Lecture 13,market as a process
General Equilibrium theory II
Content
? The,core”
? Uniqueness of equilibrium
? Stability of equilibrium
? Welfare
The,core”
? Improve upon an allocation,a group of
agents S is said to improve upon a given
allocation x,if there is some allocation x’
such that,
and
? If an allocation can be improved upon,
then there is some group of agents can do
better without market!
iii S i S w??? ???x f o r a l l i i i iS? ?xx
The,core”
? Core of an economy,a feasible allocation
x is in the core of the economy if it cannot
be improved upon by any coalition,
? If x is in the core,x must be Pareto
efficient,
See the fig,
The,core”
? Walrasian equilibrium is in core,
– Proof,let (x,p) be the Walrasian equilibrium
with initial endowment wi,
– If not,there is some coalition S and some
feasible allocation x’,such that all agents i in
S strictly prefer to,and
– But Walrasian equilibrium implies
i?x i?x
iii S i S w??? ???x
f o r i i ii S i S ww????????p x p p x p
The,core”
? Equal treatment in the core,if x is an
allocation in the r-core of a given economy,
then any two agents of the same type must
receive the same bundle,
Proof,if not,Let,
– So
– That is
– Every agent below the average will coalize to
improve upon the allocation,
1 1rAAr jxx?? ? 1
1
r
BBr jxx?? ?
1 1 1 11 1 1 1r r r rA B A j B jr r r rj j j jx x w x? ? ? ?? ? ?? ? ? ?
A B A Bx x w w? ? ?
The,core”
? Shrinking core,there is a unique market
equilibrium x* from initial endowment w,if
y is not the equilibrium,there is some
replication r,such that y is not in the r-core,
Proof,since y is not the equilibrium,there is
another allocation g improve upon A(or B) at
least,That means see the fig,
– Let (T and V are integers)
( 1 ) f o r s o m e 0AAg w y? ? ?? ? ? ?
/TV? ?
The,core”
? Replicated V times of the economy,we
have,
? So the coalition with V agents of type A
and (V-T) of type B can improve upon y,
()
[ ( 1 ) ] ( )
( ) [ ]
( ) [ ]
()
AB
A A B
A A B
A A B
AB
V g V T y
TT
V w y V T y
VV
Tw V T y y
Tw V T w w
V w V T w
??
? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ?
The,core”
? Convexity and size,
? If agent has non-convex preference,is
there still a equilibrium? See the fig,
? Replication the economy
Uniqueness of equilibrium
? Gross substitutes,two goods i and j are
gross substitutes at price p,if,
? Proposition,If all goods are gross
substitutes at all price,then if p* is an
equilibrium price,then it’s the unique
equilibrium price,
() 0 f o r j
i
z ij
p
? ??
?
p
Stability of equilibrium
? Price adjustment rule:,Gi is
some smooth sign-preserving function,
? Consider a special case,
? Then we have,
See the fig,
( ( ) )i i ip G z? p
()z?pp2( ( ) ) 2 ( ) ( )
2 ( ) ( ( ) )
2 ( )
0
i i i
i i i
d
p t p t p t
dt
p t z p t
z
?
?
?
?
??
?
pp
Stability of equilibrium
? WARP implies stability,if,and if p*
is an equilibrium,then all path of price
following WARP will converge to p*,
? Proof,Liaponov function,
()z?pp
2( ) [ ( ) ]
iiV p p p
????
()
2 ( ) ( ) 2 ( ) ( )
2 ( ( ) ( ) ) 0 ( ) 0
i i i i i i
i i i i i
d V p
p p p t p p z p
dt
p z p p z p p z p
??
??
? ? ? ?
? ? ? ? ?
??
?
Welfare
? Pareto criterion,
if and only if
? Compensation criterion (Kaldor ~),
if there is such that
and
?xx%
i?xx%
?xx% i
i? ?????xx
??x
i??xx%
Assignment
? Textbook,ex.21.2
Core of an economy
Pareto
set
Core
Endowment
Shrinking core
Pareto
set
y
Endowment
x
g
non-convex preference
Price adjustment