General equilibrium analysis:
Welfare results in pure exchange
Lectures in Microeconomic Theory
Fall 2010, Part 12
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G.B. Asheim, ECON4230-35, #12
Pareto efficiency
Edgeworth box
The initial endowment is
not Pareto-efficient,
since both consumers
can be
b made
d b
better off
ff
by moving to x  (x1 , x 2 )
21
x  (x1 , x 2 )
22 x  ( x , x )
1
2
  (1 ,  2 )
12
1
is Paretoefficient, since no consumer can be made better
off without making the
other worse off.
11
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1
Illustrating the first welfare theorem
First welfare theorem: Assume that
Any Walrasian equilibriumxis (x1 , x 2 )
Pareto-efficient. is a Walrasian
x2
x2
x1
p  ( p1 , p2 )
x1

equilibrium.
Then it is not
feasible to make
b h consumers
both
better off.
Why?
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G.B. Asheim, ECON4230-35, #12
Illustrating the second welfare theorem
Assume that
Second welfare theorem:
x  (x1 , x 2 )
Any Pareto-efficient
Pareto efficient allocation
is Pareto efficient.
can be implemented as a
If utility functions
Walrasian equilibrium.
x

are quasi-concave,
then x can be
implemented as
a Walrasian
equilibrium
How?
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2
Why is quasi-concavity assumed in
the second welfare theorem?
x  (x1 , x 2 )
is still Pareto
efficient, but the
allocation cannot be
implemented as a
Walrasian equilibrium because of
consumer 1’s nonconvex preferences.
x

Kinks — no problem.
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G.B. Asheim, ECON4230-35, #12
Weak and strong Pareto efficiency
A feasible allocation ( x 1 , x 2 )
is weakly Pareto - efficient
u2
if there is no other feasible
allocation ( x 1 , x 2 ) such that
u i ( x i )  u i ( x i ) for both i .
A feasible allocation ( x 1 , x 2 )
is strongly Pareto - efficient
(u(1u(1x(1x),1 ),
u 2u(2x(2x))2 ))
(u1 (x1 ), u 2 (x 2 ))
if there is no other feasible
u1
allocation ( x 1 , x 2 ) such that
u i ( x i )  u i ( x i ) for both i , and u i  ( x i  )  u i  ( x i  ) for one i .
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3
Equivalence of weak and strong PE
Assumption : For each i , u i is (1) continuous
(2) strictly quasi - concave, and
((3)) monotone ((i.e.,, x   x  implies
p
u i ( x )  u i ( x ) )).
Result. Under the assumption on the utility functions,
an allocation is weakly Pareto efficient if and only if
it is strongly Pareto efficient.
Proof.
If ( x 1 , x 2 ) is strongly PE, then ( x 1 , x 2 ) is weakly PE.
If ( x 1 , x 2 ) is not strongly PE, then there is ( x 1 , x 2 ) s.t.
u i ( x i )  u i ( x i ) for both i , and u i  ( x i  )  u i  ( x i  ) for one i .
By transferri ng from i , both can be made better off.
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First welfare theorem
Result. If ( x 1 , x 2 , p ) is a Walrasian equilibr.,
then ( x 1 , x 2 ) is weakly Pareto efficient.
Proof. Suppose ( x 1 , x 2 , p ) is a Walrasian equilibr.,
where ( x 1 , x 2 ) is not weakly Pareto efficient.
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
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
p  ( p1 , p2 )
x11  x12
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G.B. Asheim, ECON4230-35, #12
Second welfare theorem — sketch of proof
The set P is convex since the utility
functions are quasi-concave.
x12  x22
The monotonicity of utility functions
combined with strict quasi-concavity
leads to the existence of p  0 such
that px  p x 1  x 2  for all x  P .
The continuity of utility functions leads
to px  p x 1  x 2  for all x  P .
P
x1  x2
p  ( p1 , p2 )
x11  x12
Suppose ( x 1 , x 2 , p ) is not a Walrasian equilibrium with ( x 1 , x 2 )
as initial endowments. Then there exists i  and x i such that
u i  ( x i  )  u i  ( x i  ) and px i   px i  .




Contradiction: p x i   x i   p x i   x i  and x i   x i  P .
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5
Welfare maximization

Pareto efficiency does not yield a specific allocation.
This can be resolved by assuming the existence of a
social welfare function, W ( u1 , u 2 ) , that is
increasing in each of its arguments.
Result. If ( x 1 , x 2 ) maximizes a social welfare
function, then ( x 1 , x 2 ) is strongly Pareto efficient.
P f If it
Proof.
i would
ld have
h
b
been
possible
ibl to increase
i
ui
without decreasing the utility of the other consumer,
then it would have been possible to increase W ( u1 , u 2 ).
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Supporting a strongly PE allocation

u2
(u1 (x1 ), u 2 (x2 ))
U
(a1 , a2 )

u1
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Question: If ( x 1 , x 2 ) is
strongly PE, does there
exists a social welfare func
func

tion a1 u1 ( x 1 )  a 2 u 2 ( x 2 )
such that maximization of
this social welfare function
leads to ( x 1 , x 2 ) ?
Answer: The utility possipossi
bility set U must be convex.
Concave utility functions
ensure this in pure exchange.
G.B. Asheim, ECON4230-35, #12
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6
Determining the supporting welfare weights
a i 
1


i

1
 v i ( p , px i )
m i
u2
 v 2 (p , m 2 )
a

m 2
 slope :


 v1 ( p , m1 )
a

 m1

1

2
a1
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(u1 ( x1 ), u 2 ( x 2 ))
U
( a1 , a2 )

2

1
 v1 ( p , m1 )
 m1
 a 2
u1
Intuition?
 v 2 (p , m 2 )
m 2
G.B. Asheim, ECON4230-35, #12
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To recapitulate:

Competitive equilibria are always Pareto efficient.

Pareto efficient allocations are competitive equilibria
with quasi-concave u-functions & endowm. redistrib.

Welfare maxima are always Pareto efficient.

Pareto efficient allocations are welfare maxima with
concave u-functions for some choice of welfare weights.
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