Advanced Macroeconomics I ECON 525a - Fall 2009 Yale University

Transcription

1 Advanced Macroeconomics I ECON 525a - Fall 2009 Yale University Week 5 - Bubbles

2 Introduction Why a rational representative investor model of asset prices does not generate bubbles? Martingale property: LIE (Law of iterated expectations).

3 Introduction Why a rational representative investor model of asset prices does not generate bubbles? Martingale property: LIE (Law of iterated expectations). This is not the case with heterogeneity, since in general, average expectations fail to satisfy LIE. When private information is heterogeneous, agents rely excessively in public signals. Hence Mean price path deviates from consensus liquidation values Prices exhibit inertia.

4 Fail of LIE with heterogeneous information LIE with private information E it (E i,t+1 (θ)) = E it (θ) LIE with public information E t ( E t+1 (θ) ) = E t (θ) LIE fail in averages with asymmetric information E t ( E t+1 (θ) ) E t (θ)

5 Basics Information at all dates: θ N (y, 1 ) α Signals: x i = θ + ɛ i, where ɛ i N (0, 1 ) β Average expectation of average expectations. ( E T t t (θ) E t ( E t+1 (...E T 1 (θ) )) = See that E T t t (θ) E t (θ) = 1 ( ) ) T t ( ) T t β β y+ θ α + β α + β ( ( )) ( ) β β 1 y + θ α + β α + β

6 No learning through prices If then p t = ( 1 p t = E t (p t+1 ) ( ) ) T t ( ) T t β β y + θ α + β α + β How to obtain the equation for p t? How to deal with learning from past prices?

7 Model Single risky asset, liquidated at T + 1 but traded from 1 to T. Liquidation value θ is determined before date 1. θ N (y, 1 α ) Overlapping generation of no wealth constrained traders, each living for two periods and consuming in the second period. u(c) = e c τ Information set: {y, p 1, p 2,.., p t, x it } where x it = θ + ɛ it and ɛ it N (0, 1 β ) Each period exogenous net supply of assets s t N (0, 1 γ )

8 Path of fundamental value Beauty Contests and Iterated Expectations Figure 1 Path of fundamental value

9 result of noise traders or in terms of the subjective uncertainty facing traders on the free float of the asset that is genuinely available for sale [see Easley and O Hara (2001), footnote 9, page 52]. One potentially unsatisfactory feature of our setup is the feature that random net supplies are independent draws over time. When there are overlapping generations of traders, such an assumption is Allen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03 Private Information Figure 2 Private information

10 Price at date T Trader i s demand at date T D it = Market clearing is given by D T = Then, the price at date T is τ V it (θ) (E it (θ) p T ) τ ( ) E T (θ) p T = st V T (θ) p T = E T (θ) V T (θ) s T τ

11 Price at date t The asset price at date T 1 is p T 1 = E T 1 (p T ) V T 1(p T ) τ The asset price at a general date t is s T 1 = E T 1 E T (θ) V T 1(p T ) s T 1 τ p t = E t E t+1...e T (θ) V t(p t+1 ) s t τ

12 Proposition 1. For all t < T Allen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03 Main results E s ðjp t jþ > E s E t ðþ It is only at the final trading date, T, that we have E s ðp T Þ¼E s E T ðþ. Figure 3 Mean of time paths of p t and E t ðqþ

13 Main results Prices deviate systematically from the average expectation of the fundamental value of the asset. Inertia of prices. Intuition: Excessive weight assigned to the public signal y and previous prices.

14 Main results For risk neutral traders or infinitely precise signals, prices are fully revealing of the fundamental value. This is p t E t (θ) θ. As investors become very risk averse (τ 0), they are less aggressive and prices are less informative. This is p t q t

15 Main ideas Rational arbitrageurs may know the price of an asset exceeds the fundamental and still decide not to sell. The key is they do not know when the bubble will burst, where it is required a critical mass of speculators to do it. Main elements for this to work: Dispersion of opinions among arbitrageurs. Need for coordination.