Economics of spatial-dynamic processes.

I am honored to have the opportunity to give the AAEA Fellows Address this year. I would like to talk about a class of problems that are becoming more prevalent and yet have not received much attention by economists. These problems are a frontier area for economists--good topics for students looking for thesis topics and for the rest of us looking for new research questions. This class of problems poses challenges at the conceptual level, for empirical analysis, and at the policy design level.

Spatial-Dynamic Processes

The problems I would like to talk about I call "spatial-dynamic" problems, situations for which there is some (generally biophysical) process that generates potentially predictable patterns that evolve over space and time. The system generating such patterns may be largely exogenous. For example, the pattern of coastal area inundation around the world that we are about to witness as sea surface rises due to global warming is essentially exogenous. The biophysical forces generating the pattern are already in play, and the process is unfolding and inexorable. Alternatively, many spatial-dynamic processes are endogenous in the sense that they are influenced by individual decisions at points in space/time. A good example is a forest fire. A forest fire spreads over space in a manner influenced by some biophysical processes (e.g., wind direction and speed), but the pattern is clearly also influenced by decisions made by individuals in the landscape. Firefighters are the most obvious agents of influence. The spatial-dynamic pattern of a fire is influenced by backfires in front of the advancing front, bulldozed firebreaks, and fire retardant dropped by firefighters. In addition, however, the inhabitants of areas potentially influenced by fire also affect the pattern of spread. Most importantly, individuals who do (or do not) invest in "defensible space" around their homes influence the spatial-dynamics of fires once started.

Aside from simply being interesting systems, these kinds of processes often pose challenging economic and policy questions. Some are predictive questions such as: how will a rise in sea surface height affect individuals in low-lying coastal areas? How will various policies (e.g., dikes) alter or mitigate some of the potential impacts? How do homeowners in fire-prone areas react behaviorally to the prospect of fires? Do actions taken in a fire-prone area influence the prospective patterns of fires? Other questions raised by these kinds of processes are prescriptive or normative. How should we control a spatial dynamic process? Can we influence this process, and how many economic resources should we invest? Can we mitigate impacts? If so, how much should we spend? Is it worth building dikes all around the edge of San Francisco Bay to protect that land value in the face of rising sea surface? What parts of the areas around New Orleans should be rebuilt, and what kinds of land uses should be allowed, given the possibility of future spatial-dynamic flood processes affecting the Mississippi watershed?

One can think of many examples whereby spatial-dynamic processes link economic actors over space and time. Water in aquifers moves from areas of high to low density in a manner mediated by soil porosity. When aquifers are tapped by wells, the natural equilibrium is disturbed; water withdrawal creates "cones of depression" around the wells, which further influence relative densities and subsequent water flows. If contaminants are introduced into the groundwater system, they also flow from high-to low-density sites and are influenced by pumping rates and the spacing and depth of wells in the subsurface system. Bio-invasions are another example of spatial-dynamic processes for which patterns may be influenced by both exogenous and exogenous factors. The spread of such different organisms as the star thistle plant, the honey bee mite, and the sudden oak death causing fungus are among the kinds of bio-invasions currently experienced in the West. Each has its own propagation source, means of introduction, opportunity for establishment, and pattern of spatial-dynamics, generally mediated in some way by human activities.

Other examples of spatial-dynamic processes are the various mechanisms that link so-called metapopulations. The notion of metapopulations is a recent and important paradigm shift in biology, away from a view that depicted populations as homogeneously distributed over species' ranges to a new view of discrete subpopulations that are connected. For example, marine populations are now seen as inhabiting discrete patches or distinct aggregations (Sanchirico and Wilen 1999). These are linked with each other via adult movement, larval flow, winds, and currents. The ebb and flow of any particular sub-population is thus driven by patch-specific factors interacting with systemwide determinants of patch connectivity.

Another spatial-dynamic process of importance is that governing human and animal disease. For example, if one looks at the pattern of flu cases in a country like the United States each year, one finds a bell-shaped incidence pattern, peaking in February and flattening throughout the fall and winter, only to reappear again the next year in the same pattern. (1) Other countries experience similar patterns, with different peaks and different distributions of flu strains. If one looked only at country-specific data, it would be tempting to see these patterns as purely dynamic "local" processes. However, as epidemiologists have discovered, each country's seasonal dynamic pattern is itself part of a larger global spatial-dynamic process. In particular, virtually every annual cycle of flu originates in southeastern Asia, in high human density regions of China in proximity to domestic and wild birds and animals (Viboud et al. 2006). The source of the flu is then transmitted in a manner that reflects the dominant flow of humans along major airline routes, from China to major cities in the United States and elsewhere around the world, to regional centers and then to less-dense rural areas.

Patterns of the spread of flu are a good example of why it is important to fully understand spatial-dynamic processes. If one examines flu from a local scale, one is led to an incorrect perception that what is being observed is a dynamic phenomenon, much like technology saturation. But if one steps back and takes a global view, it is apparent that each year's local pattern is itself part of a process with a single source that then feeds multiple "sinks" around the world. The policy implications of each view are dramatically different. If one believes the local story, then one is led to reactive policies that are initiated each year once the flu strain is discovered and typed. But if one believes the global story, it is obvious that other, much more efficient policies might be envisioned. The global view leads naturally to thinking about the global public good aspects of epidemics and institutions that might be able to tackle the problem at a different scale. For example, one might envision a global fund to which potential receptor countries contribute, and that acts earlier in the flu season to quarantine early carriers before they can spread the disease globally.

These kinds of problems are especially interesting for at least two reasons. First, they are becoming more prevalent. Globalization, in particular, is a force that is linking more systems and hence increasing the opportunities for epidemic and invasion-style processes to proliferate. Second, there has been a knowledge explosion about spatial processes in the sciences, driven by new technologies such as remote sensing, GIS, and computational improvements. These technologies, remote sensing in particular, are generating vast amounts of new data on spatial patterns in the biosphere, patterns not seen before and that beg explanation. The sciences are devoting a considerable amount of attention to understanding the patterns that are newly revealed, in some cases completely revamping old concepts to focus on important spatial processes.

Despite the challenges and despite the attention that spatial-dynamic processes are attracting in the hard sciences, economists have not paid much attention to these kinds of problems. We have dynamic theories, such as the elegant analytical structures of capital theory, or the theories of renewable and non-renewable resource use that form the core of natural resource economics. And we have spatial theories, such as those espoused by von Thunen (Hall 1966), Losch (1954) and Tiebout (1956). But we have very few spatial-dynamic theories of systems whereby integrated processes drive patterns over space and time. The exceptions are work by Bhat, Huffaker, and Lenhart (1993; 1996) and Lenhart and Bhat (1999) on terrestrial pests; modeling of metapopulations I have done with my colleagues Jim Sanchirico (1999, 2005) and Marty Smith (2003); recent conceptual analysis of the optimal control of diffusion systems by Brock and Xepapadeas (2006); modeling of aquifers by Brozovik, Sunding, and Zilberman (2006); and some work on foot and mouth disease by Rich (2005) and Rich, Winter-Nelson, and Brozovik (2005).

Deconstructing Spatial-Dynamic Processes

For the remainder of my talk I would like to "deconstruct" spatial-dynamic processes by discussing features that define these kinds of problems and ways that they might be modeled. Then, I will illustrate using an example of bio-invasions. There are several features of spatial-dynamic problems with which economists are not particularly familiar. Figure 1 is an abstract representation that captures the nature of these kinds of problems. The heart of the process is the diffusion or dispersal process that governs the way something spreads over space and creates patterns. Is movement random or purposeful and behavioral? Are patterns self-generated, or are they also influenced by directional forces such as winds and currents? Does the front spread uniformly, or does it follow transportation corridors and nodes of populations? What mechanism drives movement--animal transport, human transport, spores, rhizomes, swimming, walking?

[FIGURE 1 OMITTED]

The other important aspect of spatial-dynamic problems concerns the nature of space, and in particular boundaries, geometry, and heterogeneity. What happens at boundaries, and how should we characterize it? Mathematicians use terms such as absorbing or reflecting boundaries or zero flux boundaries. When lemmings migrate to breeding grounds near the sea at normal population levels, they reach the barrier, mill around, and begin the process of breeding. But when populations are high, they reach the sea and jump in, illustrating the difference between reflecting and absorbing boundaries. Some boundaries exist for biophysical reasons, such as the transition between land and sea. Others exist because of habitat quality, and these boundaries may be species specific. Yet other boundaries exist for geopolitical reasons. For example, a forest pest spreading north from New England into Canada might spread across the border but be treated as if it stopped at the border when the U.S. Forest Service decides how much control to initiate. Another important aspect is the geometry of space. Is the relevant spatial unit like a featureless plain, or does it have corridors or choke points that influence density and flux? Is space homogeneous, or does it exhibit differentiated features that influence either the diffusion pattern or damages? Soils often exhibit widely varying character even within small scales, complicating the depiction of the flows of water, oil, or contaminants. A landscape contains a mosaic of human uses so that a bio-invasion might warrant more focused control efforts at particular points that are located in advance of certain areas with high potential damages.

Diffusion Processes

In light of the central role played by diffusion processes in spatial-dynamic problems, it is useful to elaborate how they are typically modeled (see Okubo and Levin 2002). What makes these kinds of problems interesting is the fact that patterns are generated by integrated dynamic and spatial forces. The simplest kind of spatial-dynamics process can be represented by a diffusion equation

[partial derivative]C(X, t)/[partial derivative]t = [partial derivative]/ [partial derivative]X [D [partial derivative]C(X, t)/[partial derivative]X]

= D [[partial derivative].sup.2]C/[partial derivative][X.sup.2] (1)

This is a partial differential equation that expresses a process in terms of derivatives in both time and space. This particular equation represents the most basic type of diffusion, namely random diffusion. Consider measuring the concentration C(X, t) of something like a group of particles at a point on a line X at time t. Suppose that any particle can move either right or left on the line with equal probability. Then, it can be shown that the concentration of the particle will be governed by Fick's Law, which states that the spatial diffusion at a point will be proportional to the spatial gradient at that point (Murray 2002). Here, D is the diffusion coefficient, (assumed constant) which indicates the rate of spatial flow. The essence of this idea is that particles will flow on net from high to low density areas simply because high-density areas have more particles that have a chance of ending up in low-density areas than vice versa.

A partial differential equation like equation (1) is not something with which economists are particularly familiar, and solving these kinds of equations is difficult and somewhat of an art form in mathematics (Holmes et al. 1994). As with ordinary differential equations that are expressed only as a function of time, the explicit solution must incorporate boundary conditions and initial conditions, in numbers equal to the number of derivatives in the equation. Assume that this process is started with a "point-source" injection of m particles at the origin and at time zero, and that the one-dimensional line depicting space is of infinite length in both directions away from zero. Then, this particular equation can be explicitly solved to yield

C(X, t) = m/[square root of 4[pi]Dt] exp (-[X.sup.2]/4Dt). (2)

This describes how the concentration changes over space and time. As figure 2 shows, the initial concentration spreads over space and time. Recall that this process is completely driven by random movement. At the origin initially, particles may move right or left with equal probability. But because there is a point concentration, the gradient is steep around the origin, and hence, more particles will move from the origin to adjacent low-density points in space. This is Fick's Law in action, causing the flow at a point to be proportional to the spatial gradient at that point, and causing, at the global level, particles to spread out over space.

[FIGURE 2 OMITTED]

Another useful representation of a diffusion process that fits many examples found in resource problems is one described by

[partial derivative]C(X, t)/[partial derivative]t = D [[partial derivative].sup.2]C/[partial derivative][X.sup.2] - V [partial derivative]C/[partial derivative]X. (3)

This equation is simply the random diffusion representation characterizing Fick's Law in equation (1), but modified to include what is called an advection term. The advection term contains the constant V that depicts drift of the process (Murray 2002). Advection applies whenever a diffusion process is influenced by external forcing factors such as wind or currents. If the advection process is strong enough, the process depicted in figure 2 will not only dissipate the initial infusion, but also shift the concentration over time as suggested by the solution

C(X, t) = m/[square root of 4[pi]Dt] exp (-[(X - Vt).sup.2]/4Dt). (4)

The solution to the diffusion with advection equation shows that the maximum concentration shifts over time according to the term in the exponent that acts as an axis shifter. (2)

A final representation of a diffusion process is the famous Fisher reaction-diffusion equation

[partial derivative]P(X, t)/[partial derivative]t = D [[partial derivative].sup.2]P/[partial derivative][X.sub.2] + [alpha][X.sup.2] + [alpha]P(1 - P). (5)

This equation was examined by R.A. Fisher (1937) and it depicts a process that is especially suitable to examining biological organisms. It contains the random diffusion term but also a term that represents logistic growth at a point in space. The logistic equation is a popular and useful representation of density-dependent growth of populations. At each point in space, then, a population P(X, t) will grow according to two forces: (net) diffusion and local growth. The percentage rate of local growth is highest when the population is smallest, and other things equal the population approaches a carrying capacity level at each point, where the carrying capacity is normalized at one in the above equation.

As it turns out, it is impossible to derive a closed-form solution similar to equations (2) and (4) for this equation. However, Fisher was able to show that as time gets large, the solution exhibits what is called a "traveling wave" property. A traveling wave can be envisioned as similar to a tsunami, with the sea surface height behind the wave at the same height as the crest. Kolmogorov, Petrovsky, and Piscounov (1937) provided a conjecture about the velocity of the wave front, namely that it is [square root of 4[alpha]D], or a value associated with the product of the diffusion coefficient and the growth rate of the logistic process at small population levels. This conjecture has recently been proven correct (Uchiyama 1978). The reaction-diffusion equation is a useful way of thinking about populations of plants, or animals or even humans.

A Digression: The Genius of Harold Hotelling

The notion that we might think of human populations as represented by the reaction-diffusion equation was, rather surprisingly, first hypothesized by Harold Hotelling (1921), the great statistician/economist responsible for so many imaginative and important papers over a career that began in the mid-1920s. What is not generally known is that Hotelling actually began his education not as an economist or statistician, but as a journalism student, receiving a bachelor's degree from the University of Washington in Seattle in 1919. He then decided to undertake further study for a master's degree, but in mathematics rather than journalism. His M.S. Thesis submitted to the Department of Mathematics at the University of Washington in 1921 is entitled "A Mathematical Theory of Migration." The existence of this thesis is little known, but it foreshadows Hotelling's brilliant career and elegant modeling ability. Hotelling's thesis basically proposed that we think of the westward movement in the United States as a reaction-diffusion process. As he wrote:

(if we hypothesize) that the percentage rate of

natural increase at any place is proportional

to the difference between the density of population

and a fixed saturation point, ... we have

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2][pi] + [alpha][pi]([sigma] - [pi]) (39)

This equation is not linear, and is difficult to handle. But as we have seen, we may be dealing with new and sparsely settled countries assuming a Malthusian principle that population increases in a geometric ratio; while for countries near the saturation point it is not unreasonable to assume that the number of births per unit of area is proportional to the difference between the density of population and the saturation point. For the first case, we combine equation (20) with (1); for the second we take (20) and (3). These give respectively

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2] [pi] + r[pi] (40)

and

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2] [pi] + b([sigma] - [pi]) (41)

The elegance of Hotelling's approach is admirable. He first proposes the general reaction-diffusion equation to model the westward movement (his notation for diffusion used the diffusion coefficient K, the more general gradient symbol [[nabla].sup.2] for the second partial derivative and the symbol [pi] for the population at any point). His justification for the simple Fick's Law type of diffusion process is, however, not the assumption that movement is random. Instead, he proposes that settlers move from high- to low-density areas for economic reasons (in search of low-cost land, for example). Then they will act as if they are diffusing randomly via Fick's Law, even though there is behavioral purpose driving movement. As he notes, the reaction-diffusion equation "is difficult to handle" (an understatement, since it was not really understood until sixteen years later) and hence he proposes solving two simpler problems, the solutions of which can actually be derived explicitly, and which qualitatively bracket the more general problem.

The remainder of Hotelling's thesis is empirical. He procures the U.S. census records for the 1790-1900 period and uses the data to compute the implied diffusion coefficients of the actual westward movement "wave front," thus calibrating the model to complete his analysis. Aside from the cleverness and elegance of thought that the thesis reveals in the young Hotelling, it also is the first depiction of a spatial-dynamic process driven by economic behavior depicted by a diffusion mechanism.

To summarize, the fundamental drivers of spatial-dynamic processes are diffusion or dispersal processes. These can be modeled with partial differential equation descriptions. As shown above, there is a rich variety of diffusion models available to fit circumstances ranging from simple random movement, to advection to joint diffusion/population growth. A few of these models can be solved analytically, but most cannot. This presents analysts of spatial-dynamic problems with challenges. There are essentially two ways to build understanding of these kinds of complex situations. One way is to construct numerical models of complex systems and attempt to synthesize some common understanding from the analysis of a range of cases. The other way is to start with very simple models, extracting what can be learned before adding detail. This is the method we illustrate next, using a familiar current policy problem.

An Example: Bio-Invasions

To illustrate how we might model spatial-dynamic processes in order to gain understanding about policy options, I will now turn to an example, namely the example of bioinvasions. Bio-invasions are familiar to everyone, and there are numerous cases that come to mind, including kudzu, the gypsy moth, yellow starthistle, and zebra mussels. Bio-invasions are successful colonizations of alien species that have been either accidentally or purposefully introduced into ecosystems. They may generate damages (or benefits) as they spread, and costs of control depend upon when, how intensively, and where control activities are undertaken.

Bio-invasions are becoming increasingly prominent in the public eye, and the topic has also received considerable research attention, first by ecologists but more recently by economists. For the most part, however, economists have ignored the spatial-dynamic aspects of bio-invasions, often treating them as little more than a pest-control problem. This seems to miss the most interesting aspect of bio-invasions, namely their spatial-dynamic character. While simple characterizations that address questions such as whether to spray or not, or whether to quarantine or not are illuminating, the more important questions seem to be where to spray, when, and at what intensity in a landscape setting. (3) This requires a modeling apparatus that captures essential features of spatial-dynamic problems.

The economic issues that bio-invasions raise are several. How does an uncontrolled bioinvasion unfold? What governs its rate of spread, and how fast is that spread if no actions are taken? What are the consequences to humans of the invasion? What kinds of actions are available to control a bio-invasion? At what points over time and space are instruments available to manage the bio-invasion? What is the optimal way to use controls; when, where, and how intensively should they be implemented? How is the optimal control affected by bio-economic parameters? How do basic cost/benefit parameters affect the best choice? How about the discount rate and time horizon? Does the optimal control vary with the initial size of the invasion? If an invasion is uncertain, what kinds of monitoring and detection efforts should be implemented?

Answering these kinds of questions requires some kind of modeling effort. The modeling should first describe the biophysical diffusion mechanisms that drive the bio-invasion. Diffusion may occur in a smooth and homogeneous pattern over space, driven by short-term dispersal processes such as simple local movement. For example, the classic case in biology textbooks of muskrat spread in Europe reveals a stark radial pattern with a predictable velocity of the "wave front" over several decades (Murray 2002), much like predicted by the Fisher reaction-diffusion equation. Alternatively, dispersal may involve long-distance mechanisms driven by wind and currents, or by movement of hosts including humans, animals, and birds (Hastings et al. 2005). A model of a bio-invasion should also incorporate some assumptions about boundaries and processes that occur at boundaries, as well as the geometry of the relevant space. Finally, a model of bio-invasions needs to reflect humans and define the links between humans and biophysical environment. Are humans simply passive participants, or do they influence the spatial-dynamic patterns by their actions? How are humans impacted by the bio-invasion? Once these critical components, the biophysical systems with its dispersal mechanisms and spatial character and the human linkages, are specified, the system can be simulated and optimized to generate some predictions of the consequences of various policy options. Further refinements can then be added if necessary and policy conclusions expanded.

[FIGURE 3 OMITTED]

A Simple Bio-invasion Model

Consider the simplest possible model of a bio-invasion that incorporates the above elements, albeit in the most abstract fashion. Suppose that we consider the spatial geometry of our model system to be a corridor of fixed width, with the origin at the left-hand side, and an unbounded right-hand side. As in figure 3, we assume that there is an initial invasion that is discovered after having covered an area of amount [X.sub.0]. The invasion is assumed to spread along the length of the corridor with a velocity of v. (4) The area infested at any date t will be X(t) = [X.sub.0] + vt. Assume that damages are proportional to the total area infested in each period, where the marginal damage per unit space is d. I have in mind something like star thistle, a weed that spreads through pasture land and that grazing animals will not eat. Assume that control costs are incurred each period in a manner that influences the velocity of the wave front of the invasion (Sharov and Liebhold 1998). If [bar.v] is the natural velocity of the uncontrolled bioinvasion, let control costs be quadratic in the velocity reduction associated with control activities, or:

TC(v) = a([bar.v] - v) + (1/2)b[([bar.v] - v).sup.2]. (6)

[FIGURE 4 OMITTED]

This control cost function is depicted in figure 4. As shown, a range of control options is assumed possible. One option is to simply abandon control, allowing the invasion to spread at its natural rate [bar.v]. In this case control costs are zero each period. Another possibility is to slow the invasion by choosing a control that reduces the velocity somewhat, but that ultimately allows the invasion to spread over space nevertheless. It is also possible under the assumptions made here to stop the invasion, by holding the front at a fixed point in space with some kind of razor's edge barrier control. Finally, we assume that if enough control expenditures are incurred, it is possible to actually eradicate the invading species from the original area infested, pushing it back to the origin at a negative velocity.

This simple model captures some, if not all, of the important features of a spatial-dynamic economic problem. The characterization of space is very simple, and complicated issues associated with real geometry and boundaries have been ignored. The spatial-dynamic process is likewise very simple, with a process that essentially assumes invaded space and time are proportional. We will make one more simplifying assumption, namely that the choice to be made is a single velocity choice maintained over the entire time horizon. The optimization thus will be only "quasi-dynamic" in the sense that the optimal choice will be chosen to reflect its whole effect over the complete time horizon, without actually solving for a time-varying velocity path over the horizon.

The objective for the policy problem posed here can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

or minimize, by the choice of the velocity of the invasion, the discounted sum of damage and control costs. Recall that each period's damages are proportional to the area infested, and each period's control costs are associated with the intensity of the effort devoted to reducing the invasion velocity.

[FIGURE 5 OMITTED]

As it turns out, even with all of the simplifying assumptions we have made, the solution to this problem is not fully straightforward. Space does not permit a full exposition of the solution, but it is possible to sketch the results. Most important is that there are two qualitatively different kinds of control options possible, namely infinite horizon options that ignore, slow, or stop the invasion, and finite horizon options that involve eradication of the invasion. Analysis of optimal policies thus must compare the two kinds of options under different assumptions about parameters of the problem and synthesize the results. Figure 5 synthesizes the most important features of the optimal choice of bio-invasion spread. This shows that the type of control depends, as we would expect, on the basic cost/benefit parameters. The y-axis of figure 5 shows the discounted marginal benefit d/r of a reduction in velocity over the horizon. When the marginal benefit is low, relative to the marginal costs, it is optimal to ignore the invasion at high initial invasion sizes. For example, if the marginal benefit d/r is below the marginal cost a of the first unit of velocity reduction, it does not pay to initiate any control (for most invasions sizes), and the optimal policy is an "abandon" policy. As the damages rise (or the discount rate falls), it pays to slow most invasions, and when the damages are very large, it pays to eradicate in finite time. The somewhat surprising result, although intuitive on reflection, is that the invasion size matters. It is optimal to initiate control and even eradicate invasions when they are small enough, even when simple static cost/benefit criteria suggest otherwise. But if an invasion has already gained a significant foothold before detection, the expected marginal damages must be large (vis-a-vis marginal control costs) to justify eradication.

[FIGURE 6 OMITTED]

Some Policy Implications

What else can we learn from this simple example? Consider a situation in which we have a landscape of multiple landowners, as in figure 6. Here there are two landowners A and B. Assume that each landowner can make his own individual choice about how fast the invasion spreads across his property. Then, if an invasion occurs so that A's property is partially invaded, he faces a choice much like the one outlined above. (5) There are two choices that A might make. If he stops or eradicates the invasion, then control is complete over the whole landscape and A's private decision contains the invasion. But suppose that A finds it optimal to slow or ignore the invasion. Then after some time, the invasion reaches B's border, and he must decide how to manage the problem on his property. Importantly, it is in B's interest for the invasion to be delayed as long as possible. B thus has some willingness to pay to reduce the speed of the invasion crossing A's property. But if there is no negotiation between the two parties, B's willingness to pay will go unrecorded in the calculus that determines A's control choice.

Our simple model thus illustrates a fundamental point about spatial-dynamic processes. Bio-invasions and other similar spatial-dynamic processes generally take place at landscape scales that encompass multiple landowners and decision makers. Landowners make self-interested decisions, electing to contain the problem to the extent that damage reduction justifies control costs. But early in the unfolding of a spatial-dynamic process, landowners confer uncompensated benefits to those in advance of the invading front. Early controllers thus under-control from a system-wide perspective, generating a spatial-dynamic externality.

There are potential gains from negotiation between participants across time and space. Under laissez-faire, the externality in this example could be resolved either by a once-and-for-all negotiation at the outset or by what we might call "chained bilateral" negotiation that might work as follows. Suppose that there are N landowners. Then, the owner of parcel N, the land at the farthest edge of the corridor landscape (and the last to be potentially impacted), could negotiate with his neighbor on parcel N-1, promising a payment for every unit of time that the invasion was delayed beyond its appearance at his neighbor's left-hand boundary. Then, the neighbor on parcel N-1 could negotiate a similar agreement with neighbor N-2, taking into account the promised payment from N, etc. In this way the externality would be resolved similar to solving a dynamic programming problem with backward recursion, which the problem is akin to in this simplified case.

In the real world, transactions costs are high and likely to prevent either type of solution that produces a global optimum. But it is not unreasonable to assume that some kind of second-best negotiation might emerge for invasions of this type. A second-best negotiated solution would likely unfold in a local and myopic fashion. For example, once the invasion has established, it is reasonable to assume that it becomes known not only to the landowners whose land has been invaded but also to those in the neighborhood. Suppose the neighborhood is very "local," so that landowners about to be invaded only realize it when their adjacent neighbor is invaded. Then when A is invaded, we might see negotiation between A and B, with a transfer that slows the invasion's arrival at property owner B's property. Then, once it arrives at B's property, we might expect landowner to negotiate bilaterally and sequentially with C and so on. This kind of sequentially myopic and local forward recursion negotiation would not achieve a first-best optimum, but it would achieve something better than complete atomistic behavior.

These conclusions are examples of what emerges with just a simple representation of spatial-dynamic processes. One can envision numerous extensions that would flesh out more understanding about the implications of various institutional assumptions and control options. One obvious extension would be to model alternative spatial geometries. If the landscape is a featureless plain, then an invasion would expand radially rather than linearly, as in our corridor example. Since damages increase with the square of the radius while costs increase linearly, the implication would be to expand the zone of parameters over which more intensive control (including eradication) appears optimal. Another extension might be to incorporate alternative diffusion assumptions. For example, with a dominant physical forcing mechanism causing advection, the direction of the invasion is modified, and it will matter where in the landscape particular parcels are located. Similarly, alternative invasion assumptions would influence conclusions about control. Invasions might be rare or one-shot events, periodic, continuous, or stochastic. More frequent events raise costs and reduce the likelihood that intensive controls will prove optimal, other things equal. Another extension might be to examine landscape heterogeneity, either in terms of damages and costs differing by location, or in terms of diffusion rates depending upon landscape characteristics. Landscape heterogeneity will increase the differences between finely tuned first-best policies and second-best and broad brush policies. For example, if a group of particularly susceptible parcels is located out in advance of the front, then the payoff to early control is enhanced and indicative that more intensive controls ought to be initiated early.

Summary

Spatial-dynamic processes generate complicated patterns over landscapes. In systems that impact humans and/or are mediated by humans, spatial-dynamic processes generate spatial-dynamic externalities. In complex landscapes the nature of those externalities can also be very complex. There are numerous examples of disease epidemics, for example, whereby disease incidence at any point and time and space is governed by both local factors and also global linkages between and among disease sources (Grenfell, Bjornstad, and Kappey 2001; Keeling et al. 2001). Physical landscapes are rarely homogeneous, and hence the patterns of processes such as disease and pest invasions reflect diffusion rates that differ over space, and often differ with density and the influence of forcing factors. Hence, determining exactly where to initiate controls, how intensively, and when is a difficult problem in a general and realistic setting.

Nevertheless, even our simple modeling suggests some important first principles that inform policy. As we have seen, it is generally optimal to initiate control actions at key places and points in time. Our simple bio-invasion example shows the extent to which early and intensive control near the invasion site is better than later control elsewhere. For many spatial-dynamic processes the potential scale of the biophysical system is much larger than the scales associated with property ownership and control decisions. Under laissez-faire, "keystone" agents will be uncompensated for spillover benefits they confer on others as a result of their own myopic optimization decisions. These keystone agents will thus under-control relative to an efficient systemwide solution.

In a perfect world with no transactions costs, private negotiations as Coase envisioned could eliminate all gains from trade and hence achieve a first-best optimum. But in reality, such costless negotiation is improbable, and we are more likely to witness, if any negotiation at all, only negotiation at the local scale and in a sequentially myopic fashion. It is thus an important empirical question how large the efficiency losses are between the first-best optimum, various second-best negotiated settlements, and complete laissez-faire. The answer clearly depends upon the specifics of each setting. But there is reason to believe that differences are very large in many cases of spatial-dynamic processes that we are currently facing. The reason is simply that spatial-dynamic processes generate network-like connections between many independent decision makers. For processes that expand radially or via transportation networks, the potential size of the network, and hence the ultimate scale of the problem, can be enormous. A recent monograph by McKibbin and Sidorenko (2006) analyzes the potential global cost of flu pandemics, using a reasonably sophisticated model of epidemiology coupled with a global trade model. Depending upon the severity of the impact on laborers, the global costs run from the hundreds of billions of dollars for "small" pandemics to trillions of dollars in damage for large pandemics.

The implications of these kinds of processes, whereby potential damaging impacts spread over time and space in a geometric fashion, are thus potentially enormous. In a fully globalized world where everyone is connected to everyone else, a private decision by one person has potential to have impacts that are multiplied six billion-fold. Recent experiments in economics have suggested that individuals will, under some circumstances, account for public good costs/benefits of individual actions. But these experiments are set in small group, local public good settings, rather than the vastly mismatched scale that is relevant with global spatial-dynamic processes. Large-scale linkages raise critical policy issues about how to manage processes that have the potential to generate global public goods. If we consider the manner in which various countries are considering responses to pandemics, it is clear that they are not too different from our "local and myopic" decision-making procedure discussed above in the bio-invasion example. In particular, to the extent that countries are preparing at all for events like pandemics, they are mostly taking local and contingent protective steps such as stockpiling vaccines that would be given to critical caregivers in the event of an outbreak. In contrast, as we have seen, the most efficient solution is generally to identify and treat keystone individuals or keystone countries early in the process.

The institutional design question is thus: how do we build institutions that can address the prospects of potential large-scale spatial-dynamic public bads in an efficient manner? The efficient institutional solution requires, in principle, institutions that have scale sufficient to encompass the scale of the potential pandemic or invasion. Ideally, the institution must have legitimacy and be able to collect payments, make transfers, take advantage of economies of scale, and centralize decision-making. As the potential scale of the problem grows, however, institutional solutions become problematic for all of the familiar issues associated with collective action. It is not clear whether collective action issues will be overcome for many of the global issues we are currently facing, but it is clear that with globalization, the number of these kinds of problems itself has the potential to grow in an almost geometric fashion.

The author would like to thank Josh Abbott, Julian Alston, Frances Homans, Richard Howitt, Jenny James, Becky Niell, Jim Sanchirico, Marty Smith, and Hiro Uchida for assistance and helpful comments on early presentations of this talk, as well as Paul Preckel and an anonymous editor for helpful editorial comments.

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(1) See, for example, incidence pattern data, charts and maps on the U.S. Department of Human Health, Center for Disease Control and Prevention website at http://www.cdc.gov/flu/weekly/ fluactivity.htm.

(2) Several interesting animations of diffusion-with-advection processes in 2D are given at the MIT freeware engineering course site at http://web.mit.edu/1.061/www/dream/FIVE/ch5movie.avi.

(3) Interesting animations of spatial-dynamic processes in disease transmission may be found on various websites. The paper by Keeling et al. 2001 on the outbreak of foot and mouth disease in England has an animation of the spatial-dynamics in a set of supplementary materials found at http://www.sciencemag.org/content/vol0/ issue2001/images/data/1065973/DC1/FMD_UK_Movie.gif. Another illustrative animation on rabies control in Switzerland can be found at http://www.ivv.unibe.ch/Swiss_Rabies_Center/Visualisation/ visualisation.htm.

(4) The process could be driven by a reaction-diffusion equation, with a velocity determined by the wave front velocity conjectured by Kolmogorov.

(5) It is not identical, because now A is assumed to have a finite property boundary on the right-hand side. This changes A's problem because he now must account for what occurs after the invasion either passes through the boundary, or is contained or eradicated within his property boundaries.

Fellows Address.

James E. Wilen is Director of the Center for Natural Resource Policy Analysis and Professor in the Department of Agricultural and Resource Economics at the University of California, Davis and a member of the Giannini Foundation.

This Fellows Address was presented at the annual meeting of the American Agricultural Economics Association, Portland, Oregon, July 2007. Invited addresses are not subjected to the journal's standard refereeing process.