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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