Market Equilibrium Price: Existence, Properties and Consequences

Market Equilibrium Price: Existence, Properties and Consequences

Ram Singh

Lecture 5

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 would want an increase in prices and of what type?

Does increase in prices have distributive consequences?

Individual UMP: Some Features I

Notations:

p= (p1, ...,pM)is a M-component vector inR^M.

Ifp= (p1, ...,pM)∈R^M++, thenpj >0 for allj =1, ...,M, i.e., (p1, ...,pM)>(0, ...,0).

Ifp= (p₁, ...,p_M)∈R^M+, thenp_j ≥0 for allj ∈ {1, ...,M}andp_j >0 for somej ∈ {1, ...,M}, i.e.,

(p₁, ...,p_M)≥(0, ...,0)and(p₁, ...,p_M)6= (0, ...,0).

Letx= (x1, ...,xM)andx⁰= (x₁⁰, ...,x_M⁰). Ifx⁰ ≥x, thenx_j⁰ ≥xj for all j ∈ {1, ...,M}andx_j⁰>xj for somej ∈ {1, ...,M}.

Letx= (x1, ...,xM)andx⁰= (x₁⁰, ...,x_M⁰). Ifx⁰ >x, thenx_j⁰ >xj for all j ∈ {1, ...,M}.

Individual UMP: Some Features II

Take a price vectorp= (p₁, ...,p_M)∈R^M++. That is,(p₁, ...,p_M)>(0, ...,0).

The consumeri’s OP (UMP) is to solves:

max

x∈R^J+

uⁱ(x) s.t. p.x≤p.eⁱ

Definition

uⁱ is strongly increasing if for any two bundlesxandx⁰ x⁰ ≥x⇒uⁱ(x⁰)>uⁱ(x).

Assumption

For alli∈I,uⁱ is continuous, strongly increasing, and strictly quasi-concave onR^M+

Individual UMP: Some Features III

In view of monotonicity, for givenp= (p1, ...,pM)>>(0, ...,0), consumeri solves:

max

x∈R^M+

uⁱ(x) s.t. p.x=p.eⁱ (1)

Theorem

Under the above assumptions on uⁱ(.), for every(p₁, ...,p_M)>(0, ...,0), (1) has a unique solution, sayxⁱ(p,p.eⁱ).

Note:

Existence follows from Monotonicity and Boundedness of the Budget set Uniqueness follows from ‘strictly quasi-concavity’

Individual UMP: Some Features IV

Note:

xⁱ(p,p.eⁱ)is the (Marshallian) Demand Function for individuali.

For eachi =1, ...,N,

xⁱ(p,p.eⁱ) :R^M++ 7→R^M+;

xⁱ(p,p.eⁱ) = (x₁ⁱ(p,p.eⁱ), ...,x_jⁱ(p,p.eⁱ), ...,x_Mⁱ (p,p.eⁱ)).

In general, demand forjth good depends on price ofkth good, k =1, ...,M

Demand forjth good depends on price ofkth good relative to the other prices

Individual UMP: Some Features V

Theorem

Under the above assumptions on uⁱ(.), for every(p₁, ...,p_M)>(0, ...,0), xⁱ(p,p.eⁱ)is continuous inpoverR^M++.

For all i=1,2, ...,N, we have:xⁱ(tp) =xⁱ(p), for all t>0. That is, demand of each good j by individual i satisfies the following property:

x_jⁱ(tp) =x_jⁱ(p)for all t >0.

Question

Given that uⁱ(.)is strongly increasing, isxⁱ(p)continuous overR^M+?

is the demand function x_jⁱ(p)defined at p_j =0?

Is a Cobb-Douglas utility function strongly increasing overR^M+?

Excess Demand Function I

Definition

The excess demand forjth good by theith individual is give by:

z_jⁱ(p) =x_jⁱ(p,p.eⁱ)−eⁱ_j. The aggregate excess demand forjth good is give by:

zj(p) =

N

X

i=1

x_jⁱ(p,p.eⁱ)−

N

X

i=1

e_jⁱ.

So, Aggregate Excess Demand Function is a vector-valued function:

z(p) = (z1(p), ...,zj(p), ...,zM(p)),

Excess Demand Function II

Theorem

Under the above assumptions on uⁱ(.), for anyp>>0, z(.)is continuous inp

z(tp) =z(p), for all t>0 p.z(p) =0. (the Walras’ Law)

For any given price vectorp, the individual UMP gives us p.xⁱ(p,p.eⁱ)−p.eⁱ = 0,i.e.,

M

X

j=1

p_jx_jⁱ(p,p.eⁱ)−

M

X

j=1

p_je_jⁱ = 0,i.e.,

M

X

j=1

pj[x_jⁱ(p,p.eⁱ)−eⁱ_j] = 0.

Excess Demand Function III

This gives:

N

X

i=1 M

X

j=1

p_j[x_jⁱ(p,p.eⁱ)−eⁱ_j] = 0,i.e.,

M

X

j=1 N

X

i=1

pj[x_jⁱ(p,p.eⁱ)−eⁱ_j] = 0,i.e.,

M

X

j=1

p_j

" _N X

i=1

x_jⁱ(p,p.eⁱ)−

N

X

i=1

e_jⁱ

#

= 0

That is,

M

X

j=1

pjzj(p) = 0,i.e.,

Excess Demand Function IV

So,

p₁z₁(p) +p₂z₂(p) +...+p_j−1z_j−1(p) +p_j+1z_j+1(p) + +p_Mz_M(p) =−p_jz_j(p) For a price vectorp>>0,

ifz_k(p) =0 for allk 6=j, thenz_j(p) =0 For two goods case

p1z1(p) +p2z2(p) =0., i.e.,

p1z1(p) =−p₂z2(p).

Therefore,

z₁(p) =0 ⇒ z₂(p) =0 z1(p)>0 ⇒ z2(p)<0.

Walrasian Equilibrium

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

WE: Proof I

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). Why? Letp= (p,1)andtp= (p_f,pc) We know that for allt>0:

z(tp) =z(p) that is(z_f(tp),z_c(tp)) = (z_f(p),z_c(p)) Therefore,tp.z(tp) =0, i.e.,tp_fz_f(tp) +tp_cz_c(tp) =0 implies

p.z(p) =0, i.e., pzf(p) +zc(p) =0.

Assume:

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

WE: Proof II

Note

Since utility function is monotonic,x_f(p)will explode asp_f =p→0.

Therefore,

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

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

Therefore, for a two goods case we have:

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

In general, Equlibrium price is determined byTatonnementprocess