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...terms both population and income (Shumway and Otterstrom 2001). While the increasing economic importance of environmental amenities is relatively well understood, (1) the feedback effects between economic growth and ecological changes are not. Urbanization and land development have detrimental ecological impacts, which can undermine those very environmental amenities that attract growth. Long-term economic and ecological sustainability depends on achieving a balance between economic growth and ecological change, which in turn requires an understanding of the dynamic interactions between the economic and ecological systems. These are not easily accounted for, however, because of the nonlinearities and spatial complexity that is inherent in both the ecological and land development processes.
Researchers have started to account for spatially explicit relationships between amenities and land development patterns. Brueckner, Thisse, and Zenou (1999) analyze the effect of amenities on residential location decisions and find that the location pattern of different income groups can be explained by the spatial distribution of amenities. Irwin and Bockstael (2002) examine the influence of congestion effects from neighboring development on fragmented development patterns. Lee and Fujita (1997) analyze the efficient configuration of greenbelts. Yang and Fujita (1983) and Wu and Plantinga (2003) evaluate the effect of open-space policies on urban spatial structure. Wu (2006) develops an amenity-based model to analyze the effect of spatial heterogeneity of amenities on development patterns and community characteristics. Most of these studies, however, treat environmental amenities as exogenous and none has analyzed the implications for the spatial and temporal evolution of development patterns when human-ecosystems interactions are endogenous.
On the other hand, a small but growing literature in economics has considered dynamic interactions between economic and ecological systems (e.g., Anderies 2003; Brander and Taylor 1998; Jansson, Anderies, and Walker 2004; Perrings and Walker 1997), but none have considered the endogenous interactions between environmental amenities and land development. An exception is Irwin, Jayaprakash, and Chen (2007), who develop a regional model of migration and economic growth in which endogenous environmental amenities evolve over time in response to population growth. However, the model is not spatially explicit and therefore the pattern of land development is not considered. Economic-ecological interactions have received more attention outside the economics literature, but without the economic framework necessary to clarify the individual behaviors that underpin the complex interactions and thus the implications for policy remain ambiguous (see Dendrinos (2000) for a review).
This article considers the implications of dynamic interactions between economic and ecological systems in the context of land development and water quality. The effect of interactions on the spatial pattern and time path of land use is considered within a model simple enough to be analytically tractable yet possessing sufficient richness to identify the basic forces at work. The model assumes that residential sites are differentiated by their amenities and recreational opportunities, which are provided by a nearby water body (hereafter referred to as a lake). Land development around the lake affects its water quality, which influences the level of amenities and recreational opportunities provided by the lake. The impact is cumulative and is determined by the rate and spatial pattern of development within the watershed. With this setup, we examine the efficiency loss due to the failure of private developers to consider the dynamic interaction and examine the extent to which consideration of these two-way linkages contributes to economic efficiency and improvements in water quality. We consider policy options available to a regulator seeking to achieve improvements in efficiency and water quality, including impact fees for development and a riparian buffer around the lake.
The analysis reveals a number of interesting findings. Private development is found to be more rapid and volatile to changes in economic conditions and environmental amenities than the development that a social planner would carry out. In some cases, private development causes an ecosystem crash where the lake becomes completely degraded, while a stable equilibrium with positive levels of amenities emerges under the socially optimal decision. Because private developers do not consider the endogeneity of water quality and lake amenities, they fail to account for the environmental costs of land development, which include not only the standard environmental damages from pollution, but also a loss of flexibility in adjusting the future path of pollution due to the irreversibility of development. Thus we find that the interdependence of irreversible land development and water quality generate an option value that arises even in the absence of uncertainty, which is contrary to the usual case. Finally, we show that the optimal time path and spatial pattern of land development can be implemented as a private development solution through the assessment of impact fees that internalize the present value of the pollution damages and irreversibility costs of development. Welfare can also be improved by the creation of a riparian buffer around the lake, but this policy may not achieve the first-best efficiency of the planner solution.
Our results demonstrate the importance of accounting for development irreversibility, the spatial dimension of land use, and the interdependence between land use and environmental quality in models of urbanization and amenity-driven growth. Ignoring these dimensions can lead to shortsighted decisions and behaviors that minimally will lead to inefficient land use and in the extreme, a crash of the local ecological system. Incorporating these dimensions into the analysis can help direct policies that will avoid such potentially catastrophic outcomes and result in higher levels of social welfare and environmental quality.
The Model Setup
Consider the interaction between land development and water quality in a lake ecosystem. Developed land around the lake generates pollution runoff that accumulates in the lake, which reduces water quality and clarity in the lake. Water quality degradation in the lake reduces lake-based amenities and services, which in turn induces human responses in terms of land development. In this section we first establish the nature of the interactions between land development and water quality and then discuss the potential economic and environmental impacts of water pollution in the lake.
The Spatial and Temporal Relationships
Consider first the spatial relationship between land development and water quality at a given time t; thus argument t is suppressed for the time being. Without loss of generality, the lake is assumed to be located at (0, 0) and is dimensionless. (2) All land around the lake is developable and development is irreversible. Land use before development does not cause any pollution. (3) Once a parcel of land is developed, it generates pollution runoff at a constant rate. Part of the pollution ends up in the lake, and the rest is absorbed naturally as it is transported to the lake. Let [gamma] be the rate of pollution generated per unit of developed land, and let a be the natural absorption coefficient as pollution is transported to the lake. The rate of pollution entering the lake when the radius of the developed zone is [bar.z], w([bar.z]), can be derived as (4)
(1) w([bar.z]] = [[integral].sup.[bar.z].sub.0] 2[pi]z[gamma][e.sup.-[alpha]z] dz = 1/[[alpha].sup.2] 2[pi][gamma][e.sup.[alpha][bar.z]] ([e.sup.[alpha][bar.z]] - [alpha][bar.z] - 1).
Intuitively, 2[pi]z[gamma]dz is the amount of pollution generated by development between z and z + dz, and [e.sup.[[alpha][bar.z]] is the share of the pollution that ends up in the lake. Differentiating (1) with respect to [bar.z] twice yields
(2) dw([bar.z])/d[bar.z] = 2[pi][gamma][bar.z][e.sup.-[alpha][bar.z]] >
(3) [d.sup.2]w([bar.z])/d[[bar.z].sup.2] = 2[pi][gamma] [1 - [alpha[bar.z]][e.sup.-[alpha][bar.z]] > if [bar.z] < 1/[alpha].
Equations (2) and (3) state that as more land is developed, the loading of pollution to the lake rises at an increasing rate initially, but at a decreasing rate as [??] increases beyond (1/[alpha]).
Consider next the temporal dimension of the interaction between land development and water quality. Given the rate of pollution entering the lake, the total amount of pollution accumulated in the lake at time t, W(t), equals
(4) W(t) = [[integral].sup.t.sub.0] w([??]([tau]))[e.sup.-[beta](t-[tau]] d[tau], W(0) =
where the initial level of pollution is assumed to be zero (i.e., W(0) = 0), and [beta is the natural decay rate of pollution in the lake. w([??]([tau]))[e.sup.-[beta](t-[tau])] is the amount of pollution entering the lake at time [tau] that has not been decomposed by time t. Thus, the right-hand side of (4) is the total amount of pollution entering the lake from time to t that has not been decomposed by time t. Differentiating (4) with respect to t gives the growth of pollution in the lake at time t:
(5) dW(t)/dt = w([??](t)) - [beta]W (t).
Equation (5) states that the rate of change in pollution level in the lake equals the rate of pollution entering the lake minus the rate of pollution decomposition in the lake.
The Impacts of Water Pollution
Water pollution in the lake may affect both the local residents and the general public living outside of the watershed. Local residents derive utility from the amenities and services that the lake ecosystem provides. These amenities and services may include scenic lake views, fishing, and other recreational opportunities. The level of amenities at any location is assumed to depend on its distance to the lake and water quality in the lake: a = a(W, z), where [a.sub.W] [less than or equal to] and [a.sub.z] [less than or equal to] 0, and [a.sub.W] = when W is close to zero. These conditions imply that water pollution in the lake reduces the level of amenities that the lake ecosystem provides, and that households living farther away from the lake have lower levels of amenities.
Water pollution in the lake may also cause ecosystem damages that are not fully internalized by local residents. Differences between degradation of amenities versus ecological functioning may arise, for example, due to nonlinearities or time lags in which a gradual accumulation of pollution impairs the...
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