STOCHASTIC

(c) Discuss the solution properties of the problem in (b), noting in particular when the
objective is concave.
A key assumption leading to the result of (b) is that essentially all of the functions considered
are period-by-period separable. In many cases, this assumption is not justified. For
example, decision choices in the early periods may affect the feasible alternatives in later
periods. If this happens, the result in (b) is not applicable. However, it is possible to modify
the hypotheses of the model to consider nonseparable constraints. Again, the decision
procedure in each period t is to observe the random vector , and then to choose a decision
vector z,. The decision problem is
where
£#! max
UIWL + Et1wAm&xYu2W2+ ••• + Ef, + aG]jl ,
s.t.
£ xjz -/,(»i,..-,v,-i) + v, + ct(,wl,...,wt) + b,(wi,...,wz-i)- w, + W,
= M, + st(xu,...,xm,_1,t-i,ilrd, T=l,...,r. (i)
z = (z1,...,zT) s K,
G = X d,ls,(xii,...,xmT,T)
t = T+l
+ I