The name is absent

Proposition 5 The optimal combined loading c approaches 2 [∣ — B] — r if σ_v → ∞

Proof: Recall the equation that implicitly defines c in Proposition 3 (using ∆σ2 = σ22 — σU21 )

Since 0 < Φ() < 1 and 0 < φ() < we know that c has to be bounded for σ_v ≥ 1. Hence,
X_c — c is bounded as well and the ratio Xσ^-c approaches zero for σ_v → ∞ and lim_σv→∞ c =

k Γ1 Ï2 "I      r

2 Lδ    Bi     2.

The intuition is as follows: as uncertainty in the random element of pollutant loading v in-
creases, the manager’s actions have less and less influence on which state the environmental sys-
tem will be in, and the probability that the system will be in either state approaches one half.
Thus, the optimal loading includes a term that approaches r/2, representing the expected addi-
tional pollutant loading as σv → ∞. The next period’s expected state variable again equals the
target level 2 [ɪ — B]. The right graph of Figure 1 shows that the next period’s expected state vari-
able approaches the flat horizontal line 2 [∣ — B] if σ_v increases. It is reasonable to ask whether
an increase in uncertainty σv gives less incentive for precaution and always increases the optimal
loading. As the next proposition shows, such monotonicity of behavior is not found. To the con-
trary, for all possible parameter assumptions one can find a critical value X_c such that an increase
in σv will decrease the optimal loading, increasing precaution as uncertainty increases.

Proposition 6 For all values of parameters B, k, r, δ, and σv, there exists a critical level Xc such
that an increase in the variance σV2 decreases the optimal loading c.

Proof: We will show that there exists a critical level for which -d^c- is negative. Consider X_c =
dσ_v

kσ, [¹ -Β]+2σ2-2rσ_v [1-Φ(1)]-φ(1)[Bkr∣r²∣∆σU]
2[rφ(1)∣σ, ]                       .

It is shown in the Appendix that the the optimal combined loading under these parameters is

------

cc = X_c — σ_v . This simplifies equations as Φ

Using this result in the derivative obtained after totally differentiating the equation that implic-

itly defines cc gives (for a derivation see the Appendix):

—rφ(1)

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