STOCHASTIC

168 THE AMERICAN ECONOMIC REVIEW
variety of one-period models such as, for
example, the one-period model analyzed in
detail by Hirschleifer.
But the theories of wealth allocation
most thoroughly discussed in the literature
are the one-period, two-parameter
portfolio models of Markowitz, Tobin
(1958, 1965), Sharpe (1963), and Fama
(1965, 1968a). These have in turn been
used by Sharpe (1964), Lintner (1965a,
1965b), Mossin (1966), and Fama (1968a,
1968b) as the basis of one-period theories of
capital market equilibrium. The market
equilibrium relationships between the oneperiod
expected wealth relatives and risks
of individual securities and portfolios derived
from these models, have in turn been
given some empirical support by Marshall
Blume and Michael Jensen. The remainder
of this section will be concerned with using
our model to provide a multiperiod setting
for the apparently useful results of these
one-period models.
The two-parameter portfolio models
start with the assumption that one-period
wealth relatives on asssets and portfolios
conform to two-parameter distributions of
the same general type. That is, the distribution
for any asset or portfolio can be
fully described once its expected value and
a dispersion parameter, such as the standard
deviation or the semi-interquartile
range, are known. 18 It is then shown that if
13 Since assets and portfolios must have distributions
of the same two-parameter "type," the analysis is
limited to the class of symmetric stable distributions,
which includes the normal as a special case. Properties
of these distributions are discussed, for example, in
Fama (1965) and Benoit Mandelbrot, and a discussion
of their role in portfolio theory can be found in Fama
(1968a).
Alternatively, the results of the mean-standard deviation
version of the two-parameter portfolio models
can be ohtained by assuming that one-period utility
functions are quadratic in wt+]. But strictly speaking,
since the quadratic implies negative marginal utility at
high levels of w/t-t-i, it is not a legitimate utility function.
Moreover, the empirical evidence (see Marshall Blume,
Fama (1965), Mandelbrot, and Richard Roll) that distributions
of security and portfolio wealth relatives
conform well to the infinite variance members of the
investors behave as if they try to maximize
expected utility with respect to one-period
utility functions !>t+i(ct, a'tn) that are
strictly concave in (c,, a't+i), optimal
portfolios will be efficient in terms of the
two parameters of distributions of oneperiod
wealth relatives. 14 The fact that
optimal portfolios must be efficient then
makes it possible to derive market equilibrium
relations between expected wealth
relatives and measures of risk for individual
assets and portfolios.
But these models assume somewhat
more about the utility function »t+i than
our multiperiod model. In particular, in
the multiperiod model the function »t+i
(ct, wt+i|/3t+i), which is relevant for the
consumption-investment decision of period
t, is strictly concave in (ct, iflt+i), but utility
can be a function of the state j3t+i (i.e.,
utility can be state dependent). Thus to
provide a multiperiod setting for the oneperiod
two-parameter models it is sufficient
to determine conditions under which v,+i
will be independent of /3t+i.
State dependent utilities in the derived
functions Ht+i have three possible sources.
First, tastes for given bundles of consumption
goods can be state dependent. Second,
as will be shown in Proposition 2 of the
Appendix, utilities for given dollars of
consumption depend on the available consumption
goods and services and their
prices, and these are elements of the state
of the world. Finally, the investment
opportunities available in any given future
period may depend on events occurring in
preceding periods, and such uncertainty
about investment prospects induces state
dependent utilities. Thus the most direct
way to exclude state dependent utilities
symmetric stable class casts doubt on any model that
relies on the existence of variances. Since the approach
based on general two-parameter return distributions
by-passes these problems, it seems simplest to lay the
quadratic to rest, at least for the purpose of portfolio
models.
14 This concept of efficiency was denned in fn. 5.
394 PART IV. DYNAMIC MODELS REDUCIBLE TO STATIC MODELS