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Heterogeneous expectations and the demand for money and financial assets in an exchange economy
Journal of Monetuy Economics 9 (1982)
U-Y
203-222.
North-Holland Publishing Company
Richard CASTANIAS* n,St~rle,WA 98195, Q
USA
ey as an alternative to barter in explanations fail to motivate, or this paper, information about distributed. Money is shown to It pI1 asset of wide-spread social value. Thus, this paper of Bnmner and Meltzer (1972) with the explicit
1. I. The demand for money when traders are imper$ectly informed
An e~~~tially unresolved problem in financial economics i; to explain adequately the role of money in a dynamic exchange economy.’ The principal motives generally advanced for the use of money or the 6 Gstence of a positivedemand for money are (1) lack of synchronization of -zc+eiptsand expenditures, (2) aK3ts of tran$actin~ (3) embarrassment of Jefault, (4) unccrt&ty, and (5) *he crxistsnccof productive services, such as liquidity. Of &XC~it has baen osnvincringly argued by Brunner and Me&r that the first three woti not exist without the fourth, uncertainty, or perhaps imperfect of Borne traders. In particular, Brunner and Meltzer know1 on the argue ion of information, and not the existence of nty, that induces individuals to search for, t, alternatives to barter’.* Allan Msl~r,
and an anonymous refr;reefor useful
*See Brunncr sad Mekzm (1972) far a s~m&ry of this i’:sut. 2Brunncr and Mattttr (1972, p. 786).
$02.75 @ North-Holland
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Tobin has shown that uncertainty alone can give rise to a positive demand for money on the part of risk-averse investors. His principal result is that the existence of undifferentiated uncertainty is sufficient to cause traders to hold money balances in equilibrium and, under certain circumstances, to have negatively sloped liquidity preference schedules with regard to the expected yields on risky assets.3 Rut Tobin fails to identify explicitly the productive services of money which are associated with the non-pecuniary services of motive (5) above. The Tobin analysis also fails to provide insight as to why money has remained in use in economies with sustained high levels of inflation, or accelerating inflation. Brunner and Meltzer attempt to characterize explicitly the services rendered by money and extend the theory of asset choice to include the cost of acquiring information about the market arrangements, relative prices, and exchange ratios.4 In a market where information is unevenly distributed, traders will search for sequences of transactions which minimize the costs of acquiring information and transacting. Money is shown to have ‘peculiar properties and a low marginal cost of information’ which make it an asset of wide-spread social value in constructing optimal transactions chains5 Thus, Brunner and Meltzer argue that money provides an explicit service to optimizing traders, and the use of money increases the welfare of its users, A trader may find it advantageous to allocate pa;t of his wealth to money. 1.2. Financial assets and exchanp
strategies when traders are imptvfec:ly
informed
In this paper we examine how uneusnly distributed information wiil irrductraders to adopt dynamic exchange strategies and, as W&S suggested by Brunner and Meltzer, may even induce indirect methods of exchange. Goods and assets with low marginal costs of acquiring information concerning relative prices (in terms of some or all other goods) are temporarily inventoried in quantities in excess of what will eventually be held when the trader has attained an optimal long-run portfolio. A good with such properties may be termed a medium of exchange in that it is held in excess (of long-run desired) supply only to enable the trader to make more advantageous exchanges later. We assume an exchange economy in which each trader adopts a dynamic strategy for transacting in portfolio assets and views his expected ‘cost’ of (or arbitrage gain from) purchasing an asset, given his optimal strategy, as a random variable. Thus, we can no longer simply assume that he will trade to Robin (1958). Tobin assumes that traders have quadratic utility functions and parameter values sufficient to assure that the liquidity preference schedule is downward sloping. 4Brunner and Meltzer (1972). ‘Brunner and Meltzer (1972, p. 804).
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his long-run preferreddemand level at some tatonnement generated unique price. We consider a market in which there are just two assets: a risky i’sset and a risk-k asset (money), 2ulid traders are expected to maximize $.heir qmdWic utility of Wth, TIE results are then comparable to the Liquidi:y Prt?ference model of Tob:~ and the single period capital asset pricing models of contemporary financial theory, as summarized by Jenseri.B In this market an excess dbmander of the risky asset will demand strictly less of the risky arSSeton any given transaction than in a perfectly competitive, equilibrium market, and his demand for the money asset is invc;rsely related to the expected return on the risky asset. We also find that the cost of obtaining information about the risky asset is of considerable importance to the trader. His demand for the risky asset is inversely related to thins cost. If transactions costs are sufficiently low the trader finds it optimal to transact a number of times rather than only once. That is, the trader will trade up to a desired long-run asset holding. If the information set (from which the trader derives his subjective estimates of the relevant random variables of the problem) is subject to exogenous random change, then the long-run level of asset holdings represents an upper bound rather than a limit to the sequence of actual holdings of the asset. Expressed somewhat differently, WC lind, as Tobin did, an inverse relationship between the demand for cash balances and the size of the differential yields between cash and alternative financial assets, under the more general assumptions that traders are imperfectly informed.’ What is further explained by our analysis is the portion of the spread between the yield on any two assets, one of which may be non-interest-bearing cash, due to factors other than undiflerentiated uncertainty, and how that spread is related to the trader’s optimal solution to a portfolio revision problem and an active transactions market problem when he is operating with imperfect information In a dynamic exchange economy. Additionally, the uneven distribution of information induces the traders in our model to adopt dynamic exchange strategies which may favor temporarily assets for which the marginal cost ol’ acquiring information is rela+ively low. Thus, the next section of this paper blends the imperfect information approach of the Brunner and Meltzer study with the explicit analysis of liquidity l)reference of Tobin. 2. Imperfectinformationand the analysis of liquiditypreference 2. I. The individual trader’s optimization problem Traders in the model developed in this section zre assumed to maximize @Jensen (1972). We discuss the extension of the results of .his ‘raper IO a market with many assets in se&m 2.5 and again in the conclusion. ‘Tobin (19%).
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expected utility of real wealth The traders are imperfectly informed about various features of their situation, as described below, and understand that other traders may also be imperfectly informed. Each trader has a subjective prior for the value of each asset which he might consider holding The trader will update his priors as he reeives new, exogenous information and as a function of his experiences in the market described below. He also has expectations concerning the behavior of his fellow traders and distributions of possible future transactions prices. With this information each trader can derive his long-run excess demand function. ‘Givenhis demand function, each trader optimally solves his transaction problem. This paper will focus on the analysis of the trader’s demand function, but first it is necessary to outline the simultaneous transactions problem. The soldtion to the transactions problem is a dynamic program of the type frequently encountered in optimal search theory. A typical optimal search theory result is that a dynamic, continuously functioning market composed of traders with differing expectations and asset holdings will not be characterized, necessarily, by a unique market clearing equilibrium price at any point in time.8 Furthermore, if traders are imperfectly and heterogeneously informed, then they may find it optimal to adopt dynamic strategic%to govern their trading. The dynamic trading strategies of traders will be aff,lcted by perceived arbitrage profits as well as long-run future consumption-related demmd for assets. The arbitrage profits are perceived to arise from differences of opinion concerning the ‘value’ of the assets, or from future favorable ‘terms-of-trade’. Complete agreement on the nature of present and future market relative prices and values of parameters which summarize the stoc;ha.tic variables in their environment is sufficient to insure that no traders will perceive potential arbitrage profits, and that all trad+rs will behave as long-run price takers. That is, if there is complete agreemen& the transactions problem will cease to be important, and traders will trade directly to perceived long-run optimal holdings of all assets, 2.2. The trawxactionsproblem A model of financial asset markets which does not assume price taking behaviour af unique equilibrium prices, perhaps due to ‘searching”, is inconsistent with more traditional approaches.’ These traditional models ‘Rothschild (1973).See Castanias (1978)for an extension of optimal senrch theory to financial markets. ‘An excess demander is said to be a ‘price-taker’if he believesthat he can buy all he wants of an asset at a ~@venprice and has no economic iucentive.to bid any other price for the asset. If the trader’s transactions price is a decision variable or is af@ctedby any optimizing decisions on his part then he is no longer a ‘price-taker’at some exogenously determined price. See Arrow (1974)for a good discussion of Neoclassical Economic models and applications, and a diswsian of the ‘price-taker’assumption.
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assume that a long-run equilibrium exi::s arid obtains, perhaps reached by an ‘~uction~r and recontracting” scheme, and assume that all traders base their demands on equilibrium pri~es.~~ This feature is used to close the model gat& over individual demand functions so that the resulting system be analyzed for static equilibrium relationships, say, between equilibrium prices and model parameters. The pricetaker approach to modeling financial markets outlined above has been used to construct and analyze most contemporary models of financial capital asset pricing theory and is not without considerable motivation. A wealth of insights into the workings of resource allocation in financial markets can be credited to this approach, also called the Neoclassic4 approach. On the other hand, if the auctioneer and recontracting scheme is appropriate only at ‘long-run’ equilibrium, as Arrow has su sted, and if a long-run equilibrium in the Neoclassical sense is inconsistent with imperfect, heterogeneous and costly informaticbn,” then the optimal behavior of financial asset traders should be modeled under a more general and more reaiistic set of assumptions.” Traders in imperfect markets have an incentive to become better informed, either by reprocessing existing information or gathering and processing new information. Traders also have an incentive to capitalize on the differing information of others by trading at interim and perhaps nonequilibrium price& In Castanias (1978) a particular form of dynamic trading market is studied. There, traders who randomly encounter each other for trading purposes have differing opinions covering future trading opportunities and, perhaps, concerning the Wues’ of the assts being traded. The propositions which describe those aspects of the mechanics of the dynamic trading market which have a bearing on the demand for Money problem studied here, are given below: BropaMon I. ff a @wier j is un excess demander of the ith asset and if F(P) is the disltributionof wkw’~ bids then an optimal active trading murket strategy fQr &Wkr 1, g ha j&id: if opdmd to enter the market at a& is to terminate his s@arck (buy some Q$ the ass&) the first time he encounters a svllcr’s bid price
where .K>O is his cost qf search or of” ~urti~i~atin~in the active market. The reserviation price, Pbt characterizing such an optimal strategy exists, is finite ‘%ee Gmsman (1976)for an exampk of such a model. l%ee Arm-s (1954). ; “See Roth&ild (1973) and Castanias ( 1978).
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R. Castanias, The role ofmoney in a dynamic exchunge ecwwmy
and is unique - so long as F(P) is stationary over time. Furthermore, trader j’s expected gain from following the optimal strategy is finite. Proposition 2. The reservation price, P *FJ, described in Proposition I, is decreusing in response to a mean preserving move of density toward the tuils of F(P). That is,
aP,*“/az c:0. Proposition 3. The reservation price, P *b , described above is increasing US costs qf search, K, increase. That M, aP,*b/aK > 0.
The proofs of these propositions are straight forward and can be found in Cztanias (1978). Propositions 2 and. 3 highlight two common search literature results.13 ‘Expected purchasing prices increase as costs of search increase and alecrease (under most circumstances) as the range of possible prices increase, ceteris paribus. The mincipal implications of the dynamic search solution to the transac‘ In problem are: (1) The optimal transaction strategy of each trader is characterized by a demand reservation price, above which he will not transact if the is an excess demander and conversely for an excess supplier. (2) If the information structure of’the market is such that difrerent traders generally have heterogeneous expectations then a unique long-run equilibrium price is not generally obtained.” Traders will view series of actual and potential future tmnsactions prices as having been drawn from a distribution. As will be seen below, the random transactions price property has important implications for the demand for the risky asset. (3;1 Generally speaking. each potential trader in the ith asset will (a) determine whether it is optimal to enter the acfive market for the asset (actively engage in a costly search for a favorable price). (b) upon entering the active market, encounter other traders according to a random matching scheme, (c) with each encounter, determine whether a transaction in asset i is optimal, and 13See Rothschild (1973) for a survey of this IitcratJre. 14See Rothschild (19?3) For more lengthy discussions of markets in which long-run equilibria E il to attain.
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(d) update his priors over the relevant set of random variables as new information, such as the announcement of any transaction in asset i, becomes available. Upon encountering a favorable trading possibility, a trader seeks to uce his excess dexmm&if possible, according to his excess demand nction.“’ A trader will not necessarily eliminate his excess demand upon encountering the Grst favorable trading prtssibility. The trader may instead wish to reenter the market to search for a ‘-better’ trading l bility once his excess demand has been reduced. It is this feature of he optimal search model which leads to the socially valuable, productive services of low transactionscost assets, which we call money. The buying reservation price9 Psb, which characterizes the optimal the exc&z)s: demander is an incrc;,sing function of his cost, K, ting in the transactions market, and a decreasing function of ‘t, the precision of his estiaate of the value of the asset. 2.3. Derivation of the demand function In this paper we concentrate on the solution of the trader’s demand for money and other financial assets, falling back on the results of optimal search theory in many instances to provide us with much of the dynamics and quasi-statics of price behavior, which, in a model of more completely informed traders, might have come from the equilibrium conditions. The problem of a typical trader is that of choosing his optimal holdings of money and the risky asset, given his preference relation (or utility function) and his initial wealth, which restricts him to a subset of asset space. His problem is
(1)
max E(V’(y)), where U’=Y-BY”
and
Y =Z,+Z,P,
==P,X, cx,-%,P,+Z,P,
at the time &hedecisicj~ 15made. X, and X, arc: the trader’s initial holdings of the risky as t and money, respectively, Y is end of period wealth, PI is the random value of the risky asset at the end of the holding period and P,, is the ~r~~do~nvoluble) tr~n~cti~~~, price which the trader expects to obtain as a participant in the active trading market. PO and P, nre ‘money’ prices and the value of a unit of money is fixed in real terms at $1. All variables are ‘sin the context al the optimal search problem applicable he:%, a lon~rrrn equilibrium can be uation where buying and selling reservation prices converge and where af as n desirable - that is, traders behave ‘as if ihey are “once-and-for-all” price js no Ion takers’ with respect to a single ‘price’ for an asset.
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R. Custmias, The rola of money in a dynum& txcka~
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understood to be specific to each trader and sub and super-srrript~ are deleted where the meaning is unambiguous. Demand for the risky asset, Z1, is the choice variable. Then the demand for money, Z/, can be written as Z,=P,X,
i-x,-.Z,P,.
(2)
The budget constraint for a typical trader, given that TVtr~n~~~~~ fihke~ place, is X,+pbX,=Z,+P$?J,, where PO is a realization of PO, the transaction (money) price of the risky asset. The first-order condition for (1) is EC1--2g(Pc,X1+X,c{P,
-P())z~)](P,-P*)=O,
(3)
or ( 1/(2M5
- &I)- CE(WVX, - W$)X, +(pr, - p,)X,
Solving for Zi, the demand for the risky asset,
= (1/(2g)-XX,)dPeX,Y,+X,~~-~~P,X, P~+P&2P,P,+ v, + v,
=(1/(2Q)-X,-P,X,)da+X,V, * V, + VO+(dfi2
(41
where dP=(P, -PO), V, is the variance of PO, VI is the variance of PI, the mean of PO and P, is the mean of P1.46 Let D1==Xl-Xl br: the trador’rcr excess demand for the risky asset, then
This is the trader’s, unconditionql er,ccss dtraand functi~&..~a
,/ ,’ 16We assume that PO and P, arc independent. Note that dfg futi~nsr.~ interest rate on the risky asset in this mode!.
d
have solved two tane ~usly, a trafl!5actions P,. The combined t it3 time determine the tmks
V&of the distribution of PO X8% are zmo. The zero will then be
that as Opo tmmnes positive the trader
rices if he is an excess than PI, and perhaps nd is zero) unless he a*, his ‘buying’ reservation price. inventsq of the risky asset in excess excess supplier if an appropriately high P of the excess demander when V, is greater by the V,>O graph of fig. I.
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R. C~WU&YS, The role of money in a dyiwmk exc~
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asset, ceteris paribus. An important preliminary question is how the trader% excess demand for the risky asset will be affected by an increase in ex yield. To examine this question, we differentiate (S), yielding
XrdP -VO+VI+(dp)2’ From (5) we also see that
after rearranging terms. Substituting the preceding expression into the expression for &/MI, we have aDI (K,+V,+(dfl’)D,+X,V, ap1= {VO+I$+(dp)2)dP
2dPD,+X,dP - VO+V,+(df)2
Thus, we see that the excess demander of the risky asset demands a&icJy more of the risky asset as PI increases (a variance-preserving shift in
(tiF)2s VI.That is,
for suficierltly iow expected rates of return (interest rates), The trader’s expected rate of return from an inv~~tmcnt in th is increasing in dcis;,So we have shown that the is downward-sloping for sufliciently low interest is endogenously determined given the demand Furthermore, this result is equivalent to a simil (1958), but for the more general model of this have disparate and impeffeet information. ” downward-sloping liquidity preferenw equilibrium model equivalent to the problem sol VI> (dg)‘.
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of liquidity preference addressed here, a downward doping liquidity preference
t an expected positive tiou
to fl
ek&
as an
elasticity, we have
at higher expecteci rates of q but is not likely to become
we have argued that transactions ket clearing price of competitive transactions prices to he V,>Q. Four important results d function (5). At any point in
will demaud strictly the activs: tr adin
of the risky asset as his cost of participating in et or cost of information about the risky asset, t as the precision, t, of his c for the risky asset which will be of rhc distribution of transactions ct@dttmwxtiows price Q Me Cost
of purchasing the risky in the excess a change in V,, ng in $P. Let e br: the elasticity
Ms, that is if V,>(df?‘. then the
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R. Castanias, The role ojnwney in a dynamic exchange economy
To demonstrate results (1) and (Z),we observe that ‘Eand K do not enter the trader’s excess demand function (5) diret:tly. However, Propositions 2 and 3, we know that the trader’s optimal reservation price, Qb, is affected by K and 2. In particular,
for an excess demander. The first derivative implies that if the trader’s cost of participation in the active trading market increases, he will r&e his reservation price and thus decrease his expected time of search in the mark& The second derivative imp!ies that the trader will optimally choose a higher reservation price, ceteris paribus, if he is less sure of his estimate of the teue value of the risky asset (z is the precision of his estimate of the expectd -:alue of the risky asset). Furthermore, the excess demander% buying reservation price, Po*s is his expected cost of purchase of the asset in the dynamic trading market, PO. Since the trader’s excess demand schedule is a function of zhis expected cost. (Sj will be a&ted by changes in those parameters which affeG:tthe trader’s determination of his optimal ‘buying’ reservation price, including K and r. It follows from (6) that
Differentiating (5) with respect to PC, we have
_
-(r/(2g)-XI-PiX,)+2(dB)D,
A sufficient condition for a&@&, to be negative is V,;4(djit)$ and it follows that alp&K
aD,,vkd,
215
~iffer~~tia~~~~(5) with respect to Vo,we have
t the trader% exct~~ demand for the risky asset is strictly a function of Ve j&r any yioen d~~~d~~~~~=O). Thus, the and schedule is strictly lower in the generalized case (Ve>O) than in the classical case [Ve==O)if PrzO, establishing result three above, ~o~eve~~ trader may she along the aemand curve (PO may in V,. The second term on the right-hand change) in respsn v be positive in this case and the trader% change in excess side of (9) demand in ponse to a change in V, is seen to be indeterminate without further restrictions, ving us result four above. When the trader aximixes his convex quadratic utility of future wealth, r the proportion of his investment balance that he chooses to place t, the more risk the trader assumes. At the same time, the proportion of risky assets also increases his expected return. d (2), tell us that as the trader’s cost of participating in the market increases or when he is less certain of the value of the asset, his expected return per unit of risk decreases leading to a decrease in his demand for the asset. in K or r is seen to shift the trader’s opportunity locus for risk and 2, the vertical axis represents expected future wealth, is risk of future wealth, $!. The concave curve OM, nity lscus associated with a given level of expected th, and shows that the trader can expect more wealth der’s loci of indifference between if he a~urn~ more risk.
Implicitdithmtiation yields
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R. Cmtanias, The rde ofmoney In u dynmk exchange e#~~~y
Fig. 2. Graph of utility ant! oplumunity se: loci.
combinations of risk and expected wealth can be represented, in the Jmeftnvariance plane of fig. 2, by a family of convex indifference curves such as It : for a given level of risk, CT;,a risk-averse trader always prefers a larger expected wealth, ff. The trader maximizes his expected utility by choosing the highest indiiference curve: that is, the utility curve which is tangent !Q the opportunity locus or if r1+3tangency occurs, the corner maximum associated with the utility curve passing through the origin, 0, of fig. 2. We can now see graphically the properties of the demand function for rhe risky asset. A shift in the trader’s opportunity locus from OM 1 to CPM3wilt enable the trader to achieve a higher level of utility for a given quantity of investment funds. The point of tangency Ts corresponds to a hi After substituting for V,, we have for the term in brackets
{v,x:-2x,z,v,+z:(v,+v,)J~v,+
k$)
=~~(v,+v,)(x:-2x,z,-cz:}+Y~(v~+Y~~: =v,(v,+v,)(x,-z,}“$v,(Y~cv,)z:. This last expression is always poktive, and hence so is Y’, p”= -dP(V,+
V,)(VAV,+ VI)- I/‘,V,X:)“3’2/4,
which is negative if P’ is positive and real. Therefore, the opportunity locus is WIWJIW.If XI -0, the opportunity locus will be linear and increasing,
R. Cm&
The roteof-y
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of the risky asset than the point of tangency T,. The necessary shift in the opportunity locus can ‘-e caused by an increase in the trader’s perception of frt, an increase in the trader% precision of p,, or a decrease in the trader’s cost of participating in the actiVe tmding market. equal, wiU lead to a downward shift in the c as ~8% shown ~;II z~ult three* will lead to a for the risky awt* The ambiguity of result (4) stems from abfie to say whether the opportunity Socus will shift up or in V, when we consider that Pxb the trader’s optimal ervatidn prior of the a&w trading market problem (and hence tbz wet), also is affected by V,. Pgb is der s pelception of dP to increase, putting ity locus at the same time that a shift in V, and function has a negative effect on his opportunity locus. We can observe only that, for reasonable values of all the eters determining Pzb and D,, the trader can he shown to relevant his demand for the risky asset as V, increases. In either in rd particular, the trader will be more likely to have decreasing demand $M the risky ass&: (I) the higher is l&isroostof participating in the active trading market, (2) the less precise are his priors concerning parameters relevant to either his demand or active trading market problems, (3) the larger is his excess demand, are his estimates of V. and V,, or (4) the risk werse he is. (5) the An important fatwe of Zt, tie trader’s demand function, for V,>O is its enoe on XI, the tradds initial holding of the asset. This property m~til a unique level of investment in the risky asset, XT, is reached. deternAte Xl: fiorn (4) by noting that (4) is a difference equation in 81119 (th where x,*1 Solving for X& the fixed piat of the diflerence equation
(11) ns of P, and P, and a given XT, (11) defines a unique pair
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R. Castanias, The role of morteyin a dyna& exchange economy
of long-run steady-state demands for money and the risky asset.. namely the pair (X/*, XT). Once the trader acheeves the position characterized by (X,*,X:), he will ,have no incentive to alter his holdings unless either his priors on PO or P1 change or his expected wealth, W& changes (11) is similar to the classical demand function resulting from the assumption that traders are price-takers at competitive equilibrium pricea As V,-+O in (4) and assuming that the trader’s current expected walth, WO,is the same as his expected wealth, IV& associated with the long-run steadystate,
or
(11) may be rewritten as x+ 1
(l./(2g)-XXf-XtP,,}dP =V, +(dp)? (13)
(11) gives the trader’s optimal steady-state demands (X&XT) for any givea level of current expected wealth as the limit of a sequence of transactions. It TC mains to examine how the trader’s demand function, &, compares to XT for any crsh~rpair (X,,X,) of holdings of money and the risky asset, for the same level of expected wealth. Given that the trader ma :I sta.rt at (X$,X i) with initial expected wealth WC, = X, + B,X 1 when XF > X 1 L nd W. = IV& (12) and (13) are seen to be similar but not identical since dP=Pr -Pa need not be the same in both equations. In the dynamic model, PO is the expected cost oi’ purchase when the trader uses the reservation price strategy and is equal to Pab, the excess demander’s ‘buying’ reservation price {see Proposition 1). If P1 is the ssamefor a given trader in both the perfectly competitive primtaking equilibritmi and the dynamic sea&h problem, -then the level of demand he desires will depend on how Ptb compares >to pEa,the assumed competitive equilibrium market clearing price. If the transactions market approaches an equilibrium in the longer run, it can be shown that Pgb-+Fo and long-run demand XT will approach the classical denand
R Castaiias, The de ~$mvney in a dynamic exchwe
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If V, is greater than zero, however, then
Xr, This result follows from the fact that X7 -Z1 is strictly in XI and that XT--Z,=O when X1=X!: if Vo>O, the excess will rrcde up to his optimal level of investment in the risky asset ucnce of transactions as Xt, r-+X? with each successive trana&ion. 43111~ when Ye=O will the trader trade directly to his optimal long-run level of investment in the risky asset no matter what his initial holding, X1, of the risky asset. The preceding results can be summarized as follows: For any given set of priors on PO and P, and a given level of expectea current wealth, We,
(1) as Vs-+O,the trader’s demand curve: for the ti,sky asset, Z,, converges to the competitive equilibrium demand curve of (12) in the long run as P$-+P& (2) the Excess Demander’s demand for the risky asset, %r, is strictly ler* than his optimal longer-run steady-state level of demand for the risk., asset, XT, if Ve~0. That is, the trader trades up to Xly by undertaking a tions in the risky itsset. If V+O the trader trades up transaction, no matter what his initial holding of the 8 the trader’s excess demand will be a decreasing y-state level of demand, Xt, is a decreasing in the active transactions market, an r~i~ion of his estimates of the value and an inc~asi~~~ function of his perception of the neity c~~~*~ni~ the value of the risky asset.
(3)
2.5. ~~n~Z~~~~i~~~ far Eh@ iIwnanill$i maney: Sbrt ad long rw to
the ~cy~~sia~ th Aversion, this p
y of liquidity preferen= or the Tobin
extends the analysis of liquidity
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R. Castanias,The role of moneyin * dynamice.whangeeconomy
preference to consider explicitly the non-pecuniary services provided by money, as suggested by the analysis of Brunner and Meltzer, for an exchange economy where traders are imperfectly informed. Since demand for money is defined endogenously b;t the equation Z,=P,X,
+X,+?,P,,
(1%
the short-run effect will be for the excess demander to hold a higher percentage of his wealth in money, if V,>O, than he intends to hold in the longer run. The trader will undertake a dynamic sequential strategy for gradually increasing his level of holdings of the risky asset. The tracer will attempt to capitalize on the disparity of bid prices and the dynamic nature r,C the transactions market by ‘searching’ and by ‘trading up’ to his optimal long-run preferred holdings. Both factors will result in higher cash balances in the short run than the trader would have held had he been able to trade in a competitive equilibrium market at the same price, PO. The theory of liquidity preference developed here does not lead to the result that in a strict steady-state demand for hquidity preference must be zero. However, the residual long-run demand for cash, X7, has been shown to be different from the demand for cash, Z,, when the trader is trading from a position (X,,X,) different from (X,*,X:). Then money is providing nonpecuniary services to its holders as they adjust gradually to (X$X:). ‘IIe theory, however, is more or less ambiguous concerning the relation betwa,:n the longer-run demand for money and the trader’s optima! short-term position, or between the longer-run or short-term positions and the demand for money associated wit 1 a perfectly competitive market. But the mechanics of the model developed here can be used to address these issues when questions are further defined. For example, if increased disparity of opinion concerning the value of the risky asset across traders increases the cost of maintaining a given state of information about the asset then the expected cost of purchasing the asset pnd hence the reservation price will increase, leading perhaps to a- demand for cash exceeding competitive equilibrium levels, ah else being equal. The same results concerning the demand for money can be reached if the model is extended to include multiple risky alternatives to cash. The trader’s opportunity locus for expected wealth and risk will still be concave as in fig. 2, and a downward shift in the opportunity locus will still be associated with an increase in the demand fcr money. 3. Conclusionsand comments on areasfor furtherrcesesrcb When traders are allowS& to transact in dynamic sequential trading markets for assets for which they have an excess demand, they will choose to
bold cash, the assumed QltxI;um of exchange, in quantities in excess to their preferred longer-run cash holdins Liquidity preference is seen to he a on the many factors, such as costs of ens of the heterogeneity of and demand decisions. ImpOrtantiy, the traders longer-1~x1demand for money is seen to im Itebatjvc:to the h risky assets as his uncertainty and lack distribution of information causes a bearer of specialized non-pecuniary serGces to iti fnglder, in the sense that holding money allows the trader to ns while searching for ‘bargains’ in the active asset might theoretically serve this purpose, s this function ‘best’.l9 not answered in this study are the following. If traders could participate in markets for various pairs of assets, with each market structuti as the money-risky asset market of this paper, would they choose to liquidate their excess supplies of certain commodities or assets into particular intermediate assets for which the costs of trading are particularly Iow?~~ Will traders prefer intermediate assets for which there is widespread agreement as to value and/or the vector of relative prices of all other assets? our results s st that traders should prefer intermediate assets with low K’s, high t’s low Va’s.Will traders have an incentive to ‘cash out’ of assets for which they feel they have a comparative informational disadvantage, and take up short- and intermediate-run excess deniiind positions in assets for which they have a compa-ative informational advantage? An important implication of the dynamic active trading market strategy is that the trader is sensitive to costs and the precision of his information. Thus, a result of the sort suggested above, although beyond the scope of this paper, would not be surprising. What are the implicatians of ential demand strategies for the existence dynamic trading strategies and ts? It might he argued that the existence of intermediaries in financial m of traders with “sp.yiatized trading functions’ is an sfflcient response to the disparity of information which ~hara~t~ri~~~the markets of this paper. ss the role of and existence of money in an exchange Attempts to c.,nmade by Brunner and Mcltzcr, Harris and Jones.” AI e e*Listenerof money is socially and individually advantageous when traders trade assets in u d~~~trali~~d e!;ch:+:ge. All stress that traders nse of lxst at lowering coss of acquiring information and costs of t’ in the inbrmation. *@Thecosts of trading reXerred to here arc the ‘costs of participating in the active market’. ‘lntcrmdiatc assets’are assets‘leld to be Txchangcd later for assets wanted ‘directly’, or to fulfill a de&cd long-run position. *‘Brunnar and Mel&er (1972). Harris f I j76) and Jones (1976
222
R.Cwtuei~
The role of moneyin u d.ytwm&exchungwe WKVQ
will choose ‘intermediate’as&s for their usefulness in subsequent trades as well as their direct contribution to the utility of the holders. Harris was able to establish conditions under ,which the eventual outcome of the bikteral exchanges of his model will’lead to a Pareto Optimal allocation of resou But all of the preceding studies have focused on the general role of money in the exchange process or on the evetrtual outcome of the exch and not on the micro behavior of individual traders as th adjusting portfolios actually iakes place. The results of this that the sort of model developed here, if it can be generalized to address the questions raised above, may fill this gap. References Arrow, K., 1954, Toward a theory of price adjustment, in: M. Abramovitz, ed., Alloeatian of economic resoilrces (University of Califoniia Press, Stanford, CA) 41-52 Arrow, K., 1974, Limited knowleAIgeand economic analysis, American Economic Review 64 l10. Brunner, K. and A. Meltzer, 1972, The uses of money: Money in the theory of ait exchange economy, American mnomic Review 61,748-805. Castanias, R., 1978, Essays on the behavior of asset prices under uncertainty, Ph.D. thesis (Carnegie-MellonUniversity, Pittsburgh, PA). Fama, E., 1976,Foundations of finance (BasicBooks, New York). Grossman, S., 1976, On the efficiencyof competitive stock markets where traders have diverse information, Journal of Fiance 31, May, 573-585. Harris, M., 1976, Expectations and money in a dpamic exchange economy, Cam@-Hcliun University Working Paper (Pittsburgh, PA). Hart, O., 1974, On the existenti of equilibrium in a securities market, Journal of Economic Theory 9,29J-3 11. Jensen, M., 1972,Capital markets: Theory and evidence, Beil Journal of Economies 3,X7-398. Jones, R., 1976,The origin and development of media of exchange, Journal of Political bnomy 84,757-7X. Rothschild, .M., 1973, Models of market organisation with imperfect information, Jouma? of Political Ecolrlomy52, 1283-1308. Stigler, G., 1961,The economics of information, Journal of Political Economy 69, 21.3-225. Tobin, J., 1958, Liquidity preference as a behaviour towards risk, Review of’ numie Studies 26, 65-86.
U-Y
203-222.
North-Holland Publishing Company
Richard CASTANIAS* n,St~rle,WA 98195, Q
USA
ey as an alternative to barter in explanations fail to motivate, or this paper, information about distributed. Money is shown to It pI1 asset of wide-spread social value. Thus, this paper of Bnmner and Meltzer (1972) with the explicit
1. I. The demand for money when traders are imper$ectly informed
An e~~~tially unresolved problem in financial economics i; to explain adequately the role of money in a dynamic exchange economy.’ The principal motives generally advanced for the use of money or the 6 Gstence of a positivedemand for money are (1) lack of synchronization of -zc+eiptsand expenditures, (2) aK3ts of tran$actin~ (3) embarrassment of Jefault, (4) unccrt&ty, and (5) *he crxistsnccof productive services, such as liquidity. Of &XC~it has baen osnvincringly argued by Brunner and Me&r that the first three woti not exist without the fourth, uncertainty, or perhaps imperfect of Borne traders. In particular, Brunner and Meltzer know1 on the argue ion of information, and not the existence of nty, that induces individuals to search for, t, alternatives to barter’.* Allan Msl~r,
and an anonymous refr;reefor useful
*See Brunncr sad Mekzm (1972) far a s~m&ry of this i’:sut. 2Brunncr and Mattttr (1972, p. 786).
$02.75 @ North-Holland
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Tobin has shown that uncertainty alone can give rise to a positive demand for money on the part of risk-averse investors. His principal result is that the existence of undifferentiated uncertainty is sufficient to cause traders to hold money balances in equilibrium and, under certain circumstances, to have negatively sloped liquidity preference schedules with regard to the expected yields on risky assets.3 Rut Tobin fails to identify explicitly the productive services of money which are associated with the non-pecuniary services of motive (5) above. The Tobin analysis also fails to provide insight as to why money has remained in use in economies with sustained high levels of inflation, or accelerating inflation. Brunner and Meltzer attempt to characterize explicitly the services rendered by money and extend the theory of asset choice to include the cost of acquiring information about the market arrangements, relative prices, and exchange ratios.4 In a market where information is unevenly distributed, traders will search for sequences of transactions which minimize the costs of acquiring information and transacting. Money is shown to have ‘peculiar properties and a low marginal cost of information’ which make it an asset of wide-spread social value in constructing optimal transactions chains5 Thus, Brunner and Meltzer argue that money provides an explicit service to optimizing traders, and the use of money increases the welfare of its users, A trader may find it advantageous to allocate pa;t of his wealth to money. 1.2. Financial assets and exchanp
strategies when traders are imptvfec:ly
informed
In this paper we examine how uneusnly distributed information wiil irrductraders to adopt dynamic exchange strategies and, as W&S suggested by Brunner and Meltzer, may even induce indirect methods of exchange. Goods and assets with low marginal costs of acquiring information concerning relative prices (in terms of some or all other goods) are temporarily inventoried in quantities in excess of what will eventually be held when the trader has attained an optimal long-run portfolio. A good with such properties may be termed a medium of exchange in that it is held in excess (of long-run desired) supply only to enable the trader to make more advantageous exchanges later. We assume an exchange economy in which each trader adopts a dynamic strategy for transacting in portfolio assets and views his expected ‘cost’ of (or arbitrage gain from) purchasing an asset, given his optimal strategy, as a random variable. Thus, we can no longer simply assume that he will trade to Robin (1958). Tobin assumes that traders have quadratic utility functions and parameter values sufficient to assure that the liquidity preference schedule is downward sloping. 4Brunner and Meltzer (1972). ‘Brunner and Meltzer (1972, p. 804).
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his long-run preferreddemand level at some tatonnement generated unique price. We consider a market in which there are just two assets: a risky i’sset and a risk-k asset (money), 2ulid traders are expected to maximize $.heir qmdWic utility of Wth, TIE results are then comparable to the Liquidi:y Prt?ference model of Tob:~ and the single period capital asset pricing models of contemporary financial theory, as summarized by Jenseri.B In this market an excess dbmander of the risky asset will demand strictly less of the risky arSSeton any given transaction than in a perfectly competitive, equilibrium market, and his demand for the money asset is invc;rsely related to the expected return on the risky asset. We also find that the cost of obtaining information about the risky asset is of considerable importance to the trader. His demand for the risky asset is inversely related to thins cost. If transactions costs are sufficiently low the trader finds it optimal to transact a number of times rather than only once. That is, the trader will trade up to a desired long-run asset holding. If the information set (from which the trader derives his subjective estimates of the relevant random variables of the problem) is subject to exogenous random change, then the long-run level of asset holdings represents an upper bound rather than a limit to the sequence of actual holdings of the asset. Expressed somewhat differently, WC lind, as Tobin did, an inverse relationship between the demand for cash balances and the size of the differential yields between cash and alternative financial assets, under the more general assumptions that traders are imperfectly informed.’ What is further explained by our analysis is the portion of the spread between the yield on any two assets, one of which may be non-interest-bearing cash, due to factors other than undiflerentiated uncertainty, and how that spread is related to the trader’s optimal solution to a portfolio revision problem and an active transactions market problem when he is operating with imperfect information In a dynamic exchange economy. Additionally, the uneven distribution of information induces the traders in our model to adopt dynamic exchange strategies which may favor temporarily assets for which the marginal cost ol’ acquiring information is rela+ively low. Thus, the next section of this paper blends the imperfect information approach of the Brunner and Meltzer study with the explicit analysis of liquidity l)reference of Tobin. 2. Imperfectinformationand the analysis of liquiditypreference 2. I. The individual trader’s optimization problem Traders in the model developed in this section zre assumed to maximize @Jensen (1972). We discuss the extension of the results of .his ‘raper IO a market with many assets in se&m 2.5 and again in the conclusion. ‘Tobin (19%).
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expected utility of real wealth The traders are imperfectly informed about various features of their situation, as described below, and understand that other traders may also be imperfectly informed. Each trader has a subjective prior for the value of each asset which he might consider holding The trader will update his priors as he reeives new, exogenous information and as a function of his experiences in the market described below. He also has expectations concerning the behavior of his fellow traders and distributions of possible future transactions prices. With this information each trader can derive his long-run excess demand function. ‘Givenhis demand function, each trader optimally solves his transaction problem. This paper will focus on the analysis of the trader’s demand function, but first it is necessary to outline the simultaneous transactions problem. The soldtion to the transactions problem is a dynamic program of the type frequently encountered in optimal search theory. A typical optimal search theory result is that a dynamic, continuously functioning market composed of traders with differing expectations and asset holdings will not be characterized, necessarily, by a unique market clearing equilibrium price at any point in time.8 Furthermore, if traders are imperfectly and heterogeneously informed, then they may find it optimal to adopt dynamic strategic%to govern their trading. The dynamic trading strategies of traders will be aff,lcted by perceived arbitrage profits as well as long-run future consumption-related demmd for assets. The arbitrage profits are perceived to arise from differences of opinion concerning the ‘value’ of the assets, or from future favorable ‘terms-of-trade’. Complete agreement on the nature of present and future market relative prices and values of parameters which summarize the stoc;ha.tic variables in their environment is sufficient to insure that no traders will perceive potential arbitrage profits, and that all trad+rs will behave as long-run price takers. That is, if there is complete agreemen& the transactions problem will cease to be important, and traders will trade directly to perceived long-run optimal holdings of all assets, 2.2. The trawxactionsproblem A model of financial asset markets which does not assume price taking behaviour af unique equilibrium prices, perhaps due to ‘searching”, is inconsistent with more traditional approaches.’ These traditional models ‘Rothschild (1973).See Castanias (1978)for an extension of optimal senrch theory to financial markets. ‘An excess demander is said to be a ‘price-taker’if he believesthat he can buy all he wants of an asset at a ~@venprice and has no economic iucentive.to bid any other price for the asset. If the trader’s transactions price is a decision variable or is af@ctedby any optimizing decisions on his part then he is no longer a ‘price-taker’at some exogenously determined price. See Arrow (1974)for a good discussion of Neoclassical Economic models and applications, and a diswsian of the ‘price-taker’assumption.
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assume that a long-run equilibrium exi::s arid obtains, perhaps reached by an ‘~uction~r and recontracting” scheme, and assume that all traders base their demands on equilibrium pri~es.~~ This feature is used to close the model gat& over individual demand functions so that the resulting system be analyzed for static equilibrium relationships, say, between equilibrium prices and model parameters. The pricetaker approach to modeling financial markets outlined above has been used to construct and analyze most contemporary models of financial capital asset pricing theory and is not without considerable motivation. A wealth of insights into the workings of resource allocation in financial markets can be credited to this approach, also called the Neoclassic4 approach. On the other hand, if the auctioneer and recontracting scheme is appropriate only at ‘long-run’ equilibrium, as Arrow has su sted, and if a long-run equilibrium in the Neoclassical sense is inconsistent with imperfect, heterogeneous and costly informaticbn,” then the optimal behavior of financial asset traders should be modeled under a more general and more reaiistic set of assumptions.” Traders in imperfect markets have an incentive to become better informed, either by reprocessing existing information or gathering and processing new information. Traders also have an incentive to capitalize on the differing information of others by trading at interim and perhaps nonequilibrium price& In Castanias (1978) a particular form of dynamic trading market is studied. There, traders who randomly encounter each other for trading purposes have differing opinions covering future trading opportunities and, perhaps, concerning the Wues’ of the assts being traded. The propositions which describe those aspects of the mechanics of the dynamic trading market which have a bearing on the demand for Money problem studied here, are given below: BropaMon I. ff a @wier j is un excess demander of the ith asset and if F(P) is the disltributionof wkw’~ bids then an optimal active trading murket strategy fQr &Wkr 1, g ha j&id: if opdmd to enter the market at a& is to terminate his s@arck (buy some Q$ the ass&) the first time he encounters a svllcr’s bid price
where .K>O is his cost qf search or of” ~urti~i~atin~in the active market. The reserviation price, Pbt characterizing such an optimal strategy exists, is finite ‘%ee Gmsman (1976)for an exampk of such a model. l%ee Arm-s (1954). ; “See Roth&ild (1973) and Castanias ( 1978).
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R. Castanias, The role ofmoney in a dynamic exchunge ecwwmy
and is unique - so long as F(P) is stationary over time. Furthermore, trader j’s expected gain from following the optimal strategy is finite. Proposition 2. The reservation price, P *FJ, described in Proposition I, is decreusing in response to a mean preserving move of density toward the tuils of F(P). That is,
aP,*“/az c:0. Proposition 3. The reservation price, P *b , described above is increasing US costs qf search, K, increase. That M, aP,*b/aK > 0.
The proofs of these propositions are straight forward and can be found in Cztanias (1978). Propositions 2 and. 3 highlight two common search literature results.13 ‘Expected purchasing prices increase as costs of search increase and alecrease (under most circumstances) as the range of possible prices increase, ceteris paribus. The mincipal implications of the dynamic search solution to the transac‘ In problem are: (1) The optimal transaction strategy of each trader is characterized by a demand reservation price, above which he will not transact if the is an excess demander and conversely for an excess supplier. (2) If the information structure of’the market is such that difrerent traders generally have heterogeneous expectations then a unique long-run equilibrium price is not generally obtained.” Traders will view series of actual and potential future tmnsactions prices as having been drawn from a distribution. As will be seen below, the random transactions price property has important implications for the demand for the risky asset. (3;1 Generally speaking. each potential trader in the ith asset will (a) determine whether it is optimal to enter the acfive market for the asset (actively engage in a costly search for a favorable price). (b) upon entering the active market, encounter other traders according to a random matching scheme, (c) with each encounter, determine whether a transaction in asset i is optimal, and 13See Rothschild (1973) for a survey of this IitcratJre. 14See Rothschild (19?3) For more lengthy discussions of markets in which long-run equilibria E il to attain.
R. Casran&, The ro&
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(d) update his priors over the relevant set of random variables as new information, such as the announcement of any transaction in asset i, becomes available. Upon encountering a favorable trading possibility, a trader seeks to uce his excess dexmm&if possible, according to his excess demand nction.“’ A trader will not necessarily eliminate his excess demand upon encountering the Grst favorable trading prtssibility. The trader may instead wish to reenter the market to search for a ‘-better’ trading l bility once his excess demand has been reduced. It is this feature of he optimal search model which leads to the socially valuable, productive services of low transactionscost assets, which we call money. The buying reservation price9 Psb, which characterizes the optimal the exc&z)s: demander is an incrc;,sing function of his cost, K, ting in the transactions market, and a decreasing function of ‘t, the precision of his estiaate of the value of the asset. 2.3. Derivation of the demand function In this paper we concentrate on the solution of the trader’s demand for money and other financial assets, falling back on the results of optimal search theory in many instances to provide us with much of the dynamics and quasi-statics of price behavior, which, in a model of more completely informed traders, might have come from the equilibrium conditions. The problem of a typical trader is that of choosing his optimal holdings of money and the risky asset, given his preference relation (or utility function) and his initial wealth, which restricts him to a subset of asset space. His problem is
(1)
max E(V’(y)), where U’=Y-BY”
and
Y =Z,+Z,P,
==P,X, cx,-%,P,+Z,P,
at the time &hedecisicj~ 15made. X, and X, arc: the trader’s initial holdings of the risky as t and money, respectively, Y is end of period wealth, PI is the random value of the risky asset at the end of the holding period and P,, is the ~r~~do~nvoluble) tr~n~cti~~~, price which the trader expects to obtain as a participant in the active trading market. PO and P, nre ‘money’ prices and the value of a unit of money is fixed in real terms at $1. All variables are ‘sin the context al the optimal search problem applicable he:%, a lon~rrrn equilibrium can be uation where buying and selling reservation prices converge and where af as n desirable - that is, traders behave ‘as if ihey are “once-and-for-all” price js no Ion takers’ with respect to a single ‘price’ for an asset.
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R. Custmias, The rola of money in a dynum& txcka~
ecmtmy
understood to be specific to each trader and sub and super-srrript~ are deleted where the meaning is unambiguous. Demand for the risky asset, Z1, is the choice variable. Then the demand for money, Z/, can be written as Z,=P,X,
i-x,-.Z,P,.
(2)
The budget constraint for a typical trader, given that TVtr~n~~~~~ fihke~ place, is X,+pbX,=Z,+P$?J,, where PO is a realization of PO, the transaction (money) price of the risky asset. The first-order condition for (1) is EC1--2g(Pc,X1+X,c{P,
-P())z~)](P,-P*)=O,
(3)
or ( 1/(2M5
- &I)- CE(WVX, - W$)X, +(pr, - p,)X,
Solving for Zi, the demand for the risky asset,
= (1/(2g)-XX,)dPeX,Y,+X,~~-~~P,X, P~+P&2P,P,+ v, + v,
=(1/(2Q)-X,-P,X,)da+X,V, * V, + VO+(dfi2
(41
where dP=(P, -PO), V, is the variance of PO, VI is the variance of PI, the mean of PO and P, is the mean of P1.46 Let D1==Xl-Xl br: the trador’rcr excess demand for the risky asset, then
This is the trader’s, unconditionql er,ccss dtraand functi~&..~a
,/ ,’ 16We assume that PO and P, arc independent. Note that dfg futi~nsr.~ interest rate on the risky asset in this mode!.
d
have solved two tane ~usly, a trafl!5actions P,. The combined t it3 time determine the tmks
V&of the distribution of PO X8% are zmo. The zero will then be
that as Opo tmmnes positive the trader
rices if he is an excess than PI, and perhaps nd is zero) unless he a*, his ‘buying’ reservation price. inventsq of the risky asset in excess excess supplier if an appropriately high P of the excess demander when V, is greater by the V,>O graph of fig. I.
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R. C~WU&YS, The role of money in a dyiwmk exc~
etxmmy
asset, ceteris paribus. An important preliminary question is how the trader% excess demand for the risky asset will be affected by an increase in ex yield. To examine this question, we differentiate (S), yielding
XrdP -VO+VI+(dp)2’ From (5) we also see that
after rearranging terms. Substituting the preceding expression into the expression for &/MI, we have aDI (K,+V,+(dfl’)D,+X,V, ap1= {VO+I$+(dp)2)dP
2dPD,+X,dP - VO+V,+(df)2
Thus, we see that the excess demander of the risky asset demands a&icJy more of the risky asset as PI increases (a variance-preserving shift in
(tiF)2s VI.That is,
for suficierltly iow expected rates of return (interest rates), The trader’s expected rate of return from an inv~~tmcnt in th is increasing in dcis;,So we have shown that the is downward-sloping for sufliciently low interest is endogenously determined given the demand Furthermore, this result is equivalent to a simil (1958), but for the more general model of this have disparate and impeffeet information. ” downward-sloping liquidity preferenw equilibrium model equivalent to the problem sol VI> (dg)‘.
R
Pkffd
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of liquidity preference addressed here, a downward doping liquidity preference
t an expected positive tiou
to fl
ek&
as an
elasticity, we have
at higher expecteci rates of q but is not likely to become
we have argued that transactions ket clearing price of competitive transactions prices to he V,>Q. Four important results d function (5). At any point in
will demaud strictly the activs: tr adin
of the risky asset as his cost of participating in et or cost of information about the risky asset, t as the precision, t, of his c for the risky asset which will be of rhc distribution of transactions ct@dttmwxtiows price Q Me Cost
of purchasing the risky in the excess a change in V,, ng in $P. Let e br: the elasticity
Ms, that is if V,>(df?‘. then the
214
R. Castanias, The role ojnwney in a dynamic exchange economy
To demonstrate results (1) and (Z),we observe that ‘Eand K do not enter the trader’s excess demand function (5) diret:tly. However, Propositions 2 and 3, we know that the trader’s optimal reservation price, Qb, is affected by K and 2. In particular,
for an excess demander. The first derivative implies that if the trader’s cost of participation in the active trading market increases, he will r&e his reservation price and thus decrease his expected time of search in the mark& The second derivative imp!ies that the trader will optimally choose a higher reservation price, ceteris paribus, if he is less sure of his estimate of the teue value of the risky asset (z is the precision of his estimate of the expectd -:alue of the risky asset). Furthermore, the excess demander% buying reservation price, Po*s is his expected cost of purchase of the asset in the dynamic trading market, PO. Since the trader’s excess demand schedule is a function of zhis expected cost. (Sj will be a&ted by changes in those parameters which affeG:tthe trader’s determination of his optimal ‘buying’ reservation price, including K and r. It follows from (6) that
Differentiating (5) with respect to PC, we have
_
-(r/(2g)-XI-PiX,)+2(dB)D,
A sufficient condition for a&@&, to be negative is V,;4(djit)$ and it follows that alp&K
aD,,vkd,
215
~iffer~~tia~~~~(5) with respect to Vo,we have
t the trader% exct~~ demand for the risky asset is strictly a function of Ve j&r any yioen d~~~d~~~~~=O). Thus, the and schedule is strictly lower in the generalized case (Ve>O) than in the classical case [Ve==O)if PrzO, establishing result three above, ~o~eve~~ trader may she along the aemand curve (PO may in V,. The second term on the right-hand change) in respsn v be positive in this case and the trader% change in excess side of (9) demand in ponse to a change in V, is seen to be indeterminate without further restrictions, ving us result four above. When the trader aximixes his convex quadratic utility of future wealth, r the proportion of his investment balance that he chooses to place t, the more risk the trader assumes. At the same time, the proportion of risky assets also increases his expected return. d (2), tell us that as the trader’s cost of participating in the market increases or when he is less certain of the value of the asset, his expected return per unit of risk decreases leading to a decrease in his demand for the asset. in K or r is seen to shift the trader’s opportunity locus for risk and 2, the vertical axis represents expected future wealth, is risk of future wealth, $!. The concave curve OM, nity lscus associated with a given level of expected th, and shows that the trader can expect more wealth der’s loci of indifference between if he a~urn~ more risk.
Implicitdithmtiation yields
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R. Cmtanias, The rde ofmoney In u dynmk exchange e#~~~y
Fig. 2. Graph of utility ant! oplumunity se: loci.
combinations of risk and expected wealth can be represented, in the Jmeftnvariance plane of fig. 2, by a family of convex indifference curves such as It : for a given level of risk, CT;,a risk-averse trader always prefers a larger expected wealth, ff. The trader maximizes his expected utility by choosing the highest indiiference curve: that is, the utility curve which is tangent !Q the opportunity locus or if r1+3tangency occurs, the corner maximum associated with the utility curve passing through the origin, 0, of fig. 2. We can now see graphically the properties of the demand function for rhe risky asset. A shift in the trader’s opportunity locus from OM 1 to CPM3wilt enable the trader to achieve a higher level of utility for a given quantity of investment funds. The point of tangency Ts corresponds to a hi After substituting for V,, we have for the term in brackets
{v,x:-2x,z,v,+z:(v,+v,)J~v,+
k$)
=~~(v,+v,)(x:-2x,z,-cz:}+Y~(v~+Y~~: =v,(v,+v,)(x,-z,}“$v,(Y~cv,)z:. This last expression is always poktive, and hence so is Y’, p”= -dP(V,+
V,)(VAV,+ VI)- I/‘,V,X:)“3’2/4,
which is negative if P’ is positive and real. Therefore, the opportunity locus is WIWJIW.If XI -0, the opportunity locus will be linear and increasing,
R. Cm&
The roteof-y
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economy
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of the risky asset than the point of tangency T,. The necessary shift in the opportunity locus can ‘-e caused by an increase in the trader’s perception of frt, an increase in the trader% precision of p,, or a decrease in the trader’s cost of participating in the actiVe tmding market. equal, wiU lead to a downward shift in the c as ~8% shown ~;II z~ult three* will lead to a for the risky awt* The ambiguity of result (4) stems from abfie to say whether the opportunity Socus will shift up or in V, when we consider that Pxb the trader’s optimal ervatidn prior of the a&w trading market problem (and hence tbz wet), also is affected by V,. Pgb is der s pelception of dP to increase, putting ity locus at the same time that a shift in V, and function has a negative effect on his opportunity locus. We can observe only that, for reasonable values of all the eters determining Pzb and D,, the trader can he shown to relevant his demand for the risky asset as V, increases. In either in rd particular, the trader will be more likely to have decreasing demand $M the risky ass&: (I) the higher is l&isroostof participating in the active trading market, (2) the less precise are his priors concerning parameters relevant to either his demand or active trading market problems, (3) the larger is his excess demand, are his estimates of V. and V,, or (4) the risk werse he is. (5) the An important fatwe of Zt, tie trader’s demand function, for V,>O is its enoe on XI, the tradds initial holding of the asset. This property m~til a unique level of investment in the risky asset, XT, is reached. deternAte Xl: fiorn (4) by noting that (4) is a difference equation in 81119 (th where x,*1 Solving for X& the fixed piat of the diflerence equation
(11) ns of P, and P, and a given XT, (11) defines a unique pair
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R. Castanias, The role of morteyin a dyna& exchange economy
of long-run steady-state demands for money and the risky asset.. namely the pair (X/*, XT). Once the trader acheeves the position characterized by (X,*,X:), he will ,have no incentive to alter his holdings unless either his priors on PO or P1 change or his expected wealth, W& changes (11) is similar to the classical demand function resulting from the assumption that traders are price-takers at competitive equilibrium pricea As V,-+O in (4) and assuming that the trader’s current expected walth, WO,is the same as his expected wealth, IV& associated with the long-run steadystate,
or
(11) may be rewritten as x+ 1
(l./(2g)-XXf-XtP,,}dP =V, +(dp)? (13)
(11) gives the trader’s optimal steady-state demands (X&XT) for any givea level of current expected wealth as the limit of a sequence of transactions. It TC mains to examine how the trader’s demand function, &, compares to XT for any crsh~rpair (X,,X,) of holdings of money and the risky asset, for the same level of expected wealth. Given that the trader ma :I sta.rt at (X$,X i) with initial expected wealth WC, = X, + B,X 1 when XF > X 1 L nd W. = IV& (12) and (13) are seen to be similar but not identical since dP=Pr -Pa need not be the same in both equations. In the dynamic model, PO is the expected cost oi’ purchase when the trader uses the reservation price strategy and is equal to Pab, the excess demander’s ‘buying’ reservation price {see Proposition 1). If P1 is the ssamefor a given trader in both the perfectly competitive primtaking equilibritmi and the dynamic sea&h problem, -then the level of demand he desires will depend on how Ptb compares >to pEa,the assumed competitive equilibrium market clearing price. If the transactions market approaches an equilibrium in the longer run, it can be shown that Pgb-+Fo and long-run demand XT will approach the classical denand
R Castaiias, The de ~$mvney in a dynamic exchwe
ecmwmv
219
If V, is greater than zero, however, then
Xr, This result follows from the fact that X7 -Z1 is strictly in XI and that XT--Z,=O when X1=X!: if Vo>O, the excess will rrcde up to his optimal level of investment in the risky asset ucnce of transactions as Xt, r-+X? with each successive trana&ion. 43111~ when Ye=O will the trader trade directly to his optimal long-run level of investment in the risky asset no matter what his initial holding, X1, of the risky asset. The preceding results can be summarized as follows: For any given set of priors on PO and P, and a given level of expectea current wealth, We,
(1) as Vs-+O,the trader’s demand curve: for the ti,sky asset, Z,, converges to the competitive equilibrium demand curve of (12) in the long run as P$-+P& (2) the Excess Demander’s demand for the risky asset, %r, is strictly ler* than his optimal longer-run steady-state level of demand for the risk., asset, XT, if Ve~0. That is, the trader trades up to Xly by undertaking a tions in the risky itsset. If V+O the trader trades up transaction, no matter what his initial holding of the 8 the trader’s excess demand will be a decreasing y-state level of demand, Xt, is a decreasing in the active transactions market, an r~i~ion of his estimates of the value and an inc~asi~~~ function of his perception of the neity c~~~*~ni~ the value of the risky asset.
(3)
2.5. ~~n~Z~~~~i~~~ far Eh@ iIwnanill$i maney: Sbrt ad long rw to
the ~cy~~sia~ th Aversion, this p
y of liquidity preferen= or the Tobin
extends the analysis of liquidity
220
R. Castanias,The role of moneyin * dynamice.whangeeconomy
preference to consider explicitly the non-pecuniary services provided by money, as suggested by the analysis of Brunner and Meltzer, for an exchange economy where traders are imperfectly informed. Since demand for money is defined endogenously b;t the equation Z,=P,X,
+X,+?,P,,
(1%
the short-run effect will be for the excess demander to hold a higher percentage of his wealth in money, if V,>O, than he intends to hold in the longer run. The trader will undertake a dynamic sequential strategy for gradually increasing his level of holdings of the risky asset. The tracer will attempt to capitalize on the disparity of bid prices and the dynamic nature r,C the transactions market by ‘searching’ and by ‘trading up’ to his optimal long-run preferred holdings. Both factors will result in higher cash balances in the short run than the trader would have held had he been able to trade in a competitive equilibrium market at the same price, PO. The theory of liquidity preference developed here does not lead to the result that in a strict steady-state demand for hquidity preference must be zero. However, the residual long-run demand for cash, X7, has been shown to be different from the demand for cash, Z,, when the trader is trading from a position (X,,X,) different from (X,*,X:). Then money is providing nonpecuniary services to its holders as they adjust gradually to (X$X:). ‘IIe theory, however, is more or less ambiguous concerning the relation betwa,:n the longer-run demand for money and the trader’s optima! short-term position, or between the longer-run or short-term positions and the demand for money associated wit 1 a perfectly competitive market. But the mechanics of the model developed here can be used to address these issues when questions are further defined. For example, if increased disparity of opinion concerning the value of the risky asset across traders increases the cost of maintaining a given state of information about the asset then the expected cost of purchasing the asset pnd hence the reservation price will increase, leading perhaps to a- demand for cash exceeding competitive equilibrium levels, ah else being equal. The same results concerning the demand for money can be reached if the model is extended to include multiple risky alternatives to cash. The trader’s opportunity locus for expected wealth and risk will still be concave as in fig. 2, and a downward shift in the opportunity locus will still be associated with an increase in the demand fcr money. 3. Conclusionsand comments on areasfor furtherrcesesrcb When traders are allowS& to transact in dynamic sequential trading markets for assets for which they have an excess demand, they will choose to
bold cash, the assumed QltxI;um of exchange, in quantities in excess to their preferred longer-run cash holdins Liquidity preference is seen to he a on the many factors, such as costs of ens of the heterogeneity of and demand decisions. ImpOrtantiy, the traders longer-1~x1demand for money is seen to im Itebatjvc:to the h risky assets as his uncertainty and lack distribution of information causes a bearer of specialized non-pecuniary serGces to iti fnglder, in the sense that holding money allows the trader to ns while searching for ‘bargains’ in the active asset might theoretically serve this purpose, s this function ‘best’.l9 not answered in this study are the following. If traders could participate in markets for various pairs of assets, with each market structuti as the money-risky asset market of this paper, would they choose to liquidate their excess supplies of certain commodities or assets into particular intermediate assets for which the costs of trading are particularly Iow?~~ Will traders prefer intermediate assets for which there is widespread agreement as to value and/or the vector of relative prices of all other assets? our results s st that traders should prefer intermediate assets with low K’s, high t’s low Va’s.Will traders have an incentive to ‘cash out’ of assets for which they feel they have a comparative informational disadvantage, and take up short- and intermediate-run excess deniiind positions in assets for which they have a compa-ative informational advantage? An important implication of the dynamic active trading market strategy is that the trader is sensitive to costs and the precision of his information. Thus, a result of the sort suggested above, although beyond the scope of this paper, would not be surprising. What are the implicatians of ential demand strategies for the existence dynamic trading strategies and ts? It might he argued that the existence of intermediaries in financial m of traders with “sp.yiatized trading functions’ is an sfflcient response to the disparity of information which ~hara~t~ri~~~the markets of this paper. ss the role of and existence of money in an exchange Attempts to c.,nmade by Brunner and Mcltzcr, Harris and Jones.” AI e e*Listenerof money is socially and individually advantageous when traders trade assets in u d~~~trali~~d e!;ch:+:ge. All stress that traders nse of lxst at lowering coss of acquiring information and costs of t’ in the inbrmation. *@Thecosts of trading reXerred to here arc the ‘costs of participating in the active market’. ‘lntcrmdiatc assets’are assets‘leld to be Txchangcd later for assets wanted ‘directly’, or to fulfill a de&cd long-run position. *‘Brunnar and Mel&er (1972). Harris f I j76) and Jones (1976
222
R.Cwtuei~
The role of moneyin u d.ytwm&exchungwe WKVQ
will choose ‘intermediate’as&s for their usefulness in subsequent trades as well as their direct contribution to the utility of the holders. Harris was able to establish conditions under ,which the eventual outcome of the bikteral exchanges of his model will’lead to a Pareto Optimal allocation of resou But all of the preceding studies have focused on the general role of money in the exchange process or on the evetrtual outcome of the exch and not on the micro behavior of individual traders as th adjusting portfolios actually iakes place. The results of this that the sort of model developed here, if it can be generalized to address the questions raised above, may fill this gap. References Arrow, K., 1954, Toward a theory of price adjustment, in: M. Abramovitz, ed., Alloeatian of economic resoilrces (University of Califoniia Press, Stanford, CA) 41-52 Arrow, K., 1974, Limited knowleAIgeand economic analysis, American Economic Review 64 l10. Brunner, K. and A. Meltzer, 1972, The uses of money: Money in the theory of ait exchange economy, American mnomic Review 61,748-805. Castanias, R., 1978, Essays on the behavior of asset prices under uncertainty, Ph.D. thesis (Carnegie-MellonUniversity, Pittsburgh, PA). Fama, E., 1976,Foundations of finance (BasicBooks, New York). Grossman, S., 1976, On the efficiencyof competitive stock markets where traders have diverse information, Journal of Fiance 31, May, 573-585. Harris, M., 1976, Expectations and money in a dpamic exchange economy, Cam@-Hcliun University Working Paper (Pittsburgh, PA). Hart, O., 1974, On the existenti of equilibrium in a securities market, Journal of Economic Theory 9,29J-3 11. Jensen, M., 1972,Capital markets: Theory and evidence, Beil Journal of Economies 3,X7-398. Jones, R., 1976,The origin and development of media of exchange, Journal of Political bnomy 84,757-7X. Rothschild, .M., 1973, Models of market organisation with imperfect information, Jouma? of Political Ecolrlomy52, 1283-1308. Stigler, G., 1961,The economics of information, Journal of Political Economy 69, 21.3-225. Tobin, J., 1958, Liquidity preference as a behaviour towards risk, Review of’ numie Studies 26, 65-86.