Luis Buluz
email: luis.bauluz@uni-bonn.de.

Filip Novokmet
email: filip.novokmet@uni-bonn.de

Moritz Schularick
email: schularick-moritz.schularick@uni-bonn.de

Pierre Brassac

Indu Chhatwany

Sven Eis

Tri Hoang

Timothy Meyer

Markus Schick
email: schularick@uni-bonn.de

Youpeng Zhang

Francisco Amaral

Alan Auerbach

Charlotte Bartels

Dmitry Kuvshinov

Emmanuel Saez

Joseph Stiglitz

Alan Taylor

Gabriel Zucman

Luis Bauluz
email: luis.bauluz@uni-bonn.de

Clara Martínez-Toledano

The Anatomy of the Global Saving Glut *

Keywords: JEL Classification: D31, E21, E44, N32 Income and wealth inequality, Saving, Household portfolios, historical micro data JEL classification: D31, E21, E44, N32 Income and wealth inequality, household portfolios, historical micro data

The Centre for Economic Policy Research was established in 1983 as an educational charity, to promote independent analysis and public discussion of open economies

Introduction

This paper uses household data for the world's largest economies -the U.S., Europe, and China -to study the rise in global wealth over the past four decades. We calculate asset-specific saving flows and capital gains across the wealth distribution to expose the shifts in global saving supply that have occurred in recent decades. A prominent line of thought links the fall in real interest rates since the 1980s to changes in global saving and investment behavior, but the underlying forces and motivations remain much debated. The different hypotheses include: a rise in desired saving of households and emerging markets [START_REF] Bernanke | The global saving glut and the US current account deficit[END_REF]; Rachel and Smith 2015;Mian et al. 2021a); a slowdown in growth and investment [START_REF] Gordon | The turtle's progress: Secular stagnation meets the headwinds[END_REF]; [START_REF] Sajedi | Why are real interest rates so low? the role of the relative price of investment goods[END_REF]; an increase in the demand for safe assets (Caballero et al. 2017; [START_REF] Caballero | The safety trap[END_REF] as well as financial crises and the associated deleveraging [START_REF] Lo | Secular stagnation, debt overhang and other rationales for sluggish growth, six years on[END_REF]; [START_REF] Gourinchas | Global real rates: A secular approach[END_REF].

This paper contributes to the debate by providing a micro-level view of global saving dynamics for different asset classes and across the household distribution. Linking tax data, surveys and national accounts, we construct a new data set covering asset portfolios and saving flows from 1980 until today. The data set covers the United States, Europe (France, Germany, Spain and the United Kingdom), and China (shown in Table 1). It makes it possible, for the first time, to study the role of household saving and capital gains in individual asset classes -housing, fixed income, and equity and business assets -in driving global wealth accumulation in recent decades. Wealthto-income ratios have risen substantially in the U.S., Europe, and China since the 1980s (Piketty andZucman 2014, Novokmet et al. 2018). They now stand at 500-600% of national income, up from 250-300% in 1980. With a five-fold increase, the increase in the Chinese wealth-to-income was even stronger. The macro evidence on the the rise in wealth-to-income ratios is summarized in Figure 1 and serves as the starting point for our analysis of saving and asset price dynamics.

We first disentangle the role of asset-specific capital gains and saving flows for aggregate wealth growth since 1980. We "unveil" indirect ownership of assets through pension and investment funds that has been an important driver of financial wealth growth in recent decades. This allows us to show that in the major world economies (what we label as "the G3"), capital gains Figure 1: Household Wealth-National Income Ratios, 1980Ratios, -2018 Notes: This figure displays the evolution of household wealth-to-national income ratios during 1980-2018 in three world regions: the U.S., Europe and China. Europe is the weighted average (by national income) of France, Germany, Spain and the U.K. Source: Own computations using official balance sheets for France, Germany, Spain, UK and the US. China is from Piketty et al. (2019). and saving flows each account for about half of the increase in aggregate wealth since the 1980s.

Saving flows were predominantly channeled towards financial assets, while capital gains were concentrated in housing markets, explaining about one third to half of aggregate wealth growth since 1980.

In a second step, we study how these asset-specific saving flows and capital gains were distributed across different wealth groups. We show that new saving played the key role for wealth accumulation of the top-10% of households. Across the G3 economies, we observe a rising concentration of saving flows, with the share of private saving coming from the top-10% increasing by 60% over the past decades. In 1980, the richest 10% of households accounted for 64% of total private saving in the U.S., 60% in Europe and less than 50% in China. Around 2018, the same share reached 103% in the U.S., 78% in Europe, and 90% in China. An increase in saving inequal-ity over the past four decades reflects the rise of top-10% saving and the parallel decline of middle class saving (i.e., saving of households between the 50th and 90th percentile or the middle-40%).

The "saving glut of the rich", previously documented for the U.S. (Mian et al. 2021b; [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF], is a global phenomenon, as is the dissaving of the middle-classes.

Building on our asset-specific saving data, we analyze the key drivers of rising top-10% saving. In all three economies, corporate saving (i.e., corporate retained earnings) have been the single most important source of rising saving of rich households. Growing corporate profits have been an important facet of the global economy in recent decades ( [START_REF] Karabarbounis | The global decline of the labor share[END_REF]; [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF][START_REF] Barkai | Declining labor and capital shares[END_REF]; [START_REF] Kuvshinov | The big bang: Stock market capitalization in the long run[END_REF]. As dividend growth has not kept pace with surging corporate profits, corporate saving has risen considerably (Chen et al. 2017b; [START_REF] Gruber | The corporate saving glut and falloff of investment spending in OECD economies[END_REF]. Reteined earning on corporate balance sheets are apportioned to households (and the government) as ultimate owners of the corporations. As ownership of corporate equity is heavily concentrated among rich households, top-10% saving surged in parallel with increasing corporate earnings. Our paper exposes and discusses this important link between secular trends in global profit shares, retained earnings and rising global saving inequality.

Turning to the bottom-90% of the distribution, we expose a pronounced reduction of saving by middle-class households. In the U.S. and Europe, the decline in saving of middle-class households was particularly strong. In the U.S. middle-class saving decreased from 5% of national income in the 1980s to close to zero in the past decade. In Europe, the decline from 4% to about 1% in recent years was equally stark. By contrast, there has been little movement in saving in the bottom half of the distribution. The bottom-50% stayed at zero or slightly negative saving throughout. The decline in savings rates outside the top-10% was not driven by the poorest parts of the distribution, but by the middle class.

Yet despite the sharp decline in saving, middle-class wealth increased. Capital gains, especially in housing (net of mortgage debt), accounted for close to 60% of middle class wealth accumulation. Across the G3 economies, the distribution of capital gains has been much more equal than new saving flows. Housing capital gains have been the single most important factor moderating global wealth inequality in recent decades, in line with the findings of [START_REF] Kuhn | Income and wealth inequality in America, 1949-2016[END_REF] for the U.S. All else equal, in the absence of capital gains, the growing concentration of saving flows at the top would have led to a significantly larger increase in measured wealth inequality. Whether capital gains that are the result of falling discount rates are welfare-improving is a question subject to much debate [START_REF] Moll | Comment on "Sources of U.S. Wealth Inequality: Past, Present, and Future[END_REF][START_REF] Cochrane | Wealth and Taxes[END_REF]. Our paper shows that capital gains moderated the increase in wealth inequality in recent decades. To the extent that this middle-class wealth growth was based on welfare-neutral increases in housing valuations, this would mask a much more substantial increase in welfare-relevant wealth inequality in recent decades.

We also show that the global wealth boom of the past four decades bypassed the bottom half of the wealth distribution where wealth-to-income ratios stagnated. For most households in the lower half of the distribution, capital gains were minimal as home ownership rates in this part of the distribution are low and holdings of risky financial assets small. The consequence was a marked polarization of wealth between the upper half and the lower half of the distribution, or between the "haves" and "have-nots". This divergence is not captured in widely used inequality measures such as top wealth shares or the Gini coefficient despite its potentially important social and political implications.

In a final step, we combine the savings of the different wealth groups across the G3 and the government to understand the shifts in global saving flows in recent decades. Consistent with documented within-country trends, our analysis points to corporate saving as the single most important source of rising global saving, accruing predominantly to rich households in the top-10% of the wealth distribution. Linking household-level and national accounts data therefore shows that the "global corporate saving glut" (Chen et al. 2017b) was the main driver of the measured increase in global saving since the 1980s. We discuss the factors behind increasing corporate saving and its implications for saving inequality in the last section.

Related literature. Our paper is related to four strands in the literature. The first strand seeks to measure and analyze the long-run evolution of aggregate wealth. [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF] constructed aggregate wealth and saving series for eight advanced economies. Follow-up studies construct similar series for China (Piketty et al. 2019), Sweden [START_REF] Waldenström | Wealth-income ratios in a small, developing economy: Sweden, 1810-2014[END_REF], India [START_REF] Kumar | Poor country, rich history, many lessons: The evolution of wealth-income ratios in India 1860-2012[END_REF] or Spain [START_REF] Artola-Blanco | Wealth in Spain 1900-2017: A Country of Two Lands[END_REF]. Our paper is the first to study the contribution of assetspecific capital gains and saving for the accumulation of household wealth from a cross-country perspective after unveiling the indirect ownership of assets through investment and pension funds [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF]Mian et al. 2021b).

The paper also relates to the literature studying the factors shaping the long-run dynamics of the wealth distribution (Alvaredo et al. 2018a;[START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF][START_REF] Kuhn | Income and wealth inequality in America, 1949-2016[END_REF]; [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF][START_REF] Martínez-Toledano | House price cycles, wealth inequality and portfolio reshuffling[END_REF][START_REF] Albers | The Distribution of Wealth in Germany, 1895-2018[END_REF]. We construct new series of household wealth distribution that are consistent with aggregate wealth statistics in the U.K., China and, for the 1980s, in Germany (for the U.S., France and Spain we rely on existing series from [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF][START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF], and Martínez-Toledano 2020; we rely on the German series from [START_REF] Albers | The Distribution of Wealth in Germany, 1895-2018[END_REF]Albers et al. for the period 1993Albers et al. -2018)). Furthermore, we provide a new database of saving and capital gains in specific assets (housing, business assets, fixed-income assets, equity), for different wealth groups, in all countries covered in our study. Importantly, the capital gains we compute are consistent with market data for returns on different assets [START_REF] Jordà | The rate of return on everything, 1870-2015[END_REF].

Our paper equally speaks to a rapidly growing theoretical and empirical literature on heterogeneous saving rates and asset portfolios for the dynamics of wealth inequality [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF][START_REF] Bach | From saving comes having? disentangling the impact of saving on wealth inequality[END_REF][START_REF] Benhabib | Skewed wealth distributions: Theory and empirics[END_REF][START_REF] Fagereng | Heterogeneity and persistence in returns to wealth[END_REF]Fagereng et al. 2021;[START_REF] Kuhn | Income and wealth inequality in America, 1949-2016[END_REF][START_REF] Martínez-Toledano | House price cycles, wealth inequality and portfolio reshuffling[END_REF][START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF]Mian et al. 2021b). Our study quantifies the contribution of capital gains and saving in all asset types for the wealth accumulation of different wealth groups and to evaluate their specific role for changes in the wealth distribution for the largest economies.

The final strand of the literature seeks to understand the link between the global saving supply [START_REF] Bernanke | The global saving glut and the US current account deficit[END_REF]Rachel and Smith 2015;[START_REF] Rachel | On falling neutral real rates, fiscal policy and the risk of secular stagnation[END_REF] and the fall of natural interest rates (Holston et al. 2017, Rees and[START_REF] Rees | The natural interest rate in China[END_REF]. An important literature in recent years investigated this crucial macrofinance question, investigating the role of demographics [START_REF] Eggertsson | A model of secular stagnation: Theory and quantitative evaluation[END_REF][START_REF] Carvalho | Demographics and real interest rates: Inspecting the mechanism[END_REF][START_REF] Gagnon | Understanding the new normal: the role of demographics[END_REF][START_REF] Coeurdacier | Credit constraints and growth in a global economy[END_REF][START_REF] Auclert | Demographics, Wealth, and Global Imbalances in the Twenty-First Century[END_REF], inequality (Mian et al. 2021a;Mian et al. 2021c; [START_REF] Klein | Trade Wars Are Class Wars[END_REF], and the rise of emerging countries with high demand for safe assets [START_REF] Bernanke | The global saving glut and the US current account deficit[END_REF]Caballero et al. 2008;[START_REF] Caballero | The safety trap[END_REF]. An important limitation of this literature was the absence of cross-country saving series decomposed by population groups over a longer time span that our paper provides.

The paper is organized as follows. Section 2 introduces the data, concepts and methods being used. We provide evidence on how the increase in wealth-to-income ratios was allocated across different wealth groups in section 3. Section 4 turns to the asset-specific analysis of household capital gains and saving flows. Section 5 studies the dynamics of global saving and its distribution since the 1980s. Section 6 concludes.

Concepts, data and methodology

Throughout this paper we use concepts and definitions of personal wealth and saving from the international system of national accounts (SNA 2008) and employ distributional data consistent with official balance sheets of the household sector. We follow the guidelines of the Distributional National Accounts project (Alvaredo et al. 2021), which provide a framework to construct comparable wealth inequality series from a variety of data sources (e.g., income tax returns, estate tax, survey data, etc.) which are also coherent with the official national accounts. The countries included in this study are France, Germany, Spain, the UK, China and the US. To simplify the presentation of our analysis, we group results for France, Germany, Spain and the UK (taking the weighted average of the four countries) to represent a single region, Europe.

Main concepts

Household wealth. Our concept of wealth follows standard definitions. The wealth (sometimes also referred to as the net worth) of the household sector (W H ) is the sum of non-financial assets (AN F H ) plus financial assets (AF H ) owned by households, less liabilities owed by households (L H ): W H = AN F H + AF H -L H . To decompose household wealth into different types of non- 1950-20181950-20181950-20181950-2018China 1995-20181990-20181995-20181995-2018Germany 1950-20181950-20181983-20181983-2018France 1970-2018[START_REF] Atkinson | On the measurement of inequality[END_REF]-2018[START_REF] Atkinson | On the measurement of inequality[END_REF][START_REF] Gordon | The turtle's progress: Secular stagnation meets the headwinds[END_REF][START_REF] Atkinson | On the measurement of inequality[END_REF][START_REF] Gordon | The turtle's progress: Secular stagnation meets the headwinds[END_REF]U.K. 1967-20181967-20181980-20151980-2015Spain 1980-20181980-20181984-20151984-2015 financial assets, financial assets and liabilities, we adhere to the classification of asset types in • Non-financial assets: sum of housing and business assets. Housing is the combination of the market value of dwellings (structure) and land underlying dwellings: in practice, it is generally easier to measure the sum (as in observed real estate transactions) than the two components separately. Business assets refer to non-residential fixed capital (structures, equipment, IP) and inventories, excluding durable goods. Durable goods are not considered as an asset in the SNA and, thus, we do not include them within business assets. See the description of equity for a discussion of the treatment of unincorporated businesses.

• Financial assets: sum of fixed-income assets, equity, investment funds shares, and life insurance and pension assets. Fixed-income assets are the sum of cash, deposits, bonds, and loans. Equity is the sum of both listed and unlisted shares. It also includes unincorporated businesses functioning as "quasi-corporations" (i.e., businesses that "have an economic and financial behaviour that is different from that of their owners and similar to that of corporation", ESA-2010). Investment funds shares are participations in companies pooling capital belonging to numerous investors used to purchase securities collectively. Life insurance and pension funds correspond with the financial assets of policy holders or beneficiaries and liabilities of insurers or pension funds.

• Liabilities: sum of long-term loans and other debt. Throughout the paper we treat long-term loans as "housing debt" and other debt as "non-housing debt", except in the US and China, where we distinguish housing and non-housing debt and use them directly (non-housing debt is the sum of consumer and business debt). Long-term loans are loans with maturity longer than one year. Although not all long-term loans are housing debt, they constitute its vast majority.

Household saving. To define household saving, we follow the same principles as with household wealth. Namely, household saving (S H ) is the sum of saving in non-financial assets (S

Data

Data on aggregate wealth and saving. For European countries and the U.S., we use official national accounts to construct the series of aggregate household wealth, distinguishing further by all asset classes. We cover the period 1980-2018. These data were originally developed by [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF], and later updated by [START_REF] Bauluz | Revised national income and wealth series: Australia, Canada, France, Germany, Italy, Japan, the UK and the USA[END_REF]. For Germany, we rely on corrected national accounts' estimates of housing wealth and business assets for the period 1990-2018, given that these two assets are undervalued in official balance sheets (Albers, Bartels and Schularick, 2020, pgs. 28-32). For China, we use the aggregate balance sheet of the household sector for the 1980-2015 period from Piketty et al. (2019), which we extend for the period 2016-2018 using [START_REF] Li | China's National Balance Sheet[END_REF]. The aggregate wealth data is found in the annual statistics on the financial and non-financial assets and the liabilities of the household sector. In line with SNA-2008 guidelines, all wealth components are measured at market values. The net worth of the household sector is defined as the sum of net financial wealth net of liabilities and non-financial assets.

To construct series of asset-specific saving flows, we use data on financial and non-financial saving of the household sector from official national accounts. Non-financial saving (i.e., saving in housing and in business assets) are obtained from the real accounts, while saving on financial assets (including debt acquisitions) from the financial accounts. We follow the same approach with the corporate sector and use financial flows from financial accounts and non-financial flows from real accounts to compute corporate saving. Importantly, these data include information on the portfolio composition along the wealth distribution. Different methods were used to produce the series: income capitalization (France, US, Spain), estate multiplier method (UK), combined survey and national accounts data (Germany, China). The distributional data are considered to be of high quality, and have been previously used for comparison purposes in the literature (e.g., Alvaredo et al. 2018b;[START_REF] Blanchet | Applying Generalized Pareto Curves to Inequality Analysis[END_REF]. For all analyzed countries, the distributional series are consistent with aggregate household wealth from official balance sheets.

The French series were constructed by [START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF]; the U.S. series by [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF], later updated by [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF] 3 ; the Spanish series by Martínez-Toledano (2020). We construct the German series adhering to the same procedure followed by [START_REF] Albers | The Distribution of Wealth in Germany, 1895-2018[END_REF] to estimate the German wealth distribution for the period 1993-2018. We then extend these series back to 1983 using the same data source and methods. For the UK, we rely on estate tax data and the estate multiplier method, as implemented in Alvaredo et al. (2018a) (and previously by [START_REF] Atkinson | Distribution of personal wealth in Britain[END_REF]. One important difference in our study with respect to theirs is that we include pension assets using alternative sources. Pension assets are excluded from estate taxes in the UK and, for B1).

While the main results of the paper are not significantly affected by this imputation procedure, we acknowledge the lower precision of the UK series vis à vis those of other countries.

We construct wealth distribution series for China using available wealth surveys (CHIP survey for 1995and 2002, biannual CFPS survey for 2012-2018)). We make several adjustments to raw survey data. Most importantly, we uprate each asset type to match the aggregate asset totals.

We use a uniform uprate factor estimated as the ratio between the macroeconomic asset total and the survey asset total. This approach is a well-known method to correct surveys and is widely used in the literature on wealth inequality [START_REF] Albers | The Distribution of Wealth in Germany, 1895-2018[END_REF][START_REF] Feiveson | Lifecycle patterns of saving and wealth accumulation[END_REF]Bauluz and Meyer 2021;Mian et al. 2021c). This procedure primarily affects financial assets, for which the survey totals are significantly below the balance sheet totals. The resulting distribution is more unequal than the raw survey distribution, since financial assets are strongly concentrated at the top of the distribution. Hence this adjustment also indirectly accounts for the underestimation of the rich in the survey (whose portfolio is relatively dominated by financial assets). This is the reason why our wealth inequality series for China follow closely that in Piketty et al. (2019), (Figure Appenidix B1).

Unveiling indirectly held financial assets

To understand the role of specific assets in global wealth growth, we need to unveil households' indirect asset ownership and saving through pension and investment funds. We use the asset portfolio of investment and pension funds to infer the assets owned indirectly by households through their holdings of pension assets and investment funds shares. The stock and saving flows of pensions and investment funds can be obtained from official balance sheet for financial subsectors (e.g., SNA-2008 subsectors S.124, S.128 and S.129). The "unveiling" exercise allows us to decompose the portfolio of households into three asset categories: housing, fixed-income assets (cash, deposits, bonds and loans) and equity (including unincorporated business assets). Importantly, the "unveiled" series of household assets and saving flows allows us to link undistributed corporate profits (a saving flow accrued to equity owners) with the stock of equity held directly and indirectly through investment and pension funds (see section 2.4).

We build on the work on indirect ownership through investment and pension funds in the U.S. [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF]Mian et al. 2021b), but our study is the first to provide a database of showing the proportion of investment funds and pensions funds that is invested into fixed-income assets and equity. For example, in 2018, pension funds had 64% of their assets invested in fixedincome claims, and 36% in equity. We then apply this proportions to household claims in pension funds (30% of their total assets), and obtain that U.K. households' hold 19% of their assets in fixed-income claims through pension funds (i.e., 64% of 30%), and 11% in equity (i.e., 36% of 30%). This is illustrated by the garnet dotted lines in Figure 2. The same exercise is conducted for investment funds, which in 2018 held 35% of their assets in fixed-income claims, and 65% in equity (blue dotted bars).

Wealth accumulation: saving vs. valuation effects

Aggregate wealth accumulation. To account for the dynamics of wealth accumulation, we decompose wealth growth into saving (volume effect) and capital gains (relative price effect). One important benefit of this method is that the same framework which is applied at the aggregate level (i.e., total household wealth) can be adopted at the distributional level. This allows full consistency between the macro and distributional results of our paper.

This method links the accumulation of wealth to the corresponding flow of saving, and obtains the role of asset prices as a residual [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF][START_REF] Waldenström | Wealth-income ratios in a small, developing economy: Sweden, 1810-2014[END_REF][START_REF] Artola-Blanco | Wealth in Spain 1900-2017: A Country of Two Lands[END_REF]). We go beyond previous studies and link the accumulation of each asset class and debt type to its corresponding flow of saving. 4 That is, we analyze the influence of saving and prices in the evolution of different asset classes and debt categories, and determine the relative influence of asset-specific saving and capital gains for the aggregate dynamics of wealth. As explained in sub-section 2.1, in the case of equity, saving is the sum of corporate saving accrued to households and net acquisitions of equity by households (see also Appendix A.1). Note that this method requires us to use the "unveiled" series of household assets and saving flows, which in turn allows us to link corporate saving with the stock of equity held directly and indirectly through investment and pension funds.

More precisely, for a given wealth component (housing, business assets, fixed-income assets, equity, housing debt, other debt), we decompose our series using the following transition equation:

A t+1 = (1 + q t+1,A )A t + S t+1,A (1)
where A t+1 and A t are the real value at time t+1 and t of a given wealth component from the households' wealth. S t+1,A is the net-of-depreciation saving flow in wealth component A during year t + 1. Finally, q t+1,A is the real capital gain or loss from wealth component A between time t and t + 1. In the previous equation, the capital gain part is obtained as a residual, since we observe all the other elements (i.e., annual stocks of assets or debts and flows of net saving or net dissaving).

The previous law of motion (eq. 1) can be rewritten for the accumulation of net wealth:

W t+1 = (1 + q t+1 )W t + S t+1 (2)
Where W is the sum of all asset classes, net of liabilities, and S is net-of-depreciation saving (i.e., saving in financial and non-financial assets, minus debt acquisitions). The capital gain q is the weighted average of capital gains in the different asset classes and debt, where the weights are given by the proportion of each wealth component in net wealth.5

Distributional synthetic saving. We apply the same framework to analyze the dynamics of the wealth distribution. Concretely, we use the synthetic saving method, introduced by Saez and Zucman ( 2016), and used extensively in the literature since [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF][START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF][START_REF] Kuhn | Income and wealth inequality in America, 1949-2016[END_REF][START_REF] Martínez-Toledano | House price cycles, wealth inequality and portfolio reshuffling[END_REF]Mian et al. 2021b). The method adapts the accumulation equations introduced above (equations 1 and 2) to specific wealth groups.

For a given wealth group (e.g., the top-10%), we use the following law of motion specific to each wealth component (housing, business assets, fixed-income assets, equity, housing debt, other debt):

A i t+1 = A i t (1 + q t+1,A ) + S i t+1,A (3)
where A i t+1 , and A i t denote the value of wealth component A in times t + 1 and t, for wealth group i; S i t+1,A is the synthetic saving in asset type A in year t + 1 by wealth group i; and q t+1,A is the rate of capital gains on asset type A between t and t + 1.

Synthetic saving are obtained residually, from observed data on the asset or debt values of class A of group i, and the rates of change of asset prices observed for the household sector (obtained from equation 1). Note that Equation 3 above, when applied to equity assets, provides a (residual) saving in equity that already includes corporate saving (together with direct equity acquisitions and indirect equity acquisitions through investment and pension funds) since saving in equity in Equation 1 includes retained corporate earnings.6 Importantly, the unveiling exercise is crucial since it allows us to distribute the share of corporate saving that is due to equity owners through pension and investment funds (see Section 2.3).

Accordingly, we can generalize Equation 3 to account for the accumulation of net wealth of a given group:

W i t+1 = W i t (1 + q i t+1 ) + S i t+1 ( 4
)
where W is the sum of all asset types, net of debt, of wealth group i. The capital gain becomes group specific (q i ), since it is the weighted average of each wealth component A in the net worth of group i.

Note that the synthetic saving method relies on two assumptions. First, it assumes that individuals in a given wealth group in year t remain in this group in t+1. In practice, however, there is always mobility between wealth groups over time, although inter-group persistence is relatively high, at least for broad wealth groups such as the top-10%, the middle-40% and the bottom-50%.

For example, available evidence for the US [START_REF] Kuhn | Income and wealth inequality in America, 1949-2016[END_REF], Germany [START_REF] Albers | The Distribution of Wealth in Germany, 1895-2018[END_REF], Spain (Martínez-Toledano 2020) and Sweden [START_REF] Bach | From saving comes having? disentangling the impact of saving on wealth inequality[END_REF] indicates that the probability of remaining in the top-10% of the wealth distribution in the subsequent survey wave is around 70 to 80%. For China, we also find relatively high inter-group persistence, being only slightly below the US (Table B1). Throughout the paper, we analyze three wealth groups: the bottom-50%, the middle-40% and the top-10%. Since these groups are relatively stable over longer horizons, we can follow them over time in a "pseudo panel".7

The method also assumes that different wealth groups experience the same capital gain for individual assets (e.g., housing, equity, etc.). As a result, variation in capital gains across wealth groups reflects portfolio heterogeneity. In the Appendix Figure B11, we document large portfolio differences across wealth groups, consistent with previous literature. Whether capital gains at the asset-class level differ systematically across wealth groups is an open question. However, available evidence for the U.S., Spain and Norway suggests rather small within-asset class heterogeneity of capital gains rates (Mian et al. 2021b;[START_REF] Martínez-Toledano | House price cycles, wealth inequality and portfolio reshuffling[END_REF][START_REF] Fagereng | Heterogeneity and persistence in returns to wealth[END_REF]). As a result, the well-known result that returns (i.e., the sum of the income flow and the capital gain) for individual assets differ across wealth groups [START_REF] Fagereng | Heterogeneity and persistence in returns to wealth[END_REF][START_REF] Bach | From saving comes having? disentangling the impact of saving on wealth inequality[END_REF][START_REF] Xavier | Wealth inequality in the US: the role of heterogeneous returns[END_REF], would largely reflect variation in income flows.8

The existing literature indicates that the synthetic saving method is relatively robust, whether compared to alternative methods to estimate saving rates (e.g. "income minus consumption ap- ). Moreover, it is important to recall that there is one specific saving flow -corporate saving -which does not rely on the synthetic saving method since it is allocated directly to equity owners. This is explained in great detail in Appendix A.1 (also in footnote 6). Given that equity owners are persistently over-represented at the top-end of the wealth distribution, the bulk of corporate saving is consistently allocated to the top-wealth groups. As we will see, corporate saving was a key determinant of household saving growth over the past decades. It is important to stress that this key finding regarding the central role of corporate savings does not rely on the synthetic saving method.

The distributional anatomy of the global wealth boom

We start by investigating the distributional anatomy of the surge in global wealth-to-income ratios in recent decades. In a first step, we look at the rise of wealth-to-income ratios for different wealth groups. We then turn to asset-specific saving flows and capital gains in the next section. As discussed above, the European total is constructed as the (weighted) average of Germany, France, Spain and the United Kingdom. We follow the convention in the literature and study three broad wealth groups: households in the top-10%, households in the middle-40% ("middle class"), and households in the bottom-50% of the wealth distribution (see section 2.4).

Figure 3 shows the cumulative change of household wealth-to-national income ratios in the three world regions since 1980. 9 The wealth of the top-10% and the middle 40% increased substantially between 1980 and 2018, while the bottom-50% did not participate in the wealth boom.

The group-specific accumulation patterns are presented in Table 3, in columns 1-3, which show wealth-to-national income ratios of the three wealth groups at the beginning and at the end of the period (columns 1-2) and each group's share in the total wealth growth (column 3). The top-10% saw the largest wealth increase in all regions, capturing 57% of the total wealth growth in Europe, 69% in China and 74% in the U.S. Overall, wealth growth was shared between the rich and the middle classes in the proportions of about two-thirds to one-third. The developments in Europe were the most favourable to the households outside the top-10%, where the middle class increased their wealth by almost 100pp of national income (and captured 40% of the wealth growth). In the U.S. and China, the middle-40% captured 26% and 28% of the wealth growth, respectively.

A strong fact that stands out is the absence of wealth gains in the lower half of the wealth distribution. Wealth in the bottom half decreased markedly in the U.S. and China, and stagnated in Europe. Half of the population did not participate in the global wealth boom of the past 40 years.

While wealth-to-income ratios doubled globally, about half of the population saw either decreasing 9 The graph shows the cumulative increase in household wealth (W H ) over national income (Y ) in each year n after 1980 (

W H 1980+n Y1980+n - W H 1980 Y1980
), as the sum of the wealth increase of each wealth group i relative to national income:

i W i 1980+n Y1980+n - W i 1980 Y1980 .
or stagnating wealth-to-income ratios. The upshot is that the wealth gap between the lower and the upper half sharply widened, leading to a growing wealth polarization.

Table 3: Wealth-to-national income ratio, share in total wealth growth, and share in total net wealth of three wealth groups (top 10%, middle 40% and bottom 50%) in the U.S. , Europe (1980-2018) and China (1995-2018) Columns 4-5 of Table 3 show changes in wealth inequality, looking at the development of the wealth shares of the three wealth groups. Trends were markedly different across the world regions.

[1] [2] [3] [4] [5] Wealth- income ratio b-o-p Wealth- income ratio e-o-
The surge in wealth-income ratios was accompanied by a relatively moderate increase of wealth inequality in Europe (the top-10% share increased by 3 percentage point), and a more substantial widening in the US (the top-10% share increased by 7 percentage points), but it went hand in hand However, the commonly used inequality measures as the top 10% shares (or more generally top shares or the Gini coefficent) tend to overlook the growing polarization of wealth between the upper and lower half of the distribution. Since the bottom half of the population has little wealth, changes in top wealth shares will largely reflect inequality among wealth holders (the "race between the middle classes and the rich") and will tend to overlook distributional dynamics at the bottom. This issue is even more pertinent to synthetic inequality measures such as the Gini coefficient. The fact that the bottom half of the population has been, by and large, left behind in the global wealth boom of the past decades, while wealth in the upper half rose substantially (albeit for different reasons, as we will see), has been often overlooked. 10 Generally, no single inequality measure can effectively summarize all distributional dimensions and the choice depends to a large extent on the specific research interest (e.g., [START_REF] Atkinson | On the measurement of inequality[END_REF][START_REF] Sen | On economic inequality[END_REF][START_REF] Cowell | Measuring inequality[END_REF]etc.). For example, [START_REF] Case | Rebottling the Gini: why this headline measure of inequality misses everything that matters[END_REF] have recently criticized the widespread use of the Gini coefficient and call instead for the use of inequality measures that reveal a more nuanced picture, such as percentiles ratios: "The best measures are those that match our purpose, or pick up on the places where important changes are happening." Figure 4 shows the inter-quartile ratio (75/25) of wealth and reveals the growing wealth gap between the upper and the lower half of the distribution. By this measure, wealth inequality has increased significantly more than top-10% wealth shares or trends in the Gini would suggest (see also Appendix Figure B2). 10 The growing wealth gap between the upper and the lower half of the distribution in Figure 3 and columns 1-3 in Table 3 is what is measured as "absolute" inequality, as opposed to predominantly used "relative" inequality [START_REF] Kolm | Unequal inequalities. I[END_REF]. And as pointed by [START_REF] Ravallion | Competing concepts of inequality in the globalization debate [with comments and discussion[END_REF], the absolute gap is what most people mean when talking about rising "gaps between poor and rich"; e.g. see [START_REF] Amiel | Measurement of income inequality: Experimental test by questionnaire[END_REF]). Measures of relative inequality are sensitive to proportional transformations, while absolute to additive transformations. That is, relative inequality will not change if all incomes/wealth increase proportionally, and absolute inequality will not change if all incomes/wealth increase in the same amount ("scale" versus "translation" independence).

Figure 4: Interquartile wealth ratio ratio in the U.S., Europe and China (1995=100) Note: This figure shows the evolution of the log of the ratio of the 75th and the 25th percentile's wealth relative to 1995 (=100) in the U.S. (1980-2018), Europe (1980-2018), and China (1995-2018). The levels of log(75/25) in 1995 were 2.9 in the U.S., 2.5 in Europe and 1.2 in China. Europe here is the weighted average (by national income) of France and Germany.

Unveiling global saving flows and capital gains

In this section, we will extend our analysis of the global wealth boom in the past 40 years along two important dimensions. First, we will distinguish between the role of saving and capital gains in individual asset classes. Second, we will study the contribution of saving and capital gains in individual asset classes to wealth growth across different wealth groups.

Figure 5 plots the change of different wealth components relative to national income since 1980, after "unveiling" indirectly-held assets through pension and investment funds. Figure B5 in the appendix shows the corresponding graph before "unveiling". We classify household assets into three basic types: housing, fixed-income assets, and equity (including unincorporated business assets).11 funds (i.e., the ratio of directly held fixed-income assets-to-national income in Appendix Figure B5 barely changed since 1980).

The second insight is that housing wealth was the main driving force for the wealth increase in Europe and China, experiencing an upsurge close to two times the national income since 1980.

In the U.S., in contrast, housing has been relatively less important with the exception of the housing boom of the 2000s. Instead, the sustained increase in equity values was the main factor behind the rise in the wealth-income ratio in the U.S. The increase in the value of equity and business assets is consistent with the surge in stock market capitalization in advanced economies since the 1980s ( [START_REF] Kuvshinov | The big bang: Stock market capitalization in the long run[END_REF].

Finally, we document a significant increase in household indebtedness over the last decades.

While this development was particularly strong in the U.S. and Europe until the 2008 financial crisis, it has accelerated in China since. In China, household debt increased from around 20% of national income in 2008 to close to 60% in 2018. On average, the rise in household indebtedness in the three world regions since 1980 has been equivalent to around 50% of national income. This corroborates previous findings in the literature which document the steady increase in household debt-to-national income ratios in rich countries since the 1950s (Jordà et al. 2016;[START_REF] Bartscher | Modigliani Meets Minsky: Inequality, Debt, and Financial Fragility in America, 1950-2016[END_REF].

Aggregate saving flows and capital gains

Wealth can rise because existing assets increase in value, because households acquire new assets, or because debt falls. To understand the respective role of these forces, we return to the law of motion for aggregate wealth described in section 2 (Equations 1 and 2). In Table 4, we present the results for the decomposition of household wealth accumulation into saving and capital gains over the 1980-2018 period. Overall, household wealth grew at a remarkably similar speed in Europe and the US (around 4% per year). In China, the wealth growth rate was three times faster (12.5%).

In columns 2 and 7, we show the contributions of saving and capital gains in the accumulation of household wealth for each of the G3-economies. Saving and capital gains each contributed roughly half to wealth creation in all three economies.12 In columns 3-5 (for saving) and columns 8-10 (for capital gains), we split aggregate wealth growth into the three main asset categories which, for ease of exposition, we express net of debt:

housing (net of mortgage debt), fixed-income assets (net of non-mortgage debt), and equity (including business assets). The category 'equity and business' includes the following asset types:

(i) unincorporated business assets, (ii) direct equity acquisitions, (iii) indirect equity acquisitions through investment and pension funds, (iv) and corporate saving.

Fixed-income investment have played a central role for saving accumulation in all countries.

They alone account for around 30-to-40% of overall wealth growth. This accumulation of fixedincome assets by households has mainly taken place through investment and pension funds and only becomes visible after unveiling their holdings. Saving flows into housing (net of housing debt) negatively contributed to wealth accumulation, reflecting the higher growth of housing debt relative to residential investment (in the U.S. in particular amid rising equity extraction). Saving in business equity accounts for around one-fifth of the overall wealth accumulation. Interestingly, most of the saving in equity happened through undistributed corporate profits (column 6). Corporate saving saving.

was almost negligible in the early 1980s, but contributed around one third of household saving in recent years (Appendix Table B2). These findings are consistent with other research showing the growing importance of corporate saving globally (Chen et al. 2017b; [START_REF] Gruber | The corporate saving glut and falloff of investment spending in OECD economies[END_REF].

On the capital gains side (columns 7-10), both housing and equity have registered substantial appreciation. Together, they explain about 60% of the overall increase in personal wealth. Strikingly, capital gains on net housing capture around 40-to-50% of aggregate wealth growth. Taken at face value, this result implies that wealth-income ratios would have barely increased in the absence of the gains in this particular asset. The capital gains in net housing are also due to the decrease in real debt values with inflation. This is mirrored on the asset side by negative capital gains on fixed income assets, as inflation tended to reduce their real value [START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF]). 13 The central role of housing capital gains does not mean that capital gains on equity and business assets were unimportant (column 10). They account for about one fifth of total wealth

growth. Yet it is important to recall that the measure of equity capital gains used here is net of corporate saving in the form of retained earnings in recent decades [START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF], Mian et al. 2021b, Kuvshinov andZimmermann 2021, Greenwald et al. 2019). In our estimates, more than half of the surge in stock prices is driven by the increase in undistributed corporate profits and should, therefore, not be treated as a capital gain. However, capital gains on equity were heterogeneous across world regions. In the U.S., they accounted for close to one-fourth of overall wealth accumulation, but played a smaller role in China and Europe. 14 13 The evolution of housing prices in China is notably impressive. According to our estimates, the annual capital gain in housing was slightly above 11% during 1995-2018. While this figure is substantially above official housing price indexes (which show no real increase in housing prices), it is in line with results from [START_REF] Fang | Demystifying the Chinese housing boom[END_REF] and [START_REF] Deng | Evaluating the risk of Chinese housing markets: What we know and what we need to know[END_REF], which estimate the evolution of real housing prices across a large sample of Chinese cities since the early 2000s. By construction, our housing capital gains coincide with those estimated by Piketty et al. (2019). [START_REF] Deng | Evaluating the risk of Chinese housing markets: What we know and what we need to know[END_REF] also show that rising urban land values, rather than construction costs, explain house price growth in China. See Appendix Figure B3 for a comparison of the capital gains in our study and the evolution of prices in [START_REF] Fang | Demystifying the Chinese housing boom[END_REF]. 14 Understanding the ultimate drivers of the different role of equity capital gains worldwide is beyond the scope of this paper, but one candidate explanation are regional differences in market power (e.g., De [START_REF] Loecker | The rise of market power and the macroeconomic implications[END_REF][START_REF] Loecker | The rise of market power and the macroeconomic implications[END_REF][START_REF] Philippon | The great reversal: How America gave up on free markets[END_REF].

Group-specific saving and capital gains

In this section, we study how saving flows and capital gains in individual asset classes were distributed across different wealth groups. We quantify wealth growth due to asset-specific saving and capital gains for different wealth groups. For comparison, Appendix Figure B7 shows that the resulting distribution of saving flows in the U.S. between the top-10% and the bottom-90% in this paper and in [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF] and Mian et al. (2021b) is very similar. But Table 5 goes beyond the previous literature and quantifies asset-specific wealth growth on a global scale.

We omit results for the bottom-50% wealth group because its net wealth is close to zero or even negative. As before, the Table provides a full decomposition of saving and capital gains for net housing (gross housing minus mortgage debt), net fixed income assets (gross fixed income assets minus non-mortgage debt) and equity and business assets.

The Table contains a number of important new insights. First, saving flows account for 72% of wealth growth of the top-10% in the U.S., 77% in Europe and 57% in China. In contrast, capital gains were the main source of wealth growth for the middle-40%, contributing to around two-thirds of the group's growth across the three economies. In line with the aggregate data (see Table 4), we find that saving in fixed-income assets accounted for the largest part of saving of all wealth groups, in particular of the "active" savers (the top-10% and the middle-40%).

Saving in equity and business has been concentrated in the top-10% and constitutes the second largest component of top-10% saving flows. We will discuss below that the importance of this component of saving increased substantially over time and that undistributed corporate profits account for the bulk of the saving in equity and business. Saving in net housing has largely been negative amid faster housing debt accumulation that outpaced investment in housing, especially for the middle classes in the US (see [START_REF] Bartscher | Modigliani Meets Minsky: Inequality, Debt, and Financial Fragility in America, 1950-2016[END_REF].

The table also shows that housing capital gains (column 6) constitute the largest part of total capital gains. Housing capital gains accounted for around 60% of total middle-class wealth growth in the last decades. Rising equity prices have been relatively more important in the U.S. contributing to the wealth growth of both the top and middle of the wealth distribution (around one fourth of the wealth growth of each group).

An important theme emerging from our work is that the distribution of saving flows since was much more unequal than the distribution of capital gains. This is shown in Figure 6. 15 The top-10% wealth group accounted for as much as 94% of the total saving flows in the U.S. since the 1980s, 75% in Europe and 84% in China (Figure 6, upper panel). The remaining saving were made by the middle-40%, while the bottom-50% had zero or negative saving in all three economies.

The picture is markedly different on the capital gains side where the top-10% accounted on average for around 45% of the total capital gains in the three economies (Figure 6, lower panel), with approximately the same share going to the middle-40%. The bottom-50% also obtained a positive share of capital gains that counteracted net borrowing on the saving flow side.

Overall, one of the key finding of this paper is that since the 1980s saving have been an

unequalizing force for the wealth distribution, while capital gains have been an equalizing force.

We illustrate the respective distributional roles of saving and capital gains by means of a counterfactual analysis (explained in detail in the Appendix A.2). Setting capital gains in all assets for all groups to zero since 1980, which is equivalent to assuming that all wealth groups i experienced the average capital gains of the economy (i.e., q i = q in Appendix Equation 13), and making the These findings are connected to the debate whether capital gains are welfare improving if they are driven by falling discount rates [START_REF] Moll | Comment on "Sources of U.S. Wealth Inequality: Past, Present, and Future[END_REF][START_REF] Cochrane | Wealth and Taxes[END_REF]. We showed that the middleclass was the main beneficiary of capital gains in housing that moderated the increase in wealth inequality. To the extent that some of the middle-class housing capital gains reflect welfare-neutral increases in market value, they could mask a more substantial increase in wealth inequality driven by the concentration of new saving flows at the top of the distribution.

Global saving dynamics since 1980

In this section, we study the dynamics of global saving since the 1980s. Our focus will be on shifts in the sources of saving flows across household groups. More precisely, we address two key questions: (i) which shifts have occurred in the intra-household sector saving distribution in the past 40 years?; (ii) which changes have taken places in the asset composition of saving accumulation? In a first step, we look at the dynamics of saving in each of the three major economies (U.S., Europe, 16 As explained in detail in Appendix A.2, saving have been a strong unequalizing force for the wealth distribution in all three countries. This is because the distribution of saving was more unequal than the initial wealth distribution (see Appendix, Equation 14). As a result, new saving exacerbated relative differences between the wealth groups. In contrast, capital gains had a moderating impact on the wealth distribution in all countries, as the top-10% experienced a relatively lower average real capital gains rate on its wealth (Appendix Figure B10). This is the result of the heterogeneous portfolio of the rich and the middle classes, with middle classes relatively more invested in housing and the rich in financial assets (Appendix Figure B11, left panel). The differential price effect is a result of the different price dynamics experienced by these assets. In line with the results in Section 4, housing saw the highest price increase, exceeding that on financial assets (Appendix Figure B11, right panel). China) individually. In a second step, we combine the saving of the three regions into a single area, the "G3", in an effort to trace and understand the shifts in global saving dynamics in recent decades.

Concentration of saving at the top and middle-class dissaving

We begin by tracking the saving of three wealth groups within the individual G3-economies. Fig-

ure 8 shows the evolution of saving relative to national income for the top-10%, middle-40% and the bottom-50% of households in the U.S., Europe, and China. In all three economies, we observe a growing concentration of saving at the top, reflecting a widening gap in the saving between the top group and the rest. While there is some fluctuation over time, in all three economies, the gap between saving flows coming from the top-10% and the middle-class (i.e., the middle-40%) was considerably smaller at the beginning of the period than today. In the U.S., the difference between top-10% saving and the middle-40% stood at about 3pp. of national income in the early 1980s, and about 2pp. in Europe. By 2018, the gap had more than doubled to about 8pp. in the U.S. and 5pp.

in Europe. The most remarkable surge in saving inequality took place in China. The gap between the top and the middle was around 3pp. in the mid-1990s and has widened to about 15pp. in the most recent years. This trend of widening saving disparities was driven by two interconnected developments.

First, an increase in saving of the top-10%. The exact size of the increase varies by region. For instance, in the U.S. the richest 10% of households saved about 7% of national income in the 1980s.

In the past decade, top-10% saving climbed to 9-10% of national income. The trend towards higher saving at the top was even more pronounced in China and also in Europe until the global financial crisis. Our findings here underscore that the U.S. "saving glut of the rich" discussed by Mian et al. (2021b) was indeed a global phenomenon. In line with their work, our calculations also support the idea surging saving of the top-1% played the key role for the increase in top-10% saving. This is shown in Figure B12 in the Appendix. Saving of the "next-9"% (i.e., households between the 90th and 99th percentile) remained stable while top-1% saving surged. This being said, our analysis relies on broader wealth groups to reduce the issue of intra-group mobility (see section 2.4).

The second trend is the simultaneous dissaving of the bottom-90%, and especially of the middle-40% of the distribution. In the U.S. and Europe, the decline in middle-class saving was particularly pronounced. In the U.S., middle-class saving decreased from 5% of national income in the 1980s to close to zero in the past decade. In Europe, the decline from 4% to about 1% was less pronounced, but still substantial. The bottom 50% had zero or slightly negative saving throughout, with little room to decline much further. With the exception of China, there has been little movement for the bottom half of the wealth distribution. In China, the data show a roughly commensurate decline of saving of the middle-40% and the bottom-50%.

Putting the pieces together, the years 1980 to 2018 witnessed a marked shift in the household saving distribution in all G3 economies. An increasing share of saving was supplied by households at the top of the distribution. As shown in Figure 9, around 1980 the top-10% accounted for around 60% of all household saving in the U.S. This increased to about 100% by 2018. In Europe, the share increased from about 60% to 80%, and from about 50% to 90% in China. The surge in the top-10% share in total saving reflected the combination of a trend towards larger saving flows from the top, and a collapse in middle-class saving in the U.S. and Europe (as shown in Figure 8).

The rise of corporate saving and the top-10% saving share

Using the asset-specific saving flow data constructed above, we can study the rise of top-10% saving and the decline of middle-class saving on the level of individual assets. We will distinguish three main asset categories: housing (net of mortgage debt), retained corporate earnings, and other financial assets (net of non-mortgage debt). increased alongside an increase in other financial saving at the top. On the other hand, the dissaving of the bottom-90% (and, particularly, of the middle classes) primarily occurred through lower saving in financial assets and higher debt accumulation, the latter driven by acquisition of mortgage debt (see Figure B13).

As discussed in section 2, corporate saving are allocated to the ultimate owners of the corporations in proportion to their equity holdings. The ultimate owners of companies are households and, in some cases, governments. The share of the corporate sector owned by the government is typically low today. In Europe and the U.S., corporations are predominantly owned by households while government ownership of corporations is still prevalent in China.17 Around 60% of Chinese corporate equity is in the hands of the government (Piketty et al. 2019; [START_REF] Huang | The Private Sector Advances in China[END_REF]. 18 Household equity ownership is heavily concentrated among rich households. As a consequence, corporate saving statistically accrue to the top-10% of households and reflect an increase in household saving of the owner-households (section 2).19 Our analysis links the rise in rich household saving -the "saving glut of the rich" (Mian et al., 2021b) -to a different secular macro development, namely the so-called "corporate saving their behalf (e.g., when households indirectly own equity through pension and investment funds). In practice, however, households experience an increase in the value of their equity holdings when corporations retain earnings.

glut" (Chen et al. 2017b; [START_REF] Gruber | The corporate saving glut and falloff of investment spending in OECD economies[END_REF]. The rise of corporate saving is itself linked to the secular increase of corporate profits and the capital income share (Piketty and Zucman 2014; [START_REF] Karabarbounis | The global decline of the labor share[END_REF][START_REF] Flores | The capital share and income inequality: Increasing gaps between micro and macro-data[END_REF]. As Chen et al. (2017a) out it: "The rise of corporate saving . . . emerged as the decline in the labour share pushed up corporate profits and dividend growth did not keep pace."

While the sizeable role of corporate saving for the dynamics of saving inequality since the 1980s is an important insight, identifying the causes of rising corporate profits and retained earnings is another issue and beyond the scope of this paper. The academic debate on the growing corporate profits, earnings retention and lackluster dividend growth is lively. Several factors count as potential causes for the increase in corporate profits: automation [START_REF] Holm | Saving behavior across the wealth distribution: The importance of capital gains[END_REF], rising market power [START_REF] Loecker | The rise of market power and the macroeconomic implications[END_REF][START_REF] Barkai | Declining labor and capital shares[END_REF], the decline in worker power (Stansbury and Summers 2020), and the decline in the cost of capital (Chen et al. 2017b). Dividend stickiness [START_REF] Lintner | Distribution of incomes of corporations among dividends, retained earnings, and taxes[END_REF][START_REF] Brav | Payout policy in the 21st century[END_REF][START_REF] Fama | Disappearing dividends: Changing firm characteristics or increased reluctance to pay[END_REF], cash hoarding [START_REF] Bates | Why do US firms hold so much more cash than they used to?[END_REF]) and tax incentives [START_REF] Stiglitz | Taxation, corporate financial policy, and the cost of capital[END_REF][START_REF] Auerbach | Capital gains taxation in the United States: Realizations, revenue, and rhetoric[END_REF]; [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF][START_REF] Atkinson | Distributional National Accounts (DINA) guidelines: Methods and Concepts used in the World Inequality Database[END_REF]) have also been put forward as explanation why dividends did not increase as much as profits, leading to the observed surge in corporate retained earnings.20

A global perspective

To track the evolution of global saving and its distribution in recent decades, we collapse the individual G3 economies into one. This is an intentional short-cut as distributional data are not available on a global level. The resulting G3 economy accounts for a relatively stable share of 50% in world GDP since 1980 (see Appendix Figure B14). The rise of China (from 3 to 16% of the world GDP) more than compensated the declining global GDP share of the advanced economies (from 47 to 37%; see Appendix Figure B14).

We treat the G3 as one integrated economy and aggregate national income and saving flows into a single area using market exchange rates. We assume that current accounts cancel out between these countries. Appendix Figure B16) confirms that this is a good empirical approximation.

To obtain a comprehensive coverage of global saving flows, we integrate public saving into the analysis. This allows us to compare shifts in the private saving distribution to changes in public saving.

We start from the national accounts definition of national saving i as the sum of the saving of households (S H i ), the corporate sector (S C i ), and the government (S G i ):

S N i = S H i + S C i + S G i .
The apostrophe denotes the saving of the government (S G ′ ) and households (S H ′ ) that already includes corporate saving, and express household saving as the sum of the saving of the top-10%

and bottom-90% of the wealth distribution (S

H ′ i = S T op10 i + S Bot90 i
). We rewrite national saving in country i as the sum of government saving and the saving of the top-10% and bottom-90% of the wealth distribution:

S N i = S T op10 i + S Bot90 i + S G ′ i .
Finally, we define the G3 national saving as the sum of the national saving of each region i:

S N G3 = i S N i = i S T op10 i + i S Bot90 i + i S G′ i
Appendix Figure B17 shows that trends in the G3 are a good proxy for global developments.

Gross (of depreciation) saving rates for the G3 and globally have evolved in lockstep since 1980 and increased about 4-5 percentage points. Returning to the household sector, we can show that the increase of global top-10% saving was closely linked to the secular increase in global corporate saving. This is illustrated in the upper left panel of Figure 12 that shows the decomposition of G3 top-10% saving into retained earnings and other saving. Top-10% saving, expressed as a percentage of G3 national income, increased by around 4 percentage points since the 1980s. Nearly all of this increase came from the surge in undistributed corporate profits (see the upper right panel of Figure 12). 22 The lower panel in Figure 12 shows that accompanying dissaving by the global bottom-90% primarily occurred through lower saving in fixed-income assets and higher debt accumulation (the latter being more pronounced in the late 1980s and particularly stark during the early 2000s housing boom). in corporate saving -are more intense in China than in the developed countries, but not different. In other words, the quick conversion of China into a market economy with high inequality and strong incentives for corporations to save (with the latter still more pronounced in China given strong capital market imperfections; see [START_REF] Song | Growing like china[END_REF][START_REF] Bacchetta | The demand for liquid assets, corporate saving, and international capital flows[END_REF][START_REF] Buera | Productivity growth and capital flows: The dynamics of reforms[END_REF], together with its growing share in the global economy, reinforced the global trend.

Conclusion

This paper has introduced a new international database on households' asset portfolios, capital gains and saving flows at the macro and distributional levels for the largest world economies since 1980: the U.S., Europe (France, Germany, Spain and the U.K.), and China. The data set provides cross-country series for "unveiled" indirect asset ownership through pension and investment funds.

We use the data set to expose the distributional anatomy of the global saving and wealth boom in the past four decades. We expose important shifts in the global saving and wealth distribution, and heterogeneous drivers of the global wealth boom.

First, we find that about half of the increase of aggregate wealth-income ratios during the last decades was due to capital gains, with new saving explaining the remaining half. Second, we show that saving flows have increasingly turned into an unequalizing force for the wealth distribution, while capital gains equalized the welath distribution. Importantly, housing capital gains obtained by the middle classes have been the main channel through which the middle-class wealth kept up with the top, and were the most critical factor moderating wealth inequality in recent decades. Saving played the central role for wealth accumulation of the top-10% of households, in particular saving in financial assets and rising corporate savings. Third, we present evidence that the gap between the "haves" and "have-nots" in the global wealth distribution widened as wealthto-income ratios stagnated or fell for the bottom 50% but surged in the upper half. Typically used inequality measures like top wealth shares or the Gini coefficient tend to overlook this dimension, as they mainly capture the dynamics within the 'haves' (e.g., the top-10% and the middle class).

Fourth, we expose an increase in saving inequality in major world economies since the 1980s, manifested in rising saving at the top and falling saving in the rest of the distribution. We find that the saving glut of the rich has been a global phenomenon, closely linked to the increase in global corporate saving: the "global saving glut of the rich" is the flip side of "the global corporate saving glut." The increase in retained corporate earnings accrues to equity owning households at the top of the distribution: "a global corporate saving glut of the rich" that underscores that the rise of profit shares and corporate earnings had major repercussions on global saving supply and its distribution.

Namely, a given stock of equity (directly and indirectly held) corresponds with a saving flow in three (instead of two) categories: direct equity saving, indirect equity saving through investment and pension funds, and corporate retained earnings. Corporate saving is made by corporations on behalf of equity owners and generates an increase in equity values proportional to the corporate saving flow. Since households (and governments) are the ultimate owners of corporations, the increase in equity values due to corporate saving should be treated as a saving flow and not a capital gain.

The application of the two-step approach with equity works as follows. At the aggregate level (step 1), define E t as the stock of equity (including both directly held and indirectly held through investment and pension funds) owned by households in year t. For ease of exposition, assume a closed economy (note that, in the paper, we incorporate the government and the foreign sector, as explained in section 2). In the same manner as with other assets or debt types, we decompose equity growth using Equation 1:

E t+1 = (1 + q net t+1,E )E t + S t+1,E (5)
Differently from other asset types, saving flows in equity (S t+1,E ) are the sum of two components: (i) saving flows made by households, both directly and indirectly through pension and investment funds, (S hh t+1,E ), and (ii) saving flows made by corporations, (S corp t+1,E ). From this equation we obtain capital gains as a residual. Let's refer to this capital gain as a capital gain 'net of corporate saving' (q net t+1,E ). As a result, Equation 5 can be rewritten as:

E t+1 = (1 + q net t+1,E )E t + S hh t+1,E + S corp t+1,E (6)
Next, we explain how the previous equations would be estimated if we disregarded corporate saving. Namely, we only include saving by households in Equation 1 and obtain a residual capital gains. In that case, we would obtain the following law of motion:

E t+1 = (1 + q gross t+1,E )E t + S hh t+1,E (7)
where capital gains are 'gross of corporate saving' (q gross t+1,E ). From equations 6 and 7 it follows that capital gains 'gross of corporate saving' are the sum of 'net capital gains' (q net t+1,E ) and the increase in equity values due to corporate saving (

S corp t+1,E
Et ):

q gross t+1,E = q net t+1,E + S corp t+1,E E t
In the second step, we look at the distribution. Using the synthetic saving approach, we follow the equity values of wealth group i over time. We apply the same law of motion in the case of equity as with other assets and debt types (equation 3). Since our goal is to obtain a measure of equity saving that includes both household and corporate saving, we use capital gains 'net of corporate saving' from equation 6 (q net t+1 ):

E i t+1 = (1 + q net t+1 )E i t + S i t+1,E (8)
where S i t+1,E is obtained as residual and is the sum of two components: (i) saving flows made by group i, both directly and indirectly through pension and investment funds, (S hh,i t+1,E ), and

(ii) saving flows made by corporations on behalf of group i, (S corp,i t+1,E ).

Note that the previous equation involves one key assumption. Namely, we assume that capital gains 'net of corporate saving' apply equally to different households groups i. For this to be true, however, corporate saving should be distributed proportionally to equity holdings: S corp,i t+1,E =

E i t Et • S corp t+1,E
This appears as a reasonable assumption given the analysis in Chen et al. (2017b) (and has also been used by the literature on synthetic saving before; e.g., [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF]Mian et al. 2021b). In their study, Chen et al. (2017b) carry a comprehensive analysis of corporate saving across both rich and developing countries over the last decades, using firm-level microdata (Compustat Global and Compustat North America). Importantly, Chen et al. (2017b) "do not find evidence that trends in firm saving relate significantly to firm size and age. Increases in corporate saving within industry, age, and size groups, rather than shifts... between these groups, account for the majority of the global rise of corporate saving". This result supports the assumption that corporate saving accrues to equity owners in proportion to their equity holdings.

We can accordingly write the previous equation splitting equity saving across the previous two components:

E i t+1 = (1 + q net t+1 )E i t + S hh,i t+1,E + S corp,i t+1,E (9)
Given the assumption that corporate saving is proportional to equity holdings, splitting equity saving of group i (S i t+1,E ) across the two saving types (i.e., household equity saving and corporate saving) is straightforward. It simply involves using the proportions of each saving flow at the aggregate level. Group's i household equity saving (S hh,i t+1,E ) and corporate saving (S corp,i t+1,E ) are defined, respectively, as:

S hh,i t+1,E = S i t+1,E • S hh t+1,E S t+1,E S corp,i t+1,E = S i t+1,E • S corp t+1,E S t+1,E (10)
An equivalent approach to obtaining each component of Equation 9 is to add corporate saving separately, without relying on the synthetic saving method. In a first step, total equity saving of group i (S i t+1,E ) is obtained from equation 8. In a second step, corporate saving (S corp t+1,E ) is allocated to each household group i in proportion to their equity holdings (

E i t Et ): S corp,i t+1,E = E i t E t • S corp t+1,E (11)
Finally, household equity saving of group i (S hh,i t+1,E ) are obtained as a residual of all other components in equation 9.

The two approaches are equivalent as long as corporate saving allocated to each household group i is the same in equations 10 and 11. This would be true if

E i t Et = S corp,i t+1,E S corp t+1,E
, which is precisely the assumption we use to distribute corporate saving proportionally to equity holdings: S corp,i t+1,E =

E i t Et • S corp t+1,E .

A.2 Simulations

This appendix section explains the simulation exercise conducted in section 4 (figure 7) to account for changes in wealth shares driven by saving. This simulations follow previous work using a similar framework, in particular, [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF] and [START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF].

We conduct a simulation exercise in which we first simulate the wealth shares if wealth accumulation was driven by saving flows only. For simplicity, we assume that capital gains in all assets were zero, which is equivalent to assuming all wealth groups experienced the average capital gain of the economy. As a result, capital gains are neutral for the wealth distribution.

More precisely, we start from the initial wealth holdings of each group, cumulate their saving (in fixed-income assets, housing, and equity and business assets; all components are net of debt), and estimate the wealth distribution. Subsequently allowing for differential price effect along the wealth distribution will 'return' the simulated distribution based on saving alone to the empirical wealth distribution. 23This procedure may be explicitly outlined using the wealth group-specific law of motion (Equation 4) together with the law of motion for aggregate household wealth (Equation 2, Section 4). In particular, using Equations 2 and 4, we characterize the law of motion of the share of wealth owned by a group i in total household wealth. We define sh i W,t as the share of wealth owned by group i in year t. Taking the ratio between Equations 4 and 2, we obtain:

sh i W,t+1 = (1 + q i t+1 )W i t + S i t+1 (1 + q t+1 )W t + S t+1 (12)
Defining the differential saving rate sh i S,t+1 as the ratio of group i's saving over total house-hold saving in year t + 1:

S i t+1
S t+1 , equation 12 can be expressed as:

sh i W,t+1 = 1 + q i t+1 1 + q t+1 sh i W,t + sh i S,t+1 S t+1 Wt 1 + S t+1 Wt (13)
This equation shows the dynamics of the wealth share of group i as a function of the relative capital gain 1+q i t+1 1+q t+1 , the initial wealth share sh i W,t , the differential saving sh i S,t+1 , and the aggregate wealth formation ratio S t+1 Wt (see [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF], Equation 3). Correspondingly, the simulation of the dynamics of wealth shares based on saving, is captured by the second term on the right-hand side. Note that the saving component comprises two effects: the saving differential S i t+1 S t+1 and the aggregate saving-induced wealth formation S t+1

Wt . Intuitively, this implies that the effect of relative saving rate is mediated by the aggregate importance of the new saving (relative to the initial wealth). The other term is the relative price effect 1+q i t+1 1+q t+1 , which is estimated on the real wealth holdings (thus both on the initial and new wealth). This term conveys that the relative price effect can modify the wealth distribution in the presence of differential price effects. As we look below, differential price effects along the wealth distribution stem from differences in the portfolio composition.

It follows that the condition for saving to be an unequalizing force is that the distribution of saving is more unequal than the initial wealth distribution. To see this, start from equation 12.

After applying some straightforward transformation we obtain that the conditions for wealth shares to increase when there are no differential capital gains (q i t+1 = q t+1 ) is that the proportion of new saving going to group i exceeds the initial wealth share of the group i:

ln W i t+1 W t+1 -ln W i t W t > 0 ⇔ S i t+1 W i t - S t+1 W t > 0 ⇔ S i t+1 S t+1 > W i t W t (14)
Likewise for capital gains, only that the condition refers to capital gains on the initial wealth.

As explained by [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF], in steady state, wealth shares are stable, sh i W,t = sh i W,t+1 . Assuming no relative capital gain effects (which, as pointed out by [START_REF] Garbinti | Accounting for Wealth-Inequality Dynamics: Methods, Estimates, and Simulations for France[END_REF], must ensue at one point) Equation 13 becomes sh i W = S i S , coveying that the inequality of wealth converges with the inequality of saving. (1980-2018), Europe (1980-2018), and China (1995-2018). The right panel show the log of the ratio of the 75th and the 25th percentile's wealth in the U.S. (1980-2018), Europe (1980-2018), and China (1995-2018).

Results for Europe is the unweighted average of France and Germany (for Spain and the U.K. we do not have detailed information on the wealth of different percentiles within the bottom-50% wealth group).

  households' stock of assets and saving flows on a global level. The mechanics of the "unveiling" exercise are illustrated in Figure 2, showing the example of the U.K. in 2018. The left side of the figure shows both directly and indirectly held assets of British households. The right side of the figure shows the "unveiled" household asset portfolio, simplified in three asset types: housing, fixed-income assets, and equity. Percentages refer to the share of each asset type in total household assets. Indirectly held assets as investment funds and pension funds represent 3% and 30% of household assets, respectively. The center part of the figure depicts the unveiling exercise by

Figure 2 :

 2 Figure 2: 'Unveiling' of households indirectly held assets in the U.K. in 2018

  proach": Mian et al. 2021b for the U.S., Albers et al. 2020 for Germany, Yang et al. 2012 for China) or implemented on panel data (Martínez-Toledano 2020

Figure 3 :

 3 Figure 3: Accumulation of household wealth, 1980-2018.

Figure 5

 5 Figure 5 contains three important insights. First, there has been a notable increase in fixedincome assets in all three world regions, equivalent to around 100% of national income. Both in Europe and in the US, this increase took place almost entirely through pension and investment

Figure 5 :

 5 Figure 5: Cumulative change in asset-to-income ratios since 1980 (left-axis) and household wealth-income ratio in levels (right axis).

Figure 6 :

 6 Figure 6: The distribution of saving and capital gains across wealth groups during 1980-2018 (US and European countries) and 1995-2018 (China).

Figure 7 :

 7 Figure 7: Observed and counterfactual top-10% wealth share of household wealth

(

  admittedly stark) assumptions that households do not adjust saving behavior in response to asset prices, yields a counterfactual that illustrate to what extent capital gains have dampened the rise in wealth inequality. In this counterfactual world, the wealth share of the top-10% would have risen in the period 1980-2018 by another 15-20pp. in the three economies, as suggested by Figure7. In particular, the figure reveals that capital gains in net housing played the critical role in moderating the wealth distribution since 1980.16

Figure 8 :

 8 Figure 8: Saving out of national income by wealth groups

Figure 9 :

 9 Figure 9: Distribution of household saving flows, 1980-2018

Figure 10

 10 Figure 10 presents the saving breakdown into the three asset categories (left panel), and the development of top-10% saving with and without the crucial retained earnings component (right panel). The charts reveal the central importance of retained corporate earnings for the increase in top-10% saving in the U.S. and Europe. In both economies, corporate saving account for almost the entire increase in household saving at the top. Without the contribution from corporate saving, top-10% saving would have stagnated or fallen in Europe and the U.S. In China, corporate saving

Figure 10 :

 10 Figure 10: Saving out of national income by the top-10% wealth group

Figure 11

 11 Figure 11 sums the saving of the top-10% and bottom-90% groups in each of the G3 and divides by G3 aggregate GDP. The figure conveys our core finding: the past 40 years have witnessed a substantial reshuffling of saving flows within the global household sector. The global bottom-90% reduced their saving from 5% to around 1% while global top-10% saving doubled relative to G3 aggregate income. The rise of saving by rich Chinese households played an important role for increase in the global supply of saving from the top while the decline in saving of the global bottom-90% was mainly driven by declining saving in the U.S. and Europe (Appendix Figures B18

Figure 11 :

 11 Figure 11: G3 net national saving decomposition, 1980-2018

Figure 11

 11 Figure 11 also demonstrates that the role of the public sector saving swings in the overall dynamics of G3 saving was limited. During the 1980-2018 period, public saving co-moved with economic cycles, but overall stood at average saving rates of around -1 to -2% throughout the period. There was a small shift around the year 2000 when public saving turned positive for a couple of years. From 2013 to 2015, public saving has also been slightly positive, reflecting growing saving surpluses from China's public sector and smaller deficits of Western governments (see figureB21). The substantial positive public saving of China reflects to a large extent corporate retained earnings that accrued to the government, 21 as around 60% of the Chinese corporate sector

Figure 12 :

 12 Figure 12: G3 top-10% and bottom-90% saving by asset-type

Figure 13 :

 13 Figure 13: Accumulation of saving in the G3 from 1980 until 2018

Figure 13

 13 Figure13makes clear in a rather unambiguous way which households were the main suppliers of saving to the global economy in the past four decades. 98% of all new saving since the 1980s were supplied by the richest 10% of households in the U.S., Europe, and China, while 25% came from middle-class households. The bottom-50% of households and the government have been net dissavers throughout the period. In the past four decades, global saving, the chart suggests, were first and foremost supplied by households at the top.

Figure B2 :

 B2 Figure B2: P25 and P75 wealth in the U.S., Europe and China

Figure B3 :

 B3 Figure B3: Housing real capital gains: Alternative estimates

Figure B4 :

 B4 Figure B4: Equity real capital gains: Alternative estimates

Figure B5 :

 B5 Figure B5: Cumulative change in asset-income ratios since 1980 (left-axis) and household wealth-income ratio in levels (right axis).

Figure B6 :

 B6 Figure B6: Accumulation of household wealth, 1980-2018.

Figure B7 :Figure B8 :

 B7B8 Figure B7: Cumulated household saving, in billions of U.S. dollars, 1980-2011

Figure B9 :

 B9 Figure B9: Distribution of capital gains by asset types in the U.S., Europe and China

Figure B10 :Figure B11 :

 B10B11 Figure B10: Capital gains rate

Figure B12 :

 B12 Figure B12: Household saving decomposition across wealth groups in the USA, 1970-2018

Figure B14 :

 B14 Figure B14: Share G3 in world GDP, 1980-2019

Figure B15 :Figure B16 :

 B15B16 Figure B15: G3 national income decomposition across the U.S.,China and EU-4, 1980-2019

Figure B17 :

 B17 Figure B17: World and G3 gross national saving, 1980-2018

Figure B18 :

 B18 Figure B18: G3 net saving by top 10% wealth groups, 1980-2018: regional decomposition

Figure B19 :

 B19 Figure B19: G3 net saving by bottom 90% wealth groups, 1980-2018: regional decomposition

Figure B20 :

 B20 Figure B20: China: International Investment Position, 1980-2018

Figure B22 :

 B22 Figure B22: G3 net public saving: role of corporate saving

Table 1 :

 1 A new international database on unveiled households' saving and balance sheets

		[1]	[2]	[3]	[4]
	Country	Aggregate balance sheets	Aggregate saving and capital gains	Distributional balance sheets	Distributional saving and capital gains
	U.S.

Table 2 :

 2 Household wealth

	SNA-2008 code	Concept
	AN	Non-financial assets owned by households
	AN111, AN21111	Housing
	AN -(AN111+AN21111)	Business assets
	AF	Financial assets owned by households
	AF2+AF3+AF4+AF7+AF8	Fixed-income assets
	AF51	Equity
	AF52	Investment funds shares
	AF6	Life insurance and pension funds of households
	AF	Minus: Liabilities of households
	AF41	Long-term loans (housing debt)
	AF -AF41	Other debt (non-housing debt)
		Equals: Household wealth

  2 Our saving series start in 1980 for Europe and the US, and in 1995 for China (household saving by asset class are not available before 1995 in China).

	Data on the wealth distribution. Data on the wealth distribution are available from existing
	studies or from our own construction. The time coverage of the distributional data is slightly less
	homogeneous, with series available since 1980 in France, the UK and the US, since 1983 in Ger-
	many, since 1984 in Spain, and since 1995 in China.

  Beginning and end years are 1980 and 2018 in the U.S. and Europe, and 1995 and 2018 in China. Column 3 illustrates the share of total household wealth growth captured by each group over the mentioned periods. Columns 4 and 5 show the wealth shares of three wealth groups (top-10%, middle-40% and bottom-50%) at the beginning and end years. Europe is the weighted average (by national income) of France, Germany, Spain and the U.K.

				Share in	Wealth	Wealth
				total wealth	share	share
			p	growth	b-o-p	e-o-p
	USA	301%	495%	100%	100%	100%
	Top 10%	196%	354%	74%	65%	72%
	Middle 40%	102%	140%	26%	34%	28%
	Bottom 50%	3%	1%	0%	1.0%	0.2%
	Europe	301%	578%	100%	100%	100%
	Top 10%	160%	323%	57%	53%	56%
	Middle 40%	124%	223%	38%	42%	39%
	Bottom 50%	16%	31%	5%	5.4%	5.5%
	China	242%	555%	100%	100%	100%
	Top 10%	84%	370%	69%	34%	66%
	Middle 40%	114%	160%	28%	48%	29%
	Bottom 50%	44%	25%	4%	18%	4.5%

Note: This table shows group-specific wealth-to-national income ratios at the beginning and end years. (columns 1 and 2).

Table 4 :

 4 Decomposition of aggregate household wealth accumulation in the U.S.(1980-2018), Europe (1980-2018) and China (1995-2018) The table shows annual real household wealth growth (column 1) and decomposes the accumulation of household wealth into saving-induced and capital-gains-induced wealth growth (columns 2 to 10), since 1980. Household wealth consists of housing (net of mortgage debt), fixed-income assets (net of non-mortgage debt), and equity (including unincorporated business assets). The table shows that, for example, saving in net-fixed income assets accounted for 42% of household wealth accumulation in the U.S. since 1980. Europe is the weighted average (by national income) of France, Germany, Spain and the U.K.

		[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]
		Real wealth growth	% due to saving	Net hous-ing	Net fixed-income assets	Equity and busi-ness	Of which: corpo-rate saving	% due to cap. gains	Net hous-ing	Net fixed-income assets	Equity and busi-ness
	U.S.	4.0%	55%	-6%	42%	19%	24%	45%	28%	-10%	27%
	Europe 3.9%	59%	-4%	43%	20%	18%	41%	52%	-12%	1%
	China 12.5%	48%	4%	28%	16%	11%	52%	44%	-8%	16%
	Note:

Table 5 :

 5 Decomposition of wealth accumulation of aggregate household wealth and of two wealth groups (top 10% and middle 40%) in the U.S.(1980-2018), Europe (1980-2018) and China (1995-2015)

		[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]
		Wealth				Of	Wealth
		in-crease due to	Net housing	Net fixed-income	Equity and business	which corpo-rate	increase due to cap.	Net housing	Net fixed-income	Equity and business
		saving				saving	gains
	U.S.	55%	-6%	42%	19%	24%	45%	28%	-10%	27%
	Top 10%	72 %	3%	48%	21%	26%	28%	14%	-12%	26%
	Middle 40%	37 %	-16%	41%	13%	27%	63%	49%	-13%	27%
	Europe	59%	-4%	43%	20%	18%	41%	52%	-12%	1%
	Top 10%	77 %	3%	41%	32%	24%	23%	35%	-9%	-2%
	Middle 40%	36 %	-11%	39%	8%	6%	64%	69%	-7%	2%
	China	48%	4%	28%	16%	11%	52%	44%	-8%	16%
	Top 10%	57 %	2%	32%	23%	13%	43%	36%	-9%	16%
	Middle 40%	38 %	12%	29%	-2%	2%	62%	53%	-7%	15%

Note: The table quantifies the accumulation of wealth into saving-induced and capital-gains-induced wealth growth (columns 1 to 8) for (i) all households, (ii) the top 10% wealth group, and (iii) the middle 40% wealth groups. We do not include results for the bottom 50% wealth group because their net wealth is close to zero (or even sometimes negative), which makes the interpretation of wealth growth not meaningful. Net wealth is decomposed into housing (net of mortgage debt), fixed-income assets (net of non-mortgage debt), and equity (including unincorporated business assets). The table shows that, e.g., saving in net-fixed income assets accounts for 48% of the total wealth growth of the top 10% in the U.S. since 1980. Results for Europe and the U.S. cover the period 1980-2018, and for China the period 1995-2015. Europe is the weighted average (by national income) of France, Germany, Spain and the U.K.

Table B2 :

 B2 Households: asset-specific saving decomposition

			US
	Total	Retained earnings	Other asset financial	Housing Debt
			Levels
	1980-1992 9.9%	1.6%	7.6%	6.1%	-5.5%
	1993-2005 6.3%	4.1%	7.0%	2.6%	-7.4%
	2006-2018 8.8%	4.7%	5.0%	2.1%	-3.0%
		∆ to 1980-92
	1980-1992 0.0%	0.0%	0.0%	0.0%	0.0%
	1993-2005 -3.5%	2.5%	-0.6%	-3.5% -1.9%
	2006-2018 -1.0%	3.1%	-2.6%	-4.1%	2.5%
			EUROPE
	Total	Retained earnings	Other asset financial	Housing Debt
			Levels
	1980-1992 10.0%	1.7%	10.6%	3.0%	-5.3%
	1993-2005 9.8%	2.9%	9.0%	2.7%	-4.8%
	2006-2018 8,4%	2.8%	5.9%	1.9%	-2.2%
		∆ to 1980-92
	1980-1992 0.0%	0.0%	0.0%	0.0%	0.0%
	1993-2005 -0.2%	1.2%	-1.6%	-0.3%	0.5%
	2006-2018 -1.6%	1.1%	-4.7%	-1.1%	3.1%
			CHINA
	Total	Retained earnings	Other asset financial	Housing Debt
			Levels
	1995-2005 22.7%	2.3%	17.4%	5.5%	-2.5%
	2006-2018 23.4%	5.6%	18.8%	6.1%	-7.1%
		∆ to 1995-2007
	1995-2005 0.0%	0.0%	0.0%	0.0%	0.0%
	2006-2018 0.7%	3.3%	1.4%	0.5%	-4.5%

Note: Saving expressed relative to national income. Saving in "Other financial assets" refers to saving in fixed-income assets and saving in other equity and business (excluding retained earnings). Europe is the weighted average (by national income) of France, Germany, Spain and the U.K.

When allocating corporate saving, we take into account international cross-holdings of corporations. Namely, we exclude from retained earnings of domestic corporations the share owed to foreigners but include the share of foreign retained earnings owed to resident households. We are able to implement this adjustment thanks to recent estimates of net reinvested earnings from the World Inequality Database(Alvaredo et al.

2021).

In practice, there are usually small differences between a sector's total saving in the financial and the real accounts. These differences are recorded in the national accounts as a "discrepancy". We prefer computing financial saving from financial accounts (instead of real accounts) for two reasons. First, only financial accounts disclose financial flows by asset class (equity, bonds, etc.). Second, only financial accounts measure financial saving from observed transactions. By contrast, the real accounts compute financial saving as a residual of the income of a domestic sector (e.g., households or corporations) minus its consumption and non-financial

investment.3 We use the series that were updated on September, 2020 (see: http://gabriel-zucman.eu/usdina/; last accessed on December 14, 2020). These series make some adjustments to those in[START_REF] Piketty | Distributional national accounts: methods and estimates for the United States[END_REF]. SeeSaez and Zucman (2020).

[START_REF] Artola-Blanco | Wealth in Spain 1900-2017: A Country of Two Lands[END_REF] do a similar decomposition for the accumulation of national wealth in Spain. While the focus of their study is on the accumulation of the stock of non-financial assets, in this paper we focus on the accumulation of household wealth, which includes both financial and non-financial claims.

In the appendix (FiguresB3 and B4), we compare the capital gains obtained with our method with the evolution of housing and equity prices from the Jordà-Schularick-Taylor Macrohistory Database[START_REF] Jordà | The rate of return on everything, 1870-2015[END_REF]. Overall, both series of capital gains are highly correlated although we find a lower role for equity capital gains in our series. This result should be expected since part of the rise in equity prices reflects the accumulation of corporate saving (as explained in section 2, we include as part of households' equity saving the share of corporate saving accrued to households). Moreover, equity price indices refer to listed equity, while our study covers both listed and unlisted shares.

An equivalent approach is to estimate saving in equity separately for corporate saving and other saving in equity. In this case, saving in equity in Equation 1 excludes retained corporate earnings, which are allocated directly to equity owners. In both approaches, however, corporate saving is ultimately allocated to different wealth groups in proportion to their equity holdings and, hence, its allocation does not rely on the "synthetic saving method" (see Appendix A.1).

We do not look at smaller groups within the top-10% (e.g., top-1% or top-0.1%) because intergroup mobility increases when the analysis is restricted to the top-1%[START_REF] Bach | From saving comes having? disentangling the impact of saving on wealth inequality[END_REF]).

In a recent paper,[START_REF] Fagereng | Heterogeneity and persistence in returns to wealth[END_REF] document that returns across Norwegian wealth groups vary at the asset class level. Since returns are the sum of capital gains and income flows, we asked the authors if the capital gain component drives the observed variation in returns. In an email exchange, the authors suggested that differences in asset-specific capital gains across wealth groups are fairly small, at least for housing and public equity (the two assets they could check). We are grateful to the authors for sharing this information. Martínez-Toledano (2020) looks at the variation in house prices in Spain across wealth groups (top-10%, middle-40%, bottom-50%) during the period 2005-2015, finding that capital gains varied in similar proportions across all household groups (despite the wild swings in house prices in Spain during the 2005-2015 period of a pronounced boom-bust cycle).Mian et al. (2021b), on the other hand, perform a robustness check of the synthetic saving method using variation in house prices across wealth groups, finding no substantial differences with respect to their benchmark saving rates (which use uniform capital gains for individual assets). Finally,[START_REF] Xavier | Wealth inequality in the US: the role of heterogeneous returns[END_REF] analyses returns along the wealth distribution in the U.S., but her measure of return also assumes (as we do) that capital gains in individual asset classes are the same across household groups.

Conceptually, the frontier between corporate equity and unincorporated business assets is dependent on the legal and institutional context(Alvaredo et al. 2021). All results of the paper are obtained separately for equity and unincorporated business assets, but we decided to group the two together for ease of presentation.

Appendix Figure B6 quantifies the three components: initial wealth from 1980 (absent changes in real asset prices), cumulative new saving since 1980, and cumulative capital gains since 1980 on the initial wealth and on new

Appendix FiguresB8 and B9 further decompose saving and capital gains, respectively, in specific asset categories. In line with the results in Table5, saving in fixed-income assets accounted for the largest part of overall saving (with, as said, a pronounced role of saving in equity and business for the top-10% saving), while capital gains in housing for the largest part of capital gains in all regions.

While today government ownership of equity is low in Europe, it was substantial in the 1980s in countries like France where around 50% of corporate equity was held by the government.

Note that the definition of corporate saving does not include the share of undistributed domestic profits owed to foreigners. Conversely, it includes the share of foreign undistributed profits owed to the domestic sectors (section 2).

A related interesting question is to what extent are the households aware of corporations retaining earnings on

Whatever the underlying drivers of the secular increase in corporate saving, it is important to point out that the corporate finance literature finds that households save more out of undistributed profits (i.e., corporate retained earnings) than out of other income sources, at least in the U.S. and Sweden (Di[START_REF] Maggio | Stock market returns and consumption[END_REF][START_REF] Baker | The Effect of Dividends on Consumption[END_REF]). This would imply that rising corporate saving would have led to a mechanical increase in global saving rates, were this behavior systematically present across countries and over time.

As before, we remove the "corporate veil" and allocate undistributed corporate profits to the owners (i.e., different households or the government) in proportion to their holdings of business equity.

Therefore, the documented within-country trends -the rise in top-10% saving in conjunction with the boom

We stress that this a purely 'arithmetical' exercise, which assumes no behavioral responses (or more generally, it involves no equilibrium analysis).

This figure displays the cumulative change in asset-to-national income ratios (left scale) and the household wealth-tonational income ratio in level (right scale) for the U.S., Europe and China. It shows the cumulative change in assetincome ratios before 'unveiling' the indirect asset ownership. Europe is the weighted average (by national income) of France, Germany, Spain and the U.K.

support from the European Research Council Grant (ERC-2017-COG 772332), as well as a Fellowship from the Initiative on Global Markets at the University of Chicago. The authors acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy -EXC 2126/1-39083886. Powered by TCPDF (www.tcpdf.org) support from the European Research Council Grant (ERC-2017-COG 772332), as well as a Fellowship from the Initiative on Global Markets at the University of Chicago. The views expressed herein are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of New York or the Board of Governors of the Federal Reserve System. The authors acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy -EXC 2126/1-39083886.

A Dynamics of wealth inequality: decomposition A.1 Accumulation equations and the role of corporate saving

As explained in section 2, we use asset-specific accumulation equations to decompose the real growth of different asset classes into two components: a volume effect (due to saving flows) and a valuation effect (due to capital gains). In what follows, we briefly summarize this methodology and, next, zoom in on the treatment of corporate retained earnings (i.e., corporate saving or undistributed profits) in this framework.

The general methodology follows a two-step approach. In a first step, we analyze individual asset classes (housing, equity, etc.) at the aggregate household level. Since we observe assetspecific saving flows together with the stock of individual assets, we back out asset-specific capital gains as a residual. In a second step, we analyze household groups, ranged by wealth (e.g., the wealthiest top-10%, the next-40%, etc.). Since we observe the value in specific assets of these groups (reported in our distributional series) and asset-specific capital gains obtained in the first step, we infer saving flows in individual assets at the group level as a residual (i.e., the "synthetic saving method"; [START_REF] Saez | Wealth inequality in the United States since 1913: Evidence from capitalized income tax data[END_REF].

Recall that both at the aggregate and distributional levels, our series 'unveil' the indirect assets held and the saving flows channeled through investment and pension funds (section 2.3). We identify two main categories of indirectly held assets: equity and fixed-income assets. As a result, we simplify the portfolio of households across six types of assets or debt: housing, business assets, fixed-income assets (directly and indirectly held), equity (directly and indirectly held), housing debt, and non-housing debt.

When looking within the household sector, each of the previous assets or debt types has a corresponding flow: (i) housing and residential investment, (ii) business assets and non-residential investment; (iii) fixed-income assets (directly and indirectly held) and saving in fixed-income assets (indirect and indirect investment flows); (iv) housing debt and net housing debt acquisitions;

(v) non-housing debt and net non-housing debt acquisitions. For equity, there is a particularity.

B Appendix figures and tables