These conditions are related to those obtained by Leland (1980) in an EU
equilibrium.
4.2 HARA-Based Risk Functions
In this section, we derive more specific results assuming that the risk function
F belongs to the negative HARA-class. F(e) is given by equation (2) with
7 > 2 or 7 < 1. 7 is assumed to be the same for all investors.
A necessary condition for the existence of an equilibrium in the case of a
non-exponential risk function (7 > -∞) is
∑(A+J-_) ≡ a+
i 1
ê£
> о,
V ε with A ≡ ɪɪ Ai and eε ≡ ɪɪ eiε.
This condition must hold because Ai + eiε /(1 — 7) > 0 Vi, ε is required
by the FOC (11) resp. (14). Let e ≡ ∑i ei, then eε = ε — e. Hence an
equilibrium requires A + (ε — e) / (1 — 7) > 0. For the more important case
7 < 1 this implies ε ≥ e — A(1 — 7), V ε. If Wq ≡ ∑i Wqi, then in equilibrium
Wq is the forward market value of the aggregate payoff ε, i.e. Wq = U(επ(ε)).
We first show that investor i’s sharing rule relative to that of investor j
is either concave, linear or convex.
Proposition 4 ; Consider two investors i and j who measure risk by a neg-
ative HARA-function with 7 being the same for both. Then the following
statements are equivalent;
- Investor i,s sharing rule is strictly convex [linear] [strictly concave] rel-
ative to that of investor j.
- Investor i,s sharing constant is smaller than [equal to][greater than] that
of investor j.
Proof. See Appendix D.
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