The name is absent

Which implies
δ [Bk + 2c] = k — δ-φ ——c[Bkr + 2X_cr + r² + σ²2
or equivalently

^c = ^{k 1 - b - r 1 -} Φ Γ^χ--ʌ) ^- ₂¹-^φ(^xC-^c) ^{[Bkr + 2}Xc^{r + r}2 ^{+ σ}2₂ ^{- σ}2ι^]

² L^δ _l L ∖ ^σv ) _l ^2σv ∖ ^σv )

Proof ofProposition 6 For all values of parameters B, k, r, δ, and σ_v, there exists a critical level
X_c such that an increase in the variance σ²_u decreases the optimal loading c.

Proof: First, we will show that using the above parameters the optimal loading c = X_c — σ_v. Using
the proposed c in the equation that implicitly defines c we get

+ r [1 - Φ(1)] + -¹-φ(1)[Bkr + 2^χr + r² + ∆σU]
2σv

= ^χ l^^σv ^{+ r,(1)}1 - ^σv - k Γ1 - BI ^{+ r [1} - _ф^{(1)] +} J_φ(^ι')[B^{kr +} r^{2 + ∆σ}l^]
L      σ_v      J            2 Lδ J                        '2σ,∙

^kσv [^{1 - b}] ^{+ 2σ}v ^{- 2rσ}v^{[1 - ф(1)] - φ} I ^bK^{r + r}2 + ^U^]        k Γ1   _l                  1              _{2      2}

= ----"------------------z--^σv ^- Ô U ^{- b + r [1 - ф(1)] +} — ^{,(1)[Bkr + r} + ^U^]

2σv                                         2 δ J                   2σv

= k [1 - b∖ + σ - r[1 - Φ(1)] - ɪφ(1)[bKr + r² + ∆σ^] - σ - k [¹ - B + r [1 - Φ(1)] + ɪφ(1)[Bkr + r² + ∆σ^]

2 δ                             2σv                                2 δ                        2σv

= 0

The first line is the equation that defines c. The second line uses the proposed c = ^χ_c — σ_v. The
third line factors out X_c before the fourth line uses the expression for X_c.

Second, to get ddc^, totally differentiate the above equation that implicitly defines c to obtain

∣1+ r ,, (^χc-^cʌ - ɪ φ (^χc-^cʌ [Bkr + 2X_cr + r² +∆σU]) de

I σ_v σ σ_v J 2σv ∖ σ_v J                           J

+ I ^r[Xc₂ ^c] φ (^χc--^-k - y¹2φ (^χc--^-k [Bkr + 2X_cr + r² + ∆σ'²] - ^[X; ₃ c^] φ' (^χc--^c^ [Bkr + 2X_cr + r² + ∆σU]∣ dσv

I ^σ2      ∖ ^σv  J ^2σ2  ∖ ^σv  J                                ^2σ3     ∖ ^σv J                           J

=0

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