1. Integrability Demand> expenditure function. Partial differential equation ∫x(p,e(p)=V(p) (p) The existence of solution means substitution matrix is symmetric DPe(p)=dx(p, e(p)+d x(p,e(p))x(p, e(p)) s(p, e(p))1.Integrability • Demand → expenditure function. – Partial differential equation: – The existence of solution means substitution matrix is symmetric: 0 0 0 ( , ( )) ( , ) ( ) e e u e w  =    = p x p p p p 2 ( ) ( , ( )) ( , ( )) ( , ( )) ( , ( )) T D e D e D e e p p w S e = +  = p x p p x p p x p p p p