STOCHASTIC

CONSUMPTION AND PORTFOLIO RULES 389
the preferences of the individual is made, then Eq. (28) can be solved
in closed form, and the optimal consumption and portfolio rules derived
explicitly. Assume that the utility function for the individual, U(C, t),
can be written as U(C, t) = e~ p 'K(C), where V is a member of the family
of utility functions whose measure of absolute risk aversion is positive
and hyperbolic in consumption, i.e.,
subject to the restrictions:
A(Q = -V/V = l/(-i~- + IIP) > °.
y ^ 1; JS > 0; ( ^ + ij) > 0; r, = 1 if y = -oo. (42)
All members of the HARA (hyperbolic absolute risk-aversion) family
can be expressed as
^O-IL^i (_*£_ + ,),. (43)
This family is rich, in the sense that by suitable adjustment of the parameters,
one can have a utility function with absolute or relative risk
aversion increasing, decreasing, or constant. 19
'» TABLE I
Properties of HARA Utility Functions
A(C) = —— > 0 (implies ij > 0 for y > 1)
. + 1
' -v B
A\C) = < 0 for - oo < y < 1
(1 - y) \YZ~ + i) > 0 for 1 < y < oo
= 0 for y = + co
Relative risk aversion R(C) = - V"C\ V = A(C)C
R'(C) = — >0fori)>0(-00