STOCHASTIC

ergodic theory, the book by Breiman (1968) should be consulted. Additional
results on first passage times for finite Markov chains are obtained in Exercise
ME-17, and applied to a numerical example in Exercise CR-15. The results
in Exercises ME-14-ME-17 are applied to the problem of random walk with
reflecting boundaries in Exercise ME-18. For additional material on Markov
chains, the reader may consult Bharucha-Reid (1960), Breiman (1968), Feller
(1962), Hillier and Lieberman (1967), Howard (1960), Kemeny and Snell
(1960), Kushner (1971), and Ross (1970). For renewal theory and some of its
applications, see Breiman (1968), Cox (1962), Feller (1962, 1966), and Ross
(1970).
The basic results of Miller and Orr, as generalized by Weitzman (1968),
are given in Exercise ME-19. In their original paper of 1966, Miller and Orr
assumed equal costs for transfers to or from the cash balance. Weitzman
generalized the results to unequal transfer costs, and studied the sensitivity
of the optimal policy as a function of the transfer costs' ratio. As noted in
part (e) of Exercise ME-19, the optimal policy is only slightly affected by this
ratio, for realistic values. In a recent paper, Miller and Orr (1968) have studied
in some detail the adequacy of their original assumptions. They demonstrate
that the assumption of a Bernoulli random walk can be generalized without
significant effect on the optimal cash balance policy. They show how more
general cash balance dynamics can be well approximated by mixtures of
Bernoulli random walks, and they present methods for the solution of such
problems. They discuss the possibility of more complex cost structures,
including both fixed and proportional components for cash transfers. They
also treat the interesting case of a "three asset" model, where in addition to
idle cash there exist two alternative earning accounts: a high yield, relatively
illiquid account, and a lower yield, highly liquid "buffer" account. Unfortunately,
space limitations prevent the inclusion of most of these interesting
topics. However, the results for fixed plus proportional transfer costs are
outlined in Exercise ME-20.
The Miller and Orr theory of cash balance management assumed the firm's
operating procedure to be of the simple control-limit type. In many "inventory
theoretic" cases, especially those involving complex cost structures (nonconvexities,
price breaks, etc.), such partial optimization over a set of simple
policies is the most that can be achieved, since the actual form of truly
optimal policies may be unknown. Even if optimal policies are known, they
may be sufficiently complicated from an operating point of view that suboptimization
(over simple policies) is actually preferable in practical cases.
This view, based on the paper by Karlin (1958), is the one adopted by Miller
and Orr. Other work on the cash balance problem concerns the actual
form of optimal policies. Eppen and Fama (1968) treated the cash balance
problem using linear programming. They assumed linear holding and penalty
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