23 sensitivity of our results to changes in distributional assumptions.We found that our results were essentially unchanged for very different consumption processes,provided that the mean and variances of growth rates equaled the historically observed values and the coefficient of relative risk aversion was less than ten.Using this insight on the robustness of the results to distributional assumptions from our earlier analysis we consider the case where the growth rate of consumptionis i.i.d and lognormal.We do this to facilitate exposition and because this results in closed form solutions As a consequence,the gross return on equity R.,(defined above)is i.i.d,and log-normal. Substituting U'(c )=c in the fundamental pricing relation and noting that in this exchange economy the equilibrium consumption process is y,} p,=BE{p1+y1)U@c,+i)1U@c,)} (9) we get p,=E{P+1+y+1)x (10) As p,is homogeneous of degree one in y,we can represent it as P=Wy and hence R+can be expressed as R.=(w+D).yuw+l w y W X+ (11) 6 In contrast to our approach,which is in the applied general equilibrium tradition,there is another tradition of testing Euler equations(such as equation 9)and rejecting them.Hansen and Singleton(1982)and Grossman and Shiller(1981)exemplify this approach. See Mehra and Prescott(1985)pages 156-57.The original framework also allowed us to address the issue of leverage. The exposition below is based on Abel(19).Our original analysis is presented in Appendix B23 sensitivity of our results to changes in distributional assumptions16. We found that our results were essentially unchanged for very different consumption processes, provided that the mean and variances of growth rates equaled the historically observed values and the coefficient of relative risk aversion was less than ten17. Using this insight on the robustness of the results to distributional assumptions from our earlier analysis we consider the case where the growth rate of consumption xt+1 ∫ c c t t +1 is i.i.d and lognormal. We do this to facilitate exposition and because this results in closed form solutions18. As a consequence, the gross return on equity Re t, (defined above) is i.i.d, and log-normal. Substituting ¢ = - Uc c t t ( ) a in the fundamental pricing relation and noting that in this exchange economy the equilibrium consumption process is { } yt p Ep y Uc Uc t tt t t t = + b { } ++ + ( )©( )/ ©( ) 11 1 (9) we get p Ep y x t tt t t = + { } + ++ - b a ( ) 1 11 (10) As pt is homogeneous of degree one in yt we can represent it as p wy t t = and hence Re t, +1 can be expressed as R w w y y w w x e t t t , t ( ) + + = + + ◊ = + ◊ 1 1 1 1 1 (11) 16 In contrast to our approach, which is in the applied general equilibrium tradition, there is another tradition of testing Euler equations (such as equation 9) and rejecting them. Hansen and Singleton (1982) and Grossman and Shiller (1981) exemplify this approach. 17 See Mehra and Prescott (1985) pages 156-57. The original framework also allowed us to address the issue of leverage. 18 The exposition below is based on Abel (1988). Our original analysis is presented in Appendix B