Advanced Microeconomics
lecture 6:consumption theory III
Ye Jianliang
Utility maximization
? Content,
– Integrability
– Aggregation across goods
1.Integrability
? Demand function x(p,w) (c.d.) is HD0,
satisfied Walras Law and have a
substitution matrix S(p,w) is s.n.s.d,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,
1.Integrability
? 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,
( ) {, (,) 0 }nV u e u?? ? ? ? ? ?x p x p p
(,) m i n {, ( ) }e u V u? ? ?p p x x
1.Integrability
? Demand ? expenditure function,
– Partial differential equation,
– The existence of solution means substitution
matrix is symmetric,
0 0 0
(,( ) ) (,)
()
e e u
ew
???
? ?? p
x p p p
p
2 ( ) (,( ) ) (,( ) ) (,( ) )
(,( ) )
T
p p wD e D e D e e
Se
? ? ?
?
p x p p x p p x p p
pp
2.Aggregation 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)
2.Aggregation across goods
? UMP,
? Let,and set,
? Solution,how can we
get it,
m a x ( )
,,
u
s t w??
x,z
p x q z
( ),( )p f x g??px
m a x ( )
,,
x
Ux
s t p x w??
,z
,z
qz
(,,) ( (,,) )x p w g w?? ?q x p q
2.Aggregation 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,
()pf? p
px w??qz
()xg? x
w??p x q z
2.Aggregation across goods
? Hicksian separability,no relative price
change,so,let,
? Define the indirect utility function,
? As Roy’s identity show,
0t??pp 0,p t x?? px
0
(,,) m a x ( )
,,
v p w u
s t p w
?
??
x,z
q x,z
p x qz
0
0
0
(,,) /(,,) /(,,) (,,)
(,,) / (,,) /
iv p w pv p w px p w w
v p w w v p w w
????? ? ? ? ?
? ? ? ?
? pqqq p x p q
q p q
2.Aggregation across goods
? Construction direct utility function has the
property that,
? Maximize U means maximize u,
(,,) m a x ( )
,,
x
v p w U x
s t p x w
?
??
,z
q,z
qz
2.Aggregation across goods
? Functional separability,
– Independence property
– Weakly separable utility function,
? Then the problem is
( ) ( ) ( ) ( )? ? ? ??x,z x,z x,z x,z%%
( ) ( ( ),)u U v?x,z x z
m a x ( )
,,
x
x
v
s t p w
?
?
,z
xx
x
2.Aggregation across goods
? Let be the expenditure function,then
the problem is,
? If is liner as (v is homothetic),let
and,then
(,)e p v
()
m a x ( ( ),)
,, (,)
vx
Uv
s t e p v w??
,z
xz
qz
(,)e p v ()e p v
()P e p? ()Xv? x
m a x (,)
,,
X
UX
s t P X w??
,z
z
qz
2.Aggregation across goods
? we get the same solution as problem,
? Local analysis
m a x (,)
,,
x
u
s t w??
,z
xz
p x q z
Assignment
? Text book:Ex.9.10;Ex.9.11;Ex.10.1
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
expenditure function?
preference,
x1
x2 x2
x1