SEARCH

SEARCH BY CITATION

Keywords:

  • event uncertainty;
  • groundwater;
  • renewable resource;
  • management;
  • environmental catastrophes

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[1] We study optimal management of groundwater resources under risk of occurrence of undesirable events. The analysis is carried out within a unified framework, accommodating various types of events that differ in the source of uncertainty regarding their occurrence conditions and in the damage they inflict. Characterizing the optimal policy for each type, we find that the presence of event uncertainty has profound effects. In some cases the isolated steady states, characterizing the optimal exploitation policies of many renewable resource problems, become equilibrium intervals. Other situations support isolated equilibria, but the degree of prudence they imply is sensitive to the nature of the event risk.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[2] Overexploitation of groundwater resources, when pumping exceeds recharge, is pervasive worldwide [Postel, 1999]. Such a situation involves shrinking groundwater stocks, which may lead to one of the following outcomes: (1) The extraction cost increases to a point where it is no longer beneficial to pump above recharge and the aquifer settles at a steady state. (2) The groundwater stock is depleted, and from that time onward only the recharge can be pumped. (3) An event that adversely affects future exploitation possibilities is triggered, e.g., seawater intrusion or the penetration of polluted water from nearby sources. The theory of groundwater management under the first two scenarios is well developed [e.g., Burt, 1964; Gisser and Sanchez, 1980; Feinerman and Knapp, 1983; Tsur and Graham-Tomasi, 1991], but the effects of the third type of outcome are not fully explored. This paper undertakes to characterize optimal groundwater management under the threat of occurrence of adverse environmental events.

[3] An example in mind is the exploitation of a coastal aquifer. Excessive extraction, over and above natural recharge, leads to a decline in the groundwater head, which, in turn, may result in seawater intrusion. If seawater intrusion is a gradual process that can be monitored and controlled by adjusting extraction rates, the associated damage can be avoided. Often, however, seawater intrusion occurs abruptly as soon as the fresh water head declines below some threshold level, inflicting a severe damage or rendering the aquifer useless for a long time. In such cases, seawater intrusion can be treated as a discrete event. As a result of overdraft pumping, for example, Israel's coastal aquifer interface has penetrated inland as far as 3 km in some locations during a short time period, leading to the closure of a significant number of wells. In fact, most of the western edge of the aquifer has become saline to the extent that it cannot be used. Restoring the aquifer requires moving back the interface - a costly and uncertain operation [Gvirtzman, 2002]. Furthermore, extraction from lake Kinneret (another major source of supply of Israel's freshwater) has long been controlled by the Lower Red-Line rule, based on the claim that reducing lake water level below that line might entail catastrophic deterioration in the condition of the lake.

[4] When the threshold level that triggers the event is known with certainty, it is easy to avoid the damage by ensuring that the threshold level is never reached. In most cases, however, the threshold is only partially known, due, e.g., to lacking information regarding subsurface flows. Moreover, the occurrence conditions may be affected also by stochastic environmental conditions that are not within the managers' control. Accounting for this kind of events, we enter the realm of event uncertainty.

[5] Impacts of event uncertainty on optimal exploitation policies have been studied in a variety of resource management problems, including pollution-induced events [Cropper, 1976; Clarke and Reed, 1994; Tsur and Zemel, 1996, 1998b; Aronsson et al., 1998], forest fires [Reed, 1984; Yin and Newman, 1996], species extinction [Reed, 1989; Tsur and Zemel, 1994], seawater intrusion into coastal aquifers [Tsur and Zemel, 1995], and political crises [Long, 1975; Tsur and Zemel, 1998a]. Typically, occurrence risk implies prudence, and the exploitation policies are more conservative than those obtained under certainty. In some cases, however, event uncertainty encourages more vigorous extraction policies in order to derive maximal benefit prior to occurrence. Tsur and Zemel [1998b] trace these apparently conflicting results to differences in the occurrence conditions and the damage inflicted by the events and consequently classify events as reversible or irreversible, and endogenous or exogenous.

[6] In the context of groundwater, irreversible events are those that, once occurred, render the aquifer obsolete. In other words, the cost of restoring the aquifer to usable conditions is prohibitively large (or infinite, if restoration is impossible). Reversible events, on the other hand, permit restoration (e.g., cleaning a polluted aquifer) at a cost that, although might be high, is worth bearing. Thus the difference between reversible and irreversible events in this context is attributed to the difference between the alternative cost of restoration, the actual restoration cost minus the forgone benefit if restoration is not carried out and exploitation ceases. If the alternative cost of restoration is negative (the forgone benefit exceeds the actual restoration cost), restoration is worthwhile and will be carried out to allow post event exploitation; the event is then classified as reversible. If the alternative restoration cost is positive, restoration is not worthwhile and will not be undertaken; in this case occurrence terminates exploitation and the event is classified as irreversible.

[7] Although quite common in the literature, this interpretation of “reversibility” is not universal and requires some care. Ceteris paribus, irreversible events involve higher restoration costs than their reversible counterparts. However, once the choice not to restore the aquifer has been made, the damage associated with the event reduces to the forgone benefit due to cessation of the exploitation activities, hence irreversible events can be treated as a special case of reversible events. It should be noted that both types of events involve sunk costs due to the damage caused by the event.

[8] Other interpretations of irreversibility in the context of environmental events refer to the nondegradability of pollution, or to the costs of converting back capital invested in abatement technologies [e.g., Narain and Fisher, 1998], and their analysis appeals to the general theory of investment under uncertainty. The effects of these sources of irreversibility are not directly related to the consequences of the irreversible events considered in this work.

[9] The adjective ‘endogenous’ signifies events whose occurrence is determined solely by the exploitation policy, although some essential information (e.g., the exact threshold level for seawater intrusion) is not known a priori. In contrast, exogenous events are triggered also by stochastic environmental conditions (the expansion of a nearby source of pollution), which are outside the managers' control.

[10] It turns out that the distinction among the different types of event uncertainty bears profound consequences for optimal management policies and often alters properties that are considered standard. For example, in a renewable resource context, the optimal stock process typically approaches an isolated equilibrium (steady) state. This feature, it turns out, no longer holds under endogenous event uncertainty: The unique equilibrium state expands into an equilibrium interval and the eventual steady state depends on the initial stock.

[11] In this paper we present the problem of the optimal management of groundwater resources under event uncertainty in a unified framework that accommodates all the above mentioned types of events. We begin, in the next section, by characterizing the optimal extraction policy under certainty. First we analyze the standard reference case of the nonevent problem, in which no event can ever interrupt the extraction plan, and then add certain events that occur when the groundwater stock shrinks to a known critical level. Since we show that under these conditions it is never optimal to trigger the event, it follows that the optimal policy is insensitive to the nature of the event (reversible or irreversible) or to the amount of damage it inflicts. This insensitivity, however, disappears when we deal (in section 3) with uncertain situations. We show that under endogenous uncertainty the optimal policy is to drive the stock process to the nearest edge of an equilibrium interval. The size of this equilibrium interval (which measures the degree of prudence implied by the events) turns out to depend on the expected damage from immediate occurrence. Under exogenous uncertainty, on the other hand, no extraction policy is perfectly safe and the equilibria are confined to isolated states. The effect of exogenous uncertainty is measured by the shift of these equilibrium states (relative to the nonevent counterpart) and is sensitive to the hazard and penalty associated with the events.

2. Groundwater Management Under Certainty

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[12] We consider first the management of a confined groundwater basin (aquifer) under full certainty. Let St denote the groundwater stock level at time t and R(St) the natural recharge rate (net water inflow excluding extraction), assumed decreasing and concave with R(equation image) = 0 where equation image is the aquifer's capacity. Thus recharge attains a maximal rate at an empty aquifer, diminishes with S at an increasing rate and vanishes when the aquifer is at a full capacity equation image. With xt representing groundwater extraction, the aquifer's stock evolves with time according to

  • equation image

[13] The benefit derived from consuming water at the rate x is Y(x), where Y is increasing and strictly concave with Y(0) = 0. The cost of extracting at the rate x while the stock level is S is C(S)x, where the unit cost C(S) is nonincreasing and convex. The instantaneous net benefit is then given by Y(x) − C(S)x. It is assumed that Y′(0) > C(equation image), so that some extraction is worthwhile under the most favorable conditions.

2.1. Nonevent

[14] When no event can interrupt groundwater extraction, the optimal plan is obtained by solving

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0 and S0 given. The optimal processes associated with the nonevent problem (2) will be indicated with an ne superscript. This standard problem has been treated by a variety of optimization methods [e.g., Tsur and Graham-Tomasi, 1991; Tsur and Zemel, 1994, 1995] and we summarize the main findings below.

[15] We note first that because problem (2) is autonomous, (time enters explicitly only through the discount factor), the optimal stock process Stne evolves monotonically in time [Tsur and Zemel, 1994]. Since Stne is bounded in [0, equation image] it must approach a steady state in this interval. Using the variational method of Tsur and Zemel [2001], possible steady states are located by means of a simple function L(S) of the state variable, denoted the evolution function (see Appendix A). In particular, an internal state S ∈ (0, equation image) can qualify as an optimal steady state only if it is a root of L, i.e., L(S) = 0, while the corners 0 or equation image can be optimal steady states only if L(0) ≤ 0 or L(equation image) ≥ 0, respectively.

[16] For the case at hand, the evolution function corresponding to equation (2) is given by (see equation (A3)):

  • equation image

[17] The properties of the functions Y, R and C imply that the term inside the curly brackets is decreasing while rR′(S) > 0. Moreover, the assumption that some exploitation is profitable at a full aquifer, i.e., Y′(0) > C(equation image), implies that L(equation image) = (rR′(equation image))(C(equation image) − Y′(0)) < 0. Thus either L(0) ≥ 0, in which case L(S) has a unique root in [0, equation image] or L(0) < 0. Let equation image represent the root of L(S) if L(0) ≥ 0 and equation image = 0 otherwise. The above considerations can be summarized in terms of this root by the following property.

[18] Property 1: equation image is the unique steady state to which the optimal stock process Stne converges monotonically from any initial state.

[19] The vanishing of the evolution function at an internal steady state represents the tradeoffs associated with groundwater exploitation. A steady state is optimal if any diversion from it inflicts a loss. Consider a variation on the steady state policy x = R(equation image) in which extraction is increased during a short (infinitesimal) time period dt by a small (infinitesimal) rate dx above R(equation image) and retains the recharge rate thereafter. This policy yields the additional benefit (Y′(R(equation image)) − C(equation image))dxdt. But it also decreases the groundwater stock by dS = −dxdt, which, in turn, increases the unit extraction cost by C′(equation image)dS and the extraction cost by R(equation image)C′(equation image)dS. The present value of this permanent flow of added cost is given by R(equation image)C′(equation image)dS/(rR′(equation image)). (The effective discount rate equals the market rate r minus the marginal recharge rate R′ because reducing the stock by a marginal unit and depositing the proceeds at the bank the resource owner gains the market interest rate r plus the additional recharge rate −R′ [see Pindyck, 1984].) At the root of L, these marginal benefit and cost just balance, yielding an optimal equilibrium state. The state of depletion S = 0 can be a steady state also when L(0) < 0, or equivalently when Y′(R(0)) − C(0) > −C′(0)R(0)/[rR′(0)]. This is the case when the marginal benefit exceeds the added extraction cost even when the latter is at its maximum.

[20] While property 1 implies that the stock process must approach equation image, the time to enter the steady state remains a free choice variable. Using the conditions for an optimal entry time, we establish in Appendix A that the optimal extraction rate xtne smoothly approaches the steady state recharge rate R(equation image) and the approach of Stne toward the steady state equation image is asymptotic. This implies the following property.

[21] Property 2: Initiated away from equation image, the optimal stock process Stne will not reach equation image at a finite time.

[22] Since problem (2) is autonomous, the optimal extraction can be expressed in terms of the state S alone. Let xne(S) denote optimal extraction when the stock is S. The necessary conditions for optimum give rise to the following first order, nonlinear differential equation for xne(S) (see Appendix A)

  • equation image

with the boundary condition xne(equation image) = R(equation image), implied by the smooth transition to the steady state. To allow the use of equation (4) as the basis for a numerical solution, one can remove the singularity at equation image and obtain (see Appendix A)

  • equation image

which serves as the starting step for the integration scheme.

[23] Given xne(S), the optimal stock process Stne is determined by integrating equation (1)

  • equation image

and the value function is obtained from the Dynamic Programming equation [e.g., Kamien and Schwartz, 1981, p. 242]

  • equation image

Moreover, it is shown in Appendix A that xtne = xne(Snet) is also monotonous in time. More specifically, the following property is established.

[24] Property 3: xtne decreases with time while Stne > equation image and increases with time when Stne < equation image.

[25] Observe that the decrease in the extraction rate when the groundwater stock is above the steady state takes place even though the natural rate of recharge increases as the stock declines. Thus the two flow processes that drive the stock dynamics (extraction and recharge) work together to slow down the rate of approach to the eventual steady state.

2.2. Irreversible Events

[26] Suppose now that driving the stock to some critical level Sc triggers the occurrence of some catastrophic event, e.g., the intrusion of saline water into the reservoir, rendering the groundwater useless thereafter and ceasing extraction activities. We refer to such occurrence as an irreversible event. Obviously, if Sc < equation image the event risk has no bearing on the optimal policy, because extraction falls short of the recharge rate for all S < equation image even without the event risk (property 1), hence the critical level will never be approached. We consider therefore the case S0 > Sc > equation image.

[27] Let T denote the event occurrence time (T = ∞ if the stock never shrinks to Sc to trigger the event). The certainty problem with irreversible event risk is formulated as

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0; ST = Sc and S0 > Sc given. Problem (8) differs from the nonevent problem (2) by the additional decision variable T and the additional constraint ST = Sc. Optimal processes corresponding to equation (8) are indicated with a ci superscript (c for certainty, i for irreversible).

[28] The event occurrence is evidently undesirable, since just above Sc it is preferable to extract at the recharge rate and enjoy the benefit flow associated with it rather than extract above recharge, trigger the event and lose all future benefits. Thus the event should be avoided, as the following property attests.

[29] Property 4: When the critical level Sc is known, Stci > Sc for all t and T = ∞.

[30] The certainty-irreversible event problem thus obtains the same form as the nonevent problem (2), but with the additional constraint St > Sc. The evolution function equation (3) therefore applies to this problem as well, but only roots in the range [Sc, equation image] (rather than [0, equation image]) can be feasible steady states. Being monotonous and bounded, the optimal stock process Stci must approach a steady state. However, with Sc > equation image the function L(S) is negative in the feasible interval [Sc, equation image], hence no internal steady state can be optimal. The only remaining possibility is the critical level Sc, because the negative value of L(Sc) does not exclude this corner state. These considerations establish the following property.

[31] Property 5: When the critical level Sc corresponding to an irreversible event is known and lies above equation image, the optimal stock process Stci converges monotonically to a steady state at Sc.

[32] According to property 5, in the long run Stci must lie above its nonevent counterpart Stne. It turns out that this relation holds for the complete duration of the process. Formally, we describe this relation by the following property.

[33] Property 6: Stci > Stne for all t > 0. Both processes depart from the same initial stock S0 at t = 0. According to property 6, x0ne > x0ci and the policy under event risk is always more conservative, in the sense of leaving more water in the aquifer. To see why, suppose that Sτne = Sτci at some time τ > 0. Then, Stne = Stci must hold during the entire time interval [0, τ] hence the extraction rates must also coincide during this period. This, in turn, implies, (see equation (4)) that the two stock processes must evolve together also from τ onward, violating properties 1 and 5. In fact, the extra caution due to the event risk is manifest also by the optimal extraction rates.

[34] Property 7: xci(S) < xne(S) for any S > Sc. Property 7 follows directly from property 6, when we consider the two optimization problems initiated at S0 = S.

2.3. Reversible Events

[35] Assume now that the damage inflicted by the event can be cured at some cost. For example, in some cases it may be possible to drive back the saline water by introducing large quantities of freshwater from other sources into the reservoir. Under such circumstances we refer to the event as reversible and specify the postevent value as ϕ(Sc) = W(Sc) − ψ, where

  • equation image

is the steady state value derived from keeping the extraction rate at the natural recharge rate R(S), and the penalty ψ > 0 is the (once and for all) curing cost. The postevent value ϕ thus accounts both for the fact that the stock cannot be further decreased (to avoid a second occurrence) and for the curing cost. The aquifer management problem under reversible events is modified to

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0; ST = Sc and S0 > Sc given. Optimal processes associated with equation (10) are indicated with a cr superscript.

[36] Observe that equations (8) and (10) differ only in the postevent value. It follows that an irreversible event is a special case of reversible events with a penalty that equals the steady state value W(Sc). Not surprisingly, the optimal policies for the two types of event turn out to be the same. To see this, note first that just as it is not desirable to trigger an irreversible event, it is also not desirable to do so with a reversible event because the postevent value is smaller than the steady state value that can be secured by avoiding the event occurrence. These considerations imply the following property.

[37] Property 8: When the critical level corresponding to a reversible event with any positive penalty is known, the optimal stock process Stcr > Sc and T = ∞.

[38] Note that the reversible event may not be as harmful as the irreversible event (since the penalty may be smaller than W(Sc)). Nonetheless, for both types of events, the penalty is never realized (properties 4 and 8) and its exact value (so long as it is positive) is irrelevant. It follows that the certainty policies do not depend on the nature of the event nor on the penalty it inflicts. This gives rise to the following property.

[39] Property 9: When the critical stock level at which the event is triggered is known, the optimal policies under reversible and irreversible events are the same.

[40] The lack of sensitivity of the optimal policy to the details of the catastrophic event is evidently due to the ability to avoid the event occurrence altogether. This may not be feasible (or optimal) when the critical stock level is not a priory known. The optimal policy may, in this case, lead to unintentional occurrence, whose exact consequences must be accounted for in advance. We turn, in the following section, to analyze the effect of uncertain catastrophic events on groundwater management policies.

3. Uncertain Events

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[41] Often the conditions that lead to the event occurrence are imperfectly known, or are subject to environmental uncertainty outside the planner's control. In some cases the critical level is a priori unknown, to be revealed only by the event occurrence. Alternatively, the event may be triggered at any time by external effects (such as subsurface flows of fresh and saline water) with a probability that depends on the current aquifer state. We refer to the former type of uncertainty, that due to the planner's ignorance regarding the conditions that trigger the event, as endogenous uncertainty (signifying that the event occurrence is solely due to the exploitation decisions) and to the latter as exogenous uncertainty. It turns out that the optimal policies under the two types of uncertainty are quite different. These policies are characterized below.

3.1. Endogenous Events

[42] We consider events that occur as soon as the groundwater stock reaches some critical level Sc, 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{ScS} and f(S) = dF/dS be the probability distribution and the probability density associated with the critical level Sc. 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

  • equation image

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 Sc 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

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0 and S0 given. ET 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.

[44] As the stock process evolves in time, the managers' assessment of the distributions of Sc and T can be modified since at time t they know that Sc must lie below equation image = Min0≤τ≤t{Sτ}, for otherwise the event would have occurred at some time prior to t. Thus the expected benefit in the objective of equation (12) involves equation image, i.e., the entire history up to time t, complicating the optimization task. The evaluation of the expectation in equation (12) is simplified when the stock process evolves monotonically in time, since then equation image = S0 if the process is nondecreasing (and no information relevant to the distribution of Sc is revealed) and equation image = St if the process is nonincreasing (and all the relevant information is given by the current stock St). It turns out that we can indeed make use of these simple relations, due to the following property.

[45] Property 10: The optimal stock process Sten 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 equation image 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 equation image and adds information on Sc. 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 Sc as follows:

  • equation image

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

  • equation image

subject to equation (1), xt ≥ 0; Stequation image and S0 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, equation image]. 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 equation image, hence equation image 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 Sten 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)

  • equation image

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 rψ) 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 equation image, while L(equation image) = 0, hence Laux(equation image) > 0, implying that equation image cannot be an optimal equilibrium for the auxiliary problem. Whether or not the auxiliary evolution function has a root in (equation image, equation image) (where L(S) < 0) depends on the size of the expected loss: for moderate losses, Laux vanishes at some stock level equation image in the interval (equation image, equation image), and this level is the optimal steady state for the auxiliary problem. We assume that the root equation image 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 Laux > 0 throughout, and the auxiliary process converges to a steady state at the upper bound equation image = equation image. These considerations imply the following property.

[48] Property 11: equation image is the unique steady state to which the optimal stock process Staux converges monotonically from any initial state in [equation image, equation image].

[49] Events for which equation image(S) = W(S) − ψ = 0 are denoted irreversible. Noting equation (9) and R(equation image) = 0, we see that W(equation image) = 0, while L(equation image) < 0. Thus for irreversible events Laux(equation image) < 0. It follows that the auxiliary evolution function must have a root in the interval (equation image, equation image), and the auxiliary equilibrium level for irreversible events must be an internal state.

[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 S0 < equation image, the optimal nonevent stock process Stne 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 Sten must increase. This implies that the uncertainty and nonevent processes coincide, Sten = Stne for all t, and increase monotonically toward the steady state equation image.

[52] 2. When S0 > equation image > equation image, both Stne and Staux decrease in time. If Sten is increasing, it must coincide with the nonevent process Stne, contradicting the decreasing trend of the latter. A similar argument rules out a steady state policy. Thus Sten must decrease, coinciding with the auxiliary process Staux and converging with it to the auxiliary steady state equation image.

[53] 3. When equation imageS0equation image, the nonevent stock process Stne decreases (or remains constant if S0 = equation image) and the auxiliary stock process Staux increases (or remains constant if S0 = equation image). If Sten increases, it must coincide with Stne, and if it decreases it must coincide with Staux, leading to a contradiction in both cases. The only remaining possibility is the steady state policy Sten = S0 at all t.

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

[55] Property 12: (1) Sten increases at stock levels below equation image. (2) Sten decreases at stock levels above equation image. (3) All stock levels in [equation image, equation image] are equilibrium states of Sten. The various possibilities are illustrated in Figure 1. The equilibrium interval of property 12, part 3, is unique to optimal stock processes under endogenous uncertainty. Its boundary points attract any process initiated outside the interval (as indicated by the direction of the arrows in Figure 1), while processes initiated within it must remain constant. This feature is evidently related to the splitting of the endogenous uncertainty problem into two distinct optimization problems depending on the initial trend of the optimal stock process. At equation image, the expected loss due to occurrence (represented by the second term of equation (15)) is so large that entering the interval by reducing the stock cannot be optimal even if under certainty extracting above the recharge rate would yield a higher benefit. Within the equilibrium interval, the planner can take advantage of the possibility to eliminate the occurrence risk altogether by not reducing the stock below its current level. As we shall see below, this possibility is not available under exogenous uncertainty, hence the corresponding management problem does not give rise to equilibrium intervals.

image

Figure 1. The evolution functions L(S) (equation (3)) and Laux(S) (equation (15)) corresponding to the nonevent and auxiliary problems, respectively. The arrows indicate the direction in which the optimal process Sten evolves. [equation image, equation image] is the equilibrium interval for this process.

Download figure to PowerPoint

[56] Endogenous uncertainty, then, implies more conservative extraction than the nonevent policy for any initial stock above equation image. Observe that the steady state equation image 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 Sc. The situation is depicted in Figure 2, in which the optimal stock processes corresponding to the nonevent and the endogenous uncertainty problems are compared.

image

Figure 2. A comparison between the optimal processes Stne (corresponding to the nonevent problem) and Sten (corresponding to endogenous uncertainty). The latter process is interrupted by the event at time T and the rest of the process (depicted by the dashed line process Staux) is never realized.

Download figure to PowerPoint

[57] It is also noted that under endogenous uncertainty, reversible and irreversible events differ only in the precise location of the root of the auxiliary evolution function. A large expected loss with h(equation image)rψ > −L(equation image) excludes the possibility of a root in [equation image, equation image], hence the uncertainty process will never decrease; the expected loss in this case is too high to justify taking the risk of triggering the event at any stock level. For irreversible events such a situation cannot occur: since W(equation image) = 0, the expected loss near equation image is small and the auxiliary root must lie below this state.

[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 equation image, uncertainty will never be resolved and the planner will never know that the adopted policy of approaching equation image 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{Tt}, is defined in terms of a stock-dependent hazard rate

  • equation image

as

  • equation image

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 [RIGHTWARDS ARROW] ∞. 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:

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0 and S0 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 Wex(S) associated with the steady state policy xex = R(S). Exogenous events may interrupt this policy, hence Wex(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 Wex(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 Wex(S) = W(S) − [W(S) − Wex(S) + ψ]h(S)/[r + h(S)]. Solving for Wex(S), we find

  • equation image

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 Stex holds, and the associated evolution function can be derived [Tsur and Zemel, 1998b], yielding

  • equation image

[63] For reversible events with a fixed penalty and decreasing hazard one finds that Lex(S) > L(S). Since L(S) is positive below equation image, so must Lex(S) be, precluding any steady state below equation image. Thus the root equation image of Lex(S) must lie above the nonevent equilibrium. This implies the following characterization.

[64] Property 13: The optimal stock process under exogenous uncertainty converges monotonically to the root equation image. When the hazard-rate function h(S) is decreasing, equation image > equation image and the extraction policy is more conservative than its nonevent counterpart.

[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 equation imageequation image in the equilibrium state) increases with ψ and the sensitivity to the penalty size is regained.

[66] For irreversible events, ψ = Wex(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

  • equation image

It follows that the steady state cannot lie at or above equation image: In this range, L(S) ≤ 0, implying that Y′(R(S)) − C(S) ≥ −C ′(S)R(S)/(rR′(S)) > 0. Thus both terms in the curly brackets of equation (21) are negative and Lex(S) < 0, excluding a steady state. Therefore equation image < equation image 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 equation image lies below equation image, 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 = Wex(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.

4. Concluding Comments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[70] While it is widely recognized that uncertainty may have profound effects on groundwater management, the precise manner in which the optimal extraction rules should be modified is often ambiguous. In this work we concentrate on a particular type of uncertainty, namely event uncertainty, under which the occurrence date of some catastrophe cannot be predicted in advance. The occurrence of the catastrophic event, which significantly reduces the value of the resource, might be advanced by the extraction activities. Event uncertainty therefore renders intertemporal considerations particularly relevant to the design of optimal extraction rules: Unlike other sources of uncertainty (time-varying costs and demand, stochastic recharge processes, etc.) under which the extraction policy can be updated along the process to respond to changing conditions, event uncertainty is resolved only by occurrence, when policy changes can no longer be useful. Thus the expected loss due to the catastrophic threats must be fully accounted for prior to occurrence, and the resulting policy rules are significantly modified.

[71] In this work we study optimal groundwater extraction under the threat of events that differ in the damage they inflict and the conditions that trigger occurrence. We demonstrate the sensitivity of the optimal management policy to the details of the hazard and damage specifications. The analysis is presented here in the context of groundwater resources but has wide application in a variety of resource situations involving event uncertainty.

Appendix A

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

A1. Evolution Function

[72] Tsur and Zemel [2001] consider possible equilibrium states for general infinite horizon optimization problems of the form

  • equation image

subject to equation image = g(St, xt), SStequation image, xxtequation image, S0 given, assuming that the steady state policy is feasible, i.e., there exists a “recharge” function xR(S) ≤ equation image such that setting x = R(S) in g yields g(S, R(S)) = 0. B in equation (A1) is the benefit flow, and g determines the state dynamics, while some of the inequality constraints can be relaxed by assigning infinite values to the corresponding bounds. The steady state policy x = R(S), then, yields the value

  • equation image

[73] For the nonevent problem (2), equation (A2) reduces to equation (9). The optimality of the steady state policy for a given state S is tested by comparing W(S) with the value obtained from a slight variation on this policy. It is established that an interior state S can be an optimal steady state only if it is a root (zero) of the evolution function, defined as

  • equation image

If L(S) does not vanish, a feasible variation on the steady state policy yielding a value larger than W(S) can be found, hence S is not an optimal steady state. Corner states make an exception by the possibility to qualify as optimal steady states without being roots of L(S), depending on the sign obtained by the evolution function at these states. In particular, the lower bound S can be an optimal steady state if L(S) < 0, while the upper bound can be an optimal steady state if L(S) > 0.

[74] Specializing (A3) to the nonevent and auxiliary problems (2) and (14), we obtain the corresponding evolution functions (3) and (15).

A2. Dynamics of the Nonevent Processes

[75] We assume below that the nonevent steady state equation image is internal and suppress, for brevity, the superscript ne from the associated optimal processes and steady state. Let T denote the time at which the optimal nonevent stock process St enters the steady state equation image. The nonevent problem (2) is recast in the form

  • equation image

subject to equation (1), xt ≥ 0; St ≥ 0; ST = equation image and S0 given. Denoting the current value costate variable by λt, we obtain the current value Hamiltonian

  • equation image

Necessary conditions for optimum include

  • equation image

and

  • equation image

The transversality condition associated with the free choice of the entry time T is H(equation image, xT, λT) − rW(equation image) = 0, or, noting equations (A6) and (9)Y(xT) − C(equation image)xT + [Y′(xT) − C(equation image)][R(equation image) − xT] = Y(R(equation image)) − C(equation image)R(equation image), giving

  • equation image

Recalling the concavity of Y, equation (A8) implies

  • equation image

hence the transition to the steady state extraction rate must be smooth.

[76] Taking the time derivative of equation (A6) we obtain equation image = Y″(x)equation imageC′(S)[R(S) − x]. Comparing with equation (A7), we can eliminate the co-state variable and its time derivative

  • equation image

[77] For an autonomous problem, the optimal extraction is a function of the state S alone, xt= x(St) hence equation image = x′(S)[R(S) − x]. Therefore equation (A10) reduces to a first order differential equation for x(S):

  • equation image

with the boundary condition x(equation image) = R(equation image), representing the smooth transition to the steady state established by equation (A9). When equation image is internal, both numerator and denominator of equation (A11) vanish at this state. Nevertheless, the equation can be reduced, using l'Hopital's rule, to a quadratic equation in the difference x′(equation image) − R′(equation image), yielding equation (5) [see Tsur and Zemel, 1994].

[78] Once the solution x(S) of equation (A11) is given, equation (1) is readily integrated, yielding equation (6). Since equation (5) ensures that the difference x′(equation image) − R′(equation image) is finite, the singularity of equation (6) at the steady state implies that the integral diverges when its upper limit is set at equation image, giving T = ∞ and establishing property 2. The derivation of properties 4 and 8, corresponding to known critical stocks, is simpler: the transversality condition associated with the free choice of the entry time T, H(Sc, xT, λT) − rϕ(Sc) = 0, is written in the form (see the derivation of equation (A8))

  • equation image

and the concavity of Y implies that equation (A12) cannot be solved with xTR(Sc). Thus the transversality condition cannot be satisfied in finite time and the event is never triggered.

[79] Consider now the decreasing function J(S) = −C′(S)R(S)/[rR′(S)]. From equation (A7) we deduce that J(equation image) = λ. For finite times, we use equation (1) and write equation (A7) as equation imaget = [rR′(St)]λt + C′(St)xt = [rR′(St)][λtJ(St)] − C′(St)equation imaget or

  • equation image

When the stock process St lies above equation image, it decreases, hence both C(St) and J(St) must increase. Suppose that J(St) > λt. According to equation (A13), the process λt + C(St), hence λt itself, must decrease with time. It follows that the difference J(St) − λt increases in time, violating the end condition J(equation image) = λ. Thus

  • equation image

According to equation (A6), λt + C(St) = Y′(xt) hence equations (A13) and (A14) imply that Y′(xt) increases and xt decreases with time whenever St > equation image. The same considerations show that xt increases with time when the stock process lies below equation image, establishing property 3. In fact, the extraction and stock processes always show the same trend, hence the function x(S) of equation (A11) must increase.

A3. Derivation of the Auxiliary Objective

[80] For decreasing stock processes under endogenous uncertainty, the distribution of the event occurrence time T is given by equation (13) as

  • equation image

The corresponding density and hazard rate functions are also expressed in terms of the distribution of the critical stock:

  • equation image
  • equation image

[81] Let I( ) denote the indicator function that obtains the value one when its argument is true and zero otherwise. Writing the objective of equation (12) as ET{∫0[Y(xt) − C(St)xt]I(T > t)ertdt + erTequation image(ST)∣T > 0}, we evaluate the expectation of the first term observing that ET{I(T > t)∣T > 0} = 1 − FT(t) = F(St)/F(S0). The expectation of the second term is obtained, using equation (A16), as ∫0fT(t)ϕ(St)ertdt = ∫0f(St)[xtR(St)]equation imageertdt. Collecting the terms and using equation (11), we find the expected value for decreasing processes

  • equation image

yielding the auxiliary objective equation (14).

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References

[82] This work has been supported by the Paul Ivanier Center of Robotics and Production Management, Ben Gurion University of the Negev.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Groundwater Management Under Certainty
  5. 3. Uncertain Events
  6. 4. Concluding Comments
  7. Appendix A
  8. Acknowledgments
  9. References
  • Aronsson, T., K. Backlund, and K. G. Lofgren (1998), Nuclear power, externalities and non-standard Piguvian taxes: A dynamic analysis under uncertainty, Environ. Resour. Econ., 11, 177195.
  • Burt, O. R. (1964), Optimal resource use over time with an application to groundwater, Manage. Sci., 11, 8093.
  • Clarke, H. R., and W. J. Reed (1994), Consumption/pollution tradeoffs in an environment vulnerable to pollution-related catastrophic collapse, J. Econ. Dyn. Control, 18, 9911010.
  • Cropper, M. L. (1976), Regulating activities with catastrophic environmental effects, J. Environ. Econ. Manage., 3, 115.
  • Feinerman, E., and K. C. Knapp (1983), Benefits from groundwater management: Magnitude, sensitivity and distribution, Am. J. Agric. Econ., 65, 703710.
  • Gisser, M., and D. A. Sanchez (1980), Competition versus optimal control in groundwater pumping, Water Resour. Res., 16, 638642.
  • Gvirtzman, H. (2002), Israel Water Resources (in Hebrew), Yad Ben-Zvi, Jerusalem.
  • Kamien, M. I., and N. L. Schwartz (1981), Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, North-Holland, New York.
  • Long, N. V. (1975), Resource extraction under the uncertainty about possible nationalization, J. Econ. Theory, 10, 4253.
  • Narain, U., and A. Fisher (1998), Irreversibility, uncertainty and catastrophic global warming, Working Pap. 843, Dep. of Agric. and Resour. Econ., Univ. of Calif., Berkeley.
  • Pindyck, R. S. (1984), Uncertainty in the theory of renewable resource markets, Rev. Econ. Stud., 51, 289303.
  • Postel, S. (1999), Pillar of Sand: Can the Irrigation Miracle Last?, W. W. Norton, New York.
  • Reed, W. J. (1984), The effect of the risk of fire on the optimal rotation of a forest, J. Environ. Econ. Manage., 11, 180190.
  • Reed, W. J. (1989), Optimal investment in the protection of a vulnerable biological resource, Nat. Resour. Model., 3, 463480.
  • Tsur, Y., and T. Graham-Tomasi (1991), The buffer value of groundwater with stochastic surface water supplies, J. Environ. Econ. Manage., 21, 201224.
  • Tsur, Y., and A. Zemel (1994), Endangered species and natural resource exploitation: Extinction vs. coexistence, Nat. Resour. Model., 8, 389413.
  • Tsur, Y., and A. Zemel (1995), Uncertainty and irreversibility in groundwater resource management, J. Environ. Econ. Manage., 29, 149161.
  • Tsur, Y., and A. Zemel (1996), Accounting for global warming risks: Resource management under event uncertainty, J. Econ. Dyn. Control, 20, 12891305.
  • Tsur, Y., and A. Zemel (1998a), Trans-boundary water projects and political uncertainty, in Conflict and Cooperation on Trans-boundary Water Resources, edited by R. E. Just, and S. Netanyahu, pp. 279295, Kluwer Acad., Norwell, Mass.
  • Tsur, Y., and A. Zemel (1998b), Pollution control in an uncertain environment, J. Econ. Dyn. Control, 22, 967975.
  • Tsur, Y., and A. Zemel (2001), The infinite horizon dynamic optimization problem revisited: A simple method to determine equilibrium states, Eur. J. Oper. Res., 131, 482490.
  • Yin, R., and D. H. Newman (1996), The effect of catastrophic risk on forest investment decisions, J. Environ. Econ. Manage., 31, 186197.