Advanced microeconomics lecture 6: consumption theory Ill Ye Jianliang
Advanced Microeconomics lecture 6:consumption theory III Ye Jianliang
Utility maximization Content Integrability Aggregation across goods
Utility maximization • Content: – Integrability – Aggregation across goods
1. Integrability Demand function x(p, w)(c d )is HDO, satisfied Walras law and have a substitution matrix S(p, w)is s n.s.d. for any (p, w ) if it's deduced by rational preference And if we observed an x(p, w)satisfied such conditions, can we find a preference to rationalization x(p, w ) That the integrability problem
1.Integrability • Demand function x(p,w) (c.d.) is HD0, satisfied Walras Law and have a substitution matrix S(p,w) is s.n.s.d. for any (p,w),if it’s deduced by rational preference. And if we observed an x(p,w) satisfied such conditions, can we find a preference to rationalization x(p,w)? That the integrability problem
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 fic
1.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 =
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