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Market allocation through search: Equilibrium adjustment and price dispersion
JOURNAL
OF ECONOMIC
THEORY
11,
247-262 (1975)
Market Allocation Equilibrium Adjustment YANNIS
through Search: and Price Dispersion*
M. IOANNIDES
Department of Economics, Brawn University, Providence, Rhode Island 02912 Received October 21, 1974; revised April 7, 1975
This essay discusses market allocation under uncertainty in a market for a homogeneous good. Market participants contact one another at random times to buy or sell single units of the good. Transactions are carried out at different prices simultaneously. Optimal search rules are employed to describe individuals’ behavior. Such models provide the framework of a market structure within which equilibrium adjustment processes are analyzed. These decentralized, nontatonnement processes of price and quantity adjustment are utilized to examine equilibrium. Price dispersion, which characterizes equilibrium, is essentially due to finite lifetimes of market participants.
I. INTRODUCTION This essay discusses exchange in a market for a homogeneous good under conditions of uncertainty and search. Market participants contact one another at random times to buy or sell single units of the good and transactions are carried out at different prices simultaneously. What distinguishes this work from most mainstream research in “disequilibrium” theory1 is that individuals’ behavior is integrated into a model of a market structure, for which descriptions of equilibrium adjustment processes are provided. Most disequilibrium theorists have failed to provide justification for describing individuals’ behavior by means of search models which * This essay is in part based on the author’s doctoral dissertation, and it was written while the author was an assistant professor of economics at the University of California, Riverside. Thanks go to Hayne E. Leland, Robert C. Lind, James L. Sweeney and a referee of this journal for most helpful comments and suggestions. 1 See Phelps [18] and in particular Holt [9] and Mortensen [16] therein. Also, see Diamond [3], Fisher [5J, Gronau 161, Ho&, [9] and Mortensen [17], and the survey in Rothschild [20].
247 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
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YANNIS M. IOANNIDES
are based on stationarity of price distributions2 This is attained here by modelling certain decentralized, nontbtonnement, processes of equilibrium price and quantity adjustment by means of which an appropriate equilibrium concept is investigated. It is shown how, at equilibrium, existence of consumer searchers and of inventories of unsold goods can be associated with price dispersion. Such an equilibrium analysis is accomplished, however, by retaining an awkward drawback of much previous work, namely, the assumption that decision makers know the true probability distributions that characterize the market in the present and in the future. This assumption of rational expectations3 is a very unsatisfactory one in economics. But, as Telser points out,4 the problem of deriving and characterizing optimal search rules from unknown distributions is, general, very hard. Here is an outline of the essay. In Section II, behavioral rules, which market participants are supposed to follow, are described and analyzed. Their integration into a model of market structure readily follows in Section III. The latter contains a derivation of differential equations which describe the dynamics of adjustment, a discussion of existence and of properties of equilibrium. A summary of the main results is provided in Section IV.
II. BEHAVIORAL
RULES AND MARKET
STRUCTURE
The market we are concerned with here is characterized by twofold uncertainty: sellers do not know when and at what prices they can sell and buyers do not know when and at what prices they can buy. We assume that the allocation process is carried out in two sequential stages. In a 2 Stigler, in [22], first modelled search as sampling from a probability distribution by means of a sample of fixed size-where the sample size is determined by requiring that expected search costs be minimized. Sequential decisions rules, which were employed later by McCall and others, constitute a substantial improvement but have not been integrated into models of market structure. 3 See Lucas and Prescott [ 13, p. 41. Gastwirth in an unpublished paper, “On Probabilistic examines the sensitivity of sequential Models of Consumer Search for Information,” decision rules to imperfect knowledge of the price distribution. He shows that using the wrong distribution function may lead to a dramatic increase in search costs. 4 See Telser [24, p. 441. A particular version of the problem is discussed by Yahav, in 1251. DeGroot, in [2], provides the general theory of sequential decision rules with unknown distributions. Rothschild in [21] provides a most promising new attempt to approach the problem.
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ADJUSTMENT
249
first stage, individuals have to find one another in a decentralized manner.5 No transaction, asking or reservation prices are involved in this first stage-an important assumption which deprives prices of any informational role in the matching process. At a second stage, once a buyer and a seller are in contact, both, whether a transaction occurs and at which price are decided. Being a buyer or a seller is a state through which people pass occasionally. Purchases and sales are at the single-unit level. That is, buyers and sellers are not permanent organizational entities in a market structure. As a result, price reputation or store (brand) loyalty are not applicable.6 By considering exchange of homogeneous goods we abstract from the effects of product heterogeneity on price dispersion. The Process of Contacts We assume that prospective buyers do not know in advance at what price they can buy from a particular seller. Sellers’ prices vary randomly. From a seller’s point of view, prospective buyers come in every now and then to inquire about the price. The stream of arrivals of prospective buyers can be described by means of a generalized Poisson process. The intensity function of such a process depends in general on future market conditions, and on the information acquisition and dissemination technology. In like manner, we can model the process of contacts buyers make with sellers. From a seller’s point of view, each of the prospective buyers coming in is willing to pay up to a specific amount-to be referred to as the buyer’s reservation price-for a unit of the product now, rather than wait and search in the future. Reservation prices vary randomly among buyers in the market. They can be described by means of a probability density function,f,(p; t). We assume’ that sellers set asking prices which they do 5 Market participants’ purchasing of certain kinds of information to improve the effectiveness of their search may be incorporated. See Ioannides [lo] for such an extension. 6 The approach to market structure and equilibrium adjustment taken here was originally motivated for the purpose of housing market analysis (see Ioannides [lo]). It is, however, equally applicable to job markets and certain markets for durable goods, and it can be generalized further. ’ This is, of course, not the only conceivable arrangement. A seller can ask a potential buyer to make an offer, or a potential buyer can inquire about a seller’s asking price. (See loannides, op. cit., p. 16.) An intermediate case would be for a seller to suggest a decreasing sequence of prices until the potential buyer agrees to pay one of them or until he rejects a last price that the seller is willing to suggest, in which case the potential buyer leaves and another one may come in. See Meilijson [15] for a rigorous treatment of this problem.
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YANNIS M. IOANMDES
not announce. Once a contact has been made, the seller quotes his asking price, and the potential buyer accepts the good at that price, provided that it does not exceed his own reservation price at the time. Asking prices of different sellers in the market are described by means of a probability density function, f,(p; t). Formal behavioral models corresponding to this description follow below. A Behavioral Model for Sellers Let ti , t2 ,..., t, be the time points of successive arrivals of prospective buyers. With these events are associated independent, positive random variables Pwl, Pw2,. . ., that are drawn from the reservation price distribution function, f,(p). Let P, = P,(t) denote an asking price curve. Successive buyers are told the asking price, but they do not reveal their reservation prices. Since we assumed that sellers set asking prices a seller’s policy is of the following form: a product is sold to the first individual with reservation Pw,, P,” > P,(t,), and the value to the seller is recorded as P,(t,) r(t,,), where r(t) is a discount function with r(0) = 1. If a seller does not sell the value to him is zero. The following two theorems formally characterize optimal asking price policies. THEOREM 1. Let PWI, P,% ,..., be a sequential random sample from a distribution of known density function f,(p), at time points tI , t, ,..., that are generated by a generalized Poisson process with intensity function z(t). Let r(t) be a discount function and P, = P,(t) an asking price curve, an at least piecewise continuous function of time, and define the random variable T as follows: F is equal to the smallest t, for which P,,, >, P,(t,). An optimal, in the sense of maximizing Jr, = E{P,(T) r(T)}, asking price curve, P,(t), if it exists, must satisfy the integral equation:
WaW)
r(t) [P.(t) - fx(Pa(f))
Proof
1
We shall work with the more general J,,(t) = E{P,(F)r(l”)
j p > t}.
To obtain an expression for JpO(t) we work as follows. Let G(P) = smfz(p) dp.
9
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251
ADJUSTMENT
For the seller, the probability of no acceptance during (cr.,CJ+ da) conditional on F > u is given by 1 - z(u) G(P,(a)) do, from which we can obtain: Prob {F 3 7 / p > t> = exp [-ST z(u) G@‘,(u)) da]. The expected value of P,(p) v(T), conditionally on p E (7, T + &), is r(7) P,(T), and the contribution to JP,(t) of the interval (7, T + dT) is
By integrating from J&)
T
= t to
7
-+ cc we obtain:
= ia Z(T) r(T) p&d G(&(d)
exP [-
l’ z(U) w%d)
d”] d% (2)
Let us take P,(t) as a piecewise continuous function on [t, co). The optimal policy must satisfy P, > 0 for Vt E [t, co). Let Y be the vector space of piecewise continuous functions on [0, co). J:,(t) is a real functional on Y. The Gateaux differential of Jp, , if it exists, is given by
A necessary condition8 for JPo(t) to have an interior point extremum is: 6J(P, ; h) = 0,
for
Vh E Y.
(3)
For the admissible variations we require h(t) = h(co) = 0. Equations (2) and (3) then yield:
st
*zrexp
[-
~t7zGdu]/P.G(Pa)[zf,(Pa)hdu
+ G(P,)h - P&(Pa)hl
dT = 0.
(4)
By rearrangement and integration by parts we obtain
lrn 1J,,(t) ~(~)Lif’~(d>- ~(~)Ldf’dt>)ST zrP,Gexp[*lda t + Z(T)r(T)[G(Pa(T)) - &(,)~(f%(,))l~ h(T)dT= 0, for all permissible variations. If f # 0 for all p, then the above equation Q.E.D. yields (1). B See Luenberger
[14, p. 1781.
252 Differentiating
YANNIS M. IOANNIDES
both sides of (1) with respect to t yields:
& [?xgq = -,,2%& The right-hand side of (1) is an expression for the expected value of the search process at time t, J, (t), conditionally upon p > t. This necessary condition for maximizationa is directly amenable to an economic interpretation.” Let us now assume that sellers operate with finite time-horizons of search and zero salvage values. If a seller has not sold by the end of the time horizon, the value to him is zero. If we then consider an exponential discount function and constant intensity of the process of contacts, we can easily show that a unique optimal asking price curve exists. This is summarized in the following theorem whose proof is immediate.lO THEOREM 2. Under a constant intensity of the process of contacts, z(t) = h an exponential discount function, r(t) = e-At, and a finite time horizon of search, S, the differential equation (5) with the terminal condition,
has a unique solution, the optimal asking price curve. Solving (5) with (6) yields the following results: The asking price curve is a continuous, differentiable and decreasing function of time; the terminal asking price is independent of the discount rate and the intensity of contacts and depends only upon the reservation price distribution; an increase in the discount rate shifts the asking price downward, while an increase in the intensity of contacts, search horizon, dispersion and an 9 At time t a seller, if he is still in the market, has two alternatives; he can continue with the asking price policy Pa(t) expecting to get Jp (t) or change his present policy by dP, , and hence change the probability that a prosp&tive buyer who is interested in his product will accept his asking price. Since PoG(Pa) is the expected value of a transaction with a given consumer at time t, if the asking price is Pa , the left-hand side of (1) gives the change in the expected value of the transaction for a unit change in the asking price, discounted to t = 0. For an optimal policy a change at the margin should make the individual indifferent between the two alternatives-which is precisely what (1) expresses. So we see that (1) is the familiar marginal rule applied in a dynamic search context. I0 See Ioannides, op. cit., p. 56 for a proof. The solution is facilitated by a change of variable, y = p - (G/f,). Both the general and truncated case of this problem have not previously been addressed in the literature.
EQUILIBRIUM
ADJUSTMENT
upward shift of the reservation price distribution shift of the asking price curve.
253 all cause an upward
A Behavioral Model for Buyers Let t, , t2 ,..., t, ,..., be the points in time when a prospective buyer makes successive contacts with sellers. When contacted, a seller quotes his asking price. From a buyer’s viewpoint, this can be seen as sampling from the probability distribution function of asking prices. That is, with the events t, , t, ,..., t, ,..., are associated independent positive random variables, Pal,..., P al,.... drawn from the distribution function of asking prices,f,(p). The pomt in time when an individual buys a unit of the good, T, and the price he pays, p, are random variables. The pecuniary value to the individual, net of the purchasing price, at the time of purchase is measured by the net economic surplus,11 W, = W - p. W is the value of the product to the buyer, and is determined from preferences, physical characteristics of the product, and prices of other goods and services, and is assumed to be independent of p. In order to account for psychic costs of search,12 however, we assume a utility index over W, and F, U(T, W - P”) = p(F) U,(W - P”), with p a decreasing function of time for which p(0) = 1, and U, a cardinal utility index of the von Neumann-Morgenstern type. We further assume that U, is monotonically increasing and twice differentiable. We assume that buyers conduct their search by means of policies that are optimal in the sense of maximizing the expected value from the process. It turns out13 that such a policy is in the form of a reservation price curve, Pw = P&t). That is, the “bid” of the first seller with an asking price Pa, not exceeding Pw(t,) is accepted and the utility to the buyer is recorded as p(tn) U,,(W - Pan). If no acceptable housing unit is found within T units of time-the prospective buyer’s time horizon of search-the utility to the buyer is zero. Given an intensity function for the process of contacts buyers make with sellers, Z(t), an asking price distribution, f,(p)-which is assumed to be known-and a utility function p(t) UO(W - p), an optimal reservation I1 See Sweeney[23] for a rigorous formulation in the case of housing markets. I2 Sincethe sameperson can both be a buyer and a sellerat different times, it may be argued that buyersand sellersare treated asymmetricallyin this essay.This is intended as a generalizationrather than a particular case.Most previous work has assumedriskneutrality. Diamond in [3] also employs a utility index over price and duration of search,but his analysisis restrictedto an essentiallydeterministicdescription and fails to rigorously deliver reservation price curves. I3 For a proof that this is the form of the optimal policy seeKarlin [l 11.Our presentation draws on Elfving [4], from where Theorem 3 is taken, after some modifications.
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YANNIS M. IOANNIDES
price curve can be obtained easily. It is, however, more convenient to solve for a critical utility curve, v(t), from which the reservation price can be easily obtained. The probability density function for utilities, fU , can be obtained from f, and U,( W - P). Let us then denote G(v) = s,” fU du, H(y) = J,” & du. The following theorem ensures the existence of and characterizes the optimal reservation price curve. THEOREM 3. Under the above conditions, an optimal critical utility curve exists, is unique, and is given by the solution to the differential equation
g Mt)Y(t)l = -P(t) 4tmY4(tN - c(4(1))4’(t)I,
(7)
and the terminal condition y(T) = 0. The corresponding optimal reservation price curve is uniquely obtained from P&t) = W - U;l( y(t)), where U;l is the inverse function of U, .
Certain properties of the reservation price curve can be obtained easily by solving (7). In the case of an exponential search cost function, p(t) = e-Bt, and constant intensity of the process of contacts, Z(t) = L, P&t) is continuous, differentiable and increasing. Furthermore, the following are true: an increase in the search cost factor increases the reservation price while an increase in the intensity of contacts decreases it; a longer time-horizon of search and a downward shift of the asking price distribution all cause a downward shift of the optimal reservation price curve. Furthermore, for risk-preferring and risk-neutral individuals, the reservation price curve increases at an increasing rate. Finally, for risk-neutral individuals, the optimal reservation price is higher the lower the dispersion of the asking price distribution, among distributions with the same mean.
III. EQUILIBRIUM
ADJUSTMENT
As noted by Radner,14 Hicks, in “Value and Capital,” distinguished two notions of equilibrium: temporary equilibrium, in which at a given date supply equals demand, and equilibrium over time, which he defined by “the condition that prices realized (on each date) are the same as those which were previously expected to rule at that date.“15 The latter notion, simply put as “desired equals actual” is the cornerstone of much of microeconomic as well as macroeconomic equilibrium analysis. The notion of 14Radner
[19,
p. 11.
*6Hicks [7, p. 1321.
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255
equilibrium we employ in this essay essentially derives from Hicks’ concept of equilibrium over time. Under conditions of exponential discount factors, finite time-horizons of search, and constant intensity of the process of contacts, asking prices decrease and reservation prices increase over the time sellers and buyers, respectively, stay in the market. Prospective buyers and sellers, in the market at any time, have entered randomly at different times in the past. Hence, at any point in time, asking and reservation prices vary randomly, among prospective sellers, and buyers, respectively. In this section, we integrate the behavioral models into a theory of market equilibrium adjustment. We show below that such an integration of models, which are derived on the basis of some stationarity assumptions, permits a characterization of certain states of the market as equilibria-where expectations are fulfilled and stationary distributions persist. At any point in time the state of the market is characterized by the numbers of market participants and probability distribution functions for reservation and asking prices. A major analytical problem in describing market interaction is that if contacts and transactions take place at random times, then for any distribution of say, asking prices, the distribution of reservation prices-which results from equilibrium behavior of buyers-is given by a stochastic process. Here in order to simplify matters, we consider the expected value of proportions of buyers (sellers) with given reservation (asking) prices. Let x(p; t) be such a distribution of buyers by reservation prices and ~(p; t) of sellers by asking prices. (Probability distribution functions can easily be obtained by normalization.) Such distribution functions do not denote absolute numbers of individuals but should be construed relatively to the number of people currently not in the marketthe latter assumed to be large. Since the intensities of the process of contacts for buyers and sellers were assumed independently, it is necessary for equilibrium analysis to examine how they are interrelated. A basic requirement for consistency at equilibrium is that the rate at which buyers find and buy goods be equal to the rate, as seen from the supply side, at which goods are bought. Contacts are made at random times without prior information on reservation or asking prices. Only the size of the market and the technology of information acquisition and dissemination matter. Hence, it can be easily shown that this consistency condition reduces to
In the extreme case when market participants 64211 r/z-7
are assumed to have
256
YANNIS
M.
IOANNIDES
perfect information about current market conditions and perfect foresight about future ones, then r(t) and z(t) are not endogenously determined but must satisfy (8). If, however, I is endogenously determined-initiative in search is attributed to buyers-then (8) considered simultaneously with other market equilibrium conditions, to be obtained below, functions as a condition for informational equilibrium. Dynamics of Adjustment and Equilibrium
For any distribution of asking prices the reservation price distribution is given by a stochastic process-since contacts and transactions are random. If such a stochastic process has a stationary distribution, then the latter can serve as equilibrium distribution, in that the process is found in probabilistic equilibrium. As noted earlier, to simplify matters instead of examining the dynamics of a continuous time stochastic process we shall merely consider the expected values of proportions of individuals having given characteristic prices, x(p; t) and y(p; t). Let S be the rate of arrivals of new buyers, as a proportion of the reference population. New buyers are assumed to enter the market at the bottom of the price range [p’, p”] and to stay in the market for T units of time. As the number of agents in the market is assumed to be large, many prospective buyers are simulataneously contacting sellers with different asking prices. We can thus describe the dynamic adjustment of x andy as follows: Within (pO ,po + dp) and during (to, to + dt)-and relative to the reference population-the expected number of buyers who buy goods and leave the market is equal to L dt g(pJ x(po ; to) dp. Furthermore, of all buyers entering the market at time 0, a proportion q(t) will still be in the market at time t, not having made a successful contact. Clearly, from Section II,
q(t) = exp[ - JotMpdu)) do], where g(p) is the cumulative function of f0 . Hence,
xth ; to>4 dt = -Mpo> x(po ; to) dp dt + q(P-,l(po)) x(to - f’$(p,)) - O’-,l(po
dt
+ dp)) 36 + dt - Pi%p, + dp)) dt.
By assuming the entry rate as independent of time the above yields: xt=
-Lgx+LgFexp[-J
w
“(‘) 0
Lg(P,(u))
du].
(9)
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ADJUSTMENT
The right-hand side of (9) is a function ofp alone. Let X(t) = Ji: x(p; t) dp. In the remainder of this essay we shall refer to 1
.L(Pi t) = ~T/(t) x(p; t)
as the reservation price distribution. Equation (9) describes the dynamics of adjustment on the demand side when the supply side is held at stationary conditions. If a steady-state solution exists. it will characterize the equilibrium reservation price distribution. Or,
which is the explicit solution. This is presented concisely in the following proposition. PROPOSITION 1. For any stationary distribution of sellers by asking prices and under the asumptions of identical preferences and exogenous entry rate of new buyers the adjustment process described by (9) yields: (i) a unique equilibrium (stationary) distribution of buyers by reservation prices, x0, as in (11); (ii) a unique equilibrium reservation price distribution, f,“(p),from WI andUO
In like manner we can analyze the supply side assuming a stationary distribution of buyers by reservation prices, identical preferences, and that new sellers enter the market with initial asking prices equal to p”, the upper bound of the price interval and stay in the market for S unit of time. If fi is the (constant) entry rate for new sellers a proposition analogous to previous one can be stated without proof. PROPOSITION 2. For any stationary distribution of buyers by reservation prices and an exogenous entry rate of new sellers ~(p; t) satisfies the differential equation
y’t = --hGy - hG $
exp [ - .I”‘iPi hG(P,(u)) do]. a
Equation (12) yields a unique stationary distribution prices ,u”(p) = - 2
ex p [ - ,,“““’
(12)
of sellers by asking
XG(P,(o)) du],
258
YANNIS
M.
IOANNIDES
and a unique stationary asking price distribution (14) where p” 9, s Y(P) dp-
ug = Overall Market Equilibrium
By utilizing the two adjustment models we now show that an overall equilibrium exists. Cruciall” to the approach is the following: Participants are assumed to act on expectations of steady-state equilibrium and thus stationary market conditions. At a steady-state equilibrium when expectations are fulfilled and the distributions of buyers and sellers by reservation and asking prices, respectively, and the intensities of the matching process remain stationary, the derivation of the adjustment equations (9) and (12) is still valid. Hence, the question of existence of an overall equilibrium becomes that of whether there exists a pair of distribution functions f,*(p) and f,,*(p) which satisfy simultaneously the set of equations X y’*
*-
:r ‘-,‘(‘) Lg(P,(o)) - F W exp 1- s0 t;
= -zexp
[
-
s
do],
P’1(e) hG(P,(a)) dc],
(154
(15b)
0
” L j” x* dp = h I’“?/* P’ P’
dp.
(154
It is assumed that buyers have initiative in search, that is, to find and contact sellers, with sellers being passive searchers. Hence, L is a function of x and y and (15~) determines X for equilibrium. Also, it is assumed that I6 Ideally, one would want to examine an overall market adjustment, i.e., by solving simultaneously (8), (9), and (12). Let us recall, however, that the adjustment equations have been obtained assuming constant intensity of the process of contacts and stationary price distribution. Therefore, (9) is valid, if the total number of sellers in the market and the distribution of asking prices are stationary. However, for & = 0 to be consistent with a nontrivial distribution we must also require y, = 0. Since a symmetrical argument can be made for (12), the above adjustment equations cannot be construed as describing an overall equilibrium adjustment. That is (9), (12), (8) are valid only in the context of the decomposition procedure of holding one side of the market stationary and examining the other.
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L is bounded upwards and downwards. That is, it cannot be indefinitely improved upon by the technology of information acquisition and dissemination. The following theorem guarantees the existence of stationary equilibrium solutions, x* and 9 *, but not their uniqueness. The proof makes use of the Schauder fixed point theorem.17 THEOREM 4. Under the previously stated assumptions there exist a solution pair (x*, y*) to (15), i.e., a pair of equilibrium distributions of reservation and asking prices, and of numbers of buyers and sellers.
Proof. Let Z stand for the normed vector space of continuous functions on [p’, p”], C[p’, p”] and P for the positive cone of Z. For any (x0, y”) E P x P, let (p, V) be its image, in P x P, under (15). Clearly, ,u and v are bounded, since X and 13are finite and P, and P, are bounded upwards and downwards. Let us define the sets Bx and BV in Z as follows: B” = {I”: TVE p(P)),
B” = {v; v E v(P)}:
and consider the maximaP* bounded sets Bx, P, contained in B”, B”, respectively. Then the completely continuous mapping (p, V) of Z x Z into itself has a fixed point in Bz x B”. Q.E.D. The theorem just proven shows that nondegenerate price dispersion, associated with equilibrium, can persist. We have not, however, explained how such an equilibrium may be attained. Hence, although we have not developed a model of overall adjustment, we were able to examine the existence of an overall equilibrium as follows: at stationary equilibrium, when expectations are fulfilled and market variables remain constant, the equilibrium relationships implied by the separate adjustment models, considered simultaneously, are valid. We can identify some basic reasons why price dispersion persists in our model. First, people always flow into the market-entry rates of new participants do not vanish. Second, it takes market participants time to contact one another. Third, investment in information about market and about means of contacting market participants on the other side of the I7 See Bonsall [l, pp. 22-251. I* This is possible because the mapping defined by functions with uniformly bounded derivatives and the An alternative proof would be to imbed the problem the space of (absolutely) Lebesgue integrable functions and (12) are understood to hold almost everywhere.
(15) yields uniformly bounded Arzela-Ascoli theorem applies. in the vector space L&J’, p”] on [p’, $1. Then (5), (7), (9),
260
YANNIS M. IOANNIDES
market becomes obsolete.ls Fourth, and most important, market participants stay in the market for finite periods of time. We can formulate20 the general comparative statics problem for the overall equilibrium examined above. To obtain comparative statics results is, however, extremely complicated. An important byproduct of a general comparative statics formulation would be a precise definition of the decomposition procedure employed earlier. That is, the procedure of examining changes on one side of the market by assuming that the other is held at steady-state equilibrium. In view of Samuelson’s Correspondence Principle, the validity of comparative statics results, however, is doubtful when the underlying equilibrium is not stable.21
IV. SUMMARY AND CONCLUSIONS This essay encompasses a theoretical study of the structure of an exchange economy which is characterized by what T. Koopmans refers to as “secondary uncertainty.“22 The state of the market is described by probability distribution functions of sellers’ asking prices and buyers’ reservation prices and the number of prospective buyers and sellers in the market. Buyers’ (sellers’) search is modelled as sampling at random times from the distribution of asking (reservation) prices. The process of contacts is described by a generalized Poisson whose intensity is a function of market conditions. Market participants search expecting stationary market conditions. Equilibrium adjustment of the distributions of asking and reservation prices is described by modelling the market participants’ I9 Our discussion of equilibrium is reminiscent of G. Stigler’s theory of endogenous factors that tend to preserve price dispersion. See Stigler [22, p. 2191. Also Hirschleifer notes: “Mobility of buyers and sellers in and out of the market (for example, in an intergenerational life cycle model) will provide a continuing demand for informational processes of search and advertising” (Hirshleifer [S, p. 371). 20See Ioannides [lo] for such a complete formulation and for some applications. 21Stability in the case of informational equilibrium is crucial. Since learning from experience is involved, for agents’ experience to have meaning probability distributions must remain stable. On the other hand, stability may be related to a stochastic process’ convergence to a stationary distribution. Having dealt only with expectations and lacking an overall equilibrium adjustment model we cannot pursue an examination of stability. It is nevertheless important for the meaningfulness of the decomposition procedure that the processes described by (9) and (12) be stable. Such partial stability can be shown (see Ioannides [lo]). 22See Koopmans [12, p. 1631. Koopmans argues that “... the secondary uncertainty arising from lack of communication . . . is quantitatively at least as important as the primary uncertainty arising from random acts of nature and unpredictable changes in consumers’ preferences.”
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261
“flowing through” the price range characterizing the market. Some make contacts and others do not, and some of the contacts made do not lead to transactions. Buyers and sellers set finite horizons for their search. If, at the end of the time-horizon, a seller has not sold or a buyer has not bought, they leave the market. We offer a description of the dynamics of adjustment by means of ordinary differential equations for the expectations of the underlying stochastic processes. We show that if one side of the market is held at steady-state equilibrium the adjustment process on the other side is characterized by a unique equilibrium price distribution and number of market participants. The two separate models describing the short-run dynamics of adjustment are considered simultaneously for the purpose of defining an overall equilibrium for the market. Such an equilibrium, defined as a configuration of stationary probability distribution functions of reservation and asking prices and stationary numbers of market participants, is shown to exist. It is characterized by price dispersion and unsatisfied supply and demand. That is, the market does not “clear” and no unique equilibrium price prevails. Furthermore, the adjustment process is a nont5tonnement process. The general time-dependence of behavioral models for market participants should, in future work, be explored further. Also, endogenous entry rates and the relationship between active and nonactive market participants should be considered. The structure of the matching process can be further explored in two important directions. One is the informational roles of prices. The other is the technology of information acquisition and dissemination. Among other advantages, this would allow for examination of information pooling and of the role of informational intermediaries. The formulation presented in this essay bears the advantage of having behavioral foundations and thus lends itself to incorporation of adaptive behavior and refinement of the concept of informational equilibrium. These could be accomplished best in conjunction with a study of the underlying stochastic processes.
REFERENCES 1. F. F. BONSALL, “Lectures on Some Fixed Point Theorems of Functional Analysis,” Tata Institute of Fundamental Research, Bombay, 1962. 2. M. H. DEGROOT, “Optimal Statistical Decisions,” McGraw-Hill, New York, 1970. 3. P. A. DIAMOND, A model of price adjustment, J. Econ. Theory 3 (1971), 156-168. 4. G. ELFVING, A persistency problem connected with a point process, J. Appl. Prob. 4 (1967), 77-89.
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5. F. M. FISHER, Quasi-competitive price adjustment by individual firms: a preliminary paper, J. Econ. Theory 2 (1970). 6. R. GRONAU, Information and frictional unemployment, Amer. Econ. Reu. 61 (1971), 290-301. 7. J. R. HICKS, ‘Value and Capital,” Oxford University Press, London, 1946. 8. J. HIRSHLEIFER, Where are we in the theory of information? Amer. Econ. Rev., Papers and Proceedings 63 (1973), 31-39. 9. C. C. HOLT, Job search Phillips’ wage relation and union influence: theory and evidence, in (E. S. Phelps, et al., Eds.), “Microeconomic Foundations of Employment and Inflation Theory,” Norton, New York, 1970. 10. Y. M. IOANNIDES, Equilibrium adjustment processes: the housing market case, Ph.D. Dissertation, Stanford University, 1973. 11. S. KARLIN, Stochastic models and optimal policy for selling an asset, in (K. J. Arrow et al. Eds.), “Studies in Applied Probability and Management Science,” Stanford University Press, Stanford, 1960. 12. T. C. KOOPMANS, “Three Essays on the State of Economic Science,” McGraw-Hill, New York, 1957. 13. R. E. LUCAS, JR., AND E. C. PRESCOTT, Equilibrium search and unemployment, J. Econ. Theory 7 (1974), 188-209. 14. D. G. LUENBERGER, “Optimization with Vector Space Methods,” Wiley, New York, 1969. 15. I. MEILIJSON, A bargaining problem, Ph.D. Dissertation, University of California, Berkeley, 1969. 16. D. T. MORTENSEN, A theory of wage and employment dynamics, in (E. S. Phelps, et al. Eds.), “Microeconomic Foundations of Employment and Inflation Theory,” Norton, New York, 1970. 17. D. T. MORTENSEN, Job search, the duration of unemployment, and the Phillips curve, Amer. Econ. Rev. 60 (1970), 847-862. 18. E. S. PHELPS et al., “Microeconomic Foundations of Employment and Inflation Theory,” Norton, New York, 1970. 19. R. RADNER, Market equilibrium and uncertainty: concepts and problems, Technical Report No. 16, Center for Research in Management Science, University of California, Berkeley, 1973. 20. M. ROTHSCHILD, Models of market organization under imperfect information: a survey, J. Political Econ. 81 (1973), 1283-1308. 21. M. ROTHSCHILD, Searching for the lowest price when the distribution of prices is unknown, J. Political Econ. 82 (1974), 689-711. 22. G. J. STIGLER, The economics of information, J. Political Econ. 69 (1961), 213-225. 23. J. L. SWEENEY, Housing unit maintenance and the mode of tenure, J. Econ. Theory 8 (1974), 111-137. 24. L. G. TELSER, Searching for the lowest price, Amer. Econ. Rev., Papers and Proceedings 63 (1973), 41-49. 25. J. YAHAV, On optimal stopping, Ann. Math. Statist. 37 (1966), 30-35.
OF ECONOMIC
THEORY
11,
247-262 (1975)
Market Allocation Equilibrium Adjustment YANNIS
through Search: and Price Dispersion*
M. IOANNIDES
Department of Economics, Brawn University, Providence, Rhode Island 02912 Received October 21, 1974; revised April 7, 1975
This essay discusses market allocation under uncertainty in a market for a homogeneous good. Market participants contact one another at random times to buy or sell single units of the good. Transactions are carried out at different prices simultaneously. Optimal search rules are employed to describe individuals’ behavior. Such models provide the framework of a market structure within which equilibrium adjustment processes are analyzed. These decentralized, nontatonnement processes of price and quantity adjustment are utilized to examine equilibrium. Price dispersion, which characterizes equilibrium, is essentially due to finite lifetimes of market participants.
I. INTRODUCTION This essay discusses exchange in a market for a homogeneous good under conditions of uncertainty and search. Market participants contact one another at random times to buy or sell single units of the good and transactions are carried out at different prices simultaneously. What distinguishes this work from most mainstream research in “disequilibrium” theory1 is that individuals’ behavior is integrated into a model of a market structure, for which descriptions of equilibrium adjustment processes are provided. Most disequilibrium theorists have failed to provide justification for describing individuals’ behavior by means of search models which * This essay is in part based on the author’s doctoral dissertation, and it was written while the author was an assistant professor of economics at the University of California, Riverside. Thanks go to Hayne E. Leland, Robert C. Lind, James L. Sweeney and a referee of this journal for most helpful comments and suggestions. 1 See Phelps [18] and in particular Holt [9] and Mortensen [16] therein. Also, see Diamond [3], Fisher [5J, Gronau 161, Ho&, [9] and Mortensen [17], and the survey in Rothschild [20].
247 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
248
YANNIS M. IOANNIDES
are based on stationarity of price distributions2 This is attained here by modelling certain decentralized, nontbtonnement, processes of equilibrium price and quantity adjustment by means of which an appropriate equilibrium concept is investigated. It is shown how, at equilibrium, existence of consumer searchers and of inventories of unsold goods can be associated with price dispersion. Such an equilibrium analysis is accomplished, however, by retaining an awkward drawback of much previous work, namely, the assumption that decision makers know the true probability distributions that characterize the market in the present and in the future. This assumption of rational expectations3 is a very unsatisfactory one in economics. But, as Telser points out,4 the problem of deriving and characterizing optimal search rules from unknown distributions is, general, very hard. Here is an outline of the essay. In Section II, behavioral rules, which market participants are supposed to follow, are described and analyzed. Their integration into a model of market structure readily follows in Section III. The latter contains a derivation of differential equations which describe the dynamics of adjustment, a discussion of existence and of properties of equilibrium. A summary of the main results is provided in Section IV.
II. BEHAVIORAL
RULES AND MARKET
STRUCTURE
The market we are concerned with here is characterized by twofold uncertainty: sellers do not know when and at what prices they can sell and buyers do not know when and at what prices they can buy. We assume that the allocation process is carried out in two sequential stages. In a 2 Stigler, in [22], first modelled search as sampling from a probability distribution by means of a sample of fixed size-where the sample size is determined by requiring that expected search costs be minimized. Sequential decisions rules, which were employed later by McCall and others, constitute a substantial improvement but have not been integrated into models of market structure. 3 See Lucas and Prescott [ 13, p. 41. Gastwirth in an unpublished paper, “On Probabilistic examines the sensitivity of sequential Models of Consumer Search for Information,” decision rules to imperfect knowledge of the price distribution. He shows that using the wrong distribution function may lead to a dramatic increase in search costs. 4 See Telser [24, p. 441. A particular version of the problem is discussed by Yahav, in 1251. DeGroot, in [2], provides the general theory of sequential decision rules with unknown distributions. Rothschild in [21] provides a most promising new attempt to approach the problem.
EQUILIBRIUM
ADJUSTMENT
249
first stage, individuals have to find one another in a decentralized manner.5 No transaction, asking or reservation prices are involved in this first stage-an important assumption which deprives prices of any informational role in the matching process. At a second stage, once a buyer and a seller are in contact, both, whether a transaction occurs and at which price are decided. Being a buyer or a seller is a state through which people pass occasionally. Purchases and sales are at the single-unit level. That is, buyers and sellers are not permanent organizational entities in a market structure. As a result, price reputation or store (brand) loyalty are not applicable.6 By considering exchange of homogeneous goods we abstract from the effects of product heterogeneity on price dispersion. The Process of Contacts We assume that prospective buyers do not know in advance at what price they can buy from a particular seller. Sellers’ prices vary randomly. From a seller’s point of view, prospective buyers come in every now and then to inquire about the price. The stream of arrivals of prospective buyers can be described by means of a generalized Poisson process. The intensity function of such a process depends in general on future market conditions, and on the information acquisition and dissemination technology. In like manner, we can model the process of contacts buyers make with sellers. From a seller’s point of view, each of the prospective buyers coming in is willing to pay up to a specific amount-to be referred to as the buyer’s reservation price-for a unit of the product now, rather than wait and search in the future. Reservation prices vary randomly among buyers in the market. They can be described by means of a probability density function,f,(p; t). We assume’ that sellers set asking prices which they do 5 Market participants’ purchasing of certain kinds of information to improve the effectiveness of their search may be incorporated. See Ioannides [lo] for such an extension. 6 The approach to market structure and equilibrium adjustment taken here was originally motivated for the purpose of housing market analysis (see Ioannides [lo]). It is, however, equally applicable to job markets and certain markets for durable goods, and it can be generalized further. ’ This is, of course, not the only conceivable arrangement. A seller can ask a potential buyer to make an offer, or a potential buyer can inquire about a seller’s asking price. (See loannides, op. cit., p. 16.) An intermediate case would be for a seller to suggest a decreasing sequence of prices until the potential buyer agrees to pay one of them or until he rejects a last price that the seller is willing to suggest, in which case the potential buyer leaves and another one may come in. See Meilijson [15] for a rigorous treatment of this problem.
250
YANNIS M. IOANMDES
not announce. Once a contact has been made, the seller quotes his asking price, and the potential buyer accepts the good at that price, provided that it does not exceed his own reservation price at the time. Asking prices of different sellers in the market are described by means of a probability density function, f,(p; t). Formal behavioral models corresponding to this description follow below. A Behavioral Model for Sellers Let ti , t2 ,..., t, be the time points of successive arrivals of prospective buyers. With these events are associated independent, positive random variables Pwl, Pw2,. . ., that are drawn from the reservation price distribution function, f,(p). Let P, = P,(t) denote an asking price curve. Successive buyers are told the asking price, but they do not reveal their reservation prices. Since we assumed that sellers set asking prices a seller’s policy is of the following form: a product is sold to the first individual with reservation Pw,, P,” > P,(t,), and the value to the seller is recorded as P,(t,) r(t,,), where r(t) is a discount function with r(0) = 1. If a seller does not sell the value to him is zero. The following two theorems formally characterize optimal asking price policies. THEOREM 1. Let PWI, P,% ,..., be a sequential random sample from a distribution of known density function f,(p), at time points tI , t, ,..., that are generated by a generalized Poisson process with intensity function z(t). Let r(t) be a discount function and P, = P,(t) an asking price curve, an at least piecewise continuous function of time, and define the random variable T as follows: F is equal to the smallest t, for which P,,, >, P,(t,). An optimal, in the sense of maximizing Jr, = E{P,(T) r(T)}, asking price curve, P,(t), if it exists, must satisfy the integral equation:
WaW)
r(t) [P.(t) - fx(Pa(f))
Proof
1
We shall work with the more general J,,(t) = E{P,(F)r(l”)
j p > t}.
To obtain an expression for JpO(t) we work as follows. Let G(P) = smfz(p) dp.
9
EQUILIBRIUM
251
ADJUSTMENT
For the seller, the probability of no acceptance during (cr.,CJ+ da) conditional on F > u is given by 1 - z(u) G(P,(a)) do, from which we can obtain: Prob {F 3 7 / p > t> = exp [-ST z(u) G@‘,(u)) da]. The expected value of P,(p) v(T), conditionally on p E (7, T + &), is r(7) P,(T), and the contribution to JP,(t) of the interval (7, T + dT) is
By integrating from J&)
T
= t to
7
-+ cc we obtain:
= ia Z(T) r(T) p&d G(&(d)
exP [-
l’ z(U) w%d)
d”] d% (2)
Let us take P,(t) as a piecewise continuous function on [t, co). The optimal policy must satisfy P, > 0 for Vt E [t, co). Let Y be the vector space of piecewise continuous functions on [0, co). J:,(t) is a real functional on Y. The Gateaux differential of Jp, , if it exists, is given by
A necessary condition8 for JPo(t) to have an interior point extremum is: 6J(P, ; h) = 0,
for
Vh E Y.
(3)
For the admissible variations we require h(t) = h(co) = 0. Equations (2) and (3) then yield:
st
*zrexp
[-
~t7zGdu]/P.G(Pa)[zf,(Pa)hdu
+ G(P,)h - P&(Pa)hl
dT = 0.
(4)
By rearrangement and integration by parts we obtain
lrn 1J,,(t) ~(~)Lif’~(d>- ~(~)Ldf’dt>)ST zrP,Gexp[*lda t + Z(T)r(T)[G(Pa(T)) - &(,)~(f%(,))l~ h(T)dT= 0, for all permissible variations. If f # 0 for all p, then the above equation Q.E.D. yields (1). B See Luenberger
[14, p. 1781.
252 Differentiating
YANNIS M. IOANNIDES
both sides of (1) with respect to t yields:
& [?xgq = -,,2%& The right-hand side of (1) is an expression for the expected value of the search process at time t, J, (t), conditionally upon p > t. This necessary condition for maximizationa is directly amenable to an economic interpretation.” Let us now assume that sellers operate with finite time-horizons of search and zero salvage values. If a seller has not sold by the end of the time horizon, the value to him is zero. If we then consider an exponential discount function and constant intensity of the process of contacts, we can easily show that a unique optimal asking price curve exists. This is summarized in the following theorem whose proof is immediate.lO THEOREM 2. Under a constant intensity of the process of contacts, z(t) = h an exponential discount function, r(t) = e-At, and a finite time horizon of search, S, the differential equation (5) with the terminal condition,
has a unique solution, the optimal asking price curve. Solving (5) with (6) yields the following results: The asking price curve is a continuous, differentiable and decreasing function of time; the terminal asking price is independent of the discount rate and the intensity of contacts and depends only upon the reservation price distribution; an increase in the discount rate shifts the asking price downward, while an increase in the intensity of contacts, search horizon, dispersion and an 9 At time t a seller, if he is still in the market, has two alternatives; he can continue with the asking price policy Pa(t) expecting to get Jp (t) or change his present policy by dP, , and hence change the probability that a prosp&tive buyer who is interested in his product will accept his asking price. Since PoG(Pa) is the expected value of a transaction with a given consumer at time t, if the asking price is Pa , the left-hand side of (1) gives the change in the expected value of the transaction for a unit change in the asking price, discounted to t = 0. For an optimal policy a change at the margin should make the individual indifferent between the two alternatives-which is precisely what (1) expresses. So we see that (1) is the familiar marginal rule applied in a dynamic search context. I0 See Ioannides, op. cit., p. 56 for a proof. The solution is facilitated by a change of variable, y = p - (G/f,). Both the general and truncated case of this problem have not previously been addressed in the literature.
EQUILIBRIUM
ADJUSTMENT
upward shift of the reservation price distribution shift of the asking price curve.
253 all cause an upward
A Behavioral Model for Buyers Let t, , t2 ,..., t, ,..., be the points in time when a prospective buyer makes successive contacts with sellers. When contacted, a seller quotes his asking price. From a buyer’s viewpoint, this can be seen as sampling from the probability distribution function of asking prices. That is, with the events t, , t, ,..., t, ,..., are associated independent positive random variables, Pal,..., P al,.... drawn from the distribution function of asking prices,f,(p). The pomt in time when an individual buys a unit of the good, T, and the price he pays, p, are random variables. The pecuniary value to the individual, net of the purchasing price, at the time of purchase is measured by the net economic surplus,11 W, = W - p. W is the value of the product to the buyer, and is determined from preferences, physical characteristics of the product, and prices of other goods and services, and is assumed to be independent of p. In order to account for psychic costs of search,12 however, we assume a utility index over W, and F, U(T, W - P”) = p(F) U,(W - P”), with p a decreasing function of time for which p(0) = 1, and U, a cardinal utility index of the von Neumann-Morgenstern type. We further assume that U, is monotonically increasing and twice differentiable. We assume that buyers conduct their search by means of policies that are optimal in the sense of maximizing the expected value from the process. It turns out13 that such a policy is in the form of a reservation price curve, Pw = P&t). That is, the “bid” of the first seller with an asking price Pa, not exceeding Pw(t,) is accepted and the utility to the buyer is recorded as p(tn) U,,(W - Pan). If no acceptable housing unit is found within T units of time-the prospective buyer’s time horizon of search-the utility to the buyer is zero. Given an intensity function for the process of contacts buyers make with sellers, Z(t), an asking price distribution, f,(p)-which is assumed to be known-and a utility function p(t) UO(W - p), an optimal reservation I1 See Sweeney[23] for a rigorous formulation in the case of housing markets. I2 Sincethe sameperson can both be a buyer and a sellerat different times, it may be argued that buyersand sellersare treated asymmetricallyin this essay.This is intended as a generalizationrather than a particular case.Most previous work has assumedriskneutrality. Diamond in [3] also employs a utility index over price and duration of search,but his analysisis restrictedto an essentiallydeterministicdescription and fails to rigorously deliver reservation price curves. I3 For a proof that this is the form of the optimal policy seeKarlin [l 11.Our presentation draws on Elfving [4], from where Theorem 3 is taken, after some modifications.
254
YANNIS M. IOANNIDES
price curve can be obtained easily. It is, however, more convenient to solve for a critical utility curve, v(t), from which the reservation price can be easily obtained. The probability density function for utilities, fU , can be obtained from f, and U,( W - P). Let us then denote G(v) = s,” fU du, H(y) = J,” & du. The following theorem ensures the existence of and characterizes the optimal reservation price curve. THEOREM 3. Under the above conditions, an optimal critical utility curve exists, is unique, and is given by the solution to the differential equation
g Mt)Y(t)l = -P(t) 4tmY4(tN - c(4(1))4’(t)I,
(7)
and the terminal condition y(T) = 0. The corresponding optimal reservation price curve is uniquely obtained from P&t) = W - U;l( y(t)), where U;l is the inverse function of U, .
Certain properties of the reservation price curve can be obtained easily by solving (7). In the case of an exponential search cost function, p(t) = e-Bt, and constant intensity of the process of contacts, Z(t) = L, P&t) is continuous, differentiable and increasing. Furthermore, the following are true: an increase in the search cost factor increases the reservation price while an increase in the intensity of contacts decreases it; a longer time-horizon of search and a downward shift of the asking price distribution all cause a downward shift of the optimal reservation price curve. Furthermore, for risk-preferring and risk-neutral individuals, the reservation price curve increases at an increasing rate. Finally, for risk-neutral individuals, the optimal reservation price is higher the lower the dispersion of the asking price distribution, among distributions with the same mean.
III. EQUILIBRIUM
ADJUSTMENT
As noted by Radner,14 Hicks, in “Value and Capital,” distinguished two notions of equilibrium: temporary equilibrium, in which at a given date supply equals demand, and equilibrium over time, which he defined by “the condition that prices realized (on each date) are the same as those which were previously expected to rule at that date.“15 The latter notion, simply put as “desired equals actual” is the cornerstone of much of microeconomic as well as macroeconomic equilibrium analysis. The notion of 14Radner
[19,
p. 11.
*6Hicks [7, p. 1321.
EQUILIBRlUM
ADJUSTMENT
255
equilibrium we employ in this essay essentially derives from Hicks’ concept of equilibrium over time. Under conditions of exponential discount factors, finite time-horizons of search, and constant intensity of the process of contacts, asking prices decrease and reservation prices increase over the time sellers and buyers, respectively, stay in the market. Prospective buyers and sellers, in the market at any time, have entered randomly at different times in the past. Hence, at any point in time, asking and reservation prices vary randomly, among prospective sellers, and buyers, respectively. In this section, we integrate the behavioral models into a theory of market equilibrium adjustment. We show below that such an integration of models, which are derived on the basis of some stationarity assumptions, permits a characterization of certain states of the market as equilibria-where expectations are fulfilled and stationary distributions persist. At any point in time the state of the market is characterized by the numbers of market participants and probability distribution functions for reservation and asking prices. A major analytical problem in describing market interaction is that if contacts and transactions take place at random times, then for any distribution of say, asking prices, the distribution of reservation prices-which results from equilibrium behavior of buyers-is given by a stochastic process. Here in order to simplify matters, we consider the expected value of proportions of buyers (sellers) with given reservation (asking) prices. Let x(p; t) be such a distribution of buyers by reservation prices and ~(p; t) of sellers by asking prices. (Probability distribution functions can easily be obtained by normalization.) Such distribution functions do not denote absolute numbers of individuals but should be construed relatively to the number of people currently not in the marketthe latter assumed to be large. Since the intensities of the process of contacts for buyers and sellers were assumed independently, it is necessary for equilibrium analysis to examine how they are interrelated. A basic requirement for consistency at equilibrium is that the rate at which buyers find and buy goods be equal to the rate, as seen from the supply side, at which goods are bought. Contacts are made at random times without prior information on reservation or asking prices. Only the size of the market and the technology of information acquisition and dissemination matter. Hence, it can be easily shown that this consistency condition reduces to
In the extreme case when market participants 64211 r/z-7
are assumed to have
256
YANNIS
M.
IOANNIDES
perfect information about current market conditions and perfect foresight about future ones, then r(t) and z(t) are not endogenously determined but must satisfy (8). If, however, I is endogenously determined-initiative in search is attributed to buyers-then (8) considered simultaneously with other market equilibrium conditions, to be obtained below, functions as a condition for informational equilibrium. Dynamics of Adjustment and Equilibrium
For any distribution of asking prices the reservation price distribution is given by a stochastic process-since contacts and transactions are random. If such a stochastic process has a stationary distribution, then the latter can serve as equilibrium distribution, in that the process is found in probabilistic equilibrium. As noted earlier, to simplify matters instead of examining the dynamics of a continuous time stochastic process we shall merely consider the expected values of proportions of individuals having given characteristic prices, x(p; t) and y(p; t). Let S be the rate of arrivals of new buyers, as a proportion of the reference population. New buyers are assumed to enter the market at the bottom of the price range [p’, p”] and to stay in the market for T units of time. As the number of agents in the market is assumed to be large, many prospective buyers are simulataneously contacting sellers with different asking prices. We can thus describe the dynamic adjustment of x andy as follows: Within (pO ,po + dp) and during (to, to + dt)-and relative to the reference population-the expected number of buyers who buy goods and leave the market is equal to L dt g(pJ x(po ; to) dp. Furthermore, of all buyers entering the market at time 0, a proportion q(t) will still be in the market at time t, not having made a successful contact. Clearly, from Section II,
q(t) = exp[ - JotMpdu)) do], where g(p) is the cumulative function of f0 . Hence,
xth ; to>4 dt = -Mpo> x(po ; to) dp dt + q(P-,l(po)) x(to - f’$(p,)) - O’-,l(po
dt
+ dp)) 36 + dt - Pi%p, + dp)) dt.
By assuming the entry rate as independent of time the above yields: xt=
-Lgx+LgFexp[-J
w
“(‘) 0
Lg(P,(u))
du].
(9)
EQUILIBRIUM
257
ADJUSTMENT
The right-hand side of (9) is a function ofp alone. Let X(t) = Ji: x(p; t) dp. In the remainder of this essay we shall refer to 1
.L(Pi t) = ~T/(t) x(p; t)
as the reservation price distribution. Equation (9) describes the dynamics of adjustment on the demand side when the supply side is held at stationary conditions. If a steady-state solution exists. it will characterize the equilibrium reservation price distribution. Or,
which is the explicit solution. This is presented concisely in the following proposition. PROPOSITION 1. For any stationary distribution of sellers by asking prices and under the asumptions of identical preferences and exogenous entry rate of new buyers the adjustment process described by (9) yields: (i) a unique equilibrium (stationary) distribution of buyers by reservation prices, x0, as in (11); (ii) a unique equilibrium reservation price distribution, f,“(p),from WI andUO
In like manner we can analyze the supply side assuming a stationary distribution of buyers by reservation prices, identical preferences, and that new sellers enter the market with initial asking prices equal to p”, the upper bound of the price interval and stay in the market for S unit of time. If fi is the (constant) entry rate for new sellers a proposition analogous to previous one can be stated without proof. PROPOSITION 2. For any stationary distribution of buyers by reservation prices and an exogenous entry rate of new sellers ~(p; t) satisfies the differential equation
y’t = --hGy - hG $
exp [ - .I”‘iPi hG(P,(u)) do]. a
Equation (12) yields a unique stationary distribution prices ,u”(p) = - 2
ex p [ - ,,“““’
(12)
of sellers by asking
XG(P,(o)) du],
258
YANNIS
M.
IOANNIDES
and a unique stationary asking price distribution (14) where p” 9, s Y(P) dp-
ug = Overall Market Equilibrium
By utilizing the two adjustment models we now show that an overall equilibrium exists. Cruciall” to the approach is the following: Participants are assumed to act on expectations of steady-state equilibrium and thus stationary market conditions. At a steady-state equilibrium when expectations are fulfilled and the distributions of buyers and sellers by reservation and asking prices, respectively, and the intensities of the matching process remain stationary, the derivation of the adjustment equations (9) and (12) is still valid. Hence, the question of existence of an overall equilibrium becomes that of whether there exists a pair of distribution functions f,*(p) and f,,*(p) which satisfy simultaneously the set of equations X y’*
*-
:r ‘-,‘(‘) Lg(P,(o)) - F W exp 1- s0 t;
= -zexp
[
-
s
do],
P’1(e) hG(P,(a)) dc],
(154
(15b)
0
” L j” x* dp = h I’“?/* P’ P’
dp.
(154
It is assumed that buyers have initiative in search, that is, to find and contact sellers, with sellers being passive searchers. Hence, L is a function of x and y and (15~) determines X for equilibrium. Also, it is assumed that I6 Ideally, one would want to examine an overall market adjustment, i.e., by solving simultaneously (8), (9), and (12). Let us recall, however, that the adjustment equations have been obtained assuming constant intensity of the process of contacts and stationary price distribution. Therefore, (9) is valid, if the total number of sellers in the market and the distribution of asking prices are stationary. However, for & = 0 to be consistent with a nontrivial distribution we must also require y, = 0. Since a symmetrical argument can be made for (12), the above adjustment equations cannot be construed as describing an overall equilibrium adjustment. That is (9), (12), (8) are valid only in the context of the decomposition procedure of holding one side of the market stationary and examining the other.
EQUILIBRIUM
ADJUSTMENT
259
L is bounded upwards and downwards. That is, it cannot be indefinitely improved upon by the technology of information acquisition and dissemination. The following theorem guarantees the existence of stationary equilibrium solutions, x* and 9 *, but not their uniqueness. The proof makes use of the Schauder fixed point theorem.17 THEOREM 4. Under the previously stated assumptions there exist a solution pair (x*, y*) to (15), i.e., a pair of equilibrium distributions of reservation and asking prices, and of numbers of buyers and sellers.
Proof. Let Z stand for the normed vector space of continuous functions on [p’, p”], C[p’, p”] and P for the positive cone of Z. For any (x0, y”) E P x P, let (p, V) be its image, in P x P, under (15). Clearly, ,u and v are bounded, since X and 13are finite and P, and P, are bounded upwards and downwards. Let us define the sets Bx and BV in Z as follows: B” = {I”: TVE p(P)),
B” = {v; v E v(P)}:
and consider the maximaP* bounded sets Bx, P, contained in B”, B”, respectively. Then the completely continuous mapping (p, V) of Z x Z into itself has a fixed point in Bz x B”. Q.E.D. The theorem just proven shows that nondegenerate price dispersion, associated with equilibrium, can persist. We have not, however, explained how such an equilibrium may be attained. Hence, although we have not developed a model of overall adjustment, we were able to examine the existence of an overall equilibrium as follows: at stationary equilibrium, when expectations are fulfilled and market variables remain constant, the equilibrium relationships implied by the separate adjustment models, considered simultaneously, are valid. We can identify some basic reasons why price dispersion persists in our model. First, people always flow into the market-entry rates of new participants do not vanish. Second, it takes market participants time to contact one another. Third, investment in information about market and about means of contacting market participants on the other side of the I7 See Bonsall [l, pp. 22-251. I* This is possible because the mapping defined by functions with uniformly bounded derivatives and the An alternative proof would be to imbed the problem the space of (absolutely) Lebesgue integrable functions and (12) are understood to hold almost everywhere.
(15) yields uniformly bounded Arzela-Ascoli theorem applies. in the vector space L&J’, p”] on [p’, $1. Then (5), (7), (9),
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market becomes obsolete.ls Fourth, and most important, market participants stay in the market for finite periods of time. We can formulate20 the general comparative statics problem for the overall equilibrium examined above. To obtain comparative statics results is, however, extremely complicated. An important byproduct of a general comparative statics formulation would be a precise definition of the decomposition procedure employed earlier. That is, the procedure of examining changes on one side of the market by assuming that the other is held at steady-state equilibrium. In view of Samuelson’s Correspondence Principle, the validity of comparative statics results, however, is doubtful when the underlying equilibrium is not stable.21
IV. SUMMARY AND CONCLUSIONS This essay encompasses a theoretical study of the structure of an exchange economy which is characterized by what T. Koopmans refers to as “secondary uncertainty.“22 The state of the market is described by probability distribution functions of sellers’ asking prices and buyers’ reservation prices and the number of prospective buyers and sellers in the market. Buyers’ (sellers’) search is modelled as sampling at random times from the distribution of asking (reservation) prices. The process of contacts is described by a generalized Poisson whose intensity is a function of market conditions. Market participants search expecting stationary market conditions. Equilibrium adjustment of the distributions of asking and reservation prices is described by modelling the market participants’ I9 Our discussion of equilibrium is reminiscent of G. Stigler’s theory of endogenous factors that tend to preserve price dispersion. See Stigler [22, p. 2191. Also Hirschleifer notes: “Mobility of buyers and sellers in and out of the market (for example, in an intergenerational life cycle model) will provide a continuing demand for informational processes of search and advertising” (Hirshleifer [S, p. 371). 20See Ioannides [lo] for such a complete formulation and for some applications. 21Stability in the case of informational equilibrium is crucial. Since learning from experience is involved, for agents’ experience to have meaning probability distributions must remain stable. On the other hand, stability may be related to a stochastic process’ convergence to a stationary distribution. Having dealt only with expectations and lacking an overall equilibrium adjustment model we cannot pursue an examination of stability. It is nevertheless important for the meaningfulness of the decomposition procedure that the processes described by (9) and (12) be stable. Such partial stability can be shown (see Ioannides [lo]). 22See Koopmans [12, p. 1631. Koopmans argues that “... the secondary uncertainty arising from lack of communication . . . is quantitatively at least as important as the primary uncertainty arising from random acts of nature and unpredictable changes in consumers’ preferences.”
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“flowing through” the price range characterizing the market. Some make contacts and others do not, and some of the contacts made do not lead to transactions. Buyers and sellers set finite horizons for their search. If, at the end of the time-horizon, a seller has not sold or a buyer has not bought, they leave the market. We offer a description of the dynamics of adjustment by means of ordinary differential equations for the expectations of the underlying stochastic processes. We show that if one side of the market is held at steady-state equilibrium the adjustment process on the other side is characterized by a unique equilibrium price distribution and number of market participants. The two separate models describing the short-run dynamics of adjustment are considered simultaneously for the purpose of defining an overall equilibrium for the market. Such an equilibrium, defined as a configuration of stationary probability distribution functions of reservation and asking prices and stationary numbers of market participants, is shown to exist. It is characterized by price dispersion and unsatisfied supply and demand. That is, the market does not “clear” and no unique equilibrium price prevails. Furthermore, the adjustment process is a nont5tonnement process. The general time-dependence of behavioral models for market participants should, in future work, be explored further. Also, endogenous entry rates and the relationship between active and nonactive market participants should be considered. The structure of the matching process can be further explored in two important directions. One is the informational roles of prices. The other is the technology of information acquisition and dissemination. Among other advantages, this would allow for examination of information pooling and of the role of informational intermediaries. The formulation presented in this essay bears the advantage of having behavioral foundations and thus lends itself to incorporation of adaptive behavior and refinement of the concept of informational equilibrium. These could be accomplished best in conjunction with a study of the underlying stochastic processes.
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