the MS function using condition (1) and the signs of the derivatives, Vμμ (σ, μ) = 0,
Vμσ (σ, μ) = 0 and Vσσ (σ, μ) < 0, which satisfy conditions (3-i), (3-ii), (3-iii) (4-ii) and (5-iii’).
Here, Vμσ (σ, μ) = 0 indicates that the MS function is additively separable and the
combination of condition (1) and Vμμ (σ, μ) = 0 shows that it is linearly increasing inμ . These
inferences together imply the form, V(σ,μ) =αμ+g(σ), where α denotes a positive
parameter and g(σ) denotes some function of σ . The second step of specifying the MS
function is to restrict the form so that it meets the remaining conditions, (2) and Vσσ (σ, μ) < 0.
This step may be easily carried out by restricting function g(σ) in such a way that it is
monotonously decreasing and strictly concave. Thus, the additively separable and partial
linear MS function with the restriction mentioned above,
(6) V(σ,μ) =αμ+g(σ)
(α > 0, gσ(σ)< 0 and gσσ(σ) < 0),
fully meets the imposed conditions and therefore is one of functional forms that represents
CARA under the LS condition. In order to apply form (6) to empirical work, the following
third step is necessary that specifies function g(σ) under the restrictions, gσ (σ)< 0 and
gσσ(σ)< 0. As one of candidates, let us consider specifying g(σ) to be polynomial function.
Now, expanding g(σ) by the n-th order Taylor series approximation and evaluating σ at
n g(i)(σ0)
σ0 = 0, we obtain g (σ) ≤ g (σ0) + ∑ g I ,σ1. Defining g ( σ0) = β O, g( i )( σ0) = - βi (i = 1,2,-, n )
i =1 i !
then yields
(7) g (σ ) = в0 -∑ βσi,
i =1 i !
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