The name is absent

V(σ, μ) = (μ^s — σ^δ )^δ   (1 < δ < η; μ^s — σ^δ > θ) ,

where η is the newly introduced parameter that is restricted to 1 <δ<η. Form (12) is a
nonhomothetic function whose curvature of expansion path is strictly convex, because of

(1), (2), (3-i), (3-ii), (3-iii) and (4-i), as is obvious from the following derivation coefficients,

Vμ ( σ, μ ) = μ^-1 ( μ — ■ )^1—1 > 0, Vσ ( σ, μ ) = - η σ ^-1 ( μ — σ )^{δ -1} < о,

Vμμ (σ, μ) = (1 — δ) μ^s—2■ (μ^s — σ )1—2 < 0,

V (σ, μ) = η (1 — η) -—2 (μ^s — - )^1—1 + ɪ (1 — δ) σ^2η—2 (μ^s — - ) 1—2 < 0 ,

δδ

2

Vμμ (σ, μ)Vμμ (σ, μ) — Vμσ (σ, μ)= δ-(1 — δ)(δ — η) μ^s—2σ^2η—2 (μ^s — σ^δ)^δ — > 0 and

2

—Vμσ (σ, μ) Vμ (σ, μ) + Vμμ (σ, μ) Vσ (σ, μ) = η (1 — δ) μ^s—2 σ^η—1 (μ^s — σ^η )^δ^ < 0 . Therefore form (12)

displays the case of IRRA and DARA under LS condition.⁴ Unfortunately however, the
author could not find an example that successfully modifies the MS function displaying
CRRA to the one displaying DRRA. Although the curvature condition is easy to meet, the
remaining conditions do not seem to be so (For example, the strict concave condition of
expansion path is satisfied if the restriction 1 <η<δ instead of 1 <δ<η is imposed on form
(12), it does not fulfill condition (3-iii)). This case remains for further research.

In this section, the function properties we obtained regarding MS function is summarized as

19

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