This paper models limited attention as incomplete usage of publicly available informa-tion. Informed players decide whether or not to disclose information to an audience
of observers who sometimes neglect either disclosed signals or the implications of non-disclosure. In equilibrium, we find that observers are unrealistically optimistic, disclosure is incomplete, neglect of disclosed signals increases disclosure, and neglect of a failure to disclose reduces disclosure. We also find that these insights extend to a setting in which observers choose ex ante how to allocate their limited attention. In a setting with multiple arenas of disclosure, we find that disclosure in one arena affects perceptions in fundamentally unrelated arenas, owing to cue competition, salience, and analytic interference; and that disclosure in one arena can crowd out disclosure in another.
We consider the implications of limited attention and the resulting credulity of ob-servers for disclosure regulation. Law and regulation in the U.S. require firms to reveal information in financial reports, and to disclose other relevant information. Such reg-ulation is not needed in the classic unravelling models of disclosure, in which rational observers, through appropriate skepticism, induce full disclosure, and full disclosure is a good thing. Extensions with costly disclosure can create a rationale for regulation, but the recommendation is very simple: force additional disclosure only if the social benefits exceed the costs.
In contrast, limited attention suggests that the balance of considerations is more complex, even when disclosure is costless. On the one hand, informed parties may conceal information in the hope of exploiting the inattention and credulity of observers.
This puts regulation of disclosure on the table. However, we find, paradoxically, that regulations designed to force greater disclosure can make perceptions less accurate. For example, we find that forced disclosure in one arena can crowd out disclosure in another, and thereby can reduce welfare. Thus, even if forced disclosure in one arena creates benefits to observers in that arena, there is no presumption that forced disclosure is socially desirable.
Furthermore, forcing simultaneous disclosure in multiple arenas can also reduce wel-fare, for two reasons. First, greater forced disclosure can increase what we call analytic interference, wherein a disclosed signal in one arena distracts observers from analyz-ing the reasons for a failure of a player to disclose in the other arena. Second, even if complete disclosure be enforced in both arenas, greater disclosure can cause greater cue competition between disclosures— observers have trouble attending to both signals.
Even though more information is publicly available, observer perceptions may on average be less accurate.
Thus, to determine whether forcing greater disclosure in one or in many arenas will improve welfare, policymakers have a challenging task. Evaluating such a policy
requires an assessment of the relative importance of the different arenas, the precision of the information that might be disclosed, and the salience of the different arenas.
An issue not captured in our model that deserves further exploration is that in reality an informed party can misrepresent by ‘disclosing’ an incorrect value for his information signal. This issue is highlighted by U.S. and global corporate accounting scandals such as those involving Enron, WorldCom, and Parmalat. Limits to investor attention presum-ably affect the incentives for firms to engage in fraud, or milder shading of the truth. An interesting further direction would be to analyze how different regulatory policies influ-ence the incentives for firms to exploit inattentive observers through misrepresentation as well as simply not disclosing.
We close by emphasizing that the general approach to limited attention offered here may be applicable to a variety of human transactions. As argued in Subsection 2.2, there are various interactions in which informed players seem to take advantage of inattention to manipulate the perceptions of others. Some possible directions that merit further exploration include the advertising of products to consumers, the reporting of firms’
financial condition to investors, the presentation of information by political activists, and the presentation of personal information by individuals in their everyday lives. The approach described here—in which observers have limited ability to attend to public signals or to features of the strategic environment, salience parameters influence which signals or environmental features are attended to more, and the occurence of an event is more salient than non-occurence—may be helpful in capturing parsimoniously the effects of limited attention in a range of contexts.
Appendix
Proof of Proposition 1: We now show that if αW < αD there is no equilibrium with full disclosure. The most skeptical inference in any equilibrium that could be drawn about non-disclosure would be ˆθ = θ. Thus, observers’ perception of a withholding player satisfies
θˆW ≥ (1−αW)E[θ] +αWθ. (35)
The perception upon disclosing for a type θ =θ+ǫ is
θˆD = (1−αD)E[θ] +αD(θ+ǫ). (36) Thus,
θˆW −θˆD ≥ (αD−αW) (E[θ]−θ)−αDǫ
> 0, (37)
where the last inequality holds, for given αW and αD, by choosingǫ to satisfy 0< ǫ < (αD−αW)(E[θ]−θ)
αD .
Thus, in any equilibrium there exists a set of types with θ < θ+ǫ, ǫ small that prefer not to disclose. Furthermore, so long asγ <1,θ∗ < E[θ]. Otherwise, there would be an above-average type (θ > E[θ]) who prefers not to disclose. This is impossible; if he does not disclose, the average perception of his type is below E[θ], by (3), whereas if he does disclose, he is correctly perceived as having information θ > E[θ]. k
Proof of Proposition 3:
E[ˆθ−θ] = Z θ∗
θ
{(1−αW)E[θ] +αWE[θ|θ < θ∗]−θ}f(θ)dθ +
Z θ
θ∗
{(1−αD)E[θ] +αDθ−θ}f(θ)dθ
= {(1−αW)E[θ] +αWE[θ|θ < θ∗]}F(θ∗)−E[θ|θ < θ∗]F(θ∗) +(1−αD)E[θ][1−F(θ∗)]−(1−αD){E[θ]−E[θ|θ < θ∗]F(θ∗)}
= (αD−αW)F(θ∗) (E[θ]−E[θ|θ < θ∗]). (38)
SinceαD > αW and E[θ]> E[θ|θ < θ∗], E[ˆθ−θ]>0.
Proof of Proposition 5: By (3)–(4), we can rewrite the MSE as E[(ˆθ−θ)2] =
To see how the mean squared error varies withαD, substitute θ∗ of the uniform [0,1]
distribution case from (21) into the MSE formula, (11), and differentiating with respect to αD. This yields
which can be either positive or negative: dE[(ˆθ−θ)2]/dαD =−0.0354938 when (αW, αD) = (0.4,0.5), and dE[(ˆθ−θ)2]/dαD = 0.000708395 when (αW, αD) = (0.4,0.95). k
Proof of Proposition 8: To condition on DA, we substitute αWB from (16) and αDB from (18) when θ∗A=θ (implying certainty of disclosure in arena A). This yields
αWB(DA) =.5(1−sA), αDB(DA) =.5(1 +sB−sA). (42) Similarly, to condition on WA, we substitute αWB from (16) and αDB from (18) when θA∗ =θ (implying no disclosure in arenaA). This yields
αBW(WA) =.5, αBD(WA) =.5(1 +sB). (43) From (42) and (43), the difference
αDB(DA)−αWB(DA) = αDB(WA)−αWB(WA) = .5sB
does not depend on the disclosure decision in arena A.
By Proposition 1, E[θB] > θ∗B > 0. Therefore, if θB < θB∗, E[θB]−θB > 0, which implies that the integral on the RHS of the final equation in (28) is increasing with θB∗. Thus, substituting φA =WA or DA into (28), we see that E[ˆθB|DA] > E[ˆθB|WA] if and only if θB∗(DA)> θB∗(WA). Let γB = (αDB−αBW)/(αDB). Since
γB(DA) =sB/(1 +sB−sA)> sB/(1 +sB) = γB(WA), θB∗(DA)> θ∗B(WA). Thus E[ˆθB|DA]> E[ˆθB|WA]. k
Proof of Proposition 9:
Part 1:
E[(ˆθ−θ)2]
= Z θ∗
θ
(1−αW)E[θ] +αWE[θ|θ < θ∗]−θ 2f(θ)dθ+ Z θ
θ∗
(1−αD) (E[θ]−θ) 2f(θ)dθ
=
(1−αW)E[θ] +αWE[θ|θ < θ∗] 2 Z θ∗
θ
f(θ)dθ−2
(1−αW)E[θ] +αWE[θ|θ < θ∗] Z θ∗
θ
θf(θ)dθ +
Z θ∗
θ
θ2f(θ)dθ+ Z θ
θ∗
(1−αD) (E[θ]−θ) 2f(θ)dθ. (44)
∂E[(ˆθ−θ)2]
Proof of Proposition 10, Part 1: The mean squared error in arena i is E[(ˆθi−θi)2] = derivative of mean squared error in arena i with respect toθ∼i∗ can be written as:
∂E[( ˆθi−θi)2]
By inspection of equations (15)-(18), the derivatives of αWi and αDi with respect to θ∗∼i
Therefore, in the neighborhood of a stable equilibrium (∂θ∗i/∂γi >0), the mean squared error of arena i decreases with the threshold of the other arena∼i,
∂E[( ˆθi−θi)2]
∂θ∼i∗ <0.
k
Proof of Proposition 10, Part 2:
∂E[ ˆθi−θi] The last line follows from equations (49) and (50). k
Proof of Proposition 10, Part 3: To prove Part 3, we differentiate the welfare function with respect to θ∗A when θA and θB are distributed uniformly over [0,1]. The attention probabilities are given in equations (15)-(18).
∂W Using the symmetric solution to equation (25) when salience is s, ∂W/∂θ∗A = 0.02244 when (s, λ) = (0.8,0.2), and it becomes−0.01663 when (s, λ) = (0.8,0.9). k
Calculations Underlying Figure 3: We differentiate the welfareW with respect to
the common threshold θ∗ whereθA and θB are distributed uniformly over [0,1]
∂W
∂θ∗ = 1 48
−9θ∗2−2s[2−12θ∗+ 9(θ∗)2+ 4(θ∗)3] +s2[−3 + 22θ∗−45(θ∗)2+ 24(θ∗)3+ 5(θ∗)4] . The derivative ∂W/∂θ∗ is positive for high value and negative for low values of s. For
example, ∂W/∂θ∗ = 0.0114 when s= 0.8, and∂W/∂θ∗ =−0.0079 whens = 0.2. k Proof of Proposition 11: For notational simplicity, we henceforth suppress the A subscripts of αWA, αDA, and θA. By (32) and (33) we substitute out
αB = 1−sW(σWαW)p−sD(σDαD)p (1−sW −sD) ,
and write the first order conditions of the optimization problem with respect to αW and αD as
The disclosure threshold is determined by equations (5) and (6) of the basic model applied to arena A. By (54), the ratio αW/αD does not depend on K, the weight in the observers objective on payoffs derived from attending to arena B. Therefore, we can ensure that αB > 0 by selecting K sufficiently large without affecting the equilibrium disclosure level. Also, for appropriate values of parameters σW and σD, αD > αW, ensuring that the equilibrium involves only partial disclosure. By equation (54) and since E[θ] = 0, equation (5) can be rearranged to yield
θ∗ = (1−γ)E[θ|θ < θ∗] = (1−γ)
Multiplying both sides by−1 and taking natural logs gives
Differentiating both sides of equation (57) with respect toσD, 1 Similarly, differentiating both sides of equation (57) with respect to σW gives
∂θ∗ Multiplying equation (55) by Rθ∗
θ f(θ)dθ and differentiating both sides with respect to γ gives ∂θ∗/∂γ, which was shown to be positive in (10), so The second term of the expression inside the brackets in equations (58) and (59) can be rewritten using equation (55) as
Substituting the right hand side of equation (61) for the second term inside the brackets in equations (58) and (59), the full bracketed term becomes
1
From (60) and sinceθ∗ <0, From (58), (59), and (65), Parts 1 and 2 of Proposition 11 follow immediately.
To prove Proposition 11 Part 3, we differentiate both sides of equation (57) with respect to p:
By (54), the right hand side of equation (67) can be rewritten as
− 1 By (65), the left hand side of equation (67) is negative, and the sign of the right hand side depends on whetherσWαW is greater or less thanσDαD,∂θ∗/∂p < 0 whenσWαW <
σDαD and ∂θ∗/∂p > 0σWαW > σDαD. This confirms Proposition 11 Part 3.
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0.2 0.4 0.6 0.8 1 4A 0.2
0.4 0.6 0.8 1
4B
RCA2 RCA1
RCA0
RCB 0
RCB1
RCB2
Figure 1: Reaction Curves for Disclosure in Arenas A and B
RCA0 and RCB0 are reaction curves for disclosure in arenas A and B when (sA, sB) = (0.1,0.3). RCA1 and RCB1 are the reaction curves when (sA, sB) = (0.2,0.8), and RCA2 and RCB2 are the reaction curves when (sA, sB) = (0.9,0.1).
0.2 0.4
0.6
0.8
1 s
0 0.2
0.4 0.6
0.8 1
O -0.02
0 0.02 0.04
W cccccccccccc
4A
0.2 0.4
0.6
0.8
1 s
Figure 2: Welfare effect of exogenous increase in the disclosure threshold of arena A
The graph shows the derivative of welfare as defined by equation (30) with respect to an exogenous increase in the disclosure threshold in arena A, θ∗A, as a function of the common salience of disclosure s, and the weight λ on arena A in the social welfare function.
0.2 0.4
0.6
0.8
1 s
0 0.2
0.4 0.6
0.8 1
O 0
0.01 0.02
W cccccccccc
4
0.2 0.4
0.6
0.8
1 s
Figure 3: Welfare effect of exogenous increase in the common disclosure threshold of arenas A and B
The graph shows the derivative of welfare as defined by equation (30) with respect to an exogenous increase in the common disclosure threshold for arenas A and B, θ∗, as a function of the common salience of disclosure s, and the weight λ on arena A in the social welfare function.