In this section, we derived the following properties regarding MS function under LS
condition.
Proposition 3 (applicability of additive separability)
(1) If an individual is risk averter of DARA under LS condition, then the MS function is
non-additively separable and nonlinear in both μ and σ.
(2) If an individual is risk averter of CARA under LS condition, then the MS function may be
additively separable as long as it is linearly increasing in μ and decreasing and strictly
concave in σ .
(3) If an individual is risk averter of IARA under LS condition, then the MS function may be
additively separable as long as it is increasing and strictly concave in μ and decreasing and
concave in σ .
Homotheticity
In this section, the specification of MS function is examined for three types of relative risk
aversion. The examination starts from CRRA, a frequently adopted case in empirical study
as well as CARA. If an individual is risk averter of CRRA under LS condition, then the MS
function must fully satisfy conditions (1), (2), (3-i), (3-ii), (3-iii), (4-i) and (5-ii). In this case,
the specification procedure used for forms (6) and (9) is no longer useful, because CRRA
indicates DARA (table 1) and therefore Proposition 3 (1) holds. Instead, condition (5-ii) can
play a significant role in specifying the MS function. The condition, or more apparently its
alternative expression, ∂∕∂t{-Vσ (tσ, tμ)∣Vμ (tσ, tμ)} = 0, indicates that the MS function is
homothetic and conversely a homothetic MS function satisfies the above condition (This
directly follows from Lau's lemma (1969) that a function of two or more arguments is
homothetic if and only if the ratio of the first derivatives of the function is homogeneous of
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