Market Equilibrium Price: Existence, Properties and Consequences

Market Equilibrium Price: Existence, Properties and Consequences

Ram Singh

Lecture 5

September 22, 2016

Questions

Today, we will discuss the following issues:

How does the Adam Smith’sInvisible Handwork?

Is increase in Prices bad?

Do some people want increase in prices?

If yes, who wants increase in prices and of what type?

Does increase in prices have distributive consequences?

Market Equilibrium Price I

Definition

Walrasian Equilibrium Price: A price vectorp^∗is equilibrium price vector, if for allj =1, ...,J,

z_j(p^∗) =

N

X

i=1

x_jⁱ(p^∗,p^∗.eⁱ)−

N

X

i=1

eⁱ_j =0, i.e., if

z(p^∗) =0= (0, ...,0).

Proposition

Ifp^∗is equilibrium price vector, thenp⁰=tp^∗, t >0, is also an equilibrium price vector

Ifp^∗is equilibrium price vector, thenp⁰6=tp^∗,t >0, may or may not be an equilibrium price vector

Market Equilibrium Price II

Two goods:food and cloth

Let(p_f,p_c)be the price vector. We can work withp= (^p_p^f

c,1) = (p,1). Since, we know that for allt>0:

z(tp) =z(p) Therefore, we havep_fz_f(p) +pczc(p) =0, i.e.,

pz_f(p) +zc(p) =0.

Let:

zi(p)is continuous for allp>>0, i.e., for allp>0.

Note

When utility function is monotonic,xf(p)will explode aspf =p→ ∞.

Therefore,

Market Equilibrium Price III

there exists smallp= >0 s.t.z_f(p,1)>>0 andz_c(p,1)<0 (Why?).

there exists anotherp⁰> ¹ s.t.zf(p⁰,1)<0 andzc(p,1)>0. (Why?).

Therefore, for a two goods case we have:

There is a value ofpsuch thatzf(p,1) =0 andzc(p,1) =0 That is, there exists a WE price vector.

In general, Equlibrium price is determined byTatonnementprocess

WE: General Case I

References: Jehle and Reny (2008).

Also see: Arrow and Debreu (1954), and McKenzie (2008); Arrow and Hahn (1971).

Recall, we have

For alli =1,2, ...,N, we have:xⁱ(tp) =xⁱ(p), for allt >0.

z(tp) =z(p), for allt>0.

Let

pˆ= (ˆp1, ...,ˆpM)>>(0, ...,0)be a price vector t =_pˆ ¹

1,...,ˆpM

WE: General Case II

Thenp=tpˆis such that

pj ≥>0 and

M

X

j=1

pj =1

Without any loss of generality, we can restrict attention to the following set

P=







p= (p1, ...,pM)|

M

X

j=1

pj =1 andpj ≥ 1+2M





 ,

where≥0.

Remark

Note thatPis non-empty, bounded, closed and convex set for all∈(0,1).

WE: General Case III

Theorem

Suppose uⁱ(.)satisfies the above assumptions, ande>>0. Let{p^s}be a sequence of price vectors inR^M++, such that

{p^s}converges to¯p, where

p¯∈R^M+,p¯6=0, but for some j,p¯j =0.

Then, for some good k with¯p_k =0, the sequence of excess demand (associated with{p^s}), say{z_k(p^s)}, is unbounded above.

Theorem

Under the above assumptions on uⁱ, there exists a price vectorp^∗>>0, such thatz(p^∗) =0.

WE: Proof I

Let,

¯z_j(p) =min{z_j(p),1}. (1) Note

z¯j(p) =min{z_j(p),1} ≤1.Therefore, 0≤max{¯zj(p),0} ≤1.

Next, let’s define the functionf = (f₁(p), ...,f_M(p))such that: Forj=1, ..,M, f_j(p) = +pj+max{¯zj(p),0}

M+1+PM

j=1max{z¯_j(p),0} =Nj(p) D(p),

Note thatPM

j=1fj(p) =1. Moreover, we have fj(p)≥ N_j(p)

M+1+M.1 ≥

M+1+M.1 ≥

1+2M.

WE: Proof II

Therefore,

f(p) :P7→P.

SinceD(p)≥1>0, the above function is well defined and continuous over a compact and convex domain. Therefore, there existspsuch that

f(p) =p,i.e.,

fj(p) =p_j, for allj =1, ..,M.

That is, for allj=1, ..,M,

N_j(p)

D(p) = p_j,i.e.,

p_j[M+

M

X

j=1

max{¯zj(p),0}] = +max{¯zj(p),0}. (2)

Next, we let→0. Consider the sequence of price vectors{p}, as→0.

WE: Proof III

Sequence{p}, as→0, has a convergent subsequence, say{p⁰}.

Why?

Let{p⁰}converge top^∗, as⁰→0.

Clearly,p^∗≥0. Why?

Suppose,p^∗_k =0. Recall, we have

p_k⁰



M⁰+

M

X

j=1

max{¯zj(p⁰),0}



=⁰+max{¯zk(p⁰),0}. (3)

as⁰ →0:

The LHS converges to 0, since lim⁰→0p_k⁰ =0 and term [M⁰+PM

j=1max{z¯_j(p⁰),0}]on LHS is bounded.

However, the RHS takes value 1 infinitely many times. Why?

WE: Proof IV

This is a contradiction, because the equality in (3) holds for all values of⁰. Therefore,p_j^∗>0 for allj =1, ..,M. That is,

p^∗>>0,i.e.,

→0limp=p^∗>>0.

In view of continuity ofz¯(p)overR^M++, from (2) we get (by taking limit→0):

WE: Proof V

For allj=1, ..,M

p_j^∗

M

X

j=1

max{¯z_j(p^∗),0} = max{z¯_j(p^∗),0},i.e.,

zj(p^∗)p^∗_j





M

X

j=1

max{¯zj(p^∗),0}



 = zj(p^∗)max{¯zj(p^∗),0},i.e.,

M

X

j=1

zj(p^∗)p^∗_j





M

X

j=1

max{¯zj(p^∗),0}



 =

M

X

j=1

zj(p^∗)max{¯zj(p^∗),0},i.e.,

M

X

j=1

zj(p^∗)max{¯zj(p^∗),0}=0. (4)

WE: Proof VI

Recall,z¯_j(p) =min{z_j(p),1}.Therefore,

z_j(p^∗)>0 ⇒ max{¯z_j(p^∗),0}>0;

zj(p^∗)≤0 ⇒ max{¯zj(p^∗),0}=0.

Therefore,

zj(p^∗)>0 ⇒ zj(p^∗)max{¯zj(p^∗),0}>0;

z_j(p^∗)≤0 ⇒ z_j(p^∗)max{¯z_j(p^∗),0}=0.

Suppose, for somej, we havezj(p^∗)>0, then we will have

M

X

j=1

z_j(p^∗)max{¯z_j(p^∗),0}>0. (5)

But, this is a contradiction in view of (4).

WE: Proof VII

Therefore:

For anyj=1, ..,M, we have

z_j(p^∗)≤0. (6)

Suppose,zk(p^∗)<0 for somek. From (4), we know

p^∗₁z₁(p^∗) +...+p^∗_kz_k(p^∗) +...+p^∗_Mz_M(p^∗) =0.

Sincep_j^∗>0 for allj =1, ..,M.

zk(p^∗)<0 impliesp^∗_kzk(p^∗)<0

Therefore, there must existk⁰, such that

WE: Proof VIII

zk⁰(p^∗)>0, (7)

which is a contradiction in view of (6). Therefore,

For all j =1, ..,M, we have:zj(p^∗) = 0,i.e., z(p^∗) = 0.

That is,

p^∗is a WE price vector.