Biosecurity and spread of an infectious animal disease.

[a.sup.*.sub.i] = [[lambda].sub.i] - 1 - [[lambda].sub.[i-1]] - [[lambda].sub.[i-2]] + [square root of [(1 + [[lambda].sub.i] + [[lambda].sub.[i-1]] + [[lambda].sub.[i-2]]).sup.2] + 4[[lambda].sub.i] [[lambda].sub.[i-1]] [[lambda].sub.[i-2]] / 2 + 2[[lambda].sub.[i-1]] (5)

Table 1 provides a simulation analysis, where the parameters chosen are entirely synthetic. (3) In the baseline case, Case 1, with [theta] = 0.1, potential production common at [V.sub.i] = 10, and cost coefficients common at [[alpha].sub.i] = 0.3, actions and welfare are the same across farms. Expected output from each farm is 82.7% of potential revenue, with -0.3Ln(0.242) = 0.426 lost to costs of care taking. When 18 of 30 units of potential production are allocated to farm 2 and the other farms share remaining potential production equally (Case 2), then farm 2 takes more care ([a.sup.*.sub.i] declines) and the other two farms take less care. Aggregate expected surplus increases from 24.82 to 24.91. The rationale is that farm 2 assumes a stronger incentive to biosecure because of what has to be protected. The other two farms have less to protect and free ride on farm 2 actions. Strengthening incentives to the largest producer ensures that overall surplus increases.

Notice that, when compared with baseline Case 1, if farm 2 potential production increases to 20 (Case 3) then overall expected surplus increases from 24.82 to 33.64. Thus, a 0.882 fraction of the 10-unit increment in potential production converts to incremental expected surplus. The baseline conversion is the lower 24.822/30 = 0.827. The gains accrue almost entirely to farm 2, while equilibrium surplus from farm 1 is hardly affected and farm 3 gains modestly from reduced infection on its anticlockwise neighbor. Farm 3 takes less care as it free rides on farm 2, while farm 1 takes about the same amount of care.

Cases 4 and 5 study a dramatically altered production environment--one where backyard production occurs. Here, potential production is concentrated on farm 3 (with 28 of 30 units), which is also the farm with highest protection cost. This scenario might arise because farm 3 has strong comparative advantage in a feed source that could carry infection. Case 4 is the new baseline. Case 5 involves a subsidy on the farm 1 cost through decreasing [[alpha].sub.1] from 0.03 to 0.02. This might entail a capital investment in controlling access for feed suppliers or in providing quarantine quarters for purchased livestock. Compare Case 5 with Case 4 to see that a subsidy to farm 1 leads to a small reduction in total expected production. While small, bear in mind that the cost of the subsidy has not been taken into account. A public subsidy of this form would show no gross benefit for the deadweight loss arising when raising taxes to support the subsidy.

To the extent that our model captures critical features of reality, two suggestions may be extracted. First, concentrated production internalizes disease externalities and thus may promote overall production efficiency without reference to scale economies. Second, a subsidy targeting some small farms may not be a good idea. Even though some smaller farms may practice more biosecurity, substitution effects across smaller farms may leave larger farms even more exposed than before the subsidy.

Line Topology

The intent of this section is to demonstrate the robustness and limitations of the modeling approach. To this end the circle production structure is replaced with a three-farm line topology. The farms are now located along a line segment, as illustrated in figure 1. In contrast with the circle topology, physical barriers (e.g., mountain, desert) preclude direct spread between farms 1 and 3. In contrast also with the circle case, the middle farm can infect both edge farms, n = 1 and n = 3. So farm 1 may be infected directly with probability [theta], indirectly from farm 2 with probability [theta][a.sub.1], or indirectly from farm 3 with probability [theta][a.sub.1][a.sub.2]. Farm 3 may be infected directly with probability [theta], indirectly from farm 2 with probability [theta][a.sub.3], or indirectly from farm 1 with probability [theta][a.sub.2][a.sub.3]. Farm 2 may be infected directly with probability [theta], indirectly from farm 1 with probability [theta][a.sub.2], or indirectly from farm 3 with probability [theta][a.sub.2]. (4)

Farm profits are approximately

Farm 1: [V.sub.1] - [theta][V.sub.1] - [theta][V.sub.1][a.sub.1] - [theta][V.sub.1][a.sub.1][a.sub.2] + [[alpha].sub.1] Ln([a.sub.1])

Farm 2: [V.sub.2] - [theta][V.sub.2] - 2[theta][V.sub.2][a.sub.2] + [[[alpha].sub.2]Ln([a.sub.2])

Farm 3: [V.sub.3] - [theta][V.sub.3] - [theta][V.sub.3][a.sub.3] - [theta][V.sub.3][a.sub.3][a.sub.2] + [[alpha].sub.3]Ln([a.sub.3]). (6)

With [[lambda].sub.n] = [[alpha].sub.n]/[[theta][V.sub.n]], the privately (pure-strategy) optimal responses satisfy:

Farm 1: [a.sup.*.sub.1] + [a.sup.*.sub.1][a.sup.*.sub.2] = [[lambda].sub.1]

Farm 2: 2[a.sup.*.sub.2] = [[lambda].sub.2]

Farm 3: [a.sup.*.sub.3] + [a.sup.*.sub.3][a.sup.*.sub.2] = [[lambda].sub.3] (7)

yielding the unique pure-strategy solution:

Farm n [member of] {1, 3}:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Notice the bias in the middle; if [[lambda].sub.i] = [[lambda].sub.2] = [[lambda].sub.3] = [lambda] [less than or equal to] 2, then [a.sup.*.sub.1] = [a.sup.*.sub.3] = 2[a.sup.*.sub.2]/[1 + [a.sup.*.sub.2]] [greater than or equal to] [a.sup.*.sub.2]. Farm-2 takes most care because the middle farm is immediately vulnerable to direct infection from the other two farms, whereas the edge farms are only immediately vulnerable to direct infection from the middle farm. Two failures must occur for one edge farm to infect the other.

Columns 2 and 3 of table 2 present unique Nash solutions under different parameter specifications. Case-by-case the parameter specifications are as in table 1, see column 2 in table 1. Case-by-case for all cases, edge farms 1 and 3 are better off under the line topology while farm 2 is worse off. A comparison of actions across tables shows that farm 2 takes more care under the line structure, while the other farms take less care. This occurs because farm 2 is now directly exposed to infection risk from both of the other farms. The edge farms in the line structure respond by free-riding more. Even though the critical farm under the line structure, farm 2, is more strongly motivated, for all of cases 1-3 the overall expected level of surplus is lower under the line structure than under the circle topology. This is largely because the line topology is more strongly connected in the following sense. Under the circle topology farm 3 can only infect farm 2 if two barriers are breached whereas under the line topology farm 3 can infect farm 2 if only one barrier is breached.

Unlike with the circle topology, cases 4 and 5 do not identify a loss in gross benefit due to a targeted subsidy. Cases 4-5 do, though, buck the broad generalization that the circle topology elicits larger surplus. Potential production has been loaded onto an edge farm, and critical farm 2 is better motivated under the line topology.

As for first-best with the line topology, use (6) to obtain the social optimality conditions as

Farm 1: [a.sup.*.sub.1] + [a.sub.1][a.sub.2] = [[lambda].sub.1]

Farm 2: [theta][V.sub.1][a.sup.*.sub.1][a.sup.*.sub.2] + 2[theta][V.sub.2][a.sup.*.sub.2] + [theta][V.sub.3][a.sup.*.sub.3] [a.sup.*.sub.2] = [[alpha].sub.2]

Farm 3: [a.sup.*.sub.3] + [a.sup.*.sub.3][a.sup.*.sub.2] = [[lambda].sub.3]. (9)

To illustrate, suppose that [V.sub.1] = [V.sub.2] = [V.sub.3] and [[alpha].sub.1] = [[alpha].sub.2] = [[alpha].sub.3], while X = [[alpha].sub.1]/[[theta][V.sub.1]]. Then any first-best solution must satisfy [a.sup.fb.sub.1] = [a.sup.fb.sub.3]. (5) The first-best requirement on the middle farm is [([a.sup.fb.sub.2]).sup.2] + (1 + 0.5[lambda])[a.sup.fb.sub.2] - 0.5[lambda] = 0 with unique positive solution [a.sup.fb.sub.2] = -0.5 - 0.25[lambda] + 0.5 [square root of [(1 + 0.5[lambda]).sup.2] + 2[lambda]].

Notice that d[a.sup.fb.sub.2]/d[lambda] [greater than or equal to] 0 and [Lim.sub.[lambda][right arrow][infinity]] [a.sup.fb.sub.2] = 1. In contrast with (8), where no protection is sometimes privately optimal, farm 2 should only make no effort when cost of effort becomes infinite. Comparing with Nash conjectures choice [a.sup.*.sub.2] = 0.5[lambda], we have [a.sup.fb.sub.2]/[a.sup.*.sub.2] = -0.5 - 1/(2[a.sup.*.sub.2]) + [1/(2[a.sup.*.sub.2])] [square root of [(1 + [a.sup.*.sub.2]).sup.2] + 4[a.sup.*.sub.2] [less than or equal to] 1, since [(1 + [a.sup.*.sub.2]).sup.2] + 4[a.sup.*.sub.2] [less than or equal to] (1 + 3[a.sup.*.sub.2]). Even though it takes most care, the middle farm does not protect enough.