1. Integrability expenditure function, preference Proposition6 differentiable e(p, u)is the expenditure function of sets V(u)={x∈界:p·x≥e(p,u)Vp≥0} We need to prove e(p, u)is the support function of v(u), that is e(p, u)=min(p x: XEy(u)) see the fic1.Integrability • expenditure function→ preference. • Proposition6: differentiable e(p,u) is the expenditure function of sets: • We need to prove e(p,u) is the support function of V(u) ,that is see the fig. ( ) { : ( , ) 0} n V u e u =     + x p x p pe u V u ( , ) min{ : ( )} p p x x =  