Allocating resources in an uncertain world: water management and endangered species.

The Rio Grande silver minnow (Hybognathus amarus) is exemplary of many fish species in the arid west. Its stocks have declined markedly during the past fifty years as Rio Grande water has been increasingly used to meet agricultural and municipal demands. However, any efforts to help the species recover will be done in an environment of striking uncertainty (U.S. Fish and Wildlife Service 2007). How will society protect the Rio Grande silvery minnow and other endangered species when so little is known about them, but when the costs of some mitigating measures are so much more clear? Such costs include the politically unattractive option of denying water to agriculture and municipalities to leave more water in the river. Due to extremely limited knowledge, the management problem society faces does not involve risk, where the probability distributions of interest are well understood. The problem we consider in this article is instead one involving pure or Knightian uncertainty, often deemed ambiguity (Camerer and Weber 1992; Knight 1921; Ellsberg 1961).

Water is a natural resource often taken for granted: it frequently commands such a low per-unit price that few water users even know what price they pay. However, the twenty-first century may be the era when relative water scarcity becomes a fact of life (Brown 2003). Population growth and economic development place increasing pressure on water supplies in many arid regions, especially during drought. Idaho's Snake River below Milner Dam and New Mexico's Gila River and the Rio Grande below Elephant Butte Reservoir are examples of cases where water rights holders dry up streams and rivers completely, and it is legal to do so (Benson 2004).

Societal interest in protecting endangered species creates a natural conflict between private and public interests. In the United States, public interests in such species are represented in the Endangered Species Act. This law has been the focus of many well-known legal battles over land use rights (Brown and Shogren 1998) and conflicts over water (Benson 2004). Recent controversies (Barringer 2005) pit the Columbia River salmon runs against agricultural and municipal interests. Those who want to use the river's water argue that extraction should be allowed because it cannot be proved that reductions in current flows would affect species' survival. Because of the uncertainty in the hydrology and biological situation this claim cannot be completely ignored. Even some of the biologists who believe that there is a relationship between instream flows and species survival admit that there is a fair amount of uncertainty about the exact nature of that relationship (see Shaw 2005, pp. 264-267).

The instream flow management problem explicitly needs to account for the depth of the uncertainty. We tackle this building on the ambiguity and uncertainty literature (Knight 1921; Ellsberg 1961; LeRoy and Singell 1987). Ambiguity is relevant in many natural resource problems (Shaw and Woodward 2008) and for endangered species in particular. As examples, over the period from 1991 to 1999 only about 30% of all fish stocks had known population trends (National Marine Fisheries Service 2002), and even the well-studied Columbia River Basin continues to present surprises (Barringer 2005). Even less is known about the relationship between specific environmental or habitat conditions and growth.

Our dynamic model of water allocation and fishery management explicitly introduces ambiguity and the potential for ambiguity aversion by applying robust control, an approach recently advocated by Hansen and Sargent (2001; Hansen et al. 2006). Robust control has been used to examine policies in natural resource problems including water management (Roseta-Palma and Xepapadeas 2004) and extractive fisheries (Xepapadeas and Roseta-Palma 2003). In a robust control specification, choices maximize an objective function relative to the worst-case scenario that the decision maker admits. Hansen and Sargent have argued that robust control is an appropriate representation of ambiguity-averse preferences as defined by Gilboa and Schmeidler (1989).

Some Relevant Economics Literature

In their classic article, Burness and Quirk (1979) demonstrated that the doctrine of prior appropriation (DPA), which is common throughout the arid western United States, generally will not allocate water efficiently. To fully explain allocation of scarce flows under the doctrine of appropriation several modelers consider the location of the source of flow and the distance from this by each agent who desires a diversion (Johnson, Gisser, and Werner 1981). This leads to a first-order difference equation that can be used to determine water quality or quantity (Weber 2001), and the spatial dimension allows game-theoretic equilibrium allocations. As markets for water in the United States become increasingly prevalent (Howitt and Hansen 2005), those that value instream flows, typically the residual claimants, are quite literally left with no flows with which to work (Ward 1987).

The economics literature of fisheries management relates mostly to commercial harvests. Reed (1979) and Clark and Kirkwood (1986) represent early contributions to the literature on optimal management under risk, looking at stock and measurement uncertainty respectively. Reed and Clarke (1990), Saphores (2003), and Sethi et al. (2005), allow for multiple sources of uncertainty and Xepapadeas and Roseta-Palma (2003), on which we will build below, consider the extractive fisheries management problem under both risk and ambiguity.

Though there have been important efforts (Ricker 1975; Johnson and Adams 1988; Jaeger and Mikesell 2002), the connection between instream flows and fisheries management is poorly developed. Tsur and Zemel (1994) study a situation in which water has consumptive value, but where excessive withdrawals can lead to extinction of a species. They find that risk in the form of a known probability distribution over the population leads to a cautious strategy that reduces the chance of extinction.

A Dynamic Model of Instream Flow Allocation

In this section we develop a different model of the problem of water management in the presence of an endangered species. First, we incorporate the existence value of a species directly into the benefit function. Second, the uncertainty we consider relates to the growth function of the species, not to uncertainty about water flows over time. Most importantly, we not only introduce risk with a known probability distribution, but also allow for ambiguity, which we assume arises from a lack of knowledge about the true dynamics of the relationship between species' growth and instream flows.

Assume that there is a single fish stock with a population size of [q.sub.t] living in a river at time t. The available water supply has two possible uses: it can be used for industry or agriculture, [a.sub.t], yielding benefits such as profits from farming, D([a.sub.t]), or it can be left in the river, st, yielding benefits from instream uses such as recreation or hydroelectricity, W([s.sub.t]). To focus our attention on uncertainty in the species' dynamics rather than on randomness in flows due to weather patterns, we assume that the total flow of water is constant at the rate R, so that [a.sub.1] + [s.sub.t] = R. For notational simplicity, we delete the time subscripts below except where necessary for exposition.

[FIGURE 1 OMITTED]

The growth of the fish stock is affected by both the current stock and the instream quantity of water, s. (1) Our main interest here is in a protected species, so harvesting is assumed to be illegal or negligible. We begin with a deterministic model in which the species' growth depends only on the current stock size and the current stream flow

(1) [??]/q = f(q, s).

For any value of s, we assume that the dynamics of the species is characterized by a standard biological growth model with critical depensation at [q.bar](s) and carrying capacity at [bar.q](s) as in figure 1. The top two curves in the figure, labeled high and medium flow, are typical of such dynamics. At flow levels in this range, if q < [q.bar](s) then growth will be negative and, if the flow does not change, the stock will decline to extinction. If q starts above [q.bar](s) it will tend toward the carrying capacity, [bar.q](s). We assume that an increase in the instream flow improves the species' rate of growth, shifting the growth curve upward so that [partial derivative] f(q, s)/[partial derivative]s [greater than or equal to] 0. (2)

The two lower curves are logical extensions of the idea of a flow-contingent growth. In the bottom curve in figure 1, the instream flow is so low that regardless of the species' stock level, its population will decline over time. The second-lowest curve shows the growth associated with the lowest possible flow for which it is still possible to maintain a positive stock. The flow level associated with this curve, [s.sub.L], is that which would be sought by a planner seeking to maximize withdrawals for a water course (minimizing instream flows), while at the same time ensuring that the species survives. The stock that could be maintained at this flow level is [[q.bar].sub.u].