Advanced Economics
(lecture 5,consumption theory II)
Ye Jianliang
CONTENT
? WA and demand law
? From preferences to utility
? Utility maximization
? Expenditure minimization
1.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
1.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
1.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
1.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
2.From preferences to utility
? Definition,is a utility function of
preference,if
? Can we always find a utility function of?
– Maybe
– If X is finite,there always exist utility function,
? Proposition1,only rational can be
represented by a utility function,(N.C not S.C)
? Lexicographic preference,rational but no
utility function exist,
:u ??X
% ( ) ( ),,x y u x u y x y? ? ? ? X%
%
%
2.From preferences to utility
? Continuity,
then,or x upper contour sets
and lower contour sets are
closure,
? Proposition2:If is continuous,then exist
a continuous utility function representing,
1{,},,a n d l im,l imn n n n n nn nnx y x y x x y y? ? ? ? ? ?? ? ?%
xy% {, }y y x? X %
{, }y x y? X %
%
%
2.From preferences to utility
? Desirability,preference is desirable if
– is monotone,
– is local non-satiation,there is
a
? proposition3,is strong monotone,then
it’s monotone; is monotone,it’s local
non-satiation,
%
%,,a n d,t h e n x y y x y x? X?f
i f,t h e n,i t 's s t r o n g l y m o n o t o n ey x y x? f
%,a n d > 0x ??? X
,t h a t y - x,a n d y y x?? f
%
%
2.From preferences to utility
? Convexity,x upper contour sets are convex,
– Decreasing in marginal rate of substitution,
– people like variety,
? Preferences are convex means utility
function is quasi-concave,
2.From preferences to utility
? If,then are homothetic,
? Proposition4,The are homothetic if and
only if it’s utility function is HD1,
? If
then are quasi-linear of commodity 1(it’s
called standard commodity)
? Proposition5,are quasi-linear if and only
if it’s utility function is Gorman form
,0,x y x y? ? ???:,%
%
1 1 1,( 1,0,0 ),a n d,( ) ( )x y e x e y e? ? ?? ? ? ? ? ?,L,
%
12( ) (,)nu x x x x??? L
%
2.From preferences to utility
Preference Utility function
rationality exist
X finite
Desirability increasing
Continuity continuity
Convexity Quasi-concave
Homothetic HD1
Quasi-linear Gorman form
3.Utility maximization
? Consumer’s problem (UMP),
? The solutions are called Walras’
Demand Correspondence,or function if it’s
single point,
0
m a x ( )
,, (, )
u
s t B w
?
?
x
x
x p
(, )wxp
3.Utility maximization
? Properties of
– HD0
– Satisfied Walras’ Law
– If are concave,are concave too,if
are strictly concave,is single point,
(, )wxp
(, )wxp
(, )wxp
%
%
3.Utility maximization
? If the solution,then
? Here is the shadow price,the marginal
utility of optimal consumption,
? The value function is called,indirect utility
function”,
(, ) 0w?xp? ()u ????xp
(, )w?xp
?
1
( ) ( ) (,)
(,)
ww
w
E n g e l a g g r e g a t i o n
u u D w
Dw??
??
?
? ? ? ?
? ? ? ?
x x x p
p x p1 44 2 4 43
(, ) ( (, ) )v w u w??p x p
4.Expenditure minimization
? EMP,
? The solution was called,Hicks (or
compensation ) demand correspondence”,
single point----function,
– It’s HD0 of p,
– It’s convex,as is, and single point when
are strictly convex,
0
m i n
,, ( )s t u u
?
?
?
x
px
x
(,)hup
% %
4.Expenditure minimization
? The value function:,expenditure function”
(money metric untility)
? So,when price changing,
if the wealth of consumer change
correspondently to keep the same utility
level,then is the demand changing,
Here Hicks wealth compensation is
Recall the Slutsky wealth compensation
(,) (,)e u h u??p p p
(,) (,(,) )h u e u?p x p p
(,)hup
(,)H i c k sw e u w?? ? ?p
5.Relationship
? Proposition6 (Duality),u(x) is a
continuous utility function of,
– w >0,x* is the solution of UMP,when it’s
required that,x* is the solution of
EMP,and value of EMP is w,
– If the required utility,x* is the solution
of EMP,when the expenditure is p·x*,x* is the
solution of UMP,and value of UMP is u,
( ) ( )uu ??xx
(0 )uu?
%
5.Relationship
? Hicks demand and expenditure function,
– is s.n.s.d,
? Hicks and Walras demand,
– Slutsky equation,
? Walras demand and indirect utility function,
– Roy’s identity,
(,) (,)h u e u?? pp p
2(,) (,)ppD h u D e u?p p
(,) (,) (,) (,) Tp p wD h u D w D w w? ? ?p x p x p x p
1(,) (,)
(,)
T
p
w
w v wvw? ? ???x p p p
由于 e是凹的 in p
5.Relationship
UMP EMP
x(p,w)
v(p,w)
h(p,u)
e(p,u)
duality
Slutsky equation
Roy’s
identity (,)
(,)
hu
eu
?
?p
p
p
(,(,))e p v p w w?
(,(,))v p e p u u?
(,) (,(,) )x p w h p v p w? (,) (,(,) )h p v x p e p u?
Assignment
? ex.7.4,ex.8.7,ex.8.14
WA and demand law
x2
x1
x2
x1
x2
x1
WA and demand law
x2
x1
x2
x1
Slutsky compensation
Substitution effects income effects
Slutsky and Hicks compensation
u(x)=u
(, ) (,) )w h u?x p p
(, )w?xp
(,) (, )
(, (,) )H i c k s
h u w w
eu
??? ? ?
???
p x p
x p p
(, )S lu tskyww? ??xp
Substitution effects income effects