1. Integrability Demand >expenditure function Partial differential equation X(p,e(p))=velp, u) 0/0 The existence of solution means substitution matrix is symmetric e(p=dx(p, e(p))+dx(p, elp)) x(p, elp)) 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