into Proposition 3 (1) and condition (5-ii). Thus, it transforms the additively separable,
partial linear and non-homothetic function to the non-additively separable, nonlinear and
homothetic function.
Although the homothetic property provides a useful clue to the specification of MS
function that displays CRRA under LS condition, it is not applicable to the case of
non-CRRA preference. The reason is evident from the conditions imposed on MS function
under the preference. If an individual is risk averter of type DRRA under LS condition, then
the MS function has to satisfy conditions (1), (2), (3-i), (3-ii), (3-iii), (4-i) and (5-i). On the
other hand, if an individual is risk averter of the case of IRRA which indicates DARA under
LS condition, then the MS function has to satisfy conditions (1), (2), (3-i), (3-ii), (3-iii), (4-i)
and (5-iii) (Since the cases of IRRA which indicates CARA or IARA have been already
discussed in the previous section, this section focuses on the combination of IRRA and
DARA). Here, conditions (5-i) and (5-iii), or their alternative expressions,
∂∣∂t {-½. (tσ, tμ)/Vμ (tσ, tμ)} < 0 and ∂∣∂ t {-Vσ (tσ, tμ)/Vμ (tσ, tμ)} > 0 indicate that the MS
functions are nonhomothetic (Lau’s lemma). Yet, Proposition 3 (1) still holds in both cases,
as they show DARA. Thus, the MS functions are non-additively separable and nonlinear in
μand σ as well as nonhomothetic. In specifying the non-CRRA type’s MS functions, an
MS function displaying CRRA might help, because the conditions imposed on the
non-CRRA type’s MS functions and those on the CRRA type’s MS function differ only one
point. It is that condition (5-ii) is replaced by condition (5-i) or (5-iii). Therefore, the
objective here is accomplished by modifying the CRRA type’s MS function to fit condition
(5-i) or (5-iii) with the remaining factors, conditions (1), (2), (3-i), (3-ii), (3-iii) and (4-i), still
satisfied.
In order to do that, the first thing that we have to do is to realize the functional
properties that reflect the difference between conditions (5-i) and (5-iii). As mentioned
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