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:

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)




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