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Mathematical Appendix

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Proof of Proposition 1: Under the optimal loading l_t, c(X_t, l_t(X_t)) is independent of X_t

Proof: Rewriting the Bellman equation using the fact that additive constants do not influence the
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optimal load lt we get

Note that lt only enters the maximization in the form of c = BXt + lt + b, and hence the above
problem is equivalent to

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[BX_t + b] - X_t²

X_c -c -∞

And hence the optimal solution c is independent of X_t.

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

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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