STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 315
over temporal uncertain prospects is readily seen if one contrasts extreme
situations. At one extreme, suppose that the indifference curves are nearly
linear in the vicinity of the equilibrium point: the consumer is almost
indifferent about the allocation of his total resources between Cj and c2,
which are almost perfect substitutes, the curvature is close to nil, and the
response of Cj to a compensated change in the rate of interest would be
very large. Obviously, for such a consumer, delayed uncertainty is not
appreciably different from timeless uncertainty, since the opportunity to
gear ct exactly to total resources matters little to him. At the other extreme,
suppose that the indifference curves are very close to right angles in the
vicinity of the equilibrium point: the consumer has very exacting preferences
for the allocation of his total resources between c± and c2, which
are strongly complementary, the curvature is very pronounced, and the
response of c1 to a compensated change in the rate of interest would be
negligible. For such a consumer, delayed uncertainty is very costly, due
to the imperfect allocation which it entails: the utility of a consumption
plan with given present value c^ + c2(l + r)- 1 decreases rapidly when the
allocation departs from the preferred proportions. Thus, as formula (2.9)
shows, the aversion for delayed risks grows as curvature of the indifference
loci increases, or, to use more operational terms, consumers who would
respond strongly to a (compensated) change in the rate of interest are
relatively better suited to carry delayed risks.
The role of the other factor, the marginal propensity to consume,
is again most easily understood by looking at limiting situations. If
dcjdy = 0, then the optimum Cj can be chosen without exact knowledge
of total resources, so that perfect information is worthless. At the other
extreme, a person who wants to consume all his resources now because he
derives no satisfaction from later consumption is ill-suited to bear delayed
risks: since he can only afford to consume now the resources he is sure to
own, the uncertain prospect carries no more utility for him than the
certainty of its worst outcome. In general, the inferiority of temporal over
timeless uncertain prospects will be the more severe, the larger the marginal
propensity to consume (other things being equal).
2.4. We now turn briefly to the case where y2 is given (say y2 = y2)
and r is a random variable with density {r). Our problem is to compare
timeless with temporal uncertain prospects about r. One can readily verify
that the Pratt "risk-aversion function" for timeless gambles about r is
—(yi — Ci)(VrrIVr)y . 14 Similarly, when a\ = (yy — c^f o> 2 , we see from
(2.7) that — (y1 — ^i) 2 (Co2/t/2)^ is the appropriate corresponding meas-
" Note from (A.l 1) that VT = V, (y1 — Cj) has the same sign as (y, — C[): an increase
in r affects utility positively for a lender, negatively for a borrower.
466 PART V. DYNAMIC MODELS