# **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