Assuming a normal density h(Xc) = χ 2..lσ . e
dc
dσXc
[Xc-μχc ]2
2σXc we get
- [Xc-μXc ]2
σXc ʃɪ [r [1 - φ (Xστ)] + 21vφ ■ B + 2Xcr + r2 +∆σU]][1 - 2∏σXcee "Xc X
[Xc-μχc ]2
1 + ɪ ʃ ∞ φ f X Hr + X . [Bkr + 2Xcr + r2 + ∆σ2 ]1 '---e 2σXc dXc
1 σv --∞ σ σv J [ 1 2σV l i c 1 1 ujJ √2∏σXc
Consider the numerator. The term t1 =
1-
Xc-μXc
σXc
is positive if Xc ∈ (μXc - σXc, μXc + σXc )
and negative otherwise. Furthermore, the integral /-∞c t1(Xc)dXc = f∞ t1(Xc)dXc = 0. Hence,
if most of the weight is shifted on regions where t1 is negative, the overall integral will be negative.
If μXc -
σx > c then the ratio Xc
c σv
_ ------
- will be zero for a value Xc < μxc — σxc, i.e., where t1
is negative. Furthermore, if σXc and σv are small, this implies that 1 -
1 on (-
------
∞,Xc) and 0 otherwise, while φ (xσ-c) rapidly approaches
Φ (Xσ-^c) is approximately
zero as one deviates from
Xc. Thisimpliesthatboth J-∞ r 1
2Xcr + r2 + ∆σu2]t1h(Xc)dXc < 0.
Now consider the denominator: since all terms except
easy to construct a case where the denominator is positive.
Φ (χrc) 1 tιh(Xc)dXc < 0 and ʃɪ 21-φ XX-c] [Bkr +
σv -∞ 2σv σv
Xc-'
σv
- are positive, and Xc
≤ = 0, it is
A negative numerator combined with a positive denominator implies that the optimal loading
decreases as σXc increases. The intuition is very similar to the nonmonotonicity with respect to
σv in the certainty case (as the uncertainty about the natural system is replaced with uncertainty
of the utility maximizer about the location of the threshold). An example of the nonmonotonic-
ity is displayed in Figure 2. The x-axis displays the system resilience (Xc - Xtarget), i.e. larger
positive values indicate that the target level (the desired pollution stock if the system were for
sure in either state for sure) is further below the critical value Xc . The y-axis displays the un-
certainty σXc . The graph displays contour maps of the precautionary reduction in the expected
pollution stock E[Xtarget - Xt+1]. The grey areas indicate the regions where an increase in σXc
(moving up vertically) will increase the precautionary reduction in next period’s pollution stock (or
equivalently reduce the optimal loading). The white areas, on the other hand, indicate the region
where an increase in σXc reduces the precautionary reduction in next period’s pollution stock.3 ∣
As expected ecosystem resilience and the decisionmaker’s uncertainty about threshold location
change, there is a wide variation in the degree of optimal precautionary activity (Figure 2). Even if
3We should note that similar results are obtained for various combinations of the parameters, i.e. it is not the result
of one particular set of parameter assumptions.
15