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IT TAKES t* TO TANGO:
TRADING COALITIONS IN THE EDGEWORTH PROCESS
Franklin M. Fisher
No. 446
April
1
987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
IT TAKES t* TO TANGO:
TRADING COALITIONS IN THE EDGEWORTH PROCESS
by
Franklin M. Fisher
No.
446
April
1
987
IT_TAKES_£*_TQ_TANGQi
TRADING_COALITIOHS_Iig_THE_EDGEyORTH_PBQCESS
Franklin M. Fisher
Department of Economics, E52-359
Massachusetts Institute of TechnologyCambridge, MA, USA 213 9
Abstract
the Edgeworth non-tatonnement process,
In
there
exists
some
trade occurs
coalition of agents able to make
improving trade among themselves at current prices.
that
the
size
commodities
and that,
generalized:
Let
positive endowments.
held by all agents.
tions
with
It is
there be h agents,
k
known
number
provided all agents always have
positive endowments, bilateral trade suffices.
Pareto-
a
of such coalitions is bounded by the
if
of
strictly
These results are
of whom
have
strictly
Let there be m commodities, n of which are
Then the Edgeworth-process requires
at most t* = Min
{2,
Max
(h
- k,
m -
n)
}
coalimembers.
This is a least upper bound.
JEL No. 021
Keywords: Edgeworth Process, Non-tatonnement,
Stability of General Equilibrium, t-wise optimality
7
?
?9P?
I
!.!_ Introduction
basic assumption of the Edgeworth non-tatonnement
The
cess
that trade takes place if and only if
is
coalition
pro-
exists
there
of agents able to make a Pareto-improving trade
themselves at current, disequilibrium prices.
a
among
Among other objec-
tions to this assumption is the possibility that it may require
very large number of agents to find each other
19,
1983,
observed
29-31).
pp.
(in
a
(Fisher,
a
1976, p.
In reply to this, David Schmeidler has
private communication) that such trading
coali-
tions need never involve more members than the number of commodities, while Paul Madden has shown that, if all agents always have
strictly positive endowments of all commodities, then such coalitions need never have more than two members.
(Both results can be
found in Madden, 1978).
These
are
not
problem
very reassuring answers to the
at
however, particularly if one thinks of extending the Edge-
hand,
worth process to relatively realistic settings..
If consumption
takes place at different times, then the same commodity at diffe-
dates will be treated as different commodities.
rent
easily
make
the
number of commodities much
number of agents in the economy.
result,
greater
This
can
than
the
As for Madden' s bilateral trade
it requires strictly positive endowments of all commodi-
ties for all agents,
and this is far too strong a requirement in
the context of disequilibrium trade.
It is therefore of some interest to see the extent to
the two existing results can be generalized.
possible
which
It turns out to be
to accomplish this with a very elementary
proof,
and,
while the results still do not suggest that the Edgeworth-process
assumption is free of coalition-formation problems, they may have
some intrinsic interest.
show the following under very general
I
there be h households of whom
of
all commodities.
held
number
have strictly positive endowments
k
Let there be m commodities of which n
positive amounts by all households.
in
of
trade is t*
agents who must participate in
=
Min
Max {2,
least upper bound:
Let
assumptions.
(h
- k
,
are
maximum
Then the
Edgeworth-process
an
m - n)}.
Further, this is
There exist examples requiring
a
participation
by t* traders.
It
is obvious that these results generalize and
those of Schmeidler and Madden.
Not surprisingly,
strengthen
they are also
when
quite similar to results on the closely related question of
"t-wise
trades
is
optimality"
—
non-existence
the
involving no more than
t
of
traders for some arbitrary
equivalent to full Pareto optimality.
Graham,
Rader,
difference
The
(See
Jennergen, Peterson, and Weintraub,
1968,
1976,
and,
is that,
Pareto-improving
Feldman,
t
1973,
1976, Madden, 1975,
especially, Goldman and Starr, 1982.)
in the Edgeworth process,
restricted to take place at given prices,
trading
is
so the theorems of the
t-wise optimality literature cannot be used directly.
2 J _PrelimiDaries
There
has
a
are h households and m commodities.
weakly monotonic,
dif f erentiable,
Each
household
quasi-concave utility func-
tion.
Under these assumptions,
an Edgeworth-process trade can
be
.
thought of as a circle of agents and commodities.
such trade involves
be {1,
.
take to be also {1,
hold
set of agents, which we may as well take to
a
.ft}, and
.
a set of
.
sells commodity
.,
.
sells commodity
i
i
commodities, which we may as well
such that, for
t},
1
house-
< t,
i
<;
while
to household i+1,
household
t
The question at issue is that
to household 1.
t
That is, every
of when the size of such circles can be reduced.
following
The
fairly obvious fact will be central
the
to
proofs below.
Lemma
commodities,
Suppose
selling
Consider any household,
1.
b,
a,
with H's holdings of
c,
at current prices,
that,
and buying c.
a
Then,
at the same prices,
(2)
p,
,
D
,
and p
C
,
U.
buying
c
which
b
/U
<
c
b /p c
p,
,
H
(1)
derivatives
Let the prices of the
respectively.
Then U /U
3.
<
C
three
p 3. /p_,
C
since
Evidently,
c.
in which case H would find selling b
to be utility increasing,
case
would also
utility-increasing:
could increase utility by selling a and buying
either
H
by
selling a and buying b.
subscripts in the obvious way.
goods be p
3.
H
or
of
and b both positive.
Denote H's utility function by U(.) and
Eioofj,
by
c
a
could increase utility
H
find one of the following trades to be
selling b and buying
and any triplet
H,
would find selling
a
or else
U /U,
and buying b to
<
p /P h
be
and
r
in
utility
increasing
3.._Results
I
begin with two parallel lemmata.
1982, pp.
Goldman and Starr,
597-598.)
Lemma
involving
(Cf.
2.
a
Suppose that there is an
Edgeworth-process
household with strictly positive stocks of all
trade
the
.
commodities involved in the trade.
Then there is an
Edgeworth-
process trade that involves no more than two households.
Eioofj Without loss of generality,
commodities
t
the first
>
commodity
household
{1,
.
and that household
2
1
1
t},
.,
.
Household
commodities.
t
sell
to
and
so that the assumed Edgeworth-process trade involves
households and commodities
Suppose
renumber households
has
1
(to household 2)
a
as described above.
positive
endowment
finds it utility-improving
and buy commodity t
(from
t)
Consider commodity t-1 which is being bought by household
and sold by household t-1.
increasing
If household 1 would find it utility
sell commodity t-1 and buy commodity
to
then
t,
a
Suppose, on the other hand, that this is not the case.
and t.
by Lemma 1, household
Then,
t
Edgeworth-process trade is possible between households
bilateral
1
of
sell commodity
1
would find it utility improving to
and buy commodity t-1.
1
however,
In this case,
there is an Edgeworth-process trade that involves only households
{1,
.
Repe-
t-1} and the identically-numbered commodities.
.,
.
tition of this argument proves the lemma.
Lemma
involving
by
Suppose that there is an Edgeworth-process
3.
a
amounts
commodity that is held in strictly positive
all the households involved in the trade.
Edgeworth-process
trade
trade
Then there is
that involves no more than
an
house-
two
holds.
Pioofj.
As before,
let the households and
commodities
volved in the Edgeworth-process trade be numbered {1,
Assume
that
t
>
2
and that it is commodity
1
that is
.
in-
.,
t}.
held
in
.
positive amounts by the first
Household
to sell commodity t to household
increasing
utility
households.
t
finds it
t
and
1
buy
commodity t-1 from household t-1.
Household
would
between
then
t,
households
1
1
and
it
sell
by Lemma 1, household
Then,
however,
on the other hand,
Suppose,
and t.
utility-increasing to sell commodity
In this case,
If
1.
bilateral Edgeworth-process trade is possible
a
this is not the case.
it
positive stock of commodity
a
it utility increasing to buy commodity
find
commodity
has
t
1
t
that
would find
and buy commodity
t-1.
there is an Edgeworth-process trade that
involves
only households {2,
numbered
commodities.
.
t}
.,
.
Repetition
identically-
and the
of this argument proves
the
lemma.
It is now easy to prove the main result:
Theorem
1.
Suppose that
k
of the households
positive
have
stocks of all commodities and that n of the commodities are
positive
in
t*
=
amounts
Max {2, Min
(A)
- k,
(h
held
households.
Define
Edgeworth-process trade,
then there
by
all
m - n)}.
If there exists and
exists one that involves no more than t* households.
(B)
that
t*
is a least upper bound to the number of
be required in an Edgeworth-process
can
participants
trade,
that
is,
there exist cases in which t* participants are necessary.
PlQPf..
than
t*
(A)
Any Edgeworth-process trade that involves
households
must involve either
positive stocks of all commodities or
positive
amounts
by all households.
a
a
household
that
more
has
commodity that is held in
Lemmas
2
and
3
show
that
s
.
there must then be an Edgeworth-process trade involving only
two
households
(B)
This part of the theorem can be proved by
constructing
examples in which t* participants are required.
t*
=
If
t*
Min
(h
=
- k,
m - n)
such that household
be
For
i.
1
<
i
t*
<
trade for household
ty
i
only
result
the
2,
i
t*
.
i
= h
has
a
is
-
k.
trivial.
suppose
Let households 1,
positive stock of only
suppose that the only
,
So
.
.
that
.
,
t*
commodity
utility-increasing
at current prices would be to sell commodi-
and buy commodity i-1.
For household 1,
suppose that
the
utility-increasing trade at current prices would be to sell
commodity
1
holds t*+l,
and buy commodity t*
.
.
. ,
h have no
made at current prices.
.
Finally,
suppose that house-
utility-increasing trade that can be
Then the only Edgeworth-process trade is
the obvious one involving the first t* households and commodities
and it cannot be reduced.
Theorem
results.
plainly implies both Schmeidler's
1
Let
1.
all but two of
the
strictly positive amount of every commodity.
an Edgeworth-process trade,
Corollary
2.
in
terms
have
I
a
if there is
Then,
Let all but two of the commodities be held
an Edgeworth-process trade,
closing,
households
there is one that is bilateral.
strictly positive amount by every household.
only
Madden'
Indeed, it permits us to strengthen the latter as:
Corollary
In
and
in
if there is
Then,
there is one that is bilateral.
note that the fact that Lemmas
of households and commodities
2
and
involved
3
in
speak
the
assumed Edgeworth-process trade may mean that further results are
,
possible.
do not see how to phrase such results in an
I
inte-
resting way, however.
NOTE
The assumption of differentiability can almost certainly
1.
be
weakened
to the requirement that indifference surfaces
unique supporting hyperplanes (Madden,
1978,
p.
281), but there
seems little gain in complicating the exposition to do so.
from the method of proof used,
possibility.
ing
In
Suppose that household
improving trade in which
for
regards
1
apples
1
sells carrots to
particular household
a
2
do
(1)
and
not.
Pareto-
for apples and to
Such a trade can require three participants
bananas.
though
3
the three households may have a
circumstance,
that
Apart
one needs to rule out the follow-
as perfect complements while households 2 and
bananas
have
participates in
all
3
even
transac-
This makes calculation of the minimum number of partici-
tions.
pants tedious at best, and, as the circumstance involved is quite
special,
agents
does
it
view
a
not seem worth pursuing.
if
all
given subset of commodities as perfect complements
the same proportions,
using
(Note that
then,
without loss of
generality,
that subset can be renamed as a composite commodity.)
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A.
(1973),
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mality,
40,
FISHER,
"Bilateral Trading Processes, Pairwise Opti-
463-473.
P.M.
sults
(1976),
"The Stability of General Equilibrium: Re-
and Problems"
(F.
W.
Paish Lecture delivered at the
i
Nobay,
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S.H.
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E.R.
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L.P.,
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T.
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and
(1976)
T.
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S.A.Y.
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