Web Appendix to “ Learning about Risk and Return : A Simple Model of Bubbles and Crashes ” by William A . Branch and

Proof of Proposition 1. We use the stochastic approximation results described in Evans and Honkapohja (1998, 2001) and (Marcet and Sargent 1989b). Set γ1,t = γ2,t = t−1 and define zt = pt − θt−1Xt−1 + ut = ( T (θt−1;σt−1)− θt−1 ) Xt−1 − aβσt−1vt + ut. Then (12)-(15) in the main text, for the case of exogenous supply, can be re-written as the equations (20)-(22) in the print Appendix to the paper, reproduced here: