(9) V (σ, μ) = h (μ)+ k (σ)
( hμ (μ)> 0 , hμμ (μ)< 0 , k (σ) < 0 and k (σ) ≤ 0 ),
fully meets the imposed conditions, and therefore, represents IARA under LS condition. The
third step is the specification of the functions h (μ) and k (σ). It is omitted here, since it is
easily carried out.
In contrast, MS function may not be specified as being additively separable when an
individual is risk averter of DARA under LS condition. This can be demonstrated using
conditions (1), (2), (3-i), (3-ii), (3-iii) and (4-i), which are imposed on MS function in this
case. Suppose that the MS function is additively separable, i.e., Vμσ (σ, μ) = 0 . Then,
condition (4-i) reduces to Vμμ (σ, μ) Vσ (σ, μ) < 0 . The condition reduces further to Vμμ (σ, μ) > 0,
as a consequence of condition (2). The inequality however contradicts condition (3-i).
Therefore, the MS function is non-additively separable. Besides, it is also derived that the
MS function is nonlinear in both μ and σ , as follows. Suppose that the MS function is linear
in either μ or σ , i.e., Vμμ (σ, μ) = 0 or Vσσ (σ, μ) = 0. Then condition (3-iii) reduces to
Vμ2σ (σ, μ) ≤ 0, which implies Vμσ (σ, μ) = 0. Thus, the MS function is additively separable.
However, this contradicts the result established above. Therefore, the MS function is
nonlinear in both μ and σ . These properties, non-additive separability and nonlinearity, do
not facilitate the specification of MS function by means of the procedure used for proposing
forms (6) and (9). Because they may not ‘decompose’ conditions (3-iii) and (4-i) into each
derivation coefficient, Vμ (σ, μ) , Vσ (σ, μ) , Vμμ (σ, μ),Vσσ (σ, μ) and Vμσ (σ,μ). The interaction
between them may not be ignored here. Instead of using the specification approach, the next
section considers specifying the DARA type’s MS function from different viewpoint.
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