degree zero). Therefore, candidates of the MS function can be chosen from homothetic
family that have been developed and exploited in economic analysis, and doing so fulfils
condition (5-ii). For example, consider a constant elasticity of substitution (CES) type MS
function,
(10)
V (σ, μ) = (μs — σs )δ (δ > 1; μ > σ),
where δ denotes a parameter that is restricted to δ > 1 . Since form (10) is linear
homogeneous, the homothetic property and therefore condition (5-ii) is met. Besides, it
holds Proposition 3 (1), that is, it is non-additively separable and nonlinear in μ and σ. The
remaining conditions (1), (2), (3-i), (3-ii), (3-iii) and (4-i) are also satisfied, as verified below.
Vμ (σ, μ) = μs-1 ( μs - σ )δ 1 > 0, V (σ, μ) = — σ-1 (μs - σ )δ 1 < 0,
Vμμ (σ, μ) = (1 — δ) μδ—2σ (μ — σ )1—2 < 0 , Vσσ (σ, μ) = (1 — δ) μδσδ-2 (μδ — σs )δ-2 < 0,
Vμμ (σ, μ) Vσσ (σ, P) — V (σ, P) = 0 and
—Vμσ (σ, μ) Vμ (σ, μ) + Vμμ (σ, μ) Vσ (σ, μ) = (1 — δ) μ—2 σs—1 (μs — σs )δ—2 < 0. Therefore, form (10)
displays CRRA under LS condition. Recently, Nelson and Escalante (2004) proposed the
following form,
(11)
V (σ, μ) = —(μ2 — φσ2 ) (φ > 0; μ2 — φσ2 > 0),
where φ denotes a parameter, and showed that form (11) fully meet conditions (1), (2), (3-i),
(3-ii), (3-iii), (4-i) and (5-ii). Therefore, it also displays CRRA under LS condition. Nelson
and Escalante achieved the specification of form (11) by modifying the LMV model that
displays CARA under LS condition. The modification consequently fits the LMV model
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