Differentiate equation (15) with respect to θ. Recall, fi(e⅛) > 0, f∙ (e⅛) <
0, and fz(¾) > 0. Hence it follows that ei is an increasing and convex
function in θ. As θ = 1 — π, eɪ is a decreasing, convex function in π. This
proves proposition 1 ■
An absolute sharing rule relates investor i’s payoff ei to the aggregate
payoff ε. Proposition 1 states dei∕dπ < 0. Aggregation across investors
implies dε∕dπ < 0. Hence it follows that dei∕dε > 0. The positive slope of
the absolute sharing rule does not come as a surprise. This is still in line with
EU-theory. The major difference between the risk-value and the EU-model
is reflected in the shapes of the sharing rules. Equation (15) implies for two
investors i and j in the risk-value model
—E[fJ (êj)] + f^¾⅛- . ∀ ε
d
(16)
— E[fi (e¾)] + f (e⅛)
—di
Define si := E[fi(e⅛)]/ — ηi to be investor i ,s sharing constant. Equation
(16) can be written then as
—si + f⅛- = — ¾ + f ' ; ∀ ε. (17)
—η — ηj
The new element in risk-value models as compared to expected utility
models are the sharing constants. They are generated in the case of an ex-
ogenous benchmark by the payoff constraint and, in the case of an endogenous
benchmark, by the impact of a payoff change on the benchmark. In order to
grasp the intuition behind the sharing constants, consider the inverse sharing
constant - ηi∕E[f (e⅛)]. — ηi is the efficient increase in risk due to a marginal
reduction in the initial endowment available for buying claims, holding the
required expected payoff WqR* constant. This risk increase depends on the
19
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