2 Optimal management with certain threshold location
We consider the problem of a decisionmaker choosing a value for the anthropogenic portion of
pollutant loading in each time period, lt , so as to maximize the discounted value of all future
utilities derived from the environmental system. We begin by assuming that the decisionmaker
knows the exact location of the threshold. Given a per-period discount factor of δ, the maximization
problem is then given by
V (Xt )
∞
|
ml a∞x δt klt - Xt2 {lt}t=0 t=0 |
|
BXt + b + lt + vt + u1t if BXt + b + lt + vt < Xc |
|
s.t. Xt+1 = |
|
I BXt + b + r + lt + Vt + u2t if BXt + b + lt + Vt ≥ Xc |
(2)
where f1(u), f2(u), and g(v) are the density functions of u1, u2, and v, and F1(u), F2(u), and G(v)
are the corresponding cumulative density functions. Recall that all error terms are mean zero. We
assume that v is normally distributed, but place no restrictions on u1 and u2 .
The Bellman equation of the value function that equals the discounted value of all future utili-
ties is
V (Xt )
max klt - Xt2 + δE [V (Xt+1)]
lt
max
lt
klt
Xt2 + δ
Xc
-∞
-BXt -lt -b
V (BXt
+ lt + b + v + u)f1(u)du g(v)dv +
BXt
-lt
V (BXt
+ lt + b + r + v + u
)f2 (u)du g(v)dv
(3)
For ease of notation, define c(Xt, lt) = BXt + lt + b. Note that the maximization in the Bellman
equation is with respect to loading lt , so that all other variables are constants. The next proposition
establishes that under the optimal loading lt, c(Xt, lt(Xt)) = BXt + lt(Xt) + b = c is independent
of Xt, which we use in the consecutive proof that the value function is differentiable (both proofs
are given in the Appendix).
Proposition 1 Under the optimal loading lt, c(Xt, lt(Xt)) is independent of Xt
This aries from the fact that the dynamic programming equation can be rewritten as
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 (4)
Xc -c -∞