STOCHASTIC

318 DREZE AND MODIGLIANI
income side). And we conclude that section with some remarks on the
response of current consumption to availability of market opportunities
for sharing risks.
3.2. We now introduce a general model designed to analyze simultaneous
decisions about (y2, r) and £t under perfect insurance and asset
markets. 16 These decisions are assumed to maximize expected utility over
the class of all prospects, the market value of which does not exceed that
of (y2,r).
Let there be one perfectly safe asset, yielding a rate of return r0, and
n risky assets yielding the uncertain rates of return rt, j = 1 ••• n. The
amounts invested in these n + 1 assets will be denoted by (x0, Xi ••• x„).
Let furthermore future earnings y2 be the sum of m components yi2,
/ = 1 ••• mjandletZj! be the present value of vi2 on the insurance market. 17
Denote by (1 — a,), / = 1 ••• n, thefraction ofyti that a consumer chooses
to sell on the insurance market; his current wealth and future (net)
earnings then become yl + £ i (1 — a,) za and £i a(yi2, respectively.
Given a current consumption Cj, his portfolio of assets must satisfy the
constraint
m n
Xi + 1(1-
'i - cx + X C 1 - *i) Zn - Z x i'
16
In [5, Section 6], we have used a slightly different formulation, based upon the
notion that labor income (current and future) results from activities among which the
consumer divides his time, of which a fixed quantity is available; it was also assumed
that earnings from a given activity were proportional to the amount of time devoted
to it, and that one of the activities entailed a perfectly safe income; the activities themselves
did not appear as arguments of the utility function. Under that formulation,
earnings per unit of time from the safe activity provide an implicit "insurance value"
for the earnings per unit of time from any of the risky activities.
17
This "insurance value" may be defined in a number of ways: one of them is straightforward
insurance of professional income (including unemployment and medical
insurance); another is suggested in footnote 16; another still is provided by the purchase
(or short sale) of a portfolio of assets perfectly negatively (or positively) correlated
with yi%. One might also consider a '"states of the world" model [1, 3, 7] with m states
and define:
( future earnings, if state / obtains,
0, otherwise;
^a = y>2 times the current price of a unit claim contingent on state i.
Our formal analysis is consistent with any of these interpretations, or combinations
thereof, so long as z;i is well-defined, independently of the amount of "coverage" that
our consumer buys on yi2.
1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 469