l'istituto italiano per gli studi filosofici e gli studi di economia

esources to invest in a portfolio of unit contingent contracts. Let a (x)
be the amount invested in security x. Then the investments satisfy the
constraint
S a (x) = A . (2)
x
When the actual state is realized, the individual receives, Y = a (X).
Suppose that the utility function is logarithmic. In the absence of
information, the investor maximizes
S p (x) ln a (x) ,
x
subject to (2). Then a (x) = Ap(x) for all x. The maximum expected
utility is
where, for any random variabie, X,
ln A - H(X), (3)
H (X) = – S p (x) ln p (x)
x
which happens to be the Shannon measure of information. The certainty-equivalent
(i.e., the income which, if obtained for certain,
would have the same utility as the above optimal gamble) is
YC = Ae – H (X) (4)
which is proportional to A. There is therefore essentially a constant
rate of return to investible resources.
Now suppose the assumptions of Bayesian normal sampling are
fulfilled, so that X is normally distributed and that DM can choose a
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