Tema 2

Edgeworth’s Exchange Theory

 The exchange Theory of Edgeworth. A simple
exchange model 2X2.

z 2 agents A y B and 2 goods: x 1 y x 2

z No production
z Initial endowments are given by:
w A = ( w1A , w2A ) y wB = ( w1B , w2B ) con
w1A + w1B = w1 y w2A + w2B = w2
z Each agent has well-defined preferences over
baskets of goods and can consume either her
initial endowment or exchange it with the other
agents.

©2005 Pearson Education, Inc. Chapter 16 2

 The exchange Theory of Edgeworth. A simple
exchange model 2X2.

z Let a consumption basket of A and B be:

x A = ( x1A , x 2A ) y x B = ( x1B , x 2B )
z An allocation is a pair of consumption
baskets :
x = (x A , xB )

z An allocation is feasible if:

x1A + x1B = w1A + w1B = w1 y x 2A + x 2B = w 2A + w 2B = w 2

©2005 Pearson Education, Inc. Chapter 16 3

 A simple model of Pure exchange. The
Edgeworth-Bowley box summarizes the set of all
feasible allocations.

w1B 0B
w2

w2B

w2A
w

0A A
w1
w 1
©2005 Pearson Education, Inc. Chapter 16 4
 The Exchange Theory of Edgeworth.
Pareto Rationality.
PARETO RATIONALITY: An allocation of goods is Pareto
efficient if no one can be made better off without making
someone else worse off. Formally:

Definition: A feasible allocation x is Pareto optimal (or Pareto

Efficient) if there is no other feasible allocation y such that:
1) ui(yi)≥ ui(xi) for all i, and
2) uj(yj)> uj(xj) for at least some j

 The contract curve shows all the efficient allocations of

goods between two consumers.
 To calculate the contract curve, the utility of an agent is
maximized subject to both the feasibility constraint and to
the utility level of the other agent´s constraint: Max u1(x1),
subject to u2(x2)≥u2 and subjetc to feasibility (the two FOC of
the assoc. Lagrangian →RMS1=RMS2 and feasibiity).

©2005 Pearson Education, Inc. Chapter 16 5

 The Exchange Theory of Edgeworth.
The core of an economy
z INDIVIDUAL RATIONALITY: An allocation xi satisfies
individual rationality (IR) with respect to wi if: ui(xi)≥ ui(wi)

z The core of an exchange economy is the set of

feasible allocations which cannot be improved
upon (or blocked) by any coalition of agents.

z For 2 agent-exchange economies core allocations are those

satisfying individual rationality and Pareto efficiency.

z For n agent-economies: We need to define “coalitions” of

agentes and how they can block a given allocation.

©2005 Pearson Education, Inc. Chapter 16 6

 The Exchange Theory of Edgeworth.
The core of an economy
Coalition: A coalition S is any subset of agents with mandatory
agreements.

z Any coalition S can block a proposed allocation x whenever

the agents in S can reallocate their initial endowments
among themselves and be better than under x.

z Core: RI, Pareto Rationality and rationality of all the

remaining coalitions.

z Example: three agents {A,B,C}

z Coalitions: {A},{B},{C}; {A,B,C,}; {A,B}, {B,C} y
z {A,C}

©2005 Pearson Education, Inc. Chapter 16 7

 Core of an exchange economy
z Let n be the number of agents of the economy,
z w=(w1,w2,…,wn) the vector of initial endowments,
z x=(x1,x2,…,xn) an allocation of the economy, and
z F(w)={x: ∑i xi = ∑i wi } the set of feasible allocations.

z Blocking coalition: Let S be a coalition. S blocks allocation x

in F(W), through y in F(W) if:
z 1) ui(yi)≥ ui(xi) for all i in S, and
z 2) uj(yj)> uj(xj) for at least some j in S
z 3) ∑i en S yi 6 ∑i en S wi (feasibility in S)

z The core of an exchange economy C(w):

z C(w)={x: there is no y satisfying 1),2) y 3) with x and y in
F(w)}

©2005 Pearson Education, Inc. Chapter 16 8

 Existence of the core of an exchange economy. Is the core
empty? No, whenever there exists a WE, since it belongs
to the core.
Definition (alternative): A pair allocation-price (x*, p*) is a WE:
z 1) ∑i x*i =∑i wi (x* is feasible), and
z 2) If ui (xi)> ui (x*i), then p* xi > p* wi (x is not affordable).

Proposition: If (x*, p*) is a WE for the initial endowment w, then x*

belongs to C(w).

Proof: Suppose on the contrary that x* does not belong to C(w). Then
there is a coalition S and an allocation x such that for all i in S, ui
(xi)> ui (x*i), and
∑i en S xi =∑i en S wi (x is feasible for S), →p* ∑i en S xi =p*∑i en S wi (1)
As x* is a WE, then by definition, for all i in S
p* xi > p* wi and adding over all i’s in S:
p* ∑i en S xi >p*∑i en S wi , which contradicts (1) (= ∑i en S xi )
Then x* belongs to C(w).

©2005 Pearson Education, Inc. Chapter 16 9

 Contraction of the core and
replica-economies
The core has ore allocations than the WE.
We show that if the economy increases its size, then new
coalitions will appear and more opportunities to block or
to improve upon:
The core shrinks (contracts)
We use a very simple type of growth.
Definition: 2 agents are of the same type if both their
preferences and their initial endowment are identical.
Definition: An economy is a replica of size r of another
economy, if there are r-times as many agents of each
type in the former economy as in the later.
Thus, if a large economy replicates a smaller economy, then
it will just be as a “scale up” version of the small one.
We only consider 2 types of agents: type A and type B.

©2005 Pearson Education, Inc. Chapter 16 10

 Equal Treatment in the Core
The r-core (r-C) of an economy is the core of its replica of
size r.

Lemma:Equal treatment in the Core.

Suppose that agents’ preferences are stricty convex,
continuous and strongly monotone. If x belongs to the r-core
of a given economy, then any two agents of the same type
will receive the same bundle in x.
Proof: Let
A1, A2,…..Ar and
B1,B2,……Br,
2 types of agents in the r-replica.
If all agents of the same type do no get the same allocation,
there will be one agent of each type who is the most poorly
treated.
Call these two agents: type A underdog (marginated): AM and
type B underdog: BM.
©2005 Pearson Education, Inc. Chapter 16 11
 Equal Treatment in the Core.
z (cont. proof.) Let the mean (average) allocations be:
1 r 1 r
xA = ∑xAj y xB = ∑xBj
r j=1 r j=1
z and we have that: x AM < x A , xBM ≤ x B
z (Note that if all agents receive the same, then they will get the
average allocations). By convexity of preferences AM and BM
prefer the mean allocations to their allocation in x:

u A ( x A ) > u A ( x AM ), uB ( x B ) ≥ uB ( xBM )
z Can AM and BM block core-allocation x through the average
allocations?
z They could whenever average allocations are feasible for
the coalition of them:
x A + x B = wA + wB

©2005 Pearson Education, Inc. Chapter 16 12

 Equal treatment in the Core.
z (cont. proof.) We check the feasibility of average allocations
for the underdog-coalition: by feasibility of x and given that
all agents of the same type have the same initial
endowment.
1 r 1 r
x A + x B = ∑ x Aj + ∑ xB j = x A1 +x A2 + ... + x Ar + xB1 + .... + xBr =
r j =1 r j =1
1 r 1 r
wA1 + wA2 + ... + wAr + wB1 + .... + wBr = ∑ wAj + ∑ wB j =
r j =1 r j =1
1 1
rwA + rwB = wA + wB .
r r

z Then, average allocations are feasible for the coalition of AM

and BM.

©2005 Pearson Education, Inc. Chapter 16 13

 Equal Treatment in the Core.
z (Cont). The underdog AM strictly prefers its type average
allocation to xAM and the underdog BM considers its type
average allocation at least as good as xBM.

z Strong monotonicity allows AM to remove a little quantity

from its average allocation xA − ε
z and to brive BM by offering him: x B + ε
thus forming a coalition that can improve upon allocation x.

z Then agents cannot receive a different treatment in the

r-core of an exchange economy. In the core, all agents
of the same type have to receive the same bundle.

©2005 Pearson Education, Inc. Chapter 16 14

 Contraction of the Core
z Lemma implications: To simplify the analysis of the core in
replica-economies: an allocation x in C tell us what each agent
type A and type B obtain and then we can keep on representing
core-allocations in two-dimensions (in the Edgeworth’s box).

z Any allocation x that is not a WE must eventually not be in the r-

core of the economy. Hence, core allocations of large economies
look like market equilibria.

z Proposition: Contraction of the core:

z Suppose that preferences are strictly convex and strongly
monotone and that there is a unique WE: x*, for initial
endowments w. Then if y is not a WE allocation, then, there
will exit some r-replication of the economy, such that y is not
in the r-core.

©2005 Pearson Education, Inc. Chapter 16 15

 Contraction of the Core
z Proof: Observe the following drawing:

.X*

y
. .g UA1

.
UA0 W
A

©2005 Pearson Education, Inc. Chapter 16 16

 Contraction of the Core
Since y is not a EW, the line trhough w and y must cut at least one
agent’s indifference, say uA1 through y. Then, it is possible to
choose a point such as g which A prefers to y.
We look for a replica-economy and a coalition that blocks
allocation y.
By continuity of the preferences: g=θw+(1- θ)y
Let θ=T/V<1, con T y V integer numbers.
Then: gA=(T/V)wA+(1-T/V)yA

Take the V-replica of the economy.

Form the coalition: V agents of type A and V-T agents of type B,
And consider the allocation asignación z giving gA to type A agents
and yB to those of type B:
Z: gA to type A with uA(zA)> uA(yA)
yB to type B, with uB(zB)≥ uB(yB),
Then, z is strictly preferred to y since agents’ type A can always brive
agents’ type B giving them some epsilon.

©2005 Pearson Education, Inc. Chapter 16 17

 Contraction of the Core
z In order this coalition can block allocation y through z, z has
to be feasible for the coalition. Let us check that this is the
case:

z VzA+(V-T)zB=VgA+(V-T)yB=V[(T/V)wA+(1-T/V)yA]+ (V-T)yB=
z TwA+(V-T)yA+ (V-T)yB= TwA+ +(V-T) (yA+yB)= (by feasbt y)
z TwA+ +(V-T) (wA+wB)= V wA+ +(V-T) wB, which is the initial
endowment of the proposed coalition.

z Then, the proposed coalition can block allocation y through

z, in the V-replica of the economy.

z In this way, all allocations of the core not being WE will

disappear in some replicas of the economy.
z All core-allocations in huge economies are WE.

©2005 Pearson Education, Inc. Chapter 16 18

 Contraction of the Core. Example
 2 agents: A y B
W B

g=½ w+ ½ y

W
A

©2005 Pearson Education, Inc. Chapter 16 19

 Contraction of the Core. Example
z What replica of the economy and what coalition can block y
through z, with gA for type A and y for type B?
z g=1/2 w+1/2 y then, θ=T/V=1/2,
z That is: V=2 y V-T=2-1=1
z Replicate the economy at escale V=2 (duplicate the economy: 4
agents) and form the coalition:
z V=2 agents of type A and (V-T)=1 agent of type B (coalition of 3
agents).
z This coalition can block y through z whenever z is feasible for the
coalition. Checking feasibility:
z VzA+(V-T)zB =VgA+ +(V-T)yB=2(1/2 wA +1/2 yA)+yB=
z wA +yA+yB= wA +wA+wB= 2wA+wB,
z which are the initial endowments of the coalition. Then, y can be
blocked in the duplication of the economy by the proposed
coalition of three agents and through allocation z.

©2005 Pearson Education, Inc. Chapter 16 20

 Contraction of the Core. Example

In general:
If the allocation is g=1/n w+(1-1/n)y
Then θ=T/V=1/n, that is: V=n y V-T=n-1
With gA=1/n wA +(n-1)/n yA

z Take the replica of the economy at escale V=n

z and the coalition:
z V=n agents type A and V-T=n-1 agents type B

©2005 Pearson Education, Inc. Chapter 16 21