\thrm{5} Given any two-member \po\ syndicate with heterogeneous beliefs the
following statements are equivalent:\par
\pthm{i}{There exists a \po\ \sr\ which is linear in wealth on an open
subset of $S$ where $\phi↓1(s) \neq \phi↓2(s)$.} 
\pthm{ii}{All \po\ \sr s are linear in wealth on all of $S$.}
\pthm{iii}{The members of the syndicate have HARA-class utilities with identical cautiousness.}
\prf  $(iii.)$\ $\implies$ $(ii.)$\ $\implies$ $(i).$\ is trivial.  To
show $(i.)$\
$\implies$ $(iii.)$, note that if $z↓1(x,s) = a(s)x + b(s)$, $z↓2
(x,s) = (1 - a(s))x - b(s)$.  Let $\lambda$ be the weight
associated with this sharing rule.  Then
$$a(s) = {{\rho↓1(a(s)x + b(s))} \over {\rho↓1(a(s)x +
b(s)) + \rho↓2((1-a(s))x - b(s))}}.$$
Thus
$$a(s)\rho↓2((1-a(s))x - b(s)) = (1-a(s))\rho↓1(a(s)x + b(s)).$$
Differentiating by $x$ and dividing by $a(s)(1-a(s))$ yields (note: 0 <
a(s) < 1):
$$\rho↓2↑\prime((1-a(s))x-b(s)) = \rho↓1↑\prime(a(s)x + b(s)).$$
Differentiating this by $x$ yields
$$(1-a(s))\rho↓2↑{\prime\prime}((1-a(s))x - b(s)) =
a(s)\rho↓1↑{\prime\prime}(a(s)x + b(s)),\eqno(4)$$
while differentiating by $s$ yields
$$-(a↑\prime(s)x + b↑\prime(s))\ \rho↓2((1-a(s))x-b(s))\ =\ (a↑\prime(s)x
+ b↑\prime(s))\ \rho↓1↑{\prime\prime}(a(s)x + b(s)).$$
From Theorem 4 $\texist s↓0$ s.\ t.\ $\forall x\ \
\dbyd{z↓1}{s}\rceil↓{(s↓0,x)} = a↑\prime(s↓0)x + b↑\prime(s↓0)
\neq 0$.  Thus for some $s↓0$ and all $x$ we have
$$\rho↓2↑{\prime\prime}((1-a(s↓0))x-b(s↓0)) =
-\rho↓1↑{\prime\prime}(a(s↓0)x+b(s↓0)).$$
Combining this with eqn. (4) yields 
$$\rho↓1↑{\prime\prime}(a(s↓0)x + b(s↓0)) =
\rho↓2↑{\prime\prime}((1-a(s↓0))x + b(s↓0)) = 0.$$
As $\{a(s↓0)x + b(s↓0) \relv x \in \reals \} = \reals$, we have the
desired result. \qed
\noindent {\bf COMMENT:} The ability to go from $(i.)$ to $(ii.)$ via
$(iii.)$ is, in general, quite useful.  As Theorem 8 below will
demonstrate, when using a pointwise expression of the nature of (1),
frequently one can only derive results in a certain locality.  Theorem
5 allows us to grow these results to cover all of $S$.
\cor{6} For a two-person syndicate with homogeneous beliefs, the
following two statements are equivalent (Mossin [1973]):
\pthm{i}{For all $\lambda$ in an open set the \po\ contract is linear
in wealth.}
\pthm{ii}{Both members of the syndicate have HARA-class utilities
with identical cautiousness.}
\prf Identical to the one given above but replacing the final
differentiation by $s$ with a gradient over $\lambda$.
\thrm{7} For an $N$ member syndicate, the following two statements are
equivalent:
\pthm{i}{There exists a \po\ \sr\ for the group which is linear in
wealth;}
\pthm{ii}{All the \po\ \sr s for the group are linear in wealth;}
\pthm{iii}{All of the members of the group have HARA-class utilities with
identical cautiousness;}
\yskip
\noindent and any one of these implies:
\pthm{iv}{The group forms a Wilson Syndicate.}
\prf We will show $(i.) \implies (iii.) \implies (iv.)$ and $(iii.)
\implies (ii.)$.  We begin with the latter. \par
Let $\rho↓i(x) = bx + a↓i$ and suppose $b \neq  0$ (the case of $b =
0$ is left as an exercise).  Then $U↓i↑\prime(x) = (bx +
a↓i)↑{-{1\over b}}$ (up to a positive multiplicative constant).  We
leave it as an exercise to show that in this situation (1) can be
solved (for any $\lambda$) by
$$\eqalignn{z↓i(x,s) ⊗= {{{\scriptstyle{\left( bx + \Nsum
        a↓i\right)\lambda↓i↑bf↓i(s)↑b}\over{\Nsum
            \lambda↓i↑bf↓i(s)↑b}} - a↓i}\over b}\cr
             \mu↓0(x,s) ⊗= \left({{\Nsum
\lambda↓i↑bf↓i(s)↑b}\over{bx + \Nsum a↓i}}\right)↑{\scriptstyle
                                                       1\over b}.\cr}$$ 
Thus we have $(iii.) \implies (ii.)$ and, as $\mu↓0$ is separable,
$(iii.) \implies (iv.)$.\par
We prove $(i.) \implies (iii.)$ by induction.  By Theorem 5 we know it
is true for the case $N=2$.  Suppose it is true for all $(N-1)$-member
groups.  If an $N$-member \ps\ is characterized by a linear \sr,
then the subsyndicate formed by the first $N-1$ members of the
syndicate linearly share their portion of the payoff, and thus are
equicautious-HARA (we will formalize this argument in a later
theorem).  As the order in forming the group was arbitrary, all
members of the $N$-member syndicate must also have HARA utilities with
the same cautiousness.\qed
Note that while the theorem above shows that equicautious-HARA
utilities imply that the group forms a Wilson syndicate, but not
conversely.  The counterexample given in section one demonstrates that
the converse is, in fact, not correct.  However, the following theorem
shows that the converse does hold (and thus Wilson's original result
is correct) for the case $N=2$.\par
\thrm{8} For a two-member syndicate with heterogeneous beliefs, the
following statements are equivalent:
\pthm{i}{The group forms a Wilson syndicate (for some fixed $\lambda$).}
\pthm{ii}{The \po\ \sr\ is linear in wealth (for the same $\lambda$).}
\pthm{iii}{Both members of the syndicate have HARA-class utilities
with identical cautiousness}
\prf As $f↓1 \neq f↓2$ and we have assumed these to be continuously
differentiable, the set $\{s \relv \phi↓1(s) \neq \phi↓2(s)\}$ is
nonempty and open.  By Theorem 2 we have
$$\Twosum \dbyd{z↓i}{x} \rceil↓{(x,s)} \phi↓i(s) = \phi↓0(x,s).$$
If the group is a Wilson syndicate, $\phi↓0$ is independent of
$x$, so differentiating the above expression by $x$ yields
$$\Twosum {{\partial↑2 z↓i} \over {{\partial x}↑2}} \phi↓i = 0.$$
But we also have $\Twosum \partial↑2 z↓i / {\partial x}↑2 \ = 0$.
Thus the sharing rule is linear in wealth on the set $\{ \phi↓1 \neq
\phi↓2\}$, and by Theorem 5 we have $(ii.)$ and $(iii.)$ \par
We have shown that $(i.)$ $\implies$ $(ii.)$ and $(iii.)$.  By Theorem 5 we
know that $(ii.)$ $\equiv$ $(iii.)$.  As $(iii.)$ $\implies$ $(i.)$
follows from Theorem 7, we are done.  \qed
With this groundwork in place we are in a position to prove some
results on the formation of syndicates.
\thrm{9} Let $G↑1$ and $G↑2$ be two groups which form Wilson syndicates
under the weightings $\lambda↑1$ and $\lambda↑2$, respectively.  Let
$U↑1,\ f↑1$ and $U↑2,\ f↑2$ be their respective group utilities and
group beliefs.  Suppose the two member group $\langle
U↑1,f↑1;U↑2,f↑2\rangle$ forms a Wilson syndicate under the weighting
$\langle 1,1\rangle$, with group utility and beliefs $U↓0,\ f↓0$.
Then the group $G↑1 \union G↑2$ (properly ordered) forms a Wilson
syndicate under the weighting $\langle \lambda↑1,\lambda↑2 \rangle$
with group utility and beliefs $U↓0,\ f↓0$.
\prf Let $z↑1$, $z↑2$, and $z$ denote the \po\ \sr s for the
syndicates referred to above. Then $\forall i \in G↑1$,
$$\eqalignn{\lambda↓i↑1 U↓i↑\prime(z↓i↑1(z↓1(x,s),s))f↓i(s) ⊗=
          {U↑1}↑\prime (z↓1(x,s))f↑1(s)\cr
                     ⊗= U↓0↑\prime(x)f↓0(s),\cr}$$
while $\forall j \in G↑2$,
$$\eqalignn{\lambda↓j↑2 U↓j↑\prime(z↓j↑2(z↓2(x,s),s))f↓j(s) ⊗=
          {U↑1}↑\prime (z↓2(x,s))f↓0(s)\cr
                     ⊗= U↓0↑\prime(x)f↓0(s).\cr}$$ \qed
It is now clear how one can construct a whole class of counterexamples to
Wilson's {Theorem 6}.  Form any group with equicautious HARA-class
utilities and any arbitrary set of beliefs.  By theorem 7 this group
will form a Wilson syndicate.  Let $f↓0$ be this syndicate's
consensus beliefs, and $U↓0$ its group utility.  Join to this group
any individual (or Wilson syndicate) with any arbitrary utility $U$
and with beliefs $f↓0$.  Because the two member group $\langle
U↓0,f↓0;U,f↓0\rangle$ has homogeneous beliefs, it must form a Wilson
syndicate.  Thus by Theorem 9, the group formed by adjoining the
additional individual to the original syndicate is also a Wilson
syndicate.  As the utility of the final individual was arbitrary,
the \sr\ used by the group will, in general, be nonlinear.\par 
All of the syndicates formed in the manner described above carry the
property of decomposability, \ie, each such syndicate may be divided
into two subsyndicates, which can in turn be split into subsyndicates,
and so on (note, however, that this may not be done in an arbitrary
manner).  It is clear from Theorem 8 that we have, in fact, exhausted
the set of all Wilson syndicates with this property.  It remains an
open question whether this exhausts the entire set of Wilson syndicates, \ie,
whether there exists syndicates that are not pairwise
decomposable.\par
As was noted earlier, this pairwise decomposability has implications
for the manner in which syndicates form.  If syndicates were
arbitrarily decomposable, then they could be formed in an arbitrary
order, with each subunit being able to solicit further members by
issuing a prospectus giving its group utility and beliefs.  If we
were to attempt to form in this manner one of the class of syndicates
which was decomposable only in a particular way, we would have to
follow that specific order.  If we envision the formation of syndicates
as arising from random encounters of individuals or syndicates, in
which only Wilson syndicates can absorb additional members, one would
imagine that it is rather hard to form very large syndicates of the
latter type.  Given that arbitrarily decomposable syndicates are
characterized by linear contracts under heterogeneous beliefs, one
would, therefore, expect to see large syndicates characterized by
linear \sr s.\par
The statement made above would appear to be an empirically test\- able
proposition.  As noted, one should be able to identify Wilson
syndicates by the way in which they solicit new members.  One would
therefore predict that groups which solicit new members in a syndicate
manner, but which do not have a linear \sr, to be relatively small.
While we have 
not looked into the matter, {\it a priori} it seems reasonable
that there exists data upon which one could test this proposition.\par
It should be noted that when we term a sharing rule linear in
wealth, we mean that its {\sl partial} derivative \wrt\ wealth is
independent of the payoff.  What is generally observable (in
contracts, etc.) is the {\sl total} derivative of the sharing rule
\wrt\ wealth.  For a given set of projects $s$ and $x$ are not necessarily
independent, hence these two derivatives are not the same.  Thus the
functional form of the observed sharing rule may pull some
side-betting into the dependence on wealth.  It has been shown (see
Amershi and Butterworth [1981]) that, when $\omega(\cdt,a)$ is a
sufficient statistic for the $f↓i$, all the side-betting can be pulled
into the \sr's dependence on wealth.  Thus the \po\ \sr\ may be
independent of $s$ in spite of differences in beliefs.  One must
therefore be cautious when discussing the functional dependence of
\sr s.\par
There remains the question of whether some weaker form of Wilson's
Theorem 6 may be correct.  The following theorem demonstrates one such
weaker form:
\thrm{10} Suppose the group $G = \langle U↓1,\ldots,U↓N \rangle$ forms
a Wilson syndicate for some weighting $\lambda$ and {\sl all sets} of
(Pareto-acceptable) beliefs.  Then all the $U↓i$'s are HARA-class with
identical cautiousness, and conversely.
\prf The converse of the statement follows form Theorem 7.  To prove the
statement, we need the following lemma:
\lem{} If a group forms a \po\ syndicate under the weighting
$\lambda$, then $\forall m \le N$ the first $m$ members of this group form a
syndicate under the weighting
$\langle\lambda↓1,\ldots,\lambda↓m\rangle$.
\prf Let $z$ be the \po\ \sr\ associated with the weight $\lambda$,
and let 
$$z↑m(x,s) = \sum↓{i=m+1}↑N z↓i(x,s).$$
For each $s$, the equation
$$y = x - z↑m(x,s)$$
is solvable for $x$ in terms of $y$, and as $y$ ranges over all of
$\reals$, $x$ does as well.  Hence the functions $z↓i:\reals \times S
\mapsto \reals$ given by
$$z↓i↑\prime (y,s) = z↓i(x(y,s),s) \qquad i=1,\ldots,m\ ,$$
are well defined.  We note further that
$$\eqalignn{\msum z↓i↑\prime (y,s) ⊗= \msum z↓i(x(y,s),s)\cr
                                   ⊗= \Nsum z↓i(x(y,s),s) - z↑m(x(y,s),s)\cr
                                   ⊗= x(y,s) - z↑m(x(y,s),s)\cr
                                   ⊗= y.\cr}$$
Let $\mu↓0↑\prime (y,s) = \mu↓0 (x(y,s),s)$.  Then
$$\eqalignn{\mu↓i U↓i↑\prime(z↓i↑\prime (y,s)) f↓i(s) ⊗= \mu↓i
                                        U↓i↑\prime (z↓i(x(y,s),s))f↓i(s)\cr
                     ⊗= \mu↓0(x(y,s),s)\cr
                     ⊗= \mu↓0↑\prime(y,s).\cr}$$ \qed
\noindent {\bf proof of Theorem 10:} Let the first $N-1$ members of the
group have identical beliefs and let the last member have different
beliefs.  By assumption the group forms a Wilson syndicate, and by the
lemma above the group comprised of the first $N-1$ members also forms
a Wilson syndicate.  The assumption of strict risk aversion for all
group members assures the uniqueness of the solution to the Borch
first order conditions.  From this it follows that the
two-member group composed of the $(N-1)$-member subsyndicate and the
$N↑{\th}$ individual also forms a Wilson syndicate under the weighting
$(1,\lambda↓n)$ (try the contract $z↓N↑\prime = z↓N$, $z↓1↑\prime
= \sum↓{i=1}↑{N-1} z↓i$, where $z↓i$ is the \sr\ for the $N$-member
syndicate).  Thus by Theorem 8 $U↓N$ is HARA-class.  By symmetry this
implies that all the $U↓i$ are HARA-class.\par
Equicautiousness is proved by induction.  Theorem 8 shows that it holds
for two-member syndicates.  Assume it is true for $(N-1)$-member
syndicates.  By the construction above, we can split the $N$-member
syndicates into an $(N-1)$-member syndicate and an individual, who must
in turn form a two-member syndicate.  Thus the group utility of the
$(N-1)$-member group must be HARA-class with cautiousness identical to
that of the $N↑{\th}$ individual.  By the inductive assumption all
members of the $(N-1)$-member syndicate have equal cautiousness, thus it
is clear that this is also the cautiousness of the $(N-1)$-member group.
Thus the result is true for $N$-member syndicates.  \qed
\par\vfill