Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness

Zhen Liu

doi:10.1515/bejte-2016-0027

Published by De Gruyter May 18, 2017

Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness

Zhen Liu

From the journal The B.E. Journal of Theoretical Economics

https://doi.org/10.1515/bejte-2016-0027

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Abstract

As the cost of financial information dissemination continues to decline, investors, firms, and regulators are gradually adopting the principle of fair disclosure, which requires no preferential public disclosure. We use a simple model to examine the impact of this change on information acquisition with two alternative assumptions: (1) Investors have symmetric awareness about the underlying uncertainties, or (2) this awareness is asymmetric among them. Under the first assumption, the change reduces information asymmetry among investors and induces acquisition of high-quality information. Under the second assumption, however, the reduction of information asymmetry may be limited, and information acquisition is either reduced or less efficient. Specifically, investors with high awareness may either acquire high-quality information at a higher cost or not acquire it; investors with low awareness only acquire low-quality information. The loss in overall information quality is greater when awareness asymmetry is moderate than when it is high or low; this causes information asymmetry between the insiders and outside investors as a whole. These results offer explanations for intriguing empirical findings regarding the effect of a recent accounting regulation (Regulation Fair Disclosure).

Keywords: asymmetric awareness; conference call; financial information; information acquisition, fair disclosure; regulation fair disclosure; selective disclosure; unawareness

JEL Classification: D61; G14; G38; K22; L51; D8; M4

Appendix

Derivation of the Equilibrium in the Trading Period

By Equation (3), when the quoted price is λ, the professional’s expected profit conditional on public information τU is

EU(πI∣ω)=k∫0λ(λ−x)dG(x∣τU(ω)).

Using integration by parts, we simplify it to

(8)EU(πI∣ω)=k∫0λG(x∣τU(ω))dx.

Now we are ready to prove the following lemmas.

Lemma 10.

The following is true for the profit functionEU(πI∣ω) for every ω∈Ω:

It is continuous and increases in λ.
It increases in k.
It increases when τI becomes finer.

Proof

i. and ii. The results follow from Expression (8) and that G(x∣τU(ω))∈[0,1].

iii. Let τ1,τ2 be two information structures and τ1 is finer than τ2. Let the corresponding probability distributions of the professional’s valuation conditional on public message τU be G1(x∣τU(ω)) and G2(x∣τU(ω)) for τ1 and τ2, respectively. Because E(v∣τ2(ω))=E[E(v∣τ1(ω′))∣τ2(ω)], distribution function G1(⋅∣τU(ω)) is a mean-preserving spread of G2(⋅∣τU(ω)). This implies that G2(⋅∣τU(ω)) is second-order stochastically dominant over G1(⋅∣τU(ω)) and

∫0λG1(x∣τU(ω))dx≥∫0λG2(x∣τU(ω))dx,

for every λ≥0.^[16] By Expression (8), the result follows. □

Lemma 11.

The following is true for the optimal pricing ruleλ∗(ω)for everyω∈Ω:

It exists and is unique.
It decreases when τI becomes finer.

Proof.

i. Define function d(λ)≡q[EU(v∣ω)−λ]−EU(πI∣ω).

First, d(λ) is continuous in λ by the first claim of Lemma 10. Second, if λ=0, then EU(πI∣ω)=0, and d(λ)=qEU(v∣ω)≥0. If λ=EU(v∣ω), then

d(λ)=−EU(πI∣ω)≤0.

By the Intermediate Value Theorem, there is at least one λ∈[0,EU(v∣ω)] such that d(λ)=0. Hence, by Condition (5), the pricing rule λ∗(ω) exists. By Lemma 10, EU(πI∣ω) increases in λ, therefore d(λ)strictly decreases in λ. Then the uniqueness is obtained.

ii. We show it by contradiction. Suppose λ∗(ω) increases. Then by Lemma 10, EU(πI∣ω) increases. It contradicts that in Condition (5),

q[EU(v∣ω)−λ]=EU(πI∣ω),

because the LHS strictly decreases. □

Lemma 12.

Fix any pair of messagest1, t2 in {τU(ω)}ω∈Ω and let λibe the equilibrium price corresponding toti, i=1,2. If G(⋅∣t1) is first-order stochastically dominant over G(⋅∣t2), then λ1≤λ2.

Proof.

By definition, because τU is coarser than τI, for each ω∈t∈τU and all ω′∈Ω,

EU(v∣ω)=EU(EI(v∣ω′)∣t)≡∫0∞xdG(x∣t).

First-order stochastic dominance implies ∫0∞xdG(x∣t1)≥∫0∞xdG(x∣t2). Therefore, for each ω1∈t1 and ω2∈t2,

(9)EU(v∣ω1)≥EU(v∣ω2).

FOSD also implies that G(x∣t1)≤G(x∣t2), for every x≥0. Therefore,

(10)∫0λG(x∣t1)dx≤∫0λG(x∣t2)dx.

If λ1 is the equilibrium price at ω1∈t1, then by Condition (5),

q[EU(v∣ω1)−λ1]−k∫0λ1G(x∣t1)dx=0.

Suppose λ1<λ2. By Equations (9) and (10), we have

This contradicts λ2 being the equilibrium price at ω2∈t2. Therefore, λ1≤λ2.

Next, we present a version of Chebyshev’s integral inequality by Wagener (2006). We will use a modified version of it for our proof.

Theorem 13.

(Wagener 2006) Let g,h:[a,b]→R and F:[a,b]→[0,1] be a distribution function. Suppose that g is monotonically increasing. Define HF:(a,b]→R,HF(t)=∫ath(s)dF(s)/∫atdF(s). If

HF(t)≤HF(b)

for all t∈(a,b], then

∫abg(t)h(t)dF(t)≥∫abg(t)dF(t)∫abh(t)dF(t).

The modified version of Theorem 13 is below.

Corollary 14.

Let g′,h:[a,b]→R and F:[a,b]→[0,1] be a distribution function. Suppose that g′ is decreasing and h is increasing, then

∫abg′(t)h(t)dF(t)≤∫abg′(t)dF(t)∫abh(t)dF(t).

Proof.

Because h is increasing, by definition, HF(t)≤HF(b) for all t∈(a,b]. Because g′ is decreasing, define g(t)=c−g′(t), where c is any constant. Therefore, g is monotonically increasing. By Theorem 13,

∫abg(t)h(t)dF(t)≥∫abg(t)dF(t)∫abh(t)dF(t).

This implies that

(11)∫ab(c−g′(t))h(t)dF(t)≥∫ab(c−g′(t))dF(t)∫abh(t)dF(t).

Expanding the LHS of Inequality (11), we have

∫abch(t)dF(t)−∫abg′(t))h(t)dF(t);

expanding the RHS, we have

∫abch(t)dF(t)−∫abg′(t)dF(t)⋅∫abh(t)dF(t).

Therefore, Inequality (11) implies that

∫abg′(t)h(t)dF(t)≤∫abg′(t)dF(t)∫abh(t)dF(t).

□

When public information becomes finer, the information asymmetry between the professional and others is reduced. As a consequence, the average market price increases.

Lemma 15.

Fix any pair of information structures τ1 and τ2, such that τ1 is finer than τ2. When τU=τ1, let λ1(ω) be the equilibrium price at ω∈t1∈τ1; when τU=τ2, let λ2(ω′) be the equilibrium price at ω′∈t2∈τ2. Then

λ2(ω′)≤E(λ1(ω)∣t2).

Proof.

By Condition (5), since λ1(ω) is the equilibrium price at ω∈t1∈τ1 when τU=τ1,

q[EU(v∣t1)−λ1(ω)]−k∫0λ1(ω)G(x∣t1)dx=0,

Then we integrate the LHS of this equation about ω with conditional probability f(ω∣t2). Because τ1 is finer than τ2, and because ω∈t1, EU(v∣t2)=∫ΩEU(v∣t1)f(ω∣t2)dω. Therefore,

q[EU(v∣t2)−E(λ1(ω)∣t2)]−k∫Ωf(ω∣t2)∫0λ1(ω)G(x∣t1)dxdω=0.

By Condition (5), since λ2(ω′) is the equilibrium price at ω′∈t2∈τ2 when τU=τ2,

q[EU(v∣t2)−λ2(ω′)]−k∫0λ2(ω′)G(x∣t2)dx=0.

The above two equations imply that if λ2(ω′)>E(λ1(ω)∣t2) then

(12)∫Ωf(ω∣t2)∫0λ1(ω)G(x∣t1)dxdω−∫0λ2(ω′)G(x∣t2)dx>0.

Now note that because {λ1(ω)}ω is measurable with respect to τi,

(13)∫Ωf(ω∣t2)∫0λ1(ω)G(x∣t1)dxdω=∫τ1f(t1∣t2)∫0λ1(t1)G(x∣t1)dxdt1,

where f(t1∣t2)=∫ω″∈t1f(ω″∣t2)dω″. Also, because τ1 is finer than τ2, G(x∣t2)=∫τ1G(x∣t1)f(t1∣t2)dt1. Therefore,

(14)∫0λ2(ω′)G(x∣t2)dx=∫0λ2(ω′)∫τ1G(x∣t1)f(t1∣t2)dt1dx=∫τ1f(t1∣t2)∫0λ2(t2)G(x∣t1)dxdt1.

Substituting the LHS of Inequality (12) by Equations (13) and (14), we find that

∫Ωf(ω∣t2)∫0λ1(ω)G(x∣t1)dxdω−∫0λ2(ω′)G(x∣t2)dx=∫τ1f(t1∣t2)∫λ2(t2)λ1(t1)G(x∣t1)dxdt1≤∫τ1[λ1(t1)−λ2(t2)]G(λ1(t1)∣t1)f(t1∣t2)dt1≤∫τ1[λ1(t1)−λ2(t2)]f(t1∣t2)dt1∫τ1G(λ(t1)∣t1)f(t1∣t2)dt1=(E(λ1(t1)∣t2)−λ2(t2))∫τ1G(λ1(t1)∣t1)f(t1∣t2)dt1=(E(λ1(ω)∣t2)−λ2(ω′))∫τ1G(λ1(t1)∣t1)f(t1∣t2)dt1

The first inequality is true because function G(x∣t1) increases in x. The second inequality follows from Corollary 14 by defining h(t1)≡λ1(t1)−λ2(t2) and g(t1)≡G(λ1(t1)∣t1). By Assumption 4, distributions G(⋅∣t1) can be ordered by first-order stochastic dominance. Therefore, by Lemma 12, λ1(t1) can be ranked by the opposite order. The result is that {h(t1)}t1 can be ranked by an increasing order and {g(t1)}t1 by a decreasing order. The last expression is negative if λ2(ω′)>E(λ1(ω)∣t2). That leads to a contradiction about Inequality 12. This completes the proof.

Proof of Proposition 5

It follows from Fact 3 and Lemma 11.
1. The second claim of Lemma 10 leads to the monotonicityin k. When τI becomes a finer information partition, bythe second claim of Lemma 11, the equilibrium marketprice decreases. Then by the first claim of Lemma 10,the professional’s ex ante expected profit increases.
2. Under Assumption 4, by Lemma 15, when τUbecomes finer, the expected market price increases. Then the exante expected market price Eω(λ(ω)) also increases.Market makers’ ex ante profits from small investors’ liquiditytrade, q(Eω(v(ω))−Eω(λ(ω))), mustdecrease. By Condition 5, the professional’s ex anteexpected profit decreases.
3. When τU=τI, there is no information asymmetry. Marketmakers know exactly what the professional knows. Being competitive,market makers set prices equal to the expected value of one shareat each state. Therefore, the professional’s trading profit is zero.

Proof of Proposition 7

If q(E(v)−λ(τh))≤Cs, then ΠI(k,τ0,τh)≤Cs. The professional does not profit from acquiring information τh in the pre-FD era, so she does not acquire. Because Cs<Cp, ΠI(k,τ0,τh)<Cp. The professional does not profit from acquiring τh in the FD era when public information is τ0. By Proposition 5, ΠI(k,τ0,τh)≥ΠI(k,τl,τh). Then the same no-acquisition conclusion holds when public information is τl. As to small investors, they expect the professional to earn a profit as high as ΠI(k,τ0,τl) due to unawareness. Again by Proposition 5, ΠI(k,τ0,τl)≤ΠI(k,τ0,τh)≤Cp, and they correctly expect that the professional does not acquire privately. Therefore, in the FD era, small investors are indifferent between making and not making inquiries. There is no information asymmetry in either era.

If q(E(v)−λ(τh))∈(Cs,Cp], then ΠI(k,τ0,τh)∈(Cs,Cp]. The professional uses selective forums in the pre-FD era. By the same argument used before, the professional does not privately acquire τh in the FD era. Small investors expect the professional to earn a profit as high as ΠI(k,τ0,τl) due to unawareness. Again by Proposition 5, ΠI(k,τ0,τl)≤ΠI(k,τ0,τh)≤Cp, so they correctly expect that the professional does not acquire privately. Therefore, in the FD era, they are indifferent between making and not making inquiries. There is information asymmetry in the pre-FD era but not in the FD era.

If q(E(v)−λ(τh))>Cp, then ΠI(k,τ0,τh)>Cp. The professional uses selective forums in the pre-FD era and acquires τh privately in the FD era if small investors do not make inquiries. Small investors expect the professional to privately acquire τl in the FD era if ΠI(k,τ0,τl)=q(E(v)−λ(τl))>Cp and τU=τ0, therefore they will make inquiries. Otherwise, they are indifferent between two options. When small investors make inquiries, the professional produces τh privately in the FD era if and only if ΠI(k,τl,τh)=q[E(v)−[E(λ)](τl,τh)]>Cp. There is information asymmetry in the pre-FD era. However, in the FD era, there may still be information asymmetry, and only the low-quality information is produced when there is no information asymmetry. □

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Published Online: 2017-5-18

Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness

Abstract

Appendix

Derivation of the Equilibrium in the Trading Period

Lemma 10.

Proof

Lemma 11.

Proof.

Lemma 12.

Proof.

Theorem 13.

Corollary 14.

Proof.

Lemma 15.

Proof.

Proof of Proposition 5

Proof of Proposition 7

References

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