The name is absent

3 Optimal management with uncertain threshold location

To make our problem more realistic we next assume that the decisionmaker is aware of the exis-
tence of the threshold X_c between stability domains, but is uncertain of its location. Define the
probability distribution over the critical value as h(X_c). Hence the dynamic programming equa-
tion (4) now becomes (using conditional expectations ans dropping the time subscript for ease of
notation):

δ           V (c + r + v + u)f2(u)du g(v)dv h(X_c)dX_c

Xc -c -∞

The first order condition is the same as in the previous section except that there is an additional
integration over Xc. More formally, the revised first-order condition for the optimal combined
loading becomes

And the optimal choice for c is given implicitly where

k 1          ^∞            Xc - c-       1     Xc - c-                  _{2      2}

-7 - B - r 1 - Φ I ----- ) + -—φ I ----- ) [Bkr + 2Xcr + r² + ∆σ2]

2 δ         _-∞             σ_v       2σ_v     σ_v

Note that the solution is simply a weighted average of the optimal c under certainty, where the
weight is given by beliefs over the critical threshold h(Xc). As before, if the decisionmaker be-
lieves with a high degree of certainty that the threshold is either very high or very low, then optimal
pollutant loadings approach 2 [δ — B] and 2 [ 1 — B] — r respectively; the expected pollutant stock
in the next period, E[X_t+ι], approaches X_target = k [ 1 — B] in both cases. However, if the deci-
sionmaker is uncertain about the location of the threshold, it is economically optimal to undertake
some precautionary reduction in pollutant loading. Intuitively, we are integrating over the optimal
c in the left graph of Figure 1. This explains why an increase in uncertainty can lead to both an
increase and a decrease in the optimal loading. For the former, assume that h(X_c) places all mass
on X_c where c(X_c) is at its minimum, e.g. around X_c = 0.4 for σ_v = 0.2 in the left graph of
Figure 1. An increase in uncertainty implies that more probability mass is put on Xc where it does
not pay to be cautious and c is larger. Alternatively, a decrease in the optimal loading is feasible if
initially h(X_c) places most mass on outcomes of X_c where c is large, e.g., X_c = 1.4, σ_v = 0.2 in
the left graph of Figure 1. Increasing the uncertainty will shift more weight on cases where it pays
to be cautious and reduce the loading.

Before we examine the comparative statics results with respect to an increase in the utility max-
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