STOCHASTIC

384 GORDON PTB
utility function is the realized outcome given in (2). The optimal, nonsequential,
expected utility maximizing policy will be the solution to the problem in (7).
(7) Ma,x,E{U(po+ Ei'ApizOI 0 S zT g zr-i S , • • • , S zi =S 1.
Let the expected utility of the objective in (7) be denoted by V(z) where z is the
vector, z = (zi, Zi, • • • , zT). The value of V is invariant to any renumbering of the
Zt in z. This follows because given that the Ap, are independently and identically
distributed and that the argument of U is additive in the ApiZt, any renumbering of
the zt can he written as a renumbering of the variables of integration (i.e., the Ap,).
The latter renumbering has no effect on the value of the expected utility integral.
Furthermore, if U is strictly concave, V is a strictly concave function of z. The proof
is similar to one provided in [9, p. 112].
It will now be shown that the solution to (7) must have all the values of the zt
equal to each other. Suppose that z = z maximizes V(z) for all z such that 0 ^ z S 1.
Suppose, furthermore, that at least two of the elements of z are not equal to each
other. Select two such elements and interchange their values giving z". Since z is a
renumbering of z it follows that V(z") = V(z') and also that 0 ±5 z" S 1. Consider
the vector z — \z + (1 — X)z" where 0 < X < 1. It follows that 0 S
Xz + (1 — X)z" g 1. Also, since z ^ z", it follows from the strict concavity of V{z)
that
V(Xz' + (1 - X)z") > XF(z') + (1 - X)F(z") = V(z').
Thus, Xz + (1 — X)z gives a higher value to V(z) than z . Thus, z cannot maximize
V(z) on 0 £ z g 1 unless its elements are all equal to each other. Having all the zt
equal satisfies the additional constraints in (7) that no value of z, have a larger value
than its predecessor. The maximum of V{z) subject to these additional constraints
must therefore also have the values of the Zt equal to each other. This is a necessary
condition on the solution to (7). The argument holds for any strictly concave utility
function and any arithmetic random walk.
Since the values of the zt must be equal for an optimum, dollar averaging cannot be
an optimal strategy in a multiperiod case. Dollar averaging requires that the zt strictly
decrease over time. Dollar averaging could occur only in a one period case, and then
only if by coincidence the value of zi turned out to be §. On the basis of this evidence
it seems safe to conclude that dollar averaging is a multiperiod, nonsequential strategy
based on hedging against large regrets rather than on risk averse expected utility
maximization.
3. Sequential Policies
In general, of course, the seller is perfectly free to let his future decisions depend
on the prices which will be observed before the decision actually has to be made. It is
therefore of interest to develop the sequential minimax policies. It seems to be common
experience to have second thoughts once one has embarked on a nonsequential policy
of the dollar averaging type. If the price rises subsequently one has a strong inclination
to sell less, while if the price falls one wants to sell more. This might be explained
on the basis of extrapolative expectations. Interestingly enough, however, it is also in
accord with a sequential minimax policy.
Let p, be defined as the maximum price which occurs up through time t (i.e.,
p, = Maxogigi [p,]). In terms of the notation of the last section then, pT* = p,..
Let n, be defined as the difference between the largest price which has occurred up
582 PART V. DYNAMIC MODELS