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ALFRED
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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A NOTE ON THE NECESSARY CONDITION
FOR LINEAR SHARING AND SEPARATION
t
by
Chi-fu Huang"
and
Robert Li tzenberger"
WP #1571-8^4
September I383
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
liivjfi
-^.^,
A NOTE ON THE NECESSARY CONDITION
FOR LINEAR SHARING AND SEPARATION
by
Chi-fu Huang"
and
Robert Li tzenberger""
'
WP #1571-8'+
September I583
Massachusetts Institute of Technology
Stanford University
After this note was written we learned that the result here
context.
is similar to one of Ross [5] but in a different
1
Introduction and Conclusion
.
A necessary and sufficient condition under which an optimal
is
linear, as generally accepted by the finance community,
functions be of the equicautious HARA class.
pointed out by Amershi and Stoeckenius [l] for
(a
,A„,...,A
on agents'
that utility
is
(See Mossin [^] for a discussion.)
As
)
sharing rule
a
given weighting
utilities, equicautious HARA class
sary condition when agents have heterogeneous beliefs.
indeed
is
trivia] example:
in
neces-
is
not a
This last point can easily be seen from the following
Let there be two
Then the opitmal sharing rule
endowment
a
When agents have homo-
geneous beliefs, however, the equicautious HARA class of utilities
necessary condition.
=
A
is
identical
that both get one-half of the aggregate
each and every state, which
common utility function
individuals with weighting (1,1).
is
clearly linear.
However, their
arbitrary, not necessarily of the equicautious HARA
is
class.
Note that
A,
^ A»,
In
the above example,
then the optimal
necessity part derivation
sharing rule
is
the weighting
if
not
is
=
A
(A
is
not surprising that,
the assumption that for all weightings
rule to be
HARA class.
linear
it
is
for a fixed weighting A,
A.
is
R
is
HARA utilities are necessary, however,
(cf.
A
a
an
Thus
11^).
sharing
for the optimal
equivalent to
in
the equicautious
a
g
i
ven
A
linear optimal
Thus equicautious HARA utilities are not neces-
sary for fund separation with respect to
A
pp.
Furthermore, since the fund separation property for
sharing rule for this given
all
[^],
not necessary that utilities be of
and for some endowment whose range
such that
is
)
implicit in Mossin's
linear.
open set the Pareto-opt imal sharing rules are linear (cf.
it
A
if
a
given
A
fund separation
Amershi and Stoeckenius [l]. Corollary
The equicautious
either.
is
to obtain for
2
17)-
The purpose of this short note is therefore to derive
sufficient) condition, such that the optimal sharing rule
a
Is
necessary (and
linear and
-2-
separation obtains, for a fixed weighting
This condition requires that agents'
A,
cautiousness be identical (but not necessarily determinate) along the optimal
schedule in contrast to being identical and determinate in the equicautions
HARA case.
We also demonstrate how easy it is to construct utility functions, for
a given weighting A,
such that the condition we characterize is satisfied.
The Results
2.
Instead of going through the formality,
for which we refer readers to
Amershi and Stoeckenius [l] or Wilson [7], we will speak rather loosely.
Let there be N agents
is
in
=
i
Each agent
1,2,...,N.
characterized by an increasing and strictly concave von Neumann-Morgenstern
utility function U..
It
assumed that agents have homogeneous beliefs on
is
the generic element of which is denoted by s.
the state space S,
denote the aggregate endowment
is
indexed by
the economy,
R.
A sharing rule
is
vector-va ued
a
f unct
1
S^R
We assume that the range of w
the economy.
in
Let w:
ion z =
(z, ,z_
,
.
.
.
,z,,)
R
:
-*
R
N
Z
I
such that
N
A sharing
z'
rule
z
=
= X
z. (x)
Z
i
VxeR
'
l
there
is
E(U.(z.(w(s))))
V.
said to be Pareto-opt imal
is
if
no other sharing rule
such that
E(U.(z.'(w(s))))
and with at
least a strict
of Pareto-opt imal
Remark
:
>_
inequality for some
There are an infinite number
i.
sharing rules.
Here we note that by defining
valued function on
R
loss of generality.
rather than
a
a
sharing rule to be
vector-valued function on
R X
a
vector-
S
is
without
When agents have homogeneous beliefs, optimal sharing
rules are state independent
(cf.
Wilson [6]).
Now we first record some results due to Borch [2]:
(a)
A sharing rule
z
is
Pareto-opt imal
exists constants (weight ing)
Vi,j A.U.'(z.(x))
Ill
=
,
A
,
.
'(Z.(x))
A.U
J
(A
j
'
and only if there
if
.
.
,A
)
= A such that
for all
XeR.
)
-3-
Pi(z,(x))
(b)
--V-^
=
'(x)
z
where p.(z,(x)) =
,
-
''
U.
n
is
agent i's risk tolerance, and where
'
(z. (x)
pAx)
=
is
/U
(z. (x)
''
.
p.(z.(x)).
I
1
The cautiousness o.(z.(x))
)
=
'
'
is
linear iff
1
defined to be
o.(z.(x)) = p.'(z.(x))
Our main result
is:
PROPOSITION:
Let
z
be a Pareto-opt imal
sharing rule.
Then
z
a.(z.(x)) = Oj(z^(x)) Vi,j
The sharing rule
PROOF:
linear iff
z
is
p
(z. (x))
z'.'(x)
=
Vi
and VxcR.
This
is
equivalent to
.
,
by
(b)
.
Carrying out differentiation yields:
Pq(x)o.(z.(x))z,'(x) = p.(z.(x))p^'(x)
Substituting
(b)
VeR.
gives
for z.'(x)
a.(z.(x)) =
I
Vi
'(x)
p
Vi
VxeR.
I
Thus we have:
o.(z.(x)) = a.(z.(x)) V iJ.VxcR.
Q.E.D.
Interpreting the above result
an exchange economic setting,
in
the neces-
sary and sufficient condition for the linear sharing rule to be Pareto-opt ima
and for fund separation to obtain
and the distribution of
weighting
A
initial
is
a
1
joint condition on both agents' utilities
endowments.
The latter
associated with the optimal sharing rule.
is
captured by the
Examples other than the
trivial one in the Introduction can easily be constructed to demonstrate the
above Proposition:
Given
weighting iX.,X), two scalarsa and
a
<
b
<
1,
and an increasing strictly concave utility function U.(.), we can construct
another utility function U-C.)
increasing and strictly concave such that
AjU,'(a + bx) = X^U^'i- a + (l-b)x) VxeR
This
is
easily done by defining
U2(y) = c +
where
c
.
is
^
an arbitrary scalar.
sharing rule for the group
/ U,'(
3-i^ )dt
VyeR,
Then under the weighting (A,,A„), the optimal
<U,,U2>
is
< a
+ bx,
-
a
+(l-b)x>, clearly linear.
And of course, the identical cautiousness condition as required of the
Proposition is satisfield.
Finally we should note that either when agents'
or when the optimal
sharing rules are linear over an open set of
weak aggregation (cf. Rubinstein [6])
stable with respect to
bution of endowments).
optimal
sharing rule
to a change of A.
beliefs are heterogeneous
is
a
.
X,
we have
That is, equilibrium prices are
perturoation of the weighting (or initial distriUnder the condition of the Proposition, although the
linear, equilibrium prices are not stable with respect
-5-
FOOTNOTES
1.
Here we mean for the equilibrium corresponding to the given weighting
all
individuals hold varying proportions of
risky portfolio, which
2.
the same for all
is
a
riskless asset and
a
X,
single
individuals.
Cass and Stiglitz [3] demonstrated that for monetary separation to obtain for
an
his
of
individual at varying
levels of
initial wealth,
utility function be of HARA type.
individuals, varying the weighting
bution of wealth
wealth.
in
a
HARA type.
it
is
is
necessary that
considering
a
group
just like varying the distri-
in
the group to obtain monetary separa-
necessary that his utility function
For the group as
an open set of X,
X
is
market economy and therefore the levels of initial
Thus for each individual
tion when varying X,
V/hen we are
it
a
is
of
whole to exhibit separation property for
however, agents' utility functions should be of equ
caut ious HARA class
.
i
.
1
-6-
REFERENCES
[l]
Amershi, A. H., and J. H. W. Stoeckenius, "A Unified Theory of Syndicates,
Aggregation and Separation
Borch,
(
[3]
Financial Markets," Research Paper
6lA, Graduate School of Business, Stanford University, August
No.
[2]
in
K.
"Equilibrium
,
in
a
Reinsurance Market," Econometr ca
t
Vol.
,
I98I.
30
July 1962) pp. liZk-kkk.
Cass, D.
and J. Stiglitz, "The Structure of Investor Preferences and
,
Asset Returns, and Separability in Portfolio Allocation:
A
Contribution to Pure Theory of Mutual Funds," Journal of Economic
Theory, Vol.
[A]
2
(
June 1970) pp.
122-160.
Mossin, J., Theory of Financial Markets
[5] Ross, R.,
.
New York:
Prentice-Hall,
"On the Economic Theory of Agency and the Principle of Similarity,"
in Essays on Economic Behavior Under Uncertainty Ed.
D.
[6]
McFadden, and
Rubinstein,
M.
,
S.
Wilson,
pp.
R.
by M. Balch,
Wu; New York, American Elsevier Pub.
Co., 1974.
"An Aggregation Theorem for Securities Markets,"
Journal of Financial Economics
[7]
1973.
,
Vol.
1
(September 197^) pp. 225-2^4.
B., "The Theory of Syndicates," Econometr ica
119-132.
,
Vol.
36
(
January I968)
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37S0 045
Q6D 004 485
12fi
Date Due
Lib-26-67
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