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

so that —π^z₁ (ε)/ — π₂ (ε) must decline in ε everywhere. Suppose that
—π^z₁ (ε)/ — π'₂ (ε) > 1 at the first intersection of π_i (ε) and π₂ (ε). Hence it
must be less than 1 at the second intersection and greater than 1 at the
third intersection. But the latter condition contradicts the condition that
—π^z₁ (ε)/ — π^z₂(ε) declines. Hence a third intersection cannot exist. Also
π₁ (ε) > π₂ (ε) before the first intersection and after the second intersection
and vice versa in between follows.                                       ■

Appendix G: Proof of Proposition 9

Differentiate the first order condition (15) with respect to ε. This yields

. de .

^f(⅛)-_dε = ^—%^π (ɛ); V ⁱ,ε.

Multiply this equation by -1, take logs and differentiate with respect to

ε. This yields

First, we prove the first part of equation (21). Multiply equation (31) by
de_i∕dε and aggregate across investors. This yields

^ ^fZ(e^⁾ f^deΛ²     ( ∖. y                 ∕o₉∖

Af(⅛) Ы = ^{v ε         (32)}

since ∑_i d²e_i∕dε² = 0.

In the HARA-case with 7 > —œ,

42

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