the decisionmaker believes that the ecosystem is in the desirable state (has positive resilience), it is
economically optimal to undertake precautionary reductions in loading. The relationship between
precautionary reductions in loading and uncertainty about the threshold location is nonmonotonic.
The parameter space of threshold uncertainty and expected resilience is divided into regions where
increasing uncertainty increases the level of precaution and regions where increasing uncertainty
decreases the level of precaution (Figure 2). For a given expected resilience, as uncertainty in-
creases, it may be optimal first to increase precaution and then to decrease precaution. This result
is one of our key findings, and has an intuitive explanation. If the decisionmaker is almost certain
that ecosystem resilience is very high - in either state - then it is quite unlikely that any combi-
nation of management actions and random shocks will lead to a transition between states. Thus,
a large precautionary reduction in loading is unwarranted as net benefits are almost certain to be
negligible. On the other hand, if the decisionmaker believes that ecosystem resilience is high, but
is unsure of this, then it is possible that transition between states will occur. If resilience is thought
to be negative, the possibility of transition to the desirable state is a good outcome, and thus war-
rants a precautionary reduction in loading. If resilience is thought to be positive, the possibility
of transition to the undesirable state is a bad outcome, and once again a precautionary reduction
in loading is economically justified. This explains why an initial increase in precautionary load
reduction can be optimal as uncertainty increases. However, eventually, an increase in uncertainty
will always reduce precautionary reductions in loadings (Figure 2); if the decisionmaker believes
that the threshold could be almost anywhere, a large reduction in loading is no longer optimal as
the economic benefits are no longer well defined.
We have shown that nonmonotonicity can exist for intermediate values of σXc and now extend
the analysis to the limiting case where the uncertainty again approaches infinity.
Proposition 10 dcdχc approaches zero for σXc → ∞
Proof: Consider the derivative obtained in the previous proposition
- [Xc-μXc ]2
dc σXc Γ∞ [r [1 - φ . ] + 21-φ . [Bkr + 2Xcr + r2 +∆σU]][1 - X -X X .. e 2σXc X
dσxc - [Xc-μχc ]2
1 + ɪ f' ∞ φ ( X Hr + X . [Bkr + 2Xcr + r2 ■ ∆σ ]1 '---e 2σXc dXc
1 σ- J-∞ . σ- ) \_ 1 2σ- l i c 1 1 ujJ √2∏σχ. c
If σXc → ∞ then most of the probability mass will lie where Xc differs greatly from c and
hence φ ^Xc-l')
and Xcφ (⅞-9
will be close to zero. Hence the denominator approaches one as
the integral approaches zero.
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