for an oligotrophic lake than do the green communities
n1 is the number of communities with a majority in favour of a low tax rate
n2 is the number of communities with a majority in favour of a high tax rate
n1 + n2 = n is the total number of communities that share the use of the lake
3.1 Pareto-optimal Phosphorous Loading
A benevolent politcian wishing to act on behalf of citizens will want to imple-
ment a tax that optimizes social welfare. To achieve a Pareto-optimal solution,
he needs to first find the total amount of phosphorous loading a that will maxi-
mize social welfare subject to the lake remaining in steady state. He may choose
to do this by maximizing the sum of the communities’ welfares, i.e. by solving:
max
a
— Wi + ∖' wɔ = ∑ f∞ e-ρt [lnai(t) - cιx2(t)l dt
i=1 j=1 i=1 0
— I e ρt [ln aj∙(t) — c2x2(t)] dt,
j=1 0
∞e-ρt
0
n1 n2
lnai(t) -n1c1x2(t) + ln aj (t) - n2c2x2(t)
i=1 j=1
dt,
(4)
(5)
i = 1, . . . ,n1, j = 1, . . . ,n2
x2 (t)
s.t.
rx(t) = a(t) — bx(t) + l = 0,
x2(t) + 1
n1 n2
a(t) = ai(t) + aj(t), i = 1,.. . ,n1, j = 1,. ..,n2
i=1 j=2
The current value Hamiltonian for this equation is:
n1 2 n2 2 x2(t)
Hc = Σ lnai(t)-c1n1x2(t)+y~^lnaj(t)-c2n2x2(t)+λ(t) a(t) — bx(t) +—2--—-
i=1 j=1 x (t) + 1
λ = eρtμ,
n1 n2
a(t) = ∑ ai(t) + ∑ aj(t),
i=1 j=2
i = 1,. . .,n1, j = 1,... ,n2
The first order conditions are:
dH c
dai(t)
1
ai(t)
+ λ(t) = 0, i = 1, . . . , n1
(6)