Water management in a mountain front recharge aquifer

Authors


Abstract

[1] We explore the dynamic and conflicting interaction of incentives for private versus riparian habitat water use in the context of a mountain front recharge system. A novel situation arises wherein private uses are consumptive while riparian habitat uses, although clearly consumptive, are closely related to water stocks. The hydrology of the mountain front recharge system is characterized by system lags which lead to difficulties in constructing incentive based policy tools.

1. Introduction

[2] Various aquifer-stream systems in semiarid environments have suffered irreparable riparian habitat loss as a result of persistent growth in urban and irrigation water use. The San Pedro River, a perennially gaining stream in Southern Arizona, provides an example of such a phenomenon. Agricultural and urban water have threatened the riparian habitat in the Upper San Pedro Basin, a major flyway for a multitude of avian species. Irrigated agricultural land has recently been retired from production, mitigating the crisis to an extent.

[3] The San Pedro is somewhat unique in that the river is northward flowing with the “headwaters” in Sonora, Mexico. Contrary to numerous other western rivers it is not snowpack fed, but rather the river is supplemented by underground rainwater charged aquifer flows. The riparian habitat is thus critically dependent on the uninterrupted flows of these underground supplies. Because of the public nature of riparian habitat benefits, this comprises a classic example of incompletely defined entitlements to riparian water rights, or equivalently, underground aquifer flows; see Bromley [1991, p. 31].

[4] We present an analysis of such an aquifer-stream system and then explore policy implications in general and in the context of the Upper San Pedro Basin (USPB). We begin with an overview of the hydrology of the USPB and the nature of urban and riparian habitat benefits. We then develop a hybrid economic-hydrologic model intended to highlight the trade-offs between riparian and urban uses in the context of the mountain front recharge system hydrology. Analysis provides some useful results concerning the nature of solutions and potential problems that appear in such systems. The analysis then suggests some policy options which are subsequently investigated. Fundamental results depend on the lags between urban water use and ultimate effects on riparian environment as well as the aquifer water balance conditions.

2. Hydrology, Benefits, and Costs

[5] Consistent with the discussion above, the aquifer in the USPB is termed a mountain front recharge system. This aquifer type is specific but not unique. Indeed numerous aquifers in the western US share analogous properties: the Pecos in eastern NM is similar and reflects its own potential policy problems set in that if upstream junior pumpers infringe on downstream senior rights, decreased or cessation of pumping will not rectify those downstream shortages for quite some time due to pumping/aquifer depletion lags; Albuquerque, New Mexico, sits on a mountain front recharge aquifer, but pumping/river effects lags are minimal due to the proximity of pumping to the watercourse.

[6] The cental aspect of the problem is the pumping effect lag between the initial use and subsequent alternative uses of system water. In the USPB the problem is exacerbated by the fact that the alternative use, the provision of riparian habitat, reflects a public type of benefit that may not be completely recognized in the private allocational process, regardless of lags. Moreover, as indicated above, the various benefits from water use are often closely linked to the characteristics of the hydrologic-economic interface so that the hybrid model may have to be tailored to the particular economic-hydrologic system being considered. Consequently we provide an overview of hydrology, benefits and costs in the context of the USPB.

2.1. Regional Hydrology

[7] For a more complete description of the pertinent hydrology see, e.g., Pool and Coes [1999], Corell et al. [1996], or Putnam et al. [1988]. The San Pedro River, running south to north from Northern Mexico into Southeastern Arizona, is low flow and seasonally intermittent with some perennial reaches. The San Pedro River Basin, consisting of three subbasins, the Mexican, the upper U.S. and the lower U.S. basins, is a typical Southwest mountain front recharge basin with an arid climate punctuated by a summer monsoon season. While the headwaters are north of Cananea, Sonora, Mexico, the river is not fed by snowmelt as is normally the case for most south flowing Southwestern rivers. Rather the river itself is fed entirely from underground recharge; annual precipitation in the basin provides natural recharge to the aquifer.

[8] Our focus is on the USPB, which encompasses the drainage north from the Mexico-U.S. border to The Narrows, approximately 10 miles north of Benson, Arizona The USPB lies within a major northwest structural trough that is delineated by mountain ranges, the Huachuca Mountains on the west and the lesser Mule Mountains to the east. The rock that forms these ranges, as well as the bedrock of the basin, varies in age from Precambrian to Tertiary and consists of igneous, metamorphic and sedimentary rock. The rocks have low porosity and permeability in general, resulting in minimal fluid flow except where they are fractured or faulted. (Porosity refers to the void paces found in the rock, while permeability refers to the interconnectedness of the voids. Porosity determines the amount of fluid that can be contained in rock formations, while permeability is relevant to the flow of fluid through the rock formation.) Southeast ranges are composed of consolidated rocks, which also form the bedrock of the ranges.

[9] The basin per se is composed of the Upper San Pedro River Valley and an inner valley. The Pantano Formation, a Tertiary conglomerate, overlies bedrock in the basin. The basin fill, deposited post basin and range formation, lies unconformably above the conglomerate. The basin fill is broken into lower and upper units. Within the inner valley, the upper unit is overlain by a floodplain alluvium. The Pantano conglomerate is found discontinuously on both sides of the basin. The discontinuities are caused by a series of near vertical steps faults associated with the uplifts. Locally only small amounts water in fractures and faults can be expected [Putnam et al., 1988]. The lower unit of the basin fill consists of interbedded sandstone and gravel, which varies in thickness from 250 to 500 feet. The variability of sorting and cementation results in variations of aquifer quality of the lower unit, throughout the USPB. The upper fill is composed of gravel, sand, silt and compacted clay that is weakly cemented. The floodplain alluvium consists of unconsolidated gravel, silt and sand deposited by the stream action of the San Pedro and its tributaries.

[10] The lower and upper units of the basin fill comprise the regional aquifer system and are the primary sources of groundwater in the USPB. This regional aquifer is recharged from the mountain fronts to the east and west and discharges to the San Pedro River floodplain in the center of the basin. The flood plain alluvial aquifer is recharged by streamflow as well as upward leakage from the underground confined portion of the regional aquifer and from lateral flow from the regional aquifer. The flow of water in the floodplain is parallel to the San Pedro River. In addition to the regional and alluvial aquifer, there are also a few localized hardrock aquifers, as well as some natural springs. However, their extent is not well known, nor do they substantially impact the regional water picture. Consequently, we do not consider them in our modeling. The San Pedro River exists as an intermittent flow regime along much of its course. During low water use times, the rate of groundwater discharge results in surface flow of the river. During other times of the year, the use of water by riparian vegetation exceeds the groundwater discharge and surface flow ceases. Surface flow can also occur immediately after rain events.

[11] There are consumptive and nonconsumptive water uses in the USPB. In general, consumptive uses directly deplete the regional aquifer, while nonconsumptive uses are related to the riparian areas associated with the alluvial aquifer. Strictly speaking, the “nonconsumptive” uses actually are ET based and hence consumptive, but they are stock related (to the alluvial aquifer) hence the distinction in terminology. These alternative uses and the manner of the interconnectedness of the regional and alluvial aquifers makes the incorporation of the hydrology pivotal in the context of analysis and policy development.

[12] The stock of water in the regional aquifer depends upon the thickness of the regional aquifer, the surface area of the aquifer as well as the specific yield of the aquifer, which in turn is determined in part by porosity and permeability. Changes in regional aquifer stocks are determined by additions and losses to the system. Losses include pumping for consumptive use net of return flows as well as net outflows to the alluvial aquifer. Additions to regional aquifer stocks include mountain front recharge as well as reinjected urban (treated) wastewater.

[13] The stock of water in the alluvial aquifer is affected by lateral net flows from the regional as well as upward net leakage from the confined portion of the regional aquifer. Additional changes in stocks are related to net supplements from river flows. For all of the above effects, under the appropriate circumstances the flows can represent either additions to or depletions of the alluvial aquifer. Anderson and Freethey [1995] argue that the hydrology of the regional and alluvial aquifers is such that for certain purposes it may be difficult to separate them. We thank a reviewer for pointing this out, as it greatly simplified the heuristic modeling of the aquifer.

2.2. Benefits

[14] We dichotomize benefits into two basic classes, urban benefits and habitat benefits. Urban benefits emanate directly from physical consumption of water pumped from the regional aquifer. Historically irrigated agriculture was a major player in terms of water use, but for all practical purposes has ceased to exist, ostensibly in response to the realization of looming water “shortages” vis-a-vis the desire to ensure the longevity of habitat benefits. Habitat benefits are related to riparian habitat which provides food, shelter and water or migratory song birds. While habitat benefits are dependent on consumptive use of water in the form of evapotranspirative use of water to support the riparian habitat, these consumptive uses are minor compared to the demands related to urban uses. Consequently, habitat benefits are frequently referred to as nonconsumptive, being more closely tied to alluvial water stocks. Because of the fundamental difference in the effects of urban versus habitat water demands both locationally and hydrologically, as well as quantitatively, we consider them separately.

2.2.1. Urban Benefits

[15] As a result of the demise of agricultural water use, consumptive uses of water are primarily urban consisting of domestic, industrial, commercial and institutional uses. Urban use admits to the possibility of wastewater return flow reinjection, a policy option currently underway in the USPB. Within the USPB, the main urban users are Sierra Vista and the adjacent Fort Huachuca. Roughly speaking increases in urban water benefits require increases in water pumping. Moreover urban benefits insofar as pumping is concerned may be limited more by “local” hydrological considerations (local aquifer characteristics, number and size of wells, cones of depression and other considerations). This is dealt with in more detail below.

2.2.2. Habitat Benefits

[16] Riparian habitat supports public good types of benefits. The major distinctions are (1) since habitat benefits are nonexclusive they will tend not to be provided through private decision making procedures; (2) on the other hand, since habitat benefits are nonrival the level of benefits increases as the number of users increases for any given level of water allocation (barring congestion effects).

3. A Hybrid Model

[17] Following the schematic from the previous section, we consider benefits from pumped water in the form of urban uses as well as benefits from the maintenance of riparian habitat. Benefits are measured by the usual surplus measures where net urban benefits from water use are given by

equation image

where w is water pumped from the aquifer and pU is the demand for urban water and CU(w) are the urban water production costs. Recent work, including that by Chermak and Krause [2002] and Krause et al. [2003], indicates that urban benefits tend to be heterogeneous. However, at this level of generality the effects of heterogeneous demand only tends to obfuscate and consequently is ignored. In order to focus the analysis on the private-riparian trade off we suppose that delivery and treatment costs dominate within the range of viable solutions. Thus urban water delivery costs depend only on the quantity of water delivered. We will further assume that there are constant returns to scale in urban water production costs so that dCU(w)/dw = cU and dBU(w)/dw = pU(w) − cU.

[18] Historically, wastewater was treated and channeled to drying fields, so essentially no water was returned to the aquifer. Subsequently the benefits of wastewater reinjection were realized and this practice was terminated. However reinjection is essentially limited to the amount of indoor use; i.e., outdoor uses result in evapotranspiration, evaporation or other losses that preclude reinjection. It may be possible to further disaggregated urban benefits so as to distinguish those emanating from indoor versus outdoor uses of pumped water. If β is the return flow coefficient for urban use, then α = 1 − β is the fraction of pumped water consumed, and α can be decomposed into αI and αO for indoor and outdoor use, respectively. If θ is the fraction of pumped water allocated to outdoor use, then α = θαO + (1 − θ)αI. If indoor and outdoor use values differ significantly, then it may be useful to design policy so as to target specific uses. De facto, θ becomes a policy variable so that direct or incentive based policy can be implemented so as to alter the average urban consumptive use coefficient; see Burness and Little [2003]. As urban water uses are primarily private, the level of urban benefits is more or less proportionate to the level of use.

[19] Habitat benefits are related to evapotranspiration from the alluvial aquifer

equation image

so that dBH(E)/dE = pH(E). Habitat benefits are public in nature so that the benefits attributable to riparian habitat for any given level of water use are roughly proportionate to the number of “users”. However the specification of the relevant population over which these public benefits accrue is often contentious. Moreover we ignore the possibility that the relevant population may change over time and in all likelihood be dependent on the level of habitat provision. We also ignore benefits related to option values.

[20] According to the USGS, the alluvial aquifer and the regional aquifer are hydraulically connected, so there is no need to distinguish the two analytically in our hybrid model; see Anderson and Freethey [1995] for details on groundwater hydrology for this region. However since urban benefits emanate from regional aquifer pumping and riparian benefits are determined by evapotranspiration from the alluvial aquifer, we must ensure that the modeling adequately expresses and delimits these relationships. We do this by specifying evapotranspiration, E(t), as proportional to relevant alluvial aquifer stocks, so that

equation image

where H(t) is the height of the water table above a base reference plane, equation imageA is the height of a reference plane in the alluvial aquifer above the base reference plane, equation image is the height of the bottom of the regional aquifer above the reference plane, and γ is a constant in units of AF per foot; see Figures 1a and 1b for a heuristic depiction of the hydrology. In general when we speak of “the aquifer”, we will be referring to the conjoined regional-alluvial aquifer. However in the context of equation (3) if H(t) falls below equation imageA, the riparian habitat is depleted and riparian benefits are zero. In addition there may be irreversibility issues which limit the riparian habitat's ability recover even if H(t) subsequently rises above equation imageA.

Figure 1a.

Steady state prior to pumping.

Figure 1b.

Aquifer with drawdown.

[21] The stock of water in the aquifer at time t is roughly proportional to the height of the water table and given by

equation image

where AS is the surface area of the aquifer times the specific yield. Aquifer supplies are supplemented by natural recharge to the regional aquifer, N0, resulting from rainfall within the hydrologic system. N0 is assumed to be constant and equal to historical average of observed recharge. Aquifer supplies are depleted by w(t), pumping for urban water use, and replenished by reinjection and/or natural recharge. However, the distance of pumping in the regional aquifer from the river suggests a lagged relationship between pumping and subsequent effects on the alluvial aquifer. This is reflected in the reservoir dynamics by relating the change in aquifer levels, equation image(t), to pumping from τ periods prior, w(t − τ), and independent of pumping less than τ periods prior; in terms of the Figure 1b heuristics, pumping at t − τ generates an immediate cone of depression which translates into lower aquifer level at time t. Thus aquifer levels, H(t), evapotranspiration, E(t), and hence habitat benefits at time t, are also dependent on pumping from τ periods prior. In actuality H(t) and functionally related variables will be in some way related to all pumping prior to τ. Burness and Martin [1988] consider one such representation in the analysis of a tributary aquifer. For ease of exposition in focusing on the role of lags, we maintain the more simple formulation; the more complicated formulation requires at a minimum the addition of a (dummy) state variable which effectively precludes the phase plane analysis.

[22] Just as the effects of actions in the regional aquifer are passed through to the alluvial aquifer, there is a connection between the river and the alluvial aquifer. The alluvial aquifer supplements the river by the amount of “river recharge”, R+(t) = N(t) − E(t). In the natural steady state, with steady state evaporation given by γ(H0equation imageA) = E0 we have natural steady state river recharge, R+0 = N0E0 > 0, reflecting the fact that in the natural steady state the river is perennially gaining. However, in the presence of nontrivial depletions of the regional aquifer, net inflows to the alluvial aquifer are reduced and net transfers from the alluvial aquifer to the river may be reduced or even reversed. We represent these net transfers as river recharge less aquifer based evapotranspiration losses plus a net flow related to differences in hydrostatic head of the (alluvial) aquifer vis-a-vis the river. Specifically we have “river effects”, or net flows from the alluvial aquifer into the river are

equation image

where R(t) = δ[H0H(t)], H0 is the natural steady state level of the aquifer. Letting F0 be river base flow, we require δ[H0H(t)] ≤ F0 as an upper limit on the streamflow depletions to the alluvial aquifer. The second term in equation (5) reflects the fact that if alluvial aquifer water table drops, downstream river flows will be diminished. The constraint is a physical limitation on river depletions; the coefficient δ is in AF per foot and determines the rate of river effects when the (alluvial) water table level falls below the natural steady state. This hydrological constraint on leakages from the river to the alluvial aquifer might be replaced by a more restrictive ecological-management constraint imposed relative to what is often referred to as a “safe minimum standard”, say FSMS < F0, so that FSMS replaces F0 in the restriction in equation (5). For simplicity assume dR+/dH = dN/dH = η > 0 so that dR/dH = dR+/dHdR/dH = η + δ. Similarly suppose that N(t) = 0 when H(t) = HF = H0F0/δ.

[23] The change in aquifer storage is the sum of natural recharge plus net (lagged) pumping less riparian habitat ET and river effects. Using equations (3) and (5) and simplifying this yields the (combined regional/alluvial) aquifer dynamics as

equation image

where β is the fraction of pumping returned to the aquifer as a consequence of reinjection or naturally occurring return flows.

[24] In the natural steady state, with w(t − τ) = 0, E(t) = E0 and H(t) = H0, equation image(t) = 0 in equation (6) yields R0 = N0E0 as expected. For a given pumping steady state, if w(t) = wi, then equation imagei = 0 in equation (6) leads to

equation image

Thus any steady state pumping (consumptive use) level must be sustained through reductions in stream flows. Base flow of the river is supplemented by flows from the alluvial aquifer to the river; these supplemental flows are bounded above by N0E0. Generally denote river flows are

equation image

where R(t) is as defined in equation (5).

[25] There are two limiting case steady states. One is that where (urban) pumping from the aquifer is at its largest sustainable value, the other where riparian ET is at its largest sustainable value. Clearly when riparian ET is at its largest sustainable value, urban pumping is of necessity zero. When urban pumping is at its largest sustainable value, riparian habitat will ultimately vanish. These observations raise some obvious questions about the notion of sustainability.

[26] An alternative steady state might be defined as one which maintains an “acceptable” level of habitat benefits. This level of riparian habitat benefit is often defined implicitly in terms of a “safe minimum standard” (SMS), Msms, imposed on streamflows as referred to above in footnote 11. The ofttimes expressed concern for riparian over urban water use is perhaps related to the fact that (1) riparian benefits are public and hence outside private decision making control, and (2) the total loss of riparian habitat is often viewed as being irreversible and hence purposely overvalued; the SMS is a fallback response when environmental benefits are difficult to quantify. However this line of logic can be followed to its end as follows. Given a SMS on river flows, this defines a maximum allowable/sustainable level of urban pumping via equations (5) and (7); i.e., wSMS = MSMS/(1 − β). This level of urban pumping defines a “sustainable” level of urban benefits, which in turn, is a de facto a lower bound estimate of the associated level of riparian habitat benefits implied by the choice of MSMS. Alternatively if urban pumping must be reduced to achieve wSMS, then a surrogate for the increase in riparian habitat benefits associated with this decrease in urban use is given by the absolute value of the change in urban benefits associated with this reduction in urban use.

[27] However, in the context of our inquiry this procedure begs the question. That is, if habitat benefits are known and socially optimal allocations are operationally attainable, then there is no need to impose a SMS; the notion is redundant as the optimal allocations of water are a consequence of the optimization process. The social optimum is determined by choosing w(t) and T so as to

equation image

subject to equation (6) where r is the appropriate discount rate. From equations (5) and (6) we also have w(t) ≤ (N0 + F0)/(1 − β) when H(t) ≤ HF = H0F0/δ. We suppose this constraint is nonbinding now but investigate its role in the phase plane analysis. Let the current valued Hamiltonian be

equation image

where μ is the costate variable for the transition equation (6). Necessary conditions for an interior optimum include:

equation image
equation image
equation image

as well as equation (6) above. At an interior steady state, equation (13) is redundant, so evaluating equations (11) and (12), respectively, yields

equation image

and

equation image

Equation (14) involves a variation on a familiar theme; specifically, equation (14) requires that urban water be utilized to the point where the net marginal benefits of urban consumptive use at time t is equal to the marginal user cost of having an additional acre foot of water in the aquifer at time t + τ. μ(t) is the marginal user cost of consumptive use per foot of aquifer thickness. μ(t)/AS yields marginal user cost per acre foot of consumptive use. Multiplying by (1 − β)μ(t)/AS is the marginal user cost per foot of pumped water. This is the “forward looking” aspect of intertemporal decision making referred to earlier; not only must the user cost relation capture the future effects of current decisions, but it must operationally implement those effects as to their relevance at a point τ periods advanced from the present. Evaluating equation (14) at t − τ yields

equation image

so that marginal user cost is unambiguously positive. Using equation (16) in equation (15) and rewriting yields

equation image

which describes the tradeoff between marginal habitat benefits and lagged marginal urban benefits in determining marginal user cost. Observe that the first term on the RHS of equation (17) is negative and reflects the allocational role of riparian habitat benefits while the second is positive reflecting the role of urban benefits. Marginal user cost tends to decrease or increase as the absolute value of the first term is greater or less than the second, respectively. A larger marginal user cost calls for a reduction in (τ periods) prior urban water use and enhances riparian habitat; alternatively current reductions in urban water use augments future habitat. At the steady state equation image(t) = 0 so that marginal habitat and urban benefits are balanced.

[28] Differentiating equation (16) with respect to t yields

equation image

so that consistent with the discussion above, since dpU/dw < 0, the signs of equation image and equation image are opposite and equation image > 0 results in decreased urban pumping and conversely.

[29] Next, using equation (18) in equation (17) and recalling equation (3) leads to

equation image

where D(t − τ) = equation image < 0. Alternatively, letting t = t + τ

equation image

Equation (2) and the transition equation (equation (6))

equation image

describe the optimal solution. At this level of generality, the policy implications are limited to observations on the impacts on the shape and slope of the drawdown path prior to a new steady state. The sign of the first expression on the RHS of equation (20) is unambiguously negative. Thus an increase in r, η, δ, or net marginal benefits decreases the slope of the time path w(t). The second term is clearly positive with similar interpretations. Thus the role of marginal urban benefits and marginal habitat benefits tend to offset each other regarding the magnitude of equation image(t).

[30] Kim et al. [1993] discuss a technique for solving a system with lags similar to equations (6) and (20) for the case where the benefit functions are quadratic. Their technique is a dynamic programming backward induction approach over intervals of length τ. El-Hodiri et al. [1972] demonstrate an analytical solution for a simplified special case. However, even in the light of data gaps, some insights can be gained by examining the steady state solution. Setting equation (6) equal to zero yields

equation image

Likewise, setting equation (20) equal to zero and, to ease exposition, letting pU(w) = abw(t) and dBH[γ(Hequation imageA)]/dE = deγ(Hequation imageA) yields

equation image

(Earlier assumptions on N(t) imply N(t) = η(H(t) − HF) where η = N0/(H0HF).) The last term in equation (22) is the pumping level at which the marginal benefits equal marginal pumping cost in urban water use. As long as there is nonsatiation in habitat benefits the first two terms on the RHS of equation (22) sum to a negative number for all admissible values of H(t). Thus socially optimal pumping is always less than the value of pumping which maximizes current net urban benefits (see section 4). Examining equations (21) and (22) we see that ∂w/∂Hequation image < 0 and ∂w/∂Hequation image > 0. Similarly, from equations (20) and (6) we can show that ∂equation image/∂Hequation image < 0 and ∂equation image/∂wequation image < 0. Equating equations (21) and (22) yields the socially optimal steady state (interior) solution (HS, wS) where

equation image

with Δ = b(η + δ)(rAS + η + δ) + γ2(1 − β)2e > 0 and

equation image

Figures 2a and 2b illustrate the phase plane diagrams for the socially optimal steady state for the cases of a high-flow and low-flow river, respectively.

Figure 2a.

Social optimum: Large flow.

Figure 2b.

Social optimum: Small flow.

[31] In Figure 2a, equation image = 0 is given by equation (21) and is constant at a pumping level of (N0 + F0)/(1 − β) for H(t) ≤ HF and then downward sloping. Note that for H(t) ≤ HF we have the extreme polar case of a perennially losing stream while H0 is the case of a perennially gaining stream so that a switching point occurs between HF and H0. The river effects constraint is not binding at the socially optimal steady state so the equilibrium is a saddlepath with the convergent separatices given by the arrowed curves in sectors I and III. From the natural steady state, the approach would be along the separatrix in sector III. If the riparian habitat is currently diminished but not irreparably damaged the approach is along the separatrix in sector I.

[32] Figure 2b considers the case where the interior solution for the social optimum is to the left of HF. The solution is at the intersection of equation image = 0 and the flat segment of equation image = 0. One might argue that it is somewhat misleading to catagorize this as a social optimum as one would expect that habitat benefits would be minimal or zero at HF. Nonetheless this demonstrates the forcing effect of stream-aquifer systems characterized by low streamflow.

4. Policy Analysis

[33] Policy analysis involves the comparison of an appropriate base case decision criteria with the social optimum and then evaluating the efficacy of various direct or incentive based policies vis-a-vis the achievement or approximation of the social optimum via the base case decision criteria.

[34] The base case scenario is closely tied to extant institutional arrangements and allocational distributions. As these institutional arrangements will vary by locale we provide an illustrative discussion in the context of the USP River Basin which highlights some of the issues that may arise in defining the appropriate baseline.

[35] The benchmark institutional structure for the Sierra Vista area is somewhat unique as there are a number of private water suppliers that are regulated as to water charges and are perhaps best viewed within the context of a mildly decreasing cost industry with water charges determined on a “cost of service” pricing scheme. As the concept of including user costs in rate determinations is clearly foreign to most regulatory authorities, the usual common property problems may arise and be exacerbated to the extent that marginal and average private cost of supplying water diverge. This clearly an empirical question.

[36] The other issue that arises in the construction of a baseline in this case concerns the treatment of riparian habitat. While attempts have been made in the Upper San Pedro Basin to accommodate riparian habitat concerns, these attempts have generally occurred outside the pricing system. Ultimately either scarcity value and opportunity water pricing or enforceable pumping limitations must be contemplated to assure efficient allocations of water between municipal and riparian interests. The fact that concerns arise as to the equity of allocations is really no more than a reflection of the fact that there is not now nor likely to ever be a consensus regarding a precise formulation or quantification of riparian benefits. This issue is further complicated to the extent that intertemporal efficiency considerations evolve into equity concerns.

[37] With these caveats in mind we now specify the baseline as that wherein habitat benefits are ignored but common property externalities in urban water use are considered. Thus urban pumping is chosen to maximize private benefits subject to the usual accounting constraints of aquifer hydrology; specifically,

equation image

subject to equation (6). The current valued Hamiltonian is

equation image

(Note that ignoring habitat benefits does not mean that none are generated; in fact habitat benefits generally will be present until and unless urban pumping ultimately causes H(t) to fall below equation imageA.)

[38] Utilizing the necessary conditions (11) and (12) above, we have

equation image

and

equation image

Equation (28) is similar to equation (14) except the multiplier ν(t) is more narrowly defined than μ(t). The multiplier ν(t) is the marginal user cost of current urban water use insofar as future urban water uses are concerned. The multiplier μ(t) is the marginal user cost of current urban water use insofar as future urban and habitat uses are concerned. These multipliers form the basis for any incentive based policy instrument.

[39] The dynamics of the common property problem are immediate from the more general analysis above. Setting BH(E) = 0 in equation (20) yields

equation image

while is equation image(t) still given by equation (6). Since equation image(t) is as before, equation (21) describes equation image(t) = 0 and from equation (22) with BH(E) = 0 or equation (29) we have equation image describing equation image(t) = 0.

[40] Figures 3a and 3b depict the phase plane analysis for the case where only urban uses enter the decision function. Figure 3a displays the case where river flows are large and the private optimum still provides riparian habitat albeit at a suboptimal level. The convergent separatices are equation image = 0 along from either the left or right depending on the initial value of H(t). It can be shown from equation (22) that wP > wS so that HP > HS follows directly. Thus this policy formulation considering only market benefits but internalizing common property externalities will over pump by wPwS and deplete the aquifer to a level HSHP below the socially optimal level.

Figure 3a.

Private optimum: Large flow.

Figure 3b.

Private optimum: Small flow.

[41] In Figure 3b, which may be more representative of many semiarid stream-aquifer systems, river flows are small and urban pumping drives habitat down to minimal levels. The convergent separatrix in sector II goes to the point (((N0 + F0)/(1 − β),0) on the vertical axis. Figure 4 replicates the myopic solution where habitat users are unaware or unconcerned with riparian habitat benefits and additionally are ignorant or uncaring of the effect of current pumping on future water availability. While the steady states in both Figures 3a and 3b are the same, the approach paths are different; the myopic path leads to a more rapid deterioration of riparian habitat.

Figure 4.

Myopic solution.

[42] The analysis is facilitated by considering the case where both common property externalities and habitat benefits are ignored. The steady states are identical for this and the private optimal solution above. This is the case of myopic decision making referred to in Figure 4 wherein w(t) is chosen so as to maximize equation (26) while ignoring equation (6) so that necessary conditions require that pU(w(t)) − cU = 0 or w(t) = equation image = pU−1(cU) so long as the boundary conditions on equation (6) are satisfied. (While decision makers ignore equation (6), it is an accounting relationship that must be satisfied in aggregate.) In view of this we see that ν(t) defines the Pigouvian tax necessary to internalize the common property externality and μ(t) defines the Pigouvian tax necessary to internalize both the common property and habitat externality. (Similar to earlier comments the inclusion of the coefficient (1 − β)AS transforms the tax rate to $/AF of consumptive use.) Thus μ(t) − ν(t) is the Pigouvian tax necessary to internalize the habitat externality alone.

[43] The expression in equation (28) can be integrated directly to get

equation image

where ε = (rAS + η + δ)/AS > r ≥ 0. In this case the policy instrument is easily computed and given by a tax rate which increases at the rate ε > r. Unfortunately, this case is somewhat misleading as the assumption on water production costs are unlikely to be borne out in a scenario where there are no competing uses to urban pumping. Generally in such cases the current value marginal user cost tends to display an inverted U shape; see Shah et al. [1993] and Burness and Brill [2001].

[44] The form of the multipliers ν(t) and μ(t) is generally intertemporally complex, and the task of determining these optimal taxes is both informationally and computationally demanding; see Baumol and Oates [1971]. Moreover the computation of the multiplier μ(t), for example, is dependent on knowledge of habitat benefits. Thus a policy instrument targeting private choice variables may be difficult to validate, and perhaps more importantly, difficult to implement. On the first issue, the goal is to develop such an instrument whose incentive properties mimic those of the Pigouvian tax, but which are based on private choice variables. This is at best a second best proposition. Burness and Brill [2001] illustrate such a procedure. The results of such efforts are clearly specious to the investigation at hand, but are useful in any context for determining bounds on the costs of internalizing such externalities.

[45] However a more relevant concern relates to incentive compatibility issues and cognitive problems due to bounded rationality of private decision makers. The only incentive based policy instrument in this context that private users can directly respond to is water use. According to equation (16), a proxy for current marginal user cost is net marginal urban benefits from τ periods prior. Thus the current tax rate must be tied to pumping from τ periods earlier. There would appear to be substantial uncertainty as to whether private decision makers could respond appropriately to such an incentive mechanism. Essentially current pumping determines the users' tax rate τ periods in the future. It is not clear how or even if users would respond to this incentive. Alternatively the tax rate could be tied to current pumping. In a analogous framework, Kim et al. [1993] find that current valued penalty functions tend to incentive based solutions which fall short of the optimum. (Marginal user cost in the work of Kim et al. is independent of the state variable, thus facilitating the analysis.)

5. Summary and Overview

[46] This paper develops an intertemporal hybrid economic-hydrologic model in order to analyze the normally conflicting interactions between private and social water use typical of semiarid environments. We consider a mountain front recharge aquifer-stream system wherein private water use may adversely affect the nature and extent of riparian habitat. The analysis is complicated by the fact that there are time lags between private water use and resultant impacts on social riparian benefits. This suggests that policy must be forward looking in order to mitigate potential damage that can be caused by either myopic planning or by planning that does not fully include social riparian benefits.

[47] We find, not surprisingly, the social optimal for pumping and aquifer height depends on the relative magnitudes of urban and riparian benefits as well as the nature of the stream-aquifer system. Consequently, “desired” levels of riparian habitat benefits cannot be stipulated independent of stream-aquifer hydrology and the long run steady state may result in a stream- aquifer system which may bear little resemblance to the natural steady state. For example while the natural steady state involves a perennially gaining stream, even for the large streamflow case, the phase plane analysis does not preclude the possibility of a perennially losing stream at the social optimum. Moreover, if urban benefits are not static, either due to the nature of or evolution of preferences, or as is more likely the case, as a result of expected increases in population levels, the dynamic pressures on riparian habitat and the nature of long run streamflows may be changing over time; however, analogous changes in social preferences and increased population may lead to changes in the structure of social benefits which tend to neutralize these tendencies.

[48] In the case of a low-flow river, the phase plane analysis suggests that there is a greater likelihood that the stream-aquifer system will be severely impacted by urban water use and that the socially optimal level of both private and public benefits may leave something to be desired, a reflection of the dire nature of the private-public conflict in this context. Moreover, any miscalculation in the allocation of water may involve subsequent difficulties in attempts to restore evanesced riparian habitat as in semiarid environments such habitat usually recuperates more slowly than it deteriorates. Furthermore, the lags between remedial policies and actual remediation may prolong if not preclude this process. In any event, remediation will most likely require severely curtailed urban use. As there appears to be a bias against incentive based urban water use conservation policy, direct controls may require some or all of the following: (1) the mandatory adoption and utilization of improved technology, (2) de facto drastic changes in private preferences and patterns of use, or (3) the curtailment urban growth. We note in passing that there appears to be a bias against the option of curtailing urban growth as well as that against incentive based controls; see Brookshire et al. [2002]. To wit, observe that the town of Sierra Vista, a major urban water user in the Upper San Pedro River Basin, has invested heavily in treatment and reinjection of urban wastewater and imposed use restrictions (e.g., all outdoor watering must be done by hand) in an attempt to curtail water use rather than increase water rates or seriously consider restrictive growth plans. Our analysis show that these programs, while laudable and well intentioned, may, at best, just postpone the inevitable.

[49] The points raised in the above paragraphs are empirical questions which require a good understanding of both the stream-aquifer system and the social-economic climate of the geographic region. This involves substantive hydrologic information as well as economic demographics and projections. These are areas for future research.

Acknowledgments

[50] The authors would like thank two reviewers and the associate editor for comments which greatly improved the structure and exposition of this paper. This material is based upon work supported in part by SAHRA (Sustainabilty of semi-Arid Hydrology and Riparian Areas) under the STC Program of the National Science Foundation, agreement EAR-9876800.

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