We are now equipped to show the differentiability of the value function with the help of the
ʌ
theorem by [1]. Define l(Xc) as the optimal strategy if the threshold is Xc. Now define
W(X) = k [^^(Xo) + B [X0 - X]] -X2
'--------------------------v--------------------------'
î
+δ
Xc -cî ∞
∞ -∞
V (c(X, I) + v + u) fι(u)du g(v)dv + δ
∞∞
Xc -cî -∞
V (c(X, l) + r + v + u) f2 (u)du g(v)dv
'`z"
independent of X
Thus, all perturbations in X around X0 are immediately offset in the first period by an adjustment
in the loading equal to B [X0 - X]. The advantage of W is that the payoff in future periods is
independent of X and only depends on X0, as by definition c(X, l) = BX + l + b = BX +
l(X0) + B[Xo - X] + b = BXo + l(Xo) + b.
Note that W (X) is defined on a neighborhood around X0. Clearly, W (X) is concave, dif-
ferentiable at Xo, and W(X) ≤ V (X) as W(X) uses just one feasible strategy ll out of the set of
possible strategies whose maximum yields V (X). Furthermore, by construction W(Xo) = V (Xo).
Using the above result that V (X) is concave, as well as the theorem of [1], it follows that V (X) is
differentiable at X0 as well and V'(X0) = W'(X0) = -Bk — 2X0 ∣
Corollary: The critical level Xc influences the value function only as an additive constant α
Given that V'(X) = -Bk - 2X, we also know that the value function is given by V(X) =
α — BkX — X2 I.
Proof of Proposition 3 The optimal combined loading is given by
k [1 - b∖ - r [1 - Ф X~-—c) 1 - T~φ (~—c) [Bkr + 2Xcr + r2 + σ22 - σUιI
2 δ σv 2σv σv 2 1
Proof: Maximizing the right hand side of the Bellman equation by setting the derivative with
respect to cl equal to zero we get
k — δ -1 φ
( ——cλ f v(χc + u)fι(u)du + δ f c f v'(δ + v + u)fι(u)dug(v)dv
σv
+δ -1 φ
σv
σv -∞ -∞ -∞
c—c---) V V(Xc + r + u)f2(u)du + δ [ V V'(c + r + v + u)f2(u)dug(v)dv
σ σv — ∞ — ∞ X--C — -∞
k + δ -1 φ
σv
(Xσ-c i∣∕
∞ V (Xc + r
-∞
+ u)f2 (u)du — V (Xc + u)f1(u)du
-∞
+δ / / V'(c + v + u)fι(u)du g(v)dv + δ / / V'(c + r + v + u)f2(u)du g(v)dv
- -∞ - — ∞ Jx c—c - -∞
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