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APPENDIX B. DERIVATION OF THE PROPERTIES OF THE FUNCTION F(ε) IN
THE FIRST-ORDER CONDITION.
∂F
(1) Demonstrationthat _ < 0.
∂ε
The first derivative of F with respect to ε is given by
∂F = -3U2 χ[(1 +g)(1 -β )2 + χ]2
(B.1)
d≈ σμ(1 -β )4(1+e)4
which is negative.
∂2F
(2) Demonstration that υ > 0.
∂ε2
The second derivative fo F with respect to ε is given by
∂2F = 6U2χΓ[Γ-χ]
(B.2)
dε2 (1 -β )4(1+ε)5 σ2μ
where Γ ≡ (1+ε)(1-β)2 + 2χ, (B.2) is positive.
(3) Demonstration that F(0) = [(1 ~ β) +χ] u .
(1 -β )4
This can be shown by direct examination of the right-hand side of equation (3.8) at ε = 0.
(4) Demonstration that u (1 β ) < F(ε) < [(1 β ) +χ] u
σμ (1 -β )4 σμ
Since F(0) = [(1 β) + χ] u ,
σ2μ(1 -β )4
lim = -u(1 β ) and dF < 0, F(ε) must be bounded between
e-∞ σ2 ∂ε
μ
u (1 β ) and F(0).
σ2μ
∂F
(5) Demonstration that > 0.
∂U
The first derivative of F with respect to U is given by