Number and Stability of Walrasian Equilibria
Ram Singh
Microeconomic Theory
Lecture 9
Questions
We continue to investigate the following:
Is Competitive/Walrasian equilibrium unique?
Why 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?
Gross Substitutes I
Suppose,
There are two goods
Consider three price vectors:p= (p1,p2) = (2,1),p0 = (p01,p20) = (3,1) andp¯= (¯p1,¯p2) = (3,2).
Let,xi(p), be the demand function for individuali.
Question
Suppose, the above goods are ‘gross substitutes’ for individual i.
How will x2i(p0)compare with x2i(p)?
How will x2i(¯p)compare with x2i(p)?
Letλ= maxj{¯ppj
j},j =1,2.
Note hereλ=p¯2
p2 =2
Gross Substitutes II
Also,λp≥¯p. Since(4,2)≥(3,2).
Question
What can we say about the individual demand for the two goods at these two price vectorsλp= (4,2)andp?¯
Question
What can we say about the individual demand for the two goods at the price vectorsp= (2,1)andp0= (3,2)?
Gross Substitutes III
Next, consider two price vectors
p= (p1,p2,p3) = (3,2,1)and¯p= (¯p1,p¯2,¯p3) = (5,1,4)
Question
What can we say about the excess demand at these two price vectors?
Letλ= maxj{¯ppj
j},j =1, ..,3.
Note hereλ=pp¯3
3 =4
Also,λp≥¯p. Since(12,8,4)≥(5,1,4).
Remark
z(λp) =z(p), i.e.,z(4p) =z(p).
Gross Substitutes IV
Consider the following price vectors
p= (p1,p2,p3) = (3,2,1),pˆ= (ˆp1,pˆ2,ˆp3) = (12,8,4)and p¯= (¯p1,p¯2,p¯3) = (5,1,4).
Question
What can we say about the excess demand for 3rd good at pricespˆ and p? That is,¯
How is z3(ˆp)expected to compare with z3(¯p)?
Definition
Aggregate demand function,z(.), satisfies condition of ‘Gross Substitutes’
(GS) if for allp,ˆ p¯∈RM++, such thatpˆ≥p¯and ˆp6= ¯p:
pˆj = ¯pj ⇒zj(ˆp)>zj(¯p).
GS and No of WE I
Definition
Aggregate demand function,z(.), satisfies condition of ‘Gross Substitutes’
(GS) if for allp,ˆ p¯∈RM++, such thatpˆ≥p¯and ˆp6= ¯p:
pˆj = ¯pj ⇒zj(ˆp)>zj(¯p).
Theorem
If Z(.)satisfies condition of GS, then there is unique WE.
WLOG, we can consider vectors in the set
P={p|p∈RM++, andpM =1}.
Proof: Suppose, WE is not unique. If possible, supposep,p0∈E. Moreover,p6=p0.
GS and No of WE II
Let
λ = max
j
p´j
pj
forj =1, ..,M.
= max p´1
p1
,p´2
p2
, ...,´pM
pM
Suppose, pp´k
k ≥ ´ppj
j for allj =1, ..,M.That is, λ= ´pk
pk
Clearly,λp≥p0, andpkλ= ´pk. Letp¯=λp.
This meansp¯ ≥p0andp¯k = ´pk. Hence
zk(¯p)>zk(p0).
GS and No of WE III
Butzk(p0) =0. Therefore,
zk(¯p=λp)>0, which is a contradiction. Why?
Sincep∈E, therefore
zk(p) =0.
Sincep¯=λp,
zk(¯p) =zk(p) =0.
Excess Demand Function: Basics I
Consider aN×M economy: Let,M=2 andp= (p,1)be a price vector, wherep>0.
Letz(p) = (z1(p),z2(p))be the excess demand function.
p∗ is an equilibrium price vector if and only if
z1(p∗) =0and z2(p∗) =0.
That is, iff
z1(p∗) = 0 ... = ... zM(p∗) = 0;
Clearly,
[(z1(p),z2(p)) = (0,0)]iffz1(p) =0
Excess Demand Function: Basics II
Lemma
For M=2, Price vectorp= (p,1)is equilibrium price vector of a2×2 economy iff z1(p) =0. That is, iff zM−1(p) =0
For anyN×M economy, consider a price vector sayp= (p1,p2, ...,pM) another price vectorp0 =p1
Mp= (pp1
M,pp2
M, ...,1) = (p01,p20, ...,1)
Individual and aggregate demand underp0 will be exactly the same as underp.
So, WLOG we can consider vectors in the set
P={p|p∈RM++, andpM =1}
Excess Demand Function: Basics III
Let
pvM= (p1, ...,pM−1)and zvM = (z1(p), ...,zM−1(p)) Therefore,
p= (pvM,1)and z= (zvM,zM)
Again, a price vectorp= (p1, ...,pM−1,1)is an equilibrium price vector if it solves theM×M systemz= (zvM,zM) =0, i.e., if it solves the system:
z1(p) = 0 ... = ... zM−1(p) = 0
zM(p) = 0.
Excess Demand Function: Basics IV
From Walras Law:p1z1(p) +....+pM−1zM−1(p) +pMzM(p) =0. If z1(p) = 0
... = ... zM−1(p) = 0;
thenzM(p) =0.
Proposition
A price vectorp= (p1, ...,pM−1,1)is an equilibrium price vector iff it solves the following system of M−1equations:zvM(p) =0, i.e., iff it solves the system:
z1(p) = 0 ... = ... zM−1(p) = 0.
Local Uniqueness of WE: Two Goods I
For aN×2 economy:
Definition
Anequilibriumprice vectorp= (p1,1)is calledregularifz10(p)6=0.
Definition
AnN×2 economy is regular if everyequilibriumprice vectorp= (p1,1)is regular.
Theorem
A regular equilibrium price vectorp= (p1,1)is locally unique. That is, there exists an >0such that: for everyp0= (p10,1),p06=p, andkp0−pk< , we have
z(p0)6=0.
Local Uniqueness of WE: Two Goods II
ProofSuppose,p= (p1,1)is an equilibrium price vector, i.e., z(p) =0,i.e., z1(p) =0.
Now, consider an infinitesimal change inp, saydp6=0. Letdp= (dp1,0), dp1< and
p0 =p+dp= (p1+dp1,1) Sincep= (p1,1)isregular, we havez10(p)6=0. Therefore,
dp1z10(p)6=0.
Using Taylor series approximation, we can write z1(p0)≈z1(p) +dp1z10(p)6=0.
Therefore,
z1(p0) 6= 0,i.e., z(p0) 6= 0.
That is,p0 is not WE.
Number of a WE: Two goods I
Let
E={p|p∈P, andz(p) =0}.
Note:E⊂⊂P⊆RM++. Remark
If an economy is regular, the setEis discrete.
Proposition
When ‘Boundary conditions’ onz(P)hold,Eis bounded.
Proof: Suppose,p∗= (p1∗,1)∈Eis a equilibrium price vector, i.e.,z(p∗) =0.
For a two goods Economy: Boundary conditions onz(.)imply that z1(.)>0 for very smallp1
z1(.)<0 for very highp1
Number of a WE: Two goods II
Therefore,p∗1is finite and bounded away from 0 and∞ That is,p∗is finite and bounded away from0
Therefore, the setEis bounded.
Proposition
Assuming thatzis continuous inp,Eiscompact- bounded and closed.
Hint: Consider a sequence of prices inE.
the sequence is bounded
it has a convergent sub-sequence - From Bolzano-Weierstrass Theorem, Every bounded sequence inRnhas a convergent subsequence.
Let—pbe the limit of the subsequence Sincezis continuousz(—p) =0, so—p∈E So,Eis closed.
Number of a WE: Two goods III
Next, we use the following result:
Theorem
If a set is compact and discrete, then it has to be finite.
Theorem
If an economy is regular, there are only finitely many equilibrium prices.
SinceEis bounded, closed and discrete, it is a finite set.
Theorem
If an economy is regular and the ‘boundary conditions’ onz(P)hold, then Either there will be a unique equilibrium
The number of equilibria will be odd.
