The name is absent

the next period is E[X_t+1] = c + r = 2 [∣ — B]. Note that when there is certainty whether the
additional pollutant input term r is present or not, the expected stock in the next period is 2 [∣ — B]
in each case, which we call the target stock. Furthermore, let the precautionary reduction in next
period’s expected state variable be E[X_target — X_t+1], i.e. the drop in the state variable below the
level that is desirable if the manager knows whether the system is below or above the threshold Xc.
Figure 1 shows both the optimal combined loading c (left panel) and the expected state variable in
the next period (right panel). Each graph displays the solution for various values of σ_v, while all
other parameters are taken from [25].

Second, if the variances of the error terms u1 and u2 are the same, then they do not enter the
results at all, and have no influence on optimal loading. This is the case because we model a
reversible system and hence any arbitrary shock to the system can be completely counterbalanced
in the next period (and u does not impact whether the threshold X is crossed or not). As mentioned
before, if the additional loading r is not deterministic but itself random, then σ_u²₂ > σ_u²₁, which
further reduces c.

Third, uncertainty in the form of the error term v influences the optimal loading. Recall that the
error term v is partially responsible in determining whether the additional input r is present or not:
switching to the undesirable state occurs if BX_t + l_t + b + v_t ≥ X_c . Uncertainty about whether
the threshold has been crossed and the additional input r is present induces the decisionmaker to
become more cautious so that the following period’s expected state variable is below Xtarget in
the right panel of Figure 1. We briefly establish that the precautionary reduction in next period’s
expected pollutant stock E[Xtarget — Xt+1] is nonnegative.

Proposition 4 A sufficient condition for the precautionary reduction E[X_target — X_t+1] to be non-
negative is Xc > 0

Proof: Follows directly from Xtarget = k [| ^{- B}] ^{and E[X}t+1^] = ^c + ^r
the solution for c in Proposition 3 we get

E[Xtarget ^- Xt+1^] = ~--^φ (------^∖ [B^kr + ²Xcr + ^r2 + ^σ22

2σv      σv

We now consider the relationship between uncertainty in pollutant loading and precautionary
behavior in more detail.

<<   7   8   9   10   11   12   13   >>

More intriguing information