Economics of spatial-dynamic processes.

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