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