Mathematical Appendix
ʌ ʌ
Proof of Proposition 1: Under the optimal loading lt, c(Xt, lt(Xt)) is independent of Xt
Proof: Rewriting the Bellman equation using the fact that additive constants do not influence the
ʌ
optimal load lt we get
V(Xt)
max
lt
k[BXt + lt +
Xc
b] + δ
-∞
-BXt
-lt -b
V (BXt
+ lt + b + v + u)f1(u)du g(v)dv +
BXt
-lt -b
∞ V (BXt
-∞
+ lt + b + r + v + u
)f2(u)du g(v)dv
- k[BXt + b] - Xt2
Note that lt only enters the maximization in the form of c = BXt + lt + b, and hence the above
problem is equivalent to
V(Xt)
max kc + δ V (c + v + u)f1(u)du g(v)dv +
c -∞ -∞
∞∞
δ V (c + r + v + u)f2(u)du g(v)dv - k[BXt + b] - Xt2
Xc -c -∞
And hence the optimal solution c is independent of Xt.
Proof ofProposition 2: V(X) is concave and differentiable with V'(X) = -Bk — 2X
Proof: We will first show that the dynamic programming problem constitutes a contraction map-
ping and maps concave function into concave functions. This in turn implies that the value function
itself must be concave. Finally, we use the theorem by [1] to show that the value function is differ-
entiable.
Define the operator
T(m)
max kl - X2 + δ m(c(X, l) + v + u)f1 (u)du g(v)dv
l -∞ -∞
+δ m(c(X, l) + r + v + u)f2(u)du g(v)dv
Xc-c(X,l) -∞
We can show that T constitutes a contraction mapping using Blackwell’s sufficient conditions:
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