3.3 Non-cooperative Equilibria
In the absence of management, however, each community maximizes its own
utility according to its welfare function, and therefore the agricultural commu-
nities each
∞
max
e-ρt
ln ai(t) - c1x2(t) dt, i =
1, . . . , n1
s.t. x(t) = a(t) — bx(t) H—x (^)— = 0,
x2 (t) + 1
n1 n2
a(t) = £ ai(t) + £ aj(t)
i=1 j=1
and the green communities each
max 0∞e-ρ [ln aj (t)
— c2x2(t) dt, j =
1, . . . , n2
x2 (t)
s.t.
X(t) = a(t) — bx(t) + 2 l = 0,
x2 (t) + 1
n1 n2
a(t) = £ ai(t) + £ aj(t)
i=1 j=1
Setting up a current value Hamiltonian and solving first order conditions for
these two problems yields:
a i(t)
and
a j(t)
2ai(t)2n1c1 x(t) — n1ai(t)
2aj(t)2n2c2x(t) — n2aj(t)
2x(t)
(x2(t) + 1)2
2x(t)
(x2(t) + 1)2
Solving for constant phosphorous loading, that is, dai/dt = 0 and daj /dt = 0,
one obtains the steady-state open-loop Nash equilibrium for total loading a:
a=
2x(t)
(x2(t)+1)2
2x(t)
П1 + П2
_ C1 C2 _
3.4 Graph and Analysis of the Nash Equilibrium Solution
Again, this solution is plotted in the (x,a)-plane together with the phase plot for
the steady-states of the lake when dx/dt = 0, given by equation (3). The inter-
section of the two curves gives the Nash equilibrium phosphorous loading solu-
tions for the state of the lake. Using the hysteretic lake value b = 0.6, ρ = 0.03
and n1 = 2, n2 = 2, c1 = 0.2 and c2 = 2, the curves intersect at (0.4485, 0.1016),
(0.7402, 0.0902) and (3.1832, 0.9976), as show in Figure 2. Point (0.7402, 0.0902)
is an unstable skiba point, i.e., a small variation in loading will cause the equi-
librium to shift to either the lower equilibrium at (0.4485, 0.1016) or the higher
11