lecture5-1consumption theory I(微观经济学-东南大学,周勤)

Giffen good: xl (p, w) / pl 0
See the fig.
xn ( p , w ) pn
x1 ( p , w ) pn
4.Demand function
Cournot
Aggregation: if Walrasian demand function x(p,w) is HD0 and satisfied Walras Law , then:
5.WA and demand law
Proposition5: x(p,w) satisfied only if: (p p) [x(p, w) x(p, w)] 0
WA if and and when
x(p, w) x(p, w) , (p p) [x(p, w) x(p, w)] 0
p x(p, w) w,if p x(p, w) w and x(p, w) x(p, w)
See the fig
demand function x(p,w) satisfied WA if for any (p, w) and (p, w) we have: .
For
x x(p, w) and p 0, w 0
(we will prove them next lecture)
4.Demand function
Weห้องสมุดไป่ตู้lth

effects:

Engel function: given price p ,function over wealth x(p, w) Wealth expansion path:


It’s reflexive:x X, x % x It’s transitive: x, y, z X,if x % y and y % z, then x % z
1.Preference
Propersition1:

if % is rational, then:

is reflexive and transitive and symmetric If x y % z, then x z
Prop.5
indicates, p x 0, or dp dx 0 that’s called “demand law ”, or “compensation demand law ”.
5.WA and demand law
Slutsky
matrix (substitution matrix)

x y x % y, but not y % x x y x % y,and y % x

An indifference relation “ ”:
1.Preference
Definition:

preference % on X is rational if:
It’s complete: x, y X,either x % y or y % x, or both
pDp x(p, w) x(p, w)T 0T
Engel
Aggregation: if Walrasian demand function x(p,w) satisfied Walras Law, then:
pDw x(p, w) 1
5.WA and demand law
Walrasian
Ep {x(p, w) : w 0}

Wealth effects:
(p , w) x1 w x2 (p , w) Dw x(p, w) w x (p , w) n w
4.Demand function
Wealth

is irreflexive and transitive
2.The rule of choice
We
can only observe what a man choice under a certain constrain. Is the man’s preference rational when we observed several choice of him? Or if his preference was rational, what rules will his choice take?
Choice
2.The rule of choice
Definition:
for (B , C (., % )), we say “rational preference % rationalization C (.) ( B , C (., % )) satisfied WA Propersition3: if and B contain all subsets of X that is ternary and less than ternary.

2.The rule of choice
Weak
Axiom of Revealed Preference (WARP or WA) : if x C ( B) for certainB B , and x, y B for B B and x, y B , y C( B) then must x C ( B) Revealed Preference “ % ”:
x % y B B , x, y B, and x C ( B)
2.The rule of choice
set C ( B, % ) {x B : x % y, y B} Proposition2: if % are rational, then choice construct (B , C (., % )) satisfied WA. Prop.2 indicate that if the decision-maker have rational preference, than his choice satisfied WA. Is his preference rational if his choice is observed satisfy WA?
effects
S (p, w) ( slk (p, w))nn
Substitution
slk (p, w) xl (p, w) xl (p, w) xk (p, w) pk w
S (p, w) is
n.s.d Giffen good is necessary inferior good.

example: X={x,y,z}, B ={{x,y},{y,z},{x,z}}, C({x,y})={x},C({y,z})={y},C({x,z})={z},(B , C(.)) satisfied WARP, but there is no rationalization preference relation over it.
See the fig
4.Demand function
Price

effects:
Supply curve: xl {x(p, w) : p l pl } Price effects:
(p, w) x1 p1 D p x(p, w) xn (p , w) p1
Advanced Economics
(lecture 5: consumption theory I)
Zhou Qin
Preference and Choice
Content:



Preference and rationality The rule of choice Budget Demand function WA and demanded law
petive budget
n p Consumption set X and it’s price
with the wealth w>0. Definition: Walras (competive) budget set
B(p, w) {x n : p x w}

effects:

Commodity l is normal goods if xl (p, w) / w 0 Commodity l is inferior goods if xl (p, w) / w 0 The demand is normal if xl (p, w) / w 0 l 1, n
2.The rule of choice
Choice

construct: a choice under certain
budget: (B , C(.))
Budget set: B X and B and B B ,B is a budget family, represented a limited list of all possible choice under certain system, nature, technology and other social environments that the decision-maker must take. Choice rule: C ( B) B,for B B , C (.) maybe contain more than one elements.
Proposition4:
if X is concave, then is B(p, w)
4.Demand function
any p and w are strictly positive, the corresponding demand set x(p,w) are nonempty, if x(p,w) is single point, we call it the Walrasian( Marshallian) demand function. x(p,w) is homogenous of degree zero and satisfied Walras’ law. That is px=w for all