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

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Munich Personal RePEc Archive

Recurrence of a modified random walk

and its application to an economic model

Salant, Stephen W. and Wenocur, Roberta S.

Federal Trade Commission, Drexel University

February 1981

Online at

https://mpra.ub.uni-muenchen.de/11171/

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RECURRENCE OF

A

MODIFIED RANDOM

WALK AND

ITS APPLICATION TO

AN

ECONOMIC

MODEL*

ROBERTA S. WENOCUR" AND STEPHEN W. SALANT

Abstract. Amodification ofChungandFuchs’(Mem. Amer.Math.Soc.,6(1951),pp.1-12)recurrence theorem for random walksleadstoananalogousresult foradifferentdiscrete parameter Markovprocess. Thislatter process isapplicabletoananalysisof price stabilizationprograms involving purchases andsales fromabufferstock.

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 salesfrom abuffer-stock. Let INdenotetheset ofnonnegative integers,

{0,

1,

2,...}.

At time

IN\{0},

a harvest,

Ht,

of random size, is produced, drawn independently fromanunchanging distribution; partofthe harvestisconsumedand the remainder is stored by the government or the private sector, with

0t

denoting the combined stock at the beginningofperiod t. Consumerdemandforthe commodityis assumed to increase as prices decrease. Salant shows that in any period, the market price whichinducesprofit-seekingspeculators and the governmenttosellexactlywhat consumers demand can be written as a decreasing function of the total stock atthe beginning of the period. It is shown that when 0, is in a range of a

<=

Ot

<=

b, the government can keep the market priceat the official levelP. When 0, falls below a, speculators"attack"the government stock, purchasingit in itsentirety; the government is nolonger able to defendthe ceiling, and the price rises, much as it did inthe gold marketin1968 when the

government’s

stockpilewasattacked(see,for example, Wolfe

(1976)).

If

Ot

climbs above b, buffer-stockmanagersdo not have the fundsneededto support the floor and the market price falls. LettingD denote theconsumerdemand function and

P/(.

denote the price function, the following stochastic difference equationand associated initialcondition,

00

a0,describe the evolution ofstocks"

(1’)

0t+l

Ot

"-t’-

nt+l-D(P+(Ot)).

Under the assumption that

(2’)

ix

D(P)

E(Ht),

weshallshow,bymeansof atheoremprovedin 3,thatforabuffer-stock manager, speculativeattacksandaninabilitytodefend thefloor arerecurrent events.Itis almost certainthat there is no wayfor the manager to avoid speculative attacks altogether,

howeverlarge the initial stockpile, as longas the official price is set so that

demand

equals

(or

exceeds)theexpectedharvest.

Otheraspects ofthe general problemwarrantstudy aswell, butare not investi-gated here;onesuchisthe waitingtime until the firstattack.Here,we considerthecase ofagovernment whose goalis a fixedprice(or "peg"),and whosepolicyis tomaintain this price, as was done in the gold market for muchof this century (refer to Wolfe

(1976)).

In another direction, however, we could extend our analysis to the case in which thegoalistokeep thepricewithin an interval

(or

"band"),withcorresponding policy of appropriate intervention when the price reaches either endpoint of the

*Receivedbythe editorsOctober2, 1979,and in final revised formApril 18, 1980.

-

Departmentof MathematicalSciences,DrexelUniversity, Philadelphia,PA19104. tFederalTrade Commission,Washington, DC20580.

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164 ROBERTA S. WENOCUR AND STEPHEN W. SALANT

interval.Forexample,such apolicyis nowbeingattemptedtostabilizegrain prices. The analysis of this extension requires a different approach which is being pursued separately.

2. Notation. We nowexamine a basic recurrence property ofaspecificdiscrete parameter Markov process,

{Xn,

n

N},

which we call a modified one-dimensional random walk. ’1, 2,’" are independent, identically distributed, nondegenerate, real-valuedrandom variables on aprobabilityspace (lq,4,P),with finitemean/x and common distribution functionF. to denotesanelement oflq. g’R-Ris amonotone,

nondecreasingfunctionsatisfying

x

<

a,

g(x) =t, a<- x <-b, >-I, x

>

b, forsome fixed a,b

.

Theprocess

{Xn}

is defined as

X0

a0, a

=<

ao

-<b,

(ao

constant),

Xn

Xn--1

-[-

n

_g(Xn-1),

andtheprocess

{S(

M)

}

is defined as

S(0

t)

X,

(M)

On--1-t-

CM+

n>__l.

n_>_l.

Forconvenience,

S.

S

().

Wenotethat

{S.,

n

N}

isaclassicalrandomwalk,andthat the process

{X.,

n

N}

is identical to

{Sn,

n

N}

when

g(x)=-3. Arecurrenceproperty. Now,wepresent the following result.

THEOREM.

_If

1, 2,...are i.i.d, real-valued random variables with common distribution

_function

Fand

_finite

meantx, in thesensethat

then

and

dF<o,

P{X, <

a i.o.}-- 1,

P

{X >

bi.o.}--1.

Inview of

(1’)

and (2’), setting

St

Ot,

c-t

nt,

g(x)=

D(P+(x)),

wereach the conclusionthat,indeed(inthe context of 1),speculative attacks andan inabilityto defend the floor are recurrent events.

Pro@

Let

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By

aslight modification ofatheoremofChung and Fuchs(1951),tobediscussed in 4,

(2) P(A) 1.

Forw

A,

(1)guaranteesthe existence of astoppingtime

_No

<

oo suchthat

SNo

<

a and

Si>=a foralli<No. Certainly,ifX, -->_aforalln

<

No

XNo

SN,

<

a.

Therefore, there existsN1

=<

No

suchthatX,

<

a.

Suppose

Xn

=<

b for all n ->

_N1.

Then

S(,_1)

_-<

_Xn

_-< b for all n _>-

N1.

Butthis contradictso

A,

sinceatranslationoftheS-processatthe

Nlth

step doesnot affect its recurrence.Therefore,there exists

N2 >

N1

suchthat

XN

>

b, and

X-<_b forallNl-<_n<N.

Continuingin thisfashion, employinganinduction argument,weobtain

Ac{X.<ai.o.} and A{X.>bi.o.}. An appealto

(2)

establishes our result.

4. Remarks. Given aprocess

{

Y,},

and o e1, thevalue re ff is recurrent for w

withrespectto

{

Yn}

if forevery8

>

0

Yn(o)-rl

<

i.o.

Let

:1

be anondegenerate random variable that assumes values other than integral multiples ofa fixed number. Chung and Fuchs(1951),and again, Chung andOrnstein

(1962)

have shownthat underthe aboveassumption,foreveryr6R,forevery6

>

0,

(3) P

{[Sn

r[

<

i.o.} 1.

However, in addition, it follows from

(3)

and a standard argument employing the separabilityof Randthe Archimedianprinciple that, for almostevery o f,all real values are recurrent withrespectto

{S

(M).

n

N}.

Moreprecisely, for any

M,

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[]:l)(

_f"

r.f’

{[S("M)--r]<Si’o’})

=1"

Ofcourse,(2)follows from

(4)

forthe caseofnondegenerate,nonlattice’1;in the latticecase, (2)follows from the appropriate analogueof

(4).

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166 ROBERTA S. WENOCUR AND STEPHEN W. SALANT

Acknowledgment.

We

thank G. C. Papanicolaou and to the referee for their careful readingof ourmanuscript, andforvaluable suggestions duringits revision.

REFERENCES

K. L.CHUNG (1974),A CourseinProbability Theory,secondedition,AcademicPress, NewYork.

K.L. CHUNGANDW. H.J. FUCHS(1951),On thedistribution_ofvalues_{ofsums of}randomvariables,Mem. Amer.Math.Soc.,6, pp. 1-12.

K. L. CHUNGAND D. ORNSTEIN (1962), On therecurrence ofsumsofrandom variables, Bull. Amer.

Math.Soc.,68,pp.30-32.

S.SALANT (1981), The vulnerabilityofprice stabilizationprogramstospeculativeattack, J.PoliticalEconomy,

toappear.