This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 5, number 3

Volume Author/Editor: Sanford V. Berg, editor

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm76-3

Publication Date: July 1976

Chapter Title: Microeconomics: Stochastic Models of Price Adjustment

Chapter Author: Steven Barta, Pravin Varaiva

Chapter URL: http://www.nber.org/chapters/c10479

Chapter pages in book: (p. 267 - 281)

Annals of &opunniC and Social Measurement, 5/3, 1976

MICROECONOMICS

STOCHASTIC MODELS OF PRICE ADJUSTMENT*

BY STEVEN BARTA AND PRAVIN VARAIVA

Sonic models of stochastic approximation ace presented which seek to explain how several sellers in asingle market adjust their prices and quantities in disequilibrium and when the demand for their product isimperfectly known. These adjustment schemes have the known property that they permit sellers tosimultaneously learn their demand function more accurate1v and to search for more satisfactory pricelevels wit/i little computation. Sonic subtle effects of stochastic environments are discovered which haveescaped informal discussions of the problem.

1. INTRODtJCI1ON

Several authors seeking to explain how sellers set prices or quantities outsideof equilibrium simplify their analysis by "avoid[ing] the problem of what firmsshould do when they do not know their demand functions" [4, p. 186]. Thesimplification is achieved by assuming either that sellers know very little or ignoretheir monopoly power [2, 8], or that sellers know their demand functions exactly[1, 3]. These assumptions are made in spite of the fact that in discussing theirmodels these authors often argue in terms of the uncertainty in demand.

When there is uncertainty about its demand function, the firm can experi-ment with its prices and observe the reactions of its customers, and with thisadditional information the firm may discover levels of profitable prices. Theproblem of finding the optimal sequence of prices can be posed as a problem inBayesian decision theory, and this has been done for the case of a single firm in avery simple economic environment [4]. Such a formulation has two deficiencies.First, the computational effort necessary to calculate the optimal sequence is sogreat that even its normative significance is diminished if costs of computation aretaken into account. Secondly, any attempt to extend along these directions theformulation to include several interacting firms appear to lead inevitably into theconsiderably more intractable theory of sequenial stochastic games. (For somerecent efforts in this area see [9, 10].)

It is the objective of this article to present .i family of price-adjustmentprocesses for firms in a single market which (a) are ro'iust as well as computation-ally simple, (b) exhibit the Tact that firms must experiment to discover profitableprices, and (c) possess orthodox convergence properties. From the viewpoint ofeconomic theory it is interesting to note here that the convergence of theseprocesses is determined largely by the convergence of corresponding rules (suchas those studied in [1, 2, 31]) where the firms know their demand functions inadvance. This is because the processes presented here converge if (I) the behaviorof consumers is systematic enough even while it is random so that each firm can

Research was supported by National Science Foundation under Grant GK-4 1647 and ENG 74-01551-AOl.

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e

learn, through repeated trials, the demand for its product at any set of fixed prices,and if (ii) assuming the firms know their demand functions, the adjustmentprocesses lead to prices which converge to an equilibrium. Since condition (ii) hasalready been well investigated in the economics literature, our main task is toinvestigate condition (i). As we shall see this condition captures certain subtlephenomena which elude informal discussions of price adjustment under uncer-tainty. As an example, we may mention here that prices can stabilize to an excesssupply situation simply because firms react faster to excess demand than to excesssupply.

From the mathematical viewpoint the proposed rules belong to the familyknown as "stochastic approximation" schemes following the pioneering paper ofRobbiris and Monro [7]. However, we shall follow the formulation due to Ljung[ii] not only because it is considerably more general in several respects but alsobecause it clearly points out the dual functions of learning and search mentionedabove. Motivated by the same concerns as those mentioned above, Aoki{12, 13]has already used stochastic approximation methods to model some adjustmentprocesses. The relation between his work and that presented here will be detailedin Section 4.

In the next section we state the main results of[1 1] in the form of an abstractadjustment model. In Section 3 this model is used to investigate stochasticversions of the more concrete processes proposed in [1, 2, 3].

2. AN ABSTRACT ADJUSTMENT MODELN firms produce and sell a homogeneous product. At the end of period t firmn sets certain instrumental variables (e.g. prices or quantities) denoted by thevector v. It is assumed that t; belongs to an a priori fixed, compact set B. LetB = B 1x. .. xB". Let v = (v,..., v) be the distribution of these variablesacross the market. (In the following whenever a superscript is omitted from avariable name, it designates the vector whose components correspond to thevarious firms; thus y =(y1,. . . , y") etc.) in period t+ 1 consumers search amongfirms and react to the distribution v. Their behavior as observed by the nth firm isformulated by it as the vector y'(v, where {j is a random vector sequencedefined on the probability space (Q, P). Note that y is determined by theaction of all firms v, and the random variable which does not depend on v,.Based on this observation the firm adjusts its instruments at the end of periodr + 1 according to the rule

(2.1) (w) = {v'(w) + yH(y'(v,(), (w)))]B

= [Vo)+ y,z(v,(), e,+i(w))]Here Yt >0 is a constant determining speed of adjustment, H"( ) is a functionwhich relates an observation to desirable directions of change in the variableu", z(v, 4) = H(y(v, i)), and [ jB is any function satisfying

[x]8 E Bforallx,[x] for XEB.268

For each fixed v let Z'(v) = Ez(v, ). The explicit dependence of Z1(v) issupposed to be transitory. That is, if consumers face a constant distribution v thentheir average behavior stabilizes, i.e.. there is a function Z(v) so that

Jim Z1(v) = Z(v) for each v E B.

Thus the environment is "stationary" in an important sense.Next, it is assumed that as a result of firms seeking their goals, or as a hidden

"aim" of market forces, the instrumental variables are directed to the v'' definedby the "equilibrium" condition

Z(v*)=Oi.e.Zfl(v*)=O,n=1,...,N.If each firm n could directly observe Z" (v) then v would be an equilibrium of thedifferential equation

(2.2) v=Z(v).

On the other hand, for any fixed a, Z(v) can be estimated by

(2.3) '(v, w) = + y,[z(v, ,(w)) w)J, = 0.

(For example, if Yt = t1,then (2.3) yields = f (v, which is a robustestimator.) Thus the actual adjustment rule (2.1) can be seen as a way ofcombining simultaneously the "learning" process (2.3) and the equilibrium-seeking process (2.2).

We impose the following conditions.

(Cl) O)y1, y-Oast-+D, y=o.(C2a) For each fixed VEB, the random sequence {(v)} generated by (2.3)converges to Z(v) a.s.(C2b) For each fixed , the function z,(v, ) is uniformly Lipschitz in v belongingto an open set B° D B, with Lipschitz constant k,(). Furthermore, the randomsequence {r} generated by

= r,(co) + y[k,(1(w)) r,(w)], r0 = 0,

converges to a constant r a.s.(C3a) The set B is defined by B = {v113(v) b} for some constant b where fJ is atwice continuously differentiable function, and there is a constant k so that

E[(z(v, k

for all t, and a, u in B. Here f3,,.,(u) is the Hessian of J3 evaluated at U.

(C3b) For all v E B, the boundary of B,

((v))'Z(v) <0.(C3c) v'' is an asymptotically stable equilibrium of the differential equation(2.2), and its domain of attraction contains an open set B° B.

We discuss these conditions after stating the main result.

T2.l. Consider the random sequence {v} generated by (2.1). Suppose (C1)(C3)hold. Then v converges to a" a.s.

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Proof The assertion is an immediateconsequenceof Theorem 3.!, Theorem 5.2and the subsequent remark in {1i}.

Consider the conditions in reverse order. (C3c) says that if Z(v) were directlyobservable then all solutions of (2.2) which start in B converge to v. Since thiscase is well-studied in the literature, it need not detain us further. Since (C3b) isonly slightly stronger than the statement that B is an invariant set of (2.2), it isusually satisfied whenever (C3c) is. (C3a) guarantees that the effect of thedisturbances {,} is not too large. For instance, it is easily verified if z(v, ) isContinuous in (v, ) uniformly in i and sup EJ <co.

(C2a) guarantees that it is possible to estimate Z(v) for any fixed v while(C2b) guarantees that estimates (v) converges to Z(v) uniformly for t' E B, andthis implies in particular that Z(v) is a Lipschitz function so that the differentialequation (2.2) is well-behaved.

The requirement y,>O reflects the fact that in (2.1) z is a direction ofdesirable change in v', whereas the bound Yr 1 is merely a normalizing conditionin light of the second condition y, - 0. This latter condition is necessary if learningbehavior is to be exhibited since then as time progresses new observations shouldhave decreasing importance. The condition is discussed more fully in the nextsection in the context of a specific example. The divergence of y, is obviouslyessential.

The interesting conditions therefore are (C2a) and (C3c). Roughly speaking,the former guarantees that learning is possible in principle, while the latterguarantees convergence in the absence of uncertainty. The remaining conditionslink these in such a way as to ensure that both functions tan be carried outsimultaneously.We conclude this section by giving some simple conditions which guarantee(C2a) and (C2b).

L2.1. Suppose satisfies (2.4) and y satisfies (Cl) and (2.5).(2.4) are independent if tsIM for some M<co(2.5)

and

11 for eath v in B there exists a> I such that(2.6) EIz,(v, (fl)-Z1(U)58'for some nondecreasing sequence 8, and

(2.7) y6<co, where =min(a, l+a),then (C2a) holds.

If Ek,(1) = r, converges to r, and if there exists a > 1 such that(2.8)

for 8, nondecrcasing and (2.7) is satisfied, then (C2b) holds. Furthermore, ifI <a 2, then (2.5) is not needed.

270

Proof. See Appendix A. 1.Consider (2.4). Notc that the disturbance term is a consequence of consumer

search. If periods remote in the past do not affect their search and hence theirbehavior in the present period, or if their behavior varies independently betweendistant periods, or is a new "generation" of consumers replaces the previousgeneration every so often; in all such environments (2.4) may be reasonable.' (2.5)has the following meaning. From (2.5), (v) can be expressed as a movingaverage

=c,z(v, +i)

wherec,,r='yr_i [1(1-y) ifT<tandc,, = y,,,sothatc+,c,7ifandonlyifYr )'r-IU yr). Thus (2.5) means that in the estimate '(v) recent observationsare weighted more heavily than previous observations. Condition (2.7) is moreinteresting since it exhibits a trade-off between the efficiency of search andlearning. Specifically, the slower y, converges to zero the greater is the effect ofdisturbances on the learning process, but the faster is the convergence of v, to v ifit converges at all, and (2.7) displays this conflict in the two functions.

3. SOME CONCRETE ADJUSTMENT PROCESSES

In this section we follow the abstract mode introduced above to obtainstochastic versions of some adjustment processes studied in the literature.

3.1 Fisher's Quasi-competitive Adjustment

For a discussion of this model the reader should consult [2]. At the end ofperiod t, firm n (n = 1,. . ., N), believing that it faces a flat demand curve, sets aprice p' and offers for sale the amount S(p'). p B' = [b, b]c R+, and S"(p") isjust the inverse of the marginal cost curve.

In period H- 1 consumers search among various firms and register thedemand d"(p,, 4i) at firm n. {} is a stationary sequence of random vectors. Letx"(p,, = d(p,, +i) - S"(p) be the excess demand of firm n at end of periodt+ 1. For each p fixed let D(p) = E[d(p, c,)], and let X"(p) = D(p) - S(p). Atthe end of period t 4- 1 the firm adjusts its price according to the rule

n ri ,lfl B(3.1) p,+1=[p,+y,h x (pj,4,)]where h" >0 is constant.

Assume that is bounded a.s. Let B° B be an open set such that for fixedU, f(p, ) is Lipschitz in p E B° with constant k(U) and

(3.2) k() is bounded a.s.

Assume further that for each fixed p

(3.3) x"(p, ,) is bounded a.s.

as seen from [111, (2.4) can be replaced by weaker conditions which imply that v andbecome independent as t -* cO.

271

Iersw.*,tT

Finally, assume that p E B is a unique, globally stable equilibrium of thedifferential cquation

(3.4)

with a dornainof attraction containing B°, and that(3.5) X"(p)>O ifp'b; X(p)<O fp"/.T3.1. Suppose the assumptions made above hold. Suppose Yt satisfies (C1),(25)and for some a> 1(3.6) yzco.Suppose , , are independent for 11-sIM. Then the random-sequence {p,}generated by (2. 1) converges to p" a.s.Proof. Because of T2. I we only need to verify conditions (i), (ii) of L2. 1. Becauseof (3.2), (3.3) and (3.6) these conditions hold for 8 constant.

Since Fisher has extensively discussed the stability of (3.4), we need notconsider it any further. (3.5) is reasonable in the partial equilibrium context of themodel. Hence we shall only deal with the stochastic aspects of (3.1).At first sight the stationary but myopic adjustment process

(3.7) - p'+h"f (Pt,may appear more plausible than (3.1). Similar schemes have been studied forexample in [6] and [151.2 However (3.7) is incompatible with learning in the sensethat if the firm knows it faces randomly fluctuating demand and if its intention is todiscover constant levels of price and production which are compatible withaverage conditions of demand, then this intention cannot be realized through(3.7). For suppose for simplicity that N = 1, S(p) = s0 + sp, d(p, ) = d0 - dp +and are independent with zero mean. Suppose further that h is so small thatI1h(s+d)I<1 Then Ep-p and E[ptEp,J2-o2 where so+s= d0+d15 andh2{1 - h (s + d)J2 E> U if EE>O. Thus, while the statistical average of theexcess demand tends to vanish, actual prices and production levels fluctuateconstantly with the demand.3 On the other hand, if the firm is aiming to meetaverage demand then, as it gains information about this average demand, it mustrespond less and less to instantaneous fluctuations. This accounts for 'y, -*0. Asimilar phenomenon occurs in [4] where the firm eventually stops adjusting itsprice even though demand continues to change randomly.Instead of prices adjusting in proportion to excess demand, we could consider(3.8) P+i p'+y,H[x"(p

where H is a signpreservjng function. Suppose H' has at most linear growth.Then Pr - i a.s. where EH[X" (fl, )] 0 so that j3 may not equal the competitive2Our information regarding [15] is limited to the discussion in [5].3lncidentally, a time-continuous version of this example shows that Theorem 3.3 of [61 isincorrect.

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equilibrium 1:1*. Indeed, consider the specification of the preceding example withn-1, H(x)=ax if x'>O, H(x)=x if x<0. Then is determined byEH[x(j5)+,]0 or aE[x()+J"=E[x(f)+z] where f=[V0 and f=(f) VO. It follows that if a> 1, i.e., the firm reacts faster to positive excessdemand, then X() <0 so that at the equilibrium price there is positive excesssupply. Furthermore, the greater is the randomness in demand, the larger will bethe value of X(,3)I. For instance, if is uniformly distributed over [a, a], thenX()= (1 + a)(a - a)1a. Thus if we interpret the sellers as workers supplyinglabor and if wages rise faster in conditions of excess demand than they fall insituations of unemployment, then wages will converge to an equilibrium wherethere is unemployment. Of course, the opposite tendency prevails if a < 1.

One final remark in connection with nonlinear functions H" may he ofinterest. Under the conditions on X(p) in ji2], p* is the competitive equilibriumand so, in particular, all of its components p "are equal. 'I'he equilibrium is given

by EH"[X"(j3)-F-.] = On = 1,. .., N. Even under the same conditions as in [2], it

is, of course, no more generally the case that all the 5" are equal. A similarconclusion is reached in [4, p. 201] except that these differences in adjustmentprocesses arise from different beliefs about the structure of the random demand.

3.2 Diamond's Adjustment Model

The reader should consult [1] for the model discussed here. Again there areseveral firms. In each period consumers search randomly among these firms hutthey do not discriminate between them on the basis of previous experience.Therefore each firm faces demand functions whose statistical properties areidentical and so we need consider one firm only. A consumer who entered themarket at some time 'r t stays in the market until he encounters aprice Pt whichexceeds his own cutoff price q. He then purchases the amount d(p1, +i). Thecutoff price q depends in some random way upon previous prices pr,. . . , P-I In

keeping with the spirit of the search process as described in [1], it is assumed herethat {} is a sequence of stationary, independent random vectors.

Let N(p) be the (random) number of consumers who entered the market at T,

who are still in the market at t. and whose cutoff price exceeds p. Then the demandfunction facing our firm in period t is Tt d(p, +1)N(p). The firm's unitproduction cost is constant, and we may assume it is zero, so that its profit functionis pd(p, 1)N(p) where N(p) = N(p).

Let r(p, = pd(p, and R(p) = Er(p, for each fixed p. Let p, EB =[b, b]c R+ be the price set by the firm at the end of period r. Then in periodt+ lit observes p) and '(Pt, 1+1)N(p) so that it knows r( p1,

Suppose for the moment that the firm wishes to set its price at a level p5 atwhich R (p5) equals some 'satisficing" level RY Then if p is adjusted according tothe rule

(3.9) P1+1 =[p1+y,(R5r(p1, +1))f

4We do not discuss the role of N,( p) any further since under the assumptions on the adjustmentsof the cutoff price given in [1], N,(p,)-+ N, the total (fixed) number of customers in the market,whenever Pi converges

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S

it will converge to pS as. under appropriate conditions which can he obtainedfrom the results of the previous section.

Now suppose the firm wishes to maximize R(p). Then if the maximum valueof the profit is not known, a rule such as (3.9) is clearly inappropriate. In essence.the firm needs to obtain information from which it can infer whether or not it hasreached a profit-maximizing position, and if it has not done so which direction ofchange would lead to an increase in profits. Such information is provided by themarginal revenue function M(p) (dR/dp)(p). Suppose momentarily that ateach r the firm can obtain a sample ni,( n,i) where {'i,} is a random sequencesothat, for each fixed p, Ern,(p, ) = M,( p) -* M(p) as t - cx. Then prices adjustedaccording to

(3.10) Pii '{p,+y,'n,(p,, q,1)J'1

would, under appropriate conditions, converge to p at which M(p*) =0.The sample rn,(p,, may be obtained directly iii some way not explicitlyconsidered in the model, or it can be obtained by experimentation in the followingway. Suppose each period t Consists of two subperiods labeled (t, 1) and (1, 2).Suppose at the end of period (t, 1), the firm's price is Pt,i and in period (t. 2) itobserves r(p,1, 2). At the end of period (, 2), it sets the price P1.2 = Pi.i - Q, wherea,>0 is a predetermined sequence to be specified further. In period (t+ 1, 1) thefirm observes r(pii a,, i) Define

(3.11) m1(p4I,iI)=!{r(pI,2)_T(p 1awhere Th+i = (, Suppose now that Pt+t,i is adjusted according to(3.12) P,i.i [p,.i+y,rn,(p,1, m+)]8in a way quite similar to (3.10).

We can use 12.1 and L2. I to obtain sets of conditions under anyone of whichthe sequence P,i generated by Diamond's process (3.12) converges to theprofit-maximizing price. Here is one such result whose proof can be readilyconstructed using 12.1 and L2.1.T3.2. Let B° B be an open set such that

M(b) >0, M(b) <0, M is monotonic on B°,E[r(p, )_r(p)J2o.2<c for PE B°,r(p, ) is twice continuously differentiable in p for fixed andfar dRl2Ej_(p,)__j so-;<ct for iB°

(iv) y,-0, a,-0,

Then (and hence Pv.2 also) converges to P'' a.s. where M(p*) = 0.Suppose the firm had a direct estimate of M(p) so that it could usc (3.10).Then, convergence of (3.10) to pf wou'd be guaranteed under conditions (i), (ii),274

I

(iii) and

(iv')

Note that (iv') is considerably weaker than (iv) because since a, - 0 a sequence {y,}satisfying (iv) must decrease much faster than if it had to satisfy (iv'). In turn thismeans that convergence of p, under (iv) is considerably slower than under (iv').This is a reflection of the fact that obtaining an estimate of the marginal revenuefunction M(p) from observations of random demand is a subtle process since Misnot given in parametric form. Of course, if it were parametrized, say, it is known tobe a linear function with unknown slope and intercept, then faster convergence is

possible.

3.3 Adjustment to a Nash EquilibriumFor discussion of the model introduced here see [31. In a certain sense this is

an extension of Diamond's model discussed earlier. Consumer search behaviorcauses the demand faced by a firm to depend upon the other firms' prices. Eachfirm realizes that it faces a sloping demand curve, and its objective is to exploit thismonopoly power.

If p E B" = [h, b] c R is the price set by firm n at the end of period t, then thedemand for its product during period I + 1 is d" (p,, ) where p, = (p,'.....p) isthe distribution of prices among the N firms and {,} is a stationary randomsequence. Let D"(p) = Ed"(p, ). C(q") is thc cost function of firm n. Weassume that D" and C" are twice continuously differentiable. Define the profitfunction "(p)=p"D"(p)C"[D"(p)]. We assume with Fisher [3, p. 449] thatfor fixed values of p'. i

(3.13) ir"(p) is strictly concave in p" E B".

It follows from a result due to Rosen [14] that there exists a Nash equilibrium pricevector pEB, i.e.,

n -ì n -N ti - n ii.ir(p,...,p,...,p )i(p) for p EB,fl1.....Next we assume (see [3, Theorem 3.1]) that

(3.14) is the unique Nash equilibrium in B

and for each u

(3.15) -(p)>O if p" =b,,(p)<O if P =

which is reasonable and self-explanatory.

Fisher assumes the existence of the Nash equiIibrunì when in fact it is a consequence of theconcavity assumption.

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Now, for each p = (p',.., p')a B, let p' = p"(p) E B be the price at whichfirm n maximizes prolit when all the other firms' prices are fixed at p', i i.e.,

71iL(pI,..., j5'l......... ir"(p', p,.., J)N) for j3" a B".Equivalently, in view of (3.15), ' is determined by

air'. -'i Np )=O.

Fisher imposes enough additional conditions to guarantee that the adjust-ment rule

(3.17) " =

where H" is any sign-preserving function, has an asymptotically, globally stableequilibrium j3 (see [3, Theorem 3.2]). Now for (3.17) to be a good description of anadjustment process, it presupposes that each firm n has an accurate knowledge ofits profit function ir"(p), so that it could solve for p7" from (3.16). If, however, itdoes not have this knowledge, the firm can still attempt to estimate air"/ap' fromthe observed random demand d"(p1, +i) and its own cost function, as wasproposed in regard to the Diamond model. For simplicity we assume that the firmhas directly available to it the observation rn"(p,, q11) such that

(3.18) Ein"(p, m)=,(p) for each fixed paB,

and it uses this observation to adjust p" according to the rule(3.19) p+i p+rn'1(p, m+).We can prove the following convergence result.T3.3. Assume that (3.14), (3.15) hold and that j3 is the unique asymptotically,globally stable equilibrium of (3.17) (for any sign-preserving H") with domain ofattraction containing B. Assume that {m} is a bounded, stationary process withm independent when jtsJ T for some T<OD, Let {'y,} be a sequencesatisfying

(3.20) O')l,y,-*O as t-J, )+iy,(l1), <for some a> I . Then the random sequence {pt} generated by (3.19) converges a.s,to 5.

Proof. See Appendix A.2.From our earliest discussion it is clear that if instead of (3.19) we consider themore general rule

P*-I p'+y,H"[in"(p, m+)]we still get convergence to j5 if H(x) = hx for h" a positive Constant, whereas, ifH is a nonlinear sign-preserving function then the process is likely to Converge toa different equilibrium, One additional remark which concerns the Fisher adjust-Inent process (3.17) may bc of interest. In the proof of T3.3 we show that the

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(3.16)

stability of (3.17) and the concavity assumption (3.13) together imply that thetrajectories of

(3.21)

converge to . Now, from relatively abstract mathematical considerations (see[16]), we know that Nash equilibria are unlikely to be stable equilibria of systemswith gradient dynamics as in (3.21). This consideration provides an argument (inaddition to those made by Fisher himself) that the assumptions which guaranteestability of (3.17) are very restrictive.

4. CociusioNs AND NUMERICAL RESULJSWe have tried to show how disequilibrium adjustment processes which

consider only "deterministic" environments can be modeled as stochastic approx-imation schemes so as to take into account uncertainties on the part of sellers. Indoing so we have discovered a variety of subtle phenomena which have escapedinformal discussions of the subject.

Aoki appears to be the first to have used stochastic approximation methods tomodel adjustment processes and he has been motivated by the same concerns. In[121 he has compared a RobbinsMonro scheme with three Bayesian formula-tions, all for a single firm whose demand function depends only on its own price,and shows that they are asymptotically equivalent. In [13] he considers severalinteracting firms in the same industry. In each period t each firm n adjusts itsoutput rate q (subject to an exogenous disturbance), and then learns the commonmarket clearing price Pr based on which it adjusts the next period's output rate.Thus in [13] the interacting firms are Marshallian quantity adjusting firms,somewhat similar to [17], unlike the price adjusting firms described here.

While we have given conditions which guarantee convergence to an equilib-rium, the actual rate of convergence and the behavior of the price sequenceoutside of equilibrium depends critically upon the adjustment coefficients y andthe magnitude of the disturbances. While estimates of the asymptotic behavior ofthe random price sequence are available (see [1 1H13}), these estimates provideno understanding of the "transient" behavior. Therefore we present below theresults of a numerical experiment of the scheme of Section 3.1 for a two-firm case.In terms of the notation introduced there N = 2, d" (p, E1) = D" (p) ± ' where ' is

10

lOP2 106

104

(20. 100)

94

Figure 1

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26

uniformly distributed over [a, a], , = 1, 2. The average demands D"(p) aretaken to be

12p 13 ,-f-p,pl+p2 p +p- -

D2(p)= 2p-f-p p+pwhere and 13 are positive constants. The supply functions are taken as

= a"(p" +c"),6 n 1, 2

with ce", c" as positive constants. Equation (3.1) now reads(4.1) = {p'+ y(D' (pr) + - S"(p'))]'3, n = 1, 2

Nine sample paths of (4.1) are presented corresponding to three differentvalues of {y,} and three different values of the noise parameter a, as shown inTable 4.1 below. In Figures 2a, b, c, y, 0.012, a constant. Since y is large, initialconvergence is rapid but as t increases we do not get convergence since y, does not

(C)

(b)

Figure 26The parameter values used in the numerical example are these: $ = 100, 31

= 27, 82 = 0,a2=1/18,c'=486Qc =30.

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I.

I

L

a

TABLE 4.!

0.012 5(250+ t) 0.1:

converge to zero. In Figures 3a, h, c we obtain convergence. in Figures 4a, b, cconvergence is extremely small since y, is small, in all cases t = 1,..., 1 250. It is

90(c)

Figure 3

105 lOP2- 105

(20. 100) p (20, 100) Ppt I I I I -.1 __---_-L

25 30 10 15 25 30

is(a) (b)

lop2- 105

(20. 100) pp1

(a) (b)

10 15 25 30

95

(c)

Figure 4

279

I .0 2a 3a 4a

2.5 2b 3h 4h

5.0 2c 3c 4c

lop2-

10 15

evident from these Figures that increased values of a leads to poorer convergencebehavior. Ljung fill has shown that asymptotically the behavior of the randomsequence (4.1) is similar to the behavior of the trajectories of the differentialequations

(4.2) jY'=D"(p)S(p) n=1,2and, for purposes of comparison, one trajectory of (4.2) is plotted in Figure 1.

Electronics Systems Laboratory, MITElectronics Research Laboratory,

University of California, Berkeley

APPENDIX

A.1. Proof of L2.1

Since the proofs of (i) and (ii) are identical, we only prove (i). Let E ='(v, +i)Z'(v), z(v, 1i)Z'(v). Then, by (2.3),

and it must be shown that a.s. For tn = 1.....M define the randomsequence {e" by

e if t = m modulo Me'=1(0 otherwise.

Then Eer=0 and er', e" are independent for because of (2.4). By (2.6)ô' and we have (2.7), , y'<co. It follows from Theorem 4.3 of [11]

(where condition (2.5) is used) that the random sequence i"-* 0 a.s. where= + (e - But since er = e" we have i = , 7', so that i, -+0a.s.

A.2. Proof of T.3

The only difficulty in applying T2.1 and L2.1 is to show that under thehypothesis of the theorem j3 is a globally asymptotically stable equilibrium of(A.i)

'1

.,,p

Now suppose " ="(p)p so that,i -n N nIT (p ,.. .,p ,.. . ,p )>ir (p).

It is then a consequence of (3.13) and the definition (3.16) of 3" that

according as " _pfl

280

a

Hence there is a sign-preserving function R' (which depends on p) such that

H(15 ")

and the stability of (A.1) follows.

REFERENCES

P. A. Diamond, "A Model of Price Adjustment." J. Econ. Th. 3(2), (1971), pp. 156-168.F.M. Fisher, "Quasi-Competitive Price Adjustment by Individual Firms: A Preliminary Paper,"I. &on. Th. 2 (2), (1970), pp. 195-206.F. M. Fisher, "Stability and Competitive Equilibrium in Two Models of Search and IndividualPrice Adjustment,' J. Econ. Th. 6, (1973), Pp. 446-470.M. Rothschild, "A Two-Armed Bandit Theory of Market Pricing," .1. Econ. Th. 9, (1974), pp.185-202.M. Rothschild, "Models of Market Organization with Imperfect Information: A Survey," 3. Pot.Econ. 81, (1973), pp. 1283-1308.S. J. Turnovsky and E. R. Weintraub, Intl. Econ. Rev. 12. (1), (1971), pp. 7 1-86.H. Robbins and S. Munro, "A Stochastic Approximation Method," Ann. Math. Statist. 22,(1951), pp. 400-407.S. G. Winter, "Satisficing, Selection, and the Innovating Remnant," Quart. J. Econ., (1972), pp.237-26 1.M. J. Sobel, "Non-Cooperative Stochastic Games," Ann. Math. Statist. 42, (1971), pp. 1930-1935.P. Varaiya, "N-Person Stochastic Differential Games," in J. D. Grote (ed.), The Theory andApplication of Differential Games, D. Reidel Publishing Co., (1975), pp. 97-108.L. Ljung, Convergence of Recursive Stochastic Algorithms, Lund, Sweden: Lund Institute ofTechnology, Division of Automatic Control Report 7403, (1974), 134 pp.M. Aoki, "On Some Price Adjustment Schemes," Annals of Economic and SocialMeasurement3 (1), (1974), pp.95-115.M. Aoki, "A Model of Quantity Adjustment by a Firm: A Kiefer-Wolfowitz Like StochasticApproximation Algorithm as a Bounded Rationality Algorithm," presented at the NorthAmerican Meeting of the Econometric Society. San Francisco, (Dec. 1974).J B. Rosen, "Existence and Uniqueness of Equilibrium Points for Concave N-Person Games,"Iconometnca 33 (3), (1965), pp. 520-534.3. R. Green and M. Majumdar, "The Nature and Existence of Stochastic Equilibria," Unpub-lished paper, (1972).J. D. Grote, "Solution Sets for N-Person Games," in J. D. Grote (ed.), opcit, pp.63-76.S. G. Winter, "Satisficing, Selection and the Innovating Remnant," Quart. J. Economics 85,(1971), pp. 237-261.

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