Advanced Microeconomics
lecture 7:consumption theory IV
Aggregation and Welfare
? Content,
– Consumer’s surplus
– Aggregation
– Representation consumer,
1.Consumer’s surplus
? Money metric utility function measure
the welfare of consumer in economy
changing,
? Definition,equivalent variation (EV) and
compensating Variation (CV),see the fig,
Let
(,)eup
0 1 0 1 0 0 0 1(,,) (,) (,) (,)E V w e u e u e u w? ? ? ?p p p p p
0 1 1 1 1 0 1 0(,,) (,) (,) (,)C V w e u e u w e u? ? ? ?p p p p p
0 0 1 1 0 0 1 1(,),(,),(,) (,)u v w u v w e u e u w? ? ? ?p p p p
1.Consumer’s surplus
? Consider a changing only accurse in price
of commodity 1,
0
1
1
1
0 1 0 1
0 1 1 1
1
1 1 1 1
(,,) (,)
(,) (,)
(,,)
p
p
E V w e u w
e u e u
h p u d p
?
??
??
? ?
p p p
pp
p
01
11
0 1 0
1 1 1 1(,,) (,,)
p
pC V w h p u d p?? ?p p p
01
11
0 1 0
1 1 1 1(,,) (,,)
p
pA V w x p u d p?? ?p p p
1.Consumer’s surplus
p1
x1
01 1 1(,,)h p u?p
1 1 1(,,)x p w?p
11 1 1(,,)h p u?p
11(,)x p w01(,)x p w
11p
01p
1.Consumer’s surplus
? Commodity taxation,
? The loss of the taxation,
1,
2,
1 0 11 1 1,(,)p p t T t x p w? ? ? ?
0
1
0
1
1 0 1 1 1 0 1
1 0 1
1 1 1 1 1 1 1
(,,) (,) (,)
[ (,,) (,,) ]
pt
p
L E V w T e u e u T
h p u h p t u d p
?
??
? ? ? ? ? ?
? ? ??
p p p p
pp
0
1
0
1
2 0 1 1 0 0 0
0 0 0
1 1 1 1 1 1 1
(,,) (,) (,)
[ (,,) (,,) ]
pt
p
L CV w T e u e u T
h p u h p t u d p
?
??
? ? ? ? ? ?
? ? ??
p p p p
pp
1.Consumer’s surplus
p1
x1
01 1 1(,,)h p u?p
1 1 1(,,)x p w?p
11 1 1(,,)h p u?p
01p
01pt?
011 1 1(,,)h p t u?? p 001 1 1(,,)h p t u?? p
1.Consumer’s surplus
x2
x(p1,w)
x(p0,w)
u1
u0
L1
x1
L2
2.Aggregation across consumers
? Can a society behave like a ration man?
? If a society behave like a ration man,can
the utility of this,man” represented the
welfare of society?
? Aggregate demand,
? Wealth distribution
? And total wealth
1
1
(,) (,)
I
I i i
i
w w x w
?
? ?x p pL
1()Iww?w L
1
I
i
i
ww
?
? ?
2.Aggregation across consumers
? Step1:,that means
aggregate demand are independent on the
distribution of the wealth,
? Proposition1:if and only if each consumer
has a indirect utility function of Gorman’s
aggregate demands are independent on
the distribution of the wealth,
1(,) (,)Iw w w?x p x pL
(,) ( ) ( )i i iv w a p b p w??p
2.Aggregation across consumers
? Step2,how can the AD satisfied WA?
? Definition,ULD (un-compensation law of
demand),
? Proposition2,satisfied ULD,then AD
satisfied WA,
? Proposition3,consumers have the same
preference,and their wealth non-increasing
distribution on,then AD satisfied ULD,
( ) [ (,) (,) ] 0,,i i i i ip p p w p w p p w? ? ?? ? ? ?xx
(,)iipwx
[0,]w
0( ) (,) ( )
wp p w g w d w???xx %
3.Representation consumer
? We can regard AD as a rational consumer,
? When can we measure welfare of society
by the preference of this consumer?
? Definition,PRC (positive representative
consumer),there is a rational educing
AD,then exist a PRC,
%
3.Representation consumer
? Definition,Bergson-Samuelson social
welfare function,
? A merciful government distribute the
wealth to maximize W,
(*)
,I? ? ?W
1
11
I
i= 1
m a x ( (,) (,) )
,,
I
ii
ww
i
v p w v p w
s t w w??
W
L
L
3.Representation consumer
? Porposition4,if w is the solution of (*),
then the value function of (*) is the indirect
utility function of PRC for AD,
? Definition,NRC (normative ~ ),if w is the
solution of (*),and the value function of (*)
is the indirect utility function of, The
PRC of W(.) is NRC for AD,which
%
? ? ? ?? ?
1
,,,I ii
i
w w w
?
? ?x p x p p
3.Representation consumer
? The existence of PRC does not necessary
means existing NRC,
? A,as good at least set” of RC,
? And Scitovsky counter set,
{, ( ) ( ) }nB u u?? ? ? ?x x x
{, }i i iiAi? ? ??x x x x%
Representation consumer
x A
B
x B
A ?x
Assignment
? Text boook:ex.10.2
? PPT,Proposition2,
EV & CV
x1
x2
x(p1,w)
x(p0,w)
u1
u0
x1
x2
x(p1,w)
x(p0,w)
u1
u0
EV
CV
3.Aggregation across consumers
1 2(,)wp?x 1 2(,)wpx
2 2(,)wpx
2 2(,)wp?x
122(,)wp?x
122(,)wpx