General equilibrium analysis:
Equilibrium in pure exchange
Lectures in Microeconomic Theory
Fall 2010, Part 11
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G.B. Asheim, ECON4230-35, #11
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Agents and goods
Consumer i' s initial endowment :  i  (i1 , i2 )
2
All illustrations and analysis
y will be
made with two consumers and
two goods, but can be generalized
to n consumers and k goods. 22
1
  (1 ,  2 )
12
11
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G.B. Asheim, ECON4230-35, #11
Edgeworth
box
2
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Feasible allocation
Consumer i' s consumptio n bundle : x i  ( xi1 , xi2 )
x12
Allocation :
x22
x2
x  ( x1 , x 2 )
An allocation
is feasible if
x1
x12
  (1 ,  2 )
x11  x12   1
and
x12  x22   2
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1
x
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G.B. Asheim, ECON4230-35, #11
Budget sets
If consumers take market prices, p  ( p1 , p2 ),
as given, their budget sets will be as follows :
x2
p  ( p1 , p2 )
x1
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G.B. Asheim, ECON4230-35, #11
Even if consumers choose
consumption
bundles in their
g sets, the
budget
resulting allocation need not
be feasible.
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Walrasian equilibrium
Definition : A pair of an allocation , x   ( x 1 , x 2 ) ,
and a price vector, p  ( p1 , p 2 ) , satisfying
(1) The allocation , x   ( x 1 , x 2 ) , is feasible.
(2) It holds for each consumer i that px i  p  i , and
u i ( x i )  u i ( x i ) implies px i  p  i
( x i maximizes
i i
utility
ili s.t. the
h budget
b d
constraint
i ).
)
Does a Walrasian equilibrium exist?
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G.B. Asheim, ECON4230-35, #11
Price vectors leading to excess demand
Low price of good 1 …
High price of good 1 …


… leads to excess demand.
… leads to excess supply.
Is there an equilibrium price?
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Assumption on the utility functions
x2
Assumption : For each i , u i is
(1) continuous ,
x1
(2) strictly quasi - concave, and
(3) monotone (i.e., x i  x i implies u i ( x i )  u i ( x i) ).
Assume that x i maximizes u i ( x i ) s.t. px i  m .
Part (3) implies px i  m .
Parts (2) and (3) implies that p  0 .
Parts (1) - (3) imply x i  x i ( p , m ) , where x i is point valued and continuous in positive prices and income.
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G.B. Asheim, ECON4230-35, #11
Existence of a Walrasian equilibrium
Suppose the Assumption on the utility functions holds.
Define the aggregate excess demand function:
z ( p )  x 1 ( p , p  1    1   x 2 ( p , p  2 )   2 
p is an equilibr. price vector if z (p )  0 .
Why?
Walras' law. For all p  0 , it holds that pz ( p )  0 .
Proof. Follows since px i  p  i holds for each i .
Implications given that both prices are positive:
(1) If one market clears, then so does the other.
(2) In a Walrasian equilibrium it holds that z ( p )  0 .
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Existence of a Walrasian equilibrium (cont.)
Result. There exists p  such that z ( p  )  0 .
Proof. Normalize prices so that p1  p 2  1 . Define
p i  max(( 0 , z i ( p1 , p 2 ))
g i ( p1 , p 2 ) 
1  max( 0 , z1 ( p1 , p 2p))
2  max( 0 , z 2 ( p1 , p 2 ))
Note that g 1 ( p1 , p 2 )  g 2 ( p1 , p 2 )  1 .
1
For p1 near 0 , we have that p1  g 1 ( p1 ,1  p1 )  0 .
For p1 near 1, we have that p1  g 1 ( p1 ,1  p1 )  0 .
Since p1  g 1 ( p1 ,1  p1 ) is continuous , it follows
that there exists p  such that p1  g 1 ( p1 ,1  p1 )p  0 .
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By Walras' law, this implies z1 ( p  )  z 2 ( p  )  0 .
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