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Advanced Economics
(lecture 5,consumption theory I)
Ye Jianliang
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Preference and Choice
? Content,
? Preference and rationality
? The rule of choice
? Budget
? Demand function
? WA and demanded law
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1.Preference
? An (acceptable) alternatives set (or a
consumption set )
? A (weak) preference relation,”,a binary
relationship on X,for any,means”
x is at least as good as y”,
? A strict preference,,,
? An indifference relation,,,
k???X
%
,xy? X xy%
,b u t n o t x y x y y x? %%
,a n d x y x y y x? %%
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1.Preference
? Definition,preference on X is rational if,
? It’s complete,
? It’s reflexive,
? It’s transitive,
%
,,e i t h e r o r,o r b o t hx y x y y x? X %%
,x x x?? X %
,,,i f a n d,t h e n x y z x y y z x z? X % % %
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1.Preference
? Propersition1,if is rational,then,
? is irreflexive and transitive
? is reflexive and transitive and symmetric
? If
%
,t h e n x y z x z%
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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?
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2.The rule of choice
? Choice construct,a choice under certain
budget,
? Budget set,and,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:,
maybe contain more than one elements,
(,(.))CB
B ?B a nd BB? ? ?X B
( ),f o r C B B B?? B(.)C ??
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2.The rule of choice
? Weak Axiom of Revealed Preference
(WARP or WA), if for certain,
and for and,
then must
?Revealed Preference,,,
B ?B
,x y B?
()x C B?
B??B ()y C B ??,x y B??
()x C B ??
?%
,,,a n d ( )x y B x y B x C B? ? ? ? ? ? ?B%
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2.The rule of choice
? Choice set
? Proposition2,if are rational,then choice
construct 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?
(,(,,) )C ? %B
%
(,) {,,}C B x B x y y B? ? ? ? ?%%
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2.The rule of choice
?Definition,for,we say,rational
preference rationalization
? Propersition3,if satisfied WA
and contain all subsets of X that is
ternary and less than ternary,
? 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,
(,(,,) )C ? %B
% (.)C
(,(,,) )C ? %B
B
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3.Competive budget
? Consumption set X and it’s price
with the wealth w>0,
? Definition,Walras (competive) budget set
? Proposition4,if X is concave,then is
n??p
(,) {, }nB w w?? ? ? ? ?p x p x
(,)Bwp
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4.Demand function
? For 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
(we will prove them next lecture)
(,) a n d 0,0w p w??x x p
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4.Demand function
? Wealth effects,
? Engel function,given price,function over
wealth
? Wealth expansion path,
? Wealth effects,
p
(,)wxp
{ (,), 0 }E w w??p xp
1
2
(,)
(,)
(,)
(,)
n
xw
w
xw
w
w
xw
w
Dw
?
?
?
?
?
?
??
??
??
?
??
??
????
p
p
p
xp
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4.Demand function
? Wealth effects,
? Commodity l is normal goods if
? Commodity l is inferior goods if
? The demand is normal if
See the fig
(,) / 0lx w w? ? ?p
(,) / 0lx w w? ? ?p
(,) / 0 1,lx w w l n? ? ? ? ?p
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4.Demand function
? Price effects,
? Supply curve,
? Price effects,
? Giffen good,
See the fig,
{ (,), }l l lw ????x x p p p
11
1
1
(,) (,)
(,) (,)
(,)
n
nn
n
x w x w
pp
p
x w x w
pp
Dw
??
??
??
??
??
??
?
??
??
xp
(,) / 0llx w p? ? ?p
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4.Demand function
? Cournot Aggregation,if Walrasian demand
function x(p,w) is HD0 and satisfied
Walras Law,then,
? Engel Aggregation,if Walrasian demand
function x(p,w) satisfied Walras Law,then,
(,) (,) 0TTpD w w??p x p x p
(,) 1wDw ?p x p 其实就是对 p*x=w的求
导数,
第一个是对 p求导
第二个是对 W求导
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5.WA and demand law
? Walrasian demand function x(p,w) satisfied
WA if for any we have,
See the fig,
(,) a n d (,)ww ??pp
(,),i f (,) a n d (,) (,)w w w w w w? ? ? ? ? ? ?? ? ? ? ?p x p p x p x p x p
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5.WA and demand law
? Changing in price will change wealth too,
But how can we tell the demand changing
by price changing from wealth changing?
? Given a changing from,and
people will not get worse that is
here wealth changing (compensation )
was called,Slutsky wealth
compensation” and,(Slutsky)
compensated price changing”,
(,) t o (,)ww ??pp
(,) ww????p x p
(,) ww? ? ? ?p x p
?? ? ?p p p
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5.WA and demand law
? Proposition5,x(p,w) satisfied WA if and
only if,and when
,
?Prop.5 indicates,,or that’s
called,demand law,,or,compensation
demand law,,
( ) [ (,) (,) ] 0ww? ? ?? ? ? ?p p x p x p
(,) (,)ww?? ?x p x p ( ) [ (,) (,) ] 0ww? ? ?? ? ? ?p p x p x p
0? ?? ?px 0dd??px
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5.WA and demand law
? Slutsky matrix (substitution matrix)
? Substitution effects
? is n.s.d
? Giffen good is necessary inferior good,
(,) ( (,) )l k n nS w s w ??pp
(,) (,)(,) (,)ll
l k k
k
x w x ws w x w
pw
????
??
pppp
(,)Swp
(,) (,)(,) (,) 0ll
l l l
l
x w x ws w x w
pw
??? ? ?
??
pppp
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Assignment
? If
Then
? For
then
? If Laspeyres quantity index is less than 1,
then,and if Paasche quantity
index is large than1,then
,,,,,,a n d ( ),( )B B x y B x y B x C B y C B? ? ?? ? ? ? ?B
{,} ( ),{,} ( )x y C B x y C B ???
,,a n d,( ) ( )B B B B C B C B? ? ?? ? ?B B B
( ) ( ) ( ( ) ( ) )C B C B C C B C B???
01xx?%
10xx?%
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Consumption set
x2
x1
nX ???
x2
x1
1
2
3
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Consumption set
x1
X
Leisure
24
8
x1
Leisure
X
24
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Consumption set
rice
X4
bread 4
Rice of Hangzhou
X
Bread of Shanghai
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expansion path
1(,)wxp
1(,)Bwp
2(,)Bwp
2(,)wxp
3(,)Bwp
3(,)wxp
Ep
x2
x1
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Wealth effects
Ep
x2
x1
Ep
x2
x1
Ep
x2
x1
luxury goods
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Supply curve,
2/wp
12(,,)p p wx
x2
x1
1/wp
2/wp?
12(,,)p p w?x
2/wp??
12(,,)p p w??x
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Giffen good
2/wp
12(,,)p p wx
x2
x1
1/wp
2/wp?
12(,,)p p w?x
2/wp??
12(,,)p p w??x
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5.WA and demand law
x2
x1
x2
x1
x2
x1
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5.WA and demand law
x2
x1
x2
x1
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