Allen, Morris and Shin, JFS 06Abreu and Brunnermeier, Ecta 03Allen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03Advanced Macroeconomics IECON 525a - Fall 2009Yale UniversityGuillermo L. Ordo˜nezWeek 5 - BubblesGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03IntroductionWhy a rational representative investor model of asset prices does notgenerate bubbles?Martingale property: LIE (Law of iterated expectations).This is not the case with heterogeneity, since in general, averageexpectations fail to satisfy LIE.When private information is heterogeneous, agents rely excessively inpublic signals. HenceMean price path deviates from consensus liquidation valuesPrices exhibit inertia.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03IntroductionWhy a rational representative investor model of asset prices does notgenerate bubbles?Martingale property: LIE (Law of iterated expectations).This is not the case with heterogeneity, since in general, averageexpectations fail to satisfy LIE.When private information is heterogeneous, agents rely excessively inpublic signals. HenceMean price path deviates from consensus liquidation valuesPrices exhibit inertia.Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03Fail of LIE with heterogeneous informationLIE with private informationEit(Ei ,t+1(θ)) = Eit(θ)LIE with public informationE∗tE∗t+1(θ)= E∗t(θ)LIE fail in averages with asymmetric informationEtEt+1(θ)6= Et(θ)Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03BasicsInformation at all dates:θ ∼ N (y ,1α)Signals: xi= θ + i, where i∼ N (0,1β)Average expectation of average expectations.ET −tt(θ) ≡ EtEt+1...ET −1(θ)= 1 −βα + βT −t!y+βα + βT −tθSee thatET −tt(θ) 6= Et(θ) =1 −βα + βy +βα + βθGuillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03No learning through pricesIfpt= Et(pt+1)thenpt= 1 −βα + βT −t!y +βα + βT −tθHow to obtain the equation for pt?How to deal with learning from past prices?Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03ModelSingle risky asset, liquidated at T + 1 but traded from 1 to T .Liquidation value θ is determined before date 1. θ ∼ N (y ,1α)Overlapping generation of no wealth constrained traders, each livingfor two periods and consuming in the second period. u(c) = −e−cτInformation set: {y , p1, p2, .., pt, xit} where xit= θ + itandit∼ N (0,1β)Each period exogenous net supply of assets st∼ N (0,1γ)Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03Path of fundamental valueold, they unwind their asset holding and acquire the consumption good,consume, and die. Thus, at any trading date t, there is a unit mass ofyoung traders and a unit mass of old traders.Our assumption of overlapping generations of traders is intende d toaccentuate the importance of short-run price movements for traders.Even for long-lived traders, if they have a preference for smoothingconsumption over time, they will care about short-run price movements,as well as the underlying fundamental value of the asset at its ultimateliquidation. The assumption of overlapping generations is used as adevice to throw into sharper relief this concern for short-run prices. Ourmotivation for making this assumption is to attempt to capture some ofthe intuition behind Keyne s’s beauty-contests metaphor, in which tradersare motivated to second- and third-guess other traders in order to profitfrom short-run price movements. Our assumption of overlapping genera-tions of traders makes it similar to the short-lived trader version of themodel examined in Brown and Jennings (1989).6When each ne w trader is born, they do not know the true value of .However, for a trader i born at date t, there are two sources of informa-tion. First, the full history of past and current prices are available,including the ex ante mean y of . Second, this trader observes therealization of a private signalxit¼  þ "it6In contrast, other authors such as He and Wang (1995) have examined the case where traders onlyconsume at the terminal date and trade up to that date. We discuss this case further below.Figure 1Path of fundamental valueBeauty Contests and Iterated Expectations727Guillermo L. Ordo˜nez Advanced Macroeconomics I ECON 525a - Fall 2009 Yale UniversityAllen, Morris and Shin, JFS 06 Abreu and Brunnermeier, Ecta 03Private Informationwhere "itis a normally distributed noise term with mean 0 and variance1=. We assume that the noise terms f"itg are i.i.d. across i ndividuals iand across time t. There is no other source of information for the trader.In particul ar, the private signals of the previous generation of tradersare not observable. Hence, the information set of trader i at date t isy; p1; p2; ; pt; xitfgwhere ptis the price of the asset at date t (Figure 2). As a convention, wetake pTþ1¼ . All traders have the exponential utility functionuðcÞ¼ecdefined on consumption c when they are old. The parameter is the reciprocal of the absolute risk aversion, and we shall refer to it asthe traders’ risk tolerance.Finally, in each trading period, we assume that there is an exogenousnoisy net supply of the asset, st, distributed normally with mean 0 andprecision t. The supply noise is independent over time, and independentof the fundamentals and the noise in traders’ information.The modeling device of noisy supply is commonly adopted in rationalexpectations models so as to prevent the price from being a fully revealingsignal of the fundamentals. Noisy supply is sometimes justified as theresult of noise traders or in terms of the subjective uncertainty facingtraders on the ‘‘free float’’ of the asset that is genuinely available for sale[see Easley and O’Hara