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 ∣>W_i, 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 e_b := (e—e) for the initial endowment
bW^o. Then we need to show that

E[f(e₁)] > E[f(e_b)]
—     —aη

or

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

From equation (15) it follows that V ε,

^{— E[f (e}b^)] + ^{f (e}b_S'⁾ = ^aη θ_ε = ^{a( — E[f (e}i^)] + ^{f (e}le⁾⁾-         ⁽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_bε) ≥ af (e_lεy, V ε.                          (24)

As a > 1 and f > 0, this implies

f (e_bε) > f (e_lεy, V ε.

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