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



Since fO, it follows that e < e and hence

ebε < elε,   ^ ε,

which contradicts the budget constraint (5). Therefore inequality (22)
must be true.

Second, we consider an increase in Rf so that e1 changes to ebo. Then η
changes to aoη. Hence the sharing constant decreases if inequality (22) holds
with
a and b being replaced by ao and bo. Therefore the same method by
which the first part of Lemma 1 has been proven can be applied here.

Appendix B: Proof of Proposition 2

a) Sufficiency: Suppose that F(e) is a quadratic function. Then f (e) =
a + be. Hence (17) implies linear relative sharing rules for two investors i and
j. Therefore all absolute rules are also linear.

b) Necessity: Differentiate (15) with respect to ε; this yields

ʃ//ʌ λdei
f (e) * =   ■/

(25)


Now suppose a linear absolute sharing rule for every investor: e = ai +
βi ε, so that β~ = βi. Then it follows from (25) for any two investors i and j

f (êis) η. = f> (êj£') ηl.' ’   ^ ε'

(26)


37



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