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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Valuing ecosystem services with microeconomic underpinnings presents challenges because these services typically constitute nonmarket values and contribute to human welfare indirectly through a series of ecological pathways that are dynamic, nonlinear, and difficult to quantify and link to appropriate economic spatial and temporal scales. This paper develops and demonstrates a method to value a portion of ecosystem services when a commercial fishery is dependent on the quality of estuarine habitat. Using a lumped-parameter, dynamic open access bioeconomic model that is spatially explicit and includes predator-prey interactions, this paper quantifies part of the value of improved ecosystem function in the Neuse River Estuary when nutrient pollution is reduced. Specifically, it traces the effects of nitrogen loading on the North Carolina commercial blue crab fishery by modeling the response of primary production and the subsequent impact on hypoxia (low dissolved oxygen). Hypoxia, in turn, affects blue crabs and their preferred prey. The discounted present value fishery rent increase from a 30% reduction in nitrogen loadings in the Neuse is $2.56 million, though this welfare estimate is fairly sensitive to some parameter values. Surprisingly, this number is not sensitive to initial conditions.

Valuing ecosystem services presents four challenges. First, many of the economic benefits generated are non-market. Second, ecosystem services typically contribute to human benefits indirectly. Humans may not value the service, per se, but something that it supports. For example, recreational anglers may value the fish in a stream but not necessarily the riparian habitat that support the fish population. Third, the links between ecosystem services and human values are dynamic and can be nonlinear, but economic valuation architecture is most developed for static problems. Finally, providing an empirical basis for ecosystem valuation involves multiple academic fields and profound spatial and temporal scale mismatches. To begin to address these challenges, one must simplify problems. In this paper, we highlight the strategic modeling choices that are involved in adapting a lumped-parameter bioeconomic model to study the valuation of ecosystem services.

Lumped-parameter approaches greatly simplify renewable resource dynamics by modeling a small number of states that depend on just a handful of parameters. While the overall goal of this research is to evaluate the strengths and weaknesses of lumped-parameter approaches to study the economic value of ecosystem services, we develop a particular model to highlight the research challenges that are involved in coupling economic and ecological models to answer a specific policy question. In our case, the specific policy issue is the tradeoff between nutrient loading in estuaries and the resulting effects on fisheries productivity. The simplifications in our lumped-parameter model balance conceptual knowledge about the estuarine ecology and the economics of the fishery with the data available for parameterizing each feature of the system.

We find that a lumped-parameter bioeconomic fisheries model is able to address each of the four challenges for valuing ecosystem services. First, we deal with a commercially valuable species. Hypoxia is not priced in the market, but fish that are affected by it do have market value. Our metric of economic value is fishery rent, which provides an exact welfare measure under some assumptions. Focusing on a producer problem in which we incorporate the opportunity cost of capital, we avoid many of the complications of nonmarket valuation such as substitution prospects, income effects, and identifying assumptions like weak complementarity. Implicitly, we assume that the market demand is perfectly elastic as is typical in bioeconomic models. Our work also responds to a recent National Research Council call for more research on the effects of nutrient pollution on commercially valuable coastal resources [

The paper is organized as follows. Section 2 reviews the general policy context of estuarine nutrient pollution and discusses the specific background for the Neuse River Estuary and the blue crab fishery. Section 3 describes the lumped-parameter model and places it in the context of previous work in bioeconomics. Section 4 describes the model's parameterization. Section 5 presents results, and Section 6 discusses our results and outlines modeling issues for future research on coupled ecological and economic systems. The discussion section expands on the model description by explicitly considering some of the modeling tradeoffs in this application and for future work.

Nutrient pollution poses a number of threats to ecosystem services in coastal waters. Excess nutrients have been linked to increased primary productivity, hypoxia and anoxia, changes in the composition of the planktonic community, toxic algal blooms, decreased diversity in the trophic system, and increases in disease [

Hypoxia and anoxia can have substantial consequences for a wide range of species. In the Gulf of Mexico, which is experiencing large-scale hypoxic events, hypoxia affects the spatial distribution of species including Atlantic Croaker and brown shrimp [

In an estuarine environment, nitrogen is typically the limiting nutrient to growth in primary productivity, whereas phosphorous more often is limiting in lake environments [

Nutrient pollution in North Carolina's Neuse River Watershed has generated substantial public debate. Under the United States Clean Water Act, states are required to develop Total Maximum Daily Load (TMDL) plans for water bodies that do not meet water quality criteria, and these plans “must identify the amount by which point and nonpoint sources of pollution must be reduced in order for the water body to meet its stated water quality standards” [

We examine commercial fishery productivity in the Neuse as one portion of the total economic benefits that would emerge from reduced nutrient pollution. Paerl

Blue crabs have both ecological and economic importance. Blue crabs can serve as a keystone predator in an estuary [

The blue crab fishery is the most valuable commercial fishery in North Carolina and consists of three product types sold dockside: hard shell, soft shell, and peeler. Nominal ex vessel revenues across all three categories totaled $34.44 million in 2002 and peaked at $44.96 million in 1998. The bulk of landings are hard shell, accounting for 96.6% of the 37,592,317 pounds landed in 2002. Blue crab landings in 2003 accounted for more than one third of all landed value in North Carolina commercial fisheries.

The North Carolina blue crab fishery is a large share of the total for the Eastern United States. North Carolina blue crab comprised 13.46% of total blue crab landings from 1950–1993 and 24.24% of landings from 1994–2002 [

Current and proposed management tools for the North Carolina blue crab fishery are all open access alternatives [

The lumped-parameter model is a system of ordinary differential equations that represents changes in the stock of nutrients, algae levels, spatially explicit populations of the blue crab and prey availability, total fishing effort, and the spatial distribution of fishing effort. The goal of this system is to trace changes in nutrient loadings all the way through to impacts on fishery rents, which provide a meaningful economic metric. However, changes in nutrient levels do not directly affect fishery harvest. Instead, nutrient loadings coupled with hydrodynamics alter the ecosystem that supports the fishery. Specifically, increased nutrients stimulate the production of algae in the estuary and provide a precursor to hypoxia. Hypoxic events, which tend to be spatially delineated in the estuary, affect prey availability and can induce blue crab emigration from hypoxic to oxygenated zones. Fishing pressure, in turn, responds to the overall status of the blue crab resource and its spatial distribution. Thus, changes in the fishery that are attributable to nutrient loadings can only be uncovered as they filter through complex ecological and economic dynamics. In this sense, the model aims to measure market benefits of ecosystem services that have not been measured before. Nevertheless, we acknowledge that analysts may also be interested in simpler metrics like additional catch or revenue that a fishery can sustain in the long run with a reduction in nutrient pollution. Our model measures these changes as well.

Nutrient loadings and algal growth have simple model structures. Denoting

For purposes of optimizing the model (in future analysis), it will be necessary to preserve the time-dependency of loadings. Optimizing the model, which is beyond the scope of this paper, would involve setting up and solving an optimal control problem with at least three controls, six states, and a variety of non-negativity constraints. However, to gain some qualitative insights initially, we assume that nutrient loadings are constant over time,

In this way, the long-run stock of algal blooms is limited by the nutrient stock, and the dynamics of algal growth are influenced by the dynamics of nutrient accumulation. Note that with constant nutrient loadings, the steady-state nutrient stock and algae stock are: _{∞} = _{∞} =

A potentially interesting extension of this model would be to incorporate algal biomass hysteresis. The residence time of organic matter in estuaries involves fresh and saltwater interchange as well as the physics of sediment transport [

For tractability, we model a two-patch bioeconomic system to study the effects of hypoxia on the blue crab fishery. In this lumped-parameter model, a patch can be viewed as either shallow versus deep water or estuary versus the sound. Both interpretations are consistent with the effects that we get, and the difference would be in the parameterization. We treat Patch 1 as the estuary and Patch 2 as the Pamlico Sound. Due to depth, currents, and other aspects of hydrodynamics, we assume that Patch 1 is susceptible to algal blooms and thus can experience hypoxia. As such, the algae stock in (2) is assigned to Patch 1. In contrast, we assume that Patch 2 does not have excess algae growth and hence has no hypoxia. Another way of interpreting the model is that Patch 2 has only background effects of algae, and when a hypoxic event occurs, Patch 2 is relatively more oxygenated than Patch 1. Because we limit algae growth to Patch 1, this model is inappropriate for considering doomsday scenarios in which nutrient loadings are so severe that the entire estuary is hypoxic or anoxic at all times. One can imagine that such a scenario would destroy the environmental basis for the fishery (at least the basis in the Neuse) and comparisons among policy alternatives with marginal changes in loadings would be less meaningful than global ones (in the mathematical sense). In the population dynamics below, we also assume that the patches are of equal size. This assumption could be relaxed by introducing additional parameters.

There are four state equations that describe the population dynamics: predators in each patch and prey in each patch. Blue crabs are the predators and are subject to fishing mortality and environmental limits on their abundance that include prey abundance. Prey in the model are a composite of infaunal species, mostly clams, that are unable to move across patches. Hence, we explicitly model preferred prey of adult blue crabs. We do not model potentially differential impacts of hypoxia on the various components of their diets, preferred or otherwise, and assume that hypoxia affects preferred prey but does not affect non-preferred prey. Since blue crabs are scavengers and eat a wide variety of organic material at various life stages [_{i}(t)_{i}(t)_{x}_{x}_{i}(t)_{x}_{x}_{x}_{x}

The two last components of predator population dynamics account for the direct effects of hypoxia and spatial connectivity of the patches. Both of these terms affect predator migration between the patches. Our approach is in the spirit of previous bioeconomic models of habitat dependence [

This form ensures that absolute prey differentials do not lead to migration of more than the population in a patch, and migration is limited when both predator and prey populations are scaled up or down.

The population dynamics of the preferred prey species have some similarity in structure to predator population dynamics, but the effects of hypoxia are quite different. Like in equations

Note that we have just three additional parameters to describe the prey state equations: intrinsic growth of prey (_{y}

Turning to the economics, there are two key states that we track and that together with an assumption on the production technology will close the model and determine bioeconomic outcomes. These states are the spatially-explicit levels of fishing effort (_{1}(t)_{2}(t)_{i}

The blue crab fishery in North Carolina has been essentially open access for most of its history. There are some limits on gear technology but no formal limits on entry, and though there was a temporary moratorium on new permits, we assume that it was not in effect long enough to fundamentally alter the open access dynamics of the fishery. Thus, total effort in the model follows a dynamic open access rent dissipation model originally formulated by V. Smith [_{total}(t)

Note that we also assume a constant price (

Note that in the steady-state, as in Sanchirico and Wilen [

The share of effort in each patch responds to relative density of blue crabs according to a logistic probability density function. For Patch 1, effort share is:

This approach is consistent with the empirical literature on broad fishery choice [

To close the model, we assume a simple Schaefer (1957) production function for harvest:

Using (10), we can simplify (16) to:

To summarize, we now have a description of the system that is a function of seven state variables [_{1}(t), X_{2}(t), Y_{1}(t), Y_{2}(t),E(t),_{1}(t)]

Parameterizing the model described above presents numerous challenges. For some of the parameters, empirical estimates simply do not exist or the data are of such poor quality that meaningful statistical inference is not feasible. For others, empirical estimates in the ecology literature do exist but apply to vastly different spatial and temporal scales. In many cases, the studies that provide empirical estimates involve very different model structures and not just different scales. Finally, some parameters pose units problems as well. With all of these caveats in mind, by parameterizing the model we seek to accomplish five objectives: (1) demonstrate our methodology for quantifying partial ecosystem service values; (2) provide an initial estimate of the magnitude of value (albeit one with considerable uncertainty); (3) explore qualitative insights from the model and whether parameter values affect qualitative patterns in the dynamics; (4) identify parameters to which the value of ecosystem services is sensitive; and (5) illustrate challenges in coupling ecological and economic models for policy analysis. Our actual figures for present value rents should be interpreted with caution.

For parameter values in (1) and (2) and hypoxia-related parameters in (4)–(6), we rely on work from Mark Borsuk's Ph.D. Dissertation at Duke University and related work with co-authors. Borsuk develops a Bayesian hierarchical network model that addresses a number of ecological and hydrological features of the nutrient pollution in the Neuse River and Estuary [

In (1), we rely on static models to choose two parameters: baseline loading (

The parameter μ in (2) maps steady-state nitrogen concentration into algal carrying capacity. Borsuk, Stow, and Reckhow [

We assume that algal accumulation affects dissolved oxygen concentrations instantaneously, again with the caveat that our model does not explore the episodic nature of hypoxia. Since the effect is instantaneous, it is not necessary to model algae accumulation and dissolved oxygen depletion separately. The parameters

Linking primary productivity to dissolved oxygen levels with some empirical basis inevitably confronts spatial and temporal scale mismatches as well as units problems. The spatial scale of the ecological data is often much finer than the model in this paper, so some aggregation is necessary. Also, these data are typically not collected with dynamics in mind. When dynamics do enter, they may be fast dynamics that do not match the economic dynamics. For example, Borsuk, Stow, Leuttich, Paerl, and Pinckney [

Since our primary productivity is scaled to have a maximum of approximately one, depending on the nutrient concentration, an elasticity will eliminate the units problem (at least as a first order approximation). The elasticity of SOD with respect to K (

Note that

We next consider a range of sediment oxygen demand levels from 100% of the baseline down to 50% of the baseline and we compute (in percentage terms) the corresponding decreases in primary productivity from the baseline (100%) using 1/ε. Finally, we linearize this relationship to get a marginal oxygen demand of 0.57 with a standard error of 0.03. Since oxygen demand in the system is an instantaneous response, we can simply multiply 0.57 times the blue crab responsiveness to dissolved oxygen to construct

Selberg, Eby, and Crowder [

To link dissolved oxygen to prey mortality in (6), we adapt empirical results from Borsuk, Powers, and Peterson [^{2} = 0.97) in this range of dissolved oxygen. When we include mean and median results from Borsuk, Powers, and Peterson [^{2} drops to 0.95. To summarize, 0.72 will be the mortality component of p, and we will use the intercept to guide selection of intrinsic growth of prey. Combining the two marginal effects in p, we have ρ = 0.79 × 0.72 = 0.57.

We rely on a stock assessment of North Carolina blue crab to estimate the logistic growth parameters in (4) and (5). Eggleston, Johnson, and Hightower [

Next we need to interpolate a relevant carrying capacity for the part of the NC fishery that we are studying. EJH estimate that 7% of the fishery is conducted in the Neuse River and 28% in the Pamlico Sound, which is the water body that the Neuse Estuary opens into. We assume that our model applies to this total 35% of the fishery. Thus, we assume that the range of steady-state crab population combined across the two patches is 35% of the carrying capacity range in EJH. In the absence of hypoxia, and assuming that our two patches are of equal size, half of this capacity is then allocated to each patch. This steady-state population reflects predation, but with no hypoxia, the two patches are equal and all of the migration terms in (4) and (5) drop out.

We next turn to finding values for _{x}_{x}_{x}

We have no existing literature to guide our choice of the final biological parameter,

We now turn to the economic parameters of the model. For parameterizing the model, we convert all past prices and costs to 2002 dollars using the BLS CPI South Size D (the smallest rural classification) for 1978–2002. Prior to 1978, this CPI was not available, so we use CPI All Urban Consumers. In the simulations, we use a five-year weighted average price (1998–2002) across the three product categories (hard shell, soft shell, and peeler), so

By regressing lagged CPUE and lagged NC unemployment (to account for variability in opportunity cost), we can estimate the share of average total cost that is exclusive of opportunity cost of time to be 70%. We do not use the implied values for costs from the regression because nothing is statistically significant, and we have only 7 years of trip data. We run the regression simply as an initial attempt to decompose cost components. If we assume a discount rate r = 0.05, a constant rate of depreciation of 0.1, and take the average fixed cost numbers from Rhodes, Lipton, and Shabman [

To parameterize the speed of adjustment γ, we need to make additional assumptions. First, because trip data are not available for a long enough period (and are influenced by the license moratorium), we need to backcast trips based on crab pot data. To do this backcasting, we assume total pots per trip is a stable relationship over 1984–present. We then take the overlap period of pot and trip data (1994–2002), exclude 1995–1997 because of possible over-reporting of pots, and take the average pots per trip (crab pot data was provided by Sean McKenna of NCDENR, Division of Marine Fisheries). Second, we take the implied trips from 1984–2002 and compute one-period-ahead forecasts based on the rent dissipating effort

We follow a similar one-period-ahead forecast procedure to parameterize catchability

The final economic parameter is the spatial adjustment parameter θ. Selberg, Eby, and Crowder [_{i}

We assume that the system starts at the steady-state values for the biophysical states

We choose initial population levels for predators and prey in each patch to correspond to half of the logistic growth carrying capacity parameters. For predators, we thus choose _{1}(0)_{2}(0)_{x}_{1}(0)_{2}(0)

For initial economic parameters, we choose total effort to be 35% of the total trips in the NC blue crab fishery averaged over 2000–2002. This yields _{2}(0)

Because our parameterization draws on a wide range of studies and adapts parameters from different model types, we do not have a clear roadmap for incorporating parameter uncertainty. That is, we cannot characterize the joint distribution of our parameters and draw repeatedly from that joint distribution to simulate the model. This is a limitation in comparison to the Neu-BERN model. However, what we gain from this sacrifice is the ability to incorporate dynamics explicitly on a scale that matters for human impacts.

We simulate the system in continuous time using Matlab's Ordinary Differential Equation Solver (ODE45). We consider a 30% reduction in nitrogen loadings, which is what the Neuse TMDL calls for [_{x} and implicit k_{y} parameters, and assuming a real discount rate of 2.5%, a 30% reduction in nitrogen loadings sustained over a 50-year time period generates an increases in present value rents of $2.56 million, a total catch increase of 12.4 million pounds, and a total increase in trips of 91,000. In percentage terms, these are increases of 7.7%, 3.6%, and 2.9% for rents, catch, and effort respectively. It is worth noting here that our welfare estimates assume that there are no consumer surplus gains from the policy changes. This implicitly assumes that the blue crab market demand is perfectly elastic in the range of policy-induced supply shifts. One could approximate consumer surplus changes by econometrically estimating the slope of the crab market demand, but this is beyond the scope of our paper.

Since fishery rent may not be the only metric that matters to policy-makers, we can examine the long-run impacts on catch and effort from the policy change. We take average catch and effort over the last two years of our 50-year simulation as an approximation of long-run effects. The baseline long-run catch and effort per year are 5.7 million pounds and 60,662 trips respectively. The increases from the 30% nitrogen reduction are 230,000 pounds and 2,270 trips. Trips per year is in the neighborhood of 60, which averages across full-time and part-time fishermen. This suggests that the water quality improvement would support 38 additional fishermen in the Neuse and Pamlico Sound in the long run.

Naturally, one must ask how much of the changes in rents, catch, and effort are an artifact of initial conditions. To explore this issue, we use the same parameter values to run the model out 300 years with no reduction in nitrogen. We save the last year state values (approximate steady states) as initial conditions. Then we look at 50-year scenarios with and without the 30% reduction. With these initial conditions, a 30% reduction in nitrogen loadings sustained over a 50-year time period generates an increases in present value rents of $2.75 million, a total catch increase of 13.4 million pounds, and a total increase in trips of 100,000. These numbers are similar to the simulation beginning at half of carrying capacities. The effect of nitrogen loadings in magnitude does not appear to be sensitive to initial conditions. Across the two sets of initial conditions, we obtain essentially the same estimate for welfare improvements from a 30% nitrogen reduction.

This paper develops a method to estimate the value of ecosystem services that support a commercial fishery with a solid microeconomic foundation. In particular, it uses a lumped-parameter dynamic bioeconomic model to evaluate welfare changes from reduced nutrient pollution, which enhances ecosystem function by increasing dissolved oxygen, in terms of fishery rents. We illustrate the model using the North Carolina commercial blue crab fishery and examining the proposed 30% nitrogen reduction in the Neuse River Watershed. In this discussion, we provide additional context as well as some caveats for interpreting our ecosystem service values, and we suggest future research directions linking economic and ecological models by examining our own strategic modeling decisions. Along the way, we highlight the ways in which we lump parameters and how these ways are likely to affect policy analysis.

Our best point estimate for the value of a 30% reduction in nitrogen loadings is $2.56 million, though this figure ranges from $195,000 to $7.51 million simply by varying parameters within what we consider to be a reasonable range. The TMDL plan does not require a benefit-cost analysis, so we have little information on the magnitude of total economic benefits. Still, it is worth asking why this number is so small. To understand our value of a 30% nitrogen reduction, it is essential to highlight the coexistence of other value changes from that same reduction, the importance of fisheries management institutions in driving the value, and the large degree of parameter uncertainty in our model.

The ecosystem service values that we measure are only a subset of the total that might emerge from a 30% reduction in nitrogen. Our values are partial largely because not all of the ecosystem values contribute to blue crabs, and not all values are rivals in consumption. That is, capturing value in the blue crab fishery associated with reduced nutrient pollution does not necessarily diminish values of other ecosystem services associated with the same water quality improvement. Reducing hypoxia by decreasing nitrogen loading in the Neuse may benefit other commercial fisheries as well as the blue crab fishery. Nitrogen reductions may also contribute to ecosystem values through trophic interactions that are not captured by benefits to fisheries. However, these values have not yet been quantified and arguably are more difficult to pin down.

Previous economic research has analyzed how nitrogen reductions could enhance the value of recreational fisheries and other forms of recreation in the Neuse. In fact, the hypothesis that reduced nitrogen loadings will generate recreational fishery benefits is consistent with static non-market valuation studies of the Neuse [

Perhaps the most important reason that our welfare estimate seems small is the institutional structure of the blue crab fishery. North Carolina blue crabs are managed under open access, and open access dissipates rents in the long run. As a result, any values that an environmental quality improvement generates are necessarily transitory; no sustainable value can persist under open access. In contrast, under optimal management the economic benefits of reduced nitrogen would not be dissipated. The present value welfare change could thus be considerably larger. However, without analyzing optimal fisheries management, it is difficult to speculate on how big these benefits would be. Naturally, the next step in this research agenda would be to derive the optimal policy and compute present value rents under different nitrogen reduction scenarios.

Our application illustrates the importance of institutions as a broad theme in the valuation of environmental resources. The

The interaction of institutions and value of ecosystem services naturally raises the question of how recreational blue crab fishing would be affected by different combinations of water quality changes and institutional settings. Under open access, recreators may contribute to the dissipation of gains from improved water quality by putting additional pressure on crab stocks. Similarly, potential gains from rationalizing the commercial crab fishery—and capturing more value from the crab resource itself and from the water quality improvement—could be offset to some degree by maintaining open access in the recreational sector. Optimal management would jointly consider recreational harvest, commercial harvest, and water quality. The conceptual bioeconomic literature on recreational and commercial fisheries emphasizes the jointness of recreational and commercial sectors in determining optimal management [

Just as the open access assumption is a critical determinant of the dollar figure that we find for rent changes, the assumption of a representative agent is important as well. In that sense, the economic parameters in our model are lumped; they represent averages over the population of fishermen. Since economic parameters enter nonlinearly into present value rent sums, even introducing heterogeneity through parameters with symmetric distributions could affect outcomes. Moreover, Johnson and Libecap [

Beyond the institutional considerations described above, it is worth interpreting our ecosystem values as a first order approximation purely for quantitative reasons. Since no prior study has been able to quantify the economic benefits to the blue crab fishery from reduced nutrient loading, or to any fishery for that matter, there does not exist a benchmark against which we can compare the magnitude of our result. Given that and the way that we parameterize the model, our quantitative results should be interpreted with caution, especially for non-marginal changes. First, we have only begun to explore the parameter space. Conducting a thorough sensitivity analysis will likely involve thousands of simulations. Second, all of the parameters in our empirical application involve substantial uncertainty. Most are adapted from studies that do not have a model like ours in mind; they deal with substantially different temporal and spatial scales, they involve different units, and the some of them have different functional forms. The model is quite sensitive to the parameter that maps nitrogen stock into steady-state algae. This parameter enters nonlinearly, so this sensitivity is not surprising. In terms of scientific interpretation, we need sound, statistically-based description of how much nitrogen affects chlorophyll production that, in turn, reduces dissolved oxygen levels. Most of the scientific literature that deals with this question addresses ecological dynamics that are too short (e.g., intra-seasonal) for the type of long-run welfare analysis that we are conducting here.

Units of measurement could affect the results. For two parameters, we use elasticities to overcome units problems, but these elasticities must be evaluated at particular levels of the variables. We assume mean levels, so any non-marginal change that pushes the system far from these values could also affect the elasticities. One might reasonably argue that a 30% reduction in nitrogen loadings is not a marginal change, but it is important to consider that the effects of nitrogen on blue crab filter through many ecological pathways, and nutrients are not the sole environmental basis for the fishery. To the extent that some of the ecological pathways dampen the effect of nitrogen rather than amplify it, we believe that our model works reasonably well. Overall, fruitful directions for future research thus include reducing uncertainty about key parameters and improving the match between ecological studies and economic studies in terms of spatial scale, temporal scale, and functional form.

Many strategic modeling assumptions are necessary to trace a water quality improvement all the way to changes in fishery rents. It is useful for discussion to divide them roughly into assumptions that determine three types of effects: the direction of effects, the magnitude of effects, and the timing of effects. In dynamic models in which some metric is measured in present value, magnitude and timing are interrelated. For instance, pushing back a nominal gain several periods (a timing change) ultimately reduces the present value magnitude. To the extent that we can separate these two types of effects, we focus on nominal changes to categorize them. For lumped parameters, timing and magnitude are typically inseparable.

The assumptions that determine the direction (or sign) of policy impacts are functional forms and parameter signs. Most of these assumptions are implicit parameter signs that are not controversial. Some examples include positive intrinsic growth, positive carrying capacity, positive prices, positive costs of fishing effort, and a negative predator-prey interaction in the prey state equations. We characterize most of our ecological system by linear processes. To the extent that nonlinearity is introduced, we have used predominantly convex modeling and have focused on monotonic relationships between nutrients and fishery outcomes. Dasgupta and Maler [

Predator-prey responses to hypoxia are an important determinant of the direction of policy impacts. Though we do not model it, a potentially important nonmonotonicity may exist for these responses. Taylor and Eggleston [

A related issue is our exclusive focus on the negative relationship between fishery productivity and nutrient enrichment. Formally, this restriction comes through our linear scaling of nutrient stock into algal carrying capacity. However, as Caddy [

The impact of hypoxia on fishing behavior could also affect the direction of policy impacts. As modeled, hypoxia can only reduce welfare. However, it is possible that blue crab avoidance of hypoxia could increase catchability in the non-hypoxic patch. The idea here is that since blue crabs essentially crowd into the oxygenated areas during a hypoxic event, there may an increase in catch-per-unit-effort beyond that associated with an increase in patch 2 biomass (holding effort constant). This effect would potentially create perverse outcomes: short-term rent increases probably at the expense of long-term catch. In an open access setting, whether the increase in catchability would be welfare-improving would then hinge on the discount rate, and there is at least a theoretical possibility that decreasing hypoxia by reducing nutrient pollution would be welfare-reducing. Here again the importance of institutions is critical. Under a rationalized fishery, the positive catchability effect would be offset some (possibly all) by a negative stock effect that reduces long-run profitability. With open access, long-run profitability is zero either way. Overall, the possibility of perverse welfare effects presents interesting challenges for future research on valuing ecosystem services by linking economic and ecological models.

A key determinant of the magnitude of changes from a water quality improvement is the way in which we model space. As described in Section IV, our model generates a type of natural insurance. Because patch 2 is not prone to hypoxia, there is a limit to how much nutrient pollution can affect our system. In the limiting case with severe nutrient pollution in patch 1, all blue crabs migrate to patch 2, and the entire fishery is conducted in patch 2. Even under open access, patch 2 serves as a contingent resource that insures against a complete collapse due to pollution. In an analogous common property exploitation problem, availability of groundwater can be viewed as a contingent resource when surface deliveries are stochastic [

One could generalize our model to allow for differential hypoxic effects across patches by introducing an additional parameter. This would allow for an alternative spatial interpretation that corresponds to deep versus shallow water within the estuary. Of course, all of the biological parameterization would change to reflect the fact that the estuary alone is a smaller portion of the statewide crab fishery. Given our empirical setting, we argue that our parsimonious two-patch model provides meaningful estimates for policy analysis. For developing modeling strategies more generally, the researcher must ask: does interest lie in different spatial segments of the same water body or in connected water bodies that pollution may affect differentially. If the latter, is it possible to treat one as the index water body with no (or only background) effects of pollution?

Among the most striking results from our model is the lack of sensitivity to initial conditions; that is, initial conditions only have a tiny impact on the magnitude of policy outcomes. Upon closer examination, it is a combination of assumptions that drives this result, namely the linearity of the nitrogen stock and steady-state algae, the linearity of direct hypoxic effects, and the nature of open access rent-dissipating models. For the linearity assumptions, whether stocks are big or small and whether total effort is big or small, the nitrogen reduction leads to the same increase and approximate timing in fishery productivity. The magnitude and timing do not depend on the state of the system. This is the same intuition for why the open access structure matters. For a given increase in available rents associated with a productivity increase from less pollution, there is an extra amount of effort required to dissipate these rents. The level required does not depend on the level of existing effort because the Schaefer production function restricts the exponent on effort to one. Thus, the existing level of effort does not affect the difference between revenues and costs when the stock responds to the policy change; the difference is proportional to the increase in the stock whether existing effort is high or low.

Our assumption of constant real price affects the magnitudes of several metrics that interest policy-makers. Given that blue crabs in the Neuse constitute a small share of overall east coast blue crab harvest, making price endogenous is unlikely to be worth the effort. However, we can easily introduce some exogenous growth in real prices to examine sensitivity to the constant price assumption. With 2% growth in real price, baseline rents (with no water quality improvement) are higher compared to constant prices, and the policy change generates a somewhat higher rent increase of $2.92 million. Total effort and long-run effort are also higher, but total catch and long-run catch are lower. This reflects the cost-price ratio effect in open access models.

One of the key assumptions in our model is how quickly nutrient pollution can be cycled out of the system when loadings are reduced. We assume that nitrogen decays rapidly, and as a result, gains from reduced loadings materialize quickly. In reality, this decay could be slow. The consequence for policy impacts is that benefits to the fishery from nutrient reduction materialize later. We simulate a nitrogen decay rate of 0.475 (compared to our baseline of 0.95) and find that the present value rent increase from the policy change is $146,900 less than in the baseline. One of the features of the model that limits the size of this timing effect is the bioeconomic overshooting. When the system overshoots, rents go negative for a period. Pushing the positive rents back several periods also pushes back the negative rent overshooting period, so there is a loss from positive rent delays that is partially offset by the gain from negative rent delays. There may also be hysteresis in oxygen demand. This could be modeled with hysteresis in nutrients, as in Brock and Starrett [

A related timing effect is our assumption about the intrinsic growth of algae. Since we do not specify a species of algae, our model corresponds to some generic representative algae (or representative primary productivity). By assuming an intrinsic growth rate of 1.0, we assume that algae responds rapidly to changes in nutrient levels. A smaller intrinsic growth parameter would slow responsiveness, delay benefits, and thus lower the present value benefits of the policy change. On the other hand, a smaller parameter would also delay damages from increased nutrient levels and reduce the present value benefits of avoiding more pollution. With cycling due to open access, the net effect could have either sign. A simulation in which the intrinsic growth of algae is assumed to be 0.5 demonstrates this intuition; present value rent increase from the policy change is roughly $95,000 larger than in simulations using intrinsic growth of 1.0.

Both of these timing effects illustrate a key difference between the traditional static approach to valuing water quality improvements and the dynamic bioeconomic approach in our paper. The static approach typically assumes a direct effect of water quality on behavior. When data on fish stocks are available, researchers can decompose this direct effect into the effect of water quality on the fish stock and the effect of fish stocks on behavior. The relationship between water quality and fish stocks is still a reduced-form one. Static models by construction assume that the effects of a water quality improvement are instantaneous, and only a handful of researchers are able to relax the instantaneous effect assumption because doing so requires panel data. In contrast, our approach does not separate the timing of ecological impacts from the structure of the ecology and the resulting behavioral changes. These interdependencies can interact in nonlinear ways, and treating them as a sequence of reduced-form relationships could lead to different policy inferences.

Economic speed of adjustment dramatically affects the timing and magnitude of gains from a policy change. When the harvest sector adjusts rapidly (slowly), baseline rents are small (large), and rent gains from the policy change are also small (large). Upon closer examination, this speed of adjustment parameter is also a lumped parameter. It reflects fixed costs of entry, constraints on available labor in the short run, as well constraints on available non-fishing employment in the short run. It is useful to lump these effects together in a dynamic fisheries model because they usually cannot be measured individually, whereas the speed of adjustment can be estimated econometrically as in Wilen [

Our model implicitly assumes homogenous harvesters following the bioeconomic literature on which it builds. Extending our model to account for heterogeneity would be an interesting direction for future research. Real fisheries involve fleets with heterogenous vessels, captain skill, and opportunity costs of time. In such a setting, there could be important distributional consequences of a water quality improvement. One can debate about whether rents that accrue to captain skill or other fixed inputs should be considered resource rents. So, research in this area would need to disaggregate rents in a manner that clearly separates infra-marginal rents from resource rents in the homogenous model.

As in any simulation model, the choice of a deterministic or a stochastic model can be a key strategic decision. Here we use a deterministic model for two reasons. First, since our problem is a new one in the literature, we seek to generate baseline insights with deterministic modeling. Second, point estimates are often more useful for policy-makers. One can debate the relative merits of stochastic and deterministic modeling in general or for any particular setting, but doing this is beyond the scope of our paper. Instead, we point to one aspect of the bio-physical system that raises questions about deterministic modeling for this problem. Incidence of hypoxia does not unfold on a continuous basis but rather as a series of events governed by changes in hydrodynamic conditions. As an alternative to our approach, one could model the data generating process of these events. Since the effects of hypoxia enter linearly into our model, the importance of modeling stochasticity would likely hinge on the asymmetry of distributions characterizing hypoxic events and thickness of the tails. Adapting a lumped-parameter model to deal with the episodic nature of hypoxia would be an interesting direction for future research.

The ways in which our ecological parameters lump effects could matter even if results do not appear sensitive to the particular parameters. The hypoxia-related prey mortality parameter is a good example. This parameter captures the effect of increased algae on dissolved oxygen and the effect of dissolved oxygen on prey mortality in a multiplicative fashion. If both components of this parameter are understated or overstated, the combined change could be substantial. This is a general problem of lumped-parameter models.

Our treatment of space illustrates parameter lumping in a different way. There is no parameter that explicitly governs patch size. Rather, patch size is implicitly governed by population parameters. When predator carrying capacities are equal to each other across both patches and prey carrying capacities are as well, patch sizes are implicitly of equal size. But this illustrates a broader point about logistic-growth models: that these models implicitly characterize the quantity of available habitat. Logistic growth models also implicitly capture natural mortality through the interaction of carrying capacity and intrinsic growth.

Of particular interest in a model of nutrient pollution is what the prey death parameter means when population parameters are lumped. Hypoxia can in principle lead to death and growth retardation. Over time at the population level, a lumped-parameter model cannot distinguish these effects, but it may not need to. The policy implications from a more detailed model that includes the effects of oxygen depletion on respiration of individual organisms may provide the same qualitative results and similar quantitative ones. Researchers need to ask how much would modeling the biological mechanism affect results, and is it worth the additional effort?

The most important strategic modeling assumption is the actual choice of a lumped-parameter model. A rough analogy can be drawn to an econometric choice among a structural model, a reduced form model, and a partial reduced-form. In other words, there exists a compromise choice between the two extremes. There may be ways of using structural information to provide over-identifying restrictions on a reduced-form model, and these in turn provide efficiency gains without having to estimate the structural model [

The benefits of a lumped-parameter model are straightforward, but the costs are still in question. On the benefits side, the model provides a parsimonious representation of a coupled system that is consistent with the general structure of the problem. This parsimony allows a researcher to adapt the model in transparent ways to answer policy questions directly. For modeling coupled ecological and economic systems, which parameters are lumped and how they are lumped clearly makes a difference. The strategic modeling question is the following: is it worth the costs—in terms of intellectual effort and potentially lost precision—to aggregate over different model types and data scales to arrive at values for a lumped-parameter representation? If we begin with an ecological model and attach an economic one, we may answer this question differently than if we begin with an economic model and attach an ecological one. The optimal choice along the continuum of models comprised of more or less lumped parameters will balance the benefits of additional accuracy and precision with the additional costs of developing a more detailed model in terms of time, resources, and identification challenges. Striking the proper balance between these benefits and costs may be the greatest current challenge for bringing together ecological and economic models for the purpose of valuing ecosystem services.

North Carolina blue crab fishery catch and real revenues.

Blue crab and clam populations with no nitrogen reduction and initial populations at 1/2 of k_{x}.

Time paths of effort, harvest, and rents.

Blue crab migration with no nitrogen reduction.

Time path of policy impacts on rents.

Time path of policy impacts on effort.

Sensitivity of results to the effect of nitrogen on primary productivity.

0.05 | Baseline 30% | 0.06316 | $ 39,814,000 | 373.32 | 3,432 | 6.1384 | 65.892 | ||||

0.05 | Reduction | 0.06316 | $ 40,009,000 | $ 195,000 | 0.5% | 374.21 | 0.2% | 3,439 | 0.2% | 6.1513 | 66.047 |

0.1 | Baseline 30% | 0.12632 | $ 39,077,000 | 370.04 | 3,407 | 6.0982 | 65.354 | ||||

0.1 | Reduction | 0.12632 | $ 39,512,000 | $ 435,000 | 1.1% | 372.03 | 0.5% | 3,422 | 0.4% | 6.1282 | 65.702 |

0.2 | Baseline 30% | 0.25263 | $ 37,366,000 | 362.32 | 3,349 | 5.9975 | 64.068 | ||||

0.2 | Reduction | 0.25263 | $ 38,386,000 | $ 1,020,000 | 2.7% | 367.07 | 1.3% | 3,384 | 1.0% | 6.0737 | 64.911 |

$ 33,124,000 | 342.47 | 3,199 | 5.7003 | 60.662 | |||||||

$ 35,687,000 | 354.91 | 3,291 | 5.9303 | 62.93 | |||||||

0.6 | Baseline 30% | 0.75789 | $ 28,043,000 | 317.32 | 3,007 | 5.245 | 56.223 | ||||

0.6 | Reduction | 0.75789 | $ 32,459,000 | $ 4,416,000 | 15.7% | 339.86 | 7.1% | 3,176 | 5.6% | 5.7303 | 60.415 |

0.8 | Baseline 30% | 1.01053 | $ 22,597,000 | 288.22 | 2,780 | 4.647 | 51.125 | ||||

0.8 | Reduction | 1.01053 | $ 28,813,000 | $ 6,216,000 | 27.5% | 322.14 | 11.8% | 3,039 | 9.3% | 5.4585 | 57.382 |

0.9 | Baseline 30% | 1.13684 | $ 19,930,000 | 272.84 | 2,654 | 4.3554 | 48.563 | ||||

0.9 | Reduction | 1.13684 | $ 26,872,000 | $ 6,942,000 | 34.8% | 312.37 | 14.5% | 2,963 | 11.6% | 5.291 | 55.691 |

1 | Baseline 30% | 1.26316 | $ 17,383,000 | 257.15 | 2,523 | 4.128 | 46.043 | ||||

1 | Reduction | 1.26316 | $ 24,888,000 | $ 7,505,000 | 43.2% | 302.10 | 17.5% | 2,883 | 14.2% | 5.1022 | 53.906 |

Note: Assumes all other parameter values fixed at medium levels and 2.5% discount rate. Totals are totaled over 50-year simulation. Long-run values are average per year over the last two years of each simulation.

Sensitivity to per trip costs.

125 | Baseline 30% | $44,461,000 | 226.01 | 3,370 | 4.402 | 74.165 | ||||

125 | Reduction | $47,019,000 | $2,558,000 | 5.8% | 234.22 | 3.6% | 3,464 | 2.8% | 4.3475 | 76.058 |

175 | Baseline 30% | $38,263,000 | 275.49 | 3,324 | 3.1556 | 66.856 | ||||

175 | Reduction | $40,784,000 | $2,521,000 | 6.6% | 285.22 | 3.5% | 3,415 | 2.8% | 3.3185 | 68.759 |

225 | Baseline 30% | $33,883,000 | 325.59 | 3,228 | 4.9711 | 61.197 | ||||

225 | Reduction | $36,456,000 | $2,573,000 | 7.6% | 337.45 | 3.6% | 3,319 | 2.8% | 5.2012 | 63.44 |

$33,124,000 | 342.47 | 3,199 | 5.7003 | 60.662 | ||||||

$35,687,000 | 354.91 | 3,291 | 5.9303 | 62.93 | ||||||

275 | Baseline 30% | $31,919,000 | 381.04 | 3,141 | 7.1297 | 60.614 | ||||

275 | Reduction | $34,422,000 | $2,503,000 | 7.8% | 394.59 | 3.6% | 3,233 | 2.9% | 7.3491 | 62.809 |

325 | Baseline 30% | $30,348,000 | 435.09 | 3,068 | 8.4142 | 60.876 | ||||

325 | Reduction | $32,747,000 | $2,399,000 | 7.9% | 450.10 | 3.4% | 3,159 | 3.0% | 8.6546 | 62.903 |

375 | Baseline 30% | $28,516,000 | 485.16 | 3,000 | 9.2333 | 60.189 | ||||

375 | Reduction | $30,831,000 | $2,315,000 | 8.1% | 501.63 | 3.4% | 3,089 | 3.0% | 9.5265 | 62.14 |

Note: Assumes all other parameter values fixed at medium levels and 2.5% discount rate. Totals are totaled over 50-year simulation. Long-run values are average per year over the last two years of each simulation.

Sensitivity to economic speed of adjustment.

15 | Baseline | $ 91,214,000 | 405.89 | 3,093 | 6.727 | 61.687 | ||||

15 | 30% Reduction | $ 98,192,000 | $6,978,000 | 7.7% | 424.15 | 4.5% | 3,178 | 2.7% | 6.8454 | 63.797 |

30 | Baseline | $ 51,063,000 | 367.17 | 3,175 | 5.5887 | 65.006 | ||||

30 | 30% Reduction | $ 54,812,000 | $3,749,000 | 7.3% | 380.91 | 3.7% | 3,265 | 2.8% | 5.837 | 67 |

40 | Baseline | $ 37,768,000 | 348.69 | 3,199 | 5.2503 | 62.005 | ||||

40 | 30% Reduction | $ 40,659,000 | $2,891,000 | 7.7% | 361.49 | 3.7% | 3,290 | 2.8% | 5.4859 | 64.2 |

$ 33,124,000 | 342.47 | 3,199 | 5.7003 | 60.662 | ||||||

$ 35,687,000 | 354.91 | 3,291 | 5.9303 | 62.93 | ||||||

50 | Baseline | $ 30,242,000 | 339.45 | 3,197 | 6.2566 | 60.451 | ||||

50 | 30% Reduction | $ 32,573,000 | $2,331,000 | 7.7% | 351.64 | 3.6% | 3,290 | 2.9% | 6.4771 | 62.74 |

60 | Baseline | $ 26,273,000 | 337.93 | 3,198 | 7.0963 | 62.921 | ||||

60 | 30% Reduction | $ 28,205,000 | $1,932,000 | 7.4% | 349.62 | 3.5% | 3,292 | 2.9% | 7.2763 | 65.112 |

75 | Baseline | $ 21,874,000 | 336.26 | 3,218 | 6.2273 | 66.248 | ||||

75 | 30% Reduction | $ 23,375,000 | $1,501,000 | 6.9% | 347.28 | 3.3% | 3,311 | 2.9% | 6.3746 | 68.199 |

100 | Baseline | $ 15,583,000 | 326.84 | 3,230 | 5.362 | 60.791 | ||||

100 | 30% Reduction | $ 16,682,000 | $1,099,000 | 7.1% | 337.20 | 3.2% | 3,323 | 2.9% | 5.5766 | 62.722 |

150 | Baseline | $ 11,319,000 | 325.53 | 3,237 | 6.3249 | 66.844 | ||||

150 | 30% Reduction | $ 12,100,000 | $ 781,000 | 6.9% | 335.73 | 3.1% | 3,331 | 2.9% | 6.4685 | 69.173 |

225 | Baseline | $ 7,594,000 | 320.80 | 3,235 | 6.8168 | 63.601 | ||||

225 | 30% Reduction | $ 8,167,000 | $ 573,000 | 7.5% | 330.56 | 3.0% | 3,327 | 2.8% | 7.2096 | 66.032 |

350 | Baseline | $ 5,017,000 | 318.93 | 3,246 | 6.6801 | 64.286 | ||||

350 | 30% Reduction | $ 5,453,000 | $ 436,000 | 8.7% | 328.35 | 3.0% | 3,335 | 2.8% | 7.1528 | 66.775 |

Note: Assumes all other parameter values fixed at medium levels and 2.5% discount rate. Totals are totaled over 50-year simulation. Long-run values are average per year over the last two years of each simulation.

Financial support for this projected was provided by NSF Grant # SES-0083327 and NOAA's Center for Sponsored Coastal Ocean Research Grant # NA05NOS4781197. The authors would like to thank Kerry Smith, David Finnoff, Lisa Eby, Tom Miller, Joe Ramus, Mike Orbach, Ken Reckhow, Bob Christian, Craig Stow, Doug Lipton, and Lynn Maguire for helpful comments and suggestions, Sam Cunningham for research assistance, and thank David Eggleston and Sean McKenna for access to unpublished data and reports.

Lbar | 1.2 | From Neuse TMDL (adjusted down from 1.3 to account some for stock in the system) | |

omega | 0.95 | A slow decay rate based on persistence of nutrients | |

mu | 0.4 | Borsuk, Stow, and Reckhow (2004b) | |

rx | 0.75 | Eggleston | |

kx | 25.85 | Eggleston | |

alpha | 0.37 | Backed out using Excel's nonlinear solver and imposing the steady-state condition in patch 1 (or 2) with no hypoxia | |

phi | 50 | Scale intuition based on relative carrying capacities | |

chsi | 0.5 | Allows for a wide range of responsiveness (and the potential for collapses in patch 1), less than or equal to corresponding rx to prevent negative populations | |

beta | 0.001 | Chosen to scale predation such that abs(beta*kx) < ry | |

ry | 0.97 | Clams are assumed fast growing (they are k-limited and not r-limited). Magnitude based on the survival regression in Borsuk, Powers and Peterson (2002) using minimum growth rate necessary to prevent population collapse under worst in-sample dissolved oxygen | |

rho | 0.79 | Computed elasticity from Borsuk, Higdon, Stow, and Reckhow (2001) combined with Borsuk, Powers and Peterson (2002) | |

p | 0.87 | Mean real price (weighted avg aggregated across hard, soft, and peeler, 1998–2002). In 2002 dollars using CPI Size D South | |

c + delta | 240.45 | Rhodes, Lipton, and Shabman (2001) converted to 2002 dollars | |

gamma | 45.37 | Estimated jointly with 1-period ahead forecasts on catch per trip data (1982–2002) converting pots to trips and excluding 1994–1999 due to overreporting of pots. Pots converted to trips based on 2000–2002 trip data. | |

theta | 0.4 | Smith and Wilen (2003) | |

q | 0.00003 | Using parms from Eggleston | |

Neuse share of total NC | 0.35 | Share of the fishery in the Neuse River and Estuary + the Pamlico Sound | |

N(0) | 1.26 | Loadings divided by w (start off at steady-state stock) | |

A(0) | 0.51 | Algae starts in a steady-state. | |

X_1(0) = X_2(0) | 12.92 | 1/2 of k_{x} (bundled to pop. Parameters) | |

Y_1(0) = Y_2(0) | 0.50 | 1/2 of y carrying capacity | |

pi_1(0) | 0.5 | Initial effort allocated uniformly over space. | |

E(0) | 32,077 | 35% of effort in the fishery (mean over 2000–2002) |