for λ yields
a(t)
-2x(t)a2(t)<n1 c1 ÷ 2
(ni + n2)2
α2(t)
(ni + n2)2
b+ρ-
2x(t)
(x2(t) + 1)2J [
(ni + n2)2
(ni + n2)τ--—---
a(t)
⇒ <a(t)
-2x(t)α2(t)(ni c1 ÷ '2
(ni + n2)2
α2(t)
(ni + n2)
b+ρ-
2x(t)
(x2(t) + 1)2.
[a(t)τ - (ni + n2)]
Assuming steady-state loading, i.e. <a(t) = 0:
* -2α*x(t)(n1c1 + n2c2)
(ni + n2)2 (ni + n2 )
b+ρ-
2x(t)
(x2(t) + 1)2
[a*τ - (ni + n2)]
=0
which implies a* = 0 or
2a*x(t)
(nici + n2c2)
(ni + n2)2
b+ρ-
2x(t) ' a*τ
(x2(t) + 1)2 [(ni + n2)
=0
Solving for a* gives:
*
a*
2x(t)
b+ρ-
2x(t)
(x2 (t)+i)2
(cini+c2n2)
(ni+n2)2
I__Tî____
(n1+n2)
b+ρ-
2x(t)
(x2(t) + i)2
(30)
which is the Nash equilibrium phosphorous loading when the tax is applied.
3.8 Graph and Analysis of the New Nash Equilibrium Solution
The plot of the solution is overlayed onto the Pareto-optimal steady-state load-
ing curve from section 3.1 and shown in Figure 3. Note that when the optimal
tax is applied, the Nash equilibrium loading intersects the lake dynamics in
exactly the same point (x*, a*) = (0.3472, 0.1007) as the Pareto-optimal load-
ing. Moreover, there is now only one Nash equilibrium, and it is oligotrophic
in accordance with society’s preferences. This is not surprising given that the
objective of the tax is to bring phosphorous loading to the same level as that
which would be achieved under optimal management.
16
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