文档介绍：Advanced Microeconomics lecture 6:consumption theory III
Ye Jianliang
Utility maximization
Content:
Integrability
Aggregation across goods
Demand function x(p,w) (.) is HD0, satisfied Walras Law and have a substitution matrix S(p,w) is . 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.
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.
Demand expenditure function.
Partial differential equation:
The existence of solution means substitution matrix is symmetric:
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)
across goods
UMP:
Let: and set:
Solution ,how can we get it.
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.
across goods
Hicksian separability: no relative price change, so ,let:
Define the indirect utility function:
As Roy’s identity show:
across goods
Construction direct utility function has the property that:
Maximize U means maximize u.