Existence of an Equilibrium for Lower Semicontinuous Information Acquisition Functions

We consider a two-period model in which a continuum of agents trade in a context of costly information acquisition and systematic heterogeneous expectations biases. Because of systematic biases agents are supposed not to learn from others’ decisions. In a previous work under somehow strong technical assumptions a market equilibrium was proved to exist and the supply and demand functions were proved to be strictly monotonic with respect to the price. Here we extend these results under very weak technical assumptions. We also prove that the equilibrium price maximizes the trading volume and further additional properties (such as the antimonotonicity of the trading volume with respect to the marginal information price).


Introduction
We consider a continuum of agents that act in a two-period (∈{ 0 , } ) market consisting of a single asset of value .he value is constant and deterministic but unknown to the agents.Each agent constructs an estimation for in the form of a normal random variable with known mean and variance.he numerical value of the mean, which is not necessarily and as such can be interpreted as a systematic bias, is given by the estimation method and cannot be changed.
H o w e v e r ,t h ev a r i a n c ec a nb er e d u c e da tt i m e=0 by paying a cost, which is a known deterministic function of the variance to be attained.Each agent uses a CARA utility function and constructs the function mapping each triplet (consisting of the market price, the estimation mean, and the estimation variance) to the optimal number of units to trade. he sum of all such functions from all agents results at time =0 in aggregate market demand and supply functions; the price of the asset is chosen to clear the market (we prove in particular that such a price exists and is unique).his price can be diferent from the real value and in practice it will.he agents close their position at inal time = .his paper investigates the following questions: existence of an equilibrium, continuity of supply and demand functions, and interpretation of the equilibrium price as the value maximizing the liquidity (trading volume). he paper is organized as follows.he rest of this section presents a literature overview.In Section 2 the model is explained and the fundamental Assumption 3 is introduced.In Sections 3 and 3.1, we prove the existence of an equilibrium and important properties of the liquidity (here deined as the transaction volume); in particular we prove that the equilibrium price maximizes the trading volume.We apply our results to a Grossmann-Stiglitz framework in Section .Finally, in Section 5 we show that the liquidity is inversely correlated with the marginal price of information.
1.1.Literature Overview.he model has two important ingredients: (i) the existence of heterogeneous beliefs (or expectations) biases among a continuum of agents; (ii) the fact that the information is costly (the literature refers to "information acquisition" cost).
here are many models that explain how disagreements between agent estimations generate investment decisions andtradingvolume.heimportanceoftheheterogeneityofopinions on the future value of a inancial instrument and its use in speculation has been recognized as early as Keynes (see [1]) who invokes the "beauty contest" metaphor to explain how speculators infer the future (consensus) price.
A model of speculative trading in a large economy with a continuum of agents with heterogeneous beliefs was presented in [2,3]( s e ea l s ot h er e f e r e n c e sw i t h i n ) .h e y demonstrate the existence of price ampliication efects and show that the equilibrium prices can be diferent from the rational expectation equilibrium price.It is also shown that trading volume is positively related to the directions of price changes and they explain the recurrent presence of diverse beliefs.We also refer to [ ] and references within for a survey on how heterogeneous beliefs among agents generate speculation and trading.
he diference-of-opinion approach (see [5,6]) does not consider "noise agents" but on the contrary obtains diverse posterior beliefs from the diferences in the way agents interpret common information.hey focus on the implications of the dispersion in beliefs on the price level or direction.Yet another diferent method explains diverse posterior beliefs by relaxing the assumption of a common prior distribution (see [7]); the authors also model the learning process which enables a convergence towards a common estimation when more information is available.Such a framework was invoked for modeling asset pricing during initial public oferings, but not for other speculative circumstances.Finally, Pagano [8] analyze the implications of low liquidity in a market and propose appropriate incentive schemes to shit the market to an equilibrium characterized by a higher number of transactions.
An important advance has been to recognize that the dynamics of the information gathering is important; it was thus established how the presence of private information and noise (liquidity) agents interact with market price and volume (see, e.g., [9][10][11], for recent related endeavors).More speciically it was recognized (the so called "Grossman-Stiglitz paradox") that it is not always optimal for the agents to obtain all the information on a particular asset.his remark is of importance in our paper in the following because, as explained in Section 2,ourmodelallowseachagenttochoose his level of precision related to the estimation of the true v a l u eo ft h et r a d e da s s e t .I nt h ec l a s s i c a lp a p e ro f [ 12]a n d in subsequent related works [13][14][15][16][17][18]aframeworkisproposed where the information is costly and agents can pay more to lower their uncertainty on the future value of the risky asset.Verrechia derives a closed form solution which requires some particular assumptions.hese include the convexity of the cost function with respect to the precision (the precision being the inverse of the estimate's variance).On the contrary our cost function is here only lower semicontinuous.Our approach also difers in a more fundamental way in that we suppose that heterogeneity of estimations is given but arbitrary, that is, not centered around the correct price.Moreover, the Verrecchia model relies on the heterogeneity of risk tolerances in the CARA utility function while in our work the price formation mechanism does not require such an assumption, the heterogeneity in estimations being enough.Also, in this model, the endowments of the agents do not play any role and in particular are not required to obtain an equilibrium.he paper extends a previous work [19]w h e r e stronger technical assumptions were invoked.

The Model
We consider a two-period model, =0and =,inwhicha risky security of value is traded.he value is unknown to the agents and each participant in the market constructs an estimate for at =0 , being a random variable.For simplicity, we suppose that has a normal distribution and that 1 and 2 are independent if 1 and 2 are two distinct agents (this independence assumption is motivated by the existence of an individual bias for each agent as explained below).Also, we assume that the mean and the variance of are, respectively, given by and ( ) 2 ,bo thm ea na n d variance being known to the agent .Asin [12]weworkwith the precision =1/( ) 2 instead of the variance ( ) 2 .
Many estimation procedures can output results in the form of a normal variable with known mean and variance, the most known example being a Kalman-Bucy ilter; see [20]for details.
Note that we do not model here the riskless security, but everything works as if the numeraire was the riskless security; from a technical point of view this allows setting the interest rate to zero.
An important remark is that each agent has his own bias attached to the estimate becausehehashisownprocedure to interpret the available information.It may be due to personal optimism or pessimism (e.g., the agent is a "bull" or "bear") or may be correlated with some exogenous factors, such as overall economic outlooks, commodities evolution, and geopolitical factors, which each agent interprets with a speciic systematic bias.See also the cited references for additional discussion on how agents interpret the information they obtain.We assume that the bias −of agent does not depend on the precision to be attained and only depends on the agent; the value associated to an agent is known only by him.he agent does not inluence in any way during the process of forecasting; his forecasting process is not inluenced by other agents' decisions; that is, there is no collective learning in this model.Hence, two diferent agents 1 and 2 have generically diferent biases 1 − and 2 − and thus diferent estimation averages 1 and 2 .his is not a collateral property of the model.It is instead the mere reason for which the agents trade.hey trade because they have diferent (heterogeneous) expectations on the inal value of the security.We deine () to be the distribution of among the agents; neither the law of the distribution () nor any moments or statistics is known by the agents.We also introduce the expected value with respect to (⋅),w h i c hi s denoted by E ;s e ea l s o [ 21] for related works on empirical estimation of such a distribution .W ed on o ta s s u m et h e law of to be normal or have particular properties (except technical Assumption 9).
From a theoretical point of view, it is interesting to explore the case when E () = .hi sm e a n st h a tt h ea v e r a g e estimate is , so that the agents are neither overpricing nor underpricing the security with respect to its (unknown) value.However, we will see that this does not necessarily indicate that the market price is .
he only parameter the agent can control is the accuracy of the result, that is, the precision .However,thishasacost: the agent has to pay ( ) to obtain the precision .he precision cost function : R + → R + is deined on positive numbers but if needed we set by convention () = ∞ for any <0 .S e ea l s o [ 22] for an example involving a power function and [18] for a structural model to motivate such a function.
Such a model is relevant in the case of high expense for information sources, for instance, news broadcasting fees. he expense also involves the reward of research personnel or the need for more accurate computer simulations.
Hence, each agent is characterized by three parameters: his mean estimate ,theprecision oftheestimate(that comes at a cost ( )), and the quantity of traded units, .
h ea g e n t sb u yo rs e l lt h es e c u r i t ya tt i m e=0 by formulating demand and supply functions depending on the price.he market price at time =0is chosen to clear the aggregate total demand/supply.Remark 1. he price that clears the market is also called market equilibrium price.Note however that the uniqueness of the equilibrium is, at this stage, not proved.
We set the investment horizon of all agents to be the inal time = which is the time when each agent liquidates his initial position.Each agent supposes that this inal transaction takes place at a price in agreement with his initial estimation.
In order to describe the model for the market price, we introduce for any price >0the basic notions of total supply () and total demand () deined as where for any real number we deine + = max{, 0}, − = max{−, 0}.
Ap rice * such that ( * )= ( * ) is said to clear the market.From the deinition of (⋅) and (⋅) in (1)t h i si s equivalent to saying that E () = 0; that is, at the price * , the overall (signed) demand is zero.Note that such a price may not exist or may not be unique.Hence, one of the goals of the paper is to prove existence and uniqueness of * .
hetransactionvolumeatsomeprice is the number of units that can be exchanged at that price and is deined as follows: TV () = min {(),()}. ( Ap r i c e * for which TV(⋅) reaches its maximum is of particular interest because it maximizes the total number of asset units being exchanged.Note that such a price may not exist and may also be nonunique.
(B) In addition to previous assumptions suppose that () is strictly increasing and lim →∞ () > 0,w h e r e a s () is strictly decreasing and such that (0) > 0.
hen the following statements are true.
Recall that :R + → R + ∪{ + ∞ }is said to be lower semicontinuous (denoted by "l.s.c.") if for any ( Remark . he quantity (0) <∞represents the residual cost, when precision approaches zero, to enter the market.It is notrelatedtotheprecision(becausethereisnoneinthelimit) but to the ixed costs to trade on the market (independent of the quantity).A market with ininite ixed costs is not realistic.
he assumption (0) <∞implies, by lower semicontinuity, that (0) < ∞ and is realistic in that it demands that the price of zero precision be inite.
In order to model the choices of the agents, we consider that the agents maximize a CARA-type expected utility Journal of Applied Mathematics function (see [23]);thatis,iftheoutputistherandomvariable , they maximize E(− − ).N o t et h a ti f is a normal random variable with mean E() and variance var(),then maximizing E(− − ) is equivalent to maximizing the meanvariance utility function E() − ((/2)var()).We refer to (6) for the treatment of degenerate normal variables with ininite variance.he parameter ∈R + is called the risk aversion coeicient.Note that all agents have here the same utility function; see for instance [2 , 25] who argue that diferences in preferences are not a major factor in explaining themagnitudeoftradeinspeculativemarkets.
Ofcourse,theexpectedwealthoftheagentattime=is afunctionof and .Itiscomputedundertheassumption that each agent enters the transaction (buys or sells) at time =0at the market price and exits the transaction (sells or buys) at time =at a price coherent with his estimation; that is, we condition on the available information at time = 0.hus,foragivenprice, which is not necessarily equal to the market equilibrium price P, the average expected wealth at time =of the agent denoted by is given by = ( −)−( ). he variance of the wealth, denoted by hus, for a given price (not necessarily the market equilibrium price P)t h ef a c tt h a ta g e n t optimizes his CARA utility function is equivalent to saying that he optimizes with respect to and his mean-variance utility:

Existence of the Transaction Volume
Each agent is characterized by his own bias .heagentsconsider the market price as being ixed, which means they cannot inluence it directly.hey do not know any statistics on so the market price is not directly informative, but the acquired information is.herefore, their strategy depend on two values: the bias and the market price .Under Assumption 3, the agent chooses the optimal pair of precision opt (, ; ) and demand/supply opt (, ; ), that is, the value of the pair maximizing the following expression: so that J ( opt (, ; ) , opt (, ; )) ≥ J (,), ∀, ≥ 0.
In order to prove the existence of an equilibrium we need the following auxiliary results (Lemmas 7-11).
.R e c a l lt h a t optimizes −()with respect to .hen hus, ≥ 0, which gives the conclusion.
In particular, if is ixed, then the set of such that opt (, ) is discontinuous with respect to is countable.An analogous result holds if is ixed.
his implies that 0 +is also a maximum for 0 − ().Fromthiswededucetha t 0 has at least two distinct maximums, 0 and 0 +.Let be such that has at least two distinct minimums 1  and 2 with 1 < 2 ; we associate to arationalnumber such that ∈] 1 , 2 [.T ak e and such that ̸ = ;t o ix notations suppose <. hen by the previous result 2 ≤ 1 ;m oreover < 2 ≤ 1 < ;t hatis, ̸ = .hus the set of such that has at least two distinct minimums is of cardinality smaller than the cardinality of Q,tha tis,a t most countable.Since continuity can only fail when has nonunique maximum, the conclusion follows.
Let be a sequence increasingly converging to .Forany ,thesetof such that opt (, ) is discontinuous is at most hen from the Beppo-Levi theorem, the following holds: hisprovessequentialcontinuityof() and thus its continuity. he monotonicity is a consequence of the monotonicity of opt (, ).his result also holds for the demand (), recalling that −() is increasing and lower-bounded. he property (0) = 0 is trivial.To prove lim →∞ () = 0 it is suicient to use the above upper bound for opt (, ) Recall that () is increasing on [0, +∞[ but in order to use heorem 2 we need to prove its strict monotonicity.
Proof.Note that ( (0)) + <∞implies in particular continuity of () at =0 .L e t t i n g and be such that > >≥0, Hence, Note that < <implies that (( − ) − −(− ) − )>0.So, in order to prove the strict inequality in the estimation above, it is suicient to prove that opt (, ) > 0 with in the support of .Y et opt (, ) = arg max herefore we only need to prove that there exists such that − () > 0 with in the support of .As u i c i e n t condition is that the upper limit of derivative of − () at =0be strictly positive.his means −( (0)) For the monotonicity of the demand, let and be such that >> .hen ( Since opt is decreasing for >> ,wehave Hence, Note that >> implies that ( − ) + −(− ) + <0 .Hence, demand is strictly decreasing.Previously we also proved that lim →+∞ () = 0.
hat is, as stated in [27], the function has a supporting hyperplane at 0 .S i n c e has a supporting hyperplane at 0 this implies that ( 0 )= * * ( 0 );r e c a l lt h a t * * is the convex hull of , that is, the largest convex function such that * * ≤.Hence,recallthatforanyfunction * * * = * , We thus obtained that 0 is a maximum of 0 − * * ().herefore, if is replaced by * * , the minimization problem involving gives the same solution, except possibly ac o u n t a b l es e to fv a l u e s where the maximum is attained (either for or * * )inmorethanonepoint.
For all purposes of calculating aggregate supply and demand we can thus replace by * * , that is, replace by its convex hull.herefore, one can work as if was convex.We obtain the following.heorem 18. Suppose that Assumptions 3 and 9 are satisied.hen there exists at least one price P ≥0such that (P)≥(), ∀≥0.
his price also satisies Furthermore, consider the following.
Moreover the price P satisfying (35) is unique.
In general, the price P has an implicit dependence on the cost function (⋅) with no particular properties.But when the distribution is completely symmetric around some particular value 1 ; we obtain the following result.

Remark 17 .
hisresultisparticularlyusefulwhen(0) ̸ = (0) because in this situation ( (0)) + =∞ .he no n ec a n n o t use the previous results that guarantee the uniqueness of the market clearing price.When is replaced by * * it can be shown that ( (0) ) + becomesiniteandtheresultsapplyfor * * .H o w e v e rheorem 16 allows recovering the results for the initial function and obtaining the full information on thesupplyanddemandfunctionsandonthemarketprice.