STOCHASTIC

W. T. ZIEMBA
Finally we must specify the return function. Suppose that the utility of
consumption, at stage t, discounted to the present takes the additive form
)3'" 1 Mf(c/). Then
Vs(w, i) = £ j Xr/r'H.OW,))!^! = w,s\
= «i(V.)) + £{E/ J '" 1 "1(^,,))]W / i=«'.4
^u1(c1) + pE{vt{W2,2)\1Vl = w,d}
= ul(cl) + p( < %a(z,2)[z;w,l,c1]
J - 00
is the expected value of following the policy 3 if the consumer has initial
wealth w. The reader is asked in Exercise CR-19 to verify that the monotonicity
assumption is satisfied by this return function if the u, are monotone
nondecreasing.
If \ut\ is bounded, say, by M < oo, then the return function is well defined
because
iP'-^ic.) £MJ(l-P)