A modification of Chung and Fuchs’ (Mem. Amer. Math. Soc., 6 (1951), pp. 1-12) recurrence theorem for random walks leads to an analogous result for a different discrete parameter Markov process. This latter process is applicable to an analysis of price stabilization programs involving purchases and sales from a buffer stock.speculative attack

Salant, Stephen W.

Wenocur, Roberta S.

English

Research Papers in Economics

SIAM J. APPL. MATH.
Vol. 40, No. 1, February 1981
(C) 1981 Society for Industrial and Applied Mathematics
0036-1399/81/4001-0013 $01.00/0
RECURRENCE OF A MODIFIED RANDOM WALK AND ITS
APPLICATION TO AN ECONOMIC MODEL*
ROBERTA S. WENOCUR" AND STEPHEN W. SALANT
Abstract. A modification of Chung and Fuchs’ (Mem. Amer. Math. Soc., 6 (1951), pp. 1-12) recurrence
theorem for random walks leads to an analogous result for a different discrete parameter Markov process.
This latter process is applicable to an analysis of price stabilization programs involving purchases and sales
from a buffer stock.
1. The stochastic model. By means of the following stochastic model, Salant
(1981) analyzes how speculators maximizing expected profits under "rational expec-
tations" would respond to a government price stabilization program involving
purchases and sales from a buffer-stock. Let IN denote the set of nonnegative integers,
{0, 1, 2,...}. At time IN\{0}, a harvest, Ht, of random size, is produced, drawn
independently from an unchanging distribution; part of the harvest is consumed and the
remainder is stored by the government or the private sector, with 0t denoting the
combined stock at the beginning of period t. Consumer demand for the commodity is
assumed to increase as prices decrease. Salant shows that in any period, the market
price which induces profit-seeking speculators and the government to sell exactly what
consumers demand can be written as a decreasing function of the total stock at the
beginning of the period. It is shown that when 0, is in a range of a <= Ot <= b, the
government can keep the market price at the official level P. When 0, falls below a,
speculators "attack" the government stock, purchasing it in its entirety; the government
is no longer able to defend the ceiling, and the price rises, much as it did in the gold
market in 1968 when the government’s stockpile was attacked (see, for example, Wolfe
(1976)). If Ot climbs above b, buffer-stock managers do not have the funds needed to
support the floor and the market price falls. LettingD denote the consumer demand
function and P/(. denote the price function, the following stochastic difference
equation and associated initial condition, 00 a0, describe the evolution of stocks"
(1’) 0t+l Ot "-t’- nt+l-D(P+(Ot)).
Under the assumption that
(2’) ix D(P) E(Ht),
we shall show, by means of a theorem proved in 3, that for a buffer-stock manager,
speculative attacks and an inability to defend the floor are recurrent events. It is almost
certain that there is no way for the manager to avoid speculative attacks altogether,
however large the initial stockpile, as long as the official price is set so that demand
equals (or exceeds) the expected harvest.
Other aspects of the general problem warrant study as well, but are not investi-
gated here; one such is the waiting time until the first attack. Here, we consider the case
of a government whose goal is a fixed price (or "peg"), and whose policy is to maintain
this price, as was done in the gold market for much of this century (refer to Wolfe
(1976)). In another direction, however, we could extend our analysis to the case in
which the goal is to keep the price within an interval (or "band"), with corresponding
policy of appropriate intervention when the price reaches either endpoint of the
* Received by the editors October 2, 1979, and in final revised form April 18, 1980. -
Department of Mathematical Sciences, Drexel University, Philadelphia, PA 19104.
t Federal Trade Commission, Washington, DC 20580.
163164 ROBERTA S. WENOCUR AND STEPHEN W. SALANT
interval. For example, such a policy is now being attempted to stabilize grain prices. The
analysis of this extension requires a different approach which is being pursued
separately.
2. Notation. We now examine a basic recurrence property of a specific discrete
parameter Markov process, {Xn, n N}, which we call a modified one-dimensional
random walk. ’1, 2,’" are independent, identically distributed, nondegenerate,
real-valued random variables on a probability space (lq, 4, P), with finite mean/x and
common distribution function F. to denotes an element of lq. g’R- R is a monotone,
nondecreasing function satisfying
x < a,
g(x) =t, a <- x <- b,
>-I, x > b,
for some fixed a, b . The process {Xn} is defined as
X0 a0, a =< ao -< b, (ao constant),
Xn Xn--1 -[-n g(Xn-1),
and the process {S(
M) } is defined as
S(0
t) X,
(M)
On--1 -t- CM+
n>__l.
n_>_l.
For convenience, S. S(). We note that {S., n N} is a classical random walk, and that
the process {X., n N} is identical to {Sn, n N} when g(x)=-
3. A recurrence property. Now, we present the following result.
THEOREM. If 1, 2,...are i.i.d, real-valued random variables with common
distribution function F and finite mean tx, in the sense that
then
and
dF<o,
P{X, < a i.o.}-- 1,
P{X > b i.o.}-- 1.
In view of (1’) and (2’), setting
St Ot, c-t nt, g(x)= D(P+(x)),
we reach the conclusion that, indeed (in the context of 1), speculative attacks and an
inability to defend the floor are recurrent events.
Pro@ Let
(1) A={-= lim S.< lim S.= +}.ROBERTA S. WENOCUR AND STEPHEN W. SALANT 165
By a slight modification of a theorem of Chung and Fuchs (1951), to be discussed in 4,
(2) P(A) 1.
For w A, (1) guarantees the existence of a stopping time No< oo such that
SNo< a
and
Si>=a for alli<No.
Certainly, if X, -->_ a for all n < No
XNo SN, < a.
Therefore, there exists N1 =< No such that X, < a.
Suppose
Xn =< b for all n -> N1.
Then
S(,_1) _-<Xn _-< b for all n _>- N1.
But this contradicts o A, since a translation of the S-process at the Nlth step does not
affect its recurrence. Therefore, there exists N2> N1 such that
XN> b,
and
X-<_b for allNl-<_n<N.
Continuing in this fashion, employing an induction argument, we obtain
Ac{X.<ai.o.} and A{X.>bi.o.}.
An appeal to (2) establishes our result.
4. Remarks. Given a process { Y,}, and o e 1, the value r e ff is recurrent for w
with respect to { Yn} if for every 8 > 0
Yn(o)-rl < i.o.
Let :1 be a nondegenerate random variable that assumes values other than integral
multiples of a fixed number. Chung and Fuchs (1951), and again, Chung and Ornstein
(1962) have shown that under the above assumption, for every r 6 R, for every 6 > 0,
(3) P {[Sn r[ < i.o.} 1.
However, in addition, it follows from (3) and a standard argument employing the
separability of R and the Archimedian principle that, for almost every o f, all real
values are recurrent with respect to {S(M). n N}. More precisely, for any M,
(4) []:l)( f" r.
f’ {[S("M)--r]<Si’o’}) =1"
Of course, (2) follows from (4) for the case of nondegenerate, nonlattice ’1; in the
lattice case, (2) follows from the appropriate analogue of (4).
An immediate consequence of (4) is that the process {S,; n N} almost surely
produces a countable, dense set of real numbers (see Chung (1974, p. 272)).166 ROBERTA S. WENOCUR AND STEPHEN W. SALANT
Acknowledgment. We thank G. C. Papanicolaou and to the referee for their
careful reading of our manuscript, and for valuable suggestions during its revision.
REFERENCES
K. L. CHUNG (1974), A Course in Probability Theory, second edition, Academic Press, New York.
K. L. CHUNG AND W. H. J. FUCHS (1951), On the distribution of values ofsums ofrandom variables, Mem.
Amer. Math. Soc., 6, pp. 1-12.
K. L. CHUNG AND D. ORNSTEIN (1962), On the recurrence of sums of random variables, Bull. Amer.
Math. Soc., 68, pp. 30-32.
S. SALANT (1981), The vulnerability ofprice stabilization programs to speculative attack, J. Political Economy,
to appear.
T. W. WOLFE (1976), Gold Market Report to U.S. Dept. of Treasury.

Recurrence of a modified random walk and its application to an economic model

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