metadata: authors: 0: author: Decerf,Benoit Marie A
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docna: 0: docna: Fair Inheritance Taxation
docty: Policy Research Working Paper
subtopic: Economic Assistance,Disability,Access of Poor to Social Services,Services & Transfers to Poor,International Trade and Trade Rules,Tax Administration,Public Sector Economics,Public Finance Decentralization and
            Poverty Reduction,Taxation & Subsidies,Macro-Fiscal Policy,Tax Law,Economic Adjustment and Lending,Population & Development
teratopic: Social Protections and Labor,Health, Nutrition and Population,Poverty Reduction,International Economics and Trade,Public Sector Development,Macroeconomics and Economic Growth,Law and Development
abstracts: cdata!: This paper studies the optimal taxation
            of bequests in a model in which individuals have
            heterogeneous preferences over their consumption and the
            net-of-tax bequest received by their heir. The bequest left
            by an individual depends on both her degree of altruism and
            the bequest received from her parents. First, the paper
            studies two principles that are at the heart of the debates
            on taxing inheritances: (1) children should not be penalized
            by the lack of altruism of their parents, and (2) parents
            should be free to choose their bequests. Only one social
            welfare function satisfies these two principles, together
            with Pareto efficiency and a separability principle. Second,
            the paper studies the shape of the inheritance tax scheme
            that maximizes this social welfare function. It shows that
            in the aggregate, the inheritance tax must collect money
            (redistributed through a non-negative demogrant). Moreover,
            small bequests cannot be taxed (they can potentially be
            subsidized), while bequests that are larger than those of
            the most altruistic individuals who did not receive bequests
            from their parents should be taxed as much as efficiency permits.
repnme: repnme: Fair Inheritance Taxation
display_title: Fair Inheritance Taxation
listing_relative_url: /research/2021/10/33483619/fair-inheritance-taxation
topicv3: Tax Expenditures m1326898 1122,Trade m1326924 698,Social Assistance m1327201 1466,Taxation m1327615 1121,Life Expectancy m1504363 2666
publishtoextweb_dt: 2021-10-12T00:00:00Z
dois: 10.1596/1813-9450-9804
historic_topic: Social Protections and Labor,Health, Nutrition and Population,Poverty Reduction,International Economics and Trade,Public Sector Development,Macroeconomics and Economic Growth,Law and Development
origu: DECRG: Trade & Intl. Integration (DECTI)
no_of_pages: 60
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model_extractions: Policy Research Working Paper 9804

# Fair Inheritance Taxation

_Benoit Decerf François Maniquet_ Development Economics Development Research Group October 2021
Policy Research Working Paper 9804

# **Abstract**

This paper studies the optimal taxation of bequests in a model in which individuals have heterogeneous preferences over their consumption and the net-of-tax bequest received by their heir. The bequest left by an individual depends on both her degree of altruism and the bequest received from her parents. First, the paper studies two principles that are at the heart of the debates on taxing inheritances: (1) children should not be penalized by the lack of altruism of their parents, and (2) parents should be free to choose their bequests. Only one social welfare function satisfies these two principles, together with Pareto efficiency and a separability principle. Second, the paper studies the shape of the inheritance tax scheme that maximizes this social welfare function. It shows that in the aggregate, the inheritance tax must collect money (redistributed through a non-negative demogrant). Moreover, small bequests cannot be taxed (they can potentially be subsidized), while bequests that are larger than those of the most altruistic individuals who did not receive bequests from their parents should be taxed as much as efficiency permits.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at bdecerf@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Fair Inheritance Taxation<sup>∗</sup>

Benoit Decerf<sup>†</sup> Francois Maniquet<sup>‡</sup> Originally published in the Policy Research Working Paper Series on _October 2021._ This version is updated on _January 2024._ To obtain the originally published version, please email prwp@worldbank.org.

Keywords: fairness, inheritance taxation, responsibility, compensation, tax exemption JEL: D63, D64, H21

> ∗Acknowledgments : We are grateful to Dirk Neumann and Kevin Spiritus for helpful discussions, and to Pierre Bachas, Katy Bergstrom, participants to the DEFIPP workshop of March 2017 (University of Namur), the “What is socialism today” conference (Yale University), the Louis-Andre Gerard-Varet days (Marseille, June 2022), the virtual conference on Inequality (ICEA, November 2021), the Workshop in honour of Prof. Em. Erik Schokkaert (kuleuven, September 2022) and two anonymous reviewers for their comments. Funding from the Banque Nationale de Belgique (Exercice 2017) is gratefully acknowledged. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of Executive Directors or the countries they represent. The usual disclaimer applies.

> †World Bank. bdecerf@worldbank.org

- ‡CORE/LIDAM, UCLouvain, and LISER. francois.maniquet@uclouvain.be
# **1 Introduction**

The theoretical literature on how to tax bequests has provided a long list of contradicting arguments. Some authors reach the conclusion that bequests should be subsidized, some others that they should be taxed, yet others prove that they should not be taxed nor subsidized. This lack of consensus comes from differences in, first, the modeling of the interactions between parents and children, and second, the objective that the egalitarian planner is supposed to follow. In many papers, individuals are assumed to have the same preferences but different abilities to earn income, with the consequence that the planner tries to redistribute from high to low-wage individuals. Because inheritance inequality does not reveal new information about individuals’ wages, Farhi and Werning (2010) and Kaplow (2001), following a Atkinson and Stiglitz (1976) type of argument, prove that taxing labor income is more efficient than taxing inheritance. In a similar model, Kopczuk (2013a) makes the point that a countervailing force pushing in favor of bequests taxation is that receiving a large inheritance disincentivizes labor supply. In Cremer and Pestieau (2001), parents have different wealth levels and taxing bequests reduces inequality. Kaplow (1995), Farhi and Werning (2010) and Kopczuk (2013a) all make the point that subsidizing bequests is a way to incentivize parents to internalize the positive externality of giving.

Piketty and Saez (2013) have introduced two features that have fundamentally changed the picture. First, differences in bequests do not necessarily come from differences in parents’ ability to earn income. They can come from parents’ altruism, which differs across parents. As a result, taxing inheritance can be the most efficient way to redistribute from lucky to unlucky children. Second, they do not divide individuals into parents and children. They rather consider the entire lifetime, so that all individuals are children and parents in turn. Fiscal policies then affect both the resources that individuals receive early in life and the tax they pay at the end of their life. As a result, all previous arguments based on the combined effects of giving on parents’ and children’s utilities disappear. The positive externality is now reflected in the level of sustainable tax and transfer policies. As a consequence, Piketty and Saez (2013) show that taxing bequests may end up being optimal. Optimality here is measured with respect to the maximization of some social welfare function that, following the approach developed by Saez and Stantcheva (2016), assigns social welfare weights to individuals as a function of their income. The optimality of taxing bequests is proven for some distribution of these weights.

In this paper, we solve two problems that are left unsolved in Piketty and Saez (2013). First, we show how fairness principles can be used to endogenize social welfare weights. In Piketty and Saez (2013), indeed, the formula of the optimal tax is a function of how society values the utility of different individuals, but in a world in which individuals differ in many dimensions, all of which likely to create inequality, it is not clear who should be given priority. We solve this problem by resorting to two fairness principles, which are at the heart of the debates on taxing inheritance. The first principle is that parents should be free to choose their bequest (McCaffery, 1994). This is consistent with the principle of responsibility for one’s preferences, which has inspired recent developments in optimal taxation theory (like, among others, Fleurbaey and Maniquet (2018), Lockwood and Weinzierl (2016), Piketty and Saez (2013)). The second fairness 2
principle is that children should not be penalized by the lack of altruism of their parents.

These two fairness principles may sound in conflict, but we show that they are not. This comes from the fact that each individual is both a child, who may wish to receive a part of the bequest allocated to children of other families, and a parent, who wishes to be free to bequeath the amount she prefers. As a result, there exists one social welfare function (SWF) that reconciles both principles. This is the social welfare function that we maximize to identify the shape of the optimal tax function. By identifying one and only one SWF that satisfies the two relevant fairness principles, we complement Saez and Stantcheva (2016)’s general social marginal welfare weights approach by identifying the weights that should be used with this particular SWF. These weights depend on the amount of received and left bequests. As we show in Section 4, the egalitarian nature of our SWF implies that the weights of all those who inherit something from their parents is zero, and the weights of those who did not inherit anything depends on how much they leave to their children. This distribution of social weights does not belong to the set of distributions that Piketty and Saez (2013) use to derive their results.

The two fairness principles that are central to our undertaking have already appeared in both the theoretical and the empirical literature. Interestingly, Farhi and Werning (2013) embrace the same principles. However, by sticking to a static model in which individuals are either parents or children, they reach a trade-off between these values, whereas we obtain a single social welfare function. By adopting a weighted utilitarian social welfare function with a large variety of possible weights, they justify a large range of policies, including the ones that we prove optimal for our social welfare function.

In the empirical literature, we don’t know of any experiment or questionnaire survey that directly elicits adherence to the values that our axioms capture, but many other studies indirectly justify our axioms. The view, driving our responsibility axiom, that inequalities are not unfair when they come from differences in choices has since long been proven to be backed by the public in the empirical social choice literature (see Schokkaert and Overlaet (1989) and Konow (1996) for early evidence, and Gaertner and Schokkaert (2012) for a survey). Gross et al. (2017), by showing a larger public support for inheritance taxation when the children are richer, and Bastani and Waldenstr ̈om (2021), by showing a larger public support for inheritance taxation when individuals are informed about the link between inequality in inherited wealth and inequality of opportunity, suggest a public support in favor of using inheritance taxation to compensate poorer children.<sup>1</sup> Second, and most importantly, we identify the shape of the optimal inheritance tax on low bequests, that is bequests left by parents who did not receive anything from their own parents. In Piketty and Saez (2013), indeed, the tax is assumed to be either linear, or linear after an interval of exemption. Starting

> 1In a series of contributions, Weinzierl elicit reasons why people might support taxing the rich more than the poor. The main reason he finds is the view that those who benefit more from a functioning society should also contribute more to its cost (see, for instance, Weinzierl (2014) and Weinzierl (2018)). In the labor income taxation problem, identifying those who benefit more from society as those who have a larger income is natural. In the inheritance taxation problem, we find it harder to claim that those who receive more from their parents benefit more from a functioning society. Therefore, we do not refer to Weinzierl’s work to motivate our axioms.

3
with a much broader set of admissible tax functions, we prove that the optimal function has one of the following two shapes (Proposition 2). Either it exempts low bequests, or it first subsidizes and then taxes them.<sup>2</sup> Exempting low bequests and taxing larger ones is also a candidate to be optimal in Farhi and Werning (2010). As we mention above, the authors study a static model in which individuals are either altruistic parents or their children. Their main result is that if parents transfer to their children an amount that the planner finds insufficient because the average utility of children is too low, then bequests should be subsidized to incentivize parents to give more. They also show that poorer parents should be incentivized more than richer parents because their children are also poorer so that their marginal utility is larger and they are the ones that the utilitarian planner wishes to help most, with the consequence that the tax should be progressive. Farhi and Werning (2010)’s argument in favor of exempting low bequests and taxing high ones comes from an additional argument in favor of progressivity. This argument is based on two assumptions: First, for a reason that is exogenous to the model, bequests cannot be subsidized. Second, the saving technology is concave. Under these assumptions, it may be optimal to tax large and only large bequests, because disincentivizing rich individuals from saving has a positive effect on the interest rate, which is equivalent to decreasing the cost of saving for the poor, which incentivizes them to leave larger bequests to their children, so that the utilitarian social welfare increases. Our exemption/positive tax result, on the contrary, does not follow from any a priori assumption on the sign of the possible tax. Moreover, our dynamical model `a la Piketty and Saez (2013) does not allow us to consider that children necessarily consume less than their parents. On the other hand, the argument that a decrease in total saving could make bequest less costly and help increase lowest bequests could still hold in a dynamical model, but provided it does not prevent the total stock of capital to grow sufficiently. We do not address this question here, and we assume that the interest rate is constant, which removes a possible argument in favor of taxing large bequests.

Interestingly, exemption up to some threshold and taxation above the threshold corresponds to the most frequent tax scheme among OECD countries, suggesting that the SWF we study rationalizes the social preferences that have led to this scheme. We come back to this remark in the conclusion.

The second possible shape of the optimal tax in our model is a subsidy on low bequests followed by a positive tax. In this case, the largest tax amount paid by individuals who did not receive anything from their parents is capped (it cannot exceed the maximum of the bequests that are subsidized). Optimality of a subsidy in this case has a completely different rationale than in the literature reviewed above. Low bequests are subsidized in our case because it allows us to increase the marginal tax rates on bequests left by poor but altruistic individuals, so that they are incentivized to leave lower bequests without being worse off. The larger marginal rates on these low yet not-too-low bequests are desirable only to the extend that they lead to larger average tax rates on higher bequests. We also give sufficient conditions under which subsidizing bequests (and, thereby, increasing marginal tax rates on bequests left by poor altruistic individuals) do not lead to significant increase in average tax rates on high 2We prove that tax functions that first subsidize low bequests and then tax larger bequests can be optimal. These functions are not studied by Piketty and Saez (2013), but Kopczuk (2013a) conjectures that they might be optimal.

4
bequests, yielding the exemption scheme to be optimal (Proposition 3).

An intermediary result in our analysis of the optimal tax function is that bequests should, on average, be taxed rather than subsidized (Proposition 1). This coincides with a result of Piketty and Saez (2013), except that they reach this conclusion only for some distribution of normative weights. This further illustrates that introducing fairness principles allows us to restrict the set of admissible social weights so as to reach sharper conclusions.

The model we use to reach these results has three key assumptions. First, like in Piketty and Saez (2013), all individuals are children and parents in turn. This requires that we look at a dynamic model in which parents save during their life and get some capital income. A consequence of bequests is, therefore, that the stock of wealth increases in the economy, and we focus on the long run effects of taxation. Second, all individuals have the same labor income. This strong assumption allows us to single out the effects on the tax scheme of the heterogeneity in both parent’s altruism and how much children receive when they are young. Alternatively, our results stay silent about the effects of differences in earning ability on bequest taxes. This is consistent with the assumption that bequests do not teach us anything about differences in earning ability. More generally, this is consistent with the assumption that the optimal labor income tax does not require to tax bequests. These two key assumptions allow us to focus on the trade-off between subsidizing bequests or transferring a demogrant to all. We come back in the conclusion on how to combine our results with other results on optimal labor income taxation. Third, we assume a joy-of-giving bequest motive. We discuss this assumption below.

Three policy recommendations are consistent with our results. First, taxing bequests should be viewed as a way to redistribute from individuals who inherited from their parents to those who were less lucky. That means that the tax/transfer scheme of bequests should bring some strictly positive surplus to the government, money that should be allocated to all individuals independently of how much they receive from their parents and how much they leave to their children.

Second, the trade-off between increasing the transfers to all and decreasing inheritance taxes should be solved by looking at how they affect the well-being of those who did not receive inheritance from their parents. Indeed, the most altruistic among them will prefer a decrease in inheritance tax (or an increase in inheritance subsidy) whereas the self-centered among them will prefer an increase in transfer.

Third, in spite of a positive average tax on bequests, there can be two reasons to subsidize low bequests. The first reason, reminiscent of results from the literature that we review below, is efficiency, because subsidizing may be a way to obtain a dominating distribution of bequests. The second reason is fairness, because it is a way to redistribute among the poorest individuals (the ones who did not get any bequests from their parents) from self-centered to altruistic individuals. This, however, goes with a precise condition: fairness can only justify subsidies in an interval that goes up to the bequests left by the poorest and most altruistic individuals.

As discussed in reviews by Cremer and Pestieau (2006) and Kopczuk (2013b), the efficiency and fairness implications of inheritance taxation may depend on the bequest motive. For instance, accidental bequests, which exist in the absence of a perfect annuity market when parents die before consuming all their 5
savings, can be taxed without any efficiency costs. We study inheritance taxation under a joy-of-giving motive, which explicitly acknowledges the desire that parents may hold to leave a bequest. The legitimacy of such desire is central in discussions surrounding the taxation of bequests (McCaffery, 1994). An alternative bequest motive consistent with this desire is altruism, whereby parents care for the _utility_ of their child (whereas parents care only about the net-of-tax inheritance received by their child under a joy-of-giving motive). The altruistic motive is at the center of the Barro-Becker dynastic model, which has been widely studied in the literature on optimal capital/inheritance taxation. Most centrally, Chamley (1986) and Judd (1985) conclude that the tax on inheritances should be zero in the long run. More recently, Straub and Werning (2020) overturn this early result by showing that it only holds for high values of intertemporal elasticity of substitution, but otherwise such tax is positive and significant. We do not consider altruism and it remains an open question whether our results also hold in this alternative setting. One result that is very likely to carry through is that the inheritance tax should globally collect a nonnegative amount. The reason is that any tax violating this would be dominated by Laissez-Faire, because such tax would hurt self-centered individuals whose parents are self-centered. We also note that the altruistic model of bequests has been empirically tested and rejected by Wilhelm (1996).

Some authors study the effect on the optimal tax of the fact that the number of children may differ across families and parents may decide not to give equal bequests to all their children, like Cremer et al. (2001). We could take that into account at least to some extent: the worst-off would remain the same under more general assumptions on the number of children.

Some other aspects of inheritance taxation are completely ignored in our analysis. Cremer et al. (2003) study capital income taxation as subsidiary to inheritance taxation in a world in which bequests may not be observable. Nordblom and Ohlsson (2006) study the possibilities to escape bequest taxation through inter-vivos gifts. Stantcheva (2015) studies inheritance taxation in its relationship to investments in human capital. Mirrlees et al. (2010) discuss the administrative cost implied by the collection of an inheritance tax. Golosov et al. (2003) and Kocherlakota (2005) study tax instruments that are allowed to vary over time, leaving more room for improving welfare. Fleurbaey et al. (2018) study the implications of ex-post egalitarianism for the taxation of accidental bequests resulting from premature mortality.

In Section 2, we illustrate our main results in a simple model. In Section 3, we describe the model, by insisting on the similarities and differences with the pioneer model of Piketty and Saez (2013). In Section 4, we discuss our social welfare function and the axioms that justify it. In Section 5, we study the optimal tax function and we state our main results. In Section 6, we provide some concluding comments. In Section 7, we develop the proofs of the results.

# **2 A Simple, Illustrative, Model**

In this section, we illustrate our main results in a simple model. We assume preferences are of two types: selfish or altruistic. Selfish individuals are only interested in their own consumption. Their utility function is defined as _us_ ( _c, h_ ) = _c_ , where _c_ is own consumption and _h_ is the money inherited by one’s child. Altru6
istic individuals have Cobb-Douglas preferences defined over own consumption and the money their children obtain from them. Their utility function is defined as _ua_ ( _c, h_ ) = _c_<sup>1</sup><sup>_/_2</sup> _h_<sup>1</sup><sup>_/_2</sup> . We assume there is a fraction _α_ of altruistic individuals in each generation. The type of children is independent of that of their parents, so that _α_ is also the fraction of altruistic individuals from selfish parents in each generation.

Let us assume that bequests are not taxed. We call it a laissez-faire tax system. Each individual’s budget is given by where _w_ is the (inelastic) labor income, _g_ is the money inherited from one’s parents and _R_ is the exogeneous interest rate. For an alloction to be feasible, there needs to be consistency between the _g_ ’s, which are inherited by a generation, and the _h_ ’s, which are transferred to the next generation. We define this consistency condition below, but we can skip it in this section. Selfish individuals have whereas altruistic individuals have In the long run, a fraction (1 _− α_ ) of individuals do not receive bequests from their parents, a fraction _α_ (1 _− α_ ) of individuals receive bequest<sup>_<u>Rw</u>_</sup> 2<sup>,afraction</sup><sup>_<u>Rw</u>_</sup> <u>2</u><sup><u>)</u></sup> _α_<sup>2</sup> (1 _− α_ ) of individuals receive bequest<sup>_R_</sup><sup><u>(</u></sup><sup>_w_+</sup> 2 , etc.

The resulting laissez-faire allocation has a good and a bad property. The good property is that two individuals receiving the same amount of bequest from their parents are free to choose how to allocate their wealth between consumption and bequest. To say it differently, the government treats individuals who differ only in terms of their preferences identically. The bad property is that bequests are heterogeneous, so that two individuals having the same preferences, that is two selfish or two altruistic individuals, do not end up with the same satisfaction level. The government fails to correct for the inequality arising from the birth lottery.

Our first main result is that there exists a social welfare function satisfying these two properties (same treatment for same received bequests and same satisfaction level for same preferences) together with Pareto efficiency and separability conditions. This social welfare function works by applying an extremely egalitarian aggregator (the maximin aggregator) to numbers that we could call well-being indices. The well-being index of a selfish individual consuming _c_ is equal to _c_ . The well-being index of an altruistic individual is a calibration of their utility level. The calibrated utility level associated to consuming a generic bundle ( _c, h_ ) is the number _u_<sup>_c_</sup> such that this individual is indifferent between their actual bundle and freely allocating wealth _u_<sup>_c_</sup> between consumption and 7
bequest. For an altruistic individual we have so that their calibrated utility is Observe that if an altruistic individual freely allocates a wealth of _uc_ between consumption and bequest, their calibrated utility is exactly _uc_ . Therefore, this calibration has the property that if a government wishes to equalize calibrated utility between selfish and altruistic individuals who receive the same bequest (and, therefore, have the same total wealth), this government should treat selfish and altruistic in the same way, that is there should not be any redistribution among them.

Our second main result consists in proving that the laisser-faire allocation is necessarily dominated, given this social welfare function, by an allocation in which bequests are taxed, on average, and the tax returns are redistributed equally among all individuals as a demogrant. Indeed, the individuals who do not receive any bequests from their parents are the worst-offs in such an economy (that is, they have the lowest calibrated utility), and increasing their (calibrated) utility increases social welfare. This implies that the optimal bequest tax scheme is one in which bequests are taxed, on average, at a positive rate, so that those who do not receive any bequests from their parents receive a transfer from the government.

Our next main result has to do with the non-linear shape of the optimal tax scheme. We show that a strong candidate is the tax scheme that applies a zero tax rate on bequests that are equal or lower than the bequest level of the poorest altruistic individuals, the ones who do not receive bequests themselves. Let us assume that the optimal demogrant is _D_<sup>_∗_</sup> . Then bequests should be exempted of tax up to _e_<sup>_∗_</sup> =<sup>_<u>D∗</u>_</sup> 2<sup><u>+</u></sup><sup>_<u>w</u>_</sup> .<sup>3</sup> Above that level, the tax should maximize the amount of collected taxes.

We can illustrate the difficulty of identifying the optimal tax in the case of a linear taxation above _e_<sup>_∗_</sup> . Let _t_ denote the tax rate for bequests above this threshold. The budget constraint is now When the tax rate _t_ is small enough not to generate complete bunching at _h_ = _Re_<sup>_∗_</sup> , the optimal choice of altruistic individuals reads<sup>4</sup>

> 3We assume bequests are taxed before being capitalized, as explained in footnote 10.

> 4Letting _h_  ̄ denote the maximal feasible inheritance, we have _h_  ̄ = _R_ (1 _− t_ ) ( _w_ + _D_ + _g_ ) + _Rte_  ̄<sup>_∗_</sup> . As we assume _h_<sup>_∗_</sup> _> Re_<sup>_∗_</sup> when _g >_ 0 (no bunching), we get _h_<sup>_∗_</sup> = _h/_<sup> ̄</sup> 2 and _c_<sup>_∗_</sup> = _h/_ (2 _R_ (1 _− t_ )).

8
from which we deduce that the tax collected, _w_ + _D_ + _g − c_<sup>_∗_</sup> _− h_<sup>_∗_</sup> _/R_ , is equal to This shows the dilemma of taxing bequests. On the one hand, the derivative of _T_ with respect to _t_ , which is equal to is positive for small values of _t_ , because _w_ + _D_ + _g >_ 2 _e_<sup>_∗_</sup> . On the other hand, _h_<sup>_∗_</sup> is decreasing in _t_ , and _g_ of a generation is determined by _h_<sup>_∗_</sup> of the preceding generation, so that increasing _t_ has a detrimental effect on _T_ through _g_ . Taxing bequests does not only deter bequests today, but it also brings about a dynamic effect in which wealth, and therefore bequests of the altruistic individuals, decrease over time. We do not give an analytical formula of the optimal tax below, but we are able to identify some properties of its general shape.

We claim that the exemption/positive tax scheme is a candidate to be optimal. Again, we can illustrate why such an exemption may not be optimal with the model of this section. This corresponds to the last result of the paper. Let _D_<sup>_∗_</sup> be the largest demogrant that can be sustained with an exemption/positive tax scheme. That means that the bundle of the poorest altruistic individual is _<u>w</u>_ <u>+2</u> _<u>D</u>_<sup>_∗_</sup> _,_<sup>_R_</sup><sup><u>(</u></sup><sup>_w_</sup><sup><u>+</u></sup> 2<sup>_D∗_</sup><sup><u>)</u></sup> . The key observation is that this individual can reach the � � same utility with any other bundle � _k_ <u>(</u> _w_ <u>+2</u> _D_<sup>_∗_</sup> <u>)</u> _,_<sup>_R_</sup><sup><u>(</u></sup><sup>_w_</sup> 2<sup><u>+</u></sup> _k_<sup>_D∗_</sup><sup><u>)</u></sup> �. If _k >_ 1, then this bundle can only be reached if bequests are subsidized, because the income of _k_ <u>(</u> _w_ <u>+</u> _D_<sup>_∗_</sup> <u>)</u> the selfish poor remains _w_ + _D_<sup>_∗_</sup> . So, offering � 2 _,_<sup>_R_</sup><sup><u>(</u></sup><sup>_w_</sup> 2<sup><u>+</u></sup> _k_<sup>_D∗_</sup><sup><u>)</u></sup> � to the altruistic poor has a cost, but it has the following advantages: the budget of the other, richer, altuistic individuals will be lower, which can be deduced from _k_<sup>_∗_</sup> <u>(</u> _w_ <u>+</u> _D_<sup>_∗_</sup> <u>)</u> the fact that if a poor altruistic individual consumes <u>�</u> 2 _,_<sup>_R_</sup><sup><u>(</u></sup><sup>_w_</sup> 2<sup><u>+</u></sup> _k_<sup>_∗D∗_</sup><sup><u>)</u></sup> � for some _k_<sup>_∗_</sup> _>_ 1, then no other bundle � _k_ <u>(</u> _w_ <u>+2</u> _D_<sup>_∗_</sup> <u>)</u> _,_<sup>_R_</sup><sup><u>(</u></sup><sup>_w_</sup> 2<sup><u>+</u></sup> _k_<sup>_D∗_</sup><sup><u>)</u></sup> � is affordable anymore for _k < k_<sup>_∗_</sup> . Therefore, richer individuals can be taxed more, so that a larger _D_ may be sustainable. To sum up, the optimal tax scheme can consist in subsidizing the altruistic poor individuals, not to incentivize them to let larger bequests but, on the contrary to incentivize them to leave lower bequests because such a subsidy is a way to increase the marginal tax rates without affecting the utility of these individuals.

The model we define in the next sections is more general than the one of this section, as there is a continuum of types of agents and not 2, preferences are general and not Cobb-Douglas, and the tax we look at is not necessarily linear. On the other hand, we keep the assumption that all individuals have the same income, which implies that we do not study how income and bequest taxation can complement each other, and we keep the assumption of a fixed interest rate.

# **3 The model**

We consider an economy with a discrete set of successive generations, 0 _,_ 1 _, . . ._ . Each generation contains a set [0 _,_ 1] of individuals of measure 1. We use _λ_ to 9
denote the probability measure on [0 _,_ 1], that is the mass of individuals whose names are between _i_ and _j_ ( _i, j ∈_ [0 _,_ 1], _i < j_ ) is equal to _j − i_ . We let _M_ [0 _,_ 1] denote the set of Lebesgue-measurable subsets of [0 _,_ 1] and _μ_ ( _J_ ) denotes the measure of _J ∈ M_ [0 _,_ 1].

We assume that all individuals earn an identical lifetime income of _w_ .<sup>5</sup> Therefore, differences in lifetime budgets only come from the bequests individuals get (or not) at the beginning of their life. The assumption of equal _w_ among individuals, however, is far from necessary for our results. We come back to this issue in the conclusion.

Preferences are defined over lifetime consumption, _c_ , and the inheritance received by their heir, _h_ . In addition to goods _c_ and _h_ , it is convenient to consider the quantity of money that an individual receives at the beginning of her life, _g_ , and the bequest left by an individual, _b_ , which is the quantity of money that she does not consume at the benefit of her heir. Quantities _g_ , _c_ , _b_ and _h_ will be related to each other when we model taxation below, but we don’t need to introduce these relations in the first step of our analysis, when we discuss the social welfare function.

Preferences are heterogeneous in the population and preferences of parents and children are independent of each other, even within a dynasty: a generous parent can be followed by a selfish one, whose child can be generous again, etc. For the sake of exposition, we call preferences leading to positive bequests altruistic, even if they differ from pure altruism as it is used in the literature, in which utility of the parents depends on the utility of the children.

We make three assumptions on the _joy-of-giving_ utility functions representing these preferences.

1. We assume preferences to be normal on both goods, consumption and inheritance, at all prices.

2. We assume that at each period there exist some selfish individuals, that is their utility function is A consequence of this assumption is that at each period there exist some individuals who do not receive any inheritance.

3. Finally, we assume that at each period there exist individuals exhibiting the largest level of altruism. That is, some individuals have utility function _u_<sup>**a**</sup> and for all compact opportunity set _O ⊂_ R<sup>2</sup> +<sup>,ifbundle(</sup><sup>_ca, ha_)isthe</sup> best bundle in _O_ according to _u_<sup>**a**</sup> and ( _ci, hi_ ) the best bundle according to any utility function _ui_ of another individual then _h_<sup>_a_</sup> _≥ hi_ . Observe that this assumption would be a consequence of imposing the classical single-crossing property (which amounts to assume that individuals can be ranked according to their level of altruism) and a kind of compactness of the domain of preferences. We don’t need these assumptions and limit ourselves at imposing the existence of most altruistic individuals.

5Remember that, contrary to Piketty and Saez (2013), we are not interested into the tradeoff between labor income taxation and bequest taxation. We only raise the subquestions of whether total lifetime labor income should be taxed (resp., subsidized) so as to subsidize (resp., tax) (at least some) bequest leavers.

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To sum up, the consequence of the last two assumptions is that some individuals have the lowest level of altruism, and they will never leave any bequest to their children, some other individuals have the largest level of altruism, and they will always be the ones with largest bequests among those with the same wealth, and all the other individuals with the same wealth leave bequests between these two extremes, even if there is no complete ranking of them in terms of their altruism.

Individuals live one period, after which they are replaced by the individual of the same dynasty and the next generation. Hence, any individual has only one child.

We begin by focusing on what happens to one generation. Anticipating that we will later restrict our attention to long-run equilibrium allocations, we restrict our attention to allocations in which the distribution of money received by this generation from the previous one, _g_ , is identical to the distribution of money received by the following one, _h_ . Formally, individual _i ∈_ [0 _,_ 1] consumes a bundle An _allocation z ∈ Z_ := _X_<sup>[0</sup><sup>_,_1]</sup> is a function _z_ : [0 _,_ 1] _→ X_ . An allocation _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _∈ Z_ is a _steady-state_ allocation if the distribution of the _gi_ ’s is equal to the distribution of _hi_ ’s, that is, if (ˆ _gi_ ) _i∈_ [0 _,_ 1] = ( _h_<sup>ˆ</sup> _i_ ) _i∈_ [0 _,_ 1], where (ˆ _xi_ ) _i∈_ [0 _,_ 1] denotes the permutation of ( _xi_ ) _i∈_ [0 _,_ 1] in which elements are ranked in increasing order. We let _S_ denote the set of steady-state allocations. A (one generation) _economy_ is a profile of utility functions _u_ = ( _ui_ ) _i∈_ [0 _,_ 1] _∈U_ = _U_<sup>[0</sup><sup>_,_1]</sup> , where _U_ is the set of acceptable utility functions. In particular, _u_<sup>**s**</sup> _, u_<sup>**a**</sup> _∈ U_ .

# **4 Social welfare**

In this section, we define the social welfare function (SWF) that we use in the next section to study the optimal tax and we discuss its axiomatic foundation. As it is customary in taxation theory, we do so without reference to any institutional context or constraint. As a result, a social welfare function is required to rank all allocations, independently of how they can be implemented. In the next section, on the contrary, we will take account of the fact that preferences are not observable and thus only consider allocations that can be implemented by a bequest tax function and a demogrant.

The SWF works by applying the lexicographic aggregator to individual wellbeing indices that capture the fairness principles of the planner. Any individual’s well-being index represents her preferences, which implies that the SWF is not paternalistic.<sup>6</sup> Let us define this index first. It is illustrated in Fig. 1. Individual _i_ is consuming ( _ci, hi_ ). The indifference curve through ( _ci, hi_ ) shows that individual _i_ is indifferent between ( _ci, hi_ ) and maximizing her utility over a budget set of slope _−R_ starting at ( ̃ _c,_ 0), where _R_ is the exogenous rate of return on savings per generation. Budget sets of slope _−R_ are non-distortionary, first-best, budget sets. They have the slope of the individual budget sets in the Laissez-Faire allocation, that is, in the absence of taxation. To say it differently,

> 6A property of this index representing preferences is that it does not depend on the _distribution_ of preferences in the economy, but only on the individual’s own preferences.

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this individual is indifferent between her actual consumption, ( _ci, hi_ ), and being free to allocate a wealth of _c_ ̃ between own consumption today and children’s inheritance tomorrow in the absence of taxation. We state that this individual ̃ has a current well-being of _c_ . The objective of the planner is to maximize the lowest well-being, and in case of a tie, to maximize the second lowest well-being, etc.

<!-- Start of picture text --> hi<br>ui<br>zi<br>L ( ̃ c, 0)<br>c ̃ ci<br><!-- End of picture text --> Figure 1: The _c_ -equivalent utility. Individual _i_ is indifferent between the bundle she consumes, _zi_ , and her favorite bundle in the non-distortionary budget set _L_ ( ̃ _c,_ 0). As a result, her _c_ -equivalent ̃ utility is equal to _c_ .

To define this SWF formally, we need the following notation. A _social ordering_ is a complete ordering on steady-state allocations. A _Social Welfare Function (SWF)_ is a function **_R_** associating each economy _u ∈U_ with a social ordering **_R_** ( _u_ ).

We define the non-distortionary budget set of individual _i_ associated with consumption bundle ( _ci, hi_ ) as the set of all bundles ( _c_<sup>_′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)thatindividual</sup><sup>_i_</sup> can afford with her wealth at ( _ci, hi_ ), that is with a wealth of _ci_ +<sup>_<u>h</u>_</sup> _R_<sup>_<u>i</u>_</sup> 7 These budgets are linear and of slope _−R_ .

The well-being index we are interested in, which we denote as _u_<sup>_c_</sup> and we call _c_ -equivalent utility, can be defined as follows.

> 7As mentioned above, _gi_ , _ci_ and _hi_ will be related to each other when we model taxation. Thus, even if _gi_ does not appear in the inequality defining _L_ , _ci_ and _hi_ will depend on _gi_ when we model taxation.

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**Definition 1** ( _c_ -equivalent utility) **.** _For all i ∈_ [0 _,_ 1] _, zi_ = ( _gi, ci, hi_ ) _∈ X and ui ∈ U ,_

As shown in Fig. 1, individual _i_ is indifferent between consuming _zi_ and her favorite bundle in the non-distortionary budget set _L_ ( ̃ _c,_ 0), where _i_ ’s favorite bundle in _L_ ( ̃ _c,_ 0) is different from bundle ( ̃ _c,_ 0).

The SWF **_R_**<sup>_c−lex_</sup> compares two allocations by applying the leximin aggregator to lists of _c_ -equivalent utilities associated to the allocations.

**SOF 1** ( **_R_**<sup>_c−lex_</sup> ) **.** _For all u ∈U and any two allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_</sup>

Maximizing this SWF gives a very simple formula for the Piketty and Saez (2013) social welfare weights. These weights depend on the allocation at which social welfare is measured. Therefore, at any precise allocation, the welfare weights of the worst-off individuals in terms of their _c_ -equivalent utilities are positive and all equals, and the weights of all other individuals are equal to zero. Of course, which individuals are the worst-off depends on the allocation. As we will see in the next section, when we take incentive constraints into account, the worst-off individuals in _c_ -equivalent utilities will be found among those who did not inherit anything from their parents, but, depending on the allocation, they may be found among selfish or altruistic individuals.

_c_ -equivalent utilities are equalized between two individuals if they freely allocate the same total quantity of money (arising from their labor, the demogrant and their inheritance) between own consumption and bequests in the absence of distortive taxation. This explains why exempting low bequests from taxation will play a prominent role when we study second-best taxation in the next sections.

It may be useful to contrast our SWF with a classical Rawlsian SWF. When there is only one variable of heterogeneity and preferences are identical among individuals, the Rawlsian SWF consists in maximizing the minimal utility, which is the utility of the individuals with the lowest level of the heterogeneous variable (typically, wage rate). These individuals and only them have a positive welfare weight. When there are several dimensions of heterogeneity, including preferences, the choice of the utility function representing the preferences is key to define the Rawlsian objective. This is exactly what our axiomatic analysis achieves: _c_ -equivalent utilities are a calibration of individual utility functions that makes sense of interpersonal welfare comparison. Without this axiomatic foundation, there would be as many Rawlsian SWFs as there are possible utility representations of the different preferences that we allow for. As a consequence, who are the worst-off individuals not only would depend on the policy but also on this utility representation.

_c_ -equivalent utilities have two key properties. First, they do not directly depend on _gi_ , that is the quantity of money one individual received as inheritance does not matter per se. The only quantities that matter are the money consumed by an individual and the money inherited by her child. Second, how 13
precisely an individual allocates her wealth between own consumption and bequest does not matter, provided this individual allocates it freely. As a result, more altruistic or more selfish individuals have the same _c_ -equivalent utility when they allocate the same quantity of money (that is, the same sum of own consumption and bequest) in the absence of taxation.

The combination of these two properties is the basis on which the axiomatization of this SWF is grounded. Indeed, it satisfies the following three important axioms. The first one is the classical Pareto axiom. It requires weak social preference when all individuals weakly prefer one allocation over another. In addition, it requires strict social preference when one set (of positive measure) of individuals strictly prefer the former allocation.

# **Axiom 1** (Pareto) **.**

_For all economy u ∈U and steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,ifforalli ∈_[0</sup><sup>_,_1]</sup> _then z_<sup>_′_</sup> **_R_** ( _u_ ) _z, and if, in addition, there exists a subset of individuals J ∈ M_ [0 _,_ 1] _such that μ_ ( _J_ ) _>_ 0 _and for all j ∈ J_

# _then z_<sup>_′_</sup> **_P_** ( _u_ ) _z._

The fact that **_R_**<sup>_c−lex_</sup> satisfies Pareto comes from _c_ -equivalent utility being a recalibration of the utility function.

The second axiom, compensation for children’s lack of luck, in short Compensation, encapsulates the idea that individuals should not be held responsible for the lack of altruism of their parents, that is they should be compensated for receiving low inheritance. Formally, it requires that if two individuals with identical preferences consume bundles that dominate one another (that is, one individual has both a larger consumption and a larger inheritance received by her child) then a transfer from the richer to the poorer of these individuals is a strict social improvement. The restriction that the axiom applies only when individuals have identical preferences is added to avoid a classical impossibility with Pareto. This makes this axiom a rather weak requirement, as there is no inequality restriction between the bundles of two parents as soon as they have different preferences (remember that all preferences that are normal on both goods are admissible).

# **Axiom 2** (Compensation) **.**

_For all economy u ∈U, steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,subsetsofindividualsJ, K∈M_[0</sup><sup>_,_1]</sup><sup>_suchthatμ_(</sup><sup>_J_)=</sup> _μ_ ( _K_ ) _>_ 0 _, and δ ∈_ (0 _,_<sup><u>1</u></sup> 2<sup>]</sup><sup>_,ifforallj, q∈Jandk, l∈K,_</sup> The fact that **_R_**<sup>_c−lex_</sup> satisfies Compensation comes from _c_ -equivalent utility being independent of the inheritance received _g_ . As a consequence, equalizing 14
consumption and bequest among individuals with the same preferences is a way to make individual well-being independent of how much one inherited from their parents.

The third axiom, responsibility for parents’ choices, in short Responsibility, encapsulates the idea that individuals should be considered responsible for their preferences, that is they should be free to allocate their wealth the way they wish. It requires that two individuals with the same inheritance from their parents should ideally be free to choose their preferred bundle in the same nondistortionary budget set. Rather than requiring that they should choose in the same budget set, the axiom requires that budget inequality between two such individuals should be reduced.

**Axiom 3** (Responsibility) **.**

_For all economy u ∈U, steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,subsetsofindividualsJ, K∈M_[0</sup><sup>_,_1]</sup><sup>_suchthatμ_(</sup><sup>_J_)=</sup> _μ_ ( _K_ ) _>_ 0 _, if there exists δ >_ 0 _such that for all j, q ∈ J and k, l ∈ K,_ _where_ The fact that **_R_**<sup>_c−lex_</sup> satisfies Responsibility comes from _c_ -equivalent utility assigning the same well-being to two individuals as soon as they both freely allocate their wealth in the same non-distortionary budget set.

In Online Appendix S4, we formally prove that **_R_**<sup>_c−lex_</sup> is the only SWF that satisfies these three axioms together with a consistency requirement of the SWF across economies (Separability). A number of important remarks have to be made at this stage.

1. Both Compensation and Responsibility are transfer axioms, that is, they are satisfied if a transfer of goods or of budget is implemented from the richer to the poorer individual, that is to say that they display a positive degree of inequality aversion. The SWF we come at, though, exhibits an infinite inequality aversion. The fact that the combination of Pareto, transfer and consistency axioms leads to a SWF exhibiting infinite inequality aversion is common in the literature and the underlying logics is now well understood. That **_R_**<sup>_c−lex_</sup> uses the leximin aggregator of utilities may look extreme, but the proof in Online Appendix S4 implies that any less extreme aggregator (such as Gini) would violate one of the axioms.

2. There is something less common, though, in the axiomatic foundation of **_R_**<sup>_c−lex_</sup> . Compensation and responsibility axioms, indeed, are typically incompatible with each other. It is therefore a surprise that they turn out to be compatible in this model. This comes from the fact that individuals in this model are both parents and children and each individual’s utility is influenced both when she receives inheritance, so that her wealth increases, 15
and when she is prevented from bequeathing/incentivized to bequeath money to her child. It can be illustrated through the following simple example. Assume an altruistic parent plans to bequeath some amount of money to her child, whereas a selfish parent with the same wealth does not wish to bequeath anything to her own child. As a result of the bequest, the child of the altruistic parent would turn out better-off than that of the selfish parent. On the other hand, if the altruistic parent is prevented from bequeathing, she will end up worse-off than the selfish parent. The first-best solution to this paradox, assuming preferences are observable and non-distortionary transfers can take place, would be to withdraw some wealth from both parents and to allocate it to the child of the selfish parent in compensation for the lack of inheritance, taking into account that the two children themselves can be required to give a part of their wealth in case other individuals are worse-off. That is, the ideal nondistortionary allocation would be to equalize wealth of all individuals of all generations, independently of whether this wealth comes from bequest or redistribution.<sup>8</sup> Useless to say, such an allocation is impossible to achieve through bequest taxation.

# **5 Optimal tax**

In this section, we study the allocations that maximize our SWF among those that can be implemented by a bequest tax function and a demogrant, that is, we stick to the typical assumptions in the literature that preferences are not observable whereas bequests are.

Contrary to what we did in the previous section, we now take account of the influence of the tax scheme on the transmission of bequests, as the behavior of members of one generation influences the well-being of members of the next generation. To take account of these long-run effects of the tax, we restrict our attention to tax functions and demogrants that do not depend on time and we look at the corresponding long-run allocations, that is the allocations obtained when the distribution of bequests is stabilized across generations.

More precisely, we assume that at time _t_ , each individual _it_ (from dynasty _i_ living in generation _t_ ) receives inheritance _git ≥_ 0 (which is a function of the bequest left from individual _it −_ 1) and demogrant _D_ , so that their total resources are _w_ + _D_ + _git_ . Individual _it_ chooses consumption _cit ≥_ 0 and bequest _bit_ +1 _≥_ 0 under the budget constraint Bequests are taxed according to tax function _τ_ , so that amount _bit_ +1 _− τ_ ( _bit_ +1) is transferred to individual _it_ + 1, who receives where _R_ is the interest rate. We assume that _τ_ (0) = 0, because any other value would amount to transferring the same (negative or positive) amount to all, which is exactly what _D_ achieves.

> 8Formally, using the terminology that will be introduced in the next section, if _Tit_ stands for the first-best tax or subsidy of the member of dynasty _i_ living at time _t_ , the optimal firstbest taxation requires that for all individuals _it_ , _git_ = _bit_ (children get exactly what parents leave them) and _w_ + _Tit_ + _git_ is equalized across all _it_ .

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The same process takes place at _t_ + 1, with inheritance _git_ +1 = _hit_ +1 _≥_ 0. Starting conditions at time _t_ +1 may, therefore, differ from those at time _t_ . Note that, in this model, it is the money collected at time _t_ through _τ_ that is used to fund demogrant _D_ . This captures the fact that, had we considered a model in which individuals live for many periods, with a fraction of them born and dead at each period, what an individual gets out of the redistribution system at each period of her life, _D_ , is funded by the taxes on bequests of the individuals that lived (and died) during this individual’s own life. This modeling is particularly appropriate to our objective to study the trade-off between modifying the budgets of individuals through a demogrant, funded by taxes on bequests, or subsidies to bequests, at the price of a lower or even negative demogrant.

For a given _tax-demogrant scheme_ ( _τ, D_ ), an _equilibrium_ allocation at time _t_ is an allocation _zt_ = ( _zit_ ) _i∈_ [0 _,_ 1] = ( _git, cit, hit_ +1) _i∈_ [0 _,_ 1] _∈ Z_ for which all _i ∈_ [0 _,_ 1] choose in the budget set defined by this scheme and their inheritance _git_ , i.e.

implying for all _i ∈_ [0 _,_ 1] that A _long-run equilibrium_ allocation for a tax scheme ( _τ, D_ ) is a steady-state equilibrium allocation _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _∈ S_ to which this sequence of equilibrium allocations at time _t_ may converge. That is, at a long-run equilibrium allocation, Eq. (1) holds and the profile of inheritances received, (ˆ _gi_ ) _i∈_ [0 _,_ 1], is equal to the profile of inheritances left, ( _h_<sup>ˆ</sup> _i_ ) _i∈_ [0 _,_ 1].

We need some further assumptions to guarantee that long-run equilibrium allocations exist and that we are able to apply our SWF to them.

As Piketty and Saez (2013), we assume that the stochastic transmission of preferences across generations is such that the distribution of preferences remains constant through time. In the terms of the previous section, that means that economy _u ∈U_ is constant through time, up to some (measure preserving) permutation of _i ∈_ [0 _,_ 1].

We also assume that the economy converges over time to a unique long-run equilibrium independent on the initial distribution of inheritances ( _gi_ 0) _i∈_ [0 _,_ 1]. Piketty and Saez (2012) show that this assumption is met in their framework under reasonable conditions. In particular, the average taste for bequest cannot be too strong and the stochastic transmission of preferences across generations must satisfy an ergodicity property.<sup>9</sup> Importantly, this property does NOT imply that all members of one dynasty have the same preferences. Some altruistic parents have selfish children and the converse is true as well.

Observe that ( _τ, D_ ) may yield a long-run equilibrium allocation in which not enough money is collected through tax _τ_ to fund demogrant _D_ . We need to further restrict our attention to tax schemes that meet the government budget constraint

> 9See Piketty and Saez (2013), page 1854.

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A demogrant _D_ is _sustainable_ for the tax _τ_ if the long-run equilibrium allocation associated to ( _τ, D_ ) satisfies Eq. (2).<sup>10</sup> When this is the case, we say that the tax-demogrant scheme ( _τ, D_ ) is sustainable. Observe that a given tax may admit several sustainable demogrants. For instance, both a zero demogrant and a negative demogrant are sustainable under a linear tax with rate zero.

A sustainable tax-demogrant scheme ( _τ, D_ ) is _optimal_ if there is no other sustainable tax scheme whose associated long-run equilibrium allocation is preferred by SWF **_R_**<sup>_c−lex_</sup> to that associated to ( _τ, D_ ). A tax _τ_ is _optimal_ in some domain if there is no alternative tax _τ_<sup>_′_</sup> in that domain for which the long-run equilibrium allocation associated to a sustainable scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is preferred by SWF **_R_**<sup>_c−lex_</sup> to the long-run equilibrium allocation associated to all sustainable schemes ( _τ, D_ ).

An individual is among the _worst-offs_ if all other individuals (in the longrun equilibrium generation) have a _c_ -equivalent utility at least as large as this individual.

_Laissez-Faire_ is a tax-demogrant scheme defined by a zero tax and a zero demogrant: _τ_<sup>_LF_</sup> ( _b_ ) = 0 _∀b ≥_ 0 and _D_<sup>_LF_</sup> = 0. The long-run equilibrium allocation associated to Laissez-Faire is sustainable. At this allocation individuals freely choose to allocation their wealth _w_ + _gi_ between consumption and bequest. As a result, and each individual _i_ maximizes her utility over _L_ ( _w_ + _gi,_ 0). This implies that the _c_ -equivalent utility of individual _i_ is equal to _w_ + _gi_ . As a result, 1) the worst-off individuals are those with _gi_ = 0, and 2) they all have a well-being level equal to _w_ , that is, all of them are to be considered as worst-off, independently of their preferences.

# **5.1 A positive average tax on bequests**

Our first result answers the following question: at the optimal tax, should individuals’ incomes be taxed so that bequests can be, on average, subsidized, or should bequests be taxed, on average, so as to subsidize individuals’ incomes (through a demogrant)? The answer is that bequests should be taxed: the amount globally collected by an optimal inheritance tax cannot be negative. If subsidies are provided for some bequest levels, they must be paid for by taxes collected at other bequest levels. This answer is the opposite to that of Atkinson and Stiglitz (1976), Kaplow (2001) and Farhi and Werning (2010). Our result that bequests should be, on average, taxed and not subsidized crucially depends on the fact that we take account in our model of the influence of bequest taxes on the inheritance distribution, and, therefore, the wealth of the parents. The optimal formula of Piketty and Saez (2013) is consistent with taxing bequests on average, but this happens only for some distributions of normative weights. Our result shows that the distribution of normative weights that follows from

> 10 This sustainability constraint allows us to link the government and individuals budget constraints in the following way. A sustainable ( _τ, D_ ) is optimal only if _D ≤_ � _i_<sup>_τ_(</sup><sup>_gi_+</sup><sup>_w_+</sup><sup>_D −ci_)</sup><sup>_di_and</sup><sup>_hi_=</sup><sup>_R_(</sup><sup>_w_+</sup><sup>_D_+</sup><sup>_gi −ci −τ_(</sup><sup>_w_+</sup><sup>_D_+</sup><sup>_gi −ci_)).</sup> These two equations give us the following sustainability constraint: _D ≤_ � _i_<sup>_gi_+</sup><sup>_w_+</sup><sup>_D−ci−h_</sup> _R_<sup>_<u>i</u>di⇔_</sup> � _i_<sup>_ci_+</sup><sup>_h_</sup> _R_<sup>_<u>i</u>di ≤_</sup> � _i_<sup>_gi_+</sup><sup>_wdi_.</sup>

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imposing the axioms we propose unambiguously leads to a positive average tax rate on bequests.

The proof goes by comparing the optimal tax scheme with Laissez-Faire. Under any tax-demogrant scheme ( _τ, D_ ), the _c_ -equivalent utility of any _selfcentered_ individual is also equal to her consumption ( _w_ + _D_ + _gi_ ). Provided that at least one individual who inherits nothing is self-centered,<sup>11</sup> an assumption that we impose (see assumption A1 below), her _c_ -equivalent utility is equal to _w_ + _D_ . If the demogrant is negative, then her well-being is smaller than the wellbeing of the worst-off under Laissez-Faire. The result follows from the fact that our SWF ranks tax-demogrant schemes by comparing the long-run equilibrium well-being of the worst-off. To sum up, redistribution cannot take place from the general population towards those who leave some bequests, because such a redistribution hurts those who did not receive anything from their parents and do not plan to leave anything to their children either, and these are among the The property of Laissez-Faire that all individuals for whom _gi_ = 0 are among the worst-offs and they all have the same _c_ -equivalent utility is shared by all tax schemes ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ) in which _τ_<sup>_∗_</sup> exempts the bequests left by those who inherit nothing. This suggests that if _τ_<sup>_∗_</sup> maximizes the sustainable demogrant _D_<sup>_∗_</sup> under the constraint that _τ_<sup>_∗_</sup> exempts the bequests left by those who inherit nothing, ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ) is a strong candidate to be optimal.

Indeed, our second result identifies a necessary condition on the optimal ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) to be different from ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ): it needs to be the case that _D_<sup>_′_</sup> _≥ D_<sup>_∗_</sup> . It is an easy consequence of what we already said. If ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) has _D_<sup>_′_</sup> _< D_<sup>_∗_</sup> , then the _c_ -equivalent utility of a self-centered individual who did not inherit anything is lower at ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ), where it is equal to _w_ + _D_<sup>_′_</sup> , than at ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ), where it is equal to _w_ + _D_<sup>_∗_</sup> and where this individual is among the worst-offs.

To define the largest bequest left by a zero-inheritor precisely, we impose the assumption that there are zero-inheritors with the most altruistic preferences. The needed assumption for the following proposition is, therefore: Assumption **A1** : In any long-run equilibrium allocation, there are two disjoint subsets _I_<sup>_s_</sup> _, I_<sup>_a_</sup> _⊂_ [0 _,_ 1] with _μ_ ( _I_<sup>_s_</sup> ) _>_ 0 and _μ_ ( _I_<sup>_a_</sup> ) _>_ 0 such that for all _s ∈ I_<sup>_s_</sup> and all _a ∈ I_<sup>_a_</sup> we have _ga_ = _gs_ = 0, _ua_ = _u_<sup>**a**</sup> and _us_ = _u_<sup>**s**</sup> .

Our first proposition summarizes the discussion above.

Proposition 1 makes use of the following definition. Let _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)denote</sup> the optimal bequest left by individual _a ∈ I_<sup>_a_</sup> under a scheme ( _τ, D_ ) that provides an exemption strictly larger than _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_),i.e.</sup>

**Proposition 1.** _(i) Under A1, a tax-demogrant scheme_ ( _τ, D_ ) _is optimal only if D ≥_ 0 _. (ii) Under A1, a tax-demogrant scheme_ ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ) _that provides an exemption up to b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D∗_)</sup><sup>_isoptimalifthereisnoothersustainabletax-_</sup> _demogrant scheme_ ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) _such that D_<sup>_′_</sup> _≥ D_<sup>_∗_</sup> _._

_Proof._ The proof is relegated in Online Appendix S1. ■

> 11Piketty and Saez (2013) document that about half the population in France and the US receives negligible bequests in 2010.

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The proof of (i) relies on the following argument. At the laisser-faire allocation, individuals who did not receive anything from their parents have the lowest _c_ -equivalent utility. As a result, taxing large bequests and offering a positive demogrant is a way to increase their utility, which is socially desirable given the leximin nature of the SWF. This argument, interestingly, does not depend on the specific assumption that we make on the type of altruism of the parents. If parents are interested in the total lifetime income of their offsprings or if they are interested in the utility of their children, it remains true that those who did not inherit anything from their own parents will be the worst-off, so that taxing large bequests to increase the demogrant increases their utility, with the consequence that only _D ≥_ 0 can be optimal.

The condition _D_<sup>_′_</sup> _≥ D_<sup>_∗_</sup> in (ii) is necessary for ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) to be optimal, it is of course not sufficient. In the long-run equilibrium allocation associated to ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ), all zero-inheritors are equally well-off, and the optimal tax scheme needs to make all of them at least as well-off as at ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ).

Note that the long-run equilibrium allocation associated with the optimal tax scheme depends on the preference profile considered, preferences are heterogeneous and the long-run equilibrium profile of inheritances received is _endogenous_ to the shape of the tax. Even without being able to characterize this allocation exactly, we derive in the next section some constraints on the shape of an optimal tax.

# **5.2 Tax exemption on low bequests, or limited subsidies and taxes**

In the previous section, we proved that _D_ , the demogrant, cannot be too low, and, in particular, cannot be negative. The demogrant should be thought of as the average amount transferred from those who leave a bequest for their children to the general population. In this section, we prove that _D_ cannot be too _large_ , either. The intuition for this result is the following one. Worstoff individuals are to be found among the zero-inheritors. Among them, the well-being of the self-centered individuals is entirely determined by _D_ . It is not the case for the other zero inheritors. In particular, the most altruistic among them, individuals _a_ , receive _D_ but they pay _τ_ ( _ba_ ), in which _ba_ stands for their bequest. Proposition 1 implies that _D ≥ τ_ ( _ba_ ): individuals _a_ cannot be strict contributors to the tax system. It suggests that even if _a_ end up with a lower well-being than self-centered individuals, the difference in well-being should be limited.

This has the following implication for the optimal tax scheme. Let _ba_ denote the bequest of these individuals _a_ at ( _τ, D_ ). Let _β_ be a positive bequest level smaller than _ba_ and let ∆be an amount of money smaller than _β_ . Consider the alternative tax scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) satisfying which is illustrated in Figure 2. Facing ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ), individuals _a_ do not see any difference between ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) and ( _τ, D_ ): the decrease in their lifetime income is perfectly compensated by the decrease in the tax they pay on their bequest. Zero-inheritor-self-centered individuals, on the contrary, are affected by the 20
change, as their well-being decreases by ∆. If, moreover, ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) leaves money on the table, which is quite likely because _D_ decreases, this money can be redistributed to the entire population, thereby making individuals _a_ better-off. That illustrates that decreasing _D_ and decreasing the tax below some threshold is a policy tool to increase the well-being of _a_ at the expense of _s_ (the zero-inheritor-self-centered individuals).

<!-- Start of picture text --> L ( w + D, 0)<br>hi<br>Rτ ( ba )<br>za<br>ua<br>B τ ( w + D, 0)<br>Rβ B τ ′ ( w + D ′ , 0)<br>∆ zs ci<br>w + D<br>β<br>ba<br><!-- End of picture text --> Figure 2: Tax-demogrant schemes ( _τ, D_ ) and ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) provide equal _c_ - equivalent utility to the worst-off individuals _a_ even if the latter has a smaller demogrant.

This illustration is too simple, however, as ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is typically non-sustainable. To prove Proposition 2, we do identify a truncation of ( _τ, D_ ) that is sustainable and that increases social welfare as soon as _D_ , or, equivalently, _τ_ ( _ba_ ), is too large.<sup>12</sup> As a consequence, we show in the next proposition that

1. tax functions with positive taxes on small bequest amounts are not optimal,

2. monotonically increasing tax functions are not optimal unless they provide an exemption up to the amount of bequest that would be chosen under Laissez-Faire by individuals _a_ (who inherit nothing and have the mostaltruistic preference), 3. a positive tax on the amount of bequest left by _a_ is not excluded though, at least when subsidies are provided on smaller bequest amounts, and

> 12When defining this truncation below, we give a precise value to _β_ and to ∆.

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4. even when the optimal tax function subsidizes smaller bequest amounts, the tax on the bequest amount left by _a_ must be limited (see Eq. (3) in the proposition below).

To proves these claims, we restrict our attention to tax functions _τ_ for which _−τ_ is _single-peaked_ . This domain contains all tax functions that are policy relevant. In particular, this domain is the union of the two most relevant subdomains. The first subdomain contains all (weakly) monotonically increasing tax functions _τ_ (bequests are taxed at an increasing non-negative rate), in which 0 is a peak for _−τ_ (there may be an entire interval of peaks, in which case _τ_ exempts bequests on this interval). The second subdomain contains all tax functions _τ_ that are first monotonically (weakly) decreasing (small bequests are subsidized) and then monotonically (weakly) increasing. In this subdomain the peaks are all positive. In order to simplify the exposition, we exclude tax functions that provide an exemption on small bequests and are not (weakly) monotonically increasing.<sup>13</sup> We refer to the latter subdomain as that of _positive peak_ tax functions, and to the former as that of (weakly) monotonically increasing tax functions. Observe that, by Proposition 1, any monotonically decreasing tax function is dominated because such tax cannot sustain a nonnegative demogrant. We do not comment on these functions anymore.

In this domain, the worst-off individuals are either individuals _a_ or _s_ . As illustrated in Figure 3, the reason is that the consumption of any zero-inheritor is at least as large as the consumption of _a_ , but not larger than the consumption of _s_ . As a result, their _c_ -equivalent utility cannot be smaller than both the _c_ - equivalent utilities of _s_ and _a_ . Since either _s_ or _a_ are among the worst-offs, we can restrict the normative analysis to these two individuals. When a tax function _τ_ implies a tax on the amount of bequest left by _a_ , this individual is among the worst-offs whereas _s_ is not. Welfare would be improved if it were possible to increase the well-being of _a_ while keeping the well-being of _s_ above the well-being of _a_ . The difficulty here is to make sure that such improvement materializes in the new long-run equilibrium allocation, which depends on the maximal demogrant that can be sustained.

Two additional mild assumptions are required for these results. First, when they consume at least as much as their labor income, the utility of individuals who are _not_ self-centered must be strictly increasing in the inheritance received by their child. This assumption will guarantee that there exists a sufficiently large subsidy rate that induces these individuals to leave at least a threshold amount to their children.

Assumption **A2** : For all _i ∈_ [0 _,_ 1] with _ui_ = _u_<sup>**s**</sup> we have that _ui_ ( _ci, hi_ ) is strictly monotonic in _hi_ when _ci ≥ w_ .

Second, the long-run equilibrium amount collected by a tax-demogrant scheme is continuous in the demogrant.

Assumption **A3** : For all tax _τ_ , the amount of tax collected, i.e.

> 13Hence, we exclude tax functions that first provide an exemption on small bequests, then provide subsidies on intermediate bequests and finally tax positively large bequests. All our results would remain valid if these tax functions remain included in our domain. We nonetheless exclude them in order to avoid having to treat their un-interesting case in Proposition 3.

22
<!-- Start of picture text --> ( a ) ( b )<br>hi hi ua<br>ua<br>us us<br>B τ<br>za<br>B τ za<br>L ( u c s , 0) L ( u c a , 0)<br>zs ci zs ci<br>u c s u c a w + D<br>=<br>w + D<br><!-- End of picture text --> Figure 3: (a) Individuals _s_ with _gs_ = 0 and _us_ = _u_<sup>**s**</sup> are among the worst-offs, where _u_<sup>_c_</sup> _s_<sup>denotes the c-equivalent utility</sup><sup>_uc_</sup> _s_<sup>=</sup><sup>_uc_(</sup><sup>_zs, us_).(b)</sup> Individuals _a_ with _ga_ = 0 and _ua_ = _u_<sup>**a**</sup> are among the worst-offs, where _u_<sup>_c_</sup> _a_<sup>denotesthec-equivalentutility</sup><sup>_uc_</sup> _a_<sup>=</sup><sup>_uc_(</sup><sup>_za, ua_).</sup> where _bi_ is the long-run equilibrium bequest left under ( _τ, D_ ), is continuous in _D_ .

Assumption A3 implies that, if ( _τ, D_ ) is sustainable and leaves money on the table, then for some _D_<sup>_′_</sup> _> D_ the tax-demogrant scheme ( _τ, D_<sup>_′_</sup> ) is also sustainable. Observe that assumption A3 is not necessary in the proposition below for the constraint derived on monotonically increasing tax functions.

We introduce the following notation. Let _<u>b</u> ≥_ 0 be the minimal bequest amount above which the tax _τ_ provides zero subsidy, i.e.

Obviously, amount _<u>b</u>_ is endogenous to the tax considered. For any positive peak tax, we have _<u>b</u> >_ 0 and we construct the alternative tax scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) using _β_ = _<u>b</u>_ <u>.</u> For any monotonically increasing tax, we have _<u>b</u>_ = 0 and we construct the alternative tax scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) using some _β > b_ <u>.</u> Proposition 2, our main result, provides us with the formal statements from which the four claims above are deduced.

**Proposition 2.** _Consider any tax τ for which −τ is single-peaked. Let_ _<u>b</u> be the minimal bequest amount above which no subsidies are provided under τ . Let D_<sup>_max_</sup> _be the maximal sustainable demogrant under τ . Let a be an individual who inherits nothing and holds the most altruistic preference. Let ba be the equilibrium bequest left by a under_ ( _τ, D_<sup>_max_</sup> ) _. Let b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_betheequilibrium_</sup> _bequest left by a under Laissez-Faire._

_(i) Under A1 and A2, if τ is monotonically increasing, then τ is optimal only if τ provides an exemption up to b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_._</sup> _(ii) Under A1, A2 and A3, τ is optimal only if_ 23
_Proof._ The proof is relegated in Online Appendix S2.

■ The proof of Proposition 2 is quite long. This follows from the inherent challenges summarized in the last paragraph of section 5.1. One key difficulty is that, unlike Piketty and Saez (2013), we consider non-linear taxes.

An important feature of Proposition 2 is that the shape of monotonically increasing taxes is completely characterized up to the minimal amount exempted. Importantly, this amount is _exogenous_ to the optimal tax. This amount only depends on the preference _u_<sup>**a**</sup> , the interest rate _R_ and the wage rate _w_ . This implies this shape is valid regardless of the exact preferences profile defining the economy. This contrasts with characteristics of optimal tax as derived in the literature (Piketty and Saez, 2013), which typically depend on statistics endogenous to the optimal tax. On the contrary, the bequest amount _ba_ in claim (ii) of Proposition 2 is also endogenous to the tax.

A consequence of Part (ii) of the proposition is that _c_ -equivalent utilities may not be equalized at the optimal allocation. Given the leximin nature of our SWF, this possibility only arises when individuals who did not inherit anything from their parents are all better-off than at the best allocation that provides an exemption up to _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_).</sup> To conclude this section, we note that our two propositions have an interesting corollary, namely that linear taxes are never optimal.

**Corollary 1.** _Under A1 et A2, no positive or negative linear tax is optimal._

_Proof._ Linear tax with negative rates cannot sustain non-negative demogrants. By Proposition 1, any tax scheme based on a negative demogrant is not optimal. Therefore, linear tax with negative rates are not optimal. Linear tax with positive rates are monotonically increasing and do not exempt bequests up to _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_).ByProposition2,thesetaxfunctionsarenotoptimal.</sup> ■

# **5.3 The case for a tax exemption**

A consequence of Proposition 2 is that the optimal tax either exempts inheritance up to the bequest left by the poorest most altruistic individuals under Laissez-Faire, or it subsidizes small bequests. We need to underline that the reason why subsidies can be optimal is drastically different from the arguments offered in the literature. In Kaplow (1995), Farhi and Werning (2010) and Kopczuk (2013a), subsidies are desirable because they increase the bequests left, which increases the utility of both parents and children. On the contrary, subsidies are desirable here because they incentivize poor altruistic individuals to _decrease_ their bequest below what they would leave under a non-distortionary budget set defined by the same demogrant. The advantage for society of incentivizing poor altruistic individuals to decrease their bequests comes from the fact that 1) it does not decrease the utility of these individuals, while 2) it allows the tax system to increase the tax paid by other, non-poor, altruistic individuals, the objective being to sustain a larger demogrant. This immediately suggests the drawback of these subsidies: by decreasing bequests, they impoverish future generations, so that the sustainable demogrant may fail to increase, in which case exemption is optimal.

This trade-off is illustrated in Figure 4. There is a tax function _τ_ that exempts bequests below _ba_ 1, the bequest left by individual _a_ 1 who is the most 24
altruistic individual having not received anything from her parents. Beyond _ba_ 1, bequests are taxed, so that a demogrant of _D_ is funded. Can we do better by subsidizing small bequests? Figure 4 also illustrates tax function _τ_<sup>_′_</sup> , which subsidizes bequests below _b_<sup>_′_</sup> _a_ 1<sup>whileleavingtheutilityofboththeselfishand</sup> the poor most altruistic individuals unchanged. In particular, the poor most altruistic individual now leaves a bequest _b_<sup>_′_</sup> _a_ 1<sup>_<ba_1andherchildreceivesan</sup> inheritance _h_<sup>_′_</sup> _a_ 1<sup>_< ha_1.Indeed,subsidizingbequestswhileleavingtheutilityof</sup> the poor most altruistic individual unchanged implies that this individual faces a _positive_ marginal bequest tax rate, as illustrated in the figure.

<!-- Start of picture text --> L ( w + D, 0)<br>hi ua<br>B τ za 2<br>B τ ′ za 1<br>ha 1 za ′ 2<br>h ′ a 1 z a ′ 1<br>ci<br>w + D<br>h ′ a 1<br>ha 1<br><!-- End of picture text --> Figure 4: Subsidizing small bequests allows increasing average tax rates but impoverishes future generations. Individual _a_ 2 is the altruistic child of _a_ 1 who is an altruistic individual who receives zero inheritance.

The figure also illustrates the budget facing individual _a_ 2, who is the altruistic child of the poor most altruistic individual. Following _τ_<sup>_′_</sup> , the child faces higher average tax rates than with _τ_ , but she is also poorer (because she inherited less). Here are then the consequences of shifting from a tax function with exemption on bequests left by poor individuals to a tax function that subsidizes low bequests without affecting the lowest _c_ -utility in the population, that is the _c_ -utility of the poor selfish and the poor most altruistic individuals: 1) it may increase the bequests left by poor, moderately altruistic, individuals, 2) it decreases the bequests left by poor, most altruistic, individuals, 3) it is costly to the government budget regarding the treatment of poor individuals, 4) it allows the government to tax richer individuals more, but 5) richer individuals (those who inherit something) are themselves poorer. As a result, whether subsidizing small bequests can help support a larger demogrant depends on the preferences 25
of the altruistic individuals (if the elasticity of bequest is high, a small subsidy may decrease bequests a lot, keeping utility unchanged), and the relative fraction in the population of poor and rich altruistic individuals.

We complete this section by developing an example in which no subsidy tax policy can outperform the exemption tax policy. This requires to be able to identify the long run equilibrium associated with tax functions, which can only be done under very restrictive assumptions. We indeed assume that there are only two types of individuals, selfish and altruistic. We further assume, and this is the most restrictive assumption, that preferences of the altruistic individuals are locally inelastic at their preferred bundles in all non-distortionary budgets, which goes against the result we want to prove, but also that preferences are linear over bundles involving more consumption than would be optimal under non-distortionary budgets.

Linearity yields two important simplifications. First, it allows us to precisely build incentive compatible allocations, which is a rather technical simplification. Second, and more importantly, it restricts so much the marginal rates of taxation at incentive compatible allocations (altruistic individuals having inherited less than a given amount should all face the same marginal tax rate) that we can prove that at least some individuals pay taxes _and_ leave a larger bequest than what the poorest most altruistic individuals would have left with her demogrant but in the absence of taxation.

Under these assumptions, subsidizing bequests is dominated by exempting bequests up to the bequest level of the poorest most altruistic individuals with her demogrant but in the absence of taxation while keeping the same tax scheme for bequests above that level. This comes from the fact that 1) all altruistic individuals who would benefit from the subsidy in the former policy leave larger bequests under the latter, in addition to being less costly to the government budget, whereas 2) all altruistic individuals who pay taxes in the subsidy policy pay the same amount of taxes under the latter policy while still leaving the same bequest. As a result, the exemption policy must be associated in the long run to a larger demogrant.

We consider a simplified economy with only two preference types _u_<sup>**s**</sup> and _u_<sup>_∗_</sup> _a_<sup>.Preferences of the altruistic individuals have the following properties.First,</sup> indifference curves admit a kink at bundles ( _c, h_ ) for which _h_ = _α_ + _σc_ , _σ >_ 0. The typical case would be that bequests are a luxury good, which implies _α <_ 0, but we don’t need this restriction. Second, preferences are linear below the kinks, that is with 0 _< r < R_ is the marginal rate of substitution of _u_<sup>_∗_</sup> _a_<sup>atanybundlebelow</sup> this increasing path of kinks. Preferences _u_<sup>_∗_</sup> _a_<sup>areillustratedinFigure5.This</sup> two-types economy satisfies A2.

In the two-types economy, the inheritance _gi_ received by individual _i_ only depends on the minimal number _X ∈{_ 1 _,_ 2 _, . . . }_ of generations one needs to go back in her dynasty in order to find a self-centered individual. For instance, if _i_ ’s parent is self-centered, then _X_ = 1. In that case her inheritance _gi_ is simply zero. If _i_ ’s parent is altruistic but her grand-parent is self-centered, then _X_ = 2. In that case, her inheritance _gi_ correspond to that left by an altruist who inherited nothing. And so on.

26
<!-- Start of picture text --> u ∗ a u ∗ a<br>hi<br>σ<br>- R<br>- r<br>ci<br>α<br><!-- End of picture text --> Figure 5: Indifference map of preference _u_<sup>_∗_</sup> _a_<sup>.</sup> Let _z_ be a long-run equilibrium allocation. It can be characterized by ( _zsX , zaX_ ) _X∈{_ 1 _,_ 2 _,... }_ where _sX_ (resp. _aX_ ) is a self-centered (resp. altruistic) individual whose number is _X_ .

The **incentive compatibility constraint** (IC) requires that for all _X ∈ {_ 1 _,_ 2 _. . . }_ , individual _aX_ does not prefer the bequest level selected for _aX_ + 1, which is and individual _aX_ + 1 does not prefer the bequest level selected for _aX_ , which is where in the two-types economy we have _gaX_ +1 = _haX_ for all _X ∈{_ 1 _,_ 2 _. . . }_ . Consequently, a tax scheme ( _τ, D_ ) is entirely characterized by the countable sequence of bundles ( _zsX , zaX_ ) _X∈{_ 1 _,_ 2 _,... }_ it generates, and, simultaneously, any sequence ( _zsX , zaX_ ) _X∈{_ 1 _,_ 2 _,... }_ satisfying the above incentive compatibility constraints can be generated by a set of tax systems. We restrict our attention to allocations _z_ that can be long-run equilibrium allocations generated by a tax scheme ( _τ, D_ ) such that _−τ_ is single-peaked.

**Proposition 3.** _Consider the two-types economy described above. Consider any long-run equilibrium allocation z generated by a tax scheme_ ( _τ, D_ ) _such that (i) −τ is single-peaked and (ii) such that both self-centered and altruistic individuals who inherit nothing are among the worst-offs. Let b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_betheequilibrium_</sup> _bequest left under Laissez-Faire by an altruistic individual who inherits nothing. Under A1 and A3, z is optimal only if it can be generated by a tax system that provides an exemption up to b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_._</sup>

_Proof._ The proof is relegated in Online Appendix S3.

■ 27
# **6 Conclusion**

The model that we study in this paper is designed to focus on the trade-off between subsidizing bequests or transferring a demogrant to all. A number of simplifying assumptions have been needed, to which we now come back.

We assumed away the issue of taxing labor incomes. We assume that all individuals have the same labor time and the same lifetime income, so that fairness does not require to redistribute labor income. Our result that bequests should, on average, be taxed so as to transfer a demogrant to all does not, therefore, come from the need to alleviate income inequalities, but only from the need to compensate children of selfish parents while preserving the parents’ freedom to allocate their lifetime income the way they wish. As a consequence, our conclusions are compatible with heterogeneity of wages and labor times and the existence of a labor income tax system maximizing social welfare. This claim, however, calls for two qualifications.

First, our formal analysis can be replicated in a more general model only under the assumption that individuals’ lifetime incomes are not influenced by the design of our bequest taxation system. However, this assumption is unlikely to be valid, in which case, a complete framework should study income redistribution and bequest taxation systems simultaneously. We suspect that deriving analytical results in such a complicated model is out of reach. Intuitively, though, the result of such an exercise is likely to remain that bequests are, on average, taxed, because it is the way to compensate the children of self-centered parents while maintaining a sufficiently high utility level to self-centered individuals who did not receive anything from their own parents. Identifying who are the worst-off individuals and dealing with sustainability issues, however, would become much harder.

The second qualification has to do with the identification of the worst-off individuals in the case in which lifetime incomes are not influenced by the bequest taxation system. Given that the labor income taxation system aims at redistributing from higher wage individuals to lower wage individuals, it is extremely likely that the worst-off have to be found among the minimal-wage individuals. Consequently, our assumption A1 has to be strengthened into the existence, in any long-run equilibrium allocation, of individuals who did not inherit anything from their parents, who have the minimum wage and who have either the most altruistic or self-centered preferences. As a result, Proposition 2, part (i), for instance, would become that bequests should be exempted from taxes up to the amount left by the most altruistic individuals who did not receive any bequest from their parents and worked all their life at the minimum wage.

Our main results do not give us a formula that can be calibrated, but yet they can be used to qualitatively assess current tax systems. According to a recent report (see OCDE (2021)) 12 of the 36 OECD countries do not tax bequests. Our Proposition 1, part (i) implies that this can not be optimal given our social welfare function. All the 24 countries that do tax bequests to children have a system consistent with the optimal tax system of Proposition 1, Part (ii) and Proposition 2, part (i): exemption for small bequests and a positive tax on larger ones. The interval of exemptions considerably varies across countries, from $17,133 in Belgium to $11,580,000 in the United States (numbers in 2020 USD). We note that those who work full-time (40 hours a week, 48 weeks a year) for an hourly wage of $15 and invest 15% of their income at a yearly rate 28
of 3% have accumulated more than $325,000 after 40 years, suggesting that the exemption amount in Belgium (as well as the median exemption level of $171 329, Greece) is likely to be smaller than the bequest left by the most altruistic parents whereas the exemption amount in the United States is clearly above this threshold. The money collected through bequest taxes is below 2% of the total fiscal revenues in all countries, and even below 1% in most countries, suggesting that the corresponding demogrant is not maximized. More research is needed, however, to compute the optimal tax systems in these countries.

In the model, we also assumed that the number of children is identical across households. Allowing heterogeneity among the number of children would not change the fact that the worst-off individuals have to be found among those who did not inherit anything from their parents. A new question would emerge, however, regarding the amount of exempted bequest. It would still be defined with reference to the amount left by the most altruistic individuals who did not receive any bequest from their parents, but the choice is between considering the bequests of parents of the largest number of children or with only one child. The former choice is appropriate if parents are modeled as caring about the per capita bequest received by their children. The latter choice is appropriate if parents are modeled as caring about the total bequest left.

The interest rate, _R_ , is exogenous in our model. This is typical of a small open economy. If it is endogenous, but further assumptions make it depend only on the distribution of preferences in the economy, then our analysis carries over with _R_ being replaced with the endogenous rate. If, on the contrary, the interest rate may vary across time, then our results change. The intuition is that our optimal tax scheme needs to be amended so as to redistribute further from the lucky ones who face higher interest rates towards those who face lower interested rates. Moreover the leximin nature of our SWF implies that the worstoffs belong to those who did not inherit anything from their parents. Therefore, whether an individual is lucky or not only depends on the future interest rates. So, an optimal tax system should redistribute from those who can save for their children at a high rate towards those who save at a low rate. How precisely this should be done requires additional research.

Other assumptions would be much more difficult to relax. They would require to redefine the social welfare function or the policy tools. It would be the case, for instance, if individuals are interested in the entire lifetime of their children and not only how much their children inherit, in which case an increase of the demogrant benefits the altruistic parents more than the self-centered ones, if they have unequal life expectancy, in which case the social planner may wish to subsidize the bequest of short-lived individuals, if fertility choices are constrained, in which case the social planner may wish to favor those who wanted to have children but could not, and, therefore, do not leave any bequest, if children can inherit from different adults, raising the question of whether the tax should be donor-based or recipient-based, if bequests are only partially observable, etc.

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# **ONLINE APPENDIX: Fair Inheritance taxation**

## **Benoit Decerf and Francois Maniquet**

# **S1 Proof of Proposition 1**

First, we prove claim (i). We show that any ( _τ, D_ ) with _D <_ 0 is dominated by Laissez-Faire.

We start by showing for the long-run equilibrium allocation _z_<sup>_LF_</sup> _∈ S_ associated to Laissez-Faire that _u_<sup>_c_</sup> ( _zi_<sup>_LF_</sup> _, ui_ ) _≥ w_ for all _i ∈_ [0 _,_ 1]. Under Laissez-Faire, any _i ∈_ [0 _,_ 1] choses in the budget set implying that _u_<sup>_c_</sup> ( _zi_<sup>_LF_</sup> _, ui_ ) = _w_ + _gi_<sup>_LF_</sup> . We then show for the long-run equilibrium allocation _z ∈ S_ associated to ( _τ, D_ ) that some subset _J ⊂_ [0 _,_ 1] with _μ_ ( _J_ ) _>_ 0 is such that _u_<sup>_c_</sup> ( _zj, uj_ ) = _w_ + _D_ for all _j ∈ J_ . By assumption A1, there is a subset _J ⊂_ [0 _,_ 1] with _μ_ ( _J_ ) _>_ 0 such that _gj_ = 0 and _uj_ = _u_<sup>**s**</sup> for all _j ∈ J_ . Since these individuals are self-centered, we have that _zj_ = (0 _, w_ + _D,_ 0) and _u_<sup>_c_</sup> ( _zj, uj_ ) = _w_ + _D_ .

As _D <_ 0, this shows that _u_<sup>_c_</sup> ( _zj, uj_ ) _< w_ for all _j ∈ J_ , showing that LaissezFaire is preferred to ( _τ, D_ ) by **_R_**<sup>_c−lex_</sup> .

Second, we prove claim (ii). We show that any ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) with _D_<sup>_′′_</sup> _< D_<sup>_∗_</sup> is dominated by ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ).

We start by showing for the long-run equilibrium allocation _z_<sup>_∗_</sup> _∈ S_ associated to ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ) that _u_<sup>_c_</sup> ( _zi_<sup>_∗, ui_)</sup><sup>_≥w_+</sup><sup>_D∗_forall</sup><sup>_i∈_[0</sup><sup>_,_1].</sup> In equilibrium, any _i ∈_ [0 _,_ 1] choses in the budget set _B_<sup>_τ ∗_</sup> ( _w_ + _D_<sup>_∗_</sup> + _gi_<sup>_∗,_0).Forall</sup><sup>_i, j∈_[0</sup><sup>_,_1]with</sup> _ui_ = _uj_ and _gj_<sup>_∗_=0wemusthave</sup><sup>_ui_(</sup><sup>_z_</sup> _i_<sup>_∗_)</sup><sup>_≥uj_(</sup><sup>_z_</sup> _j_<sup>_∗_)because</sup><sup>_Bτ ∗_(</sup><sup>_w_+</sup><sup>_D∗,_0)</sup><sup>_⊆_</sup> _B_<sup>_τ ∗_</sup> ( _w_ + _D_<sup>_∗_</sup> + _gi_<sup>_∗,_0).As</sup><sup>_ui_=</sup><sup>_uj_, this implies that</sup><sup>_uc_(</sup><sup>_z_</sup> _i_<sup>_∗, ui_)</sup><sup>_≥uc_(</sup><sup>_z_</sup> _j_<sup>_∗, uj_).There</sup> remains to show that _u_<sup>_c_</sup> ( _zj_<sup>_∗, uj_) =</sup><sup>_w_+</sup><sup>_D∗_.As</sup><sup>_τ ∗_providesanexemptionupto</sup> _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D∗_), any</sup><sup>_j∈_[0</sup><sup>_,_1] with</sup><sup>_g_</sup> _j_<sup>_∗_= 0 choses the same bundle</sup><sup>_z_</sup> _j_<sup>_∗_in her budget</sup> set _B_<sup>_τ ∗_</sup> ( _w_ + _D_<sup>_∗_</sup> _,_ 0), as she would chose in the non-distortionary budget set implying that _u_<sup>_c_</sup> ( _zj_<sup>_∗, uj_) =</sup><sup>_w_+</sup><sup>_D∗_.</sup> We then show for the long-run equilibrium allocation _z_<sup>_′′_</sup> _∈ S_ associated to ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) that some subset _J ⊂_ [0 _,_ 1] with _μ_ ( _J_ ) _>_ 0 is such that _u_<sup>_c_</sup> ( _zj_<sup>_′′, uj_)=</sup> _w_ + _D_<sup>_′′_</sup> for all _j ∈ J_ . By assumption A1, there is a subset _J ⊂_ [0 _,_ 1] with _μ_ ( _J_ ) _>_ 0 such that _gj_<sup>_′′_=0and</sup><sup>_uj_=</sup><sup>_u_</sup><sup>**s**forall</sup><sup>_j∈J_.Sincetheseindividuals</sup> are self-centered, we have under the long-run equilibrium allocation _z_<sup>_′′_</sup> _∈ S_ associated to ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) that _zj_<sup>_′′_= (0</sup><sup>_, w_+</sup><sup>_D′′,_0)andso</sup><sup>_uc_(</sup><sup>_z_</sup> _j_<sup>_′′, uj_) =</sup><sup>_w_+</sup><sup>_D′′_.</sup> As _D_<sup>_′′_</sup> _< D_<sup>_∗_</sup> , this shows that _u_<sup>_c_</sup> ( _zj_<sup>_′′, uj_)</sup><sup>_<w_+</sup><sup>_D∗_forall</sup><sup>_j∈J_,showing</sup> that ( _τ_<sup>_∗_</sup> _, D_<sup>_∗_</sup> ) is preferred to ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) by **_R_**<sup>_c−lex_</sup> .

# **S2 Proof of Proposition 2**

Proposition 2 provides a limit on the tax paid on the amount of bequest left by _a_ . The proof constructs another sustainable tax-demogrant scheme that dominates ( _τ, D_ ). The construction is based on a sustainable tax-demogrant scheme 1
( _τ_<sup>∆</sup> _, D −_ ∆) illustrated in Figure 6. As shown in the figure, this second scheme linearly truncates the budget set faced by individuals under ( _τ, D_ ). This truncation reduces the demogrant by an amount ∆, but the new tax _τ_<sup>∆</sup> provides large subsidies on small bequests. Provided the inheritance they receive is unchanged, individuals chose the same bundle under both schemes, at least if this bundle does not lie in the truncated part of their budget set. This implies that the wellbeing of _a_ is the same under both schemes. The smaller demogrant reduces the well-being of _s_ , but _s_ still has a larger well-being than _a_ . In the proof, we show that when ( _τ, D_ ) is sustainable, scheme ( _τ_<sup>∆</sup> _, D −_ ∆) leaves money on the table. Money is left on the table because (i) the new scheme saves on self-centered individuals who receive a smaller demogrant and (ii) the new scheme does not spend more on altruistic individuals because its subsidies on small bequests are financed by the reduction in the demogrant. Then, the money saved by the new scheme can finance a small increase in the demogrant, such that ( _τ_<sup>∆</sup> _, D −_ ∆+ _ε_ ) is sustainable for some _ε >_ 0. The larger demogrant increases the well-being of all individuals, including that of the worst-off.

<!-- Start of picture text --> L ( w + D, 0)<br>hi<br>Rτ ( ba )<br>za<br>( w + D − b, Rb )<br>L ( u c a , 0) B τ<br>B τ ∆<br>Rb<br>∆ zs ci<br>u c a b w + D<br>ba<br><!-- End of picture text --> Figure 6: The tax-demogrant scheme ( _τ, D_ ) is dominated because the tax function _τ_ taxes too much the bequest _ba_ left by _a_ . Individual _a_ is the worst-off because _u_<sup>_c_</sup> ( _za, ua_ ) _≤_ _~~u~~_<sup>_~~c~~_</sup> _a_<sup>.The</sup> sustainable scheme ( _τ_<sup>∆</sup> _, D −_ ∆) has a smaller demogrant, does not affect _u_<sup>_c_</sup> ( _za, ua_ ) and leaves money on the table.

Here is the intuition why scheme ( _τ_<sup>∆</sup> _, D −_ ∆) is sustainable. The truncation creates a kink in the budet set faced by individuals under ( _τ, D_ ). Impor2
tantly, this kink is located at an amount of bequest _<u>b</u>_ above which the tax _τ_ is non-negative and monotonically increasing. The larger the rate of subsidies on small bequests associated to _τ_<sup>∆</sup> , the more numerous the altruistic individuals who “bunch” at the kink, at least for those who used to leave a bequest smaller than _<u>b</u>_ under ( _τ, D_ ). If all of these individuals “bunch” at the kink, then the long-run equilibrium profile of inheritances under ( _τ, D_ ) would be first-order stochastically dominated by the long-run equilibrium profile of inheritances under ( _τ_<sup>∆</sup> _, D −_ ∆). We can then show that ( _τ_<sup>∆</sup> _, D −_ ∆) is sustainable if ( _τ, D_ ) is sustainable because, above the kink, the tax paid is increasing in the bequest left.

Both claims (i) and (ii) in Proposition 2 rely on the following two lemmas. Recall that we denote by _a_ an individual for whom the long-run equilibrium _ga_ = 0 and for whom _ua_ = _u_<sup>**a**</sup> , and by _s_ an individual for whom the long-run equilibrium _gs_ = 0 and for whom _us_ = _u_<sup>**s**</sup> .

**Lemma 1.** _Under A1, for any tax-demogrant scheme_ ( _τ, D_ ) _for which −τ is single-peaked, either a or s are among the worst-offs._

_Proof._ Let _z ∈ S_ denote the long-run equilibrium allocation associated to ( _τ, D_ ). By assumption A1, there exist two individuals _a_ and _s_ with _ga_ = 0, _ua_ = _u_<sup>**a**</sup> , _gs_ = 0 and _us_ = _u_<sup>**s**</sup> . We derive a contradiction when assuming that for some _k ∈_ [0 _,_ 1] we have _u_<sup>_c_</sup> ( _zk, uk_ ) _< u_<sup>_c_</sup> ( _za, ua_ ) and _u_<sup>_c_</sup> ( _zk, uk_ ) _< u_<sup>_c_</sup> ( _zs, us_ ).

Under _z_ , any _i ∈_ [0 _,_ 1] choses in her budget set _B_<sup>_τ_</sup> ( _w_ + _D_ + _gi,_ 0). This implies for all _j ∈_ [0 _,_ 1] with _uj_ = _uk_ and _gj_ = 0 that _uj_ ( _zj_ ) _≤ uk_ ( _zk_ ), and hence _u_<sup>_c_</sup> ( _zj, uj_ ) _≤ u_<sup>_c_</sup> ( _zk, uk_ ). Thus, we can assume without loss of generality that _gk_ = 0. By definition of _a_ and _s_ , the fact that _ga_ = _gk_ = _gs_ = 0 implies that _ca ≤ ck ≤ cs_ = _w_ + _D_ . Letting _u_<sup>_c_</sup> _k_<sup>=</sup><sup>_uc_(</sup><sup>_zk, uk_), we show that (</sup><sup>_ck, hk_)</sup><sup>_∈/L_(</sup><sup>_uc_</sup> _k_<sup>_,_0).</sup> There are two cases for _zs_ , which are illustrated in Figure 7.

- Case 1: ( _cs,_ 0) _∈ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0).</sup>

Let _zs_<sup>_′_=(</sup><sup>_g_</sup> _s_<sup>_′, c′_</sup> _s_<sup>_,_0)besuchthat(</sup><sup>_c′_</sup> _s_<sup>_,_0)=arg max</sup><sup>_u_</sup> _s_<sup>_L_(</sup><sup>_uc_</sup> _k_<sup>_,_0).Thiscaseis</sup> such that _us_ ( _zs_ ) _≤ us_ ( _zs_<sup>_′_).Asbydefinition</sup><sup>_uc_(</sup><sup>_z_</sup> _s_<sup>_′, us_) =</sup><sup>_uc_</sup> _k_<sup>,wehavethat</sup> _u_<sup>_c_</sup> ( _zs, us_ ) _≤ u_<sup>_c_</sup> _k_<sup>andthus</sup><sup>_uc_(</sup><sup>_zs, us_)</sup><sup>_≤uc_(</sup><sup>_zk, uk_),acontradiction.</sup>

- Case 2: ( _cs,_ 0) _∈/ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0).</sup>

As ( _w_ + _D,_ 0) _∈/ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)and</sup><sup>_gk_=</sup><sup>_gs_=0,wehave(</sup><sup>_ck, hk_)</sup><sup>_∈/L_(</sup><sup>_uc_</sup> _k_<sup>_,_0)</sup> unless _τ_ ( _w_ + _D − ck_ ) _≥_ 0.<sup>14</sup> As _−τ_ is single-peaked and _ca ≤ ck_ , this implies in turn that _τ_ ( _w_ + _D − ca_ ) _≥ τ_ ( _w_ + _D − ck_ ). Since _gk_ = _ga_ = 0, the fact that _τ_ ( _w_ + _D − ca_ ) _≥ τ_ ( _w_ + _D − ck_ ) _≥_ 0 and ( _ck, hk_ ) _∈ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)</sup> together imply that ( _ca, ha_ ) _∈ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0).Then,</sup><sup>_uc_(</sup><sup>_za, ua_)</sup><sup>_≤uc_</sup> _k_<sup>andthus</sup> _u_<sup>_c_</sup> ( _za, ua_ ) _≤ u_<sup>_c_</sup> ( _zk, uk_ ), a contradiction.

Since ( _ck, hk_ ) _∈/ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)but</sup><sup>_uc_(</sup><sup>_zk, uk_)=</sup><sup>_uc_</sup> _k_<sup>,thisimpliesthatforsome</sup> ˆ bundle _zk_ = (0 _,_ ˆ _ck, h_<sup>ˆ</sup> _k_ ) such that (ˆ _ck, h_<sup>ˆ</sup> _k_ ) _∈ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)wehave</sup><sup>_uk_(ˆ</sup><sup>_zk_)=</sup><sup>_uk_(</sup><sup>_zk_)</sup>

> 14By definition, _L_ ( _uck_<sup>_,_0) =</sup><sup>_{_(</sup><sup>_ci, hi_)</sup><sup>_| ci_+</sup><sup>_hi/R≤uc_</sup> _k_<sup>_}_.When (</sup><sup>_cs,_0) = (</sup><sup>_w_+</sup><sup>_D,_0)</sup><sup>_∈/L_(</sup><sup>_uc_</sup> _k_<sup>_,_0),</sup> we have _u_<sup>_c_</sup> _k_<sup>_<w_+</sup><sup>_D_.Sinceinequilibrium,</sup><sup>_hk_=</sup><sup>_R_(</sup><sup>_w_+</sup><sup>_D −ck−τ_(</sup><sup>_w_+</sup><sup>_D −ck_)),wehave</sup> _ck_ + _hk/R < w_ + _D_ only if _τ_ ( _w_ + _D − ck_ ) _>_ 0.

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<!-- Start of picture text --> (Case 1) (Case 2)<br>L ( u c k , 0) uk<br>hi z ˆ k hi<br>B τ ( w + D, 0) zk h ˆ k z ˆ k<br>Rτ ( bk )<br>uk<br>B τ ( w + D, 0)<br>L ( w + D, 0) za<br>zk<br>L ( u c k , 0)<br>zs z s ′ ci zs ci<br>u c k u c k w + D<br>w + D<br><!-- End of picture text --> Figure 7: Constructions used in the proof of Lemma 1.

(see Figure 7 Case 2). This implies that But by definition of the most-altruistic preferences _u_<sup>**a**</sup> , it means that _a_ would chose in _B_<sup>_τ_</sup> ( _w_ + _D,_ 0) _∪ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)abundle</sup><sup>_z_ˆ</sup><sup>_a_=(0</sup><sup>_,_ˆ</sup><sup>_ca,_ˆ</sup><sup>_ha_)suchthat</sup><sup>_h_ˆ</sup><sup>_a≥h_ˆ</sup><sup>_k_.</sup> By construction, we have (ˆ _ca, h_<sup>ˆ</sup> _a_ ) _∈ L_ ( _u_<sup>_c_</sup> _k_<sup>_,_0)and(ˆ</sup><sup>_ca,_ˆ</sup><sup>_ha_)</sup><sup>_∈/Bτ_(</sup><sup>_w_+</sup><sup>_D,_0).This</sup> shows that _u_<sup>_c_</sup> ( _za, ua_ ) _≤ u_<sup>_c_</sup> (ˆ _za, ua_ ) = _u_<sup>_c_</sup> _k_<sup>andthus</sup><sup>_uc_(</sup><sup>_za, ua_)</sup><sup>_≤uc_(</sup><sup>_zk, uk_),a</sup> contradiction. This concludes the proof of Lemma 1. ■ Let _μ_<sup>_c_</sup> _a_<sup>=</sup><sup>_uc_(</sup><sup>_za, ua_)denotethelong-runequilibrium</sup><sup>_c_-equivalentutilityof</sup> individual _a_ with _ga_ = 0 and _ua_ = _u_<sup>**a**</sup> under scheme ( _τ, D_ ). Let _ba_ denote the equilibrium bequest left by individual _a_ under scheme ( _τ, D_ ).

**Lemma 2.** _Consider any sustainable tax-demogrant scheme_ ( _τ, D_ ) _such that D ≥_ 0 _and −τ is single-peaked. Under A1 and A2, if τ is monotonically increasing, then_ ( _τ, D_ ) _is dominated if μ_<sup>_c_</sup> _a_<sup>_<w_+</sup><sup>_Dandba>_0</sup><sup>_.UnderA1,A2_</sup> _and A3,_ ( _τ, D_ ) _is dominated if μ_<sup>_c_</sup> _a_<sup>_< w_+</sup><sup>_D −_</sup><sup>_<u>b</u>_</sup> _and ba > b._

_Proof._ Let _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _∈ S_ denote the long-run equilibrium allocation associated to ( _τ, D_ ) and let ( _bi_ ) _i∈_ [0 _,_ 1] be the long-run equilibrium profile of bequests left, i.e. _bi_ = _w_ + _D_ + _gi − ci_ for all _i ∈_ [0 _,_ 1]. Let _A ⊂_ [0 _,_ 1] be the subset of altruistic individuals, i.e. _ui_ = _u_<sup>**s**</sup> for all _i ∈ A_ .

There are two cases to consider. For each case, the proof proceeds in three steps. In Step 1, we construct a particular tax-demogrant scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ). In Step 2, we show that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable if ( _τ, D_ ) is sustainable. In Step 3, we show that the long-run equilibrium allocation _z_<sup>_′_</sup> _∈ S_ associated to ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is preferred by **_R_**<sup>_c−lex_</sup> over _z_ .

**CASE 1** : for all bequest amount _b >_ 0 there is a subset _J ⊆ A_ with _μ_ ( _J_ ) _>_ 0 such that _bj < b_ for all _j ∈ J_ .

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_Step 1_ . We construct a particular tax-demogrant scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ). The construction of _τ_<sup>_′_</sup> is based on a particular bequest amount _β >_ 0, whose definition depends on the type of _τ_ .

- If _τ_ is _monotonically increasing_ , then _<u>b</u>_ = 0 and we take any _β_ such that 0 _< β <_ min( _ba, w_ + _D − μ_<sup>_c_</sup> _a_<sup>).</sup>

- If _−τ_ is _positive peak_ , then _<u>b</u> >_ 0 and we take _β_ = _<u>b</u>_ <u>.</u>

For both types, we have 0 _< β <_ min( _ba, w_ + _D − μ_<sup>_c_</sup> _a_<sup>).</sup> Given _β_ , we construct ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) from a specific member of a parametric family of “truncated” tax-demogrant schemes ( _τ_<sup>∆</sup> _, D −_ ∆) with parameter ∆ _∈_ (0 _, β_ ). The construction of ( _τ_<sup>∆</sup> _, D −_ ∆) is illustrated in Figure 8 for the case _<u>b</u>_ = 0 and in Figure 6 for the case _<u>b</u> >_ 0 (where _β_ = _<u>b</u>_ <u>).</u> All members of this family linearly truncate the budget set _B_<sup>_τ_</sup> ( _w_ + _D_ + _gi,_ 0) for bequests smaller than _β_ and differ by their associated demogrant _D −_ ∆.<sup>15</sup> Formally, we define _τ_<sup>∆</sup> as <!-- Start of picture text -->  τ ( x + ∆) − ∆ for all x ≥ β − ∆<br>τ ∆ ( x ) := (6)<br> τ ( β ) − ∆<br> β− ∆ x for all x ∈ [0 , β − ∆] .<br>L ( w + D, 0)<br>hi ua<br>Rτ ( ba )<br>za<br>( w + D − β, Rβ − Rτ ( β ))<br>L ( u c a , 0)<br>B τ<br>B τ ∆<br>∆ zs ci<br>u c a β w + D<br>ba<br><!-- End of picture text --> Figure 8: Construction of scheme ( _τ_<sup>∆</sup> _, D −_ ∆) for the case _<u>b</u>_ = 0.

The particularity of scheme ( _τ_<sup>∆</sup> _, D −_ ∆) is that any individual _i ∈_ [0 _,_ 1] for whom _bi ≥ β_ choses the same bundle under both ( _τ_<sup>∆</sup> _, D −_ ∆) and ( _τ, D_ ), i.e.

> 15Demogrant _D −_ ∆need not be the maximal sustainable demogrant under tax _τ_ ∆.

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Let _z_<sup>∆</sup> = ( _gi_<sup>∆</sup><sup>_, c_∆</sup> _i_<sup>_, h_∆</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_denotethelong-runequilibriumallocation</sup> associated to ( _τ_<sup>∆</sup> _, D −_ ∆) and let ( _b_<sup>∆</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]bethelong-runequilibriumprofile</sup> of bequests left, i.e. _b_<sup>∆</sup> _i_<sup>=</sup><sup>_w_+</sup><sup>_D −_∆+</sup><sup>_g_</sup> _i_<sup>∆</sup><sup>_−c_∆</sup> _i_<sup>forall</sup><sup>_i∈_[0</sup><sup>_,_1].Recallthat</sup> _Rβ − Rτ_ ( _β_ ) is the amount inherited by the child of any _i ∈_ [0 _,_ 1] for whom _bi_ = _β_ . Consider the subset _J_<sup>∆</sup> _⊆ A_ for whom _h_<sup>∆</sup> _j_<sup>_< Rβ −Rτ_(</sup><sup>_β_) for all</sup><sup>_j∈J_∆.</sup> We show that _μ_ ( _J_<sup>∆</sup> ) _→_ 0 when ∆ _→ β_ . The intuition for this statement is that the “truncated” budget set has a kink at bundle (0 _, w_ + _D − β, Rβ − Rτ_ ( _β_ )). Therefore, the larger is ∆, the steeper is the slope of this truncated budget sets for small bequests (this slope tends to _−∞_ when ∆ _→ β_ ), and the greater is the incentive to “bunch” at the kink for the “moderately” altruistic individuals who inherit nothing. More formally, for any altruistic preference _u ∈ U \{u_<sup>**s**</sup> _}_ , there is a ∆ _< β_ such that for any _i ∈_ [0 _,_ 1] with _ui_ = _u_ and _gi_ = 0 we have _h_<sup>∆</sup> _i_<sup>_≥Rβ −Rτ_(</sup><sup>_β_).This follows</sup> from A2, which requires that the altruistic preference _ui_ is strictly monotonic in _hi_ when _ci ≥ w_ . The latter is guaranteed because the kink is located at bundle (0 _, w_ + _D − β, Rβ − Rτ_ ( _β_ )) where _w_ + _D − β ≥ w_ .<sup>16</sup> By the binormality of preferences, we also have _h_<sup>∆</sup> _i_<sup>_≥Rβ −Rτ_(</sup><sup>_β_)forany</sup><sup>_i ∈_[0</sup><sup>_,_1]withanaltruistic</sup> preference _ui_ and _gi >_ 0, which yields the result.

We are now equipped for the definition of scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ). This definition is based on the per-capita money amount _βλs_ , which is saved by the government on the mass _λs_ of self-centered individuals when reducing the demogrant by an amount _β_ . For some<sup>_<u>β</u>_</sup> 2<sup>_λs_</sup> _>_ 0, consider the subset of altruistic individuals _<u>βλs βλs</u> J_ 2 _⊆ A_ for whom _hj <_<sup>_<u>β</u>_</sup> 2<sup>_λs_</sup> for all _j ∈ J_ 2 . In words, all altruistic individuals _<u>βλs</u>_ in _J_ 2 leave an inheritance smaller than half the per-capita amount saved on _<u>βλs</u>_ self-centered individuals. Under Case 1, we have _μ_ ( _J_ 2 ) _>_ 0. Since _μ_ ( _J_<sup>∆</sup> ) _→_ 0 when ∆ _→ β_ , there is a value ∆<sup>_∗_</sup> with ∆<sup>_∗_</sup> _>_<sup>_<u>β</u>_</sup> 2<sup>suchthat17</sup> In fact any ∆ _>_ ∆<sup>_∗_</sup> also satisfies these two properties. We define scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) as whose construction is illustrated in Figure 9.

_Step 2_ . We show that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable if ( _τ, D_ ) is sustainable. Let _z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_denotethelong-runequilibriumallocationassociated</sup> to ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) and let ( _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]bethelong-runequilibriumprofileofbequestsleft,</sup> i.e. _b_<sup>_′_</sup> _i_<sup>=</sup><sup>_w_+</sup><sup>_D′_+</sup><sup>_g_</sup> _i_<sup>_′−c′_</sup> _i_<sup>forall</sup><sup>_i ∈_[0</sup><sup>_,_1].Let(ˆ</sup><sup>_h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]betheprofileobtained</sup>

> 16Indeed, we assume that _β ≤ w_ + _D − uca_<sup>andwehave</sup><sup>_uc_</sup> _a_<sup>_≥w_(otherwisebyA1</sup><sup>_τ_is</sup> dominated by Laissez-Faire).

> 17Recall that by definition of _τ_ ∆ we have ∆ _< β_ .

> 18If the tax _τ_ is positive peak, then _τ_ ∆<sup>_∗_</sup> _subsidizes_ bequests smaller than _β −_ ∆ _∗_ for all ∆<sup>_∗_</sup> _>_ 0. In contrast, when the tax _τ_ is monotonically increasing, _τ_<sup>∆</sup><sup>_∗_</sup> _subsidizes_ small bequests when _τ_ ( _β_ ) _<_ ∆<sup>_∗_</sup> .

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<!-- Start of picture text --> hi βλs<br>2 ( w + D − β, Rβ − Rτ ( β ))<br>Rβ − Rτ ( β ) B τ ′ ( w + D ′ , 0)<br>B τ ( w + D, 0)<br>B τ ∆ ∗ ( w + D − ∆ ∗ , 0)<br>βλs β<br>2<br>βλs<br>2 zs<br>ci<br>w + D ′ w + D<br>∆ ∗<br><!-- End of picture text --> Figure 9: Construction of scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) for the case _<u>b</u>_ = 0.

from ( _h_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]whensortingdynastiesbyincreasingorderof</sup><sup>_h′_</sup> _i_<sup>,i.e.</sup><sup>_h_ˆ</sup><sup>_′_</sup> _j_<sup>_≤h_ˆ</sup><sup>_′_</sup> _k_ “ˆ” for any _j, k ∈_ [0 _,_ 1] with _j < k_ . Similarly, we use symbol to denote other bequest or inheritance profiles sorted in increasing order.

We show that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable if ( _τ, D_ ) is sustainable. If ( _τ, D_ ) is sustainable, then we have from the government’s budget constraint that where ( _bi_ ) _i∈_ [0 _,_ 1] is the long-run equilibrium profile of bequests left under ( _τ, D_ ). In order to show that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable, it is sufficient that<sup>19</sup> where ( _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]islong-runequilibriumprofileofbequestsleftunder(</sup><sup>_τ ′, D′_).</sup> Recalling that the mass of self-centered individuals is _λs_ , we have for all _i ∈_ [0 _, λs_ ] that<sup>ˆ</sup> _bi_ =<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>= 0andthus</sup><sup>_τ_(ˆ</sup><sup>_bi_) =</sup><sup>_τ_(ˆ</sup><sup>_b′_</sup> _i_<sup>) = 0.Lastinequalityholdsif</sup> the money saved by reducing the demogrant from _D_ to _D_<sup>_′_</sup> is sufficient to cover the reduction in tax collected on altruistic individuals, i.e.

In the remainder of Step 2, we first show that Eq. (7) holds in a special case. Then, we build on this special case in order to show that Eq. (7) holds

> 19Recall that the sustainability of a scheme ( _τ, D_ ) relates only to the government budget constraint under its associated long-run equilibrium allocation. Indeed, a long-run equilibrium allocation is by definition a steady-state allocation, implying that the profile of inheritances left corresponds to the profile of inheritances received.

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in general. To do that, we show that the special case is in fact the worst-case scenario.

We now show that Eq. (7) holds for the special case for which _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>=</sup><sup>_h_ˆ</sup><sup>_i≥_</sup> _Rβ − Rτ_ ( _β_ ) for all _i ∈_ ( _λs,_ 1]. When _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>_≥Rβ−Rτ_(</sup><sup>_β_),because</sup><sup>_τ ′_=</sup><sup>_τ_∆</sup><sup>_∗_</sup> ˆ we have by the definition of _τ_<sup>∆</sup> in Eq. (6) that _τ_<sup>_′_</sup> (<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>)=</sup><sup>_τ_</sup> _b_<sup>_′_</sup> _i_<sup>+ ∆</sup><sup>_∗_�</sup> _−_ ∆<sup>_∗_</sup> . If _h_ ˆ<sup>_′_</sup> _i_<sup>= ˆ</sup><sup>_hi_, which is equivalent to ˆ</sup><sup>_b′_</sup> _i_<sup>_−τ ′_(ˆ</sup><sup>_b′_</sup> _i_<sup>) = ˆ</sup><sup>_bi −τ_(ˆ</sup><sup>_bi_), the definition of</sup> �<sup>_τ ′_implies</sup> that<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>= ˆ</sup><sup>_bi−_∆</sup><sup>_∗_for all</sup><sup>_i ∈_(</sup><sup>_λs,_1].In turn, this implies that</sup><sup>_τ ′_(ˆ</sup><sup>_b′_</sup> _i_<sup>) =</sup><sup>_τ_(ˆ</sup><sup>_bi_)</sup><sup>_−_∆</sup><sup>_∗_.</sup> Replacing this expression in Eq. (7) yields Since _D − D_<sup>_′_</sup> = ∆<sup>_∗_</sup> _−_<sup>_<u>βλs</u>_lastinequalitybecomes∆</sup><sup>_∗≥_</sup><sup>_<u>β</u>_whichholdsasthe</sup> 2<sup>,</sup> 2<sup>,</sup> construction of ∆<sup>_∗_</sup> is such that<sup>_<u>β</u>_</sup> 2<sup>_<_∆</sup><sup>_∗< β_.</sup> There remains to show that Eq. (7) holds in general if Eq. (7) holds for the special case for which _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>=</sup><sup>_h_ˆ</sup><sup>_i≥Rβ−Rτ_(</sup><sup>_β_)forall</sup><sup>_i∈_(</sup><sup>_λs,_1].Weusethe</sup> following _Technical Claim_ (proved at the end of Step 2): we have _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>_≥h_ˆ</sup><sup>_i_and</sup> ˆ _<u>βλs</u> h_<sup>_′_</sup> _i_<sup>_≥Rβ−Rτ_(</sup><sup>_β_)forall</sup><sup>_i∈_(</sup><sup>_λs_+</sup><sup>_μ_(</sup><sup>_J_</sup> 2 ) _,_ 1]. From profiles ( _h_<sup>ˆ</sup> _i_ ) _i∈_ ( _λs,_ 1] and ( _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1], we show it is possible to construct two alternative profiles ( ̃ˆ</sup><sup>_hi_)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]</sup> and ( _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]thatcorrespondtothespecialcase,i.e.</sup><sup>_h_ ̃ˆ</sup><sup>_′_</sup> _i_<sup>=</sup><sup>_h_ ̃ˆ</sup><sup>_i≥Rβ −Rτ_(</sup><sup>_β_)</sup> for all _i ∈_ ( _λs,_ 1], and for which where _h_<sup> ̃ˆ</sup> _i_ = _R_<sup> ̃ˆ</sup> _bi − Rτ_ (<sup> ̃ˆ</sup> _bi_ ) and _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>=</sup><sup>_R_ ̃ˆ</sup><sup>_b′_</sup> _i_<sup>_−Rτ ′_( ̃ˆ</sup><sup>_b′_</sup> _i_<sup>)forall</sup><sup>_i∈_(</sup><sup>_λs,_1].IfEq.(8)</sup> holds, then Eq. (7) holds for (<sup>ˆ</sup> _bi_ ) _i∈_ ( _λs,_ 1] and (<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]becausetheinheritance</sup> profiles associated to (<sup> ̃ˆ</sup> _bi_ ) _i∈_ ( _λs,_ 1] and (<sup> ̃ˆ</sup> _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]correspondtothespecialcase</sup> as _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>= ̃ˆ</sup><sup>_hi≥Rβ −Rτ_(</sup><sup>_β_)forall</sup><sup>_i ∈_(</sup><sup>_λs,_1].</sup> We construct ( _h_<sup> ̃ˆ</sup> _i_ ) _i∈_ ( _λs,_ 1] from ( _h_<sup>ˆ</sup> _i_ ) _i∈_ ( _λs,_ 1] as follows (see Figure 10.a for an illustration for the case of a positive-peak tax _τ_ , for which _τ_ ( _β_ ) = 0): and construct ( _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]from(ˆ</sup><sup>_hi_)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]and(ˆ</sup><sup>_h′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]asfollows(seeFigure</sup> 10.b for an illustration for the case of a positive-peak tax _τ_ , for which _τ_ ( _β_ ) = 0): In words, profiles ( _h_<sup> ̃ˆ</sup> _i_ ) _i∈_ ( _λs,_ 1] and ( _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1]areconstructedbyreplacingthe</sup> inheritances _h_<sup>ˆ</sup> _j_ and _h_<sup>ˆ</sup><sup>_′_</sup> _j_<sup>thataresmallerthan</sup><sup>_Rβ −Rτ_(</sup><sup>_β_)by</sup><sup>_Rβ −Rτ_(</sup><sup>_β_),and</sup> then replacing inheritances _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>thatarelargerthan</sup><sup>_h_ˆ</sup><sup>_i_by</sup><sup>_h_ˆ</sup><sup>_i_.</sup> There remains to show that Eq. (8) holds. First, we show that 8
<!-- Start of picture text --> (a) (b)<br>L ( w + D, 0) L ( w + D ′ , 0)<br>Rτ ( ˆ bi ) = Rτ ( ̃ˆ bi ) hi<br>hi ˆ<br>h ̃ˆ i = ˆ hi ̃ˆ zi = z ˆ i case τ ( −β ) = τ is τ single-peaked:( ̃ˆ bj ) = 0 h ′ i ̃ˆ ′ z ˆ i ′ ̃ˆ z′i<br>h ̃ˆ j = Rβ − Rτ ( β ) ̃ˆ zj Rβ −hRτi = ( β h ̃ˆ ) i ̃ˆ z′j<br>h ˆ j z ˆ j ̃ˆ zi −Rτ ′ ( β − ∆ ∗ )<br>−Rτ ( ˆ bj )<br>ˆ z ˆ ′<br>h ′ j<br>j<br>B τ<br>B τ<br>B τ ′<br>ˆ bj<br>ci ci<br>w + D w + D ′<br>β = ̃ˆ bj<br>β − ∆ ∗<br><!-- End of picture text --> _<u>βλs βλs</u>_ Figure 10: (a) Construction of profile ( _h_<sup> ̃ˆ</sup> _i_ ) _i∈_ ( _λs,_ 1], where _j ∈_ ( _λs, λs_ + _μ_ ( _J_ 2 )] and _i ∈_ ( _λs_ + _μ_ ( _J_ 2 ) _,_ 1]. (b) Construction of profile ( _h_<sup> ̃ˆ</sup><sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_(</sup><sup>_λ_</sup> _s_<sup>_,_1], where</sup><sup>_j∈_(</sup><sup>_λs, λs_+</sup><sup>_μ′_] and</sup><sup>_i ∈_(</sup><sup>_λs_+</sup><sup>_μ′,_1].Case of a positive-peak</sup> tax _τ_ , for which _τ_ ( _β_ ) = 0.

_<u>βλs</u>_ For all _i ∈_ ( _λs_ + _μ_ ( _J_ 2 ) _,_ 1], we have _h_<sup>ˆ</sup> _i_ = _h_<sup> ̃ˆ</sup> _i ≥ Rβ − Rτ_ ( _β_ ) implying that _<u>βλs</u> τ_ (<sup>ˆ</sup> _bi_ ) = _τ_ (<sup> ̃ˆ</sup> _bi_ ). For all _j ∈_ ( _λs, λs_ + _μ_ ( _J_ 2 )], we have _h_<sup>ˆ</sup> _j < Rβ − Rτ_ ( _β_ ) and ̃ˆ _hj_ = _Rβ − Rτ_ ( _β_ ), and we show that _τ_ (ˆ _bj_ ) _≤ τ_ ( ̃ˆ _bj_ ). This is obvious if _τ_ is monotonically increasing because<sup>ˆ</sup> _bj <_<sup> ̃ˆ</sup> _bj_ = _β_ . If _τ_ is positive peak (the case illustrated in Figure 10.a), then our construction is such that _β_ = _<u>b</u>_ <u>.</u> As _−τ_ is single-peaked, this implies that _τ_ (<sup>ˆ</sup> _bj_ ) _≤_ 0 whereas _τ_ ( _β_ ) = 0. Therefore Eq. (9) holds.

Second, we show that For all _i ∈_ ( _λs_ + _μ_<sup>_′_</sup> _,_ 1], we have _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>_≥h_ ̃ˆ</sup><sup>_′_</sup> _i_<sup>_≥Rβ−Rτ_(</sup><sup>_β_)andweshowthat</sup> _τ_<sup>_′_</sup> (<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_≥τ ′_( ̃ˆ</sup><sup>_b′_</sup> _i_<sup>)(seeillustrationinFigure10.b,where</sup><sup>_−τ ′_(ˆ</sup><sup>_b′_</sup> _i_<sup>)</sup><sup>_≤−τ ′_( ̃ˆ</sup><sup>_b′_</sup> _i_<sup>)).The</sup> construction of _τ_<sup>_′_</sup> from _τ_ is such that _−τ_<sup>_′_</sup> is single-peaked when _−τ_ is singlepeaked. Also, as _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>_≥h_ ̃ˆ</sup><sup>_′_</sup> _i_<sup>_≥Rβ−Rτ_(</sup><sup>_β_)wehavethatˆ</sup><sup>_b′_</sup> _i_<sup>_≥_ ̃ˆ</sup><sup>_b′_</sup> _i_<sup>_≥β−_∆</sup><sup>_∗_.</sup> For bequest amounts larger than _β −_ ∆<sup>_∗_</sup> , _τ_<sup>_′_</sup> is monotonically increasing in bequest, which implies that _τ_<sup>_′_</sup> (<sup>ˆ</sup> _b_<sup>_′_</sup> _i_<sup>)</sup><sup>_≥τ ′_( ̃ˆ</sup><sup>_b′_</sup> _i_<sup>).Forall</sup><sup>_j∈_(</sup><sup>_λs, λs_+</sup><sup>_μ′_],wehave</sup> _h_ ˆ<sup>_′_</sup> _j_<sup>_< Rβ −Rτ_(</sup><sup>_β_)and</sup><sup>_h_ ̃ˆ</sup><sup>_′_</sup> _j_<sup>=</sup><sup>_Rβ −Rτ_(</sup><sup>_β_),andweshowthat</sup><sup>_τ ′_(ˆ</sup><sup>_b′_</sup> _j_<sup>)</sup><sup>_≥τ ′_( ̃ˆ</sup><sup>_b′_</sup> _j_<sup>).For</sup> such _j_ , we have thus<sup>ˆ</sup> _b_<sup>_′_</sup> _j_<sup>_<_ ̃ˆ</sup><sup>_b′_</sup> _j_<sup>=</sup><sup>_β −_∆</sup><sup>_∗_.As</sup><sup>_τ ′_=</sup><sup>_τ_∆</sup><sup>_∗_and ˆ</sup><sup>_b′_</sup> _j_<sup>_< β −_∆</sup><sup>_∗_,wehave</sup> _τ_<sup>_′_</sup> (<sup>ˆ</sup> _b_<sup>_′_</sup> _j_<sup>)</sup><sup>_<_0.Whatismore,theconstructionof</sup><sup>_τ ′_issuchthat</sup><sup>_τ ′_(</sup><sup>_x′_)</sup><sup>_<τ ′_(</sup><sup>_x_)</sup> for all 0 _≤ x < x_<sup>_′_</sup> _≤ β −_ ∆<sup>_∗_</sup> , i.e. the subsidy received under _τ_<sup>_′_</sup> is increasing in 9
the bequest left, when the bequest is smaller than _β −_ ∆<sup>_∗_</sup> . Therefore Eq. (10) holds.

Eq. (9) and Eq. (10) together imply that (8) holds. To concludes Step 2, there only remains to prove the Technical Claim.

_<u>βλs</u> Technicalh_ ˆ<sup>_′_</sup> _i_<sup>_≥Rβ −_</sup> _Claim:_<sup>_Rτ_(</sup><sup>_β_).20</sup> For any _i ∈_ ( _λs_ + _μ_ ( _J_ 2 ) _,_ 1] we have _h_<sup>ˆ</sup><sup>_′_</sup> _i_<sup>_≥h_ˆ</sup><sup>_i_and</sup> _Proof of Technical Claim_ : For _t ∈{_ 0 _,_ 1 _, . . . }_ , we consider successive equilibrium allocations _zt_<sup>_′_=(</sup><sup>_g_</sup> _it_<sup>_′, c_</sup> _it_<sup>_′, h_</sup> _it_<sup>_′_</sup> +1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]forwhichall</sup><sup>_i∈_[0</sup><sup>_,_1]chose</sup> in _B_<sup>_τ ′_</sup> ( _w_ + _D_<sup>_′_</sup> + _git_<sup>_′,_0)and(ˆ</sup><sup>_g_</sup> _it_<sup>_′_</sup> +1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]=(ˆ</sup><sup>_h′_</sup> _it_ +1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1],underanini-</sup> tial profile of inheritances ( _gi_<sup>_′_</sup> 0<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]=(</sup><sup>_hi_)</sup><sup>_i∈_[0</sup><sup>_,_1],whichcorrespondsto</sup> the long-run equilibrium profile associated to ( _τ, D_ ). We show for all _t ∈{_ 0 _,_ 1 _, . . . }_ that Observe that, at all _t_ , (1) and (2) compare the sorted profile of inheritances left in _t_ to the sorted profile of inheritances left under the long-run equilibrium allocation _z_ .

If (1) and (2) hold for all _t ≥_ 0, then (2) holds as well for the profile ( _h_<sup>_′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]associatedtothelong-runequilibriumallocation</sup><sup>_z′_,becausewe</sup> assume that for any given tax-demogrant scheme, the economy converges over time to a unique long-run equilibrium allocation independent on the initial distribution of inheritances.

Consider first _t_ = 0. In order to show that claims (1) and (2) hold for _t_ = 0 when the tax-demogrant scheme is ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ), it is sufficient to show that claims (1) and (2) hold for _t_ = 0 when the tax-demogrant scheme is ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ) instead of ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ). Indeed, let ( _h_<sup>∆</sup> _i_ 1<sup>_∗_)</sup><sup>_i∈_[0</sup><sup>_,_1]betheprofile</sup> of inheritances obtained in period _t_ = 0 if the tax-demogrant scheme is ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ). Since _τ_<sup>_′_</sup> = _τ_<sup>∆</sup><sup>_∗_</sup> and _D_<sup>_′_</sup> _> D −_ ∆<sup>_∗_</sup> , the binormality of preferences implies that _h_<sup>_′_</sup> _i_ 1<sup>_≥h_∆</sup> _i_ 1<sup>_∗_forall</sup><sup>_i ∈_[0</sup><sup>_,_1],whichshowsthatitis</sup> indeed sufficient to prove these claims when the tax-demogrant scheme is ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ).

Consider first claim (1) in _t_ = 0, i.e.

Consider the profile ( _hi_ 1) _i∈_ [0 _,_ 1] of inheritances obtained in period _t_ = 0 if the tax-demogrant scheme is ( _τ, D_ ) instead of ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D−_ ∆<sup>_∗_</sup> ). As the initial profile ( _gi_ 0) _i∈_ [0 _,_ 1] = ( _hi_ ) _i∈_ [0 _,_ 1], we have ( _h_<sup>ˆ</sup> _i_ 1) _i∈_ [0 _,_ 1] = ( _h_<sup>ˆ</sup> _i_ ) _i∈_ [0 _,_ 1] because _z_ is the long-run equilibrium allocation associated to ( _τ, D_ ). It is thus to show that

> 20Recall that index _i_ need not refer to the same dynasty in the two sorted distributions.

10
for Eq. (11) to hold.

Since we have _h_<sup>∆</sup> _i_ 1<sup>_∗_=</sup><sup>_hi_1=0forall</sup><sup>_i∈_[0</sup><sup>_,_1]forwhom</sup><sup>_ui_=</sup><sup>_u_</sup><sup>**s**,wecan</sup> focus on the subset _A_ = _{i ∈_ [0 _,_ 1] _|ui_ = _u_<sup>**s**</sup> _}_ of altruistic individuals. We partition _A_ into three subgroups _A_<sup>1</sup> , _A_<sup>2</sup> and _A_<sup>3</sup> , respectively defined as

**–** _A_<sup>1</sup> = _{i ∈ A|hi_ 1 _≥ Rβ − Rτ_ ( _β_ ) _}_ . By construction of ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ), any _i ∈ A_<sup>1</sup> choses the _same_ bundle in _B_<sup>_τ_</sup> ( _w_ + _D_ + _gi_ 0 _,_ 0) and in _B_<sup>_τ_∆</sup><sup>_∗_</sup> ( _w_ + _D −_ ∆<sup>_∗_</sup> + _gi_ 0 _,_ 0). This implies that _h_<sup>∆</sup> _i_ 1<sup>_∗_=</sup><sup>_hi_1forall</sup><sup>_i ∈A_1.</sup>

- _A_<sup>2</sup> = _{i ∈ A|hi_ 1 _< Rβ − Rτ_ ( _β_ ) and _h_<sup>∆</sup> _i_ 1<sup>_∗≥Rβ −Rτ_(</sup><sup>_β_)</sup><sup>_}_.</sup> By definition, we have _h_<sup>∆</sup> _i_ 1<sup>_∗> hi_1forall</sup><sup>_i ∈A_2.</sup>

- _A_<sup>3</sup> = _{i ∈ A|hi_ 1 _< Rβ − Rτ_ ( _β_ ) and _h_<sup>∆</sup> _i_ 1<sup>_∗< Rβ −Rτ_(</sup><sup>_β_)</sup><sup>_}_.</sup> By definition, any altruistic _j ∈ A_<sup>3</sup> leaves a smaller inheritance than _Rβ − Rτ_ ( _β_ ), and thus a smaller inheritance than any _i ∈ A_<sup>1</sup> _∪ A_<sup>2</sup> , i.e. _h_<sup>∆</sup> _j_ 1<sup>_∗≤h_∆</sup> _i_ 1<sup>_∗_.</sup>

_<u>βλs</u>_ We can assume without loss of generality that _μ_ ( _A_<sup>3</sup> ) _≤ μ_ ( _J_ 2 ). If it is not the case, consider for the construction of ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) a larger _<u>βλs</u>_ ∆ _∈_ (∆<sup>_∗_</sup> _, β_ ) for which we have _μ_ ( _A_<sup>3</sup> ) _≤ μ_ ( _J_ 2 ). Such larger value exists by assumption A2.

By the definition of the above partition of [0 _,_ 1], we have for all _i ∈_ [0 _,_ 1] _\A_<sup>3</sup> ,<sup>21</sup> but Eq. (13) may not hold for some _i ∈ A_<sup>3</sup> . _<u>βλs</u>_ However, we have _μ_ ( _A_<sup>3</sup> ) _≤ μ_ ( _J_ 2 ) and there is a subset of individuals _<u>βλs</u>_ of mass _μ_ ( _J_ 2 ) for whom _hi_ 1 _<_<sup>_<u>β</u>_</sup> 2<sup>_λs_.Therefore,evenif</sup><sup>_h_∆</sup> _i_ 1<sup>_∗_=0forall</sup> _i ∈ A_<sup>3</sup> , we have _h_<sup>∆</sup> _i_ 1<sup>_∗_+</sup><sup>_<u>β</u>_</sup> 2<sup>_λs_</sup> _≥_<sup>_<u>β</u>_</sup> 2<sup>_λs_</sup> for all _i ∈ A_<sup>3</sup> ,<sup>22</sup> which shows that (12) holds, and thus (11) holds.

We now turn to claim (2) for _t_ = 0 if the tax-demogrant scheme is ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ) instead of ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ), i.e.

By definition, if _i ∈ A_<sup>1</sup> _∪ A_<sup>2</sup> and _j ∈/ A_<sup>1</sup> _∪ A_<sup>2</sup> we have _h_<sup>∆</sup> _i_ 1<sup>_∗≥h_∆</sup> _j_ 1<sup>_∗_.Also,</sup> _<u>βλs</u>_ as _A_ 2 _∩ J_ 2 may be non-empty, the mass of individuals in _A_<sup>1</sup> _∪ A_<sup>2</sup> is _<u>βλs</u>_ such that _μ_ ( _A_<sup>1</sup> _∪ A_<sup>2</sup> ) _≥_ 1 _− λs − μ_ ( _J_ 2 ). Hence, it is sufficient that Eq. (14) holds for individuals in _A_<sup>1</sup> _∪ A_<sup>2</sup> . As shown when defining these two subgroups, we have _h_<sup>∆</sup> _i_ 1<sup>_∗_</sup> = _hi_ 1 _≥ Rβ − Rτ_ ( _β_ ) for all _i ∈ A_<sup>1</sup> and _h_<sup>∆</sup> _i_ 1<sup>_∗_=</sup><sup>_Rβ −Rτ_(</sup><sup>_β_)</sup><sup>_> hi_1forall</sup><sup>_i ∈A_2,whichprovesEq.(14).</sup> Together, we have shown claims (1) and (2) for _t_ = 0. We next prove these claims for _t_ = 1.

> 21We have shown that _h_ ∆ _i_ 1 _∗_<sup>=</sup><sup>_hi_1for all</sup><sup>_i ∈_[0</sup><sup>_,_1]</sup><sup>_\A_,</sup><sup>_h_∆</sup> _i_ 1<sup>_∗_=</sup><sup>_hi_1for all</sup><sup>_i ∈A_1and</sup><sup>_h_∆</sup> _i_ 1<sup>_∗> hi_1</sup> for all _i ∈ A_<sup>2</sup> .

> 22In words, even if all individuals in _A_ 3 leave no inheritances, there are enough altruistic individuals who, in the long-run equilibrium allocation _z_ , leave inheritances smaller than the additional amount<sup>_<u>βλ</u>_</sup> 2<sup>_<u>s</u>_</sup> considered in claim (1).

11
In _t_ = 1, the profile of inheritances received under ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is such that (ˆ _gi_<sup>_′_</sup> 1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]=(ˆ</sup><sup>_h′_</sup> _i_ 1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1].Byconstruction,since</sup><sup>_τ ′_=</sup><sup>_τ_∆</sup><sup>_∗_and</sup><sup>_D′_=</sup><sup>_D −_</sup> ∆<sup>_∗_</sup> +<sup>_<u>β</u>_</sup> 2<sup>_λs_,wehaveforall</sup><sup>_i ∈_[0</sup><sup>_,_1]that</sup> as can be seen in Figure 9 (for the case _gi_<sup>_′_</sup> 1<sup>=0).Therefore,theprofile</sup> ( _h_<sup>ˆ</sup><sup>_′_</sup> _i_ 2<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]obtainedunder(</sup><sup>_τ ′, D′_)for(ˆ</sup><sup>_g_</sup> _i_<sup>_′_</sup> 1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]= (ˆ</sup><sup>_h′_</sup> _i_ 1<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]isthesame</sup> as the profile ( _h_<sup>ˆ∆</sup> _i_ 2<sup>_∗_)</sup><sup>_i∈_[0</sup><sup>_,_1]thatwouldbeobtainedunder(</sup><sup>_τ_∆</sup><sup>_∗, D −_∆</sup><sup>_∗_)if</sup> the profile of inheritances received in _t_ = 1 was instead ( _h_<sup>ˆ</sup><sup>_′_</sup> _i_ 1<sup>+</sup><sup>_<u>β</u>_</sup> 2<sup>_λs_)</sup><sup>_i∈_[0</sup><sup>_,_1].</sup> ˆ ˆ From claim (1) for _t_ = 0, we have that _gi_<sup>_′_</sup> 1<sup>+</sup><sup>_<u>β</u>_</sup> 2<sup>_λs_</sup> _≥ gi_<sup>_′_</sup> 0<sup>forall</sup><sup>_i∈_[0</sup><sup>_,_1].</sup> Therefore, by the binormality of preferences, the same reasoning implies again (1) and (2) for _t_ = 1.

The same reasoning extends (1) and (2) to any _t ≥_ 2, which concludes the proof of the Technical Claim.

_Step 3_ . We show that allocation _z_<sup>_′_</sup> is preferred by **_R_**<sup>_c−lex_</sup> to allocation _z_ . By assumption A1, there is a positive mass of individuals _a_ and a positive mass of individuals _s_ . The construction of _τ_<sup>_′_</sup> from _τ_ is such that _−τ_<sup>_′_</sup> is singlepeaked when _−τ_ is single-peaked. By Lemma 1, under both _z_<sup>_′_</sup> and _z_ , either _a_ or _s_ are among the worst-offs. Individual _a_ is among the worst-offs under _z_ because _μ_<sup>_c_</sup> _a_<sup>=</sup><sup>_uc_(</sup><sup>_za, ua_)</sup><sup>_< w_+</sup><sup>_D_=</sup><sup>_uc_(</sup><sup>_zs, us_),asillustratedinFigure8.</sup> There remains to show that _μ_<sup>_c_</sup> _a_<sup>_<uc_(</sup><sup>_z_</sup> _a_<sup>_′, ua_)and</sup><sup>_μc_</sup> _a_<sup>_<uc_(</sup><sup>_z_</sup> _s_<sup>_′, us_).Wehave</sup> _μ_<sup>_c_</sup> _a_<sup>_< uc_(</sup><sup>_z_</sup> _s_<sup>_′, us_) because</sup><sup>_uc_(</sup><sup>_z_</sup> _s_<sup>_′, us_) =</sup><sup>_w_+</sup><sup>_D′> w_+</sup><sup>_D −β_and</sup><sup>_β≤w_+</sup><sup>_D −uc_</sup> _a_<sup>.23</sup> Finally, we show that _μ_<sup>_c_</sup> _a_<sup>_<uc_(</sup><sup>_z_</sup> _a_<sup>_′, ua_).Wehaveselected</sup><sup>_β_suchthat</sup><sup>_β<ba_.24</sup> When _β < ba_ , the construction of _τ_<sup>∆</sup><sup>_∗_</sup> implies that _za_<sup>∆</sup><sup>_∗_</sup> = _za_ , where _za_<sup>∆</sup><sup>_∗_</sup> is the equilibrium bundle of _a_ under scheme ( _τ_<sup>∆</sup><sup>_∗_</sup> _, D −_ ∆<sup>_∗_</sup> ). Since _τ_<sup>_′_</sup> = _τ_<sup>∆</sup><sup>_∗_</sup> but _D_<sup>_′_</sup> _> D −_ ∆<sup>_∗_</sup> , bundle _za_ lies in the _interior_ of<sup>25</sup> which is _a_ ’s budget set under ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ), as illustrated in Figure 9. This implies that _ua_ ( _za_ ) _< ua_ ( _za_<sup>_′_),hence</sup><sup>_μc_</sup> _a_<sup>_< uc_(</sup><sup>_z_</sup> _a_<sup>_′, ua_).</sup>

**CASE 2** : there exists a bequest amount _b_<sup>_∗_</sup> with 0 _< b_<sup>_∗_</sup> _<_ min( _ba, w_ + _D − μ_<sup>_c_</sup> _a_<sup>)</sup> such that for all _J ⊆ A_ with _bj < b_<sup>_∗_</sup> for all _j ∈ J_ we have _μ_ ( _J_ ) = 0.

_Step 1_ . We construct ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) from a particular tax-demogrant scheme ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ). Take _D_<sup>_′′_</sup> = _D − b_<sup>_∗_</sup> _/_ 2. The tax _τ_<sup>_′′_</sup> is constructed from _τ_ by linearly truncating the budget set _B_<sup>_τ_</sup> ( _w_ + _D_ + _gi,_ 0) for bequests smaller than _b_<sup>_∗_</sup> (as illustrated in Figure 11). Formally, we define _τ_<sup>_′′_</sup> as

> 23We have _w_ + _D′ > w_ + _D − β_ because _D′_ = _D −_ ∆ _∗_ +<sup>_<u>βλ</u>_</sup> 2 _<u>s</u>_ and ∆<sup>_∗_</sup> _∈_ ( _β/_ 2 _, β_ ) and _λs ∈_ (0 _,_ 1).

> 24If _τ_ is monotonically increasing, then _<u>b</u>_ = 0 and this case is such that _β < ba_ . If _τ_ is positive peak, then _<u>b</u> >_ 0 and this case is such that _β_ = _<u>b</u>_ and _ba > b_ <u>.</u>

> 25As _D′ > D−_ ∆ _∗_ , all bundles in _Bτ_<sup>∆</sup><sup>_∗_</sup> ( _w_ + _D −_ ∆<sup>_∗_</sup> _,_ 0) lie in the interior of _B_<sup>_τ′_</sup> ( _w_ + _D_<sup>_′_</sup> _,_ 0).

12
<!-- Start of picture text --> L ( w + D, 0)<br>b = 0<br>hi ua Rτ ( ba ) ( τ has no subsidy)<br>za<br>( w + D − b ∗ , Rb ∗ − Rτ ( b ∗ ))<br>B τ<br>B τ ′′<br>L ( u c a , 0)<br>b ∗ zs ci<br>u c a w + D<br>w + D ′′<br><!-- End of picture text --> Figure 11: The tax-demogrant scheme ( _τ, D_ ) is dominated because the tax function _τ_ taxes small bequests. Individual _a_ is the worst-off because _u_<sup>_c_</sup> ( _za, ua_ ) = _u_<sup>_c_</sup> _a_<sup>.Thesustainablescheme(</sup><sup>_τ ′′, D′′_)hasasmaller</sup> demogrant, does not affect _u_<sup>_c_</sup> ( _za, ua_ ) and leaves money on the table.

The particularity of scheme ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) is that any individual _i ∈_ [0 _,_ 1] for whom _bi ≥ b_<sup>_∗_</sup> choses the same bundle under both ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) and ( _τ, D_ ), i.e.

Let _z_<sup>_′′_</sup> = ( _gi_<sup>_′′, c′′_</sup> _i_<sup>_, h′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_denotethelong-runequilibriumallocationas-</sup> sociated to ( _τ_<sup>_′′_</sup> _, D_<sup>_′′_</sup> ) and let ( _b_<sup>_′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]bethelong-runequilibriumprofileof</sup> bequests left, i.e. _b_<sup>_′′_</sup> _i_<sup>=</sup><sup>_w_+</sup><sup>_D′′_+</sup><sup>_g_</sup> _i_<sup>_′′−c′′_</sup> _i_<sup>for all</sup><sup>_i ∈_[0</sup><sup>_,_1].Case 2 is such that the</sup> profile ( _h_<sup>ˆ</sup><sup>_′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]=(ˆ</sup><sup>_hi_)</sup><sup>_i∈_[0</sup><sup>_,_1]becauseforall</sup><sup>_J⊆A_with</sup><sup>_bj<b∗_forall</sup><sup>_j∈J_</sup> we have _μ_ ( _J_ ) = 0. This implies for all _i ∈_ ( _λs,_ 1] that<sup>ˆ</sup> _b_<sup>_′′_</sup> _i_<sup>= ˆ</sup><sup>_bi −b∗/_2andthus</sup> _τ_<sup>_′′_</sup> (<sup>ˆ</sup> _b_<sup>_′′_</sup> _i_<sup>) =</sup><sup>_τ_(ˆ</sup><sup>_bi_)</sup><sup>_−b∗/_2.As(</sup><sup>_τ, D_)issustainable,wehavethat(</sup><sup>_τ ′′, D′′_)leaveson</sup> the table an amount at least _λs_<sup>_<u>b</u>_</sup> 2<sup>_∗_,i.e.</sup> If _τ_ is positive peak, then by assumption A3 there exists a sustainable scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) with _τ_<sup>_′_</sup> = _τ_<sup>_′′_</sup> and _D_<sup>_′_</sup> _> D_<sup>_′′_</sup> . If _τ_ is monotonically increasing, then A3 is not assumed and we define scheme ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) as _Step 2_ . We show that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable if ( _τ, D_ ) is sustainable. We have already shown it using A3 in Step 1 in the case for which _τ_ is positive peak. In the case for which _τ_ is monotonically increasing, then a simplified version of 13
the argument used in Step 2 of Case 1 shows that ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is sustainable. The argument can be simplified because all altruistic individuals leave a bequest larger than _b_<sup>_∗_</sup> , implying that the partition of _A_ as _A_ = _A_<sup>1</sup> _∪ A_<sup>2</sup> _∪ A_<sup>3</sup> is such that _A_<sup>2</sup> = _A_<sup>3</sup> = ∅. We do not repeat this argument.

_Step 3_ . The long-run equilibrium allocation _z_<sup>_′_</sup> associated to ( _τ_<sup>_′_</sup> _, D_<sup>_′_</sup> ) is preferred by **_R_**<sup>_c−lex_</sup> to allocation _z_ . The argument is the same as the argument used in Step 3 of Case 1. We do not repeat this argument. This concludes the proof of Lemma 2. ■ First, we prove claim (i) of Proposition 2. Assume to the contrary that _τ_ is monotonically increasing but _τ_ does not provide an exemption up to _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_),</sup> i.e. _τ_ ( _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_))</sup><sup>_>_0.Underthiscontradictionassumption,weshowthatscheme</sup> ( _τ, D_ ) is not optimal whatever the value of _D_ . As any scheme ( _τ, D_ ) with _D <_ 0 is not optimal (Proposition 1), we consider any ( _τ, D_ ) with _D ≥_ 0. As _τ_ is monotonically increasing, we have _<u>b</u>_ = 0 and either _τ_ ( _ba_ ) _>_ 0 or _τ_ ( _ba_ ) = 0. (Recall that _ba_ denotes the equilibrium bequest left by individual _a_ with _ga_ = 0 and _ua_ = _u_<sup>**a**</sup> .) If _τ_ ( _ba_ ) _>_ 0, then Eq. (3) is violated and ( _τ, D_ ) is not optimal by Proposition 2 (ii), whose proof is given below. So assume that _τ_ ( _ba_ ) = 0, which implies that _τ_ ( _x_ ) = 0 for all _x ∈_ [0 _, ba_ ] because _τ_ is monotonically increasing. If _ba ≥ b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_),thenwehave</sup><sup>_τ_(</sup><sup>_ba_)</sup><sup>_>_0be-</sup> cause _τ_ ( _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_))</sup><sup>_>_0and</sup><sup>_τ_ismonotonicallyincreasing,acontradictiontoour</sup> assumption that _τ_ ( _ba_ ) = 0.

There remains the case for which _ba < b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_) and</sup><sup>_τ_(</sup><sup>_ba_) = 0.First, we show</sup> that any optimal ( _τ, D_ ) has _ba >_ 0. If _ba_ = 0 under ( _τ, D_ ) with _D ≥_ 0, then we can show that the economy converges to a long-run equilibrium allocation for which all inheritances are zero, i.e. _D_ = 0. Indeed, if _ba_ = 0 under a scheme ( _τ, D_ ) with _D ≥_ 0, the binormality of preferences implies that _a_ leaves no bequest under scheme ( _τ,_ 0). Therefore, any dynasty _i_ with a member _it_<sup>_′_</sup> such that _uit′_ = _u_<sup>**s**</sup> leaves no bequest for all _it_ with _t ≥ t_<sup>_′_</sup> . As there is in each generation a mass _λs_ of individuals _i ∈_ [0 _,_ 1] with _ui_ = _u_<sup>**s**</sup> , and preferences are drawn at random in each generation, all dynasties have a member _it_<sup>_′_</sup> such that _uit_<sup>_′_</sup> = _u_<sup>**s**</sup> for some _t_<sup>_′_</sup> _≤ t_ when _t_ is sufficiently large. Therefore all inheritances are zero in the long-run equilibrium allocation, which implies that the largest sustainable demogrant is _D_ = 0 when _ba_ = 0. We show that ( _τ, D_ = 0) is dominated by Laissez-Faire when _ba_ = 0. Under Laissez-Faire, the sustainable demogrant is also zero. The equilibrium bundle of _a_ under ( _τ, D_ = 0) is _za_ = (0 _, w,_ 0) because _ba_ = 0, whereas it is _za_<sup>_LF_</sup> = (0 _, w − b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_, RbLF_</sup> _a_<sup>(</sup><sup>_w_))with</sup> _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_>_0 under Laissez-Faire.As illustrated in Figure 12.a, we have</sup><sup>_ua_(</sup><sup>_za_)</sup><sup>_<_</sup> _ua_ ( _za_<sup>_LF_</sup> ) because ( _ca, ha_ ) _∈ L_ ( _w,_ 0) but _za_ = _za_<sup>_LF_</sup> where Under Laissez-Faire, any individual _i ∈_ [0 _,_ 1] allocates her lifetime resources freely in the non-distortionary budget _L_ ( _w_ + _gi,_ 0), thus we have _u_<sup>_c_</sup> ( _zi_<sup>_LF_</sup> _, ui_ ) = _w_ + _gi_ . This shows that _u_<sup>_c_</sup> ( _zi_<sup>_LF_</sup> _, ui_ ) _≥ u_<sup>_c_</sup> ( _za_<sup>_LF_</sup> _, ua_ ) for all _i ∈_ [0 _,_ 1] because _u_<sup>_c_</sup> ( _za_<sup>_LF_</sup> _, ua_ ) = _w_ as _ga_ = 0. Now, since _ua_ ( _za_<sup>_LF_</sup> ) _> ua_ ( _za_ ), we have _u_<sup>_c_</sup> ( _za_<sup>_LF_</sup> _, ua_ ) _> u_<sup>_c_</sup> ( _za, ua_ ), implying that _u_<sup>_c_</sup> ( _zi_<sup>_LF_</sup> _, ui_ ) _> u_<sup>_c_</sup> ( _za, ua_ ) for all _i ∈_ [0 _,_ 1]. By A1, there is mass of individuals _a_ with _ga_ = 0 and _ua_ = _u_<sup>**a**</sup> , showing that ( _τ, D_ = 0) is dominated by Laissez-Faire, i.e. not optimal.

14
<!-- Start of picture text --> (a) (b)<br>hi L ( w, 0) ua hi L ( w + D, 0)<br>ua<br>za LF L ( u c ( za, ua ) , 0) z ˆ a<br>L ( u c ( za, ua ) , 0)<br>B τ ua<br>za<br>B τ ua<br>za = zs zs<br>ci ci<br>u c ( za, ua ) w u c ( za, ua ) w + D<br><!-- End of picture text --> Figure 12: (a) ( _τ, D_ = 0) is dominated by Laissez-Faire when _ba_ = 0. (b) _u_<sup>_c_</sup> ( _za, ua_ ) _< w_ + _D_ when 0 _< ba < b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_).</sup> Now, for the case 0 _< ba < b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_),thebinormalityofpreferencesimplies</sup> that _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_)</sup><sup>_≤bLF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)andthus</sup><sup>_ba<bLF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_).Inwords,</sup><sup>_a_would</sup> increase her bequest if the exemption proposed by _τ_ was larger. As illustrated ˆ in Figure 12.b, because ( _ca, ha_ ) _∈ L_ ( _w_ + _D,_ 0) but _za_ = _za_ where we have _ua_ ( _za_ ) _< ua_ (ˆ _za_ ).

As _u_<sup>_c_</sup> (ˆ _za, ua_ ) = _w_ + _D_ , this implies that _u_<sup>_c_</sup> ( _za, ua_ ) _< w_ + _D_ . As _<u>b</u>_ = 0 and _ba >_ 0, we have _ba > b_ <u>,</u> and Lemma 2 implies that ( _τ, D_ ) is not optimal.

Second, we prove claim (ii) of Proposition 2. We show that, if _τ_ ( _ba_ ) _>_ _<u>b</u>_ <u>,</u> then scheme ( _τ, D_ ) is not optimal whatever the value of _D_ . As any scheme ( _τ, D_ ) with _D <_ 0 is not optimal (Proposition 1), we consider any ( _τ, D_ ) with _D ≥_ 0. As any sustainable ( _τ, D_ ) for which _D < D_<sup>_max_</sup> is dominated by ( _τ, D_<sup>_max_</sup> ), and thus not optimal, we can focus on _D_ = _D_<sup>_max_</sup> . We show that the preconditions for Lemma 2 are all met, which implies that ( _τ, D_ ) is not optimal.

By definition of _<u>b</u>_ we have _τ_ <u>(</u> _<u>b</u>_ <u>) = 0.</u> Since _τ_ ( _ba_ ) _>_ 0, we have _ba > b_ because _−τ_ is single-peaked, implying that _τ_ is monotonically increasing in _x_ for all _x ≥_ _<u>b</u>_ <u>.</u> Any individual _a_ with _ga_ = 0 and _ua_ = _u_<sup>**a**</sup> choses the equilibrium bundle _za_ = (0 _, w_ + _D − ba, Rba − Rτ_ ( _ba_ )) in the budget set _B_<sup>_τ_</sup> ( _w_ + _D,_ 0). Bundle _za_ is on the frontier of the non-distortionary budget set _L_ ( _w_ + _D − τ_ ( _ba_ ) _,_ 0), which implies that _u_<sup>_c_</sup> ( _za, ua_ ) _≤ w_ + _D − τ_ ( _ba_ ). As _τ_ ( _ba_ ) _> b_ <u>,</u> this implies that _u_<sup>_c_</sup> ( _za, ua_ ) _< w_ + _D −_ _<u>b</u>_ <u>.</u> Together, we have _D ≥_ 0 and we have shown _μ_<sup>_c_</sup> _a_<sup>_<w_+</sup><sup>_D −_</sup><sup>_<u>b</u>_</sup> and _ba >_ _<u>b</u>_ <u>.</u> Therefore, Lemma 2 implies that ( _τ, D_ ) is not optimal.

15
# **S3 Proof of Proposition 3**

Let _z_ be the long-run equilibrium allocation generated by a tax system ( _τ, D_ ), so that _D_ denote the largest demogrant for which the scheme ( _τ, D_ ) is sustainable. Let us assume _τ_ is optimal.

By assumption, both self-centered and altruistic individuals are among the worst-offs, which implies that _u_<sup>_c_</sup> ( _zs_ 1 _, u_<sup>**s**</sup> ) = _u_<sup>_c_</sup> ( _za_ 1 _, u_<sup>_∗_</sup> _a_<sup>).</sup> Graphically, bundle _za_ 1 must be on the indifference curve of preference _u_<sup>_∗_</sup> _a_<sup>thatistangenttothe</sup> non-distortionary budget set _L_ ( _cs_ 1 _,_ 0), as shown in Figure 13. Let bundle _z_<sup>_k_</sup> := ( _c_<sup>_k_</sup> _, h_<sup>_k_</sup> ) be the bundle at the kink of this indifference curve, which is the bundle that _a_ 1 would select in the non-distortionary budget set _L_ ( _cs_ 1 _,_ 0). By definition, we have where _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)istheequilibriumbequestleftunderLaissez-Fairebyan</sup> altruistic individual who receives an inheritance equal to _D_ .

Observe that the incentive compatibility constraint of _a_ 1 implies that _za_ 1 lies on the segment of this indifference curve that is below the kink, that is, _ha_ 1 _≤ h_<sup>_k_</sup> . Assume to the contrary that _a_ 1 selects a bundle _za_<sup>_′_</sup> 1<sup>onthesegment</sup> of this indifference curve that is above the kink, such that _h_<sup>_′_</sup> _a_ 1<sup>_> hk_(seeFigure</sup> 13). Individual _a_ 1 would then receive a subsidy ( _τ_ ( _b_<sup>_′_</sup> _a_ 1<sup>)</sup><sup>_<_0).Thisimplies</sup> in turn that bundle _z_<sup>_k_</sup> lies in the interior of her budget set _B_<sup>_τ_</sup> ( _w_ + _D,_ 0).<sup>26</sup> This is a contradiction to bundle _za_<sup>_′_</sup> 1<sup>beingincentivecompatiblefor</sup><sup>_a_1because</sup> _u_<sup>_∗_</sup> _a_<sup>(</sup><sup>_z_</sup> _a_<sup>_′_</sup> 1<sup>) =</sup><sup>_u∗_</sup> _a_<sup>(</sup><sup>_zk_)and</sup><sup>_zk_liesintheinteriorofherbudgetset.</sup> If _ha_ 1 = _h_<sup>_k_</sup> , then _za_ 1 = _z_<sup>_k_</sup> because _u_<sup>_c_</sup> ( _zs_ 1 _, u_<sup>**s**</sup> ) = _u_<sup>_c_</sup> ( _za_ 1 _, u_<sup>_∗_</sup> _a_<sup>).Wedon’tneed</sup> to consider this case because it is such that _τ_ admits an exemption up to bequest level _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_),where</sup><sup>_bLF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)</sup><sup>_≥bLF_</sup> _a_<sup>(</sup><sup>_w_).27Wecanthusconcentrateon</sup> the case _ha_ 1 _< h_<sup>_k_</sup> . This case is such that _τ_ ( _ba_ 1) _<_ 0 because _za_ 1 does not lie in the non-distortionary budget set _L_ ( _cs_ 1 _,_ 0).

We show that _τ_ is not optimal when _ha_ 1 _< h_<sup>_k_</sup> and _u_<sup>_c_</sup> ( _zs_ 1 _, u_<sup>**s**</sup> <u>) =</u> _u_<sup>_c_</sup> ( _za_ 1 _, u_<sup>_∗_</sup> _a_<sup>).</sup> To do this, we show there exists another sustainable scheme ( _~~τ,~~ D_ ) whose longrun equilibrium allocation is strictly preferred by SWF **_R_**<sup>_c−lex_</sup> to _z_ .

Let _~~z~~_ = ~~(~~ _~~z~~ sX ,_ _~~z~~ aX_ ) _X∈{_ 1 _,_ 2 _... }_ denote the long-run allocation associated to scheme ( _~~τ,~~ D_ ), where _D_ is the maximal sustainable demogrant for tax _τ_ . We will construct _~~τ~~_ in such a way that (i) min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ~~(~~ _~~z~~ i, ui_ ) = min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( _zi, ui_ ), (ii) _~~τ~~_ ~~(~~ _ba_ 1) _> τ_ ( _ba_ 1), and (iii) _~~τ~~_ ~~(~~ _baX_ ) _≥ τ_ ( _baX_ ) for all _X ∈{_ 2 _,_ 3 _, . . . }_ .

> 26Bundle _zk_ lies in the interior of her budget set _Bτ_ ( _w_ + _D,_ 0) because _τ_ ( _bLFa_ ( _w_ + _D_ )) _<_ 0. We must have _τ_ ( _b_<sup>_LF_</sup> _a_ ( _w_ + _D_ )) _<_ 0 because _τ_ ( _b_<sup>_′_</sup> _a_ 1<sup>)</sup><sup>_<_0and</sup><sup>_bLF_</sup> _a_ ( _w_ + _D_ ) _< b_<sup>_′_</sup> _a_ 1<sup>.Indeed,</sup><sup>_−τ_is</sup> single-peaked and our domain of tax function excludes that _τ_ provides an exemption on small bequests, then provides subsidies on intermediate bequests and finally taxes positively large bequests. Hence, if _τ_ ( _b_<sup>_LF_</sup> _a_ ( _w_ + _D_ )) _≥_ 0, then _τ_ must provide subsidies for all _b ≥ b_<sup>_′_</sup> _a_ 1<sup>.But</sup> this implies that _D_<sup>_′_</sup> _<_ 0, which contradicts that _τ_ is optimal (Proposition 1).

> 27The exact value of _τ_ ( _b_ ) for _b ∈_ (0 _, bLFa_ ( _w_ + _D_ )) does not matter because no individual selects _b < b_<sup>_LF_</sup> _a_ ( _w_ + _D_ ). In other words, as this economy only has a countable number of individuals, we only need to consider a countable number of points on the tax function, and the exact value of the tax beyond these points is irrelevant.

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<!-- Start of picture text --> hi u ∗ a<br>L ( cs 1 , 0)<br>za ′ 1<br>z k<br>Rb LF a ( w + D )<br>za 1<br>zs 1<br>ci<br>w + D<br>b LF a ( w + D )<br><!-- End of picture text --> Figure 13: Illustration of bundle _za_ 1.

We show that conditions ( _i_ ), ( _ii_ ) and ( _iii_ ) imply that there exists a sustainable scheme ( _~~τ,~~ D_ ) whose long-run equilibrium allocation is strictly preferred by SWF **_R_**<sup>_c−lex_</sup> to _z_ . Conditions (ii) and (iii) together imply that � _i_ _~~τ~~_ ~~(~~ _bi_ ) _di >_ � _i_<sup>_τ_(</sup><sup>_bi_)</sup><sup>_di_,becauseforany</sup><sup>_X_,themeasureofaltruisticindividualswithnumber</sup> _X_ is the same under both schemes. As _D_ is the maximal sutainable demogrant for _τ_ , we have _D_ = � _i_<sup>_τ_(</sup><sup>_bi_)</sup><sup>_di_.</sup> This implies that � _i_ _~~τ~~_ ~~(~~ _bi_ ) _di > D_ , and thus Assumption **A3** implies there exists some _D > D_ such that scheme ( _~~τ,~~ D_ <u>)</u> is sustainable. Let _z_ ̃ denote the long-run allocation associated to scheme ~~(~~ _~~τ,~~ D_ ). As _D > D_ , we have that min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( ̃ _zi, ui_ ) _>_ min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ~~(~~ _~~z~~ i, ui_ ). By (i), we have min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( ̃ _zi, ui_ ) _>_ min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( _zi, ui_ ), the desired result.

There remains to show that we can construct _~~τ~~_ in a way that meets conditions ( _i_ ), ( _ii_ ) and ( _iii_ ). We consider two cases:

**Case 1** : _ba_ 2 _≥ b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_).</sup> The construction of _~~τ~~_ is based on bundle _za_ 2. We start by showing that _za_ 2, whose position is illustrated in Figure 14, must satisfy the following two conditions:

(2) _τ_ ( _ba_ 2) _≥_ 0.

First, by incentive compatibility, individual _a_ 2 must prefer to leave bequest _ba_ 2 rather than _ba_ 1, which by Eq. (4) yields condition (1). As illustrated in Figure 14, condition (1) implies that bundle _za_ 2 lies in the upper contour set of the indifference curve passing through bundle ( _ca_ 1 + _ha_ 1 _, ha_ 1). Second, by incentive compatibility again, individual _a_ 1 must prefer to leave bequest _ba_ 1 rather than any other bequest _b > ba_ 1. This implies that bundle _z_<sup>_k_</sup> does not lie in the interior of her budget set _B_<sup>_τ_</sup> ( _w_ + _D,_ 0), and therefore _τ_ ( _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_))</sup><sup>_≥_0.As</sup> _−τ_ is single-peaked, the fact that we simultaneously have 17
- _τ_ ( _ba_ 1) _<_ 0 and

- • _τ_ ( _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_))</sup><sup>_≥_0,</sup>

implies that _τ_ ( _ba_ 2) _≥_ 0 because _ba_ 1 _< b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)</sup><sup>_≤ba_2.Graphically,bundle</sup> _za_ 2 must be in the non-distortionary budget set _L_ ( _cs_ 2 _,_ 0). Hence, condition (2) is also met, as desired.

<!-- Start of picture text --> hi L ( cs 2 , 0)<br>L ( cs 1 , 0) ( ca 2 − ha 1 , ha 2)<br>Rτ ( za 2)<br>Rb LF a ( w + D ) z k za 2 ( ca 1 + ha 1 , ha 1)<br>za 1<br>ha 1<br>zs 2 ci<br>zs 1<br>b LF a ( w + D )<br><!-- End of picture text --> Figure 14: Position of bundle _za_ 2 when _ba_ 2 _≥ b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)(Case1).</sup> Tax _~~τ~~_ is constructed from tax _τ_ and bundle _za_ 2. In a nutshell, _~~τ~~_ provides an exemption up to _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_),then</sup> _~~τ~~_ is defined by a constant marginal tax rate such that _~~τ~~_ ~~(~~ _ba_ 2) = _τ_ ( _ba_ 2), and then _~~τ~~_ remains equal to _τ_ for larger bequests. The implied budget sets _B_<sup>_τ_</sup> ( _w_ + _D,_ 0) and _Bτ_ ( _w_ + _D,_ 0) are contrasted in Figure 15. Formally, tax _~~τ~~_ is defined as

- _~~τ~~_ ~~(~~ _b_ ) = 0 for all _b ∈_ [0 _, b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)].</sup>

- _~~τ~~_ ~~(~~ _b_ ) = _τ_ ( _b_ ) for all _b ≥ ba_ 2.

Tax _~~τ~~_ is monotonically increasing up to bequest level _ba_ 2 by construction, because its constant marginal tax rate is non-negative because _τ_ ( _ba_ 2) _≥_ 0 (condition 2). This implies that _~~−τ~~_ is single-peaked because _τ_ is monotonically increasing for _b ≥ ba_ 2.<sup>28</sup> We now prove that tax _~~τ~~_ satisfies (i), (ii) and (iii). First, we prove (i), namely that min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ~~(~~ _~~z~~ i, ui_ ) = min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( _zi, ui_ ). (Recall that _~~z~~_ denotes the long-run allocation associated to scheme ~~(~~ _~~τ,~~ D_ ), where _D_ is the maximal sustainable demogrant for tax _τ_ .) Under tax _τ_ we have min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ( _zi, ui_ ) = _u_<sup>_c_</sup> ( _zs_ 1 _, u_<sup>**s**</sup> ) = _u_<sup>_c_</sup> ( _za_ 1 _, u_<sup>_∗_</sup> _a_<sup>)=</sup><sup>_w_+</sup><sup>_D_.Undertax</sup> _~~τ~~_ we have that _~~z~~ a_ 1 = _z_<sup>_k_</sup> , which

> 28Tax _τ_ is monotonically increasing for _b ≥ ba_ 2 because _−τ_ is single-peaked, _τ_ ( _ba_ 1) _<_ 0 and _τ_ ( _ba_ 2) _≥_ 0.

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<!-- Start of picture text --> hi<br>( ca 2 − ha 1 , ha 2)<br>ha 1 za 2<br>z k<br>Rb LF a ( w + D ) za 1 B τ ( w + D, 0)<br>B τ ′ ( w + D, 0)<br>zs 1<br>ci<br>b LF a ( w + D )<br>ba 2<br><!-- End of picture text --> Figure 15: Budget sets for individuals who ineherit zero as implied by taxes _τ_ and _~~τ~~_ ~~.~~ implies that _u_<sup>_c_</sup> ~~(~~ _~~z~~ a_ 1 _, u_<sup>_∗_</sup> _a_<sup>) =</sup><sup>_w_+</sup><sup>_D_.Giventhatwealsohave</sup><sup>_uc_</sup><sup>~~(~~</sup> _~~z~~ s_ 1 _, u_<sup>**s**</sup> ) = _w_ + _D_ , this yields again min _i∈_ [0 _,_ 1] _u_<sup>_c_</sup> ~~(~~ _~~z~~ i, ui_ ) = _w_ + _D_ (Lemma 1). This proves (i). Second, we prove (ii), namely that _~~τ~~_ ~~(~~ _ba_ 1) _> τ_ ( _ba_ 1). We have _~~τ~~_ ~~(~~ _ba_ 1) = 0 by construction of _~~τ~~_ because _ba_ 1 = _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_) as</sup> _~~z~~ a_ 1 = _z_<sup>_k_</sup> . As a result, (ii) follows immediately from the fact that _τ_ ( _ba_ 1) _<_ 0.

Third, we prove (iii), namely that _~~τ~~_ ~~(~~ _baX_ ) _≥ τ_ ( _baX_ ) for all _X ∈{_ 2 _,_ 3 _, . . . }_ . To do this, we show that _baX ≥ baX_ for all _X ∈{_ 2 _,_ 3 _, . . . }_ . If this holds, then (iii) holds because _~~τ~~_ ~~(~~ _b_ ) = _τ_ ( _b_ ) for all _b ≥ ba_ 2 and tax _τ_ is monotonically increasing for _b ≥ ba_ 2. We prove by induction that _baX ≥ baX_ for all _X ∈{_ 2 _,_ 3 _, . . . }_ . The induction basis requires us to show that _ba_ 2 _≥ ba_ 2. First, we show that _ba_ 2 _≥ ba_ 2 for the particular (counterfactual) case for which _ha_ 1 = _ha_ 1. This case implies that _a_ 2 receives the same inheritance under both schemes ( _τ, D_ ) and ~~(~~ _~~τ,~~ D_ ). As _a_ 2 selects _za_ 2 under scheme ( _τ, D_ ) rather than bundle ( _ca_ 1 + _ha_ 1 _, ha_ 1) (see Figure 14), this implies that _a_ 2 will prefer to leave a bequest of at least _b_ = _ba_ 2 under scheme ( _~~τ,~~ D_ ) when _~~g~~ a_ 2 = _ha_ 1 rather than another smaller bequest _b_<sup>_′_</sup> _< ba_ 2.<sup>29</sup> As inheritance is a normal good under preference _u_<sup>_∗_</sup> _a_<sup>,wealsohave</sup> _ba_ 2 _≥ ba_ 2 when _ha_ 1 _> ha_ 1, which is the case as _~~z~~ a_ 1 = _z_<sup>_k_</sup> . This concludes the proof of the induction basis. There remains to prove the induction step, namely that _baX_ +1 _≥ baX_ +1 when _baX ≥ baX_ . We have _~~g~~ aX_ +1 _≥ gaX_ +1 because _baX ≥ baX_ and _~~τ~~_ ~~(~~ _b_ ) = _τ_ ( _b_ ) for all _b ≥ ba_ 2. In words, individual _aX_ + 1 receives a (weakly) larger inheritance under scheme ~~(~~ _~~τ,~~ D_ ) than under scheme ( _τ, D_ ). This implies that _aX_ + 1 will leave at least as much bequest under ~~(~~ _~~τ,~~ D_ ) than under ( _τ, D_ ) because inheritance is a normal good under preference _u_<sup>_∗_</sup> _a_<sup>and</sup><sup>_τ_(</sup><sup>_b_)=</sup> _~~τ~~_ ~~(~~ _b_ ) for _b ≥ ba_ 2. This implies that _baX_ +1 _≥ baX_ +1, which proves the induction step. We

> 29Graphically, the marginal rate of substitution _−r_ of her indifference curves is (weakly) less steep than the frontier of her budget set for bequest levels in [ _b_<sup>_LF_</sup> _a_ ( _w_ + _D_ ) _, ba_ 2], which has a constant slope because the marginal rate of taxation is constant on that interval. If the frontier of _a_ 2’s budget set under scheme ( _~~τ~~ , D_ ) was strictly less steep than _−r_ , then it would imply that _u_<sup>_∗_</sup> _a_<sup>(</sup><sup>_za_2)</sup><sup>_<u∗_</sup> _a_<sup>(</sup><sup>_ca_1+</sup><sup>_ha_1</sup><sup>_, ha_1),whichwouldviolatetheincentivecompatibility</sup> constraint of _a_ 2 under scheme ( _τ, D_ ).

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have thus shown that tax _~~τ~~_ satisfies (i), (ii) and (iii), as desired.

**Case 2** : _ba_ 2 _< b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_).</sup>

The construction of _~~τ~~_ is the same as that provided for Case 1, except that here the construction is not based on bundle _za_ 2. Rather, it is based on bundle _zaX_<sup>, where</sup> _X_ is the smallest element in _{_ 3 _,_ 4 _, . . . }_ for which _baX_<sup>_≥b_</sup> _a_<sup>_LF_(</sup><sup>_w_+</sup><sup>_D_).</sup> We show there must exist such _X ∈{_ 3 _,_ 4 _, . . . }_ .

First, we show that _u_<sup>_∗_</sup> _a_<sup>(</sup><sup>_zaX_) =</sup><sup>_u∗_</sup> _a_<sup>(</sup><sup>_ca_1+</sup><sup>_haX−_1</sup><sup>_, ha_1)forall</sup><sup>_X∈{_1</sup><sup>_, . . . ,_</sup> _X −_ 1 _}_ , namely that the incentive compatibility constraints are binding for these individuals _aX_ . By the incentive compatibility constraint of _aX_ , which should prefer her bundle over ( _ca_ 1 + _haX−_ 1 _, ha_ 1), we have By the the incentive compatibility constraint of _a_ 1, which should prefer her bundle over ( _caX − haX−_ 1 _, haX_ ), we have By definition of _X_ we have _baX < b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)forall</sup><sup>_X∈{_1</sup><sup>_, . . . ,_</sup> _X −_ 1 _}_ . Therefore, we only need to consider the linear part of _u_<sup>_∗_</sup> _a_<sup>(thekinksofthe</sup> relevant indifference curves lies at larger bequest levels). Then, Eqs. (16) and (17) together imply that _u_<sup>_∗_</sup> _a_<sup>(</sup><sup>_zaX_) =</sup><sup>_u∗_</sup> _a_<sup>(</sup><sup>_ca_1+</sup><sup>_haX−_1</sup><sup>_, ha_1) for all</sup><sup>_X∈{_1</sup><sup>_, . . . ,_</sup> _X −_ 1 _}_ , as desired.

We can now show there must exist some _X ∈{_ 3 _,_ 4 _, . . . }_ such that _baX_<sup>_≥_</sup> _b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_) if</sup><sup>_τ_is optimal.If there is no such</sup> _X_ , then we have _baX < b_<sup>_LF_</sup> _a_<sup>(</sup><sup>_w_+</sup><sup>_D_)</sup> for all _X ∈{_ 3 _,_ 4 _, . . . }_ . This implies that _τ_ ( _baX_ ) _<_ 0 for all _X ∈{_ 1 _,_ 2 _, . . . }_ because _u_<sup>_∗_</sup> _a_<sup>(</sup><sup>_zaX_)=</sup><sup>_u∗_</sup> _a_<sup>(</sup><sup>_ca_1+</sup><sup>_haX−_1</sup><sup>_, ha_1)forall</sup><sup>_X∈{_1</sup><sup>_, . . . ,_</sup> _X −_ 1 _}_ , which implies that ( _caX − haX−_ 1 _, haX_ ) lies on the indifference curve passing through _za_ 1. The fact that _τ_ ( _baX_ ) _<_ 0 for all _X ∈{_ 1 _,_ 2 _, . . . }_ implies that _τ_ is not optimal because then _D <_ 0 (Proposition 1). This contradiction implies that such _X_ exists.

The argument that _~~τ~~_ (when its construction is based on _zaX_<sup>ratherthan</sup> _za_ 2) satisfies properties (i), (ii) and (iii) is a straightforward adaptation of the argument presented for Case 1, and is thus omitted.

# **S4 Proof of Proposition 4**

The following axiom is the well-known Separability axiom, according to which individuals who are assigned identical bundles in two allocations should not matter for the social ranking between these two allocations. The idea that they should not matter is captured by the requirement that the social ranking remain the same if the preferences and bundles assigned to these individuals change in such a way that the bundles assigned to these individuals remain identical in the two allocations.

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**Axiom 4** (Separability) **.**

_For all economy u ∈U, steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_,z′′_=(</sup><sup>_g_</sup> _i_<sup>_′′, c′′_</sup> _i_<sup>_, h′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_,z′′′_=(</sup><sup>_g_</sup> _i_<sup>_′′′, c′′′_</sup> _i_<sup>_, h′′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,subset_</sup> _of individuals J ∈ M_ [0 _,_ 1] _, if_ • _for all j ∈ J: zj_ = _zj_<sup>_′′andz_</sup> _j_<sup>_′_=</sup><sup>_z_</sup> _j_<sup>_′′′,_</sup> • _for all j ∈_ [0 _,_ 1] _\ J: uj_ = _u_<sup>_′_</sup> _j_<sup>_,zj_=</sup><sup>_z_</sup> _j_<sup>_′andz_</sup> _j_<sup>_′′_=</sup><sup>_z_</sup> _j_<sup>_′′′,_</sup> _then z_ **_R_** ( _u_ ) _z_<sup>_′′_</sup> _if and only if z_<sup>_′_</sup> **_R_** ( _u_<sup>_′_</sup> ) _z_<sup>_′′′_</sup> _._ The bite of this axiom is that it allows us to modify the economy in such a way that sets of individuals of positive measure have the same preferences, which is unlikely in a generic economy, whereas it is crucial to allow us to use Compensation (see Step 1 in the proof below). We now state and prove the following result, which justifies using SWF **_R_**<sup>_c−lex_</sup> .

**Proposition 4.** _If a SWF_ **(** _R_ ) _satisfies axioms Pareto, Compensation, Responsibility and Separability, then for all u ∈U, z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,ifthereexistsJ∈M_[0</sup><sup>_,_1]</sup><sup>_suchthatμ_(</sup><sup>_J_)</sup><sup>_>_0</sup><sup>_and_</sup>

_then_ _Proof._ This proof is reminiscent of similar proofs developed in models of labor income taxation in Fleurbaey and Maniquet (2006) and Fleurbaey and Maniquet (2007). The main differences are, first, that we deal here with economies with a continuum of individuals, which makes some arguments longer, whereas all individuals face the same prices (that is, the price of _h_ is equal to _R_ ), which allows us to simplify the proof.

The proof is divided in three steps. In the first step, we show that the combination of the four axioms implies a strengthening of the Responsibility axiom in which inequality aversion is infinite. In the second step, we show that the infinite inequality aversion is extended to _u_<sup>_c_</sup> ( _zi, ui_ ). In the final step, we show that this allows us to derive the desired property.

_Step 1._ We begin by defining the following strengthening of Responsibility.

**Axiom 5** (Responsibility*) **.**

_For all economy u ∈U, steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,subsetsofindividualsJ, K∈M_[0</sup><sup>_,_1]</sup><sup>_suchthatμ_(</sup><sup>_J_)=</sup> _μ_ ( _K_ ) _>_ 0 _, if there exists δ,_ ∆ _>_ 0 _such that for all j, q ∈ J and k, l ∈ K,_ With Responsibility*, we require strict social preference as soon as all individuals in _J_ gain, even if their budget gain is arbitrarily small and the budget 21
loss of members of _K_ is arbitrarily large. This is why Responsibility*, contrary to Responsibility, conveys an infinite inequality aversion.

We prove the following claim: If a SWF **(** _R_ ) satisfies Pareto, Compensation, Responsibility and Separability, then it satisfies Responsibility*.

Let _u ∈U_ , _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1], _z_<sup>_′_</sup> = ( _gi_<sup>_′, c_</sup> _i_<sup>_′, h_</sup> _i_<sup>_′_)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_,and</sup><sup>_J, K∈M_[0</sup><sup>_,_1]</sup> satisfy the conditions of Responsibility*. Let us assume, contrary to the claim, that _z_ **R** ( _u_ ) _z_<sup>_′_</sup> . We can assume, without loss of generality, that _μ_ ( _J_ ) = _μ_ ( _K_ ) _≤_ <u>1</u><sup>Ifitisnotthecase,thentheclaimisprovenbyrepeatingthisprooftwice.</sup> 4<sup>.</sup> Let _yj, yj_<sup>_′, yk, y_</sup> _k_<sup>_′∈_R+bedefinedby</sup> so that _yj < yj_<sup>_′≤y_</sup> _k_<sup>_′<yk_.ByPareto,wecanassume,w.l.o.g.,that</sup><sup>_y_</sup> _j_<sup>_′<y_</sup> _k_<sup>_′_.</sup> Indeed, if it is not the case, then we can create _z_<sup>_′′_</sup> = ( _gi_<sup>_′′, c′′_</sup> _i_<sup>_, h′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_by</sup> replacing _zk_<sup>_′_= (</sup><sup>_g_</sup> _k_<sup>_′, c′_</sup> _k_<sup>_, h′_</sup> _k_<sup>)with</sup><sup>_z_</sup> _k_<sup>_′′_= (</sup><sup>_g_</sup> _k_<sup>_′′, c′′_</sup> _k_<sup>_, h′′_</sup> _k_<sup>)suchthat</sup> for all _k ∈ K_ , so that _z_<sup>_′′_</sup> **P** ( _u_ ) _z_<sup>_′_</sup> and, by transitivity, _z_<sup>_′′_</sup> **P** ( _u_ ) _z_ and continue the proof. So, we assume _yj_<sup>_′< y_</sup> _k_<sup>_′_.Wecanevenassume,w.l.o.g.,that</sup> Indeed, if it is not the case, then the claim is proven by repeating this proof the required number of times.<sup>30</sup> Let _u_<sup>_∗_</sup> _∈ U_ and _z_ ̄ _a_ = ( ̄ _ga,_ ̄ _ca, h_<sup> ̄</sup> _a_ ) _,_ ̄ _za_<sup>_′_=( ̄</sup><sup>_g_</sup> _a_<sup>_′,_ ̄</sup><sup>_c′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′_</sup> _a_<sup>)</sup><sup>_,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′_=( ̄</sup><sup>_g_</sup> _a_<sup>_′′,_ ̄</sup><sup>_c′′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′′_</sup> _a_<sup>)</sup><sup>_,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′′_=</sup> ( ̄ _ga_<sup>_′′′,_ ̄</sup><sup>_c′′′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′′′_</sup> _a_<sup>)</sup><sup>_,_ ̄</sup><sup>_zb_= ( ̄</sup><sup>_gb,_ ̄</sup><sup>_cb,_ ̄</sup><sup>_hb_)</sup><sup>_,_ ̄</sup><sup>_z_</sup> _b_<sup>_′_= ( ̄</sup><sup>_g_</sup> _b_<sup>_′,_ ̄</sup><sup>_c′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′_</sup> _b_<sup>)</sup><sup>_,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′_= ( ̄</sup><sup>_g_</sup> _b_<sup>_′′,_ ̄</sup><sup>_c′′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′′_</sup> _b_<sup>)</sup><sup>_,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′′_= ( ̄</sup><sup>_g_</sup> _b_<sup>_′′′,_ ̄</sup><sup>_c′′′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′′′_</sup> _b_<sup>)</sup><sup>_∈_</sup> _X_ , _y_ ̄ _a,_ ̄ _ya_<sup>_′,_ ̄</sup><sup>_yb,_ ̄</sup><sup>_y_</sup> _b_<sup>_′∈_Rbesuchthat</sup> ( ̄ _ca, h_<sup> ̄</sup> _a_ ) _∈_ max _|u∗ L_ ( ̄ _ca, h_<sup> ̄</sup> _a_ ) _,_ ( ̄ _c_<sup>_′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′_</sup> _a_<sup>)</sup><sup>_∈_max</sup> _|u_<sup>_∗L_( ̄</sup><sup>_c′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′_</sup> _a_<sup>)</sup><sup>_,_</sup> ( ̄ _cb, h_<sup> ̄</sup> _b_ ) _∈_ max _|u∗ L_ ( ̄ _cb, h_<sup> ̄</sup> _b_ ) _,_ ( ̄ _c_<sup>_′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′_</sup> _b_<sup>)</sup><sup>_∈_max</sup> _|u_<sup>_∗L_( ̄</sup><sup>_c′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′_</sup> _b_<sup>)</sup><sup>_,_</sup>

> 30The fact that _yk′_<sup>_> y_</sup> _j_<sup>_′_alwaysallowsustoconstructsets</sup><sup>_A_and</sup><sup>_B_with</sup><sup>_y_ ̄</sup> _a_<sup>_′<y_ ̄</sup><sup>_a_=</sup><sup>_y_ ̄</sup> _b_<sup>_<y_ ̄</sup> _b_<sup>_′_</sup> and the proof below has to be replicated a finite number of times at least as large as _y_  ̄ _ya_<sup>_<u>′</u>_</sup> _<u>j</u>_<sup>_′−_</sup> _−_<sup>_y_</sup> _y_  ̄<sup>_<u>j</u>_</sup> _a_<sup>.</sup>

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_u_<sup>_∗_</sup> ( _za_ ) = _u_<sup>_∗_</sup> ( _za_<sup>_′′′_)</sup> _u_<sup>_∗_</sup> ( _za_<sup>_′_)</sup> = _u_<sup>_∗_</sup> ( _za_<sup>_′′_)</sup> _u_<sup>_∗_</sup> ( _zb_<sup>_′_)</sup> = _u_<sup>_∗_</sup> ( _zb_<sup>_′′_)</sup><sup>_._</sup> The construction of _u_<sup>_∗_</sup> _,_ ̄ _za,_ ̄ _za_<sup>_′,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′′,_ ̄</sup><sup>_zb,_ ̄</sup><sup>_z_</sup> _b_<sup>_′,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′′,_ ̄</sup><sup>_ya,_ ̄</sup><sup>_y_</sup> _a_<sup>_′,_ ̄</sup><sup>_yb,_ ̄</sup><sup>_y_</sup> _b_<sup>_′_isillustratedin</sup> Fig. 16.

<!-- Start of picture text --> z ′′′ a =<br><!-- End of picture text --> <!-- Start of picture text --> z ′′′ b<br><!-- End of picture text --> <!-- Start of picture text --> z ′′ a<br><!-- End of picture text --> <!-- Start of picture text --> a<br>zk<br>q<br>zq z′ z ′ b zk ′<br>za = zb<br>z ′ a zl<br>u ∗<br>zj ′ zl ′<br>zj<br>u ∗ u ∗<br>δ δ ∆ ∆<br>yj yj ′ y ′ a y ′ b yk ′ yk<br><!-- End of picture text --> Figure 16: Construction of _u_<sup>_∗_</sup> _,_ ̄ _za,_ ̄ _za_<sup>_′,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′,_ ̄</sup><sup>_z_</sup> _a_<sup>_′′′,_ ̄</sup><sup>_zb,_ ̄</sup><sup>_z_</sup> _b_<sup>_′,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′,_ ̄</sup><sup>_z_</sup> _b_<sup>_′′′,_ ̄</sup><sup>_ya,_ ̄</sup><sup>_y_</sup> _a_<sup>_′,_ ̄</sup><sup>_yb,_ ̄</sup><sup>_y_</sup> _b_<sup>_′_.</sup> The intuition of the proof and the role of the axioms can be illustrated with the figure. We need to prove that the budget increase of an amount _δ_ for individuals _j_ and _q_ at the expense of a budget decrease of an amount ∆for individuals _k_ and _l_ , with ∆possibly much larger than _δ_ , is a social improvement. Separability allows us to modify the preferences and bundles of a sufficiently large number of individuals and insert individuals of type _a_ and _b_ in the economy. The design of their preferences is key: they are indifferent between a transfer of resources, represented by bundles _z_ ̄ _i_<sup>_′′_and</sup><sup>_z_ ̄</sup> _i_<sup>_′′′_,</sup><sup>_i ∈{a, b}_, in which the beneficiary</sup> gets an amount equal to that left by the contributor, and a transfer of resources in which the beneficiary gets a different amount of resources, possible much smaller, than the one lost by the contributor, represented by bundles _z_ ̄ _i_ and _z_ ̄ _i_<sup>_′_,</sup><sup>_i∈{a, b}_.</sup> Pareto forces us to be indifferent between these two sets of transfers. Transfers are calibrated in such a way that a sequence of transfers between individuals _j_ and _a_ (using Responsibility), individuals _a_ and _b_ (using Compensation) and then individuals _b_ and _k_ (using Responsibility) allows us to reach the desired conclusion.

The following axiom, known in the literature as Pareto indifference, is a well-known consequence of Pareto.

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**Axiom 6** (Pareto Indifference) **.** _For all economy u ∈U and steady-state allocations z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S,ifforalli ∈_[0</sup><sup>_,_1]</sup>

_then z_<sup>_′_</sup> **_I_** ( _u_ ) _z._ Let _A, B ∈ M_ [0 _,_ 1] be such that _μ_ ( _A_ ) = _μ_ ( _B_ ) = _μ_ ( _J_ ) = _μ_ ( _K_ ), _A, B, J_ and _K_ are all disjoint. Since they are all disjoint, we have ( _ci, hi_ ) = ( _c_<sup>_′_</sup> _i_<sup>_, h_</sup> _i_<sup>_′_)forall</sup> _i ∈ A ∪ B_ . Let _u_<sup>_′_</sup> _∈U_ be defined by and _gi_<sup>1</sup><sup>_, g_</sup> _i_<sup>2</sup><sup>_, i ∈_[0</sup><sup>_,_1], are fixed so as to guarantee that</sup><sup>_z_1</sup><sup>_, z_2</sup><sup>_∈S_.By Separability,</sup> _z_ **R** ( _u_ ) _z_<sup>_′_</sup> _⇔ z_<sup>1</sup> **R** ( _u_<sup>_′_</sup> ) _z_<sup>2</sup> _,_ and _gi_<sup>3</sup><sup>_, i∈_[0</sup><sup>_,_1],arefixedsoastoguaranteethat</sup><sup>_z_3</sup><sup>_∈S_.ByResponsibility,</sup> _z_<sup>3</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>1</sup> , so that, by transitivity, _z_<sup>3</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>2</sup> .

and _gi_<sup>4</sup><sup>_, i∈_[0</sup><sup>_,_1],arefixedsoastoguaranteethat</sup><sup>_z_4</sup><sup>_∈S_.ByResponsibility,</sup> _z_<sup>4</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>3</sup> , so that, by transitivity, _z_<sup>4</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>2</sup> .

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and _gi_<sup>5</sup><sup>_, i ∈_[0</sup><sup>_,_1], are fixed so as to guarantee that</sup><sup>_z_5</sup><sup>_∈S_.By Pareto Indifference,</sup> _z_<sup>5</sup> **I** ( _u_<sup>_′_</sup> ) _z_<sup>4</sup> , so that, by transitivity, _z_<sup>5</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>2</sup> . Let _z_<sup>6</sup> = ( _gi_<sup>6</sup><sup>_, c_6</sup> _i_<sup>_, h_6</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈_R3[0</sup> +<sup>_,_1]</sup> be defined by ( _c_<sup>6</sup> _a_<sup>_, h_6</sup> _a_<sup>)</sup> = ( ̄ _c_<sup>_′′′_</sup> _a_<sup>_,_ ̄</sup><sup>_h′′′_</sup> _a_<sup>)</sup><sup>_, ∀a ∈A_</sup> ( _c_<sup>6</sup> _b_<sup>_, h_6</sup> _b_<sup>)</sup> = ( ̄ _c_<sup>_′′′_</sup> _b_<sup>_,_ ̄</sup><sup>_h′′′_</sup> _b_<sup>)</sup><sup>_, ∀b ∈B_</sup> ( _c_<sup>6</sup> _i_<sup>_, h_6</sup> _i_<sup>)</sup> = ( _c_<sup>5</sup> _i_<sup>_, h_5</sup> _i_<sup>)</sup><sup>_, ∀i ∈_[0</sup><sup>_,_1]</sup><sup>_\_(</sup><sup>_A ∪B_)</sup><sup>_,_</sup> and _gi_<sup>6</sup><sup>_, i∈_[0</sup><sup>_,_1],arefixedsoastoguaranteethat</sup><sup>_z_6</sup><sup>_∈S_.ByCompensation,</sup> _z_<sup>6</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>5</sup> , so that, by transitivity, _z_<sup>6</sup> **P** ( _u_<sup>_′_</sup> ) _z_<sup>2</sup> . Let _z_<sup>7</sup> = ( _gi_<sup>7</sup><sup>_, c_7</sup> _i_<sup>_, h_7</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈_R3[0</sup> +<sup>_,_1]</sup> be defined by ( _c_<sup>7</sup> _a_<sup>_, h_7</sup> _a_<sup>)</sup> = ( ̄ _ca, h_<sup> ̄</sup> _a_ ) _, ∀ a ∈ A_ ( _c_<sup>7</sup> _b_<sup>_, h_7</sup> _b_<sup>)</sup> = ( ̄ _cb, h_<sup> ̄</sup> _b_ ) _, ∀ b ∈ B_ ( _c_<sup>7</sup> _i_<sup>_, h_7</sup> _i_<sup>)</sup> = ( _c_<sup>6</sup> _i_<sup>_, h_6</sup> _i_<sup>)</sup><sup>_, ∀i ∈_[0</sup><sup>_,_1]</sup><sup>_\_(</sup><sup>_A ∪B_)</sup><sup>_,_</sup> and By Separability, so that, by transitivity, _z_<sup>8</sup> **P** ( _u_ ) _z_<sup>9</sup> . Finally, observe that _z_<sup>8</sup> = _z_<sup>9</sup> = _z_<sup>_′_</sup> , so that _z_<sup>_′_</sup> **P** ( _u_ ) _z_<sup>_′_</sup> , the desired contradiction.

_Step 2._ We now prove the following claim: If a SWF **(** _R_ ) satisfies axioms Pareto, Compensation, Responsibility and Separability, then it satisfies the following property, which amounts to requiring an infinite aversion towards inequality in _u_<sup>_c_</sup> : For all _u ∈U_ , _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1], _z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_,</sup> _J, K ∈ M_ [0 _,_ 1] such that _μ_ ( _J_ ) = _μ_ ( _K_ ) _>_ 0, if for all _j ∈ J_ and _k ∈ K_ , • _u_<sup>_c_</sup> ( _zj, uj_ ) _< u_<sup>_c_</sup> ( _zj_<sup>_′, uj_)</sup><sup>_< uc_(</sup><sup>_z_</sup> _k_<sup>_′, uk_)</sup><sup>_< uc_(</sup><sup>_zk, uk_),</sup> and _zi_ = _zi_<sup>_′_forall</sup><sup>_i∈/J∪K_then</sup><sup>_z′_</sup><sup>**P**(</sup><sup>_u_)</sup><sup>_z_.</sup> Let _u ∈U_ , _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1], _z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_, and</sup><sup>_J, K⊆M_[0</sup><sup>_,_1]</sup> satisfy the conditions of this property. Let us assume, contrary to the claim, that _z_ **R** ( _u_ ) _z_<sup>_′_</sup> .

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By Pareto Indifference, we can assume, without loss of generality, that ( _cj, hj_ ) _∈_ max _|uj L_ ( _cj, hj_ ) _,_ ( _c_<sup>_′_</sup> _j_<sup>_, h′_</sup> _j_<sup>)</sup><sup>_∈_max</sup> _|uj_<sup>_L_(</sup><sup>_c′_</sup> _j_<sup>_, h′_</sup> _j_<sup>)</sup><sup>_,_</sup> ( _ck, hk_ ) _∈_ max _|uk L_ ( _ck, hk_ ) _,_ ( _c_<sup>_′_</sup> _k_<sup>_, h′_</sup> _k_<sup>)</sup><sup>_∈_max</sup> _|uk_<sup>_L_(</sup><sup>_c_</sup> _k_<sup>_′, h′_</sup> _k_<sup>)</sup><sup>_._</sup>

Indeed, if it is not the case, then, by Pareto Indifference, we can replace bundles _zj, zj_<sup>_′_and</sup><sup>_zk, z_</sup> _k_<sup>_′_byabundleonthesameindifferencecurvethatisoptimalin</sup> the corresponding budget.

By Pareto, we can assume that for all _j, q ∈ J_ , _cj_ +<sup>_h_</sup> _R_<sup>_<u>j</u>_=</sup><sup>_cq_+</sup><sup>_h_</sup> _R_<sup>_<u>q</u>_and</sup> for all _k, l ∈ J_ , _ck_ +<sup>_<u>h</u>_</sup> _R_<sup>_<u>k</u>_</sup> = _cl_ +<sup>_<u>h</u>_</sup> _R_<sup>_<u>l</u>_.</sup> Indeed, if it is not the case, we can replace each _zj_ with _zj_<sup>_′′_such that</sup><sup>_uc_(</sup><sup>_z_</sup> _j_<sup>_′′, uj_) = sup</sup> _i∈J_<sup>_uc_(</sup><sup>_zi, ui_), each</sup><sup>_z_</sup> _j_<sup>_′_with</sup><sup>_z_</sup> _j_<sup>_′′′_</sup> such that _u_<sup>_c_</sup> ( _zj_<sup>_′′′, uj_) = inf</sup><sup>_i∈J uc_(</sup><sup>_zi, ui_),each</sup><sup>_zk_with</sup><sup>_z_</sup> _k_<sup>_′′_suchthat</sup><sup>_uc_(</sup><sup>_z_</sup> _k_<sup>_′′, uk_) =</sup> sup _i∈K u_<sup>_c_</sup> ( _zi, ui_ ), each _zk_<sup>_′_with</sup><sup>_z_</sup> _k_<sup>_′′′_suchthat</sup><sup>_uc_(</sup><sup>_z_</sup> _k_<sup>_′′′, uk_)=inf</sup><sup>_i∈K uc_(</sup><sup>_zi, ui_).By</sup> Pareto, _z_<sup>_′′_</sup> **P** ( _u_ ) _z_ and _z_<sup>_′_</sup> **P** ( _u_ ) _z_<sup>_′′′_</sup> , so that _z_<sup>_′′_</sup> **P** ( _u_ ) _z_<sup>_′′′_</sup> , and the proof continues.

By Responsibility*, _z_<sup>_′_</sup> **P** ( _u_ ) _z_ , so that, by transitivity, _z_<sup>_′_</sup> **P** ( _u_ ) _z_<sup>_′_</sup> , the desired contradiction.

_Step 3._ We now prove the claim presented in the statement of the Proposition. Let _u ∈U_ , _z_ = ( _gi, ci, hi_ ) _i∈_ [0 _,_ 1] _, z_<sup>_′_</sup> = ( _gi_<sup>_′, c′_</sup> _i_<sup>_, h′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈S_,</sup><sup>_J∈M_[0</sup><sup>_,_1]such</sup> that _μ_ ( _J_ ) _>_ 0 and Let us assume, contrary to the claim, that _z_ **_R_** ( _u_ ) _z_<sup>_′_</sup> . Let _z_<sup>_′′_</sup> = ( _gi_<sup>_′′, c′′_</sup> _i_<sup>_, h′′_</sup> _i_<sup>)</sup><sup>_i∈_[0</sup><sup>_,_1]</sup><sup>_∈_</sup> _S_ be such that _u_<sup>_c_</sup> ( _zi_<sup>_′′, ui_) =</sup><sup>_u_forall</sup><sup>_i ∈_[0</sup><sup>_,_1]and</sup> Let _N ∈N_ be an integer such that _Nμ_ ( _J_ ) _>_ 1. Let _J_<sup>_′_</sup> _⊆ J_ be such that _μ_ ( _J_<sup>_′_</sup> ) =<sup>1</sup><sup>_−_</sup> _N_<sup>_<u>μ</u>_</sup><sup><u>(</u></sup><sup>_J_</sup><sup><u>)</u></sup> . We can create a sequence _z_<sup>0</sup> _, . . . , z_<sup>_n_</sup> _, . . . , z_<sup>_N_</sup> such that _zi_<sup>0=</sup><sup>_zi_</sup> for all _i ∈_ [0 _,_ 1] _\_ ( _J \ J_<sup>_′_</sup> ), _u_<sup>_c_</sup> ( _zj_<sup>0</sup><sup>_, uj_)=</sup><sup>_u_forall</sup><sup>_j∈J\ J′_,</sup><sup>_zN_=</sup><sup>_z′′_andfor</sup> each _n ∈{_ 1 _, . . . , N }_ , there exists a set _K_<sup>_n_</sup> _∈ M_ [0 _,_ 1] such that _μ_ ( _K_<sup>_n_</sup> ) = _μ_ ( _J_<sup>_′_</sup> ), _∪n∈{_ 1 _,...,N }K_<sup>_n_</sup> _∪ J_ = [0 _,_ 1], By Pareto, _z_<sup>0</sup> **_P_** ( _u_ ) _z_ . By the property proven in Step 3, _z_<sup>_n_</sup> **_P_** ( _u_ ) _z_<sup>_n−_1</sup> . By transitivity, _z_<sup>_′′_</sup> **_P_** ( _u_ ) _z_ . By Pareto, _z_<sup>_′_</sup> **_P_** ( _u_ ) _z_<sup>_′′_</sup> , so that _z_<sup>_′_</sup> **_P_** ( _u_ ) _z_ , the desired contradiction.

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