The name is absent

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:

— Wi + ∖' wɔ  = ∑ f∞ e^-ρt [lnai(t) - cιx²(t)l dt

i=1       j=1             i=1 ⁰

— I e ^ρt [ln a_j∙(t) — c₂x²(t)] dt,
j=1 ⁰

i = 1, . . . ,n1, j = 1, . . . ,n2

x² (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)

H^c = Σ lna_i(t)-c₁n₁x²(t)+y~^lnaj(t)-c₂n₂x²(t)+λ(t) a(t) — bx(t) +—2--—-

i=1                       j=1                                               x (t) + 1

λ = e^ρtμ,

n1                  n2

^a(t) = ∑ ^ai^{(t) +} ∑ ^aj^(t),
i=1        j=2

i = 1,. . .,n1, j = 1,... ,n2

The first order conditions are:

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