#### 3.1. Endogenous Events

[42] We consider events that occur as soon as the groundwater stock reaches some critical level *S*_{c}, which is imperfectly known. The uncertainty regarding the occurrence conditions thus is entirely due to the planner's ignorance concerning the critical level rather than to the influence of exogenous environmental effects. Let *F*(*S*) = Pr{*S*_{c} ≤ *S*} and *f*(*S*) = *dF/dS* be the probability distribution and the probability density associated with the critical level *S*_{c}. The hazard function, measuring the conditional density of occurrence due to a small stock decrease given that the event has not occurred by the time the state *S* was reached, is defined by

We assume that *h*(*S*) does not vanish in the relevant range, hence no state below the initial stock can be considered a priori safe.

[43] The event occurrence time *T* is also uncertain, with a distribution that is induced by the distribution of *S*_{c} and depends on the extraction plan. Upon occurrence, the penalty ψ is inflicted and a further decrease in stock is forbidden, leaving the postevent value ϕ(*S*) = *W*(*S*) − ψ. For irreversible events, the postevent value ϕ vanishes. Given that the event has not occurred by the initial time, i.e., that *T* > 0, we seek the extraction plan that maximizes the expected benefit

subject to equation (1), *x*_{t} ≥ 0; *S*_{t} ≥ 0 and *S*_{0} given. *E*_{T} in equation (12) represents expectation with respect to the distribution of *T*. Optimal processes corresponding to the endogenous uncertainty problem (12) are denoted by the superscript *en*.

[45] Property 10: The optimal stock process *S*_{t}^{en} evolves monotonically in time. For degenerate problems that allow multiple optima, the property ensures that at least one optimal plan is monotonic. Property 10 allows confining attention to monotonic processes. Roughly speaking, the property is based on the idea that if the process reaches the same state at two different times, and no new information on the critical level is revealed during that period, then the planner faces the same optimization problem at both times. This rules out the possibility of a local maximum for the process, because remains constant around the maximum, yet the conflicting decisions to increase the stock (before the maximum) and decrease it (after the maximum) are taken at the same stock levels. A local minimum can also be ruled out even though the decreasing process modifies and adds information on *S*_{c}. However, it cannot be optimal to decrease the stock under risk (before the minimum) and then increase it (with safety, after the minimum) from the same state. In fact, at any state along the optimal process, nonoccurrence of the event cannot modify earlier decisions. Therefore prior to occurrence no need ever arises to update the original plan, and the open- and closed-loop solutions are the same (see *Tsur and Zemel* [1994] for a complete proof).

[46] For nondecreasing stock processes it is known with certainty that the event will never occur and the uncertainty problem (12) reduces to the nonevent problem (2). When the stock process decreases, the distribution of *T* is obtained from the distribution of *S*_{c} as follows:

Using equation (13), the expectation in equation (12) for decreasing processes is evaluated to give (see Appendix A)

subject to equation (1), *x*_{t} ≥ 0; *S*_{t} ≥ and *S*_{0} given. The allocation problem for which equation (14) is the objective is referred to as the auxiliary problem, and optimal processes corresponding to this problem are denoted by the superscript *aux*. The auxiliary problem could be defined for all stock levels in [0, ]. However, it turns out (see property 12, part 1, below) that this problem is relevant for the formulation of the uncertainty problem (12) only for stock levels above , hence replaces the depletion level (*S* = 0) as the lowest feasible stock for equation (14). In similarity with the previously defined problems, the optimal stock process associated with the auxiliary problem evolves monotonically with time. Notice that at this stage it is not clear whether the uncertainty problem (12) reduces to the nonevent problem or to the auxiliary problem, since it is not a priori known whether *S*_{t}^{en} decreases with time. We shall return to this question soon after the optimal auxiliary processes are characterized.

[47] Using equation (A3), we obtain the evolution function corresponding to the auxiliary problem (14)

In equation (15), *L*(*S*) is the evolution function for the nonevent problem, defined in equation (3), and *h*(*S*) is the hazard function, defined in equation (11). Occurrence of the event inflicts an instantaneous penalty ψ (or equivalently, a permanent loss flow at the rate *r*ψ) that could have been avoided by keeping the stock at the level *S*. The second term in the square brackets of equation (15) gives the expected loss due to an infinitesimal decrease in stock. Moreover, this term is positive at the lower bound , while *L*() = 0, hence *L*^{aux}() > 0, implying that cannot be an optimal equilibrium for the auxiliary problem. Whether or not the auxiliary evolution function has a root in (, ) (where *L*(*S*) < 0) depends on the size of the expected loss: for moderate losses, *L*^{aux} vanishes at some stock level in the interval (, ), and this level is the optimal steady state for the auxiliary problem. We assume that the root is unique. The case of multiple roots is discussed by *Tsur and Zemel* [2001]. This case entails some ambiguity on the identification of the steady state but contributes no further insight. Higher expected losses ensure that *L*^{aux} > 0 throughout, and the auxiliary process converges to a steady state at the upper bound = . These considerations imply the following property.

[50] We apply these results to characterize the optimal extraction plan for the endogenous uncertainty problem (12). A detailed analysis is presented by *Tsur and Zemel* [1995]. Here we outline the main considerations.

[51] 1. When *S*_{0} < , the optimal nonevent stock process *S*_{t}^{ne} increases in time. With event risk, it is possible to secure the nonevent value by applying the nonevent policy, since an endogenous event can occur only when the stock decreases. The introduction of occurrence risk cannot increase the value function, hence *S*_{t}^{en} must increase. This implies that the uncertainty and nonevent processes coincide, *S*_{t}^{en} = *S*_{t}^{ne} for all *t*, and increase monotonically toward the steady state .

[54] We summarize these considerations in the following property.

[56] Endogenous uncertainty, then, implies more conservative extraction than the nonevent policy for any initial stock above . Observe that the steady state of property 12, part 2, is a planned equilibrium level. In actual realizations, the process may be interrupted by the event at a higher stock level and the actual equilibrium level in such cases will be the occurrence state *S*_{c}. The situation is depicted in Figure 2, in which the optimal stock processes corresponding to the nonevent and the endogenous uncertainty problems are compared.

[58] A feature similar to both the certainty and endogenous uncertainty processes is the (planned) smooth transition to the steady states. When the initial stock is outside the equilibrium interval, the condition for an optimal entry time to the steady state implies that extraction converges smoothly to the recharge rate and the planned steady state will not be entered at a finite time. Thus property 8 extends also to endogenous uncertainty. It follows that when the critical level actually lies below , uncertainty will never be resolved and the planner will never know that the adopted policy of approaching is indeed safe. Of course, in the less fortunate case in which the critical level lies above the steady state, the event will occur, resolving uncertainty at a finite time (see Figure 2).

#### 3.2. Exogenous Events

[59] Random catastrophic events can be triggered by exogenous environmental conditions that are not within the resource managers' control. The current groundwater stock level can affect the hazard of immediate occurrence, but whether the latter will actually take place is determined by stochastic exogenous conditions. This type of event uncertainty was introduced by *Cropper* [1976] and analyzed by *Clarke and Reed* [1994], *Tsur and Zemel* [1998b], and *Aronsson et al.* [1998] in the context of environmental pollution and nuclear waste control. Here we consider the implications of this kind of uncertainty on groundwater resource management. Under exogenous uncertainty, the knowledge that a certain stock level has been reached in the past without triggering the event is not a safeguard from occurrence at the same stock level sometime in the future, lest the exogenous conditions turn out to be less favorable. Therefore the mechanism that gives rise to the safe equilibrium intervals under endogenous uncertainty does not work here, and we shall show below that such intervals do not characterize the optimal processes under exogenous uncertainty.

[60] As above, the postevent value is denoted by ϕ(*S*), which vanishes identically for all *S* under irreversible events. The expected value from an extraction plan that can be interrupted by an event at time *T* is again given by the objective of equation (12), but for exogenous events the probability distribution of *T*, *F*(*t*) = Pr{*T* ≤ *t*}, is defined in terms of a stock-dependent hazard rate

as

We assume that no stock level is completely safe, hence *h*(*S*) does not vanish and the integral in equation (17) diverges for any feasible process as *t* ∞. We further assume that *h*(*S*) is decreasing, i.e., filling the aquifer reduces the occurrence hazard.

[61] Using equation (17) to evaluate the expected value derived from any feasible process we obtain the exogenous uncertainty problem:

subject to equation (1), *x*_{t} ≥ 0; *S*_{t} ≥ 0 and *S*_{0} given. Unlike the auxiliary problem (14) for endogenous events, equation (18) provides the correct formulation for the exogenous uncertainty problem regardless of whether the stock process decreases or increases.

[62] To characterize the steady state, we need to specify the value *W*^{ex}(*S*) associated with the steady state policy *x*^{ex} = *R*(*S*). Exogenous events may interrupt this policy, hence *W*^{ex}(*S*) differs from the value function *W*(*S*) of equation (5) obtained from the steady state policy under certainty or endogenous uncertainty. An occurrence inflicts the penalty, but does not affect the hazard of future events. The postevent policy, then, is to remain at the steady state and receive the postevent value *W*^{ex}(*S*) − ψ. Under the steady state policy, equation (17) reduces to the exponential distribution *F*(*t*) = 1 − exp[−*h*(*S*)*t*], yielding the expected value *W*^{ex}(*S*) = *W*(*S*) − [*W*(*S*) − *W*^{ex}(*S*) + ψ]*h*(*S*)/[*r* + *h*(*S*)]. Solving for *W*^{ex}(*S*), we find

where the second term represents the expected loss over an infinite time horizon. The explicit time dependence of the distribution *F*(*t*) of equation (17) does not allow to present the optimization problem (18) in an autonomous form. Nevertheless, the argument for the monotonicity of the optimal stock process *S*_{t}^{ex} holds, and the associated evolution function can be derived [*Tsur and Zemel*, 1998b], yielding

[65] Property 13 is due to the second term of equation (20) which measures the marginal expected loss due to a decrease in stock. The latter implies a higher occurrence risk, which in turn calls for a more prudent extraction policy. Indeed, if the hazard is state- independent, the second term of equation (20) vanishes, implying that the evolution functions of the nonevent and exogenous uncertain event problems are the same and so are their steady states. In this case, extraction activities have no effect on the expected loss hence the tradeoffs that determine the optimal equilibrium need not account for the penalty, no matter how large it may be. For a decreasing hazard, however, the degree of prudence (measured by the shift − in the equilibrium state) increases with ψ and the sensitivity to the penalty size is regained.

[66] For irreversible events, ψ = *W*^{ex}(*S*) and equation (19) implies that ψ = *rW*(*S*)/(*r* + *h*(*S*)), hence the second term of equation (20) becomes −[*h*(*S*)*rW*(*S*)/(*r* + *h*(*S*))]′ which is usually of indefinite sign because *W*(*S*) can increase with *S* at low stock levels. The case of a constant hazard, (*h*(*S*) = *h* > 0) is of particular interest. In this case, we use equations (3) and (9) to reduce equation (20) to

It follows that the steady state cannot lie at or above : In this range, *L*(*S*) ≤ 0, implying that *Y*′(*R*(*S*)) − *C*(*S*) ≥ −*C* ′(*S*)*R*(*S*)/(*r* − *R*′(*S*)) > 0. Thus both terms in the curly brackets of equation (21) are negative and *L*^{ex}(*S*) < 0, excluding a steady state. Therefore < and the uncertainty policy is *less* conservative than its nonevent counterpart.

[67] Property 14: When the hazard of irreversible exogenous events is constant, the optimal steady state lies below , and uncertainty induces higher extraction rates.

[68] The intuition behind property 14 is clear: with a stock-independent hazard rate, the extraction policy does not affect the occurrence probability. However, since the postevent value vanishes, the planners wish to accumulate as much benefit as possible prior to occurrence, speeding up the extraction activities and reducing the equilibrium stock. In terms of equation (20), we see that the penalty ψ = *W*^{ex}(*S*) increases with the stock, hence reducing the latter is equivalent to reducing the expected loss, encouraging vigorous extraction. Similar results have been derived by *Clarke and Reed* [1994] for catastrophic environmental pollution.

[69] The results presented in this section highlight the sensitivity of the optimal uncertainty processes to the details of an interrupting event. The type of uncertainty determines the equilibrium structure: endogenous uncertainty gives rise to equilibrium intervals while exogenous uncertainty implies isolated equilibrium states. In most cases, the expected loss due to occurrence encourages prudent extraction policies, but the opposite behavior is optimal under constant hazard of irreversible exogenous events.