## 1. Value Functions of the Investors' Optimization Problem

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---

### Introduction — key motivation and results
- Empirical puzzle: asset prices often deviate persistently from unbiased expectations of fundamentals (persistent over- and undervaluation).
- Behavioral mechanism introduced: cognitive dissonance (Festinger, 1957) causes investors to potentially ignore a public signal ε revealed at time 2 in order to protect a positive self-image.
- Heed-or-ignore decision rule: investors observe ε at time 2 and heed it only if the conditional expected utility with the signal exceeds the unconditional expected utility without the signal.
- Endogenous quasirational behavior: identical agents (preferences and endowments) can behave differently across equilibria—some heed, others ignore—due to general equilibrium effects.
- Equilibrium price implications (as stated in source):
  - Equity prices can both underreact and overreact to a publicly observable signal ε.
  - Prices overreact to moderately negative information; prices underreact to extremely positive and extremely negative information.
  - For slightly worse information, prices can become totally insensitive to the news.
  - Mispricing is bounded: if information is sufficiently negative, some investors revert to heeding (rational behavior) because potential abnormal profits compensate for dissonance discomfort.
  - Excessive price volatility and excessive price stability can both arise endogenously.
  - Distortions persist even as the signal becomes asymptotically uninformative.

### The asset-pricing model — setup and timing
- Economy:
  - Three periods.
  - Continuum of identical agents I = [0,1].
  - Consumption only at time 3.
- Preferences and wealth:
  - Utility: U(c) = -exp(-βc), where β is the constant degree of absolute risk aversion; c is final wealth at time 3.
- Assets:
  - Risk-free asset (numeraire price normalized to 1) with fixed rate r in periods 2 and 3; perfectly elastic supply.
  - Risky asset (shares) with dividend d = γ~ + ε~, where γ~ and ε~ are stochastically independent, E[γ~] > 0, E[ε~] = 0.
  - ε~ is observed at time 2; γ~ is revealed only at time 3.
- Endowments and timing:
  - At time 1 each agent is endowed with one unit of the risky asset and e > 0 units of the safe asset.
  - Period 1: investors choose portfolios and trade.
  - Period 2: signal ε is observed; investors decide whether to heed or ignore ε and may reshape portfolios.
  - Period 3: dividends and consumption.

### Budget constraints and wealth (preserved source expressions)
- Period 1 budget constraint (equation (1)):
  - e + p1 = s1 + p1 λ1.
- Period 2 portfolio-change constraint (equation (2)):
  - s2 + p2 λ2 = (1 + r) s1 + p2 λ1.
- Wealth (consumption) at period 3 (equation (3)):
  - c = (1 + r) s2 + λ2(d) + (1 + r) s1 + λ1 p2 − λ2 p2 [source text expression: .)()]()1)[(1( 22121 λεγλλ++−+++=psrrc — preserved formulaic structure from source].

### Benchmark Case 1 — All investors heed ε (fully rational)
- Time-2 optimization (Stage I): max_{λ2} E_{γ~}[U(c2(λ2))] with first-order condition implying (equation (4)) — preserved source symbolic structure:
  - )1)}(('{ )}~)(('{
  - rcUE cUE
  - +
  - + = γ γ εγ ε
- Equilibrium facts stated in source:
  - Market clearing: s*1 = 1 in bond market, λ*2 = λ*1 = 1 in stock market, and (1 + r) s*2 = s*1 holds by (2).
  - Equilibrium stock price at time 2 (equation (5)) expressed in source as a function of cUE, covariances, and ε: p*2(ε) = [ (1 + r) cUE / cUE' + cov_{γ~}(... ) ] — (preserved structure and dependence on cUE and covariances as in source).
- Time-1 equilibrium price p*1 (equation (6)): p*1 expressed via expectations and covariances in source; stock price at time 1 is lower than the discounted expected stock price at time 2 because cov[.,.] is negative (as stated in source).

### Benchmark Case 2 — All investors ignore ε (no-information)
- Time-2 no-information optimization (equation (7)): max_{λ2} E_{γ~,ε~}[U(c2(λ2))] with ε ignored.
- First-order condition gives the equity price when no information is used (equation (8)) — preserved source structure:
  - pD2(ε) = pD2 = )1()}('{
    )}
    ~
    )(('{
    )(~ .
- Since no new information enters individual decisions between time 1 and time 2, pD = p1 holds (ppD = 1 in source notation).

### Asset pricing with cognitive dissonance — value functions and equilibrium concept
- Time-2 value functions (equations (9) and (10)):
  - Full-information value function Ψ(p, ε):
    - Ψ(p, ε) = max_{λ2} E_{γ~}[U(c2(λ2))] subject to price p when ε is heeded; solution λH2(p, ε) is per-capita demand when ε is heeded.
  - No-information value function Φ(p):
    - Φ(p) = max_{λ2} E_{γ~,ε~}[U(c2(λ2))] when ε is ignored; solution λD2(p) is demand when signal is ignored.
- Heed vs ignore decision rule (explicit):
  - For a given signal ε and observed price p, an investor ignores ε iff Φ(p) > Ψ(p, ε).
- Nash equilibrium definitions for a realized ε′ (Definition 1) — preserved source statements:
  - (a) Pure-strategy full-information equilibrium at ε′: price p*(ε′) and λ*2,H such that λ*2,H = 1 at p*(ε′) and Φ(p*(ε′)) ≥ Ψ(p*(ε′), ε′).
  - (b) Pure-strategy no-information equilibrium at ε′: price p such that λD2(p) = 1 and Ψ(p, ε′) ≥ Φ(p).
  - (c) Mixed-strategy equilibrium at ε′: exists μ in (0,1) such that aggregate demand mixes λD and λH and Ψ = Φ at the mixing price p̂.
- Interpretation: equilibrium conditions ensure agents do not deviate; they heed only if heeding yields higher welfare; they ignore only if ignoring yields higher welfare.
- Proposition 1 (lead-in): the value functions Ψ(.) and Φ(.) have properties to be derived (figure referenced in source).

### Equilibrium statement and implications (preserved propositions)
- If p*(ε) is a pure-strategy equilibrium equity price for realization ε and all agents heed the information:
  - The full-information value function Ψ(p, ε) possesses a global minimum at p = p*(ε).
  - Ψ(p, ε) is strictly monotone on the intervals (-∞, p*(ε)) and (p*(ε), ∞).
- If p is a pure-strategy equilibrium asset price when all agents ignore ε:
  - The no-information value function Φ(p) has a global minimum at p = p and is strictly monotone on the intervals (-∞, p) and (p, ∞).
- In equilibrium all investors hold the same asset portfolio at time 1: (s1, λ1) = (e, 1).

### Closed-form desired holdings (multivariate normal case) — preserved source forms
- With (γ~, ε~) jointly normal, var(γ~) = γσ, var(ε~) = εσ, first-order conditions yield closed-form desired holdings (source equations (19) and (20)):
  - Desired equity holdings when information is heeded (equation (19)) — preserved source symbolic form:
    - H2λ(p, ε) = prE{p} + γεεβσ2 )1( +−+?  [source presents (19) as: γ β σ γε ε ε λ prE p H )1(}~{),(2 +−+ = ]
  - Desired equity holdings when information is ignored (equation (20)) — preserved source symbolic form:
    - D2λ(p) = prE{p} + ( )1(}~{)(2 εγσσβ γ λ + +− = )  [as per source form (20)]

### IV. How information affects asset prices — regional characterization and equilibrium price functions (preserved source regions and key equations)
- The equilibrium equity price function is piecewise by realization ε. Key thresholds: ε1, ε0, ε2, ε3, and ε̂ (epsilon-hat).

A. Full-information equilibria (ε ∈ [ε1, ε0])
- Market-clearing H2λ = 1 and (19) imply efficient share price p*(ε) (equation (21)) — preserved source form:
  - p*(ε) = (1/(1 + r)) [ prE + γ β σ γε ε ]  [preserve form from source: r E p + − + = 1 }~{ )( * γ β σ γε ε ε ]
- Slope of p*(ε) (equation (26)):
  - ∂p*/∂ε = 1/(1 + r).
- Proposition 2: For any ε ∈ [ε1, ε0] there exists a unique full-information equilibrium (type (a)), price p*(ε) given by (21).

B. Mixed strategies with steeper response (ε ∈ (ε2, ε1))
- No pure-strategy equilibrium; mixed-strategy equilibria exist with fraction μ ∈ (0,1) heeding.
- Mixing price p̂(ε) solves equality Ψ = Φ (equation (30)); differentiation yields (31) and (32) giving:
  - ∂p̂/∂ε > (1/(1 + r)) [expression in (32)] — p̂(ε) is linear between ε2 and ε1 but steeper than p*(ε).
- Intuition: rational agents who heed ε perceive lower risk and hold more than one share (H2λ > 1); quasirational agents sell; aggregate effect increases price sensitivity.

C. No-information equilibria (ε ∈ [ε3, ε2])
- All investors ignore information; market-clearing price p equals the no-information equilibrium price p given in (29) — preserved source expression:
  - p = r−1 E{ ... } as in (29): [] )(}~{ 1 1 : εγσσβγ +− + = E r p
- ε3 is the indifference point between heeding and ignoring; ε̂ is where p*(ε) meets p̂.
- For ε > ε̂ equities are undervalued relative to efficient price; for ε < ε̂ equities are overvalued.

D. Mixed strategies for extreme signals (ε ∈ (−∞, ε3) and ε ∈ (ε0, ∞))
- Pure-strategy equilibria do not exist; unique mixed-strategy equilibria exist where some heed and some ignore.
- In these regions (equation (33)):
  - ∂p̂/∂ε < (1/(1 + r)) [expression in (33)] — price derivative condition reversed relative to (32).
- Implications:
  - For very unfavorable ε (ε < ε3): shares are overvalued relative to p*(ε); rational investors sell short.
  - For very favorable ε (ε > ε0): shares are undervalued relative to p*(ε); quasirational investors short-sell.

E. Upper bound on fraction of rational traders for extreme signals
- Market-clearing equation for share α(ε) of rational traders (equation (34)) — preserved source form:
  - 1 = [ ] (1)(,) (22 = −+ ελεαεελεαpp DH )
- Limit for extreme signals (equation (35)):
  - lim_{ε→∞} α(ε) = α, where α ∈ [0, 1/2] and I := γεε/(σσσ + ) measures information content. (Equation (35) preserved in form.)
- Paradox noted in source: the less informative the signal (smaller I), the more attention it ultimately receives (more agents become rational as ε → ∞), but α is bounded above by 1/2.

### V. Informativeness and magnitude/direction of mispricing (preserved comparative statics)
- Informativeness I defined in source as: I := σσ ε / (γε σ σ + ) (preserve source definition).
- Comparative statics (equations (36) and (37)):
  - ∂ε1/∂I > 0 (equation (36)): as signal becomes less informative (I decreases), ε1 moves accordingly (undervaluation region shrinks).
  - ∂ε3/∂I < 0 (equation (37)): as informativeness decreases, ε3 moves left (region where all ignore information enlarges).
- Limits as informativeness → 0 (as stated in source):
  - 0 ε → ∞, and ε2, ε1 → 2/ε̂  (source provides limit expressions and L’Hôpital application).
- Implications highlighted in source:
  - With nearly irrelevant signals (low I), equilibrium price function is nearly vertical near ε = 2/ε̂, so small less-favorable news can cause sharp price drops despite signals being almost uninformative.
  - Region where every investor ignores information grows as informativeness falls; overvaluation extent decreases with greater informativeness.
  - As I → 1 (signal fully resolves uncertainty), pricing converges to efficient price for all ε: ∀ε → p(ε) = p*(ε).

### Summary of equilibrium pricing behavior (qualitative, from source)
- Risky asset priced efficiently for ε in [ε1, ε0].
- Asset undervalued (and risk overvalued) for ε ∈ (ε̂, ε1] and for ε > ε0.
- Asset overvalued for ε < ε̂.
- Price volatility relative to efficient benchmark:
  - Excessive sensitivity (excessive volatility) between ε2 and ε1.
  - “Right amount” of price volatility between ε1 and ε0.
  - Excessive stability (insensitivity) for ε below ε2 or above ε0.
- Counterintuitive outcome emphasized in source: almost irrelevant (low-information) news can produce large price corrections and persistent mispricing; cognitive dissonance provides a mechanism linking such behavior to sunspot-like effects.

### VI. Concluding remarks — implications and avenues for further research (from source)
- The model endogenizes quasirational behavior via cognitive dissonance: investors may ignore unfavorable information post-decision, generating persistent over- and undervaluation.
- Asset prices can overreact or underreact depending on ε; portfolio decisions are efficient only for ε near ex ante expectations (ε ∈ [ε1, ε0]).
- When mispricing is large enough, rational investors exploit distortions, earning arbitrage profits that compensate for cognitive dissonance costs.
- Potential empirical and theoretical consequences (as suggested in source):
  - May shed light on financial anomalies and on stock market crashes via sudden shifts in behavior once accumulated adverse information becomes recognized.
  - Suggests research directions: regulation, market microstructure, disclosure requirements, multi-asset generalizations, cross-market spillovers from mispricing induced by cognitive dissonance.

*Source: _wp0509 - 1. Value Functions of the Investors' Optimization Problem (PDF chapter/section)*

### 1. Value Functions of the Investors' Optimization Problem ....................................................25

### 1. Value Functions of the Investors' Optimization Problem

### Introduction — key motivation and results
- Growing evidence questions that asset prices equal investors’ unbiased expectations of fundamentals; persistent over- and undervaluation occurs.
- Cognitive dissonance (Festinger, 1957) is introduced into an asset-pricing setting: investors may ignore signals that conflict with their ex ante portfolio choice to protect a positive self-image.
- Main mechanism: investors decide to heed a public signal ε revealed at time 2 only if the conditional expected utility with the signal exceeds the unconditional expected utility without the signal; otherwise they ignore ε.
- Endogenous emergence of quasirational behavior: identical agents (preferences and endowments) may behave differently across equilibria—some heed information, others ignore it—due to general equilibrium effects.
- Equilibrium implications for prices:
  - Equity prices can both underreact and overreact to a publicly observable signal ε.
  - Prices overreact to moderately negative information; prices underreact to extremely positive and extremely negative information.
  - For slightly worse information, prices can become totally insensitive to the news.
  - Mispricing is bounded: if information is sufficiently negative, some investors revert to heeding (rational behavior) because potential abnormal profits compensate for dissonance discomfort.
  - Excessive price volatility and excessive price stability can both arise endogenously.
  - Distortions persist even as the signal becomes asymptotically uninformative.

### The asset-pricing model — setup and timing
- Economy:
  - Three periods; continuum of identical agents I = [0,1].
  - Consumption only at time 3.
  - Utility: U(c) = -exp(-βc), where β is the constant degree of absolute risk aversion; c is final wealth at time 3.
- Assets:
  - Risk-free asset (numeraire price normalized to 1) with fixed rate r in periods 2 and 3; perfectly elastic supply.
  - Risky asset (shares) with dividend d = γ~ + ε~, where γ~ and ε~ are stochastically independent, E[γ~] > 0, E[ε~] = 0.
  - ε~ is observed at time 2; γ~ is revealed only at time 3.
- Endowments and choices:
  - At time 1 each agent is endowed with one unit of the risky asset and e > 0 units of the safe asset.
- Sequence of events (periods):
  - Period 1: investors choose portfolios and trade.
  - Period 2: signal ε is observed; investors decide whether to heed or ignore ε and may reshape portfolios.
  - Period 3: dividends and consumption.

### Budget constraints and wealth
- Period 1 budget constraint (equation (1)):
  - e + p1 = s1 + p1 λ1.
- Period 2 portfolio-change constraint (equation (2)):
  - s2 + p2 λ2 = (1 + r) s1 + p2 λ1.
- Wealth (consumption) at period 3 (equation (3)):
  - c = (1 + r) s2 + λ2(d) + (1 + r) s1 + λ1 p2 − λ2 p2 [source text expression: .)()]()1)[(1( 22121 λεγλλ++−+++=psrrc — preserved formulaic structure from source].

### Benchmark Case 1 — All investors heed ε (fully rational)
- Stage I (after observing ε at time 2): agent solves
  - max_{λ2} E_{γ~}[U(c2(λ2))] with first-order condition implying (equation (4)):
    - )1)}(('{ )}~)(('{
    - rcUE cUE
    - +
    - + = γ γ εγ ε
  - In equilibrium market clearing: s*1 = 1 in bond market, λ*2 = λ*1 = 1 in stock market, and (1 + r) s*2 = s*1 holds by (2).
  - Equilibrium stock price at time 2 (equation (5)):
    - p*2(ε) = [ (1 + r) cUE / cUE' + cov_{γ~}(... ) ] — (preserved structure and dependence on cUE and covariances as in source). With r c + (1)2 εγ~ + = erc (preserved source expression: .~
    ~)1(2 εγ+++=erc ).
- Stage II (time 1 equilibrium price p*1) (equation (6)):
  - p*1 = expression involving expectations and covariances: )1)}(('{ )]~(),('cov[ 1 )}~({ )1)}(('{ )}~()('{
  - The stock price at time 1 is lower than the discounted expected stock price at time 2 because cov[.,.] is negative (as stated in source).

### Benchmark Case 2 — All investors ignore ε (no-information)
- If investors ignore ε, they solve the no-information optimization at time 2 (equation (7)):
  - max_{λ2} E_{γ~,ε~}[U(c2(λ2))] with ε ignored.
- First-order condition gives the equity price when no information is used (equation (8)):
  - pD2(ε) = pD2 = )1()}('{
    )}
    ~
    )(('{
    )(~ , preserved from source
  - Since no new information enters individual decisions between time 1 and time 2, pD = p1 holds (ppD = 1 in source notation).

### Asset pricing with cognitive dissonance — value functions and equilibrium concept
- Definition of value functions at time 2 (after ε observed and price p observed):
  - Full-information value function Ψ(p, ε) (equation (9)):
    - Ψ(p, ε) = max_{λ2} E_{γ~}[U(c2(λ2))] subject to price p when ε is heeded; solution λH2(p, ε) is per-capita demand when ε is heeded.
  - No-information value function Φ(p) (equation (10)):
    - Φ(p) = max_{λ2} E_{γ~,ε~}[U(c2(λ2))] when ε is ignored; solution λD2(p) is demand when signal is ignored.
- Heed vs ignore decision rule (explicit):
  - For a given signal ε and observed price p, an investor ignores ε iff Φ(p) > Ψ(p, ε).
- Nash equilibria definitions (pure and mixed strategies) for a realized ε′ (Definition 1):
  - (a) Pure-strategy full-information equilibrium at ε′: price p*(ε′) and λ*2,H such that λ*2,H = 1 at p*(ε′) and Φ(p*(ε′)) ≥ Ψ(p*(ε′), ε′).
  - (b) Pure-strategy no-information equilibrium at ε′: price p such that λD2(p) = 1 and Ψ(p, ε′) ≥ Φ(p).
  - (c) Mixed-strategy equilibrium at ε′: exists μ in (0,1) such that aggregate demand mixes λD and λH and Ψ = Φ at the mixing price p̂.
- Interpretation of equilibrium conditions:
  - Conditions ensure agents do not deviate: they heed only if heeding yields higher welfare; they ignore only if ignoring yields higher welfare.
- Proposition 1 (lead-in): The value functions Ψ(.) and Φ(.) have properties to be derived (figure referenced in source).

*Source: _wp0509 - 1. Value Functions of the Investors' Optimization Problem*

### 1.   Let p*(

### _wp0509 - 1.   Let p*(

### Equilibrium statement and implications
- Let p*(ε) be a pure-strategy equilibrium equity price for the realization ε and assume that all agents heed the information.
- Then, as a function of p, the full-information value function, Ψ(p, ε), possesses a global minimum at p = p*(ε).
- Ψ(p, ε) is strictly monotone on the intervals (-∞, p*(ε)) and (p*(ε), ∞).

*Source: _wp0509 - 1.   Let p*( (PDF chapter/section))*

### 2.  Let

### 2.  Let

### Model setup and key equilibrium conditions
- p be a pure-strategy equilibrium asset price when all agents ignore the information ε. The no-information value function Φ(p) has a global minimum at p = p and is strictly monotone on the intervals (-∞, p) and (p, ∞).
- Investors observe a signal ε (or ignore it by symmetric argument). Individual preferences: U(c) = -exp(-βc). In equilibrium all investors hold the same asset portfolio at time 1: (s1, λ1) = (e, 1).
- With multivariate normal (γ~, ε~) and var(γ~) = γσ, var(ε~) = εσ, expected utilities and first-order conditions yield:
  - Desired equity holdings when information is heeded (equation (19)):
    - H2λ(p, ε) = prE{p} + γεεβσ2 )1( +−+?  [preserve exact expression as presented in source:  
      γβσ γε εελ prE p H )1(}~{),(2 +−+ = ]
      (the source presents (19) as: γ β σ γε ε ε λ prE p H )1(}~{),(2 +−+ = )
  - Desired equity holdings when information is ignored (equation (20)):
    - D2λ(p) = prE{p} + ( )1(}~{)(2 εγσσβ γ λ + +− = )  [as per source form (20)]

(Note: the source gives closed-form expressions (19) and (20) for H2λ and D2λ as functions of p; the exact algebraic forms are preserved in the source text.)

### IV. How information affects asset prices — Regional characterization and equilibrium price functions
- The equilibrium equity price function depends on realization ε and is piecewise with distinct regions characterized by investor responses to ε. Key thresholds: ε1, ε0, ε2, ε3, and ε̂ (epsilon-hat).

A. Full-information equilibria: all investors heed the information
- If almost all agents heed information, market clearing H2λ = 1 and (19) imply the efficient share price p*(ε) (equation (21)):
  - p*(ε) = (1/(1 + r)) [ prE + γ β σ γε ε ]  [preserve form from source: r E p + − + = 1 }~{ )( * γ β σ γε ε ε ]
- The slope of p*(ε) (equation (26)):
  - ∂p*/∂ε = 1/(1 + r)
- Define thresholds ε1 (negative solution) and ε0 (positive solution) via (25):
  - ε1 = [ ( ... ) ] (negative solution)
  - ε0 = [ ( ... ) ] (positive solution)
  - Explicitly (25) in source:
    - ε1 = [ (γ/(1+γ)) (σσ/(βσ+...)) ... ] and ε0 analogous (source displays exact formulas).
- Proposition 2: For any ε ∈ [ε1, ε0] there exists a unique full-information equilibrium (type (a)), price p*(ε) given by (21).

B. Slightly less favorable information: mixed strategies and steeper price response
- For ε in (ε2, ε1), pure-strategy Nash equilibrium does not exist; mixed-strategy equilibria exist with some agents heeding and others ignoring information.
- Market-clearing price p̂(ε) solves equality of expected utilities (equation (30)). Differentiation yields (31) and combining gives (32):
  - ∂p̂/∂ε > (1/(1 + r)) [ expression in (32) ] — i.e., p̂(ε) is linear between ε2 and ε1 but steeper than p*(ε).
- Intuition: rational agents who heed ε perceive lower risk and hold more than one share (H2λ > 1); quasirational agents sell. Therefore equilibrium price sensitivity to ε is higher than under full information in this interval.

C. Sufficiently bad news: all investors ignore information
- For ε ∈ [ε3, ε2], information is ignored by all agents. Market-clearing price p equals the no-information equilibrium price p given in (29):
  - p = r−1 E{ ... } as in (29): [] )(}~{ 1 1 : εγσσβγ +− + = E r p
- At ε3 agents are indifferent between heeding and ignoring; ε̂ is where efficient p*(ε) meets p̂. For ε > ε̂ equities are undervalued relative to efficient price; for ε < ε̂ equities are overvalued.

D. Very unfavorable and very favorable news: mixed strategies again
- For ε ∈ (−∞, ε3) and ε ∈ (ε0, ∞) pure-strategy equilibria do not exist; unique mixed-strategy equilibria exist where some heed and some ignore.
- For these regions (equation (33)):
  - ∂p̂/∂ε < (1/(1 + r)) [ expression in (33) ] — price derivative condition reversed relative to (32).
- For very unfavorable ε (ε < ε3) shares are overvalued relative to p*(ε); rational investors sell short. For very favorable ε (ε > ε0) shares are undervalued relative to p*(ε); quasirational investors short-sell.

E. Upper bound on fraction of rational traders for extreme signals
- The share α(ε) of rational traders satisfies market-clearing (34):
  - 1 = [ ] (1)(,) (22 = −+ ελεαεελεαpp DH ) — exact equation (34) from source.
- In the limit ε → ∞ (extremely unfavorable region), the share α converges to α ∈ [0, 1/2] determined by (35):
  - limε→∞ α(ε) = α, where α ∈ [0, 1/2] and I := γεε/(σσσ + ) measures information content. (Equation (35) preserved in form in source.)
- Paradox: the less informative the signal (smaller I), the more attention it ultimately receives (more agents become rational as ε → ∞), but α is bounded above by 1/2: at most half the investors heed extreme favorable or unfavorable information.

### V. Informativeness and the magnitude/direction of mispricing
- Informativeness I = εσ / (σ + γε?) — as defined in source: I := σσ ε / (γε σ σ + ) (preserve source definition).
- Perturbation preserving p*(ε) (hold position of p*(ε) unchanged) implies dEd = γσβ/γ ? (source states: }~{ γσβ γ dEd = .)
- Key comparative statics (equations (36) and (37)):
  - ∂ε1/∂I > 0 (equation (36)): as signal becomes less informative (I decreases), ε1 moves accordingly (undervaluation region shrinks).
  - ∂ε3/∂I < 0 (equation (37)): as informativeness decreases, ε3 moves left (region where all ignore information enlarges).
- Limits as informativeness → 0:
  - 0 ε → ∞, and ε2, ε1 → 2/ε̂  (source provides limit expressions and L’Hôpital application).
- Implications:
  - With nearly irrelevant signals (low I), equilibrium price function is nearly vertical near ε = 2/ε̂, so small less-favorable news can cause sharp price drops despite signals being almost uninformative.
  - Region where every investor ignores information grows as informativeness falls; overvaluation extent decreases with greater informativeness.
  - As I → 1 (signal fully resolves uncertainty), pricing converges to efficient price for all ε: ∀ε → p(ε) = p*(ε).

### Summary of equilibrium pricing behavior (qualitative)
- Risky asset priced efficiently for ε in [ε1, ε0].
- Asset undervalued (and risk overvalued) for ε ∈ (ε̂, ε1] and for ε > ε0.
- Asset overvalued for ε < ε̂.
- Price volatility relative to efficient benchmark:
  - Excessive sensitivity (excessive volatility) between ε2 and ε1.
  - “Right amount” of price volatility between ε1 and ε0.
  - Excessive stability (insensitivity) for ε below ε2 or above ε0.
- Counterintuitive result: almost irrelevant (low-information) news can produce large price corrections and persistent mispricing; the model provides a mechanism linking cognitive dissonance to sunspot-like effects.

### VI. Concluding remarks — implications and avenues for further research
- The model endogenizes quasirational behavior via cognitive dissonance: investors may ignore unfavorable information post-decision, generating persistent over- and undervaluation.
- Asset prices can overreact or underreact depending on ε; portfolio decisions are efficient only for ε near ex ante expectations (ε ∈ [ε1, ε0]).
- When mispricing is large enough, rational investors exploit distortions, earning arbitrage profits that compensate for cognitive dissonance costs.
- Potential empirical and theoretical consequences:
  - Could shed light on financial anomalies (e.g., equity premium puzzle) and on stock market crashes via sudden shifts in behavior once accumulated adverse information becomes recognized.
  - Suggests directions for research on regulation, market microstructure, disclosure requirements, multi-asset generalizations, and cross-market spillovers from mispricing induced by cognitive dissonance.

*Italic: Source: _wp0509 - 2.  Let (PDF chapter/section).*

### References

### _wp0509 - References

### Behavioral finance and investor psychology
- Akerlof, George A., and W.T. Dickens, 1982, “The Economic Consequences of Cognitive Dissonance,” American Economic Review, Vol. 72, pp. 307–19.
- Barberis, Nicholas, and Richard Thaler, 2002, “A Survey of Behavioral Finance,” NBER Working Paper No. 9222 (Cambridge, Massachusetts: National Bureau for Economic Research).
- Barberis, Nicholas, Andrei Shleifer, and Robert Vishny, 1998, “A Model of Investor Sentiment,” Journal of Financial Economics, Vol. 49, pp. 307–343.
- Bernatzi, Shlomo, and Richard H. Thaler, 1995, “Myopic Loss Aversion and the Equity Premium Puzzle,” Quarterly Journal of Economics, Vol. 110, pp. 73–92.
- Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 1998, “Investor Psychology and Securities Market Under- and Overreaction,” Journal of Finance, Vol. 53, No. 6, 1839–1885.
- Daniel, Kent, David Hirshleifer, and Siew Hong Teoh, 2002, “Investor Psychology in Capital Markets: Evidence and Policy Implications,” Journal of Monetary Economics, Vol. 49, No. 1, 139–209.
- Festinger, Leon, 1957, A Theory of Cognitive Dissonance (Stanford, CA: Stanford University Press).
- Harmon-Jones, Eddie, and Judson Mills, 1999, “An Introduction to Cognitive Dissonance Theory and an Overview of Current Perspectives on the Theory,” in: Cognitive Dissonance: Progress on a Pivotal Theory in Social Psychology, ed. by E. Harmon-Jones and J. Mills (Washington, D.C.: American Psychological Association).
- Hirshleifer, David, 2001, “Investor Psychology and Asset Pricing,” Journal of Finance, Vol. 56, No. 4, 1533–97.
- Odean, Terrance, 1998, “Volume, Volatility, Price, and Profit When All Traders Are Above Average,” Journal of Finance, Vol. 53, No. 6, 1887–1934.
- Rabin, Matthew, 1998, “Psychology and Economics,” Journal of Economic Literature, Vol. 36, pp. 11–46.
- Rabin, Matthew, and Joel L. Schrag, 1999, “First Impressions Matter: A Model of Confirmatory Bias,” Quarterly Journal of Economics, Vol. 114, pp. 37–82.
- Tversky, Amos, and Daniel Kahneman, 1974, “Judgment Under Uncertainty: Heuristics and Biases,” Science, Vol. 185, pp. 1124–1131.
- Yariv, Leeat, 2002, “I’ll See It When I Believe It—A Simple Model of Cognitive Consistency,” mimeo, UCLA.

### Asset pricing, market efficiency, and long-run risks
- Aiyagari, S. Rao, and Mark Gertler, 1999, “Overreaction of Asset Prices in General Equilibrium,” Review of Economic Dynamics, Vol. 2, pp. 3–35.
- Bansal, Ravi, and Amir Yaron, 2004, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles,” Journal of Finance, Vol. 59, pp. 1481–1509.
- Bakshi, Gurdip S., and Zhiwu Chen, 1996, “The Spirit of Capitalism and Stock Market Prices,” American Economic Review, Vol. 86, pp. 133–57.
- Campbell, John Y., and John H. Cochrane, 1999, “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior,” Journal of Political Economy, Vol. 107, pp. 205–51.
- Campbell, John Y., and Robert J. Shiller, 1988, “Stock Prices, Earnings, and Expected Dividends,” Journal of Finance, Vol. 43, pp. 661–76.
- Fama, Eugene F., 1991, “Efficient Capital Markets: II,” Journal of Finance, Vol. 46, pp. 1575–1617.
- Fama, Eugene F., 1998, “Market Efficiency, Long-Term Returns, and Behavioral Finance, Journal of Financial Economics, Vol. 49, pp. 283–306.
- Fama, Eugene F., and Kenneth R. French, 1988, “Permanent and Temporary Components of Stock Prices,” Journal of Political Economy, Vol. 96, pp. 246–273.
- Grossman, Sanford J., and Joseph Stiglitz, 1980, “On the Impossibility of Informationally Efficient Markets,” American Economic Review, Vol. 70, pp. 393–408.
- LeRoy, Stephen F., 1989, “Efficient Capital Markets and Martingales,” Journal of Economic Literature, Vol. 27, pp. 1583–1621.
- Malkiel, Burton G., 2003, “The Efficient Markets Hypothesis and Its Critics,” Journal of Economic Perspectives, Vol. 17, pp. 59-82.
- Mehra R., and Edward C. Prescott, 1985, “The Equity Premium: A Puzzle,” Journal of Monetary Economics, Vol. 15, pp. 145–61.
- Shiller, Robert J., 1981, “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?” American Economic Review, Vol. 71, pp. 421–36.
- Shiller, Robert J., 1998, “Human Behavior and the Efficiency of the Financial System,” NBER Working Paper 6375 (Cambridge, Massachusetts: National Bureau for Economic Research).
- Shiller, Robert J., 2003. “From Efficient Markets Theory to Behavioral Finance,” Journal of Economic Perspectives, Vol. 17, pp. 83-104.
- West, Kenneth D., 1988, “Bubbles, Fads and Stock Price Volatility Tests: A Partial Evaluation,” Journal of Finance, Vol. 43, pp. 639–60.

### Overreaction, underreaction, momentum, and market anomalies
- Barberis, Nicholas, Andrei Shleifer, and Robert Vishny, 1998, “A Model of Investor Sentiment,” Journal of Financial Economics, Vol. 49, pp. 307–343.
- Chopra, Navin, Josef Lakonishok, and Jay R. Ritter, 1992, “Measuring Abnormal Performance: Do Stocks Overreact?” Journal of Financial Economics, Vol. 31, pp. 235–68.
- Cohen, Randolph B., Paul A. Gompers, and Tuomo Vuolteenaho, 2002, “Who Underreacts To Cash-Flow News? Evidence From Trading Between Individuals and Institutions,” Journal of Financial Economics, Vol. 66, pp. 409–62.
- Cutler, David M., James M. Poterba, and Lawrence H. Summers, 1991, “Speculative Dynamics,” Review of Economic Studies, Vol. 58, pp. 529–46.
- DeBondt, Werner F.M., and Richard H. Thaler, 1985, “Does the Stock Market Overreact?” Journal of Finance, Vol. 40, pp. 793–808.
- DeBondt, Werner F.M., and Richard H. Thaler, 1995, “Financial Decision-Making in Markets and Firms: A Behavioral Perspective,” in Handbook in Operations Research and Management Science, Vol. 9, Finance, ed. by Robert Jarrow and others (Amsterdam: Elsevier).
- Grinblatt, Mark, Sheridan Titman, and Russ Wermers, 1995, “Momentum Investment Strategies, Portfolio Performance, and Herding: A Study of Mutual Fund Behavior,” American Economic Review, Vol. 85, pp. 1088–1105.
- Hong, Harrison, and Jeremy C. Stein, 1999, “A Unified Theory of Underreaction, Momentum Trading, and Overreaction in Asset Markets,” Journal of Finance, Vol. 54, pp. 2143-84.
- Jegadeesh, Narasimhan, and Sheridan Titman, 1993, “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” Journal of Finance, Vol. 48, pp. 65–91.
- Rouwenhorst, K. Geert, 1998, “International Momentum Strategies,” Journal of Finance, Vol. 53, pp. 267–84.

### Theory, decision-making, and bounded rationality
- Carrillo, Juan D., and Thomas Mariotti, 2000, “Strategic Ignorance As a Self-Disciplining Device,” Review of Economic Studies, Vol. 67, pp. 529-44.
- Fiske, Susan T., and Shelley E. Taylor, 1991, Social Cognition (New York: McGraw-Hill).
- Gigerenzer, Gerd, and Reinhard Selten, 2001, Bounded Rationality: The Adaptive Toolbox (Cambridge, Massachusetts: MIT Press).
- Hess, Patrick J., and Marc R. Reinganum, 1979, “Efficient Capital Markets,” Handbook of Financial Economics (Amsterdam: North-Holland).
- Rabin, Matthew, 1998, “Psychology and Economics,” Journal of Economic Literature, Vol. 36, pp. 11–46.
- Zou, Heng-fu, 1994, “ ‘Spirit of Capitalism’ and Long-Run Growth,” European Journal of Political Economy, Vol. 10, pp. 279–93.

*Source: _wp0509 - References*

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