STOCHASTIC

242 THE REVIEW OF ECONOMICS AND STATISTICS
one-period portfolio problem. In our terms,
this becomes
Ji(WT-i) = Max U[CT-i]
+ E(1+P)-W[(WT_1-CT_1)
{(l-wT_,)(l+r)
-HWT-i.Zr-1}- 1 ]. (12)
Here the expected value operator E operates
only on the random variable of the next period
since current consumption Cj._! is known once
we have made our decision. Writing the second
term as EF(ZT), this becomes
EF(ZT) = /F(Zj.)(Zr|Z3._1,ZT_2,... ,Z0)
•'o
= / F(ZT)dP(ZT), by our independence
•'o
postulate.
In the general case, at a later stage of decision
making, say t = T— 1, knowledge will be available
of the outcomes of earlier random variables,
Z,_2,... ; since these might be relevant
to the distribution of subsequent random variables,
conditional probabilities of the form
P(ZT-I\ZT-2, • • • ) are thus involved. However,
in cases like the present one, where independence
of distributions is posited, conditional
probabilities can be dispensed within
favor of simple distributions.
Note that in (12) we have substituted for
CT its value as given by the constraint in (11)
or (10).
To determine this optimum (Cj._1( Wr-i),
we differentiate with respect to each separately,
to get
0 = £/'[Cr_,] - (1+p)- 1 EU'[CT]
{(l-uiT-OO+r) +»T_1Zr_i} (12')
0 = EV lCT](WT-i -CT_i)(Zi.-i-l-r)
"u'UWr-i-Cr-i)
{(1 —ieij._i(l+r) - K)7._iZT_1)]
(H^T-i-Cr-i) (ZT-i-l-r)dP(ZT-i)
(12")
Solving these simultaneously, we get our
optimal decisions (C*T_!, a»*T_i) as functions
of initial wealth W r T_1 alone. Note that if
somehow C*r_x were known, (12") would by
itself be the familiar one-period portfolio optimality
condition, and could trivially be rewritten
to handle any number of alternative
assets.
Substituting (C*T_1, w*T-i) into the expression
to be maximized gives us J1(WT_1) explicitly.
From the equations in (12), we can,
by standard calculus methods, relate the derivatives
of U to those of /, namely, by the
envelope relation
/,'(»',-,) = [/' [C,_,]. (13)
Now that we know /I[WT-IL it is easy to
determine optimal behavior one period earlier,
namely by
•MWr-a) = Max U[CT-2]
{CT-2,WT_2}
+ E(.l+p)-iJ1[(WT-1-CT-I)
{(l-a»T_2)(l+r) +wT-!iZT_2}].
(14)
Differentiating (14) just as we did (11)
gives the following equations like those of (12)
0 = v [cv_2] - (I+^-'EA' [wT-2]
{(l-wr-i)(l+r) +WT-2ZT-2} (15')
0 = £/,' [WT-!] (WT-2 - CT-2)(ZT-2 - l-r)
£ Ji [(Wr-2 -CV_2){(l-a)r_2) (1+r)
+ a>r_2Zr_2}] (WT_2 —CT-2)(ZT-2 — 1— r)
dP(ZT^).
(15")
These equations, which could by (13) be related
to U'\CT-i\, can be solved simultaneously
to determine optimal (C*T_2, ai*T_2) and
h(WT_2).
Continuing recursively in this way for T-3,
T —4,..., 2, 1, 0, we finally have our problem
solved. The general recursive optimality equations
can be written as
0= U'[C0] - (1+p)-* EJ'r-i[W,]
{(l-a>0)(l-t-r) +w0Zo)
0 = EJ'T.1[W1](W(I - Co)(Z0 - l-r)
\
0 = U'lCr-i] - (l+p)-*EJ'T-t[W,]
{(l-w,_1)(l+r)+a>,_1Z,_I} (16')
0 = £/'r_,[W,_, -C.-iXZ,-! - l-r),
(t=l,...,T-l). (16")
In (16'), of course, the proper substitutions
must be made and the £ operators must be
over the proper probability distributions. Solving
(16") at any stage will give the optimal
decision rules for consumption-saving and for
portfolio selection, in the form
C*, = )[Wt;Zl_1,...,Z0]
= ;']•_( [Wi] if the Z's are independently
distributed
520 PART V. DYNAMIC MODELS