The name is absent



Examining terms on the right hand side and using the definitions of V(X) and Vt(X), we get

Xc —cc ∞                                  Xc —cc ∞

I   V,(c + v + u)fι(u)du g(v)dv = /       [   [—Bk — 2[c + v + u]]fι(u)du g(v)dv

∞ √—∞                        √—∞ √—∞

Xc —cc

∞


'∞


—Bk — 2c — 2v — 2


Zufι(u)du g(v)dv = /

∞                  - —


4.


Xc —cc


-∞


[-Bk — 2c — 2v] g(v)dv


— [Bk + 2c]Φ Xc—--\+2^συφ(Xc—^-
σ σv J σ σv J

as well as

∞∞

X Xc—c — -∞


∞∞

V,(c + r + v + u)f2(u)du g(v)dv = / I [—Bk — 2[c + r + v + u]]f2(u)du g(v)dv
X Xc—c — -∞

∞

Xc


—cc


'∞


—Bk — 2c — 2v — 2r — 2


Zuf2(u)du g(v)dv = /

∞                     ∙ΛX


∙∞


4.


X.


[—Bk — 2c — 2r — 2v] g(v)dv
c —cc


[Bk + 2c+2r] 1

—*i ⅛ )l

— 2σv φ(JXσV(1 )

and finally

V V (Xc + r + u)f2(u)du — f
— -∞                    — —∞

V (Xc + u)fι(u)du


∖ ∖α — Bk[Xc + r + u] — [Xc + r + u]2] f2(u)du — f ∖α — Bk[Xc + u] — [Xc + u]2] fι(u)du
-∞                               -∞

[α — BkXc


z∞                    fi∞)               ∞∞

f2(u)du —[Bk + 2Xc] / [r + u]f2(u)du — / [r + u]2f2(u)du

∞                J—∞            J—∞


4.


4.


v

σ22 +r2


— [a — BkXc


4.


∞∞

fι(u)du +[Bk + 2Xc] / ufι(u)du + /
∞                 —-∞         ——


u2fι(u)du


4.


4.


σ≡ι


—[Bk + 2Xc ]r —


r2


— σ;


.2


U2


+ σ'.


.2


U1


Putting things together, the first-order condition becomes

k—


δ—φ f —c ^ ∖Bkr + 2Xcr + r2 + σ;
συ ∖ συ J


U2


— σ(


U1


] — δ[Bk + 2c]Φ χx- c


+2σ.δφ(XcσvC )


δ[Bk + 2c + 2r] 1


— 2avδφ(Xσ^ )


k—


δ—φ f —c---ʌ ∖Bkr + 2Xcr + r2 + σ,

σ21 ] — δ[Bk + 2c] — 2δr 1


συ \ συ y

30



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