STOCHASTIC

44 NEAVE
Let g(x) = In x; h(x) = \x\(x + 5);f = g + /;.
Then r,*{x) = (x^ 1 + x(x + 5)-*)l(x- 1 + (x + 5)- l ),r,*(l) = 55.5/63,
and rf*(2) = 53/63.
The next lemma can be used to show that increasing or decreasing
relative risk aversion is preserved under expectation operations for
sufficiently large values of x.
LEMMA 6. Let f be a random variable defined on any finite closed
interval C (0, oo), let k be a real number, and let f(x) — E{g(k + £.v)},
where g is a strictly increasing function. Then if lim^* r,*(x) exists, for
each e > 0 there is x, such that | rf*(x) — r„*(x)\ < e for x ~^ x, .
Proof. Let e > 0. For each fixed value of £, define xtiC to be a number
such that
\ttx/(k + £*)) r,*(k + £x) - r,*(x): < e for x > .ve.£ . (2.2)
Finite numbers xE,£ exist because lim,,x (t,xj(k -- £.r)) = 1 and because
limj_,„ r„*(x) exists. Then let xf = lub {.v,c}. Now the function r,* can
be written as
r,*(x) = £{(?*/(£ + £x))g'(k + Kx)r„*(k + lx)}/E{g'(k -r- £*)£}. (2.3)
The results of the lemma can now be established by multiplying both
sides of inequality (2.2) byg'^ + £.v)£ > 0 and taking expectations to give
| E{g'(k + lx)lKrf*(x) - r,*(x))\ < E{g'(k + £*)£}« for x > x,, (2.4)
and finally, dividing both sides of inequality (2.4) by E{g'(k + £*)£} > 0.
Q.E.D.
The results of Lemma 6 can be employed to argue that if r„* is
increasing, then rf* is increasing for sufficiently large values of x.
Some useful properties of the risk aversion measures now having been
established, we turn to a consideration of the multiperiod decision problem
sketched in the Introduction. The behavior of the risk aversion
measures under maximization will be explored in the context of this
consumption-investment allocation problem.
3. MULTIPERIOD CONSUMPTION-INVESTMENT DECISIONS
AND RISK PREFERENCE
This section considers the multiperiod risk preferences of a consumer
faced with recurring consumption-investment decisions. At the beginning
of each period the consumer is required to allocate his current wealth
2. OPTIMAL CAPITAL ACCUMULATION AND PORTFOLIO SELECTION 505