Is Competitive Equilibrium Unique?
Ram Singh
Lecture 7
October 05, 2016Questions
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)
Multiple WE: Example
Example Two consumers:
u1(.) =x11−18 1
(x21)8 andu2(.) =−18 1 (x12)8 +x22 e1= (2,r)ande2= (r,2);r =289 −219 The equilibria are solution to
p2 p1
−19
+2+r p2
p1
− p2
p1
89
=2+r,i.e.,
there are three equilibria:
p2 p1 = 1
2,1, and 2.
Multiple WE: Example I
Consider
Two goods: food and cloth Let(pf,pc)be the price vector.
We know that for allt >0:z(tp) =z(p).
Therefore, we can work with p= (pf
pc,1) = (p,1).
From Walras’s Law, we have
pzf(p) +zc(p) =0.
Assumptions for existence of WE:
zi(p)is continuous for allp>>p, i.e., for allp>0.
Multiple WE: Example II
there exists smallp= >0 s.t.zf(,1)>0, and there exists anotherp0> 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:
zf0(p)<0 for allp>0.
Normal Goods and Number of Equilibria I
Let,
there be two goods - food and cloth.
e1= (e1f,e1c)ande2= (e2f,e2c)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 allp>>0, i.e., for allp>0.
there exists smallp= >0 s.t.zf(,1)>0 and anotherp0 >1 s.t.
zc(p0,1)>0.
Normal Goods and Number of Equilibria II
By definition:
zf(p) = zf1(p) +zf2(p)
= [xf1(p)−ef1] + [xf2(p)−e2f]
Since endowments are fixed, we get
∂zf(p)
∂p =
∂xf1(p)
∂p
du1=0
−(xf1(p)−ef1)
∂xf1(p)
∂I
+
∂xf2(p)
∂p
du2=0
−(xf2(p)−ef2)
∂xf2(p)
∂I
(1)
Let
p∗be an equilibrium price vector. We know thatp∗exists. Why?
Normal Goods and Number of Equilibria III
WLOG assume that at in equilibrium Person 1 is net buyer of food; i.e., xf1(p∗)−e1f >0.
At equilibrium price,p∗, we have
∂zf(p∗)
∂p =
∂xf1(p∗)
∂p
du1=0
−(xf1(p∗)−ef1)
∂xf1(p∗)
∂I
+
∂xf2(p∗)
∂p
du2=0
−(xf2(p∗)−ef2)
∂xf2(p∗)
∂I
(2)
In equi. (food) market clears. So,
xf2(p∗)−e2f =−[xf1(p∗)−ef1].
We can rearrange (2) to get
Normal Goods and Number of Equilibria IV
∂zf(p∗)
∂p =
∂xf1(p∗)
∂p
du1=0
+
∂xf2(p∗)
∂p
du1=0
+ (xf1(p∗)−ef1)
∂xf2(p∗)
∂I −∂xf1(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.
WARP and no of WE I
Let,
x(p)denote the bundle demanded at pricep.
x0 =x(p0)denote the bundle demanded at pricep0. Therefore,
x=p.x(p)is the expenditure incurred at pricep.
p0.x0=p0.x(p0)is the expenditure incurred at pricep0.
p.x0≤p.ximplies that the bundlex0 was affordable at pricep.
p0.x>p0.x0implies that the bundlex=x(p)is strictly more expensive (thanx0=x(p0)) at pricep0.
WARP and no of WE II
Definition
The demand satisfies WARP, if
px0≤px ⇒ p0.x0<p0.x.
p(x(p0)−x(p))≤0 ⇒ p0(x(p0)−x(p))<0.
Theorem
If the aggregate demand satisfies the WARP, then the WE is unique.
WARP and no of WE III
Proof: Suppose, WE is not unique. If possible, supposep,p0∈E, andp6=p0. Note for any price vectorspandp0, we have:
p.z(p) = 0,i.e., p.(x(p)−e) = 0.
Sincep0is 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,i.e.,
p.(x(p0)−x(p)) = 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. (5)
WARP and no of WE IV
Similarly, we get:
p0.z(p0) = 0,i.e., p0.(x(p0)−e) = 0
p0.(x(p0)−x(p)) = 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?
WARP and no of WE V
Note:
Aggregate demand functionx(.)will exist iff if the individual demand function, i.e.,xi(.), exists for alli =1, ..,N.
xi(.)exists if the underlying utility function satisfies the assumption of continuity, strict monotonicity and strict quasi-concavity.
