Is Competitive Equilibrium Unique?
Ram Singh
Lecture 7
October 05, 2016
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Questions
Is Competitive/Walrasian equilibrium unique?
Why is a unique equilibrium helpful?
If WE is not unique, how many WE can be there?
What are the conditions, for a unique WE?
Do these conditions hold in the real world?
Readings: Arrow and Hahn. (1971), Jehle and Reny* (2008), MWG*(1995)
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Multiple WE: Example
Example
Two consumers:
u 1 (.) = x11 −
1 1
8 (x 1 )8
2
and u 2 (.) = − 81
1
8
(x12 )
8
+ x22
1
e1 = (2, r ) and e2 = (r , 2); r = 2 9 − 2 9
The equilibria are solution to
p2
p1
− 19
+2+r
p2
p1
−
p2
p1
89
= 2 + r , i.e.,
there are three equilibria:
1
p2
= , 1, and 2.
p1
2
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Multiple WE: Example I
Consider
Two goods: food and cloth
Let (pf , pc ) be the price vector.
We know that for all t > 0: z(tp) = z(p).
Therefore, we can work with
p=(
pf
, 1) = (p, 1).
pc
From Walras’s Law, we have
pzf (p) + zc (p) = 0.
Assumptions for existence of WE:
zi (p) is continuous for all p >> p, i.e., for all p > 0.
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Multiple WE: Example II
there exists small p = > 0 s.t. zf (, 1) > 0, and
there exists another p0 >
1
s.t. zc (p0 , 1) > 0.
We can ensure above by assuming the utility functions to be continuous and
strictly monotonic.
Question
Do the above assumptions guarantee unique WE?
Under what conditions the WE be unique?
Additional assumption for uniqueness of WE:
0
zf (p) < 0 for all p > 0.
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Normal Goods and Number of Equilibria I
Let,
there be two goods - food and cloth.
e1 = (ef1 , ec1 ) and e2 = (ef2 , ec2 ) be the initial endowment vectors
p = (p, 1) be a price vector.
utility functions be continuous, strictly monotonic and strictly
quasi-concave
From Walras Law we have
pzf (p) + zc (p) = 0.
Assume:
zi (p) is continuous for all p >> 0, i.e., for all p > 0.
there exists small p = > 0 s.t. zf (, 1) > 0 and another p0 >
zc (p0 , 1) > 0.
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General Equilibrium Analysis
1
s.t.
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Normal Goods and Number of Equilibria II
By definition:
zf (p)
=
zf1 (p) + zf2 (p)
=
[xf1 (p) − ef1 ] + [xf2 (p) − ef2 ]
Since endowments are fixed, we get
∂zf (p)
∂p
∂xf1 (p)
∂p
∂xf2 (p)
∂p
=
+
−
(xf1 (p)
−
(xf2 (p)
−
ef1 )
−
ef2 )
∂xf1 (p)
∂I
∂xf2 (p)
∂I
du 1 =0
du 2 =0
(1)
Let
p∗ be an equilibrium price vector. We know that p∗ exists. Why?
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Normal Goods and Number of Equilibria III
WLOG assume that at in equilibrium Person 1 is net buyer of food; i.e.,
xf1 (p∗ ) − ef1 > 0.
At equilibrium price, p∗ , we have
1 ∗ 1 ∗ ∂xf (p )
∂xf (p )
∂zf (p∗ )
=
− (xf1 (p∗ ) − ef1 )
∂p
∂p
∂I
1
du =0
2 ∗ 2 ∗ ∂xf (p )
∂xf (p )
+
− (xf2 (p∗ ) − ef2 )
∂p
∂I
du 2 =0
(2)
In equi. (food) market clears. So,
xf2 (p∗ ) − ef2 = −[xf1 (p∗ ) − ef1 ].
We can rearrange (2) to get
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Normal Goods and Number of Equilibria IV
∂zf (p∗ )
∂p
=
+
∂xf1 (p∗ )
∂p
+
du 1 =0
(xf1 (p∗ ) − ef1 )
∂xf2 (p∗ )
∂p
∂xf2 (p∗ )
∂I
−
du 1 =0
1 ∗ ∂xf (p )
∂I
,
Now, even if both goods are normal,
Person 2 might have large income effect that can offset the negative
substitution effects.
∂zf (p∗ )
∂p
< 0 might not hold.
So, we cannot be sure of uniqueness of WE.
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WARP and no of WE I
Let,
x(p) denote the bundle demanded at price p.
x0 = x(p0 ) denote the bundle demanded at price p0 .
Therefore,
x = p.x(p) is the expenditure incurred at price p.
p0 .x0 = p0 .x(p0 ) is the expenditure incurred at price p0 .
p.x0 ≤ p.x implies that the bundle x0 was affordable at price p.
p0 .x > p0 .x0 implies that the bundle x = x(p) is strictly more expensive
(than x0 = x(p0 )) at price p0 .
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WARP and no of WE II
Definition
The demand satisfies WARP, if
px0 ≤ px
0
p(x(p ) − x(p)) ≤ 0
⇒ p0 .x0 < p0 .x.
⇒ p0 (x(p0 ) − x(p)) < 0.
Theorem
If the aggregate demand satisfies the WARP, then the WE is unique.
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WARP and no of WE III
Proof: Suppose, WE is not unique. If possible, suppose p, p0 ∈ E, and p 6= p0 .
Note for any price vectors p and p0 , we have:
p.z(p)
p.(x(p) − e)
= 0, i.e.,
= 0.
Since p0 is an equi. price vector, z(p) = x(p0 ) − e = 0, i.e., x(p0 ) = e.
Therefore, the above gives us
p.(x(p) − x(p0 ))
0
p.(x(p ) − x(p))
= 0, i.e.,
= 0.
(3)
From WARP, we know that
p(x(p0 ) − x(p)) ≤ 0 ⇒ p0 (x(p0 ) − x(p)) < 0.
(4)
(3) and (4) give us,
p0 .(x(p0 ) − x(p)) < 0.
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(5)
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WARP and no of WE IV
Similarly, we get:
p0 .z(p0 )
0
0
p .(x(p ) − e)
0
0
p .(x(p ) − x(p))
= 0, i.e.,
= 0
= 0
(6)
which is a contradiction, since in view of (5), we have
p0 (x(p0 ) − x(p)) < 0.
Question
What are the assumptions needed for the aggregate demand function to
exist?
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General Equilibrium Analysis
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WARP and no of WE V
Note:
Aggregate demand function x(.) will exist iff if the individual demand
function, i.e., xi (.), exists for all i = 1, .., N.
xi (.) exists if the underlying utility function satisfies the assumption of
continuity, strict monotonicity and strict quasi-concavity.
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