STOCHASTIC

588 NILS H. HAKANSSON
aversion index, — u"(c)/u'(c), is a positive constant for all c ^ 0, i.e., u(c) = c y ,
0 < y < 1, u(c) = — c~ y , y > 0, u(c) = log c, and u(c) = —e~ yt , y > 0.
Section 4 is devoted to a discussion of the properties of the optimal consumption
strategies, which turn out to be linear and increasing in wealth and in the present
value of the noncapital income stream. In three of the four models studied, the
optimal consumption strategies precisely satisfy the properties specified by the
consumption hypotheses of Modigliani and Brumberg [9] and of Friedman [5].
The effects of changes in impatience and in risk aversion on the optimal amount to
consume are found to coincide with one's expectations. In response to changes in
the "favorableness" of the investment opportunities, however, the four models
exhibit an exceptionally diverse pattern with respect to consumption behavior.
The optimal investment strategies have the property that the optimal mix of
risky (productive) investments in each model is independent of the individual's
wealth, noncapital income stream, and impatience to consume. It is shown in
Section 5 that the optimal mix depends in each case only on the probability distributions
of the returns, the interest rate, and the individual's one-period utility
function of consumption. This section also discusses the properties of the optimal
lending and borrowing strategies, which are linear in wealth. Three of the models
always call for borrowing when the individual is poor while the fourth model
always calls for lending when he is sufficiently rich. The effect of differing borrowing
and lending rates is also examined.
Necessary and sufficient conditions for capital growth are derived in Section 6.
It is found that when the one-period utility function of consumption is logarithmic,
the individual will always invest the capital available after the allotment to current
consumption so as to maximize the expected growth rate of capital plus the present
value of the noncapital income stream. Finally, Section 7 indicates how the preceding
results are modified in the nonstationary case and under a finite horizon.
2. THE MODEL
In this section we shall combine the building blocks discussed in the previous
section into a formal model. The following notation and assumptions will be
employed:
Cj: amount of consumption in period j, where Cj ^ 0 (decision variable).
U{cl,c2,ci,...}: the utility function, defined over all possible consumption programs [cuc2,c3,...).
The class of functions to be considered is that of the form
(1) f(c„ c2, c3,...) = u(ci) + M(c2, c3, c4,...)
= £ x'-'u(Cj), 0 < a < 1.
j= I
It is assumed that u(c) is monotone increasing, twice differentiate, and strictly concave for c ^ 0.
The objective in each case is to maximize £[[/(cl( c2,...)], i.e., the expected utility derived from consumption
over time. 3
Xj; amount of capital (debt) on hand at decision pointy (the beginning of the;th period) (state variable).
v: income received from noncapital sources at the end of each period, where 0 < v < oo.
3 While we make use of the expected utility theorem, we assume that the von Neumann-Morgenstern
postulates [12] have been modified in such a way as to permit unbounded utility functions.
PART V. DYNAMIC MODELS