Number and Stability of Walrasian Equilibria

Number and Stability of Walrasian Equilibria

Ram Singh

Microeconomic Theory

Lecture 10

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, ...,p_M) another price vectorp⁰ =_p¹

Mp= (_p^p1

M,_p^p2

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∈R^M++, andp_M =1}

Excess Demand Function: Basics III

Let

p_vM= (p1, ...,pM−1)and z_vM = (z₁(p), ...,z_M−1(p)) Therefore,

p= (p_vM,1)and z= (z_vM,z_M)

Again, a price vectorp= (p1, ...,pM−1,1)is an equilibrium price vector if it solves theM×M systemz= (z_vM,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) +p_Mz_M(p) =0. If z₁(p) = 0

... = ... zM−1(p) = 0;

thenzM(p) =0.

Proposition

A price vectorp= (p₁, ...,p_M−1,1)is an equilibrium price vector iff it solves the following system of M−1equations:z_vM(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⊆R^M++. 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.

Local Uniqueness of WE: M Goods I

Define theM−1×M−1 matrix of first order derivatives:

Dz_vM(p) =









∂z1(p)

∂p1

∂z1(p)

∂p2 · · · _∂^∂z_p^1(p)

M−1

... ... . .. ...

∂zM−1(p)

∂p1

∂zM−1(p)

∂p2 · · · ^∂z_∂^M−1_p ^(p)

M−1









Definition

An equilibrium price vectorp= (p₁, ...,pM−1,1)isregularif the M−1×M−1 matrix,Dz_vM(.), is non-singular atp= (p₁, ...,pM−1,1).

Definition

An economy is regular if everyequilibriumprice vectorp= (p1, ...,pM−1,1) is regular.

Local Uniqueness of WE: M Goods II

Theorem

A regular equilibrium price vectorp= (p₁, ...,pM−1,1)is locally unique. That is, there exists an >0such that: for everyp⁰= (p₁⁰, ...,p⁰_M−1,1),p⁰6=p, and kp⁰−pk< , we have

z(p⁰)6=0.

ProofSuppose,p= (p₁, ...,pM−1,1)is an equilibrium price vector, i.e., z(p) =0.

Now, consider an infinitesimal change inp, saydp6=0. Let dp= (dp1, ...,dpM−1,0), and

p⁰ =p+dp= (p₁+dp₁, ...,pM−1+dpM−1,1)

Local Uniqueness of WE: M Goods III

Since,Dz_vM(p)is non-singular, we have

Dz_vM(p)dp_vM=









∂z1(p)

∂p1

∂z1(p)

∂p2 · · · _∂p^∂z1(p)

M−1

... ... . .. ...

∂zM−1(p)

∂p1

∂zM−1(p)

∂p2 · · · ^∂z_∂p^M−1(p)

M−1

















 dp1

dp2

... dpM−1









 (1)

Dz_vM(p)dp_vM 6=0. Why?

z_vM(p⁰)≈z_vM(p) +Dz_vM(p)dp_vM 6=0.

Therefore,

z_vM(p⁰) 6= 0,i.e., z(p⁰) 6= 0.

That is,p⁰ is not WE.

Number of a WE I

Let

E={p|p∈P, andz(p) =0}.

Remark

Note:E⊂⊂P⊆R^M++. Also, if an economy is regular, the setEis discrete.

Proposition

When ‘Boundary conditions’ onz(P)hold,Eis bounded.

Proof: Suppose,p^∗= (p₁^∗, ...,p^∗_M−1,1)∈Eis an equilibrium price vector, i.e., z(p^∗) =0.

In general, if the ‘boundary conditions’ hold, then there existsr >0 such that:

For allj=1, ..,J

1

r <p_j^∗<r.

Number of a WE II

Therefore, the setEis bounded.

Assuming thatzis continuous inp,Eis closed.

Hint: Consider a sequence of prices inEand use continuity ofz.

As earlier, 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.

Unique WE: Conditions? I

For an individual consumer. Let

u(.)is a continuous, strictly increasing and strictly quasi-concave utility function

u^∗=v(p,I), and

x:R^M+1++ 7→R^M+ be the (Marshallian) demand function generated byu(.).

x^H :R^M+1++ 7→R^M+be the associated Hicksian demand function.

∂xj(p,I)

∂pk = ^∂x

H j (p,u^∗)

∂pk −x_k(p,I)^∂xj_∂I^(p,I), i.e.,

∂x_j^H(p,u^∗)

∂pk =^∂x_∂p^j(p,I)

k +x_k(p,I)^∂xj_∂I^(p,I).

Unique WE: Conditions? II

We know that

∂x_j^H(p,u^∗)

∂p_j =

∂xj(p,u^∗)

∂p_j

du=0

=∂xj(p,I)

∂p_j +x_j(p,I)∂xj(p,I)

∂I <0.

Let,

D(x^H) =









∂x₁^H(p,u)

∂p1 · · · ^∂x1_∂p^H(p,u) .. M

. · · · ...

∂x_M^H(p,u)

∂p1 · · · ^∂xM_∂p^H(p,u)

M









=











∂²E(p,u)

∂p²₁ · · · ^∂_∂p^2E(p,u)

M∂p1

... · · · ...

∂²E(p,u)

∂p1∂pM · · · ^∂2E(p,u)

∂p²_M











We can write

D(x^H) =









∂x1(p,I)

∂p1 +x1(p,I)^∂x1_∂I^(p,I) · · · ^∂x_∂p^1(p,I)

M +xM(p,I)^∂x1_∂I^(p,I)

... . .. ...

∂xM(p,I)

∂p1 +x1(p,I)^∂xM_∂I^(p,I) · · · ^∂x_∂p^M(p,I)

M +xM(p,I)^∂xM_∂I^(p,I)









Unique WE: Conditions? III

We know that

MatrixD(x^H)is negative semi-definite. Why?

Moreover, for every pair(p,I), there is unique ‘equilibrium’ for the consumer. Why

Theorem

Ifx:R^M+1++ 7→R^M+, is a demand function generated by a continuous, strictly increasing and strictly quasi-concave utility function,then xsatisfies

budget balancedness, symmetry and

negative semi-definiteness.

Unique WE: Conditions? IV

Theorem

If a function,x, satisfies budget balancedness, symmetry and negative semi-definiteness, then it is a demand functionx:R^M+1++ 7→R^M+

generated by some continuous, strictly increasing and strictly quasi-concave utility function.

That is, if matrix

D{x}=









∂x1(p,I)

∂p1 +x₁(p,I)^∂x1_∂I^(p,I) · · · ^∂x_∂p^1(p,I)

M +x_M(p,I)^∂x1_∂I^(p,I)

... . .. ...

∂xM(p,I)

∂p1 +x₁(p,I)^∂xM_∂I^(p,I) · · · ^∂x_∂p^M(p,I)

M +x_M(p,I)^∂xM_∂I^(p,I)









is symmetric and negative semi-definite, then

Unique WE: Conditions? V

There is a continuous, strictly increasing and strictly quasi-concave utility functionu(.)that induces demand functionx.

The ‘equilibrium of the consumer’ is unique.

Moreover, if goodj is normal, then ^∂x_∂^j_p^(p)

j <0.

Question

Does a similar result hold for the aggregate demand function? That is, does a similar result hold at the aggregate level?

Let

D{z(p)}=

∂z_j(p)

∂pk

, j,k =1, ...,J−1.

At aggregate level, we have:

Unique WE: Conditions? VI

Theorem

WE is unique, if the matrix

D{z(.)}is negative semi-definite at allp∈E, i.e., D{−z(.)}has a positive determinant at allp∈E. Note, now Slutsky equation is:

∂x_jⁱ(p^∗)

∂pk

!

duⁱ=0

= ∂x_jⁱ(p^∗)

∂pk

+ x_kⁱ(p^∗)−eⁱ_k ∂x_jⁱ(p^∗)

∂I

!

dp=0

We have seen that even for 2×2 economy,

Unique WE: Conditions? VII

Remark

D{z(.)}is not necessarily negative semi-definite, even ifD(x^H)is negative semi-definite.

Moreover, even if the goods are all normal, ^∂z_∂p^j(p)

j <0 may not hold.