core - university of california, los angeles

Core

Ichiro Obara

UCLA

December 3, 2008

Obara (UCLA) Core December 3, 2008 1 / 22

Core in Edgeworth Box




Motivation

How should we interpret the price in Walrasian equilibrium?

The notion of core provides a different way to understand

competitive equilibria.



Example

Consider a pure exchange economy with two consumers and two

goods.

Contract curve is the set of Pareto efficient allocations which are

better than the initial endowments for both consumers.



Example

1

2

e

Contract Curve



Example

Consider a pure exchange economy with four consumers and two

goods where two consumers are type 1 and two consumers are type 2

(type i : (�i , ei )). This economy is called a (2-)replica economy of

the previous pure exchange economy with two consumers.

Consider a feasible allocation x ′ ∈ A (the same type consumes the

same amount). x ′ can be blocked by a coalition of one type 1

consumer and two type 2 consumers. Type 1 consumes at y”1 and

type 2 consumes at y ′2 where y”1 + 2y ′2 = e1 + 2e2.



Example

1

2

e

x’

y’

y’’

2 consumers for each type



Example

Consider only symmetric allocations (witin each type). Then we can

use the same Edgeworth box to represent allocations in r -replica

economies for r = 2, 3, ...

Every inefficient allocation can be blocked by the grand coalition.

Pareto efficient allocations which are not on the contract curve are

blocked by just one consumer.

More and more (symmetric) allocations on the contract curve are

blocked for r -replica economies as r increases.

In the limit, every alocation other than the CE allocation in the

two-consumer economy is blocked.



Example

1

2

e

x’

y’

y’’

With more consumers….



Example

1

2

e

…in the limit…


Core

Core


Core

Assumptions

We consider a pure exchange economy Epure .

Assume that, for every i ∈ I ,

I Xi = <L+,

I �i is locally nonsatiated, continuous and strictly convex,

I ei � 0.


Core

Definitions

A coalition S ⊂ I blocks x∗ ∈ A if ∃(xi )i∈S ∈ <|S|×L+ such that xi � x∗i for

every i ∈ S and∑

i∈S xi ≤∑

i∈S ei .

Core

x∗ ∈ A has the core property if there is no coalition that can block it. The

set of feasible allocations that have the core property is the core of the

economy.


Core

Remark.

We know that every Walrasian equilibrium allocation has the core

property.

Since there exists an equilibrium in this economy, the core is not

empty.


Core

Replica Economy

Fix one pure exchange economy Epure =({Xi ,�i , ei}i∈I

).

An r -replica economy of Epure , denoted by Epurer , is a pure exchange

economy where, for each i ∈ I , there are r consumers whose

preference is �i and endowment is ei .

A consumption vector of the q th consumer of type i is denoted by

xi ,q.

The set of feasible allocations in Epurer is denoted by Ar .


Core

Equal Treatment Property

First, it is without loss of generality to focus on symmetric allocations.

Equal Treatment Property

Suppose that x∗ ∈ Ar has the core property for Epurer . Then x∗i ,q = x∗i ,q′ for

all q, q′ and i ∈ I .


Core

Proof

Step 1. Pick the worst guy for each type. Suppose not. Without

loss of generality, assume that x∗i ,q �i x∗i ,1 for q = 2, ..., r for every

i ∈ I and x∗i ,q �i x∗i ,1 for some (i , q).

Step 2. The coalition of the worst guys can block x∗.

I Let x∗i = 1r

∑rq=1 x∗i,q be the average consumption of type i consumers

given x∗.

I Then x∗i � (�)x∗i,1 for every (some) i ∈ I and∑

i∈I x∗i ≤∑

i∈I ei .

I With a slight redistribution from (x∗i )i∈I , the coalition of consumer 1s

can block x∗ ∈ A.


Core

Remark

For any x ∈ A of Epure , let xr ∈ Ar be (x1, ..., x1︸︷︷︸r times

, ...., xI , ..., xI︸︷︷︸r times

). Since

a competitive equilibrium is in the core, the equal treatment property

holds for CE as well, i.e. if (x , p) is a CE for Epurer , then there exists

x ′ ∈ A such that x = x ′r . Also note that (x ′, p) is a competitive

equilibrium for Epure .

A corollary of ETP: the core of Epurer+1 is smaller than the core of Epure

r .

Hence the core is shrinking as r is increasing.

It turns out that the core “converges to” the set of competitive

equilibrium allcations as r →∞.


Core

Core Convergence

Core Convergence Theorem

Suppose that, for some x∗ ∈ A, x∗r ∈ Ar is in the core of Epurer for

r = 1, 2, .... Then there exists p∗ ∈ <L+ such that (x∗, p∗) is a competitive

equilibrium for Epure .

Remark. (x∗r , p∗) is a competitive equilibtium for Epurer for every r .


Core

Proof

Step 1. Z = the (average) resources required to “block” x∗

Suppose that x∗r is in the core of Epurer for r = 1, 2, ... for some

x∗ ∈ A. Define

Zi ≡{

zi ∈ <L : zi + ei ∈ <L+ and zi + ei �i x∗i

}Define a convex (check!) set Z by

Z ≡

{z ∈ <L : ∃a ∈ ∆I & ∃zi ∈ Zi , i ∈ I s.t. z ≥

∑i∈I

aizi

}

(Interpretation: if rz is at your disposal for some z ∈ Z , you can find

a coalition which consists of “rai” consumers of type i and blocks x∗

in Epurer ).


Core

Proof

Step 2. 0 6∈ Z

Suppose 0 ∈ Z (This immediately leads to a contradiction if ai , i ∈ I

are all rational numbers). Then there exist a ∈ ∆I and zi ∈ Zi , i ∈ i

such that zi + ei �i x∗i and 0 ≥∑

i∈I aizi = 0. For each i ∈ I , let ari

be the smallest integer such that rai ≤ ari . Define z r

i = raiarizi . Then

I z ri + ei ∈ <L

+ for r = 1, 2, ...,

I z ri → zi (thus z r

i + ei �i x∗i for large r), and

I 0 ≥∑

i∈I ari z

ri for r = 1, 2, ....

→ this contradicts to x∗r being in the core for every r .


Core

Proof

Step 3. Apply SHT and the standard trick.

By SHT, there exists p∗( 6= 0) ∈ <L such that p∗ · z ≥ 0 for all z ∈ Z .

Clearly p∗ > 0.

By local nonsatiation, you can show that p∗ · xi ≥ p∗ · ei for any

xi �i x∗i (set aj = 0 for any j 6= i).

Since∑

i∈I x∗i ≤ r , we have p∗ · x∗i = p∗ · ei .

Since p∗ · x∗i = p∗ · ei > 0, cost minimization implies utility

maximization, i.e., x∗i � xi for any xi such that

p∗ · xi ≤ p∗ · x∗i = p∗ · ei .


core - university of california, los angeles

Documents