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Therefore:


*   (nɪ + n - 1)

(27)


τ =--------

a*

is the constant tax that will achieve the Pareto-optimal amount of phosphorous
loading when each community acts to maximize its welfare in a non-cooperative
way.

3.7 Private Equilibrium with Tax

To determine the impact of the tax on the non-cooperative phosphorous loading
in steady state and on the state of the lake, λ(t), λ
i(t) and λj∙(t) is substituted
into the steady-state tax equation. Therefore substituting (14), (24) and (25)
back into (23) one obtains:

-(nɪ + n )α(t)
α
2(t)


1        n1                                        2x(t)

2---------Г I 2cix(t) + λi(t) b + p — ------------2 I

(nɪ + П2) ЁЛ              L       (x2(t) + 1)2J√

+ g 2cCx x(t) + λj (t) b + P 2 2/,ʌ(,) -, x2 λ)
j=Λ               L        (x2(t) + 1) )

. ./,ʌ          2 /,ʌ 2/,λ(n1c1 + n2c2)

a(t) = -Cx(t)α (t)---------

(nɪ + n2)2

a2(t) Γ,           Cx(t)    n n^ ʌ   ^2   z Λ z ʌ

2-------πr b + P--λ   g λi(t) + g λj(t)  (C8)

(nɪ + n2)2[       (χ2(t) + i)2J y⅛ iu  j=i jɪ !

From (C0)

λi(t) =τ - oi(tyand λj(t) =τ - aj⅛t              (C9)

By substituting into (C8) one obtains:

a(t) = -Cx(t)a2(i)(nιc1 + n2,c2>

(nɪ + n2)2

- √a2(i[b+P —Cx':/I 2
(nɪ + n2)2           (x2(t) + 1)2

<j∙(t) = -Cx(t)tt)(nn1+^>

- Γja2(t[b+P -   Cx(t) 2

(nɪ + n2)2           (x2(t) + 1)2


n1

∑(

i=1 v


⅛) )+g (τ - a⅛)


n1

i=1


1
α
i(t)


n2
+ g

j=1


Recognizing the last two terms as -λ(n1 + n2) and substituting -(nɪ + n2)∕α(t)

15



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