Biosecurity and spread of an infectious animal disease.

Diminishing infectious animal disease prevalence amounts to a global public good. This has in part motivated the long tradition of public involvement in infectious animal disease control. But appropriately designed public intervention requires a clear understanding of failure in private incentives.

Two important features of communicable disease are spatial spread and the possibility of costly private actions to reduce spread. In addition, since routine biosecurity actions by growers are typically costly to verify by veterinary authorities, voluntary compliance is often essential. To this end, growers must recognize the consequences of their actions and must have adequate incentives to take socially desirable actions. The intent of this article is to provide a better understanding of private incentives to protect against the spread of infectious animal disease. We do so by developing a model that emphasizes spatial relations in infection, prevention technology, and externalities across agent payoffs.

The economics literature on public goods aspects of animal diseases is limited. Indeed, a surprisingly small body of work exists on the economics of infectious human diseases (Gersovitz and Hammer 2004). Animal diseases have been the subject of formal models (Chi et al. 2002), but the issue generally addressed is that of internal costs and not how farms inter-relate. A strongly related theme is that of controlling invasive species. Economic perspective on this issue is expanding, but has been confined largely to quarantine (Mumford 2002) and other public behaviors given an assumed exogenous stochastic dynamic process for infection (Olson and Roy 2002).

The direction in Hennessy, Roosen, and Jensen (2005) is closest to that in the present work. While theirs is not a spatial model, the biosecuring decisions involve whether to trade in young stock and the extent of production. Private benefits from trade are shown to lead to socially excessive losses from an endemic communicable disease. Furthermore, communicable disease is shown to alter the format and scale of production.

This article emphasizes protecting a farm's borders in a spatial model of private behavior to prevent the spread of infection. For farms arranged on a circle, we show how biosecuring actions are local substitutes and explain what this means for behavioral patterns under simultaneous-moves Nash equilibrium. Two insights obtained are that losses from disease-spread externalities are smaller when production is concentrated, and subsidies to small producers may exacerbate overall disease losses. We also consider a line topology for farms in order to show how the model can be adapted, and how locational asymmetries can affect incentives to protect farm boundaries. We find that more centrally located farms, which are more vulnerable, will take more care than other farms, all else equal. But they may not adopt enough protective measures for the social good. More isolated farms take less care, all else equal, but over protect.

Circle Topology

A region has N [greater than or equal to] 3 farms labeled n [member of] {1, 2, ..., N} = [[OMEGA].sub.N], where each seeks to protect potential production to the value of [V.sub.n] > 0. The N [greater than or equal to] 3 assumption on the extent of the outbreak is convenient, but could be relaxed at only the cost of substantially more tedious algebra. An infected farm loses all of [V.sub.n]. The farms are located on a circle (see figure 1). The circle topology was chosen because farms are located symmetrically on it. It enables a consideration of many of the article's main points but not the role of location asymmetry. Location asymmetry issues are examined in a separate section using a line spatial structure.

[FIGURE 1 OMITTED]

Infection is rare and can enter the region at some farm with probability [theta], where each farm is equally likely to be the first infected. By "rare" we mean it is almost certainly true that at most one farm inside the region becomes infected from outside the region at any time. The first farm to be infected within a region is labeled as the "originating farm." It will also be assumed that public authorities intervene to suppress a disease outbreak after the disease spreads to no more than the most proximate two farms (clockwise), if indeed it spreads at all. We allow infection to occur only in one (arbitrarily, the clockwise) direction to simplify algebra. Condition N [greater than or equal to] 3 was imposed to avoid double-counting on the circle, where the disease spreads clockwise back to the originating farm. The case where infection occurs in both clockwise and anticlockwise directions is available upon request.

Farm-level care taking is modeled through actions taken at the farm border. If infection has reached its direct anticlockwise neighbor, the nth farm will become infected with probability [a.sub.n]. The grower can change this probability at a cost. The nth farm is said to take comparatively less care when the value of [a.sub.n] is comparatively high. Before representing the cost of reducing probability [a.sub.n], consider the expected revenue loss.

Farm 1 may be the first infected, where the probability of first infection is [theta]. Or it may contract the disease from its neighbor. Anticlockwise is farm N. If farm N is infected first, then farm 1 becomes infected through farm N with probability [theta][a.sub.1]. The originating farm's probability does not enter the calculation because we assume a farm has no incentive to try preventing the disease from exiting the farm. Farm N - 2 may also be the source of infection to farm 1, where the probability this occurs is [theta][a.sub.1][a.sub.N]. Since infection is rare, the overall probability that farm 1 is infected is approximately [theta] + [theta][a.sub.1] + [theta][a.sub.1][a.sub.N].

In order to develop a general expression for each farm's infection risk, define [n + i] = n + i - zN where z is an integer chosen such that [n + i] [member of] [[OMEGA].sub.N]. That is, clock algebra (also called modular algebra) is used. For the nth farm, the infection probability is approximately

[[omega].sub.n] = [theta] + [theta][a.sub.n] + [theta][a.sub.n][a.sub.[n-1]]. (1)

The overall expected loss in revenue to the region is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, a prevention technology exists. Farms differ in their capacity to protect themselves, and the protection cost at entry probability level [a.sub.n] is [C.sup.n]([a.sub.n]), a decreasing function. Private profit to a farm is [L.sub.n] = [V.sub.n] - [V.sub.n] [[omega].sub.n] - [C.sup.n]([a.sub.n]), while the overall expected profit to the region is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The region is assumed to produce a small share of overall market output, so that consumer surplus may be ignored. Thus, L represents social surplus. Since actions by farm [n - 1] enter [L.sub.n] through [[omega].sub.n], externalities exist and one should not expect market competition to support the maximization of L.

In order to better understand protection incentives, the nth farm's cost of protection is posited as -[[alpha].sub.n]Ln([a.sub.n]) where [[alpha].sub.n] > 0. This ensures that the cost of not protecting at all is -[[alpha].sub.n]Ln(1) = 0, while the cost of complete protection--where [a.sub.n] = 0--is infinite. This, we believe, reflects reality to the extent that not protecting at all requires no expenditure, complete protection is prohibitively expensive, and the protection cost increases with the extent of protective action.

Private Incentives and Responses

The nth farm's profit is

[L.sub.n] = [V.sub.n] - [V.sub.n][[omega].sub.n] + [[alpha].sub.n] Ln([a.sub.n]), n [member of] [[OMEGA].sub.N]. (2)

Substituting (1) into (2) and differentiating, we obtain [[partial derivative].sup.2][L.sub.n]/[partial derivative][a.sub.k] [partial derivative][a.sub.s] [less than or equal to] 0 [for all] k, s [member of] [[OMEGA].sub.N], k [not equal to] s. Thus, an increase in care by the nth farm reduces the marginal net benefit of own care by farms other than the nth. This leads to the following:

PROPOSITION 1. If the probability of infection is as given in equation (1), then farm biosecurity actions to prevent the spread of infection are strategic substitutes.

This observation shows that the game being played is not of the type involving global strategic complementarities (Vives 2005). (1) Given substituting strategic interactions, the possibility exists for a public intervention to do harm by indirectly discouraging important actions while directly encouraging less important actions.

With Nash conjectures on payoffs (2), protective actions are chosen as solutions to

[partial derivative][L.sub.n]/[partial derivative][a.sub.n] = - [theta] [V.sub.n] (1 + [a.sub.[n-1]]) + [[alpha].sub.n]/[a.sub.n] = 0. (3)

Solutions are denoted by [a.sup.*.sub.n]. The system is illustrated for N = 3, so that profits are

Farm 1: [V.sub.1] - (1 + [a.sub.1] + [a.sub.1][a.sub.3])[theta][V.sub.1] + [[alpha].sub.1] Ln([a.sub.1])

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

Farm 3: [V.sub.3] - (1 + [a.sub.3] + [a.sub.2][a.sub.3])[theta] [V.sub.3] + [[alpha].sub.3]Ln([a.sub.3]). (4)

From (3), ith farm Nash private conjectures are given by [a.sup.*.sub.i] + [a.sup*.sub.i][a.sup.*.sub.[i-1]] = [[lambda].sub.i] where [[lambda].sub.i] = [[alpha].sub.i]/[[theta][V.sub.i]]. The unique pure strategy interior solution is (2)

[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.

On the edge farms under [V.sub.1] = [V.sub.2] = [V.sub.3] and [[alpha].sub.1] = [[alpha].sub.2] = [[alpha].sub.3], first-best actions are [a.sup.fb.sub.1] = [a.sup.fb.sub.3] = 2[lambda](1 - 0.5[lambda] + [square root of 1 + 3[lambda] + 0.25[[lambda].sup.2]), so that [a.sup.fb.sub.1] [less than or equal to] [a.sup.*.sub.1] if 2.25[[lambda].sup.2] [less than or equal to] 0.25[[lambda].sup.2], a false statement. Thus, the edge farms protect too much even though they take less care than the middle farm. Also, observe that [a.sup.fb.sub.1] and [a.sup.fb.sub.3] are increasing in [lambda] along [[lambda].sub.1] = [[lambda].sub.2] = [[lambda].sub.3] = [lambda], with [a.sup.fb.sub.1] = [a.sup.fb.sub.3] = 1 at [lambda] = 4/3. By contrast with the middle farm, it may be socially optimal for edge farms to make no effort because the middle farm's action is a substitute, and the middle farm takes appropriate care in first-best.

Columns 4 and 5 of table 2 provide actions and social welfare under first-best, to be compared with those given in columns 2 and 3. In cases 1-3, welfare is marginally larger under first best. In cases 4-5, the welfare gap is larger because the consequences of poor incentives for farm 2 are more pronounced. In all cases, farm 2 is worse off under first-best than under Nash behavior while the other two farms are better off. This is because farms 1 and 3 can afford to take less care and farm 2 is no longer allowed to free-ride. First-best does not Pareto dominate the unique pure strategy Nash equilibrium, and farm 2 may resist attempts to achieve first-best unless it is provided with additional compensation.

Discussion

A terse illustrative model of agricultural biosecurity actions under spatial disease spillovers was presented, where only farm location and production scale were articulated. Three observations were made. The nature of spatial interactions matters as it determines the extent of incentives to free ride on neighbors' actions. Intensive production on some farms could reduce the proportion of potential production lost to disease in a region by strengthening private incentives to protect. Subsidies targeted to smaller production lots may, depending upon circumstances, reduce overall surplus.

References

Chi, J., A. Weersink, J.A. VanLeeuwen, and G.P. Keefe. 2002. "The Economics of Controlling Infectious Diseases on Dairy Farms." Canadian Journal of Agricultural Economics 50:237-56.

Gersovitz, M., and J.S. Hammer. 2004. "The Economic Control of Infectious Diseases." Economic Journal 114:1-27.

Hennessy, D.A., J. Roosen, and H.H. Jensen. 2005. "Infectious Disease, Productivity, and Scale in Open and Closed Animal Production Systems." American Journal of Agricultural Economics 87:900-17.

Mumford, J.D. 2002. "Economic Issues Related to Quarantine in International Trade." European Review of Agricultural Economics 29:329-48.

Olson, L.J., and S. Roy. 2002. "The Economics of Controlling a Stochastic Biological Invasion." American Journal of Agricultural Economics 84:1311-16.

Vives, X. 2005. "Complementarities and Games: New Developments." Journal of Economic Literature 43:437-79.

(1) By contrast, it can be shown that actions to protect against the entry of a disease into a region are strategic complements.

(2) Write [a.sup.*.sub.3] = [[lambda].sub.3]/(1 + [a.sup.*.sub.2]), [a.sup.*.sub.2] = [[lambda].sub.2]/(1 + [a.sup.*.sub.1]), and [a.sup.*.sub.i] = [[lambda].sub.1]/(1 + [a.sup.*.sub.3]). Substitute in successively to eliminate all but [a.sup.*.sub.1], and simplify to obtain a quadratic equation. Just one root is positive, that in (5). Solve for [a.sup.*.sub.2] and [a.sup.*.sub.3] in the same manner.

(3) Due to confidentiality concerns, datasets identifying both spatial proximity and biosecurity actions taken may be difficult to obtain for estimation purposes.

(4) In the circle topology with N = 3, there are also nine ways that farms can become infected. Each can become infected directly, or from its anticlockwise neighbor, or from its clockwise neighbor through its anticlockwise neighbor.

(5) To demonstrate this, sum the payoffs in (6). Fixing [a.sub.2] at any admissible value notice that the sum of surpluses--from (6)--is symmetric and concave in the choices of [a.sub.1] and [a.sub.3]. This means that any admissible choice ([a.sup.fb.sub.1], [a.sup.fb.sub.3]) such that [a.sup.fb.sub.1] [not equal to] [a.sup.fb.sub.3] delivers lower expected welfare than ([[??].sup.fb.sub.1], [[??].sup.fb.sub.3]) = (([[??].sup.fb.sub.1] + [[??].sup.fb.sub.3])/2, ([[??].sup.fb.sub.1] + [[??].sup.fb.sub.3])/2), a contradiction since convexity of the action space ensures the average is admissible.

David A. Hennessy is Professor at the Department of Economics and Affiliate of the Center for Agricultural and Rural Development, Iowa State University. Comments and suggestions from Paul Preckel and Glenn Sheriff are appreciated.

This article was presented in a principal paper session at the AAEA annual meeting (Portland, OR, July 2007). The articles in these sessions are not subjected to the journal's standard refereeing process. Table 1. Actions and Welfare for Three Farms on Circle, [theta] = 0.1

([V.sub.1], [V.sub.2], [V.sub.3])

([[alpha].sub.1], [[alpha].sub.2], Case [[alpha].sub.3]) 1. Baseline (10, 10, 10)

(0.3, 0.3, 0.3) 2. More dispersion (6, 18, 6)

(0.3, 0.3, 0.3) 3. Add production (10, 20, 10)

(0.3, 0.3, 0.3) 4. Backyard production (1, 1, 28)

(0.03, 0.03, 2) 5. Subsidy, farm 1 (1, 1, 28)

(0.02, 0.03, 2)

Nash Actions

([a.sup.*.sub.1], [a.sup.*.sub.2], Case [a.sup.*.sub.3]) 1. Baseline (0.242, 0.242, 0.242) 2. More dispersion (0.346, 0.124, 0.445) 3. Add production (0.237, 0.121, 0.268) 4. Backyard production (0.191, 0.252, 0.571) 5. Subsidy, farm 1 (0.128, 0.266, 0.564)

Nash Welfare

([L.sub.1], [L.sub.2], Case [L.sub.3]), L 1. Baseline (8.274, 8.274, 8.274)

L = 24.822 2. More dispersion (4.782, 15.273, 4.857)

L = 24.912 3. Add production (8.267, 17.067, 8.304)

L = 33.639 4. Backyard production (0.82, 0.829, 22.078)

L = 23.727 5. Subsidy, farm 1 (0.839, 0.830, 22.055)

L = 23.724 Note: Column 2 provides value parameters and cost technology coefficients for the case at hand. Column 3 identifies equilibrium private actions, as represented by entry probabilities at the farm border. Column 4 states farm and total profits. Table 2. Actions and Welfare for Three Farms on Line, [theta] = 0.1

Nash Actions

([a.sup.*.sub.1], [a.sup.*.sub.2], Case [a.sup.*.sub.3]) 1. Baseline (0.261, 0.15, 0.261) 2. More dispersion (0.462, 0.083, 0.462) 3. Add production (0.279, 0.075, 0.279) 4. Backyard production (0.261, 0.15, 0.621) 5. Subsidy, farm 1 (0.174, 0.15, 0.621)

Nash Welfare

([L.sub.1], [L.sub.2], Case [L.sub.3]), L 1. Baseline (8.297, 8.131, 8.297)

L = 24.725 2. More dispersion (4.868, 15.155, 4.868)

L = 24.891 3. Add production (8.317, 16.923, 8.317)

L = 33.557 4. Backyard production (0.83, 0.813, 22.248)

L = 23.89 5. Subsidy, farm 1 (0.845, 0.813, 22.248)

L = 23.906

First-Best Actions

([a.sup.fb.sub.1], [a.sup.fb.sub.2], Case [a.sup.fb.sub.3]) 1. Baseline (0.268, 0.118, 0.268) 2. More dispersion (0.466, 0.072, 0.466) 3. Add production (0.281, 0.066, 0.281) 4. Backyard production (0.296, 0.014, 0.705) 5. Subsidy, farm 1 (0.197, 0.014, 0.705)

First-Best Welfare

([L.sub.1], [L.sub.2], Case [L.sub.3]), L 1. Baseline (8.305, 8.123, 8.305)

L = 24.734 2. More dispersion (4.871, 15.152, 4.871)

L = 24.894 3. Add production (8.32, 16.92, 8.32)

L = 33.56 4. Backyard production (0.833, 0.768, 22.5)

L = 24.102 5. Subsidy, farm 1 (0.848, 0.769, 22.5)

L = 24.116