manager more careful. In these latter papers, it is never desirable to cross the threshold, and once
it has been located, it is never crossed again. Note that in this strand of literature, an increase in
uncertainty corresponds to an increase in the hazard rate. In our model this need not be the case
as the threshold separates domains with distinct system dynamics rather than representing an in-
stantaneous penalty function. Thus, depending on current state, an increase in uncertainty in our
model can increase or decrease the probability of switching between states.
A smaller body of literature considers thresholds not in terms of penalty functions but as points
or regions in which system behavior switches between alternative states, where one state is viewed
as more ‘desirable’ than the other, either in terms of economic or ecologic benefits. Most economic
models of environmental systems with reversible thresholds and multiple dynamic states assume
perfect knowledge of system dynamics and focus on target trajectories to optimal steady states
[2, 14, 19]. The majority of these studies have analyzed lake ecosystems, where excess nutrient in-
puts can cause switching from oligotrophic to eutrophic states. Such environmental systems have
been modeled in two ways. First, some studies use continuous nonconvex equations of motion
that show a rapid change in system behavior over a small interval (e.g. [2, 14, 16, 19]); to date,
these types of system have only been solved numerically. Second, some studies use multiple equa-
tions of motion with switches occurring when a threshold is crossed (e.g. [5, 18, 25]). In general,
these studies use numerical approximation methods and suggest that optimal policy choices are
insensitive to threshold proximity. An exception is Naevdal [23], who uses a deterministic optimal
control model with a jump equation at the threshold to obtain a mix of analytical and numerical
solutions and shows that for at least some parameter values, the optimal control ‘chatters’ around
the threshold.
Finally, a broad definition of a catastrophic event can include extinction of a renewable re-
source. Analysis of the conditions under which extinction may be optimal goes back to the de-
terministic model of Clark [7], who showed that if the resource growth rate is below the discount
rate, immediate extinction of the resource is economically rational. More recent work shows that
in stochastic systems, it is also necessary to consider characteristics of the welfare function, non-
concave biological growth functions, and the initial stock size in determining optimal outcome
[12, 21, 22]. Olson and Roy [21] find that the choice between conservation and extinction may
be complex: for example, an increased but uncertain productivity can reduce the range of ini-
tial stocks for which conservation is efficient, and therefore increase the likelihood of extinction.
There is also an analogous literature on optimal nonrenewable resource extraction, where extrac-