General Equilibrium Analysis: Lecture 4

General Equilibrium Analysis: Lecture 4

Ram Singh

Course 001

September 22, 2014

Market Exchange I

Let us introduce ‘price’ in our pure exchange economy. Let, There beNindividuals andM goods

eⁱ = (e₁ⁱ, ...,eⁱ_M)denote endowment for individuali

p_i denote the ‘price’ ofith good;p_i >0 for alli =1, ..,M. So the price vector isp= (p₁, ...,p_M).

Assume

each good has a market and each individual is a ‘price-taker’.

p= (p1, ...,pM)>>0.

For each individual,

Total value of the initial endowment depends on the price vector However, the total value of the bundle bought cannot exceed the total

Market Exchange: 2 × 2 economy

For person 1, the set of feasible allocations/consumptions is the set of y¹= (y₁¹,y₂¹)such that:p₁y₁¹+p₂y₂¹≤p₁e¹₁+p₂e¹₂.

Assuming monotonic preferences, Person 1 maximizes utility by choosing bundlex¹= (x₁¹,x₂¹)s.t.

p1x₁¹+p2x₂¹=p1e¹₁+p2e¹₂ Person 2 maximizes utility s.t.p1x₁²+p2x₂²=p1e²₁+p2e₂².

Recall, within the Edgeworth box, for each allocation(x¹,x²), we have x₁¹+x₁²=e¹₁+e₁², andx₂¹+x₂²=e₂¹+e²₂.

Also,

p1e₁²+p2e²₂ = p1x₁²+p2x₂²,i.e.,

p₁e₁²+p₂e²₂ = p₁(e¹₁+e₁²−x₁¹) +p₂(e¹₂+e²₂−x₂¹),i.e., 0 = p1(e¹₁−x₁¹) +p2(e¹₂−x₂¹),i.e.,

p1x₁¹+p2x₂¹ = p1e¹₁+p2e₂¹, which is the budget line for the 1 person.

Preferences and Utilities: Assumptions

We assume:

Preference relations to be continuous, strictly monotonic, and strictly convex

The utility functions to be continuous, strictly monotonic and strictly quasi-concave

However, several of the results will hold under weaker conditions:

Competitive Equilibrium: 2 × 2 economy I

An allocation isˆx= (ˆx¹,ˆx²)along with a price vectorp= (p₁,p₂)is competitive equilibrium, if

1 xˆ¹maximizesu¹(.)subject top₁x₁¹+p₂x₂¹=p₁e₁¹+p₂e¹₂

2 xˆ²maximizesu²(.)subject top1x₁²+p2x₂²=p1e₁²+p2e²₂

3 xˆ₁¹+ ˆx₁²=e₁¹+e²₁

4 xˆ₂¹+ ˆx₂²=e₂¹+e²₂ For ‘well-behaved’ utilities:

1. Implies : In equi. IC of person 1 will be tangent to her budget line.

2. Implies : In equi. IC of person 2 will be tangent to his budget line We know that: both of the demanded bundles, i.e.,ˆx¹andˆx²lie on the same line.Why?

3 and 4 imply that the demanded bundles, i.e.,xˆ¹andˆx²coincide.Why?

Competitive Equilibrium: 2 × 2 economy II

Therefore, the ICs are tangent to each other

Therefore, the eq. allocationˆx= (ˆx¹,xˆ²)is Pareto Optimum.

Question

Does the eq. allocationˆx= (ˆx¹,ˆx²)belong to the core?

Competitive Equilibrium: N × M economy I

Consider a general(uⁱ(.),e),N×M economy.

An allocationˆx= (ˆx¹, ...,ˆx^N)along with a price vectorp= (p₁, ...,p_M)is a competitive equilibrium, if the following conditions are satisfied:

First:ˆxⁱ maximizesuⁱ(.). That is,ˆxⁱ solves max

xⁱ

{uⁱ(xⁱ)} (1) subject top.xⁱ =p.eⁱ, i.e.,p1x₁ⁱ +...+pMx_Mⁱ =p1e₁ⁱ +...+pMeⁱ_M.

Second: For allj =1, ...,M

N

X

i=1

ˆx_jⁱ =

N

X

i=1

eⁱ_j (2)

Definition

(ˆx,p)is called a Competitive or Walrasian equilibrium, if(ˆx,p)together satisfy (1) and (2) simultaneously.

Competitive Equilibrium: N × M economy II

Definition

The set of Walrasian/Competitive Equilibria,W(uⁱ(.),eⁱ)_N×M, is given by

W(uⁱ(.),eⁱ)_N×M ={x| ∃psuch that(x,p)satisfy (1) and (2), simultaneously.}

Remark

Walrasian/Competitive equilibrium may not exist. However, If utilities fns are continuous, strictly increasing and strictly quasi-concave, there does exist at least one equilibrium.

In general there can be more than one Competitive equilibrium.

Walrasian/Competitive equilibrium depends on the vector of initial endowments, i.e.,e.

Some Observations I

Letxˆ= (ˆx¹, ...,ˆx^N)be a Competitive equilibrium allocation.

Proposition

Suppose,(ˆx,p)is a competitive equilibrium. Then,xˆ= (ˆx¹, ...,ˆx^N)is a feasible allocation.

Proposition

Suppose,(ˆx,p)is a competitive equilibrium. If uⁱ(yⁱ)>uⁱ(ˆxⁱ), then p.yⁱ >p.eⁱ. Formally,

uⁱ(yⁱ)>uⁱ(ˆxⁱ) ⇒ p.yⁱ >p.eⁱ uⁱ(yⁱ)>uⁱ(ˆxⁱ) ⇒





J

X

j=1

pjy_jⁱ >

J

X

j=1

pjeⁱ_j





Some Observations II

Proposition

Suppose,(ˆx,p)is a competitive equilibrium, and the individual preferences are monotonic, i.e., uⁱ is increasing. If uⁱ(yⁱ)≥uⁱ(ˆxⁱ), thenp.yⁱ ≥p.eⁱ. Formally,

uⁱ(yⁱ)≥uⁱ(ˆxⁱ) ⇒ p.yⁱ ≥p.eⁱ i.e., p.yⁱ <p.eⁱ ⇒ uⁱ(yⁱ)<uⁱ(ˆxⁱ)

Competitive Equilibrium and Core I

Let

W(uⁱ(.),eⁱ)_N×M denote the set of Walrasian/competitive allocations.

C(uⁱ(.),eⁱ)_N×Mdenote the set of Core allocations.

For a 2×2 economy, suppose an allocationˆx= (ˆx¹,xˆ²)along with a price vectorp= (p₁,p₂)is competitive equilibrium. Then,

Individuali prefersxⁱ at least as much aseⁱ

Indifference curves of the individuals are tangent to each other Allocationˆx= (ˆx¹,ˆx²)is Pareto Optimum

In view of the above, allocationxˆ= (ˆx¹,ˆx²)is in the Core.

Competitive Equilibrium and Core II

So, for a 2×2 economy,

x∈W(uⁱ(.),eⁱ)⇒x∈C(uⁱ(.),eⁱ).

Theorem

Consider an exchange economy(uⁱ(.),eⁱ)N×M, where individual preferences are monotonic, i.e., uⁱ is increasing. Ifxis a WEA, thenx∈C(uⁱ(.),eⁱ)_N×M. Formally,

W(uⁱ(.),eⁱ)_N×M⊆C(uⁱ(.),eⁱ)_N×M.

Proof: Take anyxWEA. Let,xalong with the price vectorpbe a WE.

Suppose

x6∈C(e).

Therefore, there exists a ‘blocking coalition’ againstx. That is, there exists a setS⊆Nand an ’allocation’ sayy, s.t.

Competitive Equilibrium and Core III

X

i∈S

yⁱ =X

i∈S

eⁱ (3)

Moreover,

uⁱ(yⁱ)≥uⁱ(xⁱ)for alli ∈S (4) and for somei⁰∈S

uⁱ(yⁱ⁰)>uⁱ(xⁱ⁰). (5) (3) implies

p.X

i∈S

yⁱ =p.X

i∈S

eⁱ (6)

(4) implies

p.yⁱ ≥p.xⁱ =p.eⁱ, for alli ∈S (7) (6) implies: for somei⁰∈S

p.yⁱ⁰>p.xⁱ⁰ =p.eⁱ⁰. (8)

Competitive Equilibrium and Core IV

(7) and (8) together give us:

p.X

i∈S

yⁱ >p.X

i∈S

eⁱ (9)

But, (4) and (9) are mutually contradictory. Therefore, x∈C(e).

Theorem

Consider an exchange economy(uⁱ,eⁱ)_i∈I, where uⁱ is strictly increasing, for all i =1, ..,I.

Every WEA is Pareto optimum.

Competitive Equilibrium: Merits and Demerits I

Question

Is the price/market economy better than the barter economy, in terms of its functioning?

Is the price/market economy better than the barter economy, in terms of the outcome achieved?

Question

What are the limitations of a market economy?

Can these limitations be overcome?