparison between our calculations and experiments indicates that eigenvalue-or scissor-self-consistent GW calculations starting from the PBE functional lie well within the experimental range, while the other options explored here tend to slightly underestimate the experimental values. We should also keep in mind that the measured optical gaps might include non-negligible renormalization from excitonic effects and electron-phonon interactions. These effects will further lower the theoretical band gaps in Fig. 5. In particular, in Sec. 5.2 we will show that the electron-phonon interactions reduce the band gap by as much as ∼ 0.2 eV at room temperature.

Fig. 3(c) shows how quasiparticle corrections modify the curvature of the bands near the band extrema. To quantify this effect, we report in Fig. 6 the effective masses calculated at various levels of theory. Unlike for the band gaps, we find that the effective mass renormalization is not very sensitive to the DFT starting point for the GW calculations, and that HSE masses are similar to GW masses.

The band edges of Cs 2 InAgCl 6 can be described via an electron mass, a light hole mass, and a heavy hole mass, as it can be seen in Fig. 3(c). We find that both electron and hole effective mass tensors are isotropic, therefore in Fig. 6 we report only the isotropic average.

Taking ∆ k -scGW calculations with PBE starting point as a reference, we find that the electron mass increases by 15% from 0.27 m 0 (PBE) to 0.31 m 0 (GW ), while the light hole mass decreases by 20% from 0.36 m 0 to 0.30 m 0 , and the heavy hole mass decreases by 17% from 0.90 m 0 to 0.77 m 0 . We note that the determination of the heavy hole mass is less reliable owing to the non-dispersive nature of the top of the valence bands along the Γ-X direction.

To the best of our knowledge, no measurements of the effective masses in Cs In Fig. 8 we show the momentum-resolved spectral function and the electronic DOS of Cs 2 InAgCl 6 including electron-phonon interactions at 300 K, as obtained via the special displacement method. [START_REF] Zacharias | Theory of the special displacement method for electronic structure calculations at finite temperature[END_REF] Electron-phonon couplings are seen to shift and broaden the noninteracting DFT bands, which are also shown in Fig. 8 as gray lines for comparison.

Electron-phonon coupling

Phonon-induced renormalization is most pronounced for the valence bands. Indeed, the VBM acquires a finite linewidth of 143 meV and blueshifts by 255 meV. In contrast, the CBM acquires a finite linewidth of 9 meV, and blueshifts by 68 meV. To rationalize this significant difference between VBM and CBM, in Fig. 9 we show the electron density around the metalhalide octahedra, for both the ground state structure and the perturbed ZG structure at 300 K, evaluated at the Γ point. From this comparison we see that the diffuse and isotropic nature of the Ag/In-5s states makes the CBM wavefunction relatively insensitive to lattice fluctuations. On the other hand, the more localized and directional nature of the Ag e g states (4d x 2 -y 2 and 4d z 2 ) makes the VBM wavefunction more responsive to a breaking of octahedral symmetry.

The uneven temperature shift of VBM and CBM leads to a band gap renormalization of 187 meV at 300 K. Combined with our self-consistent GW calculations in Table 2 (starting from PBE), this result implies that our most accurate theoretical gap of Cs 2 InAgCl 6 , including both many-body electron-electron and electron-phonon interactions, should be in the range 3.07-3.19 eV. It is worth noting that the electron-phonon coupling evaluated at the PBE level is expected to be slightly too weak (by a few tens of percent) as a result of the band gap underestimation. [START_REF] Antonius | Many-Body Effects on the Zero-Point Renormalization of the Band Structure[END_REF] Therefore, our calculations of the band gap renormalization likely constitute a lower bound to the true effect, and our final band gap range is probably overestimated by a few tens of meV.

Given that the conduction band near the CBM is relatively sharp [see Fig. 8(a)], it is possible to extract the renormalized electron effective mass from the spectral function using a simple parabolic fit. This procedure yields the effective mass 0.33 m 0 at 300 K. Since the bare mass is 0.27 m 0 , we conclude that phonons induce a mass enhancement of 22 %.

Assuming that the most significant contribution arises from longitudinal optical phonons as for the related CH 3 NH 3 PbI 3 and Cs 2 AgBiCl 6 , 135,136 we can use the mass renormalization to extract an approximate polaron coupling constant α. 137 The mass enhancement in Feynman's model 138 is given by 1 + α/6 + 0.025α 2 , yielding a coupling constant α = 1.1 for electrons in Cs 2 InAgCl 6 . This value indicates a weak polaron coupling regime for the electrons. We also estimate the size of polaronic carriers using Feynman's expression, r p = (3.4 h/m * α ω) 1/2 . 139

Using hω = 32.7 meV from Fig. 7 and m * = 0.27 m 0 , we find r p = 52 Å, indicating large electron polarons. This analysis suggests that electron carriers in Cs 2 InAgCl 6 undergo standard diffusive transport. A similar parabolic fit for the light hole carriers (along the Γ-L direction) yields a much more pronounced mass enhancement, from 0.36 m 0 to 1.2 m 0 , therefore in this case we have α = 6.9 and r p = 18 Å. As expected, in this simple model the holes form intermediate polarons, hence thermal activation may play an important role in hole transport.

Conclusions

In this work we reported a detailed investigation of the quasiparticle band structure and electron-phonon couplings in the lead-free halide double perovskite Cs 2 InAgCl 6 . We performed a thorough sensitivity analysis of the GW quasiparticle corrections by considering several DFT exchange and correlation functionals as starting point, and several strategies for self-consistency. In addition, we computed the band gap renormalization arising from electron-phonon interactions at room temperature via the special displacement method.

Upon including both electron-electron and electron-phonon interactions, our most accurate estimate for the band gap is 3.07-3.19 eV. These values are well within the experimental band gap range as determined via UV-Vis absorption spectroscopy. In comparison, we found that both the one-shot G 0 W 0 and HSE band gaps underestimate the band gap by as much as an eV, hence they do not provide a reliable description of the band structure of Cs 2 InAgCl 6 . The present study indicates that the luminescence of this compound around 2 eV is not related to band-to-band recombination, which occurs at a much higher energy. Remaining options to explain the observed PL are the formation of strongly bound polarons, defect-induced 2: Bandgap of Cs 2 InAgCl 6 calculated using DFT, GW, and HSE. (a) Calculations perfomed using the optimized structures for each functional, as described in Table 1. (b) Calculations performed using the experimental lattice constant and the fractional coordinates reported in Table 1 for each functional. (c) Difference between the values obtained in (b) and in (a). In all cases we report results including or excluding In semicore states. The labels on the leftmost column indicate eigenvalue-self-consistent GW (ev-scGW ), self-consistent scissor GW with a k-independent scissor (∆ Γ -scGW ), and self-consistent scissor GW with a k-dependent scissor (∆ k -scGW ). HSE calculations were performed using the structure optimized via the PBE functional. Figure 5: GW band gaps calculated starting from various DFT functionals. We report the one-shot G 0 W 0 gaps (orange), the eigenvalue-self-consistent GW gaps (blue), and the scissor-self-consistent GW gaps (∆ Γ -scGW, red; ∆ k -scGW, green). For comparison we also report the HSE gap (purple), and the experimental data (pink

LDA

W K U L X W K L U W L K E-E VBM (eV)

  with 76 or 100 Ry cutoffs are pratically indistinguishable. The band gap red-shifts at 300 K obtained from 3 × 3 × 3, 4 × 4 × 4, and 5 × 5 × 5 supercells are 123 meV, 187 meV, and 165 meV, respectively [c.f. inset of Fig.8(a)]. Based on these calculations, we expect the converged value of the electron-phonon renormalization to fall in the interval 176±11 meV.

Fig. 4 .

 4 Fig. 4. This plot shows how, upon performing self consistency on the eigenvalues starting from PBE, the band gap increases from 2.6 eV to 3.4 eV after five iterations. Similar trends

Figure 7 2

 72 Figure 7 shows the calculated phonon dispersion relations of Cs 2 InAgCl 6 , as well as the phonon density of states (DOS) projected onto Cs, Ag/In, and Cl displacements. The lack of soft modes confirms that Cs 2 InAgCl 6 is dynamically stable in the F m3m structure. We can see three main groups of vibrations: a low-energy group around 5-10 meV, which is dominated by the vibrations of Cs and Ag, an intermediate-energy group of modes around 15 meV, associated with the bending motion of the Ag-Cl/In-Cl bonds, and high-energy modes in the range 25-30 meV associated with Ag-Cl and In-Cl stretching vibrations. The phonon dispersions and the mode assignment are similar to the related double perovskite Cs 2 BiAgCl 6 . 135

Figure 1 :

 1 Figure 1: Ball-and-stick model of (a) conventional cell, and (b) primitive unit cell of Cs 2 InAgCl 6 . The color code is as follows, Cs: cyan; In: purple, Ag: grey, and Cl: orange.The structure belongs to the F m3m space group, and the experimental lattice parameter is 10.47 Å.[START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] The PBEsol Ag-Cl and In-Cl bond lengths are 2.659 Å and 2.533 Å, respectively, see Table1.

Figure 2 :

 2 Figure 2: Convergence test for the GW calculations. The one-shot GW band gap of Cs 2 InAgCl 6 is calculated starting from PBE, as a function of the number of bands (energy window) and of the kinetic energy cutoff for the dielectric matrix. The energy window referred to the top of the valence bands. inset is a zoom near the region where convergence is achieved.

X

Figure 3 :

 3 Figure 3: Band structure of Cs 2 InAgCl 6 computed at various levels of theory. (a) Brillouin zone of Cs 2 InAgCl 6 and high-symmetry points used in the band structure plots. (b) Comparison between the PBE band structure (gray) and the ∆ k -scGW band structure starting from PBE (green). The GW bands are obtained with the PBE functional and ONCVPSP pseudopotentials including In semicore states. (c) Comparison between the GW band structures obtained using G 0 W 0 (orange), ev-scGW (blue), ∆ Γ -scGW (red), ∆ k -scGW (green) and HSE (purple) near the band edges.

  [a][b][c][d][e] [f] [g][h] [i] [j]

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Calculated quasiparticle effective masses of Cs 2 InAgCl 6 . We provide the electron mass (m e ), light hole mass (m lh ), and heavy hole mass (m hh ). All calculations employ the PBE functional as a starting point. The acronyms are the same as in Fig. 5.

Figure 9 :

 9 Figure 9: Wavefunction isosurfaces of the VBM and CBM of Cs 2 InAgCl 6 in the unperturbed structure and after introducing thermal displacements. (a),(c) Close-up view of the CBM wavefunction (Γ point) without/with thermal displacements, respectively. (b),(d) Same as above, but for the VBM. Blue/green indicates a positive/negative sign, respectively. The VBM is mainly derived from Ag-4d /Cl-3p states, thus, only the AgCl 6 octahedron is shown.

  The much heavier carrier effective masses of Cs 2 InAgCl 6 as compared to the prototypical halide perovskite CH 3 NH 3 PbI 3 (∼ 0.1 m 0 , Ref.[ 134]) suggest that this compound will exhibit poor transport properties. Indeed, the calculated carrier mobility of the related double perovskite Cs 2 BiAgCl 6 , which exhibits effective masses similar to Cs 2 InAgCl 6 , lies in the range 6-11 cm 2 /Vs depending on direction, 135 therefore for Cs 2 InAgCl 6 we can expect mobilities within this same range. This tentative estimate is in line with both experimental measurements of the carrier mobility in Cs 2 InAgCl 6 , yielding 2.3-3.3 cm 2 /Vs,[START_REF] Luo | Cs 2 AgInCl 6 Double Perovskite Single Crystals: Parity Forbidden Transitions and Their Application For Sensitive and Fast UV Photodetectors[END_REF] and theoretical estimates based on a Fröhlich polaron model, yielding 2.34 cm 2 /Vs.[START_REF] Manna | Lattice Dynamics and Electron-Phonon Coupling in Lead-Free Cs 2 AgIn 1-x Bi x Cl 6 Double Perovskite Nanocrystals[END_REF] Clearly further work will be needed to clarify the conduction mechanisms in Cs 2 InAgCl 6 and related materials, and to understand how the transport properties of this compound can be improved via doping.[START_REF] Manna | Lattice Dynamics and Electron-Phonon Coupling in Lead-Free Cs 2 AgIn 1-x Bi x Cl 6 Double Perovskite Nanocrystals[END_REF]

2 InAgCl 6 are available. However, given that GW calculations of quasiparticle effective masses in the related halide CH 3 NH 3 PbI 3 closely reproduce experimental effective masses obtained from magnetic measurements of exciton binding energies, 134 we expect our calculations to carry reasonable predictive power.

Table 1 :

 1 emission, and self-trapped excitons. Regarding polarons, our simplified analysis based on Feynman's model indicates that electron carriers are in the large polaron coupling regime, while hole carriers are moderately coupled to phonons due to the d x 2 -y 2 and d z 2 character of the VBM. It is likely that hole carriers in this compound form intermediate polarons, although a precise determination of the polaron formation energies will require a separate, detailed investigation. In summary, this work establishes clear guidelines for future ab initio calculations of the electronic and optical properties of Cs 2 InAgCl 6 , and provides a firm basis for quantitative studies of the photophysics of this promising light-emitting material. This work is supported by the Robert A. Welch Foundation under award number F-1990-20190330 (V.H., calculations and data analysis) and the Computational Materials SciencesProgram funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020129 (H.L., Wannier functions, and F.G., project supervision). Optimized structure of cubic Cs 2 InAgCl 6 with space group F m3m. We report the Wyckoff site and fractional coordinates of each atom in the primitive cell obtained using the LDA, PBE, and PBEsol functionals, as well as the bond lengths and lattice constants. For reference, the measured lattice constant is 10.47 Å.[START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] Only data obtained by including In semicore states are reported; the bond lengths and lattice constants calculated without semicore states are within 0.05% (0.01 Å) of the values reported below.

	7 Acknowledgments
	G.V. acknowledges the Chaire de Recherche Rennes Métropole. The authors acknowledge
	the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for
	providing HPC resources, including the Lonestar5 system and Frontera under LRAC award
	2007638, that have contributed to the research results reported within this paper. URL:
	http://www.tacc.utexas.edu. This research also used resources of the National Energy Re-
	search Scientific Computing Center, a DOE Office of Science User Facility supported by
	the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-
	05CH11231.

Table

Table 1 .

 1 [START_REF] Volonakis | Cs 2 InAgCl 6 : A New Lead-Free Halide Double Perovskite with Direct Band Gap[END_REF] The PBEsol Ag-Cl and In-Cl bond lengths are 2.659 Å and 2.533 Å, respectively, see

	20

  Convergence of eigenvalue-self-consistent (ev-scGW, blue) and scissor-selfconsistent (∆ Γ -scGW, red, and ∆ k -scGW, green) calculations of the band gap of Cs 2 InAgCl 6 . At each iteration, the starting band gap on the horizontal axis is obtained from the GW gap at the previous iteration on the vertical axis. The inset shows the change in band gap between successive iterations.

	2.8 3.0 3.2 3.6 3.4 E QP g (eV) 3.4 3.2 2.6 E 2.8 3.0 g (eV)	QP correction (eV)	1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.4 1.2	1	2	3 Iteration 4 ev-scGW@PBE 5 -scGW@PBE k-scGW@PBE	6	QP correction
	2.6 2.4	G 0 W 0			band gap shift
	2.2	1.0 LDA		1.5 Corrected band gap (eV) 2.0 2.5 HSE PBEsol PBE	3.0 Exp.

G0W0

Figure

4

:

  ). All GW calculations include In semicore states. The experimental references are as follows: a) Powder samples, Ref. 26, b) Single crystal, Ref. 33, c) Single crystal, Ref. 47, d) Single crystal, Ref. 28, e) Nanocrystal, Ref. 79, f) Nanocrystal, Ref. 80, g) Nanocrystal, Ref. 49, h) Nanocrystal, Ref. 51, i) Film, Ref. 28, j) Film, Ref. 81.

	Effective mass (m 0 )	0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9	0.27	0.31 HSE 0.31 k-scGW 0.31 -scGW ev-scGW DFT G0W0	0.31	0.3	0.36	0.32	0.29	0.3	0.3	0.33	0.9	0.72	0.72	0.74	0.77	0.82
		0.1		m e					m lh					m hh


Marie-Noëlle Lefebvre
email: m.lefebvre@groupe-espi.fr.

Eddy Zanoutene
email: eddy.zanoutene@u-paris2.fr.

Wealth and Income Responses to Dividend Taxation: Evidence from France

Keywords: Dividend taxation, Wealth Tax, Efficiency, Difference-in-Difference JEL Code: H21, H24, H31, C23

Introduction

In developed economies, capital income is highly concentrated, especially compared to labor income. France is of no exception, as according to [START_REF] Garbinti | Accounting for wealth-inequality dynamics: Methods, estimates, and simulations for france[END_REF], the top 1% wealth group owns up to 35% of total capital income, while their share of total labor income does not exceed 4.5%. Besides, this concentration of capital income is significantly increasing since the 80s, while in the meantime the share of labor income of the wealthy is declining.

Hence if one is concerned about making the "extremely wealthy pay their fair share"1 , shifting the tax burden towards capital income rather labor income can be appealing. This was actually the policy route followed by the French government in 2012 as they decided to increase capital income taxation. 2 Although such a policy can have important consequences for inequality, as documented for instance by [START_REF] Paquier | Effets des réformes 2018 de la fiscalité du capital des ménages sur les inégalités de niveau de vie en france: une évaluation par micro-simulation[END_REF], the main rationale was to actually increase government revenue in order to balance French budget and comply with European fiscal rules. In that respect, short run behavioral responses are of first order importance as it will eventually determine the revenue raised by the tax hikes. Estimating these behavioral responses and its consequences for government revenue is the main goal of this paper.

Using the French reform of dividend taxation implemented in 2013, we document how wealthy households can respond to capital income tax hikes. Our objective is not only to unravel the direct response of dividend to its own marginal tax rate but also to detect so-called "cross-base responses", i.e changes in other tax bases that could be attributed to the reform.

Focusing on wealthy households allows us not only to identify cross-base responses of other incomes, from labor income to rents, but also to elicit changes in taxable wealth. Indeed, at the time of reform, France levied a comprehensive wealth tax so that top wealth owner had to report their wealth to French authority, with the wealthiest ones having to decompose between financial and real estate assets. Thanks to the CASD3 , we have an access to both income and wealth tax returns of all French households that allows us to construct a 10-year long panel documenting individual-level evolution of taxable incomes and wealth.

The 2013 reform did not affect all taxpayers as it consists in removing a flat-tax option, called Prélévement forfaitaire libératoire (PFL), which was not used by all households earning dividends. The end of PFL therefore provides an exogenous variation of dividend marginal tax rate for only a subgroup of taxpayers, allowing us to implement a difference-in-difference strategy to elicit the causal impact of the reform. We consider as treated taxpayers that have always used the flat-tax option for their dividends and are thus the most likely to be affected by the reform. To find a counterfactual similar to the treated, we choose households with only an intermittent use of PFL for their dividends as our control group. As they have not always chose the PFL option for their dividends, they are likely to be less affected by the reform. Since the PFL option for dividends was created in 2008, this imply that treated households have chosen the flat tax for 5 consecutive years. The average flat tax rate for dividend during this period was of 31.86% and can move up to 40.2% after the removal of PFL, so that the reform provide a salient increase of dividend taxation.

Our main results are the following. First, the increase in dividend tax rates led to a significant drop in dividends reported to tax authorities. This drop is persistent as it amounts to a 44% decrease in dividends 5 years after the reform. Second, we do not find significant responses of other income types, be it labor incomes, fixed incomes or rents. Third, we find a significant cross-base response of taxable wealth, with a more than 11% increase in reported gross wealth. In particular, we identify an important response of financial wealth, especially when excluding liquidity. This non-liquid financial wealth increases by 19% because of the dividend tax hike. This is our main contribution as this cross-base response of taxable wealth to capital income taxation has not been yet documented to our knowledge. Behaviors such as retained earnings could both explain the drop in dividend distribution and the increase in non-liquid assets' value after the reform. Eventually we compute the impact of such cross-base responses on government revenue. Without cross-base responses, the increase in dividend taxes for our treated households led to a shrinking tax base, eventually resulting in a loss of more than 46 million euros for the government. Including the increase of taxable wealth can compensate for up to 24% of the direct loss in tax revenue. Hence wealth cross-base responses mitigate strong direct behavioral responses of dividends but are not sufficient to offset the loss in government revenue caused by the tax hike.

To our knowledge this paper is the first to provide evidences of cross-base responses of wealth to capital income taxes, namely dividend taxation. Our work first relates to the literature estimating the responses of dividend to taxes. Using firm-level data, [START_REF] Chetty | Dividend taxes and corporate behavior:evidence from the 2003 dividend tax cut[END_REF] and [START_REF] Yagan | Capital tax reform and the real economy: The effects of the 2003 dividend tax cut[END_REF] exploit a decrease of dividend tax rates in the US in 2003 to unravel the elasticity of dividend to its tax rate. [START_REF] Yagan | Capital tax reform and the real economy: The effects of the 2003 dividend tax cut[END_REF] also provides an analysis of investment reactions to the tax cut and find no significant impact of the policy. Making a similar exercise on French data, [START_REF] Boissel | Higher dividend taxes, no problem! evidence from taxing entrepreneurs in france[END_REF] document a strong decrease in dividend payments after a rise in social contribution requirements for a specific category of firm owner. Combining French household level data with firm-level data, [START_REF] Bach | Follow the money! combining household and firm-level evidence to unravel the tax elasticity of dividends. Combining Household and Firm-Level Evidence to Unravel the Tax Elasticity of Dividends[END_REF] not only estimate the elasticity of dividend to its marginal tax rates but also link it to changes in firms payout policy. The increase in treated firms' equity they identify can explain part of the rise in taxable wealth that we observe at the household level. Exploiting a vast reform of both capital income and wealth taxation in the Netherlands, [START_REF] Zoutman | The elasticity of taxable wealth: Evidence from the netherlands[END_REF] deliver estimates of the elasticity of taxable wealth with respect to the after tax gross rate-of-return. More precisely, the author combines changes in wealth and capital income taxation to compute the overall variation of the marginal tax rate of returns induced by the reform. In this sense, wealth and capital income taxes are treated as equivalent instruments for taxing returns to savings. Our exercise is different as we study variation of both wealth and income resulting from tax changes of only capital income taxation, namely dividend taxation. In particular, we show that responses of wealth and dividend differ strongly and have actually opposite signs : increasing the marginal tax rate of dividends leads to a decrease in dividends (direct response) and to an increase of taxable wealth (cross-base response).

The rest of the paper is organized as follows. Section 2 provides an overview of the French capital tax system and explains the 2013 reform. Section 3 details our data source and the construction of our panel of French households paying the wealth tax. We present our empirical strategy in Section 4, by providing a definition along with descriptive statistics of the treatment and control groups and by describing our estimation equation. Section 5 gives the theoretical expectation one can have regarding household's behavioral responses to the reform.

Our results are presented in Section 6. Section 7 concludes.

Institutional Setting

A brief overview of capital taxation in France before 2013

There exists three ways to tax an asset, be it a financial or a real estate one, in France.

First, the government can directly tax the ownership of an asset through property and wealth taxation. Second, the income generated by an asset, be it rents, interests, dividends or capital gains, can also expose a household to taxes. Third, the French government levies taxes on asset transmission through bequest taxation. The relation between these three dimensions of capital taxation can vary across asset types. For instance, the vast majority of French households is not liable for taxes on the ownership of corporate stocks. However, most of them would have to pay taxes on dividends, i.e on an income associated to stock ownership. Therefore, a basic knowledge of the whole capital tax system before the 2013 reform is needed to understand behavioral responses of household, as their tax profile can vary in a more or less complicated fashion along the three dimensions of capital taxation.

Capital income taxation. Before the 2013 reform, households could choose between two options for the taxation of their dividends, interests and capital gains. First, they could decide to include these incomes within the French personal income tax, called Impôt sur le Revenu (IR). In this case, capital incomes are taxed according to a progressive income tax with four brackets until 2012 (5.5%, 14%, 30% and 41%). In 2012, a fifth bracket at 45% is created for income above 150,000 euros. As this paper focuses on dividends, it is worth mentioning that dividends taxed at the IR benefit from a 40% tax rebate, lowering the top marginal tax rate on dividends roughly to 27% (60% of 45%). Before 2013, there was a second option for capital income taxation called the Prélèvement Forfaitaire Libératoire (PFL). This option, originally limited to some specific capital incomes but extended to dividends in 2008, granted access to a flat tax on capital income instead of the standard progressive income tax. Basically, every year taxpayers can decide whether they use or not the flat tax option for their dividends. The PFL rate for dividends has gradually increased from 18% in 2008 to 21% in 2012, before the removal of this option with the 2013 reform. Eventually, note that on top of the IR or the PFL, capital incomes are submitted to social contributions, with a flat-rate ranging from 12% in 2008 to 15,5% in 2012. For taxpayers, the choice between the PFL or the IR was not necessarily obvious, as it depends on the amount of dividends but also family composition4 , tax credits, labor income, pensions and other taxable income.

Wealth taxation. Until 2017, the French government levied a wealth tax on rich households named Impôt de Solidarité sur la Fortune (ISF). The wealth threshold above which one had to pay the ISF was 800,000 euros in 2008, with a tax base including real estate, luxury goods and financial assets. Work-related properties are not included in taxable wealth. 5 Although there has been some reform of the ISF between 2008 and 2013, with for instance a raise in the threshold from 800,000 to 1,3 million euros, it remained a progressive wealth tax with 6 brackets in 2013, ranging from 0.5% to 1.5%. In 2018, financial wealth has been excluded from taxable wealth and the ISF turned into a tax on housing wealth, becoming an Impôt sur la fortune immobilière (IFI).

The 2013 reform

In 2013, capital income taxation is modified with the removal of the PFL option for most of capital incomes (with the exception of life insurance products and other very specific saving products). Therefore dividends became necessarily taxed at the personal income tax rate while still benefiting from the 40% tax rebate. For instance, the marginal tax rate on dividends of a household in the 45% bracket in 2013 rises from 36,5%6 to 40,2%7 after the reform.

Data

Data Source

We use two data set provided by the French General Direction of Public Finance (DG-FIP) on the CASD : the personal income tax files and the wealth tax files.8 . Personal Income Tax Records : We use the POTE9 files to have access to all the information, regarding income, age or family composition, filled by taxpayers on their personal income tax report. 10 The data set is exhaustive and provides an encrypted identifier so that we can follow taxpayers for several years and create a panel data. Besides, note that personal income tax records are pre-filled by fiscal authorities using information directly transmitted by employers, public administration or banks. It therefore appears as as particularly reliable data source.

ISF/IFI. We also use the ISF/IFI files to get information on the wealth of taxpayers, be it financial or housing wealth. These files contain all the items of the wealth tax declaration, with the same encrypted identifier as the POTE files so that we can match the wealth tax record of an household with its personal income tax report. However, not all households subject to wealth taxation have to precisely fill the wealth tax report since those below a certain thresh-old11 only report their total net wealth. Since we examine the responses of different component of wealth in this paper, we focus on taxpayers who have to report separately real estate and financial wealth, using the 2725 wealth tax report.

Data Construction

The main difficulty when using fiscal data is to create stable income and wealth aggregates, despite the various changes in tax boxes resulting from changes in legislation. To create stable income aggregates that we can track year upon year, we use the Tax and Income Survey (ERFS) produced by the French National Institute of Statistics and Economics (INSEE). Wealth variables are less prone to tax box changes so that we only use the handbook provided by the fiscal administration to construct our aggregates.

Once the various income aggregates for each year constructed, a cylindrical panel of tax returns is build using the encrypted tax identifiers corresponding to household and tax filer "1". This technique excludes from our study households that experienced divorce, death, PACS or marriage between 2008 and 2017.

Empirical Approach

Sample of analysis

In this paper we use a difference-in-difference strategy to examine the behavioral responses of rich taxpayers to an increase in dividend taxation. More precisely we are interested in the cross-base response of wealth aggregates, especially financial ones, to an exogenous increase in dividend taxation, namely the end of PFL in 2013. To do so we compare a group of household that is likely to be strongly affected by the end of PFL to a group that is likely to be less affected by the 2013 reform. We start from two balanced panel. The first one is composed of all households liable for the ISF (i.e the wealth tax) every year between 2008 and 2017. We use this panel to track the responses of dividends and other incomes to the removal of PFL in 2013. The second one is a subsample of the first one, as we focus on households liable for the ISF who have to decompose their holdings of assets between financial and real estate ones. We often refer to the first sample as the "large panel" and to the second one as the "constrained panel".

Definition of treatment and control group. Our treatment group is composed of all households who have used the PFL for their dividend every single year between 2008 and 2012, which is the period where the PFL was available for dividends. These households, either because of their marginal income tax rate, their dividend level or their family composition, have always used the PFL and are therefore likely to be strongly affected by the removal of this option in 2013. We compare this treated households to taxpayers that had an intermittent use of PFL. Therefore our control group is composed of households who have used the PFL between 2008 and 2012, but not every year. These households, as they have been interested in the PFL at some point, are likely to be similar to our treated households. However, as they did not always use the PFL for their dividends, they are likely to be unaffected or less affected by its removal in 2013.

Descriptive Statistics

In table 1 we provide descriptive statistics on the treatment and the control groups in both large and constrained panels. The large panel is composed of 18 880 households which represent all taxpayers liable for the ISF who have used the PFL at least once between 2008 and 2012. 3 271 have declared their dividends at the PFL every year between 2008 and 2012 and therefore belong to the treatment group while 15 609 have had only an intermittent use of the PFL and belong to the control group. We report population average and standard of deviation of several income categories, family composition and age in both groups. Total income is the sum of all income perceived by the members of an household during a given year. 12 Here labor income is understood in a broad way as it includes not only wages but also pensions, unemployment benefits and business incomes of liberal professions.

The objective is to provide a first comparison between the treatment and control group. Households in the treatment group have on average higher incomes than those in the control group, be it labor incomes, dividends or fixed capital income. In particular, treated households declared three times more dividends in 2011 compared to those in the control group. This is not surprising as both higher dividends and higher total income, implying higher marginal tax rates, can explain the choice of the PFL instead of the regular personal income tax. Note however that on average households in the control group are larger with 2,3 tax units against 2,1 in the treatment group, given that a couple has 2 fiscal units while a couple

Estimation

We want to estimate the following equation for all households in our sample over the period 2008-2017.

ln (y i,t ) = α + ∑ k =2011 β k 1 t=k × 1 {i∈ Treated } + ∑ k =2011 δ k × 1 {t=k} + ω i + u i,t (1)
where y i,t is our variable of interest for household i at time t and 1 {i∈ Treated } is a dummy equal to 1 for household i belonging to the treatment group. ω and δ are household and time fixed effect, respectively. ω captures the time-invariant difference between households while δ reflects changes over time that affect all households. Standard errors are clustered at the household level. Instead of directly estimating equation (1), we get rid of the household fixed effect ω i by normalizing ln(y i,t ) in 2011 :

ln y i,t y i,2011 = α + ∑ k =2011 β k 1 t=k × 1 {i∈ Treated } + ∑ k =2011 δ k × 1 {t=k} + u i,t (2)
The coefficients we are interested in are the β k that capture the difference between the treatment and the control groups in a given year k relative to the year 2011. We chose 2011 as our baseline year since the reform is announced in 2012 and could therefore affect dividends earned in 2012. 2011 is therefore the last year unaffected by the end of PFL. The identification hypothesis is that absent the suppression of PFL in 2013, household in the treatment group would have evolved in the same way as households in the control group. Although this assumption cannot be verified, the dynamic specification of (2) allows us to assess the credibility of this so-called "parallel trend" hypothesis by checking if coefficients β k are not statistically significant for k < 2011.

Theoretical prediction

Theoretically, an increase of the marginal tax rate on dividends could trigger ambiguous responses, depending on the relative strength of income and substitution effects. Indeed, for a given level of before-tax income, the reform decreases after-tax income. If income effects dominate, household will compensate this decrease in after tax income by increasing their dividends. However, some income sources have not been affected by the reform. Therefore agents could compensate the decrease in after-tax income after the reform by substituting dividend payments with these unaffected income types. Interests have also been concerned by the end of PFL so one can think of at least three income sources that have become relatively more profitable after the reform :

• Labor income : in France, labor income has always been taxed at the progressive income tax schedule so the end of PFL does not affect labor income marginal tax rate. When thinking about substitution between labor income and dividends, it is important to distinguish responses in labor supply from pure income shifting. Indeed, some households who own private businesses can to some extent switch their compensation from dividends to wages. This pure income shifting is only driven by fiscal considerations and is not related to actual changes in labor supply. On the other hand, households who do not have the leeway of choosing their compensation scheme could still substitute dividends with labor income by increasing their labor supply.

• Rents : Income from real estate investments, be it rents or taxable gains, could not be taxed at the PFL so were not affected by the 2013 reform. Rents could therefore provide an alternative to dividends although the taxation of real estate property might have changed during the period 2013-2017, depending on the location of the good.13

• Capital gains : Before 2013, capital gains were taxed at the PFL so this source of income has been affected by the 2013 reform. However, depending on the type of assets sold or the date of acquisition of the asset before the sell, capital gains taxation can decrease substantially. These tax deductions for capital gains were not removed by the 2013 reform so that realizing such gains could provide an alternative to dividends.

A rise in dividend taxation can be seen as a tax increase of the underlying asset paying dividends. Therefore behavioral responses to dividend taxation can affect households' asset portfolio, be it through changes in asset value or portfolio composition. For instance, firms owned or partially controlled by households using the PFL might reduce dividends distribution because of the 2013 reform. Such retained earnings could therefore increase the value of the firm so that the value of assets owned by treated household would increase because of the end of PFL. This would be coherent with a substitution from dividends to capital gains compensation. In this sense, households could also change their portfolio allocation by substituting stocks from firms compensating their shareholders with dividends to firms with rising stock prices, compensating shareholders through capital gains. On the other hand, a strategy substituting dividends with rents would imply a reallocation of households' portfolios from financial to real estate assets.

Results

Responses of dividends and other incomes to the 2013 reform

As the 2013 reform induces an increase of the marginal tax rate of dividends for treated households relative to the control group, we first display the evolution of dividends between 2008 and 2017 in both groups. It appears from Figure 1 that dividends follow a similar pattern in both groups until 2012 before diverging from 2013 to 2017. Indeed, although dividends falls significantly in both groups when the PFL is removed in 2013, the treatment group experiences a more severe drop than the control one. This divergence is persistent since the gap between the treated and the control household does not reduce after the choc, while both groups declare far less dividends in the post-reform period compared with pre-reform years. It is not surprising though that both groups experience a decline of dividends, since households in the control group might have been directly affected by the end of PFL as well, although in a less severe fashion than the treated. Besides, firm-level decisions to decrease dividend payments affect all shareholders, independent of their exposure to the end of PFL. For instance, if households in the control group have stocks from a firm controlled by treated households, then they are likely to be affected by a decision of the firm to decrease dividends.

As discussed in section 5, the end of PFL in 2013 could have impacted not only dividends but also other types of income. To explore the potential responses of other incomes to the reform, we display in Figure 2 the evolution of labor incomes, fixed incomes and rents in the treatment and in the control group between 2008 and 2012. It appears from this comparison exercise that labor income have not been affected by the reform, with the two groups experimenting a quite flat labor income dynamic. At this stage, a substitution between labor income and dividends, either due to pure income shifting or to an increase in labor supply, seems unlikely. Regarding fixed income and rents, the two groups exhibit similar dynamics before the reform and a small divergence in 2013 and 2014, that disappear at the end of the period. So far, the end of PFL does not seem to have really impacted other incomes than dividends.

To rigorously establish the causal impact of the reform, we graph in Figure 3 the estimates of the β k coefficients of equation ( 2). It first appear that the two group experiment similar trend before the reform as the β k are not statistically significant before 2013, except for dividends in 2009, which is encouraging regarding our identifying assumption. As expected, the increase in dividend taxation induced by the reform has triggered an important drop in dividends, with treated households declaring 30% 14 less dividends in 2013 compared to the control group. This drop is persistent and becomes even more severe by the end of the period, with dividends eventually falling by 44% 15 in 2017. Besides, it seems that other income types have not been used as substitute of dividend payments, as labor income for instance appear 14 exp(β 2013 ) -1 exp(-0.35)-1 -0.3 15 exp(-0.58) -1 -0.44 completely unaffected by the reform. Regarding fixed income, we can detect a statistically significant increase in 2013 but this effect vanishes quickly. Besides, looking at Figure 2 shows us that this small positive effect is due to a decline in fixed income of the control group and not to an increase in the treatment group.

Figure 3: Estimates of the β k coefficients in equation ( 2), for dividends, labor income, rents and fixed income. We display confidence interval at the 95% level with standards errors clustered at the household level.

Responses of wealth to the 2013 dividend reform

The end of PFL and the induced increase of the marginal tax rate of dividends has therefore generated a massive and persistent drop in dividends declared to fiscal authorities.

However, this reform does not seem to have triggered significant cross-base responses of other incomes. In particular labor incomes of wealthy French households appear quite unaffected according to our estimates, ruling out income shifting behaviors. Nevertheless, as explained in section 5, changes in dividend taxation could not only impact incomes but also wealth, through changes in asset choices or in asset values. In particular, financial wealth, which includes assets that pay dividends, could be affected by the reform.

As discussed in section 3, not all households paying the wealth tax have to fill a detailed report with a decomposition between financial and housing wealth. Throughout this section we therefore focus on the constrained panel composed of households that had to precisely fill a wealth tax report between 2008 and 2017. To gauge the effect of the reform on financial wealth, we start by looking at the evolution of financial wealth in the treatment and in Our difference-in-difference analysis, depicted in Figure 5 indicate a positive, statistically significant and persistent impact of the reform on financial wealth. We do not detect a statistically significant pre-trend, especially when we exclude liquidity from financial wealth with a point estimate being really close to 0, so our identifying assumption appear plausible.

Using our estimates for 2017, the increase in dividend taxation led to a 19%16 increase in nonliquid financial wealth. It appears from this analysis that the end of PFL did not alter the financial wealth accumulation process of rich French households, on the contrary. Such an increase in assets value is coherent with a substitution from dividends to capital gains compensation, be it either through a portfolio reallocation towards growing businesses or through a retained earnings behavior that mechanically augments assets' value. Note that if financial assets rise mainly because of retained earnings, then our estimates is likely downward biased as stocks from a company mainly owned by the household is excluded from taxable wealth. In any case, it seems that part of the lost dividends can be found, and actually can be taxed, by looking at the asset side of households' tax profile.

While the end of PFL does not seem to have reduced investment in financial assets, some taxpayers might still have re-balanced their investment towards real estate. Although a look at Figure 3 indicates an absence of responses of rents to the reform, with even a slight decrease in 2014, it is still possible that households increase their real estate investment, as the taxation of such assets is left unchanged by the reform. This is why we look at the evolution of 2), for financial wealth, with and without liquidity. We display confidence interval at the 95% level with standards errors clustered at the household level.

housing wealth, as opposed to financial wealth, in both groups in Figure 6.

As one could have expected from descriptive statistics of Table 1, both groups display similar pattern regarding their housing wealth profiles before the reform. It appears from the left graph that the growth of housing wealth is slightly more important in the treatment group from 2014 to 2017, a divergence that is more striking when we exclude primary residences and therefore focus on a more investment-oriented definition of real estate. Our differencein-difference estimates displayed in Figure 7 indicate that there is no statistically significant impact of the reform on total housing wealth, while we detect a slightly positive impact of 5.5% 2), for gross wealth. We display confidence interval at the 95% level with standards errors clustered at the household level.

Responses in terms of government revenue

It appears that the 2013 reform has triggered several significant responses of households' dividends and wealth. Among the many consequences such responses can have on welfare, we focus here on the impact of the reform on government revenue. We can quantify this impact using the following formula18 :

∆GovRevenue = Dividend × ∆τ div + ∆Dividend × τ div + ∆Wealth × τ w (3)
We can decompose the right hand side of equation (3) as the following :

• The first term embeds the mechanical effect of the increase of dividend taxation, absent behavioral responses of households.

• The second term reflects the direct behavioral responses of dividends resulting from the reform on dividend taxation.

• Eventually our identification of significant responses of taxable wealth leads us to include a third term with the cross-base response of wealth to the increase of dividend taxation.

Hence our calculation of the impact of the reform on government revenue depends on the value we attribute to these three terms of equation ( 3). million euros of government revenue. Hence by looking only at the direct effect of the reform, we find that the increase of dividend tax rates actually led to a loss of more than 46 million euros for the government. Now, there exists several ways to include the cross-base response of wealth. Both theory and our estimation indicates that non-liquid financial wealth is the main component of taxable wealth to react to dividend taxation. On our sample, taking the average during the prereform period, non-liquid financial wealth amounts to approximately 4.5 billion euros. Besides, we have estimated than non-liquid financial wealth has increased by 19% because of the reform.

Assuming a 1% marginal tax rate on wealth, this would imply a gain of 8.4 million euros for the government. Hence the total effect on government revenue, accounting for both direct and cross-base responses, of the 2013 reform would amount to a loss of roughly 38 million euros.

Note however that other components of wealth are likely to have responded to the dividend tax hike. Therefore, if instead of using our estimate for non-liquid financial wealth, we use our 11% estimated hike in gross wealth, applied to the 6.4 billion euros of gross wealth reported by the treated households, then government losses decrease to roughly 35 million euros.

Including the cross-base response of wealth therefore diminishes the direct loss (mechanical + behavioral response of dividends) imputed to the dividend tax hike. This mitigating effect lies between 18 and 24% of the direct loss, which is not sufficient to offset the strong behavioral response of dividends but is significant enough to be taken into account.

Conclusion

This paper provides estimates of behavioral responses of taxpayers to changes in dividend taxation using a panel of French taxpayers. It appears that dividends react strongly to their own marginal tax rate with an important decrease in dividends reported to tax authorities after a hike in dividend taxation. This result is in line with findings of the existing literature on dividend responses to taxes. We however suggest that in presence of a wealth tax, estimating only the responses of dividends is not sufficient as wealth is also likely to respond to the tax hike. Indeed, we show that while dividends significantly decline after the reform, wealth and especially non-liquid financial wealth, increases. Although this rise in taxable wealth does not compensate the loss in government revenue attributed the decrease in dividends, it mitigates the negative impact of the reform. This cross-base responses and the link between dividend payments and taxable wealth advocates for a comprehensive approach when measuring the impact of tax reforms.

Figure 1 :Figure 2 :

 12 Figure 1: Evolution of the average log dividend normalized in 2011, in the treatment and in the control group -large panel.

Figure 4 :

 4 Figure 4: Evolution of average log financial wealth, with (left) and without (right) liquidity, normalized in 2011, in the treatment and in the control group -constrained panel

Figure 6 :Figure 8 :

 68 Figure 6: Evolution of average log housing wealth, with (left) and without (right) primary residences, normalized in 2011, in the treatment and in the control group -constrained panel

Table 1 :

 1 Descriptive Statistics on the treatment and control groups in the large and constrained Panels in 2011 with 1 child has 2,5 fiscal unit. Unsurprisingly households in this part of the wealth distribution are relatively old, although they are on average one year older in the control group (62 years old) than in the treatment one (61 years old). In the constrained panel we only consider the richest households who therefore had to fill an ISF form and especially to distinguish between financial and real estate assets. Within this subsample, 1 521 have always used the PFL while 5 219 only used it occasionally. Although the treated still earned higher income in 2011, they declared 2,5 times more dividend than the control group so that the gap between treated and untreated is less important. Regarding wealth levels in 2011, the treatment and control group appear quite similar, be it in real estate or financial wealth. Financial wealth, which include corporate share, bonds, treasury bill, cash and deposit, is slightly larger in the control group (around 4,6 million euros) than in the treatment one (around 4,4 million euros) while real estate is almost identical in both groups (around 1,8 million euros).

			Large Panel -18 880 household
	Variable	Treatment group -3 271 households	Control group -15 609 households
		Mean	SE	Mean	SE
	Total Income	785 252	1 997 100	431 724	1 257 250
	Labor Income	202 884	258 215	156 466	217 019
	Dividend	439 034	1 670 092	142 576	639 948
	Fixed Income	25 131	61 614	21 182	99 407
	Nb Tax Units	2.130	0,84	2.29	0,86
	Age	60,99	10,83	61,99	10,78
			Constrained Panel -6 750 households
	Variable	Treatment group -1 521 households	Control group -5 129 households
		Mean	SE	Mean	SE
	Total Income	1 149 506	2 831 269	683 990	1 704 453
	Labor Income	241 346	342 680	186 958	308 393
	Dividend	657 207	2 383 620	256 677	981 539
	Fixed Income	39 903	84 549	43 514	163 177
	Total Gross Asset	6 994 264	5 865 503	7 669 037	9 333 424
	Real Estate Asset	1 825 363	1 854 535	1 826 352	1 759 709
	Financial Asset	4 386 195	5 253 482	4 561 150	8 606 755
	Nb Tax Units	2.257	0,80	2,257	0,84
	Age	63,47	11.06	63,54	10,97

  the control group between 2008 and 2017. Looking at the left graph of Figure4, it appears that financial wealth evolves in a similar fashion in both groups between 2008 and 2011. Besides, descriptive statistics in Table1indicates that in 2011, average financial wealth level amounts to a similar amount in both groups. However the two groups start diverging in 2012 with a widening gap between 2014 and 2017. Although dividends did fall in 2012, the divergence between the two groups only occur when the PFL is removed in 2013. This divergence in financial wealth in 2012 could be interpreted as a reaction of the increase of PFL in 2012, with the flat tax rate on dividends increasing from 32.5% to 36.5% between 2011 and 2012. The gap between the two group is even more striking when we exclude liquidity, i.e cash, deposit and liquid savings account, from financial wealth as shown in the right graph of Figure4. Excluding liquidity allows us to focus on assets that are likely to pay dividends, such as corporate stock. While the value of such assets remains quite flat in the control group between 2014 and 2017, it increases substantially in the treatment one.

Table 2 :

 2 The result depends on which behavioral responses are taken into account and are summed up in Table2. Impact of the 2013 tax hike on dividends on government revenue, in million of euros As we do not observe wealth of taxpayers in the large panel, our calculation is based on the the 1 521 households in the treatment group of the constrained panel. To account for the volatility of dividends and the changes of the flat tax rate between 2008 and 2012, we average both dividends and τ div between 2008 and 2012. On average, the 1521 households in the treatment group have declared 537 116 euros of dividends between 2008 and 2012. Hence our tax base of dividend before the reform amounts approximately to 817 million euros. The average flat rate on dividends in the pre-reform period is of 31.86%. Assuming that all treated households would have been in the last bracket of the personal income tax after the reform, absent any behavioral responses, we use a 40.2% marginal tax rate on dividend after the reform. This yields a ∆t div of 0.0834. This therefore implies a mechanical increase of government revenue after the reform of 68,13 million euros, as displayed in the first column of Table2. Using our estimated 44% drop in dividends, direct behavioral responses induce a loss of more than 114,5

	Mechanical Effect	Behavioral Responses	Total Effect
		Dividend	-114,52	-46,39
	68,13	NonLiq-FinWealth	8,48	-37,91
		GrossWealth	11,05	-35,34

We quote US senator Bernie Sanders here.

According to François Hollande's program, these tax hikes were needed to bring capital income taxation closer to labor income taxation.

Centre d'accès sécurisé aux données

In France, taxable income depends on the number of tax units called part fiscale, with 1 unit per adult, 0.5 unit for the two first children and 1 unit for the other children. Roughly speaking, taxable income = income/total number of units.

A taxpayer who owns assets of a firm she is working in can qualify these assets as Work-related properties, under certain conditions. For instance, she has to prove that these assets are needed for the activity of the firm and that her activity in the firm is her main activity. In particular, a household has to own at least 25% of the firm equity to qualify their closely-held-stock as Work-related property.

36,5% is the sum of the 21% PFL rate and the 15,5% social contribution rate in 2012.

Taking into account the 40% tax rebate and the tax deduction of 5.1% of social contributions when dividends are taxed at the IR, we have : 40,2% = 0,6*0,45 + 0.155-0.45*0.051

Centre d'accès sécurisée aux données

Fichier permanent des occurences et des traitements

More precisely this includes all the items of the 2042 and 2042 complementary tax returns.

For instance in 2017, households submitted to wealth taxation but with net taxable wealth below 2,57 million euros did not need to fill a complete wealth tax record.

More precisely it corresponds to the Revenu Fiscal de Référence (RFR) which is more or less the French equivalent of the American Adjusted Gross Income (AGI).

Property taxes are decided at the municipality level and could have increased is some places at the same time as the PFL ended. Besides, some cities like Paris introduced rent caps that could also have affected the profitability of real estate investments

exp(0,18)-1 0,19

when we exclude primary residences. Hence we cannot rule out an increase of investment in real estate after the reform, although the magnitude of this effect is not of the same order as the increase in financial wealth.To get a sense of the overall impact of the dividend tax hike on wealth, one can aggregate all the responses of taxable wealth by directly looking at the evolution of gross wealth, as displayed in Figure8. Unsurprisingly the evolution of gross wealth is aligned with our previous findings on financial and housing wealth so that treated households experience a stronger growth than control ones. Hence our difference-in-difference estimates for gross wealth, dis-17 exp(0,053)-1 0,055

We do not identify responses of other incomes than dividends so we only include the cross-base response of taxable wealth in equation (3).

* This research benfits from funding by France Stratégie to which we are deeply grateful. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-EQPX-17 -Centre d'accès sécurisé aux données -CASD) † ESPI2R Research Unit and CRED (TEPP) Université Paris II Panthéon-Assas.