Grass Substitutes and Number of Equilibria

Grass Substitutes and Number of Equilibria

Ram Singh

Lecture 8

October 6, 2015

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{^¯p_p^j

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¯= (3,2)?

Question

What can we say about the individual demand for the two goods at the price vectorsp= (2,1)andλp= (4,2)?

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=max_j{^¯p_p^j

j},j=1, ..,3.

Note hereλ=max{⁵₃,¹₂,⁴₁}= ^¯p_p³

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 zj(¯p)?

Note:

pˆ=λpandpˆ ≥p¯ pˆ3=λp3= ¯p3.

GS and No of WE I

Definition

Aggregate demand function,z(.), satisfies condition of ‘Gross Substitutes’

(GS) if for allp,ˆ p¯∈R^M++, such thatpˆ≥p¯and ˆp6= ¯p:

pˆ_j = ¯p_j ⇒z_j(ˆp)>z_j(¯p).

Theorem

If Z(.)satisfies condition of GS, then there is unique WE.

WLOG, we can consider vectors in the set

P={p|p∈R^M++, andp_M =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

p_j

forj=1, ..,M.

= max p´1

p1

,´p2

p2

, ...,´pM

pM

Suppose, ^p_p^´k

k ≥ ^´p_p^j

j for allj =1, ..,M.That is, λ= ´p_k

pk

Clearly,λp≥p⁰, andp_kλ= ´p_k. Letp¯=λp.

This meansp¯ ≥p⁰andp¯k = ´pk. Hence

zk(¯p)>zk(p⁰).

GS and No of WE III

Butzk(p⁰) =0. Therefore,

z_k(¯p=λp)>0, which is a contradiction. Why?

Sincep∈E, therefore

zk(p) =0.

Sincep¯=λp,

z_k(¯p) =z_k(p) =0.

Normal Goods and Number of Equilibria I

Let,

there be two goods - food and cloth.

e¹= (e¹_f,e¹_c)ande²= (e²_f,e²_c)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

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

Assume:

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

there exists smallp= >0 s.t.z_f(,1)>0 and anotherp⁰ >¹ s.t.

z_c(p⁰,1)>0.

Normal Goods and Number of Equilibria II

By definition:

z_f(p) = z_f¹(p) +z_f²(p)

= [x_f¹(p)−e_f¹] + [x_f²(p)−e²_f]

Since endowments are fixed, we get

∂zf(p)

∂p =

∂x_f¹(p)

∂p

du¹=0

−(x_f¹(p)−e_f¹)

∂x_f¹(p)

∂I

+

∂x_f²(p)

∂p

du²=0

−(x_f²(p)−e_f²)

∂x_f²(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., x_f¹(p^∗)−e¹_f >0.

At equilibrium price,p^∗, we have

∂z_f(p^∗)

∂p =

∂x_f¹(p^∗)

∂p

du¹=0

−(x_f¹(p^∗)−e_f¹)

∂x_f¹(p^∗)

∂I

+

∂x_f²(p^∗)

∂p

du²=0

−(x_f²(p^∗)−e_f²)

∂x_f²(p^∗)

∂I

(2)

In equi. (food) market clears. So,

x_f²(p^∗)−e²_f =−[x_f¹(p^∗)−e_f¹].

We can rearrange (2) to get

Normal Goods and Number of Equilibria IV

∂z_f(p^∗)

∂p =

∂x_f¹(p^∗)

∂p

du¹=0

+

∂x_f²(p^∗)

∂p

du¹=0

+ (x_f¹(p^∗)−e_f¹)

∂x_f²(p^∗)

∂I −∂x_f¹(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.