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.

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