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



so that πz1 (ε)/ π2 (ε) must decline in ε everywhere. Suppose that
πz1 (ε)/ π'2 (ε) 1 at the first intersection of πi (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
πz1 (ε)/ πz2(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)- = %π (ɛ); V i,ε.

(30)


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

ε. This yields

fz z(⅛)
fl (⅛)


de,∙

-j→


d2ei∕dε2
dei / dε


= —c&;


V i, ε.


(31)


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

^ fZ(e^) fdeΛ2     ( . y                 ∕o9

Af(⅛) Ы = v ε         (32)

since i d2ei∕dε2 = 0.

In the HARA-case with 7 > —œ,

42



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