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working paper
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of economics
ON THE EFFICIENCY OF KEYNES IAN EQUILIBRIUM
by
Eric Maskin
Jean Tirole
Number 258
June 1980
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ON THE EFFICIENCY OF KEYNESIAN EQUILIBRIUM
by
Eric Maskin
Jean Tirole
Number 258
June 1980
Introduction
Theorems characterizing equilibrium in economics that fail to
satisfy some of the strictures of the Arrow-Debreu model have recently
abounded.
In particular,
Grossman and Hart
[8],
papers by
Grossman [7],
and Hahn [10] have studied the efficiency
properties of equilibrium with incomplete market structures and have
established analogues of the two principal theorems of welfare economics.
In this paper we undertake a study of the efficiency of a model with
a
More precisely, we
different kind of imperfection, fixed prices.
start by showing (Proposition 2) that Grandmont's
[5]
notion of
K-equilibrium in fixed-price models, which embraces both the Dreze
and Benassy
[1]
[3]
equilibrium concepts, is equivalent to a kind of
Social Nash Optimum
2
,
in which optimization is incompletely coordinated
across markets and where the control variables are quantity constraints.
Viewing K-equilibria as Social Nash Optima, we believe, permits a
better understanding of the structure of the set of equilibria (see Froposition 3)
K-equilibria possess two important properties:
order (the requirement
that at most one side of the market can be quantity-constrained) and
voluntary exchange (no one trades more of any good than he wants to)
After studying K-equilibria, we examine order and voluntary exchange
and their connection with the two principal concepts of optimality
in a fixed-price economy:
constrained Pareto optimality (optimality
relative to trades that are feasible at the fixed prices! and implementable
Pareto optimality (optimality relative to feasible trades that satisfy
voluntary exchange).
We show first (Proposition
definition of order (c.f., Hahn
[9]
4)
that the usual
and Grandmont [5]), in fact,
-2-
implies that exchange is voluntary (assuming that preferences are convex
and dif f erent iable)
.
We,
therefore, introduce a less demanding notion
of order, weak order, which is distinct from voluntary exchange.
(Corollary
6)
that, with convexity and differentiability, order is equivalent
to the conjunction of implementability (voluntary exchange)
We then demonstrate (Proposition
although
x>7eakly
7)
and weak order.
that constrained Pareto optima,
orderly, are, except by accident, non-implementable and, hence,
Furthermore,
non-orderly.
We prove
(Proposition
8)
implementable Pareto optima
need not even be weakly orderly (although in an interesting special case
absence of "spillover" effects
In section
1
— they
— the
will be)
we introduce the notation
and
definitions.
2
treats the existence and characterization of K-equilibria.
3
studies
Section
Section
the relationships among order, weak order and voluntary exchange.
We consider optimality in section 4.
Finally, in section 5, we compute
the dimension of the various sets of optima.
1.
Notation and Definitions
Consider an economy of m +
1
goods indexed by h(h = 0,1,..., m)_, whose
price vector p is fixed (p„ = 1), and n traders indexed by i(i = l,...,n)
where trader
X
i
has a feasible net trade set X
c= R
.
We assume that
is convex and contains the origin (so that trading nothing is possible)
and that trader i's preferences (denoted
strictly convex on this set.
dif ferentiable as well.
by>.) are continuous
We will at times require preferences' to be
Following Grandmont [5],
for such an economy as follows:
and
we define an equilibrium
-3-
Definition
1
A K-equilibrium is a vector of net trades (t ,...,t
:
associated with the vector of quantity constraints
(with Z
1
1
1
Z
0,
<_
1
>_
Z
0,
=
and ZT =
-°°,
1
is feasible at prices p:
(b)
quantity constraints are observed
(c)
exchange is voluntary:
),..., (Z
,
Z
))
^
.
'vl
t
^_
=0}
|p-t
f) it
<_
Z,
:
is the
t
= X
X
e
t
t
Z
,
such that, for all i,
+°°)
(a)
((Z_
)
Z
maximal element among
-
net trades satisfying (a) and (b)
(d)
exchange is orderly:
l
1
>
t
t
and t
j
>
t
j
1
Y
h
(Z,Z)
-
= {t
1
X
e
then (t^ - zj)
h
h
,
'
i
1
1
IZ
'—k
aggregate feasibility:
(e)
for some commodity h, (t
if,
<
-
k
<
-
St
—h
n
1
1
t,
(t{ - zj)
>
—
0,
)ey^(Z_,Z)y-y^(Z,Z)
t
,
where
Vk^O.h}.
Z,
k
=0.
i
For any trade
"canonical" rations
then
if t} >
h
—
*
Z^t 1 )
h
by trader i, we may confine our attention to the
t
_Z(t
)
and Z(t ), associated with that trade:
= tj and Z^Ct
h
—h
1
=
)
0;
if
'
<
—
t""
h
0,'
then
zV
1
)
for h 4 0,
=
and
zNt
—h
Voluntary exchange implies that agents are not forced to trade
more of any good than they want to.
An allocation characterized by
voluntary exchange is said to be implemen table.
Definition
(t
,...,t
2
)
:
Formally, we have
An implementable allocation is a vector of net trades
satisfying conditions (a),
(b)
,
(c)
,
and (e) for the canonical
rations associated with these trades.
A market is orderly if buyers and sellers are not both constrained
on that market.
The next two definitions represent alternative attempts
to capture the idea of order.
First we introduce property (d'), which is
equivalent to (see Proposition
1)
but somewhat easier to work with than (d)
1
)
= t^
h
(d'):
A vector of net trades (t ,...,t
)
satisfies property (d') if,
for all markets h, there exists no alternative vector (t
lly,
and
(Z(t ),Z(t ))
1
I
1
t,
= 0.
such that
£
t
,
...,t )e
(with at least one strict preference)
t
3
h
The following definition is standard:
Definition
3
An orderly allocation is a vector of net trades (t
:
satisfying (a),
(b)
,
(d'), and (e)
,
...,t
)
for the canonical rations associated
with those trades.
The problem with the above definition of an orderly allocation, if
one is attempting to distinguish between the notions of order and
voluntary exchange, is that it itself embodies elements of voluntary
Indeed, we will show below (Proposition 4) that, with
exchange.
differentiability, the above concept of order implies
Heuristically, this is because, under the definition of order,
exchange.
the trade
voluntary
in Y,(Z(t ), Z(t )) could be preferred to t
t
forced trading on
on market h.
a
market k 4 h and not because
t
simply because
involves
t
relaxes a constraint
Therefore, we propose a new notion of order that
is free from the taint of voluntary exchange.
We first define property (d")
(d")
:
A vector of net trades (t ,..,,t
)
satisfies property (d") if, for
~1
~n
all markets h, there exists no alternative vector (t ,...,t )
...
such that, for each i,
and
E
i
tj =
0.,
t
where yJCtVft
.
e
—i
11
Y,
(with at least one strict preference)
t
1
P
i=l
.
>
e
X
1
|
tj = tj, k f 0, h
}
i
(t
)
-5-
Notice that properties (d') and (d") are identical except that the latter
requires that alternative net trade vectors be identical to the original
trades in all markets other than h and 0.
Definition
,...,t
(t
4
A weakly orderly allocation is a vector of net trades
;
satisfying property (a),
)
(b)
(d")
,
,
and (e) for the canonical
rations associated with the trades.
An orderly allocation is obviously weakly orderly.
Below we shall be interested in the Pareto-maximal elements in the
sets of K-equilibria, implementahle allocations, orderly allocations
and weakly orderly allocations, which will be called K-Pareto optima
(KPO)
implementable Pareto optima (IPO), orderly Pareto optima (0P0)
,
and weakly orderly Pareto optima (WPO)
respectively.
,
A still stronger
notion of optimality, selecting Pareto-maximal elements in the set of
all feasible allocations, is constrained Pareto optimality:
Definition
5
A constrained Pareto optimum (CPO) is a Pareto optimum of the
:
economy for feasible consumption sets X
= X f\ {t
p*t
= 0)
I.e., it
.
solves the program
max
1
subject to
1
and Y.t~ = 0,
i=l
for some choice of non-negative A 's, where the u 's are utility functions
(*)
Z
X
u
(t
)
t e
X
representing preferences over net trades.
Characterization and Existence of K -equilibrium
We first check the consistency of the definitions:
P roposition 1
{K-equilibrium allocations} = {Implementable allocations} C\
2.
:
{Orderly allocations}.
Proof:
is not satisfied for
(d')
n
X
(t
?
We need just check that (d) and (d') are equivalent.
.
.
.
,t
)e n
1=1
Y^UCt
h
1
)
(t
^(t
,...,t
1
))
)",
4
If
there exist h and
such that
(t
,...,t
n
)
Pareto-dominates
(t
,...,t
n
)
-6-
Because
(t
...,t
,
)
maintains equilibrium on market h, we can infer that
at least one agent is demand-constrained and one supply-constrained in
(t
,...,t ), which contradicts (d)
satisfied by
such that
,...,t ), there exist h,
(t
(t\
t
j
t
,
EY^CZCt^ZCt
)
1
(t
on the other hand,
If,
.
J
)
1
))
x
Y
Pareto-dominates
J
h
i,
j
,
(d)
is not
and
J
(Z(t j ), Z(t ))
(t
1
t
,
J
and
)
(t^"
1
-
t,
h
h
)
(t^
h
- t?
)
<
h
0.
Therefore, if the constraints on market h are relaxed by the amount
min
{|t?" -
t^"
|
,
|t^ - t^|},
Q.E.D.
property (d') is contradicted.
We now turn to the characterization of K-equilibria in terms of
social Nash optima.
As indicated above, the idea behind a social
Nash optimum is to consider m uncoordinated planners, one for each
market h (h^O)
,
who choose quantity constraints to maximize a weighted
sum of consumers' utilities, subject to keeping equilibrium on their
own market and given the rations chosen by the other planners.
A
social Nash optimum is then defined as a Nash equilibrium of that "game."
Formally:
Definition
6
:
For each i, let
t
(p
Z_
,
,
Z
)
solve trader i's preference
maximization problem, given prices p and rations
i
the indirect utility function V (p,
—i
i
Z_
,
Z
i
)
i
Z_
°
= u (t
x
is a utility function representing i's preferences.
for fixed positive weights
{\
}
,
—
Z
—
and Z
i —
(p,Z_ ,Z
Define
.
)),
where u
Suppose that,
the manager of market h chooses Z^ and
n
i
i
i
—
A
I
V (p, Z , Z ) subject only to
for each i so as to maximize
—
.
h
n
i= 1
°i
i
—
the constraint £ t
Z ) =
and taking as given the rations
(p, Z_
i=1
i
-i
Z
and Z in each market k 4 h, Q.
The allocation corresponding to the
h
.
.
.
.
,
"
K.
<
K.
equilibrium of such a "game" is a
social Nash optimum
.
By definition, a social Nash optimum is an implementable allocation.
From the equivalence between (d) and (d'), it is also orderly.
Conversely,
-7-
a K-equilibrium is a social Nash optimum.
We have thus characterized
the set of K-equilibria:
Proposition
2
;
{K-equilibria} = {Social Nash optima}
The set of weakly orderly and orderly allocations can be characterized
similarly.
In particular, a weakly orderly allocation is equivalent
to a social Nash optimum where the instruments of planner h are the
trades {t
}
on his own market.
The characterization of orderly allocations,
although straightforward, is less natural because of the hybrid nature of
these allocations:
an orderly allocation is a social Nash optimum where
each planner chooses trades on his own market given the canonical rations
associated with the allocations chosen by the other planners.
In other
words, each planner assumes that the others have the power only to
choose rations, whereas, in fact, they choose the actual trades.
Under differentiability, Corollary
6
below guarantees that
this kind of social Nash optimum is identical to that of Definition 6.
As a by-product of the character izaton of K-equilibria as social
Nash optima, we obtain a straightforward proof of the existence of
a
K-equilibrium at prices p based on the Social Equilibrium Existence
Theorem of Debreu
Proposition
3
:
[2]
Under the above assumptions, for any vector of positive
prices p and any choice of positive weights {A
Nash optimum
and, hence, a
},
there exists a social
K-equilibrium, associated with those prices
and weights.
Proof
:
We must show that each planner faces a concave, continuous
objective function and that his feasible strategy space is a convex- and
compact-valued correspondence of the strategies of the other planners.
-8-
To see the concavity of the objective function, consider two alternative
((Z 1
choices of constraints,
1
For any a,
a
/
)
, .
.
.
feasible for the constraints (aZ
1
,
7
(Z
,
Z
n -n
.
Z
,
+ (1-a)
Z_
aZ
1
£L
1
((Z_
~n
),..., (Z
Z
,
+ (1-a) t
)
,
..
and
,Z ))
1
the trade at (p
l,
<_
^ls
Z1
,
1
X
(p,Z_,
+ (1-a)
Z
z"
£ik
Z )).
,
is
)
The
).
concavity of the utility functions then implies the concavity of the
Continuity follows immediately from the
planner's objective function.
The set of feasible strategies for planner
continuity of preferences.
defined by
h is
z£»| j.tjCp,
{((zj, zj;),...,(z£,
and is denoted by
r,
restrict
Z.,
r,
h
(Z
h
,,
—)h('
N
,
(Z.,
Z.,
— )h(.,
.)
Without loss of generality, we can
.
)h(
to canonical quantity
j constraints.
i
.)
)h(
is not empty because it contains
(0,0).
the preferred vector in y (Z\,
Zv
<
—
< max
—
Z,
h
{ t,
0}
,
h
To see that
i
(?£.
for
Zjj))
<
and
a <
CZ_
h
1.
,
If
T
h
Z
h
(Z
—
.,
.
min/t
,
,
0l< Z .
)h(
r
is convex, choose
,.)
)h(
Z.
,
in
)
(Z
Z
h
for example,
t,
n
because constraints are canonical,
and tJ;(p,aZ^ + (1-a) Z
,
h
(Z N
Z.,
,,
—)h('
.
,)
)h(
It is bounded because if t
is
and
<
—i—l
_
°
t.'s.
,)
r,
is closed because of the continuity of the
It
.
,,
5
= o>
z£, zj
h(i z£, zjh( )
Z
fe
)h(
,a Z
J;
)
(p,
t,
n
Z,
—
h
(p,
Z,) =
—h'
,
h>
Z., ,
7 h(
,
Z,
h
,
Z.,
Z,
n
Z.,
; n
,
.,
)h(
Z^)
Z
n
.)
=
Z
,),...
h
Z^
+ (1-a) (Z^,
Z,
then,
n
v.
,
((Z, ,
h'
h
and consider a(Z_
+ U-«) Z*,
Similarly, for the upper constraints.
( Z,
Z.
.
)n{.
,
=
)
—Z,h
;
= aZ^ + (1-a) z£.
That the correspondence
T
is upper
hemi-continuous follows from the continuity and convexity of preferences.
It
is also immediate that
restrict the domain of
be canonical
I\
h
constraints.
bounded and convex.
I\
h
to
is lower hemi-continuous.
only those rations
(Z., ,,
-)h('
Finally,
Z N1 ,)
)hC
we can
which could ever
This domain is obviously closed,
We can thus apply the Social Equilibrium Existence
Theorem to conclude that a social Nash optimum exists.
Q.E.D.
,
Z^
-93
Order and Voluntary Exchange
.
We can now demonstrate that if preferences are differentiable and there
are at least three markets, order implies voluntary exchange.
For convenience,
we shall, from now on, delete the component of trade vectors corresponding to good
Thus
corresponds to trades t.,...,t on markets
m
1
m
on market h.
implicit trade t n = -£ p t
h h
h=l
Proposition 4:
If preferences are differentiable and m>
zero.
t
to m, with an
1
an orderly
2,
allocation is implementable.
Proof
Consider an orderly allocation
:
is not implementable,
in particular,
then,
if
If this allocation
,...,t ).
(t
(t*
...,t*
,
is the vector
)
of feasible, preference-maximizing net trades associated with the canonical
rations for
h
and
(t
.
,
.
.
such that
t,
h
,t
t*
,
)
4
= t,
t,,t.
,t., .)
h
)h(
t
feasible and
is
= 0)
and since
j
or
t
(ii)
and t,, =
t
t
choose h' 4 h.
,
1
t,
•
h
(A)
t
<
h
If
0.
(i)
+
At?"
n
=
t
E
holds, then, since
+ (1-A) t*
By convexity,
=
n
t?"
!
X
1,
(A) X-
F1
)
is not orderly, afterall.
combination of
t,
h
assumption above.
Because (0,
t
,
t Xl ,)
)h(
and
If
t,
h
|t
If
.
>
|
t,
h
= 0,
If
(ii)
then
t,
h
It*
L
(t
h
I
n
,...t ).
h
t,
_±
t
.
,
•
= A t
1
)
X
t
(A)
Therefore
is a convex
which violates our
is not orderly.
>
—
+ (l-A)t*
then, by strict convexity, (0, tv, ,)S-.
,...,t
t,
h
h
there exists
holds, then
= t*
and
1
Furthermore,
satisfies the canonical constraints
we again conclude that
1,
,
I
n
t" (A)
.
satisfies the canonical constraints associated with
1
'
Let
.
>
1
>
—
I
j^i
,...,t
(t
(Z(t ),Z(t )),
,
'h
I
h
h
h
t
1
t,
.
/
(t
ey
t
contradicts the order of
This, in turn, implies that either (i)
.
Then
cannot satisfy the canonical constraints associated
t
scalar Ae[0,l] such that
Then
t?",, t
'
=l
Therefore,
with
t~, =
E
(otherwise,
(ot
t
.
l
satisfies the canonical
If t
.
quantity constraints associated with
r
t
maximizes i's preferences among
by differentiability and coivexity, t
all net trades satisfying p«t
For this i, there exist
for some i.
t
associated with
Thus,
—
t
1
.
-10-
in all cases,
if
(t
,...,t
is not implemen table, we conclude that it is
)
Thus order implies implementability.
not orderly, a contradiction.
Q.E.D.
That there be at least three markets and that preferences be
diff erentiable are hypotheses essential for the validity of the
preceding proposition.
Consider, for example, a two-market economy as
represented in the Edgeworth box in Figure
Point A represents the
1.
initial endowment;
Figure 1
the line through A, prices;
indifference curves.
and the curves tangent to the line,
Any allocation between B and C is clearly
orderly, but not implementable since it involves forced trading by the
agent whose indifference curve is tangent at B.
is crucial,
To see that differentiability
consider a two-person three-good economy where agents have
preferences of the form log min {x
ferences, we can treat goods
1
and
,
2
x
}
+ log x
.
Given these pre-
together as a composite commodity,
since traders will always hold goods 1 and
2
in equal amounts.
economy is, in effect, reduced to two goods, and so Figure
becomes applicable.
1
Thus the
again
-11-
We next show that voluntary exchange and weak order together imply
To do so, we make use of the following:
order.
Definition
trade
6
Agent
:
is said to be constrained on market h for feasible
i
if there exists
t
feasible and
Proposition
(t
5
,
t,
t,
h
V-
,)
.
with It,
h
'
t
>
I
'
such that
t.
h
'
h
is
,)
t-.
,
)h\
.
If preferences are dif ferentiable,
:
(t.
{implementable allocations}!
{Weakly orderly allocations}^: (Orderly allocations}.
Proof
Suppose that (t ,...,t
:
vector of
)
is a weakly orderly and implementable
If it is not orderly, there exists market h
net trades.
and feasible net trades (t ,...,t
)
that Pareto dominate (t ,...,t
)
n
=
I
t,
and, for all i, t e Y. (Z(t
There are
Z(t ) )
)
h
h ,
1=1
two cases, depending on whether all agents i are constrained on market
such that
,
.
.
h for trade
Case
I
t
.
Every agent
:
i
constrained on market h for
t
.
Either everyone is constrained from buying or all are constrained
from selling - a mixture is not possible because of weak order.
Suppose,
without loss of generality, that everyone is constrained from buying.
Then
t.
h
=
for all i, and there exists
all i then (t ,...,t
with
(t ",..., t
),
)
j
J
such that t^ <
h
a contradiction of implementability)
h
(t^,
h
=
t,
for
h
satisfies the canonical constraints associated
constrained from buying on market h, there exists t^ >
is feasible and
(if
t ^
)n
.)
(
Vf j
,
tJ
•
.
Because
j
such that
is
(t,
n
,
t
;
n
)
{
Choose Ae(0,l) such that At J + (l-X)tr' =
h
h
0.
1
-12-
Let
t
J
of
= X (t^
t.^
,
+ (l-A)f3
)
By strict convexity
.
t
,...,t )'s implement ability.
(t
,1
that
,...,t
(t
Case II
n.
,
1
t
/^.
I
J
and therefore £t,
i
such that
,
yet
,
violation
,
t,
h
and |t
j
Let
S
be the
From strict convexity and implementability, we
Thus if tt = t* for all ieS, there exists
h
h
.
1
t
J
for all i, with strict inequality if and only if
t
1
2l
1
t
a
,
t
Not all agents are constrained on market h.
:
t,
t
.
is orderly.
)
set of all sellers.
I
(A)
Therefore, in this case, we conclude
Suppose, in fact, that the sellers are not constrained.
have
J
satisfies the canonical constraints associated with
(A)
t
(A)
>
0,
<
—
0.
a contradiction.
jg!S
J
such that
Therefore, there exists
<
t,
<
|t
h
h
S
'
Assume, for the moment, that
[}.
1
=
H^tf)
For each k e H,
.
1
1
t
(k)
(such a choice is possible since i is
t
-constrained on market k)
|
>
tjM
|
tjM
convexity, we can choose the
=
i £
J
t,
Define H = {k i is constrained on market k for
and, if
)
h
.
choose t* so that
A(t -t
>
t?
(1-A)
of elements in H.
|h|
t,
(t -t^)
'
s
(tJ\ t.
/
),
t
(k))^-.
so that there exists Ae(0,l)
for all keH, where
|h|
~
~
(1 A) ti5
°
t
By
.
for which
is the number
Define
S
A Z V-Qt)
l± =
+
1
if
,H|
>
rlf
1
t
,
if
|h|
=
is strictly preferred to t
=
By convexity,
t
for all keH.
But by differentiability, convexity, and implementability,
,
and, by construction
t,
t,
-13-
t
is the best trade that i can make if forced to trade
t,
k
on each market
Thus, in this case too we conclude that (t ,...,t
keH, a contradiction.
is orderly.
Corollary
6
Q.E.D.
If m > 2 and preferences are dif ferentiable then
:
{implementable allocations}
allocations}
4.
f)
{Weakly orderly allocations} =
{Orderly
{K-equilibrium allocations}
=
Optimality
We next turn to constrained Pareto optimality.
a
)
We show that although
constrained Pareto optimal allocation is weakly orderly, it is
ordinarily neither implementable nor orderly, at least when preferences
are dif ferentiable.
Proposition
A constrained Pareto optimum (CPO) is
7:
(i)
weakly
orderly (which implies that {Constrained Pareto optimal allocations} =
{weakly orderly Pareto optimal allocations} and
preferences, neither implementable nor (when m
(ii) with diff erentiable
>
2)
orderly, if it is not
a Walrasian equilibrium allocation, if each trader is assigned a
strictly positive weight in the program (*)
,
and if there is some (i.e.,
non-zero) trade on every market.
.
Proof
:
Let (t
n
,
.
.
.
,
)
be a CPO.
If it were not weakly orderly,
then trades could be altered on some
market h, leaving trades on other markets undisturbed, in a Paretoimproving way, a contradiction of optimality.
Therefore,
(i)
is established.
-14-
Suppose that
,...,t
(t
is not a Walrasian equilibrium allocation,
)
that preferences are dif f erentiable, that there is non-zero trade on
every market, and that all traders have positive weight.
establish that
(t
,...,t
is not implementable.
)
We will
Because it is not
Walrasian, there exists at least one market h and one agent
prefer a trade different from
k ^ 0, h.
If,
say, trader i
t,
h
,
,...,t
(t
)
is a net buyer of h, either he would like
follows immediately.
like to buy more.
who would
given his trades on other markets
to buy more or to buy less of good h.
of
i
If less,
the non-implementability
Assume, therefore, that he would
Because, by assumption, there is non-zero trade on
market h, there are traders who sell positive quantities of good h.
among these traders, there exists an agent
j
who would like to sell
less of good h, the proof is, again, complete.
sell more than -t,
than
and h)
,
If
If
j
would like to
units of good h (given his trades on markets other
i and j
can arrange a mutually beneficial trade at
prices p, contradicting constrained Pareto optimality.
that all sellers on market h are unconstrained.
Therefore, assume
From differentiability,
forcing them to sell a bit more of good h does not change their utility
to the first order but does increase i's utility.
allocation assigns positive weight to
trading.
4
Thus
(t
,...,t
)
:
in (*)
,
is not implementable.
implies it is not orderly.
Remark
i
Therefore, if the
it involves forced
If
m
> 2,
Proposition
Q.E.D.
This proposition demonstrates that Hahn's Proposition 2.2
(see [9]), which asserts the existence of a K-equilibrium that is also
constrained Pareto optimal, is false.
-18-
By the absence of spillovers, we mean that a change in a constraint on a market
does not alter net trades in any of the other markets, except the uncon-
strained market.
A sufficient condition to obtain no spillovers is
that traders' utility functions take the form U
= t_
U
+
£ $,
(t,
h
h
)
.
In
the no spillover case, the only change in Figure 3 is that the IPO
set shrinks to coincide with the KPO and 0P0 sets.
Proposition
Proof
:
9
:
In the case of no spillovers,
An IPO must be orderly.
We have
(iPO) =
{KPO} =
{OPO}
Otherwise, slightly relaxing the
constraints in market h for one demand- cons trained and one supply-
constrained agent would be implement able (since it would not disturb
the other markets) and Pareto improving.
Q.E.D.
-15-
The hypothesis of differentiability in Proposition
is,
7
as in
Crucial too is the assumption that all
previous results, essential.
traders have positive weight in the program (*).
To see this, refer
Point B is both constrained Pareto optimal and
again to Figure
1.
implementable.
However, the trader whose indifference curve is tangent
to C has zero weight.
(Note,
incidentally, that all the other CPO's -
the line segment between B and C - are non-implementable.
)
Finally,
the hypothesis of non-zero trade on each market is necessary.
for example, to the Edgeworth box economy of Figure 2.
Figure
Refer,
Initial endowments
2
are given by A, which is also a constrained Pareto optimum relative to
the price line drawn.
Although A does not involve forced trading, it
does not violate the Proposition, as it involves no trade at all.
Although differentiability is a restrictive assumption, the
non-zero weight and trade assumptions rule out only negligibly many
CPO's.
On the basis of Proposition
7,
we may conclude that, with
differentiability, CPO's are generically non-implementable and non-orderly.
-16-
We now consider the set of Pareto optima among implementable
the Implementable Pareto optima.
allocations:
As opposed to a Social
Nash Optimum where an uncoordinated planners choose the rations on their
own market according to their own weights, an IPO is a Pareto optimum
for a unique planner choosing all the rations.
Obvious questions are
whether IPO's are necessarily orderly or even weakly orderly.
The
following proposition demonstrates that this is not the case.
Proposition
8
Implementable Pareto optima need not be weakly orderly,
:
a fortiori
(nor,
Proof
orderly).
,
The proof takes the form of an example.
:
Consider a two-trader,
three-good economy in which trader A derives utility only from good
and has an endowment of one unit each of goods
1
Trader B has
and 2.
a utility function of the form
U(x ,x ,x
Q
1
)
2
=
—x
15
3
+ — x
- 3x
- 3x x
2
2
- x
2
2
+ x
Q
,
where x. is consumption of good i, and an endowment of one unit of good
0.
All prices are fixed at
It can verified that trader B's
1.
unconstrained demands for goods
respectively.
trader
B.
and
1
at these prices are
2
is
3
-rrj-
16
.
and
—
o
This is an IPO in which all the weight is assigned to
In this IPO, trader A is constrained on both markets.
suppose that trader B is constrained from buying more than
of good 1.
—
12
—
13 11
—
r-r-
Now
units
It can be verified that his corresponding demand for good 2
Because
-7-7-
24
+ — 16
>
——
12
-
H
-
—
8
,
trader A obtains more of
.
-17-
of good 3 when B is constrained on market
1
than when unconstrained.
more
Therefore the allocation in which A is constrained from selling
than the vector
good
1
(~
is an IPO.
,-A-
)
and B from buying more than
units of
Furthermore, it is not weakly orderly, because given
a purchase of 3/16 units of good 2,
of good A,
24
trader B would like to buy -^-
units
Q.E.D.
and 5/96 > ~j^~
this
We can summarize the results (with differentiability) to
point in a schematic diagram (Figure
Figure
3)
3
One "unappealing" feature of Figure
3
is the fact that the set of
IPO's is neither completely within nor without the set of weakly orderly
allocations.
With an additional hypothesis often made in the disequilibrim
literature - the absence of spillover effects - this unaesthetic property disappears,
•19-
5
.
Structure of K-Equilibrla, Constrained Pareto Optima and Keynesian
Pareto Optima Sets
.
In this section we undertake a study of the local dimension of
the different sets.
5 .1
(h =
K-Equilibria
Assume that at a K-equilibrium the m markets
.
l,...,m) comprise at least one binding constraint (a constraint
is binding if the derivative of the indirect utility function with
respect to the constraint is different from zero).
Let us show that,
the number of degrees of freedom is equal to the number
on market h,
of binding constraints
(b
)
,
minus one; for that let us remember that
a K-equilibrium is a Social Nash Optimum.
The first order condition
for a Social Nash Optimum with a binding ration Z
h
yields
1
i
A,
h
3V
=
~-i
u,
h
,
where
u,
h
is the multiplier associated with market
3Z,
1
1
A
h's equilibrium constraint.
Let
A,
h
=
— and
%
F =
—
A,
h
:
1
I
It is easy to see that the jacobian of F is of rank
is a function of
of dimension (b
thus
E
(2b
-1).
)
Z,
[b
variables, the inverse image F
1
(p.z /^
+1].
(0)
1
)
Because F
is a manifold
The local dimension, of the set of K-equilibria is
- 1) = b - m,
(b
1
1
3Z,
where b is the total number of binding constraints,
h
5.2
Constrained Pareto Optima
.
The local dimension of this set is (n-1) since a Constrained Pareto
Optimum is
a
Pareto Optimum of the economy with consumption sets X
induced preferences.
and
-20-
5.3
K eynesian Pareto Optima
.
The only (direct) way to change the
weight between two traders is to change their rations for a market
on which they are. both constrained.
constrained on
Call T = {(i,j)
at least one market}.
the redundant pairs; more precisely,
at most one sequence of pairs:
|
and
i
j
are both
T is obtained from T by eliminating
in T,
(i,j),
starting from
(j ,k)
,
.
.
.
,
(£,i)
i
there can be
leading back
to i.
The local dimension of the set of Keynesian Pareto Optima is
then:
Min
[
I
T
,
|
n - 1]
FOOTNOTES
Benassy's equilibrium is a K-equilibrium if preferences are convex.
2
The term is due to Grossman [7].
Grossman's SNO, however, is related
only in spirit to our own.
3
Grandmont, Laroque, and Younes [6] call property
(d
'
)
market-by-market
efficiency.
4
This equivalence is demonstrated by Grandmont, Laroque, and Younes [6].
If x is an m-dimensional vector,
vector (x
,
.
.
.
,x,
deleting component
,
h.
x,
...... ,x
),
denotes the
the notation x N1
)h(
,
i.e.,
the vector obtained by
REFERENCES
[1]
Benassy, J.P.
"Neo-Keynesian Disequilibrium Theory in a Monetary
,
Economy," Review of Economic Studies
[2]
42,
,
1975.
Debreu, G., "A Social Equilibrium Existence Theorem," Proceedings
of the National Academy of Sciences of the USA
[3]
,
38,
Dreze, J., "Existence of an Equilibrium under Price Rigidity and
Quantity Rationing," International Economic Review
[4]
1952.
Dreze, J. and Miiller, H.
,
,
46,
1975.
"Optimality Properties of Rationing
Schemes," DP7901 Core 1979.
[5]
Grandmont, J.M.
,
"The Logic of the Fixed-Price Method," Scandivanian
Journal of Economics
[6]
Grandmont, J.M.
,
,
1977.
Laroque, G. and Younes, Y.
Allocations and Recontradicting,
[7]
Grossman,
S.
,
"
,
"Disequilibrium
Journal of Economic Theory
"A Characterization of the Optimality of Equilibrium
in Incomplete Markets," Journal of Economic Theory
[8]
Grossman,
S.
Hahn, F.,
45,
[10]
1-18,
Hahn, F.
,
,
15,
1-15, 1977.
and Hart, 0., "A Theory of Competitive Equilibrium in
Stock Market Economies," Econometrica
[9]
.
,
42,
293-330, 1979.
"On Non-Walrasian Equilibria," Review of Economic Studies
1978.
Comment on Grossman-Hart, 1979.
,
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