tion occurs while the ultimate size of the resource is unknown (in this case, the ‘threshold’ event
is exhaustion of the resource). Cropper [11] showed that when reserves are uncertain, the optimal
path of planned extraction is no longer necessarily monotonic.
The paper is laid out as follows. In Section 1, we present the basic model we use in our
analysis. In the following section (Section 2), we derive results and analyze the case when there
is stochastic pollutant loading but the threshold location is known. In Section 3, we extend this
analysis to the case where threshold location is also uncertain. Following this, we reconcile the
differences between our results and those of previous studies in Section 4. Finally, we explain the
policy implications of our results.
1 Modeling framework
We begin by presenting a minimal model for the management of a multistate ecosystem with a
reversible threshold that describes the dynamics of an undesirable ecosystem pollutant or charac-
teristic, Xt , through time:
BXt + b + lt + vt + u1t if BXt + b + lt + vt < Xc
Xt+1 = (1)
^ BXt + b + r + lt + vt + U2t if BXt + b + lt + Vt ≥ Xc
The parameter B ∈ [0, 1] represents the proportion of the pollutant X that carries over from one
period to the next, b represents the mean natural input of pollutant to the environmental system, and
lt is the anthropogenic pollutant input. Uncertainty about the system dynamics is captured by the
parameters vt, u1t, and u2t, which are error terms with means μv = μu1 = μu2 = 0 and standard
deviations σv, σu1 and σu2 .2 We assume that Vt is normally distributed, but place no restrictions
on u1t and u2t. Two interpretations of our model are possible: if pollutant levels must be greater
than zero, then Xt can be taken to represent the logarithm of the amount of pollutant at time t, so
that the stochastic input terms follow a lognormal distribution. Alternatively, if we take Xt to be
the pollutant level relative to some baseline, and negative levels are allowed, then Xt can represent
the pollutant level relative to that baseline, and stochastic inputs are normally distributed. Either
of these interpretations is consistent with the model presented.
2Our baseline model assumes σu1 = σu2, but the above setup incorporates the case where the additional loading r
is random. Since the sum of two normal variables is normal again, such a case is equivalent to choosing a non-random
r and σu22 = σu21 + σr2.