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.
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Gersovitz, M., and J.S. Hammer. 2004. "The Economic Control of
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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
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(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)
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