Heterogeneity of Investors and Asset Pricing in a Risk-Value World



Appendix A: Proof of Lemma 1

First, we consider an increase in Wq to >Wi, holding the required expected
return
R* constant; b > 1. Then η changes to ; a> 0. Define eɪ := (e e)
for the initial endowment
Wq and define eb := (ee) for the initial endowment
bW^o. Then we need to show that

E[f(e1)]E[f(eb)]
—     —

or

aE[f(e1)] >E[f(eb)].                           (22)

From equation (15) it follows that V ε,

— E[f (eb)] + f (ebS') = θε = a( — E[f (ei)] + f (ele))-         (23)

As the mean absolute deviation between payoffs across states has to grow
with
Wq , the monotonicity of f implies that also the mean absolute deviation
I E[f (e)] f (e)] I has to grow. Hence a > 1. Now assume, by contradiction,
that inequality (22) is not true. Then equation (23) implies

f (e)af (ey, V ε.                          (24)

As a > 1 and f > 0, this implies

f (e) > f (ey, V ε.

36



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