Externalities, Nonconvexities, and Fixed Points

Letting E 1 =[E 1 \(X × U 2 )] ∪ £ E 1 ∩ (X × U 2 ) ¤ ,wehaveforall(y, x) ∈ E 1 \(X × U 2 ),ρ w ∗ ×w ∗((y, x),E∗ )= ρ w ∗ ×w ∗((y, x), [E1 \(X × U 2 )] ∪ [E 2 \X × U 1 )])≤ ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )] ∪ [E 2 ∩ (X × U 1 )])= ρ w ∗ ×w ∗((y, x),E2 ).Moreover, we have for all(y, x) ∈ E 1 ∩ (X × U 2 ),ρ w ∗ ×w ∗((y, x),E∗ )= ρ w ∗ ×w ∗((y, x), [E1 \(X × U 2 )] ∪ [E 2 \(X × U 1 )])and= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )]),ρ w ∗ ×w ∗((y, x),E2 )Thus, for all (y, x) ∈ E 1 ,= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )] ∪ [E 2 ∩ (X × U 1 )])= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )]).ρ w ∗ ×w ∗((y, x),E∗ ) ≤ ρ w ∗ ×w ∗((y, x),E2 ),implying that e w ∗ ×w ∗(E1 ,E ∗ ) ≤ e w ∗ ×w ∗(E1 ,E 2 ). Together,e w ∗ ×w ∗(E1 ,E ∗ ) ≤ e w ∗ ×w ∗(E1 ,E 2 )ande w ∗ ×w ∗(E∗ ,E 1 ) ≤ e w ∗ ×w ∗(E2 ,E 1 )imply thatThus, we haveh w ∗ ×w ∗(E∗ ,E 1 ) ≤ h w ∗ ×w ∗(E2 ,E 1 ) < 2δ 0 .h w ∗ ×w ∗(E∗ ,E 0 ) ≤ h w ∗ ×w ∗(E∗ ,E 1 )+h w ∗ ×w ∗(E1 ,E 0 )≤ h w ∗ ×w ∗(E2 ,E 1 )+h w ∗ ×w ∗(E1 ,E 0 )< 2δ 0 + δ 0 < 3δ 0 .56