哈佛大学：《中级微观经济学》（英文版）Lecture 2 Choice and Utility Functions

Differentiating p.x=w with respect to pk yields ∑p 2x;(P,v) +xk=0 opk or multiplying the whole expression by ph Sx(p-")B+②=∑m+mk=0 opk where again ni pixi

Differentiating p  x  w with respect to pk yields:  i1 L pi xip,w pk  xk  0 or multiplying the whole expression by pk w :  i1 L xip,w pk pk xi pixi w  pkxk w   i1 L pk i i  k  0, where again i  pixi w

This means that cross price elasticities sum to-1 times the budget share of the relevant good. Overall, these elasticities have to sum to a negative number

This means that cross price elasticities sum to -1 times the budget share of the relevant good. Overall, these elasticities have to sum to a negative number

MWG Definition 2. F 1 The Walrasian Demand function satisfies the weak axiom of revealed preference if the following property holds for any two price-wealth situations(p, w)and(p, w) fp·x(p’,w)≤ w and x(P,W)≠x(p,形) then p·x(p,y)>w. In words-if the goods that are chosen with budget set (a) are affordable at budget set (b), and not the same as the goods that are chosen at budget set( b), then the goods that are chosen at budget set (b) are not affordable at budget set(a) Just like in the last lecture. WarP means

MWG Definition 2.F.1 The Walrasian Demand function satisfies the weak axiom of revealed preference if the following property holds for any two price-wealth situations p,w and p ,w  : If p  xp ,w   w and xp ,w   xp,w then p  xp,w  w . In words– if the goods that are chosen with budget set (a) are affordable at budget set (b), and not the same as the goods that are chosen at budget set (b), then the goods that are chosen at budget set (b) are not affordable at budget set (a). Just like in the last lecture, WARP means

that if bundle(b)is preferred to bundle(a in one setting, it will be preferred in all other settings

that if bundle (b) is preferred to bundle (a) in one setting, it will be preferred in all other settings

a property that follows from WARP: price changes that are fully income compensated make consumers weakly better off Take any (p, w) and let p=p+ Ap Compensate the consumer with an income change so that the old bundle is exactly affordable at the new prices, i.e v′=W+△w=x(P,)·△ The consumers new consumption level at

A property that follows from WARP: price changes that are fully income compensated make consumers weakly better off. Take any p,w and let p  p  p Compensate the consumer with an income change so that the old bundle is exactly affordable at the new prices, i.e.: w  w  w  xp,w  p The consumer’s new consumption level at