MS function is restricted by Proposition 2. For example, Properties 1 and 2 make it
monotonously increasing in μ and decreasing in σ, respectively. Property 4 stipulates that
the relevant Hessian matrix with respect to μ and σ is negative semi definitive. Properties 5
and 6 restrict the slope of the indifference curve when the individual is risk averter. Property
5 makes S(σ,μ) to be decreasing (constant, increasing) in μ when the individual’s absolute
risk aversion is decreasing (constant, increasing), while Property 6 makes it to be decreasing
(constant, increasing) along rays through the origin when the individual’s relative risk
aversion is decreasing (constant, increasing).3 For later convenience, they are summarized as
follows (This study focuses on risk averter’s case).
(1) Vμ (σ, μ )> 0,
(2) V ( σ, μ )< 0 ,
(3-i) Vμμ (σ, μ)≤ 0 ,
(3-ii) Vσσ (σ, μ)≤ 0 ,
(3-iii) Vμμ (σ, μ) Vσσ (σ, P)- V2 (σ, P)≥ 0 ,
(4-i: DARA) -Vμσ (σ, μ) Vμ (σ, μ) + Vμμ (σ, μ) ∖' (σ, μ) < 0 ,
(4-ii: CARA) -Vμσ (σ, μ)Vμ (σ, μ) +Vμμ (σ, μ)Vσ (σ, μ) =0,
(4-iii: IARA) -Vμσ (σ, μ)Vμ (σ, μ) +Vμμ (σ, μ)Vσ (σ, μ) >0,
(5-i: DRRA)
{-Vμσ(σ,μ)Vμ(σ,μ)+Vμμ(σ,μ)Vσ(σ,μ)}μ+{-Vσσ(σ,μ)Vμ(σ,μ)+Vμσ(σ,μ)Vσ(σ,μ)}σ<0,
(5-ii: CRRA)
{-Vμσ(σ,μ)Vμ(σ,μ)+Vμμ(σ,μ)Vσ(σ,μ)}μ+{-Vσσ(σ,μ)Vμ(σ,μ)+Vμσ(σ,μ)Vσ(σ,μ)}σ=0,
(5-iii: IRRA)
{-Vμσ(σ,μ)Vμ(σ,μ)+Vμμ(σ,μ)Vσ(σ,μ)}μ+{-Vσσ(σ,μ)Vμ(σ,μ)+Vμσ(σ,μ)Vσ(σ,μ)}σ>0.