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 aη; a> 0. Define eɪ := (e — e)
for the initial endowment Wq and define eb := (e—e) for the initial endowment
bW^o. Then we need to show that
E[f(e1)] > E[f(eb)]
— —aη
or
aE[f(e1)] >E[f(eb)]. (22)
From equation (15) it follows that V ε,
— E[f (eb)] + f (ebS') = aη 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 (ebε) ≥ af (elεy, V ε. (24)
As a > 1 and f > 0, this implies
f (ebε) > f (elεy, V ε.
36
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