Supplement to “ Strategy - proofness and efficiency with non - quasi - linear preferences : A characterization of minimum price Walrasian rule

Hall’s theorem (Hall 1935). Let N ≡ {1 n} and M ≡ {1 m}. For each i ∈N , let Di ⊆ M . Then there is a one-to-one mapping x′ from N to M such that for each i ∈ N , x′(i) ∈Di if and only if for each N ′ ⊆N , |i∈N ′ Di| ≥ |N ′|. Fact 4 (Mishra and Talman 2010). Let R ⊆ RE and R ∈ Rn. A price vector p is a Walrasian equilibrium price vector for R if and only if no set is overdemanded and no set is underdemanded at p for R.