\vskip 1.00cm
\noindent{\bf II.  FORMAL DEVELOPMENT}\par
\vskip 1.00cm
As mentioned above, the essence of the question we wish to consider is
the nature of equilibria in an
economy involving uncertainty but in which all commodities are
tradable.  Within this section we will suppose the
commodity space to be one-dimensional.  In section III we will treat
the more general case and demonstrate conditions under which this
restriction is insignificant.  We consider the economy as populated by
$N$ Savage
rational decision makers, and will use the following conventions:\par
\yskip
\noindent 1.\quad $X = \reals =$ commodity space, and $x$ is a generic
element of $X$.\par
\yskip
\noindent 2.\quad $U↓i:X \mapsto \reals$ is the \ith individual's utility,
and we will assume $U↓i↑\prime > 0$, $U↓i↑{\prime \prime}$
continuous and $U↓i↑{\prime\prime}\le 0$.
For convenience, we will also assume $U↓i(0) = 0\ \ \forall i$.\par
\yskip
\noindent 3.\quad $S$ is the relevant state space, and the beliefs of the
\ith individual will be denoted by a probability measure $\Pscr↓i$ on a
$\sigma$-algebra $\sigma(S)$.  All the $\Pscr↓i$ are assumed absolutely
continuous\footnote{A measure $\mu$ is absolutely continuous \wrt\
another measure $\nu$ iff $\forall\, B\ \in\ \sigma(S),\ \ (\nu(B) = 0)
\implies (\mu(B) = 0)$.} \wrt\ each other (otherwise \po ity could not
be achieved---see Amershi and Butterworth [1981]) and $f↓i:S \mapsto
\reals $ is the Radon-Nikodym derivative\footnote{The Radon-Nikodym
derivative $f$ of a measure $\mu$ \wrt\ another measure $\nu$ is the relative
density of the first measure with respect to the second (\ie,
$\forall\,B\,\in\,\sigma(S), \mu(B) = \int↓B f\,d\mu$ ).  The
Radon-Nikodym theorem states that such a function exists if the first
measure is absolutely continuous \wrt\ the second.  For further
information see Royden [1968].} of $\Pscr↓i$ \wrt\ some common
measure $\Pscr$ (\eg, $\Pscr = {1 \over N} \Nsum \Pscr↓i$).
For ease of exposition we will assume $S \subset \reals$,
and we will assume each $f↓i$ continuously
differentiable.\footnote{The reader should note that none of these
restrictions are significant, for one can always model a given economic
situation with many different state spaces.  While we have not gone
through the details, and thus can only conjecture at the moment, we
believe that all of the results given in this paper would go through
without differentiability assumptions on $f↓i(s).$} \par
\yskip
\noindent 4.\quad $A$ denotes the set of possible projects (joint actions)
the group may undertake.  Associated with this set is a technology
$\omega :S \times A \mapsto X$ such that $\omega (s,a)$ is the payoff
to project $a$ in state $s$.  Of course, each of these objects (and
the state space as well) is essentially meaningless without the other,
and many different combinations of $A$ and $\omega$ give rise to
economically identical situations.\par
\yskip
\noindent 5.\quad $F(x,s\!\relv\!a)$ is the distribution function of
the probability measure induced on $X$ by $\Pscr$ via the random
variable $\omega (\cdt,a)$.\par
\noindent 6.\quad A {\bf sharing rule} is a function 
$z:X \times S \mapsto \reals↑N$
which obeys the constraint $\Nsum z↓i(x,s)=x$.  The \ith component of $z$
is the ``share'' of the \ith player in the event that the group's
payoff is $x$ and the state is $s$.  A {\bf contract} is an ordered pair
consisting of a sharing rule and a project (element of $A$):
$(z,a)$.\par
\yskip
\noindent 7.\quad A contract $(z,a)$ is defined to be
{\bf Pareto-superior} to a contract $(z↑\prime , a↑\prime )$ iff $\forall i$
$$\Escr↓i[\Uz] \ge \Escr↓i[U↓i(z↓i↑\prime(\omega(s,a↑\prime),s))],$$
with the strict inequality holding for some $i$ ($\Escr↓i$ is the
expectation operator \wrt\ $\Pscr↓i$).  A {\sl contract} is defined to be
{\bf \po\ }iff there does not exist a Pareto-superior contract.
A {\sl \sr\ } is defined as \po\ for
some $a$ iff there does not exist a $z↑\prime$ s.\ t.
$$\Escr↓i[\Uz] \le \Escr↓i[U↓i(z↓i↑\prime(\omega(s,a),s))],$$
with the strict inequality holding for some $i$.\par
\yskip
\noindent8.\quad A project $a$ is {\bf Pareto-preferred} to a project
$a↑\prime$ iff $\forall z↑\prime\ \ \texist z$ such that
$$\Escr↓i[\Uz] \ge \Escr↓i[U↓i(z↓i↑\prime(\omega(s,a↑\prime),s))].$$
We shall term the order induced by this relation to be the {\bf
Pareto-order} on projects (while the relation of Pareto-superiority among
contracts will be the Pareto-order on contracts).\par
\yskip
Our interest in this section will clearly center on the Pareto-order
over projects defined in item 8 above.  The simplest approach to this
problem will, however, require us to first consider \po\ \sr s.\par
Fix some $a \in A$.  We term an $N$-vector $\langle y↓1, \ldots ,y↓N
\rangle$ feasible iff there exists a sharing rule $z$ such that
$y↓i \le \Escr↓i[\Uz] $.  It is fairly easy to see that, for a fixed
$a$, the set of feasible points is convex.  Hence,
for some positive $N$-vector $\lambda$ (which we will term a
{\bf weighting}), a \po\ \sr\ will solve the following problem:
$$\max↓z \Nsum \lambda↓i \Escr↓i[\Uz]$$
$$\hbox{s.\ t.} \qquad \Nsum z↓i(x,s) = x.$$
As demonstrated in Borch [1962], a \sr\ $z$ will solve
this problem iff $\forall i,j$:
$$\eqalignn{\Upz ⊗= \lambda↓j U↓j ↑\prime (z↓j(\omega(s,a),s)) f↓j(s)\cr
                ⊗\ \ \hbox{almost everywhere (a.\ e.) \wrt\ }\Pscr,\cr}$$
and the constraint is satisfied .  This
set of equations is equivalent to (Zahl [1963])
$$\eqalignn{\Upzx ⊗= \mu↓0 (x,s) \quad \forall↓i \cr
                ⊗  \quad \hbox{a.\ e.\ \wrt\ } F(x,s\!\relv\!a),\cr}
                                                           \eqno(1)$$
where $\mu↓0(x,s)$ is the Lagrange multiplier of the budget constraint.
We will use the following theorem without proof:
\par\eject
\thrm{1} If eqns.\ (1) are solvable a.\ e.\ \wrt\ some probability
measure, they are solvable everywhere.\par
\yyskip
\noindent Hence, for a given $\lambda$, the intersection of the
equivalence classes of solutions of (1) over all $a \in A$ is
nonempty (if (1) is solvable for at least one a), and any element of
this intersection is a \po\ \sr\ for all a.
By limiting consideration to these functions, we can speak of the
\po ity of a \sr\ without referring to any particular
action.  Henceforth in this paper, the term \po\ \sr\ will designate
one of these sharing rules.  If such a \sr\ exists for a given group (\ie,
if (1) is solvable for some $\lambda$), the group is said to form a
{\bf \ps.}\footnote{Not all syndicates are \ps s, even under the assumption
$U↑\pprime < 0\ \forall i$. It is true that, even without this
assumption, all syndicates with homogeneous beliefs are \ps s.  To see
that this is not the case for heterogeneous beliefs, let $S=(0,1)$,
$f↓1(s) = 1$, $f↓2(s) = 1/2\sqrt{s}$ and
\ftdisp{U↓1(x) = U↓2(x) = (2\pi + arctan(x))x + {1\over2}\ln(x↑2 + 1).}
\hbox to 14pt{}In general, under the assumption $U↑\pprime < 0$, (1) is
solvable for an open and connected, but possibly empty, set of
$\lambda$.  Under the assumption of homogeneous beliefs this set is
nonempty, while in the case of HARA-class utilities this set is
$\reals↑N$.} \par
We term a complete order on $A$ as derivable from an {\bf evaluation
measure} (see Wilson [1968])
iff $\texist\ M:X\times S \mapsto \reals$ such that $a$ is preferred
to $a↑\prime$ iff
$$\int↓S M(\omega(s,a),s)\, d\Pscr \ge \int↓S
                                 M(\omega(s,a↑\prime),s)\,  d\Pscr$$
We will now show that for any \ps\ there always exists a
pareto-inclusive complete order (see above) which is derivable from an
evaluation measure.\par
Fix a set $A$ of projects, and suppose the group must choose a \po\
contract from this set (using any \sr).  Let $(z,a)$ be such a
contract.  It is clear that $z$ must be a \po\ \sr, and we must have:
\footnote{the set $\argmax(q)$ is the set of all maximal elements \wrt\
the relation $q$.}
$$a \in \argmax↓A \Nsum \lambda↓i \Escr↓i[\Uz].\eqno(2)$$
By the definition of $f↓i$ we have
$$\eqalignn{\Nsum \lambda↓i \Escr↓i[\Uz] ⊗= \Nsum \lambda↓i \int↓S \Uz
                                                           f↓i(s) \, d\Pscr\cr
                ⊗= \int↓S \Nsum \lambda↓i \Uz f↓i(s) \, d\Pscr\cr
                ⊗= \int↓S \int↓0↑{\omega(s,a)} \Nsum \Upzx 
                         {{\partial z↓i} \over {\partial x}} \, dx\, d\Pscr\cr
                ⊗= \int↓S \int↓0↑{\omega(s,a)} \mu↓0 (x,s)
                       \Nsum {{\partial z↓i} \over {\partial x}}\, dx\, d\Pscr\cr
                ⊗= \int↓S \int↓0↑{\omega(s,a)} \mu↓0 (x,s) \, dx\, d\Pscr\cr
                ⊗= \int↓S M(\omega (s,a),s) \, d\Pscr,\cr}$$
where $M(x,s) = \int↓0↑x \mu↓0 (x,s) \, dx = \Nsum \lambda↓i \Uzx
f↓i(s)$, using the assumption $U↓i(0) = 0$
(Note: one could run through an essentially identical derivation for a
contract involving randomization over several acts).\par
What the derivation given above demonstrates is that to each \po\ \sr,
and therefore to each weighting $\lambda$, there corresponds a
Pareto-inclusive complete order derivable from an evaluation measure.
Furthermore, for any set $A$, if the group chooses an optimal contract
involving the \sr\ $z$, then the optimal project must be the most
preferred element of $A$ \wrt\ this order.  This is what we eluded to
earlier when we said that, \wrt\ a certain set of projects, the group
acts as if it possessed a particular complete order.  For any contract
$(z,a)$ to be \po\ \wrt\ a set $A$, it must be that $\forall
a↑\prime\in A$:
$$\int↓S M↓z(\omega(s,a),s)\,d\Pscr \ge \int↓S
M↓z(\omega(s,a↑\prime),s)\,d\Pscr.$$
This property will be used extensively in Section III.\par
It is important to note, however, that the set of orders given by
evaluation measures formed in the above manner does not exhaust the
set of Pareto-inclusive complete orders, nor the subset of these
derivable from evaluation measures.  There may, in general, exist
other evaluation measures.  If a syndicate were to be endowed with
one of these orders, then we would expect to see the sharing rule
associated with an optimal contract over a set $A$ to change as $A$
changes, and we may see \po\ contracts involving randomization over
contracts.\par
At this time there is little that is known about Pareto-inclusive
orders that are not derivable from the given class of evaluation
measures; all published results we are aware of fail to deal with this
``undesirable'' set or orders.  One can get around consideration of
these orders in two ways.  One is to
limit consideration to those groups for which all Pareto-inclusive
orders are derivable from the given set of evaluation measures.  This
condition, which is alluded to in Wilson [1968] and studied more
extensively by Rossing [1970] and Wallace [1974], is equivalent to the
assertion that the \po\ surface (in the space of expected
utilities, $\reals↑N$) for any given project does not
intersect that of any other project.  This requirement turns out to be
entirely too strong.  We will demonstrate in section III below that
such a condition is equivalent to the aggregation property, which has as
trivial consequences linearity of \sr s and unanimity.\par
The other method of overcoming the ``undersirable'' orders problem is
to limit consideration to those groups which utilize one of the
``desirable'' orders.  This is the technique actually employed by
Wilson, and it leads to interesting results.
This constraint also has a significant economic interpretation.  Note
that if a group's total order is derivable from one of the given
evaluation measures, then the group will employ the same \sr\
regardless of the set of projects from which it must choose, and vice
versa.  Thus we are limiting consideration to those syndicates which
can decide upon a \sr\ without regard to the nature of the risks they
will be sharing.\footnote{If we were to extend the model to include
disutility for acts (though not moral hazard), we would generalize the
notion of determining a sharing rule to include determination of
reimbursement for any disutility an individual suffers.  The key
requirement is that this compensation would be invariant across
actions which give rise to the same disutility.\lbreak
\hbox to 14pt{}It should also be noted that the requirement of
precommitment to a given sharing rule does not rule out the
possibility of \po\ contracts involving randomization. What is
necessary is that this randomization be strictly over acts and not
over \sr s.}\par
\eject
Let 
$$\eqalignn{\rho↓i (x) ⊗= -{{U↓i↑\prime} \over {U↓i↑{\prime\prime}}}\cr
           \phi↓i(s) ⊗= {{f↓i↑\prime (s)} \over {f↓i (s)}}\cr}
\qquad\eqalignn{\rho↓0 (x,s) ⊗= {{\mu↓0} \over {\partial \mu↓0 /
                                    \partial x}}\cr 
                \phi↓0 (x,s) ⊗= {{\partial \mu↓0 / \partial s}
                                          \over {\mu↓0}}.\cr}$$
The following results are well known (see Wilson [1968]):\par
{\lineskip 7pt\save1\hbox{\bf THEOREM 2:\hskip 5pt}
\noindent\hangindent 1wd1 \box1
a.  $\Nsum \rho↓i (z↓i (x,s)) = \rho↓0 (x,s)$.\lbreak
b.  ${{\partial z↓i} \over {\partial x}}\rceil↓{(x,s)} =
     {{\rho↓i (z↓i (x,s))} \over {\rho↓0 (x,s)}}$.\lbreak
c.  $\Nsum {{\partial z↓i} \over {\partial x}}
     \phi↓i (s) = \phi↓0 (x,s)$.\lbreak
d.  The following conditions are equivalent:\lbreak
\hbox to 24pt{\hss $i$. }The syndicate behaves in a Savage-rational
        manner.\lbreak
\hbox to 24pt{\hss $ii$. }$M(x,s)$ is separable.\lbreak
\hbox to 24pt{\hss $iii$. }$\mu↓0 (x,s)$ is separable.\lbreak
\hbox to 24pt{\hss $iv$. }${{\partial \rho↓0} \over {\partial s}} = 0.$\lbreak
\hbox to 24pt{\hss $v$. }${{\partial \phi↓0} \over {\partial x}} = 0.$\lbreak}
\par\yyskip
\noindent A \ps\ obeying any of the conditions $(i.)$\ - $(v.)$\ will be
termed a {\bf Wilson syndicate} or merely a syndicate.\par
The following result is straightforward.
\thrm{3} Suppose all but one member of a \ps\ are strictly
risk-averse, and that member is risk-neutral.  Then the group will form a
Wilson syndicate, the \po\ \sr\ will be independent of wealth for all
the risk-averse members, and the group's utility and beliefs will be
that of the risk-neutral member.\par
\yyskip
\noindent The case of syndicates with more than one risk-neutral
member is
more complicated but equally uninteresting.  Because of this we will
henceforth assume all syndicate members are strictly risk-averse
($U↑\pprime < 0$).\par
To this point we have closely followed the ``classical'' development of
the theory of syndicates.  From this point onward we will deviate
from this path 
somewhat.  We begin with a number of results concerning two-person
($N=2$) syndicates.
\thrm{4} For a two-person \ps\
$$\{ s \relv \phi↓1 (s) \neq \phi↓2 (s)\} = \{ s \relv {{\partial
z↓i} \over {\partial s}} \rceil↓{(x,s)} \neq 0 \ \forall x,i\}.$$
\prf As the syndicate is composed of only two members, $z↓1 (x,s) = x
- z↓2 (x,s)$.  Thus for some $\lambda$,
$$\lambda↓1 U↓1↑\prime (z↓1(x,s))f↓1(s) = \lambda↓2U↓2↑\prime
(x-z↓1(x,s))f↓2(s). \eqno(3)$$
Therefore
$$\lambda↓1 U↓i↑{\prime\prime} (z↓1)f↓1(s) \dbyd{z↓1}{s}
+ \lambda↓1U↓1(z↓1)f↓1↑\prime(s)= 
-\lambda↓2U↓2↑{\prime\prime}
(x-z↓1) f↓2(z)\dbyd{z↓1}{s} +
\lambda↓2U↓2↑\prime(x-z↓1)f↓2↑\prime(s).$$
Dividing by (3) and setting $\dbyd{z↓1}{s} = 0$ yields $\phi↓1(s) =
\phi↓2(s)$.  Conversely, setting $\phi↓1 = \phi↓2$ yields
$$\lambda↓1 U↓1↑{\prime\prime}(z↓1)f↓1(s)\dbyd{z↓1}{s} =
-\lambda↓2U↓2↑{\prime\prime}(x-z↓1)f↓2(s)\dbyd{z↓1}{s}.$$
As $\lambda↓i > 0$, $U↓i↑{\prime\prime} < 0$, $f↓i > 0$, we must
have $\dbyd{z↓1}{s} = 0$. \qed
We term a sharing rule linear in wealth on a set $\Oscr \subset S$ iff
$\forall i$ 
$$z↓i(x,s) = a↓i(s)x + b↓i(s),$$
for $z \in \Oscr$.  For the case of homogeneous beliefs, it is easy to
see from the Borch first order conditions that $z$ is independent of
$s$, so this reduces to
$$z↓i(x) = a↓ix + b↓i.$$
It is well known (Mossin [1973]) that if a group's \po\ \sr\ is linear
{\sl for an open set of $\lambda$}, then the utilities of each of the
members of this group are HARA-class with the same cautiousness.  The
following results show that for the case of nonhomogeneous beliefs the
same result holds even if the \sr s are linear for only one
$\lambda$.\par