## _wp1133

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### Model setup and assumptions
- Market structure:
  - Two building blocks: (i) a market for information where J financial analysts determine information sold to investors; (ii) a game among a continuum [0,1] of infinitesimal investors whose investment return is endogenous.
  - Timing: analysts offer messages m_j on the return R; investors choose which messages to acquire; then investors choose investment sizes x_i.
- Return specification:
  - R = α + φX, where φ ∈ [0,1], α ~ Normal(0, precision τ_α), and X = ∫ x_i di is aggregate investment.
- Analysts’ information:
  - Each analyst j observes private signal s_j = α + η_j with precision τ_s (η_j i.i.d., mean 0).
  - All analysts observe a common signal y = α + η_y with precision τ_y (η_y mean 0).
  - Define ̄η = ∑_j η_j / J (precision J τ_s) and ̄s = α + ̄η.
- Messages and market for information:
  - Analyst j commits to a single message m_j sold at price q_j to proportion N_j of investors; production cost is zero.
  - Assumption A (normality): (R, y, s_1, ..., s_J, m_1, ..., m_J) is Gaussian.
  - Implicit market assumptions: no price discrimination, no resale, analysts do not buy other analysts’ messages; analysts “truthfully commit” to distributional properties of m_j.
- Investors:
  - Investor i maximizes R x_i − 1/2 x_i^2 − ∑_{j∈I_i} q_j.
  - Given normality, optimal investment is x_i(I_i) = E(R | I_i). (Property 1)
  - x_i is linear in messages and does not depend on message prices q_j.

### Equilibrium definition, properties, and messaging
- Four equilibrium properties (in symmetric linear equilibrium):
  1. x_i(I_i) = E(R | I_i) (linear in messages; independent of q_j).
  2. Every message m_j is sold to all investors.
  3. Every analyst sells all of his information: distribution of R | m_j equals R | (s_j, y). (Assumption B: analysts treat joint distribution of (R, s_j, y) as exogenous.)
  4. Price q_j equals the maximum price an investor is willing to pay given equilibrium information set I_i.
- Linear symmetric equilibrium (Definition 3.3):
  - Investors buy all messages and invest E(R | m_1, ..., m_J).
  - Analysts sell m_j = μ y + (1−μ) s_j at price q_j = 1/2 (Var(R | all messages except j) − Var(R | all messages)).
- Equilibrium message representation:
  - ∀j, m_j = μ y + (1−μ) s_j, with μ independent of j and weights normalized to sum to 1.
  - Equivalent: m_j = α + μ η_y + (1−μ) η_j.
- Pricing equation (Lemma 3.2):
  - q_j = 1/2 (Var(R | m_1, ..., m_{j−1}, m_{j+1}, ..., m_J) − Var(R | m_1, ..., m_J)).
- Return and sufficiency:
  - The average message ̄m is a sufficient statistic for investors.
  - R = α + φ E(R | ̄m). (Equation (6))
  - E(R | ̄m) = (1/(1−φ)) E(α | ̄m) and R = α + λ ̄m. (Equation (7))
  - λ = φ/(1−φ) · cov(α, ̄m)/Var(̄m) = φ/(1−φ) · [1/τ_α] / [1/τ_α + μ^2 (1/τ_y) + (1−μ)^2 (1/(J τ_s))]. (Equation (8))

### Equilibrium behavior and conformism
- Analysts’ objective trade-off:
  - Analysts balance giving a good prediction about the fundamental α and conforming to other analysts’ predictions (degree of “conformism” = weight on the average message λ).
- Components of conformism:
  - Exogenous component: related to φ/(1−φ), i.e., the correlation between return and aggregate investment.
  - Endogenous component: related to cov(α, ̄m)/Var(̄m), which depends on μ and information precisions.

### Multiplicity of equilibria — main results and patterns
- Existence:
  - Equilibrium μ solves a cubic polynomial derived from consistency condition μ = μ′/(μ′+ν′).
- Proposition 3.4 (structure):
  - Assume φ ∈ [0,1]. There exists a threshold φ^* such that:
    1. If φ > φ^*, then there are 3 linear symmetric equilibria.
    2. If φ < φ^*, then there is 1 linear symmetric equilibrium.
  - One equilibrium always has μ ∈ [τ_y/(τ_s + τ_y), 1], while the two others (when they exist) satisfy μ ≤ −τ_y/τ_α.
  - The threshold φ^* = 1/(J−1) J A^* + 1 ∈ (0,1), where
    - A^* = sup_{μ ≤ −τ_y/τ_α} (μ−1) (τ_y + μ τ_α) / (τ_y − (τ_s + τ_y) μ) · (1 + μ^2 τ_α/τ_y + (1−μ)^2 τ_α/(J τ_s)) ∈ (0, +∞).
- Corollary 3.5 (limit behavior):
  - As φ → 0:
    - Unique equilibrium; λ → 0; μ → τ_y/(τ_y + τ_s).
  - As φ → 1:
    - There are 3 equilibria; λ → +∞ for the 3 equilibria.
    - One μ → 1, another μ → −τ_y/τ_α, the third μ → −∞.
- Equilibrium patterns:
  - “Regular” equilibria: μ ∈ [0,1], overweight public information (standard beauty-contest behavior).
  - “Inverted” equilibria: μ < 0, characterized by:
    - Self-fulfilling misinterpretation of the public signal (m_j reacts negatively to y when μ < 0).
    - Overconfidence in private information: weight (1−μ) on s_j > 1.
    - These inverted equilibria can produce bubble-like dynamics where collective coordination validates an initially counterintuitive recommendation.

### Information transmission and loss
- Precision available in the economy about α:
  - τ_{α | y, s_1, ..., s_J} = τ_α + τ_y + J τ_s.
- Precision transmitted through messages:
  - Given ̄m = μ y + (1−μ) ̄s and ̄m = α + μ η_y + (1−μ) ̄η,
  - τ_{α | m_1, ..., m_J} = τ_{α | ̄m} = τ_α + 1 / [μ^2 (1/τ_y) + (1−μ)^2 (1/(J τ_s))].
- Proposition 3.6 (information loss):
  - τ_{α | m_1, ..., m_J} < τ_{α | y, s_1, ..., s_J}. (Equation (11))
  - Intuition: correlation across analysts’ messages induces an information loss because investors cannot separate correlation coming from the fundamental from correlation coming from message similarity. The loss disappears only if analysts’ information sources were uncorrelated conditional on α (e.g., no public y).

### Conclusion — implications and central messages
- Endogenous private information supplied by profit-maximizing analysts together with R correlated with aggregate investment (φ > 0) generates a beauty contest among analysts.
- Main findings:
  - Multiplicity: there can be either 1 or 3 linear symmetric equilibria depending on φ, τ_α, τ_y, τ_s, and J.
  - Equilibrium types: regular (public information overweighted) and inverted (collective misinterpretation μ < 0 and over-reliance on private signals).
  - Information inefficiency: investors receive strictly less precision about α than is available in the economy because of correlated messages, even though each analyst individually reveals all his information.
- Economic interpretation:
  - High φ amplifies conformism and can lead to extreme and self-fulfilling collective behaviors, including bubble-like outcomes where analysts’ coordinated signals validate counterintuitive investment recommendations.

*Source: Extracts from "I. INTRODUCTION", "II. The model", "III. Equilibrium", and "IV. Conclusion" in the provided PDF content.*

### References .............................................................................................................

### References — I. INTRODUCTION (excerpt)

### Model setup and assumptions
- Investors buy information (messages) from financial analysts to make investment decisions; the return of the investment is unknown and endogenous and is correlated with aggregate investment.
- Two building blocks:
  - Analysts determine messages based on public and private information.
  - A game among informed investors whose information consists of messages purchased from analysts.
- Analysts hold two signals:
  - A private signal (privileged information and their estimate of the impact of commonly observed information whose interpretation differs across analysts).
  - A public signal (commonly observed and identically interpreted by all analysts).
- Investors cannot directly observe analysts’ signals; only analysts process these signals and sell messages.
- Analysts’ objective: maximize profit. Investors’ objective: maximize expected final wealth (net value of investment minus cost of acquiring messages).
- Equilibrium restricted to linear and symmetric equilibria characterized by four properties:
  1. The amount invested by investor i is linear in messages and does not depend on the prices of the messages.
  2. Every message is sold to all the investors.
  3. Every analyst sells all of his information.
  4. The price of a message is the maximum price that an investor is willing to pay, given his equilibrium information set.
- Equilibrium messages are linear combinations of the analyst’s public and private signals.
- The correlation between return and aggregate investment generates a beauty contest among analysts: analysts must predict other analysts’ messages to forecast returns.

### Equilibrium behavior and "conformism"
- An analyst’s equilibrium behavior balances:
  - Giving a good prediction about the fundamental.
  - Conforming to other analysts’ predictions.
- The weight on the average message is interpretable as a degree of “conformism”.
- Conformism has two components:
  - Exogenous component: related to the correlation between the return and the aggregate investment.
  - Endogenous component: related to the precision of the messages sold by analysts.

### Main results
- Multiplicity:
  - There are either one or three linear symmetric equilibria.
  - If the correlation parameter tends to 0, there is a unique equilibrium; the degree of conformism of this equilibrium tends to 0.
  - When the correlation parameter tends to 1, there are 3 equilibria; the degree of conformism of the 3 equilibria tends to infinity.
- Two different patterns of behavior across equilibria:
  - Standard “beauty contest” equilibria (à la Allen, Morris and Shin (2006)): weight on public information is excessive to provide information on others’ messages.
  - “Inverted” equilibria: the beauty contest effect is exacerbated, displaying self-fulfilling misinterpretation of the public signal and overconfidence in private information.
    - Example of misinterpretation: a high public signal a priori corresponds to a high return, yet messages sold by analysts may be negatively correlated with the public signal due to endogeneity of the correlation between messages and the return; this negative correlation can be self-fulfilling.
- Collective dynamics:
  - For some fundamentals, conformism (the magnitude of the beauty contest) can be very strong, allowing analysts to coordinate on different interpretations of signals; each interpretation can be self-fulfilling, leading to “collective manipulation”.

### Information transmission and loss
- Correlation among analysts’ information sources and their role in providing information to investors imply a loss in information transmitted to investors, even when each analyst transmits all of his information to all investors.
- Messages bought by investors are correlated; this correlation comprises:
  - Information conveyed about the fundamental.
  - Pure noise.
- Investors cannot separate these components, creating an information loss that disappears only if analysts’ sources of information were uncorrelated.

### Relation to literature
- Closest in spirit to Allen, Morris and Shin (2006); similarity: overweighing of the public signal relative to private signal, but driven here by analyst conformism rather than forward-looking asset pricing.
- Connects with literature on beauty contests, public news, and social welfare (Morris and Shin (2002), Hellwig (2005), Angeletos and Pavan (2007)).
- Distinct from literature on reputational effects and verifiability of analysts’ messages (Trueman (1994), Crawford and Sobel (1982), Prendergast (1993), Ottaviani and Sorensen (2001))—this paper assumes investors rely completely on analysts’ messages and does not model analysts’ information acquisition (Admati and Pfleiderer (1986, 1988, 1990), Barlevy and Veronesi (2000), Verrecchia (1982)).
- Links to literature on herding, inefficiency, and volatility (Brunnermeier (2001), Chamley (2003), Ottaviani and Sorensen (2000), Abreu and Brunnermeier (2003), Prat and Dasgupta (2006), Prat, Dasgupta and Verardo (2006)).
- Uses an investment technology related to models with strategic complementarities and extensions of binary action games (Farmer 1999; Morris 2002).

*Source: Extract from "I. INTRODUCTION" in the provided PDF content.*

### Section 2 presents the model. Section 3 describes the equilibrium. The pricing equation, the

### _wp1133 - Section 2 presents the model. Section 3 describes the equilibrium. The pricing equation, the

### II. The model — setup and timing
- Two building blocks:
  - (i) A market for information where J financial analysts determine information sold to investors.
  - (ii) A game among a continuum [0,1] of infinitesimal investors whose investment return is endogenous.
- Timing:
  - First: analysts offer messages m_j on the return R; investors choose which messages to acquire.
  - Then: investors choose investment sizes x_i.
- Return specification:
  - R = α + φX, where φ ∈ [0,1], α ~ Normal(0, precision τ_α), and X = ∫ x_i di is aggregate investment.
  - This links R to aggregate investment and generates a “beauty contest” among analysts because X depends on analysts’ messages.
- Analysts’ information:
  - Each analyst j observes private signal s_j = α + η_j with precision τ_s (η_j i.i.d., mean 0).
  - All analysts observe a common signal y = α + η_y with precision τ_y (η_y mean 0).
  - Define ̄η = ∑_j η_j / J (precision Jτ_s) and ̄s = α + ̄η.
- Messages and market for information:
  - Analyst j commits to a single message m_j (one report) sold at price q_j to proportion N_j of investors; production cost of information is zero.
  - Assumption A (normality): (R, y, s_1, ..., s_J, m_1, ..., m_J) is Gaussian — ensures linearity of outcomes and tractability.
  - Implicit market assumptions: no price discrimination, no resale of messages, analysts do not buy other analysts’ messages; analysts “truthfully commit” to distributional properties of m_j.
- Investors’ objective and optimal action:
  - Investor i maximizes Rx_i − 1/2 x_i^2 − ∑_{j∈I_i} q_j.
  - Given normality, optimal investment is x_i(I_i) = E(R | I_i). (Property 1)
  - x_i is linear in messages and does not depend on message prices q_j.

### III. Equilibrium — definition, properties, and characterization
- Four equilibrium properties (in symmetric linear equilibrium):
  - Property 1: x_i(I_i) = E(R | I_i) (linear in messages; independent of q_j).
  - Property 2: Every message m_j is sold to all investors (symmetry and zero production cost).
  - Property 3: Every analyst sells all of his information: distribution of R | m_j equals R | (s_j, y).
    - Assumption B: each analyst treats joint distribution of (R, s_j, y) as exogenous (return-taking behavior).
  - Property 4: Price q_j equals the maximum price an investor is willing to pay given equilibrium information set I_i.
- Linear symmetric equilibrium definition (Definition 3.3):
  - Investors buy all messages and invest E(R | m_1, ..., m_J).
  - Analysts sell m_j = μ y + (1−μ) s_j at price q_j = 1/2 (Var(R | all messages except j) − Var(R | all messages)).
- Equilibrium message form (Definition 3.1):
  - ∀j, m_j = μ y + (1−μ) s_j, with μ independent of j and weights normalized to sum to 1.
  - Equivalent representation: m_j = α + μ η_y + (1−μ) η_j.
- Pricing equation for q_j (Lemma 3.2):
  - q_j = 1/2 (Var(R | m_1, ..., m_{j−1}, m_{j+1}, ..., m_J) − Var(R | m_1, ..., m_J)).
- Return equation using average message ̄m (because ̄m is sufficient statistic):
  - R = α + φ E(R | ̄m). (Equation (6))
  - Hence E(R | ̄m) = (1/(1−φ)) E(α | ̄m) and R = α + λ ̄m (Equation (7)).
  - λ = φ/(1−φ) · cov(α, ̄m)/Var(̄m) = φ/(1−φ) · [1/τ_α] / [1/τ_α + μ^2 (1/τ_y) + (1−μ)^2 (1/(J τ_s))]. (Equation (8))
- Beauty contest interpretation:
  - Equation (9): ∀j, E(R | s_j, y) = E(α | s_j, y) + λ E(̄m | s_j, y).
  - Analysts trade off predicting fundamentals α vs. conforming to others’ predictions (degree of conformism = λ).
  - Conformism depends on exogenous φ/(1−φ) and endogenous cov(α, ̄m)/Var(̄m).

### Multiplicity of equilibria — main results and patterns
- Equilibrium μ solves a cubic polynomial derived from consistency condition μ = μ′/(μ′+ν′) (details in Proposition 3.4 and proof).
- Proposition 3.4:
  - Assume φ ∈ [0,1]. There exists a threshold φ^* such that:
    1. If φ > φ^*, then there are 3 linear symmetric equilibria.
    2. If φ < φ^*, then there is 1 linear symmetric equilibrium.
  - One equilibrium always has μ ∈ [τ_y/(τ_s + τ_y), 1], while the two others (when they exist) satisfy μ ≤ −τ_y/τ_α.
  - The threshold φ^* = 1/(J−1) J A^* + 1 ∈ (0,1), where
    - A^* = sup_{μ ≤ −τ_y/τ_α} (μ−1) (τ_y + μ τ_α) / (τ_y − (τ_s + τ_y) μ) · (1 + μ^2 τ_α/τ_y + (1−μ)^2 τ_α/(J τ_s)) ∈ (0, +∞).
  - Existence of 1 or 3 equilibria follows from solving a degree-3 polynomial in μ.
- Corollary 3.5 (limit cases):
  - As φ → 0:
    - Unique equilibrium; λ → 0; μ → τ_y/(τ_y + τ_s).
  - As φ → 1:
    - There are 3 equilibria; λ → +∞ for the 3 equilibria.
    - One μ → 1, another μ → −τ_y/τ_α, the third μ → −∞.
  - Interpretation:
    - φ = 0: R = α exogenous; no beauty contest; analysts reveal E(α | s_j, y).
    - φ ≈ 1: multiplicity robust; equilibria correspond to extreme conformism and possibly inverted behavior.

- Equilibrium patterns and interpretation (Comment 2):
  - “Regular” equilibria: μ ∈ [0,1], overweight public information (standard beauty-contest behavior).
  - “Inverted” equilibria: μ < 0, collective misinterpretation of public signal and overconfidence in private information:
    - Self-fulfilling misinterpretation: m_j reacts negatively to y (μ < 0) — collective voluntary misinterpretation becomes self-fulfilling.
    - Overconfidence: weight (1−μ) on s_j > 1, implying exaggerated reliance on private signals.
  - “Inverted” equilibria can be interpreted as bubble-like stories: e.g., negative public signal y leads analysts to recommend investing due to coordination motives, producing higher aggregate X and higher realized return R, validating the inverted recommendation.

### Information precision and aggregate information loss
- Precision in economy:
  - Precision available about α in the economy (public y plus private s_j): τ_{α | y, s_1, ..., s_J} = τ_α + τ_y + J τ_s.
- Precision transmitted through messages to investors:
  - Given ̄m = μ y + (1−μ) ̄s and ̄m = α + μ η_y + (1−μ) ̄η, the precision is
    - τ_{α | m_1, ..., m_J} = τ_{α | ̄m} = τ_α + 1 / [μ^2 (1/τ_y) + (1−μ)^2 (1/(J τ_s))].
- Proposition 3.6 (information loss):
  - τ_{α | m_1, ..., m_J} < τ_{α | y, s_1, ..., s_J}. (Equation (11))
  - Not all information available on α is transmitted to investors despite each analyst individually revealing all his information: correlation across analysts’ messages creates an information loss because investors cannot fully disentangle signal correlation from noise correlation.
  - If analysts’ information sources were uncorrelated conditional on α (for example, no public y), the inequality would be an equality.

### IV. Conclusion — implications and central messages
- Endogenous private information (messages sold by profit-maximizing analysts) together with R correlated with aggregate investment generates a beauty contest among analysts.
- Main result: multiplicity of equilibria can arise (1 or 3 linear symmetric equilibria) depending on φ and the information precisions (τ_α, τ_y, τ_s) and J.
- Equilibrium types:
  - Regular beauty-contest equilibrium where public information is overweighted.
  - Inverted equilibria where analysts collectively misinterpret public signal (μ < 0) and over-rely on private signals — these can be self-fulfilling and resemble bubble dynamics.
- Information inefficiency: the presence of analysts and correlated information sources induces an information loss in what investors ultimately receive, even when each analyst transmits all his information.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1133.pdf*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1133.pdf_