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IT TAKES t* TO TAUGO:
TRADING COALITIOH IB THE EDGEVOHTH PROCESS
(Revision)
by
FranJclin M. Fisher
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
IT TAKES t* TO TAUGO:
TRADING COALITIOS IH THE EDGEVOETH PROCESS
(Revision)
by
Franklin M. Fisher
Ko. 436
March 1988
Revised February, 1988
TB^DIBS_CQ^^LITIDUS_IK_THB_EDCEI:/DBTH_£BDCE2S
Franklin M. Fisher
Department of Economics, E52-359
Massachusetts Institute of Technology
Cambridge, HA, USA 02139*
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
in
known
number
of
strictly
These results are
so that the maximum required coalition size is given
terms of the number of agents holding at least m
and the number of commodities held by at least
JSL No.
Keyv.'orcs
It is
provided all agents alv;ays have
positive endov;ments, bilateral trade suffices.
Pareto-
a
of such coalitions is bounded by the
if
:
k
21
Edgeworth Process, Non-tatonnement,
commodities
agents.
that trade takes place if and only if
is
cess
pro-
basic assumption of the Edgeworth non-tatonnement
The
coalition
exists
there
a
among
of agents able to make a Pareto-improving trade
Among other objec-
themselves at current, disequilibrium prices.
tions to this assumption is the possibility that it may require a
very large number of agents to find each other
12,
1983,
observed
29-31).
pp.
(in
a
1976, p.
(Fisher,
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.
consumption
If
takes place at different times, then the same commodity at diffe-
dares will be treated as different commodities.
rent
easily
make
rhe
numoer of commodities much
number cf 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
I
the context cf disequilibrium trade."
It is therefore of some interest to see the extent to
the two existing results can be generalized.
'Dossible
to ace omo"'
"^s^
t^*^s
w"^***^
-^
tto^-it
which
It turns out to be
did—ip^^a-s^^tt
o^'oc^'f
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 as limiting the number of traders needed
for a
(within trading coalition) Pareto-improving trade at
prices in
trade
that
barter economy.
a
obtained depend on
results
The
takes
form that
a
Edgeworth-process
whether
shall call "simple trade" or
I
form
a
involves
Simple trade
shall call "compound trade".
I
given
a
circle of transactions in which each household sells one commodity
buys another
and
Compound
trade
(weakly)
involves
increasing
thereby.
utility
its
transactions in which
some
household
sells one commodity and buys another even though it would
prefer
not to do so, because the sale involved induces another household
transaction that eventually leads to an increase
to
enter into
in
the original household's utility.
this
below
a
shall argue that simple trade
and
assumption in
a
natural
the
the results are
fairly
show the follov/ing under very general assumptions. In a
I
pure
exchange
ties.
rcr any
holding
is
noncooperative, competitive setting.
When only simple trade is involved,
rich.
about
shall be precise
I
let there be h households and n con.modi-
econo.-ny,
-.,
1
<
-;
<
n,
let x
(-.)
be the nu-oer of households
amounts.
at least m conunodities in positive
Then
the
existence of a (simple) Edgeworth-process trade implies the existence of such
n - m
-!-
2
}
=
trade with no more than
participants.
In other words,
pants need not exceed the largest of
holding
less
fewer than m commodities,
(m - 2)
t. (-)
2,
=
Max {2, h - x
(m)
the number of partici-
the number of households
and the number of commodities
for any k,
Similarly,
commodities
held
- k
h
participants
k
^ h,
held
households less
(k
a
need
In other words,
Evidently,
if
t*
by fewer than
- 2)
>
number
k
Max
=
{2,
the number of
number
of
households and the
number
of
2,
Min
=
t*
Edgeworth-process coalition
{Min
m
t, (m)
t„(k)}.
Min
,
k
then t* is no larger than the number
of
positive stock of all commodities or
a
of commodities osi held in positive amounts
households.
When
implies
.
households who do uq^ hold
the
households.
the
not exceed the largest of 2,
exceed
not
be the number of
trade with no more than t^{k)
Putting these results together,
size
let y(k)
Edgeworth-process trade
+ 2} participants.
need
commodities
(simple)
a
existence of such
n - y(k),
<
in positive amounts by at least k
Then the existence of
the
1
by
all
results
are
There are other results as well.
trade
can be compound,
hov/ever,
fev/er
available.
Here the only new result is that coalition size need
not
the
exceed
larger of
and the number
2
of
households
Qpt
holding all commodities.
These
results ger.eralize and srrengrhen those of Sch.~eidler
and Madden.
on
They are, however, considerably weaker than results
the related question of when "r-wise optimality"
existence
traders
cf
for
optimality.
Weintraub,
ly,
Goldman
—
?areto-i-.proving trades involving no r.ore
some arbitrary
(See Feldman,
t
—
is equivalent
to
non-
the
full
than
t
Pareto
1973, Graham, Jennergen, Peterson,
and
1976, Kadden, 1975, Rader, 1968, 1976, and, especial-
and Starr,
1982.)
The reasons for that difference
are instructive and are best understood after an example.
Consider Theorem 1.1 in Goldman and Starr
theorem originally due to Rader.
is
(1982,
It states that, provided there
trader holding positive quantities of all goods,
a
597), a
p,
then
the
absence of any mutually-improving bilateral trade implies Paretooptimality,
that there are no mutually-improving trades
so
any number of traders.
mutually-improving
the existence of
Put differently,
trade
implies the existence of
a
for
some
mutually-
improving bilateral trade.
prices
The proof of this theorem consists in observing -that
corresponding
goods
all
to the marginal utilities of the trader who
holds
optimum
since
trader
(say
1)
must support
a
Pareto
otherwise some mutually-improving bilateral trade would be possiTrue enough. If, for trader 2, the marginal rate of substi-
ble.
tution
trader
between some pair of goods were different than it is
for
then a mutually-improving bilateral trade between them
1,
would be possible at some other set of prices.
however,
Note,
because
is
that
this
it is assumed that no
possiole
present
improving
would
contradiction
only
mutually-improving bilateral trade
The equivalent assumption
at ^Dy prices.
case
leads to a
be the much weaker one
that
no
in
the
mutuallyprices.
bilateral trade is possible at a siygD set of
That this does not lead to the same result can be seen by observing
thar,
utilities,
trace
if
prices happen to be equal to trader I's
trader
1
will not wish to trade.
between traders
1
and
2
r.arginal
Hence, the possible
will not be possible at the
=iv£3
(without further assumptions)
,
there is nothing to prevent there
from being a mutually-improving trade involving several
traders other than trader
all)
(or
1.
The general point is as follows.
In showing
that the exis-
of some mutually-improving trade implies the existence
tence
such
trade with no more than
a
literature
effectively
t
considers
traders,
the t-wise optimality
the case in which
improving trade is possible at aoy set of prices.
trade be possible at given prices,
no
This is
assumption than the condition that no t-wise
stronger
of
t-wise
a
much
improving
and it is therefore not
sur-
prising that it leads to much stronger results.
Since
prices,
trade in actual economies often takes place at
however,
it
given
is interesting to know hov; many traders are
required with prices fixed.
^^-SfgliiniDarigSi-Siinple-Tigdgs.SD^-CCIIiPPyDd.Tiadgs
There
Each
are h households and n commodities.
household
has a dif f erentiable, locally-nonsatiated, strictly quasi-concave
utility
Prices
function
that
is
non-decreasing
are assumed to be strictly positive.
convenience.
its
in
arguments.
(This is mainly
2
a
)
Definition
1.
An Edgeworth-process trade is
prices
such that,
budget
constraints,
a
trade at given
with all participants in the trade
no
on
their
participant's utility decreases and
at
least one participant's utility increases.
For later purposes,
observe that the strict quasi-concavity
of the utility functions i-plies that an Edgeworth-process
remains
trade
an Edgeworth-process trade if the amounts of each ccmmc-
traded by each participant are all multiplied by
dity
scalar,
/X
^
,
^
:$
1
same
the
We can thus consider very small trades
.
and work in terms of marginal rates of substitution.
Lemma
In an Edgeworth-process trade,
1,
it
possible
is not
to partition the participating households into two sets, A and B,
such that some household in A sells some commodity to a household
in
household
B but no household in B sells any commodity to any
in A.
Then the total wealth of households in
Suppose not.
CiCfif*
A would be greater after the trade than before, contradicting the
fact that all households must remain on their budget constraints.
consider
Now
household
Edgeworth-process
an
selling commodity
i
j
trade
involves
which
to household i'.
Household
i'
must sell to some other household or households, and they in turn
must sell to others,
and so on.
said to buy commodity
j
All of these households will be
from household
directly or indirectly.
1,
Denote the set of such households as B(i, j).
Lemma
Household
2.
i
is a member of B(i,
j).
Suppose not. Take A as the set of households involved
El<?p£^
in the Edgeworrh-process trade that are not in 3(i,
as 3(i,
Then Lem.~a
j).
Hence every sale
1
(or
Take B
j).
is contradicted.
purchase) of
a
commodity by
a
household
in an Edgeworth-process trade involves a circle of households and
commodities,
with
each household in
a
circle buying
from the preceding one and selling a commodity to the
a
com.-odity
succeeding
household
one.
We
sells
more than one commodity to another as involving more
can
think of transactions in which a given
than
one circle
(possibly with all the same households and almost
all
the same commodities)
Definition
if
2.
An Edgeworth-process trade is called "simple"
at least one of the circles composing it is itself
worth-process
trade.
is
not
trade,
the
Edgeworth-process trade that
An
Edge-
an
simple will be called "compound".
in a simple Edgeworth-process
other words,
In
households
participating in at least one circle v/ould be willing
cir-
do so even if they were not also participating in other
to
In a compound Edgeworth-process trade,
cles.
on the other hand,
least one household participating in any circle only does
at
because
such
participation
bring
is required to
so
different
a
circle into existence.
example will help here.
An
ting
of two "circles".
Figure
Nodes in the diagram
dities, indicated by letters.
2,
household
household
3
selling
"circle", household
sells ccrTm.odity
dity
c
But
»»-^-^>-iJ
represent
house-
e
2
selling commodity
selling comjnodity b to
co.TJTiodity
1
1
c
household
to household 1.
sells co-u-odity
to household 3,
d
to house-
a
and
3,
In the left-hand
to household
and household
3
household
^. ,
sells commo-
to household i.
This
cles"
trade consis-
a
Thus, in the diagrammed trade, the
right-hand "circle" has household
4
shows
indicated by numbers, while arrov;s denote sales of commo-
holds,
hold
1
-rade would be simple if at least one of
were (weakly) utiliry-improving for all irs
suppose that the situation is as follows.
1-j.cwfc
i^aK-^s
yj.ai-c,
iit-iui^ii'-'
j.
— ;;
^
ci^iv^
i
j.i»;Cl
these
"cir-
participants.
At rhe prices
^ZlSj.i.
^
es^Cw
at
<_ j.
v
=
hold
House-
in the diagrammed trade to be utility increasing.
roles
however, would not be willing to participate in the left-
1,
hand circle standing alone.
That is, at the given prices, house-
would not be willing to sell
hold
1
the
other
be
hand,
d and
buy c.
happy to engage in the
standing alone, selling
a
would,
It
(selling
c
By contrast, household
and buying c.
and buying
e)
Edgeworth-process
buying b)
standing alone,
nei.ther circle,
trade.
c and
Nevertheless,
In
the entire transaction
a
hold
agreeing to participate in the left-hand circle
change
.
to
would be an
taken as
1
3
standing
but would not be willing so
,
participate in the right-hand one (selliing
this circumstance,
circle
right-hand
would be willing to participate in the left-hand circle
alone
on
whole can be an Edgeworth-process trade,
house-
with
in
for household 3's agreement to participate in the
ex-
right-
hand one.
Without
the strong assumption that all commodities are held
in positive amounts by all households,
there is nothing to
pre-
vent the possibility that the only utility-im.proving trades
pos-
sible
at
given prices are compound.
4
The assumption that
rraces will nevertheless take place seems very
anc
SG~e'w~.Er
cu"
ci place in
a
strong,
non-cooperative,
sucn
however,
competitive
.ng
Where only utiliry-improving circle trades are involved, one
can
imagine
listing
rhe
prices
announced and
each
household
co-imodiries that it would like ro buy and
would like to sell.
circle
being
Someone (the "market") then arranges
trades accordingly.
that
then
it
(small)
If each household understands
that
any one of its commitments to purchase may be paired with any one
8
of
commitments to sell,
its
then
place,
takes
offers will be made in such
tion of compound trades,
preference
prices)
on the other hand,
At the very least,
mation.
way
any
that
The construc-
requires more infor-
requires households to specify
it
ordering for goods,
the
a
utility-improving trade is simple.
(small enough)
pairing
other
whether or not any
in the sense that
(at the
given
household is willing to exchange any good lov;er
in
the ordering for any good higher up (at least in small amounts)
To put it another way,
a
.
the construction of a compound trade
requires considerably more information about preferences than the
construction of
a
This is particularly so for trades
simple one.
more complex than that of Figure
in which many more than two of
1
the households participate in one or more circles as the gyi^ piQ
gys for obtaining participation in another one.
In
does
the
present state of Edgeworth-process
not matter.
With stability proofs depending
commodity
holdings by all participants,
construct
compound trades.
If that
tion is ever to be relaxed,
however,
1:0
that
have
it
on
this
positive
is not necessary
to
unreasonably strong assumpit
would be very desirable
proof of Sdgeworth-process stability that assumes only
a
trade
possible
analysis,
and
takes
place if
does
si.T.ple
utility-ir.proving
not require participants
to
find
trade
is
compound
trad-es.
In
context,
any event,
it
is
in or out of the specific
obviously
Edgewcrth-process
interesting separately
simple and compound trades, and
I
shall do sc.
to
consider
begin
I
simple,
is
then
which
least one of the circles of
at
trade
If an Edgeworth-process
simple trades.
v;ith
Hence, in consi-
composed is itself an Edgeworth-process trade.
the maximum number of participants required for a
dering
Edgeworth-process
trade,
it
is
it
simple
trade
suffices to assume that the
involved is itself just a single circle.
Furthermore,
circle,
then
a
given commodity occurs twice in
households
2,
and
Finally,
and,
1
can be removed from the trade.
and
can be removed.
2
can all be removed,
3
x
=
in any case,
from the trade.
It
trades,
then households 1,
possible.
making bilateral trade
5
then
If x = b,
trade,
gains from
implies that household
= d
x
If x = a,
can be removed
5
There is nothing special about this example.
in considering simple
that,
follov7S
it
If x = c,
is impossible if household
d
be
Here, a, b,
2,
is the same as any one of the other four commodities.
1
can
a
Suppose that commodity x
and d are all different commodities.
then household
such
number of participants in the circle
the
Consider the trade diagrammed in Figure
reduced.
c,
if
Eagev;orth-process
suffices ro looK at circles with the same number
of
notation as foliov/s.
Definition
which we
r.ay
as well take to be
comm.ocities,
such
that,
household
which we
for
i-i-1,
is a circle of households,
A standard t-trade
3,
1
i
{1,
.
.,
.
miay
as well take to be also
i
t,
<
while household
household
t
1.
10
i
and
t},
{1,
sez
a
.
.
sells commodity
sells commodity
t
to
.,
i
of
z]
to
household
dard
when considering a stan-
shall adopt the convention that,
I
commodity
t-trade,
each household
dity t-1.
= 1,
i
is taken to be comnodity t,
denote {i-1,
I
sells commodity
t
.,
.
.
that
so
and buys commo-
t
by S(i).
i}
The following fairly obvious fact is central to the analysis
of simple trades.
Lemma
Consider any household,
3.
commodities, a, b, c, with H's holdings of
Suppose
selling
at current prices,
that,
a
and buying c.
£l0Pij As before,
(2)
selling
U_/U
p /p^
<
and buying c.
would
.else
Evidently,
This
<
in
p_/p,_^,
leads
would also
utility-increasing:
a
,
p,
by
(1)
and buying b.
and p
,
Let
respectively.
,
since H could increase utility by selling
,
either
find selling b and buying
U_/U.^
H
denote H's utility function by U(.),
the prices of the three goods be p
Then
and b both positive.
at the same prices,
Then,
or
c
a
of
could increase utility
H
find one of the following trades to be
selling b and buying
and any triplet
H,
c
U.
7U
<
p.
/p
in which case H
,
increasing,
to be utility
which
or
a
and
almost
all
which case H would find selling
to rhe following lemma from
a
later results are derived.
Leruna
4.
S'J.ppcse
is possible with r
S
-)
holds
a
>
2.
tr.ar
an Edceworrh-process standard r-rrade
Suppose furrher thar household
positive ar.ount cf so-e co.TJnodity
11
j
(1
i
i
j
(1
£
^
i
t)
,
involving no more than t-1 households.
Without loss of generality, we can take
£l2of^
which certainly holds commodity
household
1,
dity
where
j,
<
willing to sell commodity
That is,
ble.
household,
j
households {1,
selling commodity
1
.
.
a
1
is
it
is
commodity
standard j-trade possican trade with each
j}
.,
to household g+1 and household
g
to household 1.
j
or else
j
is willing to sell
Then there is
selling commodity
g,
either household
3,
and buy commodity t,
j
and buy commodity j.
also holds commo-
1
and buy commodity
1
Suppose first that household
I
1
By Lemma
t.
<
j
sell commodity
to
willing
1
Then
= 1,
i
Since
<
j
t,
there are at
most t-1 households involved in this trade.
Now suppose that household
and buy commodity t.
that
household
1
households
is willing to sell
In this case,
can trade with household
except
1
household
g
t
households {j+1,
selling commodity
sells commodity
sells couunodity
j
Theorem
1.
(A)
t,
1}
.
.
.,
to household
+ 1
-
g+1,
and
1,
The number
of
this
is
j),
and
1.
>
j
(t
j
to household
to household j+1.
involved in this trade is
less than t, since
t
g
coinmodity
Per any -,
1
<
m i n, lez x
(-)
be ihe
nu.TJDer
of households holding ar lea.sz m com^modities in positive amounts.
If
there
exists
=
sirr.ole
exists one wizh at -ost
(5)
For any k,
Edceworth— i^rocess
t, (m)
^ h,
= .Max
{h - x (m)
then
there
n - - + 2}
parti-
trsce,
,
let y(k) be the number of commodi-
1
<
k
ties held by at least
k
households in positive amounts.
12
If there
exists
with at most t2(k)
Eieofj
exists
Since
t
)
,
h - k + 2)
Then
n -
>
there
suppose that
t
t, (m)
>
2)
i
,
goods not in
2)
must hold some good involved
trade that is not in S(i).
Since
t>n-m+22
in
household
Let that
holds at least (m -
i
-
(m
participants.
standard t-trade with
must hold at least m goods.
i.
}
at least one of the households involved
h - x(m),
>
household
Since
- y (k
Edgeworth-process
trade
the
{n
Without loss of generality,
(A)
an
t
Max
=
one
then there exists
simple Edgeworth-process trade,
a
2,
be
S{i).
the
in
Lemma
4
now
yields the desired result.
(B)
Again
suppose that there exists
standard t-trade with
one
k
>
Since
t_(k).
t
>
y(k),
n -
least
at
of the commodities involved in the trade is held by at least
households.
by at least
h -
t
Edgeworth-process
an
(k
Let that commodity be commodity j.
(k
- 2),
-
2)
households,
i,
with
j
Then
not in S(i).
j
is held
Since
t
>
at least one such household must be involved in the
t> h-k-r22
trade. Since
from Lemma
4.
Corollary
1.
If
there exiscs
2,
a
the desired result again follows
siniple
Edgeworth-process
trade,
then rhere exiscs one wirh ar most
z* = Min
{Kin t^(x). Kin t^lk)}
participants (where rhe notat-icn is as in Theorem
^^^^-
Corollary
one
<-\
2.
1)
.
— ..•^,,_
If a simple Edgeworth-process trade
exists with no more than Kax {2,
participants
13
Kin
(h
-
exists,
x(n),
then
n - y(h))}
participants.
Set m = n and
ElCSfji
Corollary
2
= h
k
states that, if
in Theorem 1.
a
simple Edgeworth-process trade
require
it need not
more than two participants,
requires
more
the number of households net holding all commodities or the
than
number of commodities oQi held by all households.
Corollary
3
(Schmeidler)
Edgeworth-process
If a simple
.
trade
exists, then one exists with no more than n participants.
EiQQij.
Follows from Corollary
Corollary
4.
2
and the fact that y(h) ^ 0.
Suppose that at least h-2 households hold m
commodities (not necessarily the same ones)
Edgeworth-process trade exists,
such
a
Then,
,
^
2
if a simple
trade exists with no more
than n-m+2 participants.
Eiopf^ In Theorem
Corollary
k
^
2
simple
5.
(A),
1
x(m)
^ h-2.
Suppose that at least n-2 commodities are held
households (not necessarily the same
Edgeworth-process trade exists,
Then,
ones.
such
a
by
if
a
trade exists with
no more than h-k+2 participants.
In Theorem
riSPfj.
1
(B), y(k)
2 n-2.
These results coviously imply:
Corollary
6.
held
commodities
all
Suppose that either
(a)
at least
in ocsitive am.ounts cr
h-2
(b)
at
households
least
ccm-.odiries are held in posirive amiOunts by all households.
simple
Edgeworth-process trade exists,
worth-orocess trade exists.
14
then
a
bilateral
n-2
If a
Edge-
This is
Corollary
7
a
slightly stronger version of:
(Madden)
.
Suppose that all households hold positive
If a simple
amounts of all commodities.
exists, then
a
Edgeworth-process trade
bilateral Edgeworth-process trade exists.
5^_2iippl£-Tie£3£Si_£sD_Eyiibei_E£5yl£s_D£_Dbi§iDed2
The number of participants required for an Edgeworth-process
depends- on the distribution of commodity holdings and,
trade
course,
on the distribution of tastes.
of
ob-
The results so far
for simple trades have made no assumptions on the distri-
tained
distribution of
bution of tastes and have only characterized the
commodity holdings by the two functions,
tively,
x(.)
and y(.),
(respec-
the number of households holding at least a given number
commodities and the number of commodities held by at least
of
a
given number of households)
Since that information does not completely characterize
the
holding of commodities by households, it is easy to see that more
on the pattern of such holdings can make a consider-
information
able difference.
Assume
(A)
exists
h = n
with n even.
2,
only the ccTu-cdities in S(i)
and i, with commodity
while x(m)
for
K
>
1
k
i h.
<
>
2.
Suppose
that
Edceworrh-process standard n-trade with
an
holding
To see this, consider the following example:
=
for m
2.
Similarly,
This means that
Evidently, t*
= n
t^ (m)
y(2)
.
= n,
5 n = t2(k)
in Corollary 1,
household
comr.ocities
(that is,
taken to be commodity n)
>
i
i-1
Then x(2) = h,
while y(k)
for
and,
oovicus that the standard n-trade cannot be reduced.
15
there
1
<
=
C
m i n and
indeed, it is
nov/
holds only the commodities
i
instead of
and i+1,
are the same as in
and y(.)
however,
In this case,
in
the
and
i-1
Then the functions
(with commodity n+1 taken to be commodity 1),
x(.)
i
so that t* = n, as before.
(A),
every household,
i,
owns
standard n-trade that is not in S(i),
good involved
a
Lemma
so that
4
shows the existence of an Edgeworth-process trade with fewer than
n
participants.
it is not hard to show
fact,
In^
there
that
exists such a trade with n/2 participants, since, along the lines
of
the proof of Lemma 4,
every odd-numbered participant in
the
standard n-trade can bypass participant i+1.
more
Somewhat
Corollary
1
surprising than this is the fact that t*
trades
need not be the least upper bound on required
given only the information in x(.) and y(.).
of
To see this, consi-
der the follov;ing example.
Suppose that there exists an m*
3,
2
with n
- m*
+
2
m*
>
such that:
for m i m*
h
(1)
x(m)
=f
for m
L-
In other words,
n
;
m*
>
every household owns exactly
In this case.
(2)
t-
y(k)
{=}=>
•^
h for
'''
h - k + 2 f or k ^ -
IT.
>
r*
and, similarly.
r
n for k
>
16
m*
z-*
for
k
^ m*
for
k
>
m*
,
a:
=
cc.T~odities
Then
t*
>
=
t*
m*
so that y(t*)
,
- m*
= n
t, (m*)
=
+
Note
2.
In other words,
0.
assumption,
by
that,
there is no commodi-
ty ovmed by as many as t* households.
Now consider any simple Edgewor th-process trade involving t*
Without loss of generality,
households.
the
standard t*-trade.
such a trade.
Each household,
n
- m*
+ 2 goods
owns at least
i,
involved
(m*
-
2)
in
goods
In order for none of these goods to be involved in
not in S(i).
the
There are
we may take this to be
trade,
those
participants.
-
(m*
Since
goods must be the same for
2)
we know that this is impossible.
all
t*
Lemma
4
tells us that there is an Edgeworth-process trade with fev/er than
participants.
t*
It remains to show that the functions x(.)
(1)
i
identified
commodity
{i-1,
(1
j
^
i,
.
.
com^T^odity n and
v,'ith
j
<
given in
This is easily done by having household
can actually occur.
own commodities
and y(.)
m*-2-)
.,
i+m*-2},
commodity n+j
with commodity
with
identified
.
Evidently, further work along these lines can produce stronger
lower bounds on the numoer of required participants than t^
number
co-jnodities owned by sets cf the households
of
and
the
total number cf households that own sets cf
r
ecuirements
household
even en rhe nuroer cf cozniocities held bv a
already
disequilibrium
single
strain what we can reasonably assume aoout
process,
I
do net believe that such further
17
a
re-
however,
this,
of
the analysis of this problem may
and
be
a
suitable subject for further research.
now
I
to the more complex case of
turn
The principal reason for
Here results are not so easily come by.
this
is
that the result of Lemma
trades.
To
household
1
does not hold
4
onsider Figure
see this,
trades.
compound
again and
1
compound
for
recall
that
participates in the left-hand circle only in order to
participate in the right-hand one, while the opposite is true for
household
Suppose that household
3,
by Lemma 3,
household
else willing to buy
trade
between households
the former case applies,
purchasing either
3
Then,
In the latter case, bilateral
and
4
is possible,
and that household
4
but suppose
that
has no interest in
If the left-hand circle were itself an
or b.
a
owns commodity c.
is either willing to sell c and buy d or
4
and sell e.
c
4
Edgeworth-process trade, bilateral trade between households
4
would
be possible,
merely to sell
willing
but now it is not.
d
and buy c;
it
that gives it the opportunity to sell
replace rhe lefr-hand circle by
hclcs
1
household
and 4,
3
a
a
Household
is
1
is doing so only
and buy c.
and
not
because
If we try to
bilarerai rrade berween
will no longer
1
receive
e.
houseSince
household 3's participarion in rhe right-hand circle is conditional
on its getting e,
z.he
righr-hand circle.
no
reason to sell
d
household
3
will no longer participate in
But, in rhai: case, household
and buy c,
and the whole trade
1
will have
will
break
d own
Moreover,
it
is not true
(as
IS
it is in the case
of
sirriple
that the existence of
trades)
commodity involved in the
a
held by all participants implies that the number of partici-
and
consider Figure
pants
can be reduced.
there
are two circles each involving the ggaie three
(Household
household
circle
To see this,
2
has been exhibited twice for clarity.)
2
gains utility from participation in
3
1
household
owns a and d,
owns a and c.
Thus,
For convenience,
Denoting
a
2
1
ov/ns
households.
Suppose that
right-hand
the
while
Assume that house-
and 3.
a,
Here
3.
and loses from participation in the left-hand one,
the opposite is true for households
hold
trade
and e,
b,
and household
is owned by all households.
assume that all prices are equal to unity.
the utility function of household
i
by
U'''
and
marginal
utilities by subscripts, the information given implies:
(4)
U^
>
U^^
>
5)
U^
>
U^
;
U^
>
U^
>
a
b
U^
,
U^
d
<
\P-
e
t
and
(6)
e
c
U,^
o
Suppose thar, in addirion.
(~)
Ut
>
U"
/Ri
"2
.
-2
/c^
"3
.
..3
a
e
Inequalities
(4)
>
U'
„2
^
„2
..3
^
..3
D
'
and
a
(7)
imply that household
19
1
will
not
sell either a or d in order to purchase b or e, while
that household
will not sell
2
a
in order to purchase
bilateral trade between households
inequalities
Similarly,
not
will
sell
household
2
2
(9)
2
and
3
purchase c,
while
(6)
and
(9)
not sell a in order to purchase d.
will
between households
and
1
3
Hence
(8)
3
that
implies
Hence bila-
cannot take place.
implies that household
(4)
d.
imply that household
while
e,
implies
cannot take place.
will not sell a in order to purchase c.
Finally,
to
and
purchase b or
a to
teral trade between households
order
(7)
and
1
(8)
1
will not sell
in
a
imply that household
3
trade
Hence bilateral
cannot take place.
Thus, no bilateral trade is possible, and the compound trade
shown in Figure
2
participants.
Basically,
result
in
the
cannot be reduced even though a is ov;ned by all
the reasoning that led to a different
case of simple trades breaks
because the trades are all interdependent.
that
owns
household
a
c,
a
and sell c,
1
and
3
and,
ro replace household 2 and deal
the
if
trade,
3
have no direct interest in such
a
this
directly
trade of
a
and enga-ginc in it would renove household 2's reason for
This does not m.ean that it is impossible to obtain
results,
That
3.
In the compound case being examined, however,
with household 1.
for
the household
circle were itself an Edgeworth-process
would allow household
households
Thus,
before)
(as
good that it is not trading is household
would be glad to purchase
right-hand
down
however.
positive
In fact, the parallelism, between comumodities
and households breaks down in the case of compound trades, fo;
20
remains true that the presence of
a
multilateral trade permits
reduction in the number of parti-
a
consider the following lemma which gives
To see this,
cipants.
household owning all goods in
a
the result parallel to (and v;eaker than)
that of Lemma
the
for
4
case of compound trades.
Lemma
with
t
households,
set
participating in the trade, such that the
and i',
commodities
of
owned in positive amounts
all commodities being traded
includes
hold
i
by
household
i
(bought or sold) by house-
with
Then there exists an Edgeworth-process trade
i.
two
Suppose further that there exist
participants.
2
>
exists
trade
Suppose that an Edgev;orth-process
5.
no
more than t-1 participants.
Household
ErSPfi
site
good
another.
a
(a
i'
linear combination of ordinary goods)
(For example,
combination of
household
i'
household
i
household
and d.)
a
buys,
and
S
1
in Figure
buys
1
c
If
household
trade berveen households
increasing
replace household
i'
would
i
to sell B and buy S,
in the original trade,
i
and
2,
£,
Then
i'
a
find
selling
is possible.
S
i
can
and buying
(parts of)
Corol-
and 7, above, to the case cf compound trades.
21
it
bilateral
then ho-jsehcld
This leads im.-ediarely zo zhe extension of
laries
and sells
the com.posite good that it sells.
strictly utility increasing to sell 3 and buy 3, then
urility
selling
and
Let B denote the composite good that
owns both B and S.
Edgewor rh-prccess
compo-
can be thought of as buying one
(As
ties,
and x
(n)
the number of households holding positive amounts
of all commodities.)
Theorem
then
one
Suppose that at least h-2 households hold
all
If an Edgeworth-process trade exists,
2.
exists with no more than Max {2, h - x(n)} participants,
ElQQZj. Obvious from Lemma 5.
Corollary
8.
commodities in positive amounts.
If an Edgeworth-process
trade
exists, then a bilateral Edgeworth-process trade exists.
This is a slightly stronger version of:
Corollary
positive
9
amounts
trade exists, then
(Madden)
.
Suppose that all
of all commodities.
a
If an
households
hold
Edgeworth-process
bilateral Edgev/orth-process trade exists.
22
5
NOTES
*
I
am indebted to a referee for comments and to Peter
Diamond
for
error.
I
helpful discussion but
A.
for
responsibility
retain
aunt
wish to dedicate this paper to the memory of my
and dancing teacher, Ethel Fisher Korn.
Note, however, that existing proofs of stability in the
1.
process require the positive endowment assumption any-
Edgeworth
1962, Uzawa, 1962, Arrow and Hahn, pp. 328-337). Thus
way (Hahn,
Madden's result formally answers the criticism that large numbers
of traders may be required. That answer will not be satisfactory,
however,
the analysis is ever to be advanced beyond
if
such
a
strong assumption.
Strict quasi-concavity is not to be interpreted to
2.
out
possibility of satiation in one or more (but
the
goods,
rule
not
all)
that indifference surfaces can become parallel to one
so
or more of the axes.
The assumption of dif f erentiabilitv can almost certainly
weakened
to
the
requirement that
unique supporting hyperplanes
1978,
281), but there
p.
Apart
lirrle gain in co-plicaring the exposiricn to zo so.
seerr.s
as
(Madden,
have
surfaces
indifference
the following.
bananas
Suppose that househc"'d
1
2
and
3
for
bananas.
though
tions.
a
Such
a
1
sells carrots to
2
(1)
s
^^
^— ^_
for apples anc to
trade can require three participants
particular household
not.
do
;
wnicn
and
re'^ards apples
as perfect complements while households
be
participates in all
j
even
transac-
This makes calculation of the minimum number of partici-
pants tedious at best, and, as the circumstance involved is quite
special,
it
(although it might
be
possible
to handle it along the lines of the treatment given
to
does not seem worth pursuing
"compound"
trades below)
subset
commodities
of
.
(Note that if ell agents view a given
as perfect complements
using
the
same
proportions, then, without loss of generality, that subset can be
renamed as a composite commodity.)
The
3.
'
principal complication avoided is that
of
keeping
of gifts in which one household gives a free good to
track
ano-
ther without getting anything in return.
4.
bilateral
It
is easy to see,
Edgeworth-process
that the existence
of
a
implies the existence
of
a
however,
trade
simple bilateral Edgeworth-process trade.
5.
Schmeidler
'
s
result,
while true,
does not seem readily
provable for compound trades along the lines here developed.
24
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FIGURE
1
FIGURE
2
FIGURE
3
/
5582 096
<5
-</'
wn aBi!«™^
Jill
3 cioao
5Mb
33b 3..
QQ^,.-33,