The name is absent

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, l_t , 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

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

max klt - X_t² + δE [V (Xt+1)]
lt

For ease of notation, define c(Xt, lt) = BXt + lt + b. Note that the maximization in the Bellman
equation is with respect to loading l_t , so that all other variables are constants. The next proposition
establishes that under the optimal loading l_t, c(X_t, l_t(X_t)) = BX_t + l_t(X_t) + 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 l_t, c(X_t, l_t(Xt)) is independent of X_t

This aries from the fact that the dynamic programming equation can be rewritten as

V (X_t) = 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] - X_t² (4)

X_c -c -∞

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