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

In the traditional state preference-model, the investor maximizes his ex-
pected utility E [u (e)] subject to the budget constraint (5). Let μ denote the
Lagrange-multiplier of the budget constraint. Then the first order conditions
imply for two investors i and j

^ui ^(eiɛ)        ^uj^(ejɛ)    ; ^{V ε}∙

μ_i     ^μ     μ_j

Cass and Stiglitz (1970) have shown that in the EU-model all investors
can have linear sharing rules only if u(e) belongs to the HARA class with the
exponent 7 being the same for all investors. Since (26) is formally the same
as (27), it follows for the risk-value model that all investor can have linear
sharing rules only if f (e) belongs to the HARA class with (7 — 1) being the
same for all investors. Therefore we can rewrite (17) as

Suppose that s_i = s_j. Then, the last equation can hold for every ε only
if 7 = 2. Hence F(e) must be quadratic for every investor.

Appendix C: Proof of Proposition 3

Differentiating equation (16) with respect to e_j∙ yields

Hence dei∕dej is a constant if and only if fj(êf/f(eβ) is a constant.
Then investor i’s sharing rule is linear relative to that of investor j. In-
vestor i’s sharing rule is strictly convex [concave] relative to that of investor
j if f (ef/f (e_i) is strictly increasing [decreasing] in e_j∙ and, hence, in the

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