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

This proves the second part of equation (21). The proof of proposition 9
is the same for 7 = -∞.

Appendix H: Proof of Proposition 10

For a HARA-based risk function with 1 > 7 > -∞ the FOC (11) resp.

(14) yields

7-1

or

. e,∙^              ,          .                            ,, i_ . .                                               . ,

A + -----= (—% K — 1 + ^si ])⁷^¹; V i,ε-           (43)

1 — 7

Aggregating across investors yields A + e_ε /(1 — 7). Dividing the aggregate
equation by A_i + e_iε∕(1 — 7) yields 1∕g_iε and hence,

so that

d(1∕g_ig) = ^  1   / —% λ ¹-⁷ / - — 1 + S_i ∖ ¹-7 ¹    S_j — s_i        )

^dε ʌl ^— 7 ∖~^l∣jJ ∖π_ε ^{— 1 + s}J     K ^{— 1 + s}3^{)2 π ε}'

(45)

Assume 7 < 1 and s⅛ ≤ s_j Vi. Then π^z(ε) < 0 implies d(1∕g⅛_ε)∕dε < 0 so
that dg_hε∕dε > 0.

45

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