The name is absent

The last term of the denominator becomes

The proceeding proposition implies that for all values of the parameters B, k, r, δ, and σ_v ,
there is a critical threshold level for which the regulator becomes more cautious as uncertainty
in pollutant loading increases. While one can always find a threshold level Xc such that dc is
negative, it is also true that for fixed parameter values and threshold Xc , this derivative becomes
positive as σ_v increases, so that the level of precaution ultimately decreases.

Proposition 7 For given parameters including X_c, the optimal loading is increasing in σ_v once σ_v
becomes large.

Proof: From Proposition 5 we know that X_σ-^c approaches zero as σ_v increases.

→ 0 in the derivative derived in the previous proposition

dc   φ (Xσ^-c) [Bkr + 2cr + r² + ∆σU] - [X-c] [Bkr + 2Xcr + r² + ∆σU]

^dσv         2σ² + 2σvφ (X-c) r + ^χ-c [Bkr + 2Xcr + r² + ∆σU]
we see that ddc- > 0 as all terms that could potentially be negative approach zero at a faster rate. ∣

Extremely large values of σv correspond to the case where the manager’s actions have almost
no effect on the probability of a threshold crossing occuring. Thus, if anthropogenic pollutant
loading has almost no influence on whether a threshold crossing will occur, a reduced level of
precaution and increased loading are optimal, as any reduction in loading is too costly compared
to the negligible reduction in the probability of crossing the threshold.

We have shown that the total deterministic pollutant loading in any period, cc, consisting of
the sum of carry-over from the previous period, natural background inputs (not including r), and
(optimal) anthropogenic inputs, does not depend on the current pollutant level. The optimal pollu-
tant loading cc changes nonlinearly as the threshold X_c between stability domains changes (Figure
1). As Xc becomes very large, it becomes very unlikely that transition into the undesirable state

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