Lecture 12,Exchange
General Equilibrium theory I
Content
? Pure exchange system
? The existence of Walrasian equilibrium
? The first theorem of welfare economics
? The second theorem of welfare economics
? Exchange with production,
Pure exchange system
? An allocation A= (x1,…xI) is feasible if
? is the endowments of commodity l
? Edgeworth box,2× 2
1
,f o r a n y 1,
I
l i l
i
x l L?
?
???
l?
Pure exchange system
? Walrasian equilibrium,
– An allocation and price are
a Walrasian equilibrium,if,
? Utility maximization
? Feasible see the fig,
1(,)Ixx?? L? ??p
m a x ( )
,,
i
i i ixX
ii
x u x
s t x ?
?
?
??
?
? p p
1
I
li l
i
x ??
?
??
The existence of Walrasian
equilibrium
? Given an endowment w,is there any
walrasian equilibria?
? Assumption1:The demand function is HD0
of price,
The aggregated extra-demands
? Assumption2,Walras’ law,
1
( ) [ (,) ]I i i i
i
z p x p p w w
?
???
( ) 0p z p ?
The existence of Walrasian
equilibrium
? Assumption3,market clearing
? Desirability,
? lemma1,free goods,if p* is a walrasian
equilibrium price,and,then
? lemma2,Equivalent of demand and
supply,all the commodities are desirable
and p* is a walrasian equilibrium price,
then
( ) 0 f o r 0iz p p??
( ) 0jzp ? ? 0
jp? ?
( ) 0jzp ? ?
The existence of Walrasian
equilibrium
? Brouwer’s fixed-point theorem,continuous
function,( is a unit simplex)
and there exist an x,that,
? Scarf,1973,
? when n=1;let g(x)=f(x)-x,
,nnf s s?
()f?xx
( 0 ) ( 0 ) 0 0 ; ( 1 ) ( 1 ) 1 0g f g f? ? ? ? ? ?
,( ) 0 ( )x g x f x x? ? ? ?
ns
The existence of Walrasian
equilibrium
? If is a continuous function witch
satisfied Walras’ law,,then there
exist some in such that,
– Construct a unit simplex,
– Defined
1,kkzs ? ??
( ) 0z ?pp
?p 1ks ? ( ) 0z ? ?p
1
i
i K
jj
pp
p?
? ?
1
1{ in, 1
kkk
iisp
?
?? ? ??p
11,kkg s s???
1
m a x( 0,( ) )()
1 m a x( 0,( )
ii
i K
jj
pzg
z?
??
? ?
pp
p
The existence of Walrasian
equilibrium
– By brouwer’s fixed-point theorem,there is a
p*,such that
– So
()g???pp
1
m a x ( 0,( ) )()
1 m a x ( 0,( )
ii
ii K
jj
pzp g p
z
??
??
?
?
???
? ?
p
p
1[ m a x (0,( ) ] m a x (0,( ) ) f o r 1
K
i j ijp z z i k? ? ?? ??? pp
The existence of Walrasian
equilibrium
– And
– Sum up with i
– So,we have,
? Example,C-D economy
1( ) [ m a x (0,( ) ] ( ) m a x (0,( ) )
K
i i j i ijz p z z z? ? ? ? ?? ??p p p p
1 11[ m a x (0,( ) ] ( ) ( ) m a x (0,( ) )
kkK
j i i i ij
ii
z p z z z? ? ? ? ??
??
?? ? ?p p p p
11
( ) m a x (0,( ) ) = 0 f o r ( ) 0
kk
i i i i
ii
z z p z? ? ? ?
??
???p p p
( ) 0iz ? ?p
The first theorem of welfare
economics
? Pareto efficient allocations,
? The solution is Pareto sets and also
called contract curve,See the fig,
11
2 2 2
1 2 1 2
m a x ( )
., ( )
u
s t u u
ww
?
? ? ?
12x,x
x
x
xx
12(,)??xx
The first theorem of welfare
economics
? If is a Walrasian equilibrium,then x is
Pareto efficient,
()x,p
The second theorem of welfare
economics
? If x* is a Pareto efficient allocation
and,suppose that preference are
convex,continuous and monotonic,then
x* is a Walrasian equilibrium for the initial
endowment for i=1……n,
? Proof1,upper counter set,
0? ??x
=iiw ?x
{:} ki i i i iU ?? ? ?x x x
{, }i i i iU U z z a n d U? ? ? ??? xx
The second theorem of welfare
economics
? U is convex,
? Let,is Pareto efficient,so
? Separating hyper-plane theorem,there
exists a such that,
? Nonnegative
? If,then
iw ??? x i?x i U??x
0?p izx ?? ?pp
0?p
j j j?yx
jjyx??pp
The second theorem of welfare
economics
? Proof2,if there exist a Walras’ equilibrium
(p’,x’),for the endowment,then
? Because is Pareto efficient,means
so if is the optima,so is,
iiw ?? x i i i
??xx%
i i i??xxi?x
i?x i?x
Exchange with production
? An allocation A= (x1,…xI;y1……yJ) is a
combine of consumption vector x and
production vector y,A is feasible if
See the fig,
11
,f o r a n y 1,
IJ
l i l l j
ij
x y l L?
??
? ? ???
IJL??
Exchange with production
? Pareto Optimal ( Pareto efficient ),
? An allocation is Pareto
efficient ( optimal ) if there isn’t any the
other feasible allocation,
made for any i and for
some i,
11(,;,)IJx x y y? ? ? ?
11(,;,)IJx x y y
( ) ( )iiuu? ?xx ( ) ( )iiuu? ?xx
Exchange with production
? Walrasian equilibrium,
– An allocation and price
are a Walrasian equilibrium,if,
? Profit maximization
? Utility maximization
? Market clearing
11(,;,)IJx x y y? ? ? ? L? ??p
m a x
jjjyY
y p y j???? ? ?
1
m a x ( ),,
i
J
i i i i i i j jxX
j
x u x i s t p x p p y??? ? ? ? ?
? ?
? ? ? ? ? ??
11
IJ
l i l l j
ij
xy???
??
????
Exchange with production
? Existence of the equilibrium
– Preferences are convex,continuous,
monotone,
– Technologies are cloused,convex,no free-
lunch and free disposal,free entrance,
? (Debrue 1959)
Exchange with production
? The first theorem of welfare economics,
? The second theorem of welfare economics
Edgeworth box
Commodity 1
0
0
Co
mmo
dit
y 2
B’s
co
mm
od
ity
2
A’s
co
mm
od
ity
2
A’s commodity 1 B’s commodity 1
endowment
B
A
Walrasian equilibrium
B
A
endowment
A’s
Brouwer’s fixed-point theorem
Pareto sets
B
A
Exchange with production
Assignment
? Ex.17.4