Crises and Prices: Information Aggregation, Multiplicity and Volatility

Transcription

1 : Information Aggregation, Multiplicity and Volatility Reading Group UC3M G.M. Angeletos and I. Werning November 09

2 Motivation Modelling Crises I There is a wide literature analyzing crises (currency attacks, bank runs, nancial crashes...): "Coordination Failures" I If there is common knowledge about the fundamentals, multiple equilibria arise (I withdraw because I think others will withdraw...) I Multiplicity: Good or Bad? 1. Good: because there is heterogeneity in real-world and allows to explain seemingly irrational behavior. 2. Bad: because it gives little predictory power and it shows that the model is incomplete. I Solution: Fundamentals are not Common Knowledge, Morris and Shin (98, 99, 00)

3 Motivation This Paper I Morris and Shin showed that if there is no Common Knowledge about the fundamentals but the noise of the private signal is not too large, only one equilibrium survives. I Uncertainty di culties coordination and so, "coordination failures" dissapear. I Their information structure was simple: Two exogenous noisy signals; one private, the other public I However, it seems that in real-world applications, the precision of the signals is endogenous Example Argentinean Currency Crise (00) - bank positions and forward/peso rate closely monitored by media

4 Motivation This Paper I Use a version of Grossman-Stiglitz rational expectations economy with unobserved supply shocks I The price is the endogenous public signal. I Whenever, the private signal becomes more precise, the public signal will also become more precise. Intuition The more precise the signal, the more will trading strategies depend on it. Since the price aggregates those decisions, it will become a better estimator of the fundamental value. Implication Agents are able to forecast better the actions of the others and the coordination motive comes back Result Multiplicity obtains whenever the private signals are very precise

5 Morris Shin I There is a continuum of traders i 2 [0, 1] I Each trader trader may take two actions: attack a i = 1 or mantain the status quo, a i = 0 I Payo s are U(a i, A, θ) = a i [1 Aθ c] I θ is the main parameter, θ 2 R I A = R a i di measures the mass of attackers (if many people attacks, the attack succeeds)

6 Morris Shin I Suppose θ is common knowledge (standard literature) θ > 1 )No attack is the unique equilibrium θ 0 )Attack is the unique equilibrium θ 2 (0, 1] )Multiplicity I Assume now that θ is a random variable I Each agent observes (x i, z) where x i = θ + σ x ɛ i and z = θ + σ x ζ

7 Morris Shin I We shall look at PBE with the monotone property (attack if and only if x i x (z)) I In such an equilibrium A(θ, z) = Pr (x x (z) j θ) = Φ( 1 σ x (x (z) θ)) The Status quo is abandoned if θ θ(z), where θ(z) solves x (z) = θ(z) + σ x Φ 1 (θ(z)), so that A(θ(z), z) = θ Since at that thresold, traders must be indi erent whether to attack or not, we have Pr (θ θ(z) j x, z) = c I Together, these two conditions characterize the equilibrium

8 Morris Shin I There is only one solution to that system if 0 < σ x σ 2 z p 2π In the area where uniqueness obtains, private information is used more heavily, so that coordination cannot be achieved and uniqueness obtains. I Notice, that there is a unique equilibrium for arbitrarily small noise, leading to a discontinuity of the model

9 Financial Markets: Endogenous Information I Two stage game: First they trade a risky asset with dividend f at price p I Agents have CARA utility, so that V (ω i ) = e γω i and ω i = ω 0 + (θ p)k i I The supply of the asset is uncertain, K s (ɛ) = σ ɛ ɛ. This noise impedes perfect revelation of information through prices, needed for tractability I The second stage is the game played before.

10 Equilibrium De nition An Equilibrium for this economy is a price P(θ, ɛ), a pair of strategies k(x, p) and a(x, p), and their aggregates K (θ, p), A(θ, p) such that 1. k(x, p) 2 arg max E [V (ω 0 + (θ p)k) j x, p] 2. K (θ, p) = E [k(x, p) j θ, p] 3. K (θ, P(θ, ɛ)) = K s (ɛ) 4. a(x, p) 2 arg max a E [U(a, A(θ, p), θ) j x, p] 5. A(θ, p) = E [a(x, p) j θ, p] 6. The regime changes if and only if A(θ, P(θ, ɛ)) > θ

11 Equilibrium Analysis I Following GS, we look to linear price functions. It follows, that all random variables will also be normal and so, strategies will be linear (CARA) I In particular, demand for the asset will be k(x, p) = x p θ σ 2 ) K (θ, p) = p x γ σ 2 x γ Market clearing imposes that P(θ, ɛ) = θ σ ɛ σ 2 x γɛ And so the precision of the price, to identify θ depends positively on the precision of the private signal. I Hence, using the results in Morris-Shin, if the noise is small, there is multiplicity

12 Comparison I Comparison between the two models Whenever private information is more precise, public information is more precise, so that there is multiplicity.

13 Extensions: Multiple Prices I Price determined in the coordination game (the dividend depends on whether people attacked). Then, there are multiple equilibrium prices. I There may be equilibria in which high price leads to agents less willing to attack and so, prices signal good outcomes leading to upward sloping demands

14 Conclusions I Tractable and simple model of endogenous information in asset markets (usually a very hard problem) I Intuition regarding the continuity of the model to perturbations in information I Analysis of the feedback between coordination game and pricing I No clear intuition or policy recommendation: The examples used in the motivation are very di erent from the modelling assumptions.