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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).

We consider a continuum of agents that act in a two-period (

The paper is organized as follows. The rest of this section presents a literature overview. In Section

The model has two important ingredients:

the existence of heterogeneous beliefs (or expectations) biases among a continuum of agents;

the fact that the information is costly (the literature refers to “information acquisition” cost).

There are many models that explain how disagreements between agent estimations generate investment decisions and trading volume. The importance of the heterogeneity of opinions on the future value of a financial instrument and its use in speculation has been recognized as early as Keynes (see [

A model of speculative trading in a large economy with a continuum of agents with heterogeneous beliefs was presented in [

The difference-of-opinion approach (see [

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., [

We consider a two-period model,

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 filter; see [

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

We define

From a theoretical point of view, it is interesting to explore the case when

The only parameter the agent can control is the accuracy of the result, that is, the precision

Such a model is relevant in the case of high expense for information sources, for instance, news broadcasting fees. The expense also involves the reward of research personnel or the need for more accurate computer simulations.

Based on his estimations the agent

Hence, each agent is characterized by three parameters: his mean estimate

The agents buy or sell the security at time

The 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 final time

In order to describe the model for the market price, we introduce for any price

A price

The transaction volume at some price

A price

Let us recall the following result (see [

Suppose that functions

If

In addition to previous assumptions suppose that

There exists a unique

There exists a unique

Moreover

Recall that

For any function

Let us introduce an important assumption of this paper.

We say that a function

The quantity

The assumption

In order to model the choices of the agents, we consider that the agents maximize a CARA-type expected utility function (see [

Of course, the expected wealth of the agent at time

Thus, for a given price

Each agent

Under Assumption

Let

Under Assumption

Since

Note that

When

Let

Note that the formula

In order to prove the existence of an equilibrium we need the following auxiliary results (Lemmas

Under Assumption

Let

Under Assumption

Let

Since

Let

We say that

Let

To prove that

Let

Let

Then from the Beppo-Levi theorem, the following holds:

The property

Recall that

Under Assumptions

if

The assumption

Note that

We have already seen that

For the monotonicity of the demand, let

Recalling that

Since

Hence, demand is strictly decreasing. Previously we also proved that

The above results can be summarized in the following.

Under Assumptions

there exists at least a

suppose that

the functions

there exists a unique

Note that the results of [

If

Illustration of Remark

Since we assume the distribution

The critical value

lowering the perception of risk, that is, lower the

making

making

In this section we relax the assumption

Let

except for a countable set of values

as a consequence

To prove point

To prove point

Therefore, if

Point

For all purposes of calculating aggregate supply and demand we can thus replace

This result is particularly useful when

We obtain the following.

Suppose that Assumptions

Furthermore, consider the following.

If there exists

Suppose now that

(alternative 1) suppose that

the functions

the price

(alternative 2) if on the contrary one supposes that

then

We prove first point (I). If

The point (IIa) follows from the discussion above.

To prove (IIb) we need to analyze in greater detail the values of

If

From (

In general, the price

Suppose that Assumptions

The proof follows from the remark that, except possibly for a null measure set of values

We follow [

In the Grossman-Stiglitz model agents can either pay nothing and have a precision

The unsigned demand is

We describe in the following the relationship between the cost function

Suppose that

Assume that

In particular if

(all are lateral derivatives) then

Note that if

If

(A) We first show that, except for a countable set of values

The demand and supply of the agents are monotonic and given for

Let

It has been proved that

Let

Similarly we prove that

(B) We prove that (

Suppose that

Alternative

We only prove

Illustration of the proof of Theorem

The main focus of this work is to establish the existence of an equilibrium and its optimality in terms of trading volumes for the model in Section

The authors declare that there is no conflict of interests regarding the publication of this paper.