STOCHASTIC

CONSUMPTION UNDER UNCERTAINTY 319
and his future consumption is defined by
m n
c2 = X a.>'i2 + *o(l + r0) + Y. x i( l + O)
= X « + >i - c i + X < J - a -) z *iK' + r o) + X x s( r >
i-i V 1-1 / j=i
= z2 + O'I - Ci)(l + r„) + X a i(>'*2 - z,2) + X x A r i — r u)- (3-2)
>-i i-i
where
z,2 =det z.iU + ^o) and z2 =det X z >2 •
Given the joint density (yl2 ••• ym2, rx--- rn), the simultaneous choice
of an asset portfolio {xx ••• xn), an insurance portfolio (1 — at ••• 1 — am)
and a consumption level cx, is then arrived at,by solving the following
problem:
max (/ )C!, z2 + (>! - cOCl + r0) + X