The name is absent

CARA under LS condition, then S (σ, μ) has to be not only constant in μ but also increasing
along rays through the origin. Thus, condition (5-iii) in addition to condition (4-ii) is
imposed.

Combining the three kinds of restrictions discussed above, the complete set of
conditions that MS function has to fully meet under LS condition is obtained. For example,
when an individual is risk averse of CARA, then Proposition 2 and table 1 indicate that the
MS function has to fully satisfy conditions (1), (2), (3-i), (3-ii), (3-iii), (4-ii) and (5-iii) (Other
cases are summarized in table 2). Besides, Proposition 1 indicates that if the MS function is
transformed, it needs to be linear transformation. In the following two sections, we consider
the specification of MS function for each type of risk aversion, taking into full consideration
the conditions shown in table 2.

Additive separability

In this section, the specification of MS function is examined for three types of absolute risk
aversion. The examination proceeds in order of CARA, IARA, and DARA. If an individual
is risk averter of CARA under LS condition, then the MS function must fully meet conditions
(1), (2), (3-i), (3-ii), (3-iii), (4-ii) and (5-iii) as shown in table 2. Since condition (4-ii)
indicates that the first term in left-hand-side of condition (5-iii) is zero, (5-iii) is reduced to:

^(5-iii’)                       V (^σ, μ)Vμ (^σ, μ) ⁺ V₁*, (^σ, μ)Vσ (^σ, μ)> ⁰ .

Therefore, condition (5-iii) is replaceable to (5-iii’). The specification of the MS function
may be carried out using the sign of its differential coefficients that fully satisfy these
conditions. Although the procedure relies on a trial-and-error method, it allows the objective
to be accomplished in the following three steps. The first step is to draw a rough outline of

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