After collecting terms
{ι+ɪ rφ
+ {-2⅛J '
f ∖ c∖ — 1φ /∖ c∖ [Bkr + 2Xcr + r2 + ∆σ2]1 l
σv 2σv σv
( —c---) [Bkr + 2cr + r2 + ∆σ2 ] +
[Xc —
συ
dc
φ' (—c) [Bkr + 2Xcr + r2 + ∆σ2] I
dσv
r + Xc 2 c [Bkr + 2Xcr + r2 +
2σv2
dc
+ I-Λ φfXc-^ )
I 2σV ∖ σv J L
Bkr + 2cr + r2 + ∆σ2 —
[Xc —
σ2
σv
— [Bkr + 2Xcr + r2 + ∆σ,2] j>
dσv
The second line uses φ'(x) = -xφ(x). If Xc is as defined above, we know that φ (⅜-i) = Φ(1)-
Using this in the derivative we get
dc
—2 [Bkr + 2cr + r2 + ∆σ2 — Bkr — 2Xcr — r2 — ∆σ2 ]
2σv l u ut
dσv
1 I φ(1) I
1 + _ r +
σv
ɪ [Bkr + 2Xcr + r2 + ∆σ2 ]
2σv l u-∣
1 + φ(1) r + ɪ [Bkr + 2Xcr + r2 + ∆σ2]
σv 2,σv l c uj
—rφ(1)
σv + φ(1) r + ɪ [Bkr +
kσv [ J -B]+2σV-2rσv[1-Φ(1)]-φ(1)[Bkr+r2+∆σ2 ]
[rφ(1)+σv ]
r + r2 + ∆σ2 ]
The last term of the denominator becomes
1
r + ---[Bkr +
1
kσv [1 - Bl + 2σ2 - 2rσυ [1 - Φ(1)] - φ(1)[Bkr + r2 + ∆σ2 ] 2 2
------------------------------------------------------------r + r2 + ∆σ* ]
[rφ(1) + συ ]
2σv [rφ(1) + σv ]
1
2σv [rφ(1) + σv ]
^2συ r[rφ(1) + συ ] + Bkr [rφ(1) + συ ] + krσυ - - B j + 2rσ2 - 2r2 συ [1 - Φ(1)] - rφ(1)[Bkr + r2 + ∆σj ] + [r2 + ∆σj ] [rφ(1) + συ ]j
Bkrσv + krσv - b] + 4rσV + 2r2σ^[Φ(1) + φ(1) - 1] + σv [r2 + ∆σ^ ]] > 0
It is positive as 0 < δ,B < 1 implies that ∣ — B > 0 and Φ(1) + φ(1) — 1 > 0.
Proof of Proposition 9 An increase in the variance σ2χc can result in nonmonotonic behavior in
precautionary reductions, e.g., it can first decrease and then increase the optimal loading c.
The optimal loading when there is uncertainty about Xc was defined as
k rɪ
z∞
-∞
• (⅛c )l
+ —— φ (— cʌ) [Bkr + 2Xcr + r2 + ∆σ2] h(X,.)dX,. = 0
2σv ∖ συ Jl-
32