The name is absent

Setting up a current value Hamiltonian and solving first order conditions for
each of these two problems yields:

τ(t) = aɪ) + λi(t), i = 1,...,nι                   (20)

τ (t) = _a⅛+ λj (t)’j' = 1>∙∙∙-"2                 (21)

From these two results, the following holds:
n1    1 n1

nι^{τ (t)} = § —+ ⁺ § Mt)
_i=1 ai (t)    _i=1

and
n2    1 n2

n^{2τ (t)}=§   ■ ⁺ § ^{λ (t)}

which added together gives

(nι+n2)τ (t) = § ɪ + § ɪ + § λi(t) + § λj (t)
i=1 aⁱ(t) j =1 a^j(t) i=1          j =1

Recognizing the first two terms on the right hand side as -(n1 + n2)λ(t) from

Section 3.1 results in

(nɪ + n2)τ(t) = -(nɪ + n2)λ(t) + § λi(t) + § λj(t)
i=1          j=1

⇒ τ(t) = -λ(t) + ₇---¹---ʌ § λi (t) + ----¹---- § λj (t)        (22)

(nɪ + n2) ʌɪ i^v7 (nɪ + n2) ⅛ ^{jV>        V}

i=1                        j=1

Note that in parallel with Maler et al.’s result in the single welfare function case,
the optimal tax bridges the gap between society’s shadow cost of phosphorous
loading and each community’s private cost.

3.6 Optimal Constant Tax Rate

As noted by Maler et al., it is not practical to implement a time-variable tax
and a constant tax is preferable, i.e., a tax such that dτ /dt = 0. Using this
condition and combining with equation (22), one can solve for constant λ* as
follows to then derive the constant tax rate.

τ (<   ¹ ʌ § λi(t)^ + τ ( z   ¹ ʌ § λj(t∕∣ = 0

dt ∖ (nɪ + n2)          / dt ∖ (nɪ + n2) ʌ  ³ I

i=1                                j =1

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