STOCHASTIC

Second, operating policies can involve decreases as well as increases of cash
inventory (for example, by transferring a quantity of cash to an earning account
or productive asset). Because of these differences, the cash balance problem
is mathematically more complex than the ordinary inventory problem, even if
the cost structures are similar in the two cases.
The similarity between the cash balance problem and the standard inventory
problem was originally exploited by Baumol (1952), who applied to cash
holdings the classical "lot-size" model of inventory management. This model
was extended by Miller and Orr (1966) to include uncertainty. Miller and Orr
assumed the following costs: (1) a holding cost proportional to the average
cash balance level and (2) a fixed cost for transfers to or from the cash balance.
They assumed that changes in the cash level could be adequately described
by a Bernoulli random walk, with known probability p of an increase by a
fixed amount m, and probability (1 —p) of a decrease by m. The firm was
assumed to employ the following simple policy for cash balance management.
Let the cash balance wander freely within the control limits 0 and h > 0, and
return it instantly to a fixed, intermediate value z upon reaching either control
limit. The objective was to determine the upper control limit h and return
point z so as to minimize the long-run average cost of the cash balance.
In the Miller and Orr theory the behavior of the cash balance level may be
described as a random walk with reflecting boundaries. A number of useful
methods for random walk problems are given by Feller (1962). To make the
present treatment as self-contained as possible, several of the necessary mathematical
tools have been developed in the exercises. In the cash balance problem
the random walk process starts anew after each transfer from the control
limits to the return point. The times between successive cash transfer form a
sequence of nonnegative iid random variables. Significant economies can often
be achieved by studying general properties of systems which "restart" themselves
in this manner, and Exercise ME-14 presents a number of simple but
useful results pertaining to such renewal processes. For a given (stationary)
operating policy, the cash balance levels form a stationary, finite Markov chain.
Exercise ME-15 introduces some definitions relating to Markov chains, and
proves a number of important facts about recurrent events and the limiting
behavior in time of chains having finitely many states. For applications to the
cash balance problem, it is necessary to know the asymptotic behavior of
average reward functions defined on a Markov chain. This topic is discussed in
Exercise ME-16. It is shown that the long-run average cost is given by the
expected value of the cost function, taken with respect to the asymptotic
probability distribution. This is a typical example of a so-called ergodic
theorem, and is true under more general circumstances. Although the general
treatment of such problems can become highly technical, the results for finite
Markov chains are reasonably simple to obtain. For a general introduction to
444 PART V DYNAMIC MODELS