Core
Ichiro Obara
UCLA
December 3, 2008
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Core in Edgeworth Box
Core in Edgeworth Box
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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.
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Core in Edgeworth Box
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.
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Core in Edgeworth Box
Example
1
2
e
Contract Curve
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Core in Edgeworth Box
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.
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Core in Edgeworth Box
Example
1
2
e
x’
y’
y’’
2 consumers for each type
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Core in Edgeworth Box
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.
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Core in Edgeworth Box
Example
1
2
e
x’
y’
y’’
With more consumers….
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Core in Edgeworth Box
Example
1
2
e
…in the limit…
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Core
Core
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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.
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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.
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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.
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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 .
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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 .
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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.
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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 →∞.
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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 .
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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 ).
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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 .
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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 .
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