The name is absent

above, conditions (5-i) and (5-iii) indicates that the MS functions are nonhomothetic. In
other words, an expansion path, a locus which links the points that give the same slope of
indifference curve in σ - μ axis, is nonlinear. Then, examining the curvature of the
expansion path reveals the difference between conditions (5-i) and (5-iii). Formally,
consider an expansion path, S = -V_σ (σ, μ)V_μ (σ, μ), where S denotes an arbitral slope of
indifference curve. Total differentiation of S yields

dи        -V (σ, μ) V, (σ, μ) + V (σ, μ) V (σ, μ)    , . ,                ,    ,       „ ,

— =--;————^σ----;—ʌ ^μ _r—ʌ, which expresses the slope of the expansion

^dσ S     ^-Vμσ (^σ, μ) Vμ ^(σ, μ⁾⁺ V ^(σ, P) Vμμ ^(σ, P)

path. In the case of DRRA, this is less than ^μ because of conditions (4-i) and (5-i). And in
σ

the case of IRRA and DARA, this is more than ^μ because of conditions (4-i) and (5-iii).

σ

dμμ  dμμ

< and > , derive,
dσ s σ     dσ s σ

)₁ d² μ       1 ( d μ      μ ^             ...     _rτ,..                _{1     1}

and —y > — — - — , respectively. These mean that the expansion
dσ s   σ ( dσ s   σ J

path is strictly concave in the case of DRRA and strictly convex in the case of IRRA and
DARA. Conversely, condition (4-i) and the strict concavity of expansion path derive
condition (5-i) while condition (4-i) and the strict convexity of expansion path derive
condition (5-iii). Therefore, conditions (5-i) and (5-iii) are replaced by each property of
expansion path. If it is possible to modify MS function displaying CRRA in such a way that
each property of the curvature of expansion path is satisfied with the remaining conditions
(1), (2), (3-i), (3-ii), (3-iii) and (4-i), then the objective here is achieved. A successful
example of the modification is obtained in the case of IRRA and DARA. Form (10) is
modified by introducing a new parameter as follows.

18

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