Crises and Prices: Information Aggregation, Multiplicity, and Volatility by George-Marios Angeletos and Iv´ an Werning (AER, 2007) Pau Roldan NYU February 26, 2014 1 / 23
Motivation Exogenous versus endogenous information ◮ Crises are times of high nonfundamental volatility in which informative variables are being closely monitored. ◮ This model : ◮ Introduction of a financial market in a coordination game with imperfect information. ◮ Information will be endogenous (through equilibrium outcomes). ◮ Asset price aggregates dispersed private information and will act as a public noisy signal. ◮ Results : ◮ Uniqueness may no longer obtain as a perturbation from perfect information ◮ Sunspots will deliver nonfundamental volatility in crises. ◮ Nonfundamental volatility may arise when agents use sunspots to coordinate on different equilibria. 2 / 23
Motivation Exogenous versus endogenous information ( ctd. ) ◮ Aim is to study role of endogenous information during crises: ◮ Investigate nonfundamental volatility arising from sunspots (multiplicity of equilibria). ◮ Examine, when there is uniqueness, sensitivity of outcomes to nonfundamental disturbances (noise in public signals). ◮ Main insight: ◮ Precision of endogenous public information increases with the precision of exogenous private information. ◮ If private signals are more precise, asset demands are more sensitive and equilibrium prices react more to fundamental than to non fundamental variables, conveying more precise information. 3 / 23
Motivation Coordination: Uniqueness versus multiplicity ◮ More precise public information allows individuals to forecast one another’s actions and allows easier coordination. ◮ This gives rise to multiplicity when information (private or public) has little noise: 4 / 23
Outline Motivation Model Exogenous Information Endogenous Information Extensions Extension 1: Price Multiplicity Extension 2: Observing One Another Conclusions 5 / 23
Outline Motivation Model Exogenous Information Endogenous Information Extensions Extension 1: Price Multiplicity Extension 2: Observing One Another Conclusions 6 / 23
Baseline: Exogenous Information Basic Setting ◮ Measure-one continuum of agents, i ∈ [0 , 1]. ◮ Choice of action a i ∈ { 0 , 1 } (attack status quo or not). ◮ Cost c ∈ (0 , 1) of attacking. � 1 ◮ Let A := 0 a i di be size of attack, θ an exogenous fundamental. ◮ Status quo is abandoned (attack is successful) if A > θ . ◮ Individual payoff: U ( a i , A , θ ) := a i ( 1 [ A >θ ] − c ) ◮ Coordination: U (1 , A , θ ) − U (0 , A , θ ) increases with A . 7 / 23
Baseline: Exogenous Information Information structure ◮ State θ is not common knowledge (imperfect information). ◮ Common diffuse prior : Nature draws θ ∼ unif ( −∞ , + ∞ ). ◮ Private signals : Mutually independent, uncorrelated with θ . ξ i ∼ N (0 , 1) x i = θ + σ x ξ i , ◮ Exogenous (for now) public signal : z = θ + σ z ε, ε ∼ N (0 , 1) ◮ Denote precisions by: α x := σ − 2 α z := σ − 2 and x z 8 / 23
Baseline: Exogenous Information Symmetric Perfect Bayesian Equilibrium ◮ As in Morris and Shin, focus is on monotone PBE. ◮ For each z , agent i will attack iff x i ≤ x ⋆ ( z ). ◮ Aggregate size of attack is A ( θ, z ) = P [ x ≤ x ⋆ ( z ) | θ ]. ◮ Status quo is abandoned if θ ≤ θ ⋆ ( z ), where θ ⋆ ( z ) solves A ( θ, z ) = θ , or 1 Φ − 1 [ θ ⋆ ( z )] x ⋆ ( z ) = θ ⋆ ( z ) + √ α x (1) ◮ x ⋆ ( z ) solves indifference condition P [ θ ≤ θ ⋆ ( z ) | x , z ] = c , or � √ α x + α z θ ⋆ ( z ) − α x x ⋆ ( z ) + α z z � �� Φ = c (2) α x + α z ◮ Equilibrium is the joint solution to (1) and (2). 9 / 23
Baseline: Exogenous Information Symmetric Perfect Bayesian Equilibrium ( ctd. ) Proposition (Morris and Shin) The equilibrium is unique iff private noise is small relative to public noise: √ σ x ≤ 2 π σ 2 z ◮ Intuition : ◮ When private information is more diverse, it is more difficult for individuals to predict the actions of others. ◮ When this effect is strong enough, multiplicity breaks down. 10 / 23
Outline Motivation Model Exogenous Information Endogenous Information Extensions Extension 1: Price Multiplicity Extension 2: Observing One Another Conclusions 11 / 23
Endogenous Information Motivation ◮ Until now, private and public noise were independent from each other because information structure was exogenous. ◮ Any small noise away from perfect information (where σ x = σ z = 0) ensured uniqueness (Morris and Shin). ◮ Thus, non fundamental volatility disappears when private noise vanishes. ◮ Now, introduce a financial market where agents can trade prior to the coordination game. ◮ Dividends will depend on aggregate attack and equilibrium prices will convey information. 12 / 23
Endogenous Information Setting ◮ Nature draws θ ∼ unif ( −∞ , + ∞ ). ◮ Private signal for agent i : σ x > 0, ξ i ∼ N (0 , 1) x i = θ + σ x ξ i , ◮ First stage : ◮ Trade over risky asset with dividend f ( θ ) of price p . ◮ Utility of agent i : − e − γ w i , V ( w i ) = γ > 0 w i = w 0 − pk i + fk i where k i is investment in risky asset. ◮ Stochastic asset supply (prices are not fully revealing): K s ( ε ) = σ ε ε, σ ε > 0, ε ∼ N (0 , 1) ◮ σ ε is exogenous noise in aggregation process. 13 / 23
Endogenous Information Setting ( ctd. ) ◮ Second stage : ◮ After trade, agents observe stage-1 price p and choose whether or not to attack status quo. ◮ Utility for agent i : U ( a i , A , θ ) = a i ( 1 [ A >θ ] − c ) � 1 where A := 0 a i di . ◮ Regime outcome, asset’s dividend and payoffs from both stages realized at the end of stage 2. ◮ Equilibrium definition combines rational-expectations equilibrium (first stage) and PBE (second stage). 14 / 23
Endogenous Information Equilibrium definition Definition 1. An equilibrium is a price function P ( θ, ε ), individual strategies for investment and attacking, k ( x , p ) and a ( x , p ), and their corresponding aggregates, K ( θ, p ) and A ( θ, p ), such that: ∈ E [ V ( w 0 + ( f ( θ ) − p ) k ) | x , p ] k ( x , p ) arg max k ∈ R K ( θ, p ) = E [ k ( x , p ) | θ, p ] K s ( ε ) K ( θ, P ( θ, ε )) = a ( x , p ) ∈ arg max a ∈{ 0 , 1 } E [ U ( a , A ( θ, p ) , θ ) | x , p ] A ( θ, p ) = E [ a ( x , p ) | θ, p ] 2. The equilibrium regime outcome is R ( θ, ε ) := 1 [ A ( θ, P ( θ,ε )) >θ ] . 15 / 23
Endogenous Information Equilibrium analysis ◮ Use f ( θ ) = θ . ◮ Guess a linear price function that is not perfectly revealing. ◮ Price is a normally distributed public signal with precision α p . ◮ By Bayes’ rule: � α x x + α p p 1 � θ | x , p ∼ N , α x + α p α x + α p ◮ Equilibrium asset demand (Grossman and Stiglitz (1980)): k ( x , p ) = E [ f ( θ ) | x , p ] − p = α x ( x − p ) γ V [ f ( θ ) | x , p ] γ ◮ Equilibrium price: P ( θ, ε ) = θ − σ p ε 16 / 23
Endogenous Information Equilibrium analysis ( ctd. ) ◮ Linear price guess is verified for: σ p = γσ ε σ 2 x ◮ Thus, unlike before, public information improves with private information (lower σ x means lower σ p ). ◮ Before : ◮ Precision of public information was fixed, so that sufficiently precise private information ensured uniqueness. ◮ Now : ◮ Better private information improves public information and reduces strategic uncertainty to ensure multiplicity. ◮ There is multiplicity even for a small deviation of σ x or σ p from zero (the perfect-information benchmark). ◮ Crises (periods of high nonfundamental volatility) can be addressed even when σ x → 0. 17 / 23
Endogenous Information Equilibrium analysis ( ctd. ) Proposition There are multiple equilibria if either source of noise is small is 1 small, so that σ 2 ε σ 3 x < γ 2 √ 2 π . Proposition As either source of noise vanishes ( σ x → 0 or σ ε → 0 ), 1. There exists a passive equilibrium such that R ( θ, ε ) → 0 , ∀ θ ∈ ( θ, θ ) 2. There exists an aggressive equilibrium such that R ( θ, ε ) → 1 , ∀ θ ∈ ( θ, θ ) ◮ That is, unlike in the baseline Morris and Shin, regime outcome is fully sunspot-driven! 18 / 23
Outline Motivation Model Exogenous Information Endogenous Information Extensions Extension 1: Price Multiplicity Extension 2: Observing One Another Conclusions 19 / 23
Price Multiplicity ◮ Before, financial market (first stage) endogenously provided information to coordination game (second stage) through prices. ◮ Now, feedback is in the opposite direction as well. ◮ This will give multiplicity also in the equilibrium price. ◮ Setting : ◮ Same as before, except now dividend is endogenous, a function of the aggregate state: f ≡ f ( A ) = − Φ − 1 ( A ) ◮ Linear monotone equilibrium features: ◮ σ p = γσ ε σ x , i.e. public info improves again with private info. ◮ Backward-bending asset demand K ( θ, p ), thus price multiplicity in a non-empty parameter region of ( θ, ε ). 20 / 23
Outline Motivation Model Exogenous Information Endogenous Information Extensions Extension 1: Price Multiplicity Extension 2: Observing One Another Conclusions 21 / 23
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