Repetition of
General equilibrium analysis
Existence
Pareto efficient allocation (welfare theorems)
Lectures in Microeconomic Theory
Fall 2010, Part 22
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G.B. Asheim, ECON4230-35, #22
1
Pure exchange

All economic agents are consumers.
Consumption goods
Consumer 1
Consumer 2
Payments
Given the market prices and initial endowment of
consumption goods, consumers choose the best
vector of consumption goods, given that positive
net demand of some goods must be financed by
positive net supply of other goods.
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G.B. Asheim, ECON4230-35, #22
Agents and goods
Consumer i' s initial endowment : i  (i1 , i2 )
2
All illustrations and analysis
will be
made with two consumers and
two goods, but can be generalized
to n consumers and
d k goods.
d 22
1
  (1 ,  2 )
12

1
1
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G.B. Asheim, ECON4230-35, #22
Edgeworth
box
3
1
Feasible allocation
Consumer i' s consumptio n bundle : x i  ( xi1 , xi2 )
x12
x22
x2
Allocation :
x  (x1 , x 2 )
An allocation
is feasible if
x1
x12
  (1 ,  2 )
x11  x12   1
and
x  x22   2
2
1
x11
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G.B. Asheim, ECON4230-35, #22
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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
Even if consumers choose
consumption
mpti
bundles in their
budget sets, the
resulting allocation need not
be feasible.
G.B. Asheim, ECON4230-35, #22
5
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
h ld for
f each
h consumer i that
th t px i  p  i , andd
u i ( x i )  u i ( x i ) implies px i  p  i
( x i maximizes utility s.t. the budget constraint ).
Does a Walrasian equilibrium exist?
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2
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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G.B. Asheim, ECON4230-35, #22
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, #22
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 .

Implications
given
thatpx
both
prices are positive:
Proof. Follows
since
i  p  i holds for each i .
(1) If one market clears, then so does the other.
(2) In a Walrasian equilibrium it holds that z ( p )  0 .
Result. There exists p  such that z ( p  )  0 .
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3
Pareto efficiency
Edgeworth box
The initial endowment is
not Pareto-efficient,
since both consumers
can be made better off
by moving to x  (x1 , x 2 )
 21
22 x  ( x , x )
1
2
x  (x1 , x 2 )
  (1 ,  2 )
12
is Paretoefficient, since no consumer can be made better
off without making the
other worse off.
11
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Illustrating the first welfare theorem
First welfare theorem: Assume that
Any Walrasian equilibriumxis (x1 , x 2 )
Pareto-efficient. is a Walrasian
x2
x2
x1
p  ( p1 , p2 )
x1

equilibrium.
Then it is not
feasible to make
both consumers
better off.
Why?
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First welfare theorem
Result. If ( x 1 , x 2 , p ) is a Walrasian equilibr., then
it is not feasible to make both consumers better off
Proof. Suppose ( x 1 , x 2 , p ) is a Walrasian equilibr.,
and it is feasible to make both consumers better off.
Then there is a feasible allocation ( x 1 , x 2 ) such that
u i ( x i )  u i ( x i ) for both i . Hence,
 1   2  x 1  x 2 and p x i  p  i for both i .
This leads to a contradict ion :
p  1   2   p x 1  x 2   p x 1  p x 2  p  1  p  2 .
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4
Illustrating the second welfare theorem
Assume that
Second welfare theorem:
x  (x1 , x 2 )
Any Pareto-efficient allocation
is Pareto efficient.
can be implemented as a
If utility functions
Walrasian equilibrium.
x
are quasi-concave,
i
then x can be
implemented as
a Walrasian
equilibrium

How?
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G.B. Asheim, ECON4230-35, #22
Why is quasi-concavity assumed in
the second welfare theorem?
x  ( x1 , x 2 )
is still Pareto
efficient, but the
allocation
ll ti cannott be
b
implemented as a
Walrasian equilibrium because of
consumer 1’s nonconvex preferences.
x

Kinks — no problem.
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Second welfare theorem
Result. Suppose the assumption on the utility functions
holds. If the feasible allocation ( x 1 , x 2 ) is PE, then there
is a p such that ( x 1 , x 2 , p ) is a Walrasian equilibr.
x12  x22
Set of aggregate consumption vectors that
lead to a Pareto
improvement.
P
x1  x2
2
p  ( p1 , p2 )
x11  x12
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5
With production

Some economic agents are consumers, other agents
are firms.
Consumption goods
and payments
Consumers
Profits
Firms
Labor (and capital) and
Given the market
Given the market
prices, each consumer wages (and interest)
prices and the
chooses a best combination of
technological constraints,
labor supply and consumption
each firm chooses a
good demand, given that he
combination of consumption
must pay for the consumption
good supply and factor demand
that maximizes profits.
goods with his labor income.
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G.B. Asheim, ECON4230-35, #22
Robinson Crusoe economy
Consumpt.
( p, w)
Output
Profitmaximizing
output
UtilityUtilit
maximizing
consumption
Labor
input
Leisure
Profit in
terms of
output
Utility-maximizing leisure Profit-maximizing labor input
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G.B. Asheim, ECON4230-35, #22
Equilibrium with production
Consumpt.
( p  , w )
Profitmaximizing
output
UtilityUtilit
maximizing
consumption
Labor
input
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Output
Utility-maximizing leisure
Profit in
Leisure terms of
Profit-maximizing output
labor input
G.B. Asheim, ECON4230-35, #22
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6
x12  x22
Main existence result of
general equilibrium theory
P
x1  x2
p  ( p1 , p2 )
x11  x12

Result. (Arrow-Debreu-McKenzie, 1951–54)
If production sets are convex, and
utility functions are quasi-concave,
then an equilibrium exists.
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G.B. Asheim, ECON4230-35, #22
1st welfare theorem holds …
Consumpt.
Output
Labor
input
Leisure
… even if the production set is not convex.
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G.B. Asheim, ECON4230-35, #22
2nd welfare theorem need not hold …
Consumpt.
Output
Labor
input
Leisure
… if the production set is not convex.
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7