imizer’s uncertainty about the critical threshold Xc, we briefly show that precautionary reductions
in loadings are equivalent to reductions in the expected state variable in the next period below the
target level. This is necessary as the expected pollutant stock in the next period is a function both
of the loading and the endogenous probability that the threshold is crossed.
Proposition 8 An increase in σXc increases the precautionary reduction in the state variable
E [Xtarget — Xt+1] if and only if it decreases the optimal loading c.
Proof: Taking the derivative and noting that Xtarget is a constant we get
dE[Xtarget — Xt+1] dE[Xt+1] d{c+ [1 - φ (⅛) ] r}
------------ —--—--
dσXc dσXc dσXc
1+φ
X Xc — C λ r^
σv σv
de
dσXc
>0
How important is threshold uncertainty in inducing the decisionmaker to reduce pollutant load-
ings as a precautionary measure? In the context of our model, expected ecosystem resilience can
be defined as the expected difference between the threshold and the pollutant stock in the next
period, E[Xc — Xt+1]. With this definition, a large positive value implies high resilience of the
desirable state and a large negative value implies high resilience of the undesirable state. When
the decisionmaker is almost certain that the ecosystem will not switch between states - whether
it is currently in the desirable or the undesirable state - the expected pollutant stock approaches
the previously defined target level Xtarget . A measure of the extent to which uncertainty about
the location of the threshold induces reductions in pollutant loading in the economic optimum is
thus given by the difference between the target level and the pollutant stock that is expected in the
next period when an optimum policy is followed, namely E[Xtarget — Xt+1]. This follows because
expected ecosystem resilience E[Xc — Xt+1] can be decomposed into the difference between the
critical level and a constant target level if resilience is high (Xc — Xtarget), as well as additional
precautionary reductions below this target level (E[Xtarget —Xt+1]). This enables us to look at what
happens to the optimal loading and precautionary reduction in the next period’s pollution stock.
Proposition 9 An increase in the variance σX2 c can result in nonmonotonic behavior in precau-
tionary reductions, e.g. it can first decrease and then increase the optimal loading cc.
Totally differentiating the equation that defines the optimal cc in case the threshold is unknown gives
(for a derivation see the Appendix)
dc J-∞∞ [r [1 - φ v '] + ⅛φ ■ IBkr + X + r2 + δ't2 ] hX
c cc
dσX 1 + ɪ ∕-∞∞ φ (Xv-c) [r + X- IBkr + 2Xcr + r2 + ∆σ2]] h(Xc)dXc
14