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\noindent{\bf III.  AGGREGATION AND SEPARATION}\par
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The question of what are sufficient conditions on an economy to insure
that the prices of risky securities and invariant under changes in
initial allocation, and the related question of when the optimal
portfolio of such securities is the same for all individuals, has long
been of interest to the finance literature (see, for example
Rubinstein [1974], Brenan and Kraus [1978], and Breeden and
Litzenburger [1978]).  The authors in this field have noted the
similarity between their results with those of Wilson [1968], but the
formal connection between the Theory of Syndicates and the problems of
stability of prices and optimal portfolios has never been made.  In
this section we will establish this formal connection, and thereby
provide alternate, and we believe highly elegant, derivations of the
of some results in this area.\par
\Df{1} Consider an exchange economy comprised of $N$ Savage-rational
individuals with utilities $\langle U↓1,\ldots,U↓N \rangle$ and
beliefs $\langle f↓i,\ldots,f↓N \rangle$, respectively.  Let the
endowment of this economy in state $s$ be given by $x(s)$.  This
economy has (linearly independent) risky securities $\langle
X↓1,\ldots,X↓M\rangle$ with prices $\langle
P↓1,\ldots,P↓M\rangle$ and a riskless security $X↓0$ with price
$P↓0$.  This economy is said to have the
{\bf aggregation property} \wrt\ $x$ iff all equilibrium endowments
(over individuals) give rise to the same prices.  If the economy has
this property \wrt\ all (feasible) $x:S\mapsto X$, then it is said to
simply have the aggregation property.\par
\Df{2} Given the same setting as above, the economy is said to have
the (two fund) {\bf separation property} \wrt\ $x$ iff for all equilibrium
endowments the portfolio proportions of all risky securities are
identical across individuals.  As above, when this is true for all $x$
we say the economy has the separation property.\par
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In the previous section we always dealt with a one-dimensional
commodity space.  However, the following theorems show that for two
types of separable utilities the situation is almost identical for the
case of a finite dimensional commodity space.\par
\thrm{11} Let $X$ be an $n$-dimensional commodity space.  Assume that
$\forall i$,
$$U↓i(x) = U↓i(\scriptstyle{\sum↓{l=1}↑n x↓l}).$$
Then a \sr\ $z:X↑n\times S\mapsto X↑{N\ast n}$ is \po\ iff the \sr\
$z↑\prime (\scriptstyle{\sum↓{l=1}↑n x↓l},s) = \sum↓{l=1}↑n
z↑l(x,s)$ is \po\ in the one-dimensional sense.\par
\thrm{12} Given the same setting as above, assume
$$U↓i(x) = \sum↓{l=1}↑n U↓i↑l(x↓l).$$
Then a \sr\ $z$ is \po\ only if each of its components $z↑l$
is in equilibrium in the one dimensional sense.\par
\yyskip
\noindent The proof of either of these results is straightforward and
will not be given here.  Henceforth we will assume that the one or the
other hypotheses
of these theorems are satisfied, and will not distinguish between the
case of one or many commodities.\par
Assume the set of securities is complete.  Then, as they both satisfy the
same constraint condition, every endowment is also a \sr.  Clearly if
an endowment is to be in equilibrium it must be \po\ as a \sr.  We
showed earlier that to each \po\ \sr\ there corresponds a complete
order over projects, and that if a contract $(z,a)$ is \po\ \wrt\ a
certain set of projects $A$, then a is the optimal element of $A$
\wrt\ the order associated with $z$.
\par