Thomas Piketty
email: piketty@pse.ens.fr

Emmanuel Saez
email: saez@econ.berkeley.edu

Gabriel Zucman
email: zuc-man@berkeley.edu.

Rethinking Capital and Wealth Taxation

Keywords: optimal capital taxation, wealth taxation, inheritance taxation JEL classification: H21

This paper reviews recent developments in the theory and practice of optimal capital taxation. We emphasize three main rationales for capital taxation. First, the frontier between capital and labor income flows is often fuzzy, thereby lending support to a broadbased, comprehensive income tax. Next, the very notions of income and consumption flows are difficult to define and measure for top wealth holders where capital gains due to asset price effects dwarf ordinary income and consumption flows. Therefore the proper way to tax billionaires is a progressive wealth tax. Finally, as individuals cannot choose their parents, there are strong meritocratic reasons why we should tax inherited wealth more than earned income or self-made wealth for which individuals can be held responsible, at least in part. This implies that the ideal fiscal system should also include a progressive inheritance tax, in addition to progressive income and wealth taxes. We then confront our prescriptions with historical experience. Although there are significant differences, we argue that observed fiscal systems in modern democracies bear important similarities with this ideal tryptic.

Introduction

This paper reviews a number of recent developments in the theory of optimal capital taxation and confronts them with tax practice. The equity-efficiency tradeoff is at the heart of optimal tax theory. For capital taxation, this tradeoff is especially marked as capital ownership is much more concentrated than labor income. While the top 1% labor income share is generally below 10%, the top 1% wealth share is typically several times higher, ranging from 25% to 40% in advanced modern economies. While the top 10% labor income share generally ranges from 25% to 50%, the top wealth share is usually around 60-80%. Even more strikingly, while the bottom 50% labor income ranges around 20-25% in some of the most advanced countries in the world, the bottom 50% wealth share is below 5% in pretty much every country on the planet (see [START_REF] Chancel | The World Inequality Report 2022[END_REF], for a recent presentation of such statistics across the world). Capital taxation is also more complex than labor taxation as capital not only generates income flows but is also a stock variable.

There are a variety of taxes on capital: property and wealth taxes assessed on asset stocks, individual income taxes on many forms of capital income received by individuals, corporate income taxes on corporate profits, and inheritance (or estate) taxes on transfers at death. It is useful at the outset to show how tax progressivity looks like when including all taxes collected at all levels of government, using the recently developed Distributional National Accounts methodology which is being applied to a growing number of countries (e.g., Piketty, Saez andZucman 2018, Blanchet et al., 2021). Figure 1 depicts the average tax rate by income groups in the United States, France, and the Netherlands. Income is defined to match total national income as recorded in the national accounts, following internationally-agreed standard and methods, and thus maximizing comparability across countries. National income includes all forms of labor income (salary and benefits) and capital income (including profits retained in corporations).

Labor taxes are assigned to corresponding workers, capital taxes to owners, and consumption taxes to consumers. In all countries, tax progressivity, if it exists at all, is modest: all income groups, including those at the bottom of the distribution, pay about the same fraction of their income in taxes. At the very top, the tax system becomes regressive, particularly so in the Netherlands where the data are of highest quality for the very top (see below). Existing taxes on capital income-income which is highly concentrated at the top of the distribution-are not sufficient in practice to maintain progressivity at the upper end of the scale. This pattern of regressivity at the top seems to be true no matter whether countries have large governments (like France), medium-size governments (like the Netherlands) or relatively small governments (like the United States). 1Naturally, progressivity touches only upon the equity aspect of the optimal tax problem.

Figure 1 is silent on the efficiency effects of taxation. From a purely logical perspective, it is conceivable that lower tax rates on the rich (due to a more favorable treatment of capital income than labor income) is actually beneficial for the economy and in the interest of people with lower incomes. The literature on optimal capital taxation has developed models to capture the efficiency costs of capital taxation and analyze this tradeoff.

From this rich literature, we emphasize three main rationales for capital taxation that strike us as most important and hence most relevant in practice. First, the frontier between capital and labor income flows is often fuzzy, thereby lending support to a broad-based, comprehensive income tax to reduce tax avoidance opportunities. Next, the very notions of income and consumption flows are difficult to define and measure for top wealth holders. For the ultra-wealthy, capital gains due to asset price effects dwarf ordinary income and consumption flows. Furthermore, wealth at the very top is primarily in the form of business stock, which is by definition divisible and hence taxable. Therefore the proper way to tax billionaires is a progressive wealth tax. Finally, because individuals do not choose their parents, there are strong meritocratic reasons why we should tax inherited wealth more than earned income or self-made wealth, for which individuals are at least in part responsible for. This implies that the ideal fiscal system should also include a progressive inheritance tax, in addition to progressive income and wealth taxes.

We confront our prescriptions with historical experience. Although there are significant differences, in particular regarding the wealth tax, we argue that observed fiscal systems in modern democracies bear important similarities with this ideal tryptic. There is still a long way to go toward a socially optimal system, however. We argue that this tryptic should be reinforced and made more systematic and consistent, both at the domestic and global level. This would require much more intensive international coordination than what has been achieved until now, as well as a more active role by major international organizations and a modernization of existing fiscal doctrine (a task to which the present paper attempts to contribute). In the longer run, the wealth tax and inheritance tax components could and should be extended substantially if we want to reach a more equal distribution of wealth and economic power. In particular, progressive wealth and inheritance taxes could be used to finance a minimum inheritance to all-arguably one of the most promising avenues in order to equalize wealth and opportunities. This will certainly take time, but we feel that it is important and useful to take a long-run perspective on these important issues and to set broad targets for the future.

We should make clear from the outset that we do not attempt in the present paper to cover all possible rationales for capital taxation. In particular, we do not cover time-inconsistency arguments. 2 Nor do we cover rationales that are based upon redistribution between different age groups in the presence of inter-temporal market failures. 3 More generally, capital market imperfections offer a large variety of motives and implications for capital taxation and wealth redistribution, which we cover only partially in the present paper. 4 At a more modest level, our objective in this paper is to show that the theory of optimal capital taxation has made some progress, in the sense that we now have a number of simple, tractable economic models that allow us to think about the pros and cons of existing systems of capital taxation. Needless to say, more research is needed in order to reach a more complete understanding of this important issue.

We are not the first ones to review the recent literature on optimal capital taxation and confront it with practice. [START_REF] Boadway | From Optimal Tax Theory to Tax Policy: Retrospective and Prospective Views[END_REF] is a comprehensive survey connecting optimal taxation to practice. [START_REF] Diamond | The Case for a Progressive Tax: From Basic Research to Policy Recommendations[END_REF] is a shorter take on the same issue. [START_REF] Scheuer | Taxation and the Superrich[END_REF] provide a recent survey on the literature on taxing the rich covering both theory and empirics. [START_REF] Kaplow | Optimal Income Taxation[END_REF] provides a theoretically oriented survey on optimal income taxation with a significant focus on the rich and issues of market power. Stantcheva (2020) presents a survey on optimal dynamic taxation from a theoretical angle. Saez and Zucman (2019) discuss 2 Once capital is created, it is tempting to tax it, even if it would have been preferable to commit ex-ante not to do so (see e.g. [START_REF] Farhi | Nonlinear Capital Taxation Without Commitment[END_REF] for a model along these lines). In our view, this is a relatively weak rationale for capital taxation. If this was the main reason why capital is taxed, then the right policy response would be to create an independent tax authority with a zero-capital-tax mandate (or a low-capital-tax mandate), similar to the low-inflation mandate of independent central banks.

3 With uninsurable income risk and borrowing constraints, taxing capital income can be a way to shift the tax burden onto older cohorts and to alleviate the liquidity constraints faced by younger cohorts. For a model along these lines, see [START_REF] Conesa | Taxing Capital? Not a Bad Idea after All![END_REF]. In principle, this could also be achieved by using age-dependent taxation (which to some extent public pension systems do). 4 We refer below to a particular form of capital market imperfection, namely uninsurable idiosyncratic shocks to rates of return. Other imperfections, e.g., borrowing constraints, also matter a great deal for optimal capital taxation and redistribution. See, e.g., [START_REF] Chamley | Capital Income Taxation, Wealth Distribution and Borrowing Constraints[END_REF].

prospects for progressive wealth taxation mostly from a practical angle. [START_REF] Scheuer | Taxing Our Wealth[END_REF] provide a recent overview on wealth taxation. This contribution provides a very selective review focusing on the theoretical aspects that we think are the most relevant in practice, some of which have not yet been modeled and analyzed fully.

The Rationale for a Comprehensive Income Tax

In the real world, the frontier between capital and labor income flows is often fuzzy-or at least more difficult to draw than what is generally assumed in theoretical models. Typically, self-employed individuals and business owners can (at least partly) decide how much they get paid in wages vs. dividends. This also applies to corporate executives, whose compensation packages often involve a complex and diverse set of income flows. Sometimes it is not at all obvious to decompose these flows into a pure labor component (payment for labor services) and a pure capital component (compensation for capital ownership). For example, if individual wage bargaining power is influenced by one's equity position, or if there is collusion between employees and owners so as to minimize tax burden, the frontier might be fuzzy.

In our view, the fuzziness of the capital vs. labor frontier is the simplest-and the most compelling-rationale for a comprehensive income tax (i.e., an income tax treating labor and capital income flows alike) or, at least, for taxing capital and labor income flows at rates that are not too different.

Take the extreme case where the frontier is entirely fuzzy, i.e., each individual can costlessly convert labor income into capital income and vice versa. That is, each individual i receives total income y i = y li + y ki , where y li is labor income and y ki is capital income, but the government can only observe total income y i (the division between the two components can be manipulated at no cost). Then the only possible tax policy is a comprehensive income tax, i.e., a tax τ (y) on total income. Consider now the case where it is costly to shift income flows between tax bases. The government can now try to impose a dual income tax system, with different tax schedules τ l (y l ) and τ k (y k ) applying to labor and capital income flows. However if the tax rates differ widely, then individual taxpayers may choose to pay the cost and shift their income to the most favorable tax base. If we note e s the relevant income shifting elasticity, one can easily show that the optimal tax differential τ l -τ k is a declining function of e s . That is, the higher the income shifting elasticity, the closer the tax rates on labor and capital should be.5 Empirically, a large body of work has shown that tax avoidance responses can be large when there are tax avoidance opportunities. These avoidance responses often dwarf real responses (see Saez, Slemrod, and Giertz 2012 for a survey), as posited by the influential hierarchical model of behavioral responses to taxation of [START_REF] Slemrod | The Economic Impact of the Tax Reform Act of 1986[END_REF][START_REF] Slemrod | Income Creation or Income Shifting? Behavioral Responses to the Tax Reform Act of 1986[END_REF].

It is worth stressing that the fuzziness rationale also applies in economic environments where there is otherwise no reason at all to tax capital income. Consider the benchmark [START_REF] Atkinson | The Design of Tax Structure: Direct Versus Indirect Taxation[END_REF] model where individuals live during two periods t = 1, 2 and are born with zero inherited wealth. In period t = 1, nobody owns any wealth, so that income is simply equal to labor income, which individual i allocates to consumption and saving: y 1i = y 1li = c 1i + k 2i . In period t = 2, income is equal to the sum of labor and capital income: y 2i = y 2li + y 2ki , with

y 2ki = R • k 2i where R = 1 +
r is the exogenous rate of return on savings. Under standard separability assumptions on preferences for consumption vs. leisure, a well-known result in this class of models is that taxing capital income is useless: it creates a pure intertemporal distortion between periods 1 and 2 consumption decisions (just like differential commodity taxation) and brings no welfare gain. The efficient tax policy in this setting is to tax solely labor income flows (i.e. τ = τ (y l )).6

But if the government can only observe total income (or if individuals can easily convert labor into capital income and vice versa, so that the income shifting elasticity is large), then there is no choice but using a comprehensive income tax (τ = τ (y), with y = y l + y k ), or a dual income tax with limited tax differentials between income categories. Conservatives often advocate for a consumption tax on the grounds that it is equivalent to a labor income tax and hence exempts the return on capital from taxation. This is true in the basic two-period model just described, but the progressive tax on consumption would have to be defined on the present discounted value of life-time consumption c 1i + c 2i /(1 + r). Even assuming that c 1i and c 2i are measurable, this would require measuring r which again leaves the door open for manipulation, i.e., claiming that labor income is small and capital income is large, so that r is larger and c 1i + c 2i /(1 + r) is smaller. The most common form of progressive consumption tax proposed in practice is a progressive tax on annual consumption which cannot replicate the optimal labor income tax in the standard Atkinson-Stiglitz model.

As is common in optimal tax theory, a lot hinges on the relative magnitude of different elasticities. If the cross-sectional income shifting elasticity e s is large compared to the intertemporal substitution elasticity (as suggested by available empirical estimates), then comprehensive income taxes are desirable and create little intertemporal distortions. Conversely, if the shifting elasticity is small compared to the intertemporal elasticity, then the intertemporal distortion induced by capital taxation generates significant welfare costs, so that it is better to have a dual system with low tax rates on capital income. In economic environments where there are other reasons to tax capital (e.g., the existence of inheritance, as discussed below), then other parameters play a role. In any case, the income shifting elasticity e s plays an important role for the determination of the optimal tax system.

At the upper end of the distribution, the most important area where income shifting considerations are relevant involves shifting between the individual vs. corporate income tax bases.

The logic of comprehensive income taxation laid out here implies that corporate profits should be included in the individual income tax base, just like the profits of unincorporated businesses.

The US experience since the Tax Reform Act of 1986 shows that taxing profits at the individual level is possible even for large and complex businesses. Currently, S-corporations and partnerships (many of which are large and some of which are even publicly traded) are taxed directly and solely on the individual tax returns of their owners. Applying this regime to all corporations would be a key step towards a comprehensive income tax system. The corporate income tax would become a mere withholding tax at source that could be credited back to owners for individual tax purposes, just like the taxes withheld on wage earnings are. This comprehensive income tax would make it impossible for business owners to avoid taxation by shifting income from the individual to the corporate base and retaining profits in corporations; see [START_REF] Saez | The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay[END_REF] and our discussion below.

3 The Rationale for a Progressive Wealth Tax One important limitation of income taxes is that income flows are often difficult to define and measure for top wealth holders. In particular, owners of very large fortunes typically receive personal, taxable income flows that are much smaller than their full economic income. Their wealth portfolio is generally a large stake in a business and/or managed through a holding company, a private foundation or other bodies, and most of the return is being accumulated within this vehicle. The individual owners then choose to receive an annual personal income flow that is sufficient to pay for their private consumption-which can be a very small fraction of their wealth if they are very wealthy. Although we do not have systematic data on this issue, there is much anecdotal evidence suggesting that the personal income reported by top billionaires can indeed be a tiny fraction of their true economic income.7

In other words, income flows themselves-and not only their division into capital and labor income components-are often non-observable for top wealth holders. Assume for simplicity that there is tiny group of billionaires-making a fixed fraction λ of the population-for whom the government can only observe the evolution of their net wealth k ti , k t+1i , etc. In principle, one could try to recover the full economic income y ti (in the Haig-Simons sense) by using the following accounting equation:

k t+1i = k ti + y ti -c ti i.e. y ti = ∆k ti + c ti , with ∆k ti = k t+1i -k ti
The problem is that the consumption flow c ti of top wealth holders might be as difficult to define and estimate as the income flow itself y ti . Should we include the private jet used by Bill Gates or his collaborators as part of his private consumption, or as part of the income flow that is being re-invested by his foundation in order to promote new projects? It can be quite difficult-and cumbersome-to decide.

The net wealth sequence k ti , k t+1i , etc., is generally easier to observe than y ti and c ti . Generally, billionaires' wealth is tied to large business stakes (e.g., Tesla stock for Elon Musk, currently the richest person in the world). If businesses are publicly traded, wealth is easy to measure.

In the United States for example, large equity stakes in publicly traded businesses have to be reported to the Security Exchange Commission. These data form the backbone of the rich lists compiled by Forbes and Bloomberg (see [START_REF] Saez | Top Wealth in America: A Reexamination[END_REF]. Using the global billionaires list compiled by Forbes over the 1987-2013 period, we can see that top global wealth holders have risen at a very fast pace over the past three decades. The average, real yearly growth rate ∆k ti /k ti appears to be of the order of 6 -7% over the 1987-2013 period (or even higher at the very top of the billionaires' list). 8 These high wealth growth rates primarily reflect asset price effects rather than ordinary capital income earned by the asset (such as profits in the case of corporate stock). Indeed, corporate stock values surge when a business is expected to become eventually profitable well before actual profits materialize (e.g., Amazon or Google). Therefore, for billionaires, wealth is the most relevant economic variable, not income flows derived from wealth.

For most billionaires the consumption flow c ti is likely quite small compared to ∆k ti . For instance, with net wealth k ti equal to $3 billion (roughly the average wealth of billionaires), average ∆k ti is of the order of $180-210 millions (6-7% of k ti ), so that a consumption flow c ti of (say) $10 million would correspond to about 5% of ∆k ti only.

One possibility would then be to ignore the consumption flow and to tax billionaires by applying the regular income tax τ (y) to their implicit income y ti = ∆k ti (a lower bound for their Haig-Simons income), or maybe to y ti = max(∆k ti , y pti ), where y pti is their conventionally measured personal income, generally much smaller than ∆k ti . Indeed, some economists have advocated for the taxation of unrealized capital gains in addition to regular income (see Saez, Yagan, Zucman 2021 for a recent discussion in the US context). For the ultra rich, this corresponds approximately to taxing wealth gains ∆k ti . Such proposals have gained traction in the US tax policy debate in recent years and the Biden administration proposed a version of this idea in its 2023 budget, in the form of a minimum tax on unrealized gains for taxpayers with wealth in excess of $100 million (US Office of Management and Budget, 2022).

Taxing increments in wealth is relatively arbitrary, however. Most billionaires seem to derive direct utility from the wealth they own (and the power, prestige and influence conferred by their wealth or by the control of a large business), at least as much as from their private consumption (probably because of consumption satiation). 9 Indeed, popular rich lists such as the Forbes billionaires list focus on wealth rather than wealth gains. 10 Moreover, ∆k ti can be highly volatile (strongly negative in stock market downturns, and strongly positive in booms), which raises practical implementation difficulties. 11 8 See Piketty, 2014, chapter 12, table 12.1. 9 Saez and Stantcheva (2018) show that introducing wealth in the utility in the standard dynamic model overturns the classic result by [START_REF] Chamley | Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives[END_REF] and [START_REF] Judd | Redistributive Taxation in a Simple Perfect Foresight Model[END_REF] that capital taxes should be zero in the longrun.

10 To be sure, these lists do mention sometimes the largest gains or the largest losses in wealth but the primary focus is on wealth itself.

11 The recent US proposal to tax the unrealized gains of the ultra-wealthy (US Office of Management and

Standard optimal capital tax theory-such as the one coming out of the [START_REF] Atkinson | The Design of Tax Structure: Direct Versus Indirect Taxation[END_REF] life-cycle model or the [START_REF] Chamley | Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives[END_REF] and [START_REF] Judd | Redistributive Taxation in a Simple Perfect Foresight Model[END_REF] infinite horizon modelabstracts from price effects on wealth and falls short for understanding the value of wealth taxes, particularly at the very top of the distribution. To take a simple illustration, consider an annual tax on wealth τ (k) and assume that the population includes a fixed fraction 1 -λ of workers with fixed labor income y lti = y l (who do not save), and a fixed fraction λ of billionaires with the following stochastic wealth process:

k t+1i = R(e ti ) • (k ti -τ (k ti )), (1)
Where R(e ti ) is a stochastic rate of return (reflecting both price effects on assets and income earned on assets), which in general might depend on individual effort decision e ti such as entrepreneurial effort.

It is important to note that a tax on capital income cannot replicate the wealth tax for two reasons. First and theoretically, suppose that income is defined according to Haig-Simons as the full return on wealth including price effects, i.e.,

y ti = ( R(e ti ) -1) • k ti . With a tax on Haig-Simons income at rate τ Y , the transition equation is k t+1i = R(e ti )k ti -τ Y (( R(e ti ) -1) • k ti )
which is no longer equivalent to (1) if returns are heterogeneous across individuals. The Haig-Simons income tax hits harder those with particularly high returns than the wealth tax. Hence, the wealth tax favors the wealthy with high returns (typically new self-made billionaires) and disfavors the wealthy with low returns (typically older or inheriting billionaires). [START_REF] Allais | L'impôt sur le Capital et la Réforme Monétaire[END_REF] famously advocated for a wealth tax on such efficiency grounds and [START_REF] Guvenen | Use It or Lose It: Efficiency Gains from Wealth Taxation[END_REF][START_REF] Guvenen | Taxing Wealth and Capital Income when Returns are Heterogeneous[END_REF] provide a modern modeling. Second and practically, no income tax system to date has been able to tax the full return on wealth which includes unrealized capital gains. Actual income tax systems have a much narrower base limited to income produced by the asset (such as rent, interest, or dividends) and realized capital gains (i.e., gains measured solely when the assets are sold). Hence, annual wealth taxation has been so far the only-and still quite imperfect-tool to go after unrealized returns on a continuous basis (see our discussion in Section 5).

Coming back to our model, the long-run ergodic distribution of top wealth will depend on the wealth tax schedule τ (k) both through mechanical effects-the fact that the wealth tax directly reduces next period wealth in equation ( 1)-and through behavioral effects-the fact that the Budget, 2022) is akin to a pre-paid minimum tax on the stock of unrealized gains that builds up over time, rather than an attempt to tax annual unrealized gains in full year after year, precisely to smooth volatility.

wealth tax may reduce the incentives to put effort e ti toward building or maintaining wealth.

Take a simple example with a linear wealth tax at rate τ . Assume that billionaires choose effort e ti so as to maximize

U = Ek t+1i -V (e it ) • k ti = (1 -τ )E R(e ti ) • k ti -V (e it )
• k ti (effort costs are assumed to be proportional to portfolio size). This produces a first order condition for effort

(1 -τ )E R (e ti ) = V (e it ) so that effort will decrease with 1 -τ under standard assumptions of concavity of R(e) and convexity of V (e). As a result, the long-run wealth distribution depends on the net-of-tax rate 1 -τ through this behavioral effort channel. reacts to the net-of-tax rate on wealth in the long run, which nests insights from several existing models, and can be calibrated using estimable elasticities. In his benchmark calibration, the revenue-maximizing wealth tax rate at the top is high (slightly above 10%), but the revenue collected from the tax is lower than in the static case as the wealth tax has a large long-run impact on top wealth. 12

These recent contributions depart from the standard micro-founded model of optimal intertemporal consumption smoothing and focus instead on understanding the forces shaping the wealth distribution and the fiscal consequences of wealth taxation, which are significant given the high and rising wealth-to-income ratios and concentration of wealth (see, e.g., [START_REF] Saez | The Desirability of Commodity Taxation under Nonlinear Income Taxation and Heterogeneous Tastes[END_REF]Zucman, 2016, 2020 To capture the relevant trade-offs involved in taxing top-end wealth, this new approach is in our view more relevant and robust, because it seems unlikely that consumption smoothing considerations-the heart of standard dynamic models-play a large role in the creation and destruction of great fortunes.

The Rationale for a Progressive Inheritance Tax

Inherited wealth is usually perceived-and taxed-differently than earned income or self-made wealth. Most normative theories of distributive justice put a strong emphasis on individual responsibility and merit, and share the view that life opportunities should be equalized as much as possible (in particular between individuals with different levels of inherited wealth). From an equal-opportunity viewpoint, it seems to make sense to tax earned income or self-made wealthfor which individuals can be held responsible, at least in part-less heavily than inherited wealthfor which individuals can hardly be held responsible. 14 This merit-based argument implies that the ideal fiscal system should also include a progressive inheritance tax, in addition to progressive income and wealth taxes.

There is substantial controversy, however, about the proper level of taxation of inherited 12 Jakobsen et al. (2020) estimate wealth tax elasticities leveraging the high-quality administrative wealth data of Denmark and also find the wealth tax has significant effects on long-run wealth levels at the top.

13 A 1% marginal wealth tax above the top 1% wealth threshold raises 1% (instead of 1.5%) as it exempts wealth below the threshold which is 1/3 of the base (as average wealth above a given threshold is approximately 3 times the threshold).

14 For instance, according to the compensation principle of fair taxation, individuals should be compensated for inequality they are not responsible for-such as bequests received-but not for inequality they are responsible for-such as labor income [START_REF] Fleurbaey | Fairness, Responsability and Welfare[END_REF].

wealth. The public debate centers around the equity vs. efficiency trade-off. In the economic debate, there is a disparate set of models and results on optimal inheritance taxation. These models differ primarily in terms of preferences for savings/bequests and the structure of economic shocks. One central conceptual difficulty is that each individual is at the same time-at least potentially-a bequest receiver and a bequest leaver. Even individuals who received zero bequest might prefer not to tax inheritance too heavily, because they themselves value the possibility of leaving a bequest to their own children. At the same time, if the tax burden falls entirely on labor income and inheritance is not taxed at all, then it might be more difficult for zero bequest receivers to accumulate wealth out of their labor income. The challenge is to be able to take into account these different effects in a tractable manner. Piketty and Saez (2013) consider such a model and show that optimal inheritance tax formulas can be expressed in terms of estimable "sufficient statistics" including behavioral elasticities, distributional parameters, and social preferences for redistribution. 15 Those formulas are robust to the underlying primitives of the model and capture the key equity-efficiency trade-off in a transparent way. They apply to a large class of models where inequality is two-dimensional: individuals differ both in terms of earnings (e.g., due to productivity shocks and labor taste shocks) and in terms of inherited wealth (e.g., due to their ancestors' productivity shocks and bequest taste shocks). 16Consider a simplified version of the Piketty and Saez (2013) model with a measure one of individuals indexed by i, who are both bequests receivers and bequest leavers. A linear tax on bequests at rate τ B funds a lumpsum grant E. The life-time budget constraint of individual i is

c i + b i = R • (1 -τ B ) • b r i + y Li + E where c i is consumption, b i is bequest left, y Li is inelastic labor income, b r i is pre-tax bequest received with R = 1 + r the generational rate of return on bequests received. Individual i has utility V i (c, b) with b = R(1 -τ B )b net-of-tax bequests left and solves max b i V i (y Li + E + R(1 -τ B )b r i -b i , Rb i (1 -τ B )) ⇒ V i c = R(1 -τ B )V i b .
The government budget constraint is E = τ B b with b aggregate (=average) bequests. The government chooses τ B to maximizes a social welfare function of the form:

SW F = i ω i V i (y Li + τ B b + R(1 -τ B )b r i -b i , Rb i (1 -τ B ))
with ω i ≥ 0 Pareto weights.

Let us consider a meritocratic Rawlsian criterion where the government maximizes welfare of those receiving no inheritances with uniform social marginal welfare weight ω i V i c among zero-receivers. In this case the optimum bequest tax rate takes the simple form:

τ B = 1 - b 1 + e B , (2)
V i b Rb i dτ B = -b i dτ B V i c /(1 -τ B
) (using the envelope theorem as b i maximizes utility). Therefore, the net effect on social utility is

ω i V i c bdτ B • [1 -e B τ B /(1 -τ B ) -(b i /b)/(1 -τ B )]
. Summing across all zero receivers (using the fact that the welfare weights ω i V i c are assumed to be uniform in this group) and setting this equal to zero at the optimum leads to 1

-e B τ B /(1 -τ B ) -b/(1 -τ B ) = 0
which can be rewritten as formula (2).

The optimal linear bequest tax rate τ B in this formula refers to what we label the "zerobequest receivers" optimum, or "Meritocratic Rawlsian" optimum. This is the tax rate maximizing the welfare of individuals who received zero bequests. Note that about half the population in France or the US-or in any country for which data is available-receives negligible bequests. 17

Hence, this " Meritocratic Rawlsian" optimum has relatively broad appeal. 18

17 The bottom 50% of the distribution of received bequests typically receives less than 5% of the aggregate inheritance flow, while the top 10% generally receives at least 60-70% (see Piketty 2011 for a comprehensive study for France) 18 For more general formulas applying to the case with any level of received bequest (and with positive elastic-

The elasticity e B in the formula is the long-run elasticity of the aggregate bequest flow with respect to the net-of-tax rate 1 -τ B . This parameter reflects how much individuals respond to bequest taxation by accumulating less wealth. Available estimates using tax changes suggest that the elasticity e B is moderately positive (say e B 0.1 -0.2, see [START_REF] Kopczuk | The Impact of the Estate Tax on Wealth Accumulation and Avoidance Behavior[END_REF]. However this is really an empirical issue, and one certainly cannot exclude the possibility of higher elasticities. 19 Note that if b = 0, i.e. zero bequest receivers also leave themselves negligible bequests (relative to average), then the formula boils down to the classic inverse elasticity τ B = 1/(1+e B ) that maximizes bequest tax revenue. Conversely, if zero bequest receivers expect to leave very large bequests, say above average with b > 1, they would favor bequest subsidies (i.e.

τ B < 0). One can see the crucial role of wealth mobility-and beliefs about wealth mobility-for the determination of optimal inheritance tax rates.

Empirically in France or the United States, b 0.5, i.e. zero receivers leave themselves bequests that are perhaps around half of the average. In such a case, it is in their interest to tax bequests at a rate of "only" τ B 50% even if the elasticity of bequests is zero. This is because zero receivers want to leave bequests, so that they face a true trade-off between raising inheritance tax revenue (in order to receive a larger lumpsum grant) vs. not taxing their own children.

The model just presented assumes that wealth accumulation is motivated solely by bequest motives. However, according to available estimates, there is wide variety of motives for wealth accumulation in the population: some accumulate wealth primarily due to bequest motives, others accumulate for precautionary reasons, or for the prestige, power or social status that sometime goes with wealth. If only a fraction ν of bequests are due to bequest motives, then b should simply be replaced by ν • b in formula (2). In principle, one can estimate ν using wealth surveys. An average value around ν = 0.5 might be realistic (see e.g. Kopczuk and Luton 2007).

Unsurprisingly, the optimal bequest tax rate τ B decreases sharply with ν. In the extreme case ν = 0, then the formula boils down to the standard inverse-elasticity formula τ B = 1/(1 + e B ).

ities of labor supply, which are assumed to be zero in the formula presented here), see Piketty and Saez (2013).

In the paper we also provide simulations of optimal inheritance tax rates using wealth survey data from France and the U.S.

19 Goupille-Lebret and Infante (2018) propose a particularly well identified study in the case of France where there was an abrupt change in inheritance taxation for a specific savings vehicle ("Assurance Vie"). They obtain an elasticity of .4 but this elasticity includes both avoidance and real effects so that the real elasticity e B could be substantially lower.

That is, if zero-bequest receivers do not care at all about leaving a bequest, then the only force limiting the taxation of bequests (from their viewpoint) is the elasticity effect. In case e B = 0, then they want to tax bequests at confiscatory rates: τ B = 100%. Consistent with this discussion, many countries, such as France or Germany, have lower inheritance taxes for bequests to children than for less related or unrelated heirs, presumably because bequest motives are stronger when bequests are left to children rather than less related individuals.

Finally, note that these results about optimal inheritance taxation also have implications about lifetime capital taxes. That is, if one introduces capital market imperfections, then it might be optimal to split the inheritance tax burden between a tax paid at the time of inheritance and a tax paid during the inheritor's lifetime (either in the form of a tax on the flow income from capital or a property or wealth tax levied on the stock). For instance, with uninsurable idiosyncratic risk about the future returns to capital, one does not know at the time of inheritance what the capitalized bequest value will be, so it is more efficient to spread the tax burden. As a consequence, depending on the specific parameters (e.g., the effort elasticity of future rates of return, the share of inheritance in total wealth, etc.), the optimal tax rate on capital income flows might be either higher or smaller than the optimal tax rate on labor income flows (see [START_REF] Piketty | A Theory of Optimal Capital Taxation[END_REF].

The recent model of stochastic wealth by [START_REF] Blanchet | Uncovering the Dynamics of the Wealth Distribution[END_REF] shows that, given the substantial mobility in wealth over a lifetime, bequests taxes are much less powerful than annual wealth taxes to reduce wealth concentration. To see this, note that even if bequest taxes were 100%, there would still be substantial wealth inequality due to self-made wealth. For example, a significant fraction of the Forbes 400 richest Americans are self-made and hence would still be billionaires even in presence of complete bequest confiscation. In contrast, wealth taxes at sufficiently high rates can quickly compress the distribution of wealth (as discussed and proposed by Piketty 2020). Given the level of wealth mobility estimated in [START_REF] Blanchet | Uncovering the Dynamics of the Wealth Distribution[END_REF] for the United States, a 100% bequest tax has approximately the same impact as a 3% annual wealth tax on reducing wealth concentration. This point connects with our previous section on the need for wealth taxation even on top of optimal inheritance taxes.

Comparing Actual Tax Systems with the Ideal Tryptic

Our analysis so far suggests that the ideal fiscal system should include a comprehensive income tax, together with an annual progressive wealth tax, and a progressive inheritance tax. We now briefly confront our prescriptions with historical experience. Although there are significant differences, in particular regarding the wealth tax, we argue that observed fiscal systems in modern democracies bear important similarities with this ideal tryptic.

The comprehensive income tax cum inheritance tax consensus

( We start with the comprehensive income tax. When the modern income tax was created, in the late 19th and early 20th centuries, all developed countries decided to institute a comprehensive income tax. In every country, the progressive tax schedule-and in particular the top marginal rate (depicted on Figure 2 for the United States, the United Kingdom and France)-applied to the sum of labor and capital income. The tax base was defined in a comprehensive manner, particularly for capital income that included all business profits, rents (sometimes even including imputed rent on homeowners), as well as all dividends and interest paid by corporate stocks and fixed-income claims. A separate corporate income tax was often added as a way to tax corporate profits at source. Some countries such as the United States effectively imposed a double taxation of corporate profits (with first the corporate income tax, and then the individual income tax on dividends). Most other countries-notably in Western Europe-developed ways to alleviate this double taxation by providing a credit for corporate taxes paid for dividend income reported on individual tax returns, effectively making the corporate tax like a withholding tax at source and respecting the logic of the comprehensive income tax we discussed in Section 2.

It is unclear the extent to which the choice to opt for comprehensive income taxes was due to concerns about income shifting. In the standard Haig-Simons writings about the comprehensive income tax, one finds for the most part rationales expressed in terms of ability to pay (all forms of income should be treated alike, because they reflect similar ability to pay taxes). 20 There was probably some concern about income shifting, but the main concern seems to have been about

horizontal equity (across income sources) and vertical inequality (across income levels). Given the huge concentration of wealth prevailing at the time-the highest incomes were mostly made 20 See e.g. [START_REF] Seligman | The Income Tax: A Study of the History, Theory and Practice of Income Taxation at Home and Abroad[END_REF], [START_REF] Haig | The Concept of Income -Economic and Legal Aspects[END_REF], [START_REF] Simons | Personal Income Taxation: the Definition of Income as a Problem of Fiscal Policy[END_REF].

of capital incomes (see [START_REF] Atkinson | Top Incomes in the Long Run of History[END_REF], it was obvious to everybody that the income tax should tax capital income at least as much as labor income. In practice, effective tax rates on capital (taking into account all capital taxes) typically exceeded effective tax rates on labor in developed countries in the 1960s and 1970s [START_REF] Pierre | Globalization and Factor Income Taxation[END_REF].

An illustration of this reality is that a number of countries applied tax surcharges for capital income flows. During the interwar period, capital income flows were taxed more heavily than labor income flows pretty much everywhere. In the United States and in the United Kingdom, the top rate applying to so-called "earned income"-i.e. labor income-was at times somewhat lower than the top rate applying to so-called "unearned income"-i.e. capital income. In particular, in the 1960s-1970s, the top rates reported on Figure 2 were those applying to capital income (the rates applying to earned income were often about 10 points lower). This is also confirmed by the fact that the rise of comprehensive income tax during the 20th century came together with the development of steeply progressive inheritance taxes, particularly in the United States and in the United Kingdom (see Figure 3). Inheritance taxes had long been advocated by a number of economists and philosophers as one of the most desirable forms of taxation (at least since Thomas Paine and John Stuart Mill). In the 1910s-1920s, when modern progressive inheritance taxes were created, the chief concern was to limit the perpetuation of large wealth disparities across generations. In his famous 1919 presidential address to the American Economic Association, Irving Fisher expressed strong concerns about the rising concentration of wealth in America (which in his view was becoming as unequal and "undemocratic" as in Old Europe), and called for steeply progressive taxes on inheritance and capital incomes as the proper way to restore equality of opportunities.21

The treatment of capital gains generated a conundrum for these progressive tax systems.

Countries typically taxed capital gains only upon realization, and generally at lower rates than ordinary income. For example, when individual tax rates on ordinary income were at or above Other countries, notably the United States, never implemented direct progressive wealth taxes, perhaps because of constitutional constraints. 22 However, the estate tax was seen as a backstop to go after unrealized gains accumulated over a lifetime at the time of bequest.

Similarly, the corporate tax was also a backstop to go after business profits even if these profits were retained within the corporation and not distributed to shareholders in the form of dividends. Furthermore, regulations were put in place to prevent corporations from excessive retained earnings and from distributing these earnings through share repurchases instead of regular dividends. 23 The estimates by [START_REF] Saez | The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay[END_REF] show that the high effective tax rates on high earners in the post-war decades were achieved thanks to a large (as high as 50%) corporate tax on profits at source.

Obviously, such progressive tax systems combining highly progressive individual income and inheritance taxes along with heavy corporate taxation had to rely on strong enforcement to work as designed. This enforcement encompassed both direct enforcement from tax authorities but also more generally strong social norms against tax avoidance and evasion [START_REF] Saez | The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay[END_REF].

The decline of tax progressivity and the vanishing capital tax base (since 1980)

Starting around 1980, one can observe in most developed country a sharp decline in tax progressivity, with the United States under Reagan and the United Kingdom under Thatcher leading the way. Top tax rates on large income flows and large bequests were reduced, especially in English-speaking countries (see Figures 23). In many countries, a growing fraction of capital income was gradually left out of the progressive income tax base and either taxed at flatter

22 US states have a long tradition of proportional but comprehensive wealth taxation (on both real estate and financial assets); see [START_REF] Saez | The Desirability of Commodity Taxation under Nonlinear Income Taxation and Heterogeneous Tastes[END_REF]Zucman, 2019b, chapter 2, and[START_REF] Dray | Wealth and Property Taxation in the United States[END_REF]Stantcheva (2022).

23 For example, in the United States, share repurchases were largely banned through regulation until 1982 when they were almost completely deregulated. Since then, share repurchases have become the major form of distribution of profits to shareholders.

preferential rates or sometimes fully exempt for some asset classes (such as life insurance in France). As a result, in many OECD countries, the progressive income tax has almost morphed into a progressive labor income tax, sometimes with an explicit dual income tax system as in Scandinavian countries or the Netherlands. [START_REF] Saez | The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay[END_REF] show that this reduced nominal progressivity translated into real reduced progressivity in the case of the United States. When taking into account all taxes at all levels of government, the US had a very strongly progressive tax system in the decades just after World War II with high-income groups paying a much larger fraction of their income in taxes than lower income groups. Today, by contrast, the US tax system looks like a giant flat tax with all income groups paying fairly similar tax rates, with some regressivity at the very top.

As shown on Figure 1, recent studies show that this is also the pattern in European countries such as France [START_REF] Bozio | Predistribution vs. Redistribution: Evidence from France and the U.S[END_REF]) and the Netherlands [START_REF] Bruil | Inequality and Redistribution in the Netherlands[END_REF]. Regressivity for European countries is more pronounced than in the United States and starts around percentile 95th while it is confined to the top 0.01% in the United States. The study for the Netherlands is the most sophisticated, as it links individual business owners to their businesses and hence can assign business-level taxes to individual owners with great precision (something that cannot be done systematically in the United States, due to the lack of administrative data on the owners of businesses subject to the corporate income tax).

In all countries, regressivity at the top is driven by three main factors. First, the large payroll and consumption taxes that generate a substantial fraction of total tax revenues are regressive at the top, as income at the top does not come primarily from labor (and payroll taxes are often capped above some earnings levels) and is to a large extent saved rather than consumed (so that consumption is a small fraction of income). Second, the individual income tax is regressive at the top-end (in spite of nominally progressive schedules) because a large fraction of the income of the very rich is sheltered from the individual income tax (separate taxation at preferential flat rates for various capital income categories, or retained profits within corporations or shells).

Third, the corporate income tax, which is the main tax for the very rich (who derive most of their income from business profits) has shrunk over the last 50 years in most OECD countries.

In the United States, the federal corporate tax rate was in excess of 50% in the mid-20th century while it is only 21% today (see Zucman 2014 for a longer discussion).

One can think of several explanations for the demise of progressive taxation. To some extent, this can be viewed as a rational collective response to changes in the nature of wealth. That is, one can observe in the postwar period a decline in inherited wealth, a relative rise of life-cycle wealth, and a compression of wealth inequality [START_REF] Piketty | Capital in the 21st Century[END_REF]). In the extreme case where there is zero inherited wealth and pure lifecycle accumulation, then under preference separability and perfect capital markets assumptions it can indeed be optimal not to tax capital income flows [START_REF] Atkinson | The Design of Tax Structure: Direct Versus Indirect Taxation[END_REF].

This can be only a partial explanation, however. While it is true that inheritance flows were historically low in the 1950s-1960s (at the time Modigliani formulated the pure lifecycle model), this was largely a transitory state due to war shocks, and inheritance flows are now back to higher levels [START_REF] Piketty | On the Long-Run Evolution of Inheritance: France 1820-2050[END_REF]. It is also important to note that the historical decline in wealth concentration has been less spectacular than what some observers tend to imagine. The top 10% wealth shares used to be as much as 80-90% of aggregate wealth in developed countries at the beginning of the 20th century; in the late 20th and early 21st centuries, it is about 60-70% [START_REF] Chancel | The World Inequality Report 2022[END_REF]. The bottom line is that wealth is so concentrated that from a social welfare viewpoint distributional effects are very much likely to dominate intertemporal distortions effects-unless one is ready to assume very high intertemporal elasticities. 24 A more promising line of explanation is a change in the balance of political power. For instance, according to the optimal inheritance tax formulas described above (calibrated with plausible distributional parameters and elasticities), the top inheritance tax rates observed in the United States until the 1970s-1980s were close to optimal from the viewpoint of the bottom two-thirds of the population, while those observed in the 2000s-2010s are closer to the optimum from the viewpoint of the top 10-20% of the distribution. Why and how this change in political power-and also the change in perceptions and beliefs about expected wealth mobility-came about is a complicated and fascinating political science question, and which is indeed attracting growing attention. 25 Finally, there is little doubt that financial globalization and international tax competition have contributed to the decline in capital taxation (and possibly to the shift in the balance of political power). With free capital flows and little reporting of cross-border assets and income, 24 [START_REF] Lucas | Supply-Side Economics: An Analytical Review[END_REF] views the zero-capital-tax result obtained in the zero-shock, infinite-horizon model of [START_REF] Chamley | Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives[END_REF] and [START_REF] Judd | Redistributive Taxation in a Simple Perfect Foresight Model[END_REF] as the "largest genuinely free lunch" brought by economic science. However there is little evidence supporting the infinite long-run elasticity of capital supply implicitly assumed in this class of models.

25 See e.g. [START_REF] Bonica | Why Hasn't Democracy Slowed Rising Inequality?[END_REF].

20

each country is in effect facing a highly elastic capital tax base. This is particularly true for small European countries, such as Scandinavian countries where dual income tax systems were adopted in the 1990s-2000s and in some cases where the inheritance tax was abolished (in spite of strong egalitarian values, such as in Sweden). From a single-country perspective, it might indeed be optimal with perfect capital mobility to opt for zero capital taxes, even though every country would attain higher social welfare from tax coordination and positive capital taxation. From this viewpoint, it is particularly striking to compare the conclusions of the Mirrlees ( 2011) report (which takes for the most part a single-country, U.K. perspective on the optimal tax system, and therefore recommends to pursue corporate tax cuts and favors a very moderate approach on tax progressivity and inheritance taxation) and the previous British reports on the ideal tax system (e.g., [START_REF] Kaldor | An Expenditure Tax[END_REF][START_REF] Meade | The Structure and Reform of Direct Taxation, Report of a Committee chaired[END_REF], who take a much more progressive perspective.26

As narrated in [START_REF] Saez | The Triumph of Injustice: How the Rich Dodge Taxes and How to Make Them Pay[END_REF] in the case of the United States, the demise of progressive taxation took place in two steps. First, there was a weakening of enforcement that led to an explosion of tax avoidance. This rise in tax avoidance (e.g., the boom of tax shelters in the late 1970s and early 1980s) was then used to argue that "progressive taxation does not work" and to advocate for lower statutory taxes on the rich.

The return of the wealth tax and the future of tax coordination

It is unclear at this stage whether rising tax competition or increased tax coordination will prevail in the future. According to the estimates of [START_REF] Tørsløv | The Missing Profits of Nations[END_REF], close to 40% of multinational profits-profits booked by corporations outside of the country of their headquarter-are shifted to tax havens each year. In spite of ambitious reforms-such as the Base Erosion and Profit Shifting initiative launched under the auspices of the OECD in 2015, and the US tax reform of 2017-there is no sign that this shifting is abating [START_REF] Wier | Global Profit Shifting, 1975-2019[END_REF][START_REF] Garcia-Bernardo | Did the Tax Cuts and Jobs Act Reduce Profit Shifting by US Multinational Companies?[END_REF]. In this context, there is growing recognization that more tax coordination-e.g. in the form of a common minimum corporate tax-might be desirable. In 2021, 136 countries and territories agreed to impose a 15% minimum tax on the country-by-country profits of multinational companies. However, as of the end of 2022, no large countries had yet enacted this tax, so it remains to be seen whether this change would happen.

The agreement also has a number of limitations. The 15% rate is a low tax rate (relative to, e.g., typical tax rates on wage income). It applies to a narrow base that allows for significant carveouts for payroll and physical capital deployed abroad. If implemented, this international agreement would undermine artificial profit shifting (moving paper profits to territories with no real economic activity) but would keep tax competition (moving workers and physical capital to low-tax places) very much alive [START_REF] Barake | Minimizing the Minimum Tax? The Critical Effects of Substance Carveouts[END_REF].

There has also been growing awareness of the fact that a rising fraction of household wealth is located in tax havens [START_REF] Zucman | The Missing Wealth of Nations: Are Europe and the U.S. Net Debtors or Net Creditors?[END_REF][START_REF] Zucman | The Hidden Wealth of Nations: The Scourge of Tax Havens[END_REF], most of which owned by households at the top of the wealth distribution [START_REF] Alstadsaeter | Who Owns the Wealth in Tax Havens? Macro Evidence and Implications for Global Inequality[END_REF]. To address this issue, the United States moved unilaterally in 2010 by passing the Foreign Account Tax Compliance Act (FATCA) that requires all foreign financial institutions to report to US tax authorities balances and income information for all US individual clients they may have. Failure to comply carries heavy penalties that the United States can credibly enforce as it is the key player in the financial international system, and leak risks for financial institutions are real. The unilateral move by the United States was later followed by an international agreement, the Common Reporting Standard for the Automatic Exchange of Information on financial accounts between tax authorities, which was developed by the OECD in 2014 and which about 100 countries and territories signed in. The list of signatories include many tax havens such as Switzerland which used to have strict bank secrecy rules that in effect protected foreign tax evaders. Even though implementation has been slow-many tax authorities may not yet have used the reported information aggressively, and important forms of wealth such as real estate are not yet covered by this agreement [START_REF] Alstadsaeter | Who Owns Offshore Real Estate? Evidence from Dubai[END_REF]-it is clear that offshore tax evasion now carries a higher risk (or perceived risk) of detection than before 2010. This change is perhaps the starkest example of how new forms of international cooperation, long deemed utopian, can materialize relatively quickly and change the practicality of taxing capital in a globalized world.

Proposals in favor of a coordinated international registry on financial securities (e.g., as described in [START_REF] Zucman | Taxing across borders: Tracking personal wealth and corporate profits[END_REF][START_REF] Zucman | The Hidden Wealth of Nations: The Scourge of Tax Havens[END_REF] are also becoming increasingly popular. In this context, some form of annual wealth tax-or registration duty-would be a natural way to establish individual property rights and contribute to such a registry. Given the fast growth rates in wealth observed among billionaires, a coordinated wealth tax would also be a logical response (see above). This is particularly evident in Europe, where aggregate wealth-income ratios have been rising steeply over the past decades, so that the wealth tax base is quite dynamic as compared to the income tax base [START_REF] Piketty | Capital is Back: Wealth-Income Ratios in Rich Countries 1700-2010[END_REF][START_REF] Chancel | The World Inequality Report 2022[END_REF].

More generally, it should be noted that annual wealth taxes have been much more present historically in Europe than in the United States (or United Kingdom). Annual progressive wealth taxes have been applied since the early 20th century in countries like Germany, Switzerland or Sweden, and were introduced in the last third of the 20th century in countries like

France or Spain. The top rate was as large as 4% in Sweden in the early 1980s and came in addition to progressive income and inheritance taxes (though these two taxes were less steeply progressive than in English-speaking countries).

Annual wealth taxes experimented in Continental Europe during the 20th century suffered from four main weaknesses, however (see [START_REF] Saez | The Desirability of Commodity Taxation under Nonlinear Income Taxation and Heterogeneous Tastes[END_REF]Zucman 2019, 2022 for detailed discussions).

First, they did not have a comprehensive base, creating tax avoidance opportunities. The biggest issue was the exemption of the business wealth of owner-managers (broadly defined), making it easy for the ultra-rich to avoid the tax. Some asset classes such as real estate were

often not valued at market prices, creating horizontal inequities. These issues led to the repeal of the wealth tax in Germany and Sweden in the 1990s and 2000s. Second, the wealth taxes were relatively easy to avoid by moving abroad and easy to evade by putting wealth in offshore accounts. Available estimates suggest that in the 2000s, more than 20% of the wealth of the top 0.01% richest Scandinavians was hidden in tax havens [START_REF] Alstadsaeter | Tax Evasion and Inequality[END_REF]. Third, the European wealth taxes had relatively low exemption thresholds (typically $1 million or even less) creating financial hardships for some illiquid millionaires that could be exploited by politicians to repeal the tax. Last, these taxes were based on self-reported wealth (as opposed to pre-populated returns) and enforcement was often weak. In France, tax authorities stopped requiring a detailed tax return from most taxpayers after 2011; taxpayers could simply report their estimate of their net wealth with no further details, increasing opportunities for tax evasion [START_REF] Garbinti | Tax Design, Information, and Elasticities: Evidence from the French Wealth Tax[END_REF].

These issues could be remedied by making the tax base more comprehensive (as Switzerland does for example), taxing expatriates (if not for life as in the United States, at least for a number of years), using the new common reporting standard to tax offshore wealth, increasing the exemption level so that the wealth tax hits only the very rich where liquidity issues are minimal, and strengthening information reporting requirements and enforcement. The recent wealth tax proposal by presidential candidate Elizabeth Warren in 2019 had these characteristics. [START_REF] Saez | Progressive Wealth Taxation[END_REF] estimate that such a tax, if successfully enforced, would dramatically increase the progressivity of the US tax system at the very top.

The redistribution of inheritance and the redistribution of wealth

Let us conclude by stressing that the taxation of capital and wealth has two main purposes: the first one is the raising of fiscal revenue in the most efficient and fair manner; the second one is the redistribution of wealth. While this paper has largely focused upon the first objective, it is important to stress that the second objective is also very important.

If the objective of wealth taxation is not only to raise revenue but also to have a significant impact on the long-run distribution of wealth, then going beyond revenue maximization can make sense. For instance, the 2019 tax plan made by presidential candidate Elizabeth Warren initially proposed a wealth tax at a rate of 2% above $50 million and 3% above $1 billion.

While this is enough to raise very significant tax revenue, this might not be sufficient to affect the wealth distribution in a significant manner, first because top billionaire wealth has been rising much faster in recent decades (typically around 6-8% per year), and next because the proposal did not include a wealth transfer to lower wealth classes. In contrast, Senator Bernie Sanders proposed a more graduate wealth tax with rates of up to 8% for fortunes above $10 billion. Elizabeth Warren later proposed a higher marginal tax rate of 6% surtax above $1 billion instead of 3% in her original plan. With top tax rates around 6-8% it becomes possible to reduce top-end wealth concentration, or at least to keep it under control (Saez and Zucman 2019 provide a simple simulation of impact based on the Forbes 400 richest since 1982). The complicated question from a normative perspective is to decide what the target should be in terms of long run distribution of wealth. E.g. is it desirable to return to the wealth distribution of 1980 or 1950, or do we aim for a more equal distribution of wealth? The models and simulation tools that were recently developed by [START_REF] Blanchet | Uncovering the Dynamics of the Wealth Distribution[END_REF] make it possible to simulate the longrun impact of any progressive wealth tax schedule, an important step toward tackling these complicated questions.

In order to analyze how progressive taxation of capital and wealth can help us reach a more equal distribution of wealth and economic opportunities, it is also critical to analyze how tax revenues can be used to finance wealth transfers to the benefit of lower social groups. The bottom 50% wealth share is extremely small (less than 5%) in pretty much every country and region of the world: it is currently about 4% in Europe, 2% in the US and 1% in Latin America (see [START_REF] Chancel | The World Inequality Report 2022[END_REF]. Given that the distribution of wealth among decedents is approximately the same as among the living (as a first approximation), the distribution of inherited wealth is similarly skewed: bottom 50% children typically receive less than 5% of the total, while top 10% children receive 60-80%. Proposals have been made in order to use progressive wealth and inheritance taxes in order to have a system of minimum universal inheritance equal to 60% of per adult wealth at age 25. E.g. the minimum inheritance would be equal to e120 000 if average per adult wealth is equal to e200 000 (as is approximately the case in Western Europe currently). The total annual cost would be about 5% of national income and could be paid for by a mixture of progressive wealth tax (raising about 4% of national income) and progressive inheritance tax raising an additional 1% of national income (see Piketty 2022, Table 2, p.161).

Needless to say, a fully satisfactory analysis of such redistributive schemes would require substantial progress in terms of theoretical modeling and normative thinking. In particular, the long-run efficiency impact of wealth redistribution requires a proper modeling of credit constraints and capital market imperfections. In principle having bottom 50% children receive around 20-30% of total inheritance (which would be approximately the case with a minimum inheritance equal to 60% of per adult wealth) could make a big difference in terms of investment opportunities as compared to the current situation where they receive less than 5% of total inheritance. But putting numbers on such efficiency gains is certainly a very complicated task.

It is worth pointing out however that one would need a minimum inheritance of at least this amount in order to make a substantial difference to the long-run wealth distribution. E.g. if the minimum inheritance is only equal to 10% of per adult wealth, then this is very unlikely to make a big difference regarding the long-run evolution of the bottom 50% wealth share. Finally, the practical implementation of such redistributive schemes would obviously require major shifts in the balance of political power between social groups. It is worth noting however that the magnitude of the tax revenues required for this transformation to take place (5% of national income) are not enormous as compared to the historical rise of public spending in advanced countries (from less than 10% of national income before World War I to about 30-50% today).

In any case, we feel that it is important that scholars working in this area contribute to develop a long-run perspective on these important issues and to set broad targets for the future. P 0 -1 0 P 1 0 -2 0 P 2 0 -3 0 P 3 0 -4 0 P 4 0 -5 0 P 5 0 -6 0 P 6 0 -7 0 P 7 0 -8 0 P 8 0 -9 0 P 9 0 -9 5 P 9 5 -9 9 P 9 9 -9 9 . 9 P 9 9 . 9 -9 9 . 9 9 P 9 9 . 9 9 -9 9 . 9 9 9 P 9 9 . 9 9 9 -1 0 0

France

Notes: This figure depicts the top marginal tax rate for the inheritance (or estate) tax in the United States (Federal income tax only), the United Kingdom, and France from 1900 to 2020. The tax rate for bequests to children is direct line is reported (when rates for other inheritors are higher in France). Source is [START_REF] Piketty | Capital in the 21st Century[END_REF] and updates.

  . The elasticity created by pure mechanical effects is equal to the average number of years billionaires remain billionaires and hence are subject to the billionaire wealth tax. The wealth tax has less impact on top wealth when there is a lot of mobility among billionaires and when new fortunes are created and destroyed quickly. Conversely, in a dynastic environment with little wealth mobility, a progressive wealth tax will have much larger long-run effects as it hits fortunes for a long time, possibly several generations. Using a simple calibration based on the Forbes 400 rich list for the United States since 1982,[START_REF] Saez | Progressive Wealth Taxation[END_REF] find that billionaires have been on the list for fifteen years on average, implying a long-run elasticity of around 15. The corresponding average wealth tax rate that maximizes wealth tax revenue is then given by the standard inverse elasticity rule τ * = 1/(1 + e) = 1/(1 + 15) = 6.25%.[START_REF] Blanchet | Uncovering the Dynamics of the Wealth Distribution[END_REF] considers a much more general model of the wealth distribution using a continuous time stochastic model calibrated using US wealth distribution data, allowing one to assess the short-and long-run effects of wealth taxation. The model can incorporate both mechanical effects and behavioral effects. He derives simple formulas for how the tax base

  with e B = 1-τ B b db d(1-τ B ) the elasticity of aggregate bequests with respect to the net-of-tax rate 1 -τ B and b = E[b i |b r i =0] b the average bequest left by zero-receivers relative to the population average b. The proof is easy to understand intuitively. Suppose the government increases τ B by dτ B . This mechanically raises bdτ B extra in taxes but it loses τ B db = -τ B e B bdτ B /(1 -τ B ) through behavioral responses so that net government new revenue funding the lumpsum grant is dE = bdτ B [1 -e B τ B /(1 -τ B )]. There are two welfare effects on zero receivers utility V i (y Li + Eb i , Rb i (1 -τ B )). First, there is the mechanical effect of dE equal to V i c dE. Second, there is the effect of dτ B as the change in τ B hurts those who leave bequests by -

Figure 1 .

 1 Figure 1. Average tax rates by income group: US, France, Netherlands

Figure

  Figure 3: Top Inheritance Tax Rates 1900-2020

  [START_REF] Saez | Progressive Wealth Taxation[END_REF] consider a simple model of billionaires' wealth that includes only mechanical effects and no behavioral responses. Mechanical effects still produce a long-term elasticity of top wealth with respect to the net-of-tax wealth tax rate that can be large. To

see this, if consumption is negligible for billionaires, after T years of taxation at average rate τ each year, the wealth of a billionaire is mechanically reduced by a factor (1 -τ ) T

  in the case of the United States, and Zucman, 2019, for a review of the global evidence). In advanced economies, wealth to national income ratios have reached levels about five or more today (up from 2-3 in mid-20th century). If the top 1% owns 30% of wealth, a 1% average wealth tax on the top 1% raises about 5 • .3 • 1% = 1.5% of national income which is quite large.13

  90% in the United States in the 1950s, the tax rate on capital gains was only 25%. Realized capital gains are often lumpy, making it challenging to apply the ordinary progressive tax schedule on gains that may represent a lifetime of wealth accumulation (e.g., a founder selling his or her business). The United States recognized this problem as early as 1922 when it first gave preferential treatment to capital gains by introducing an alternative tax rate of 12.5 percent for gains on assets held at least two years. Taxing unrealized capital gains is the Haigs-Simons theoretical solution, sometimes proposed (most recently by the Biden administration in 2022, US Office of Management and Budget 2022), but never implemented in practice. Instead, some countries, mostly in continental Europe, adopted annual progressive wealth taxes to indirectly tax unrealized gains.

Of course, taxes fund transfers and transfers are progressive, so that the net effect of the tax-and-transfer system is almost always progressive (seeChancel et al.,

2022).

See Piketty, Saez and Stantcheva (2014) for a formal modeling for the top rate rate andPiketty and Saez (2013b) for a simpler linear tax rate analysis.

See Atkinson and Stiglitz (1976). See also[START_REF] Saez | The Desirability of Commodity Taxation under Nonlinear Income Taxation and Heterogeneous Tastes[END_REF].

See Saez and Zucman (2019b) and the billionaire tax data leaked to Propublica(Eisinger et al. 2021) for evidence in the case of the United States.

[START_REF] Decerf | Fair Inheritance Taxation[END_REF] use a similar model but using fairness principles instead of standard social welfare maximization. They consider more general inheritance taxation and obtain qualitatively similar results.

The bi-dimensionality of inequality-labor and inheritance-is the key element that breaks the classical Atkinson-Stiglitz result of zero capital taxation (which is obtained in a life-cycle model with only labor income inequality).

Fisher recommended to apply the Rignano principle, according to which the entire bequest should be taxed if it has been transmitted for at least three generations. See[START_REF] Fisher | Economists in Public Service: Annual Address of the President[END_REF].

Note that the steeply progressive consumption tax advocated by Kaldor has never been implemented in any country (in part because the proper measurement of individual consumption levels requires the measurement of both income and wealth, i.e., a progressive consumption tax requires the existence of a progressive income tax and a progressive wealth tax).