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
2. Aggregation across goods Why local analysis is rational? That is, it's rational to model consumer choice by partial maximization What's the restriction of the preference that we can do like that Separability: partitioning consumption bundle into two“sub- bundles”.(x,z)and price vector(p, g)
2.Aggregation across goods • Why local analysis is rational? That is, it’s rational to model consumer choice by partial maximization. • What’s the restriction of the preference that we can do like that. • Separability: partitioning consumption bundle into two “sub-bundles”. (x, z) and price vector (p, q)
2. Aggregation across goods UMP: max u(x, z) S.t. px+ gz=w Let p=f(p),x=g(x) and set max U(x, z) S.t. px t qz=w Solution x(p, q, w)=g(x'(p, q, w), how can we get it
2.Aggregation across goods • UMP: • Let: and set: • Solution ,how can we get it. max ( ) . . u s t w + = x,z px qz p f x g = = ( ), ( ) p x max ( ) . . x U x s t px w + = , z , z qz x p w g w ( , , ) ( ( , , )) q x p q =
2. Aggregation across goods Two ways Aggregate prices p=f(p)at first, and then maximize u on the budget px+qZ=w this is called Hicksian separability Maximize u on budget px+qz=w at first then aggregate quantities to get x=8(x) it's called functional separability
2.Aggregation across goods • Two ways: • Aggregate prices at first, and then maximize U on the budget , this is called Hicksian separability. • Maximize u on budget at first, then aggregate quantities to get , it’s called functional separability. p f = ( ) p px w + = qzx g = ( ) x px qz + = w
2. Aggregation across goods Hicksian separability: no relative price change, so p=t plet: p=t, x=px Define the indirect utility function v(p, q, w)=max u(x, Z) s.t. ppx+gZ=w As Roy' s identity show: x(p, q, w)= Ov(p, q, w)/Op >ov(pp, q, w)/Op, px(p, q, w) av(p, q, w)/aw av(pp, q, w)/aw
2.Aggregation across goods • Hicksian separability: no relative price change, so ,let: • Define the indirect utility function: • As Roy’s identity show: 0 p p = t 0 p t x = = , p x 0 ( , , ) max ( ) . . v p w u s t p w = + = x,z q x,z p x qz 0 0 0 ( , , ) / ( , , ) / ( , , ) ( , , ) ( , , ) / ( , , ) / i v p w p v p w p x p w w v p w w v p w w = − = − = q p q q p x p q q p q
2. Aggregation across goods Construction direct utility function has the property that: v(p, g, w)=max U(x, Z) s.t. px+ gz Maximize u means maximize u
2.Aggregation across goods • Construction direct utility function has the property that: • Maximize U means maximize u. ( , , ) max ( ) . . x v p w U x s t px w = + = , z q , z qz
