**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