and
maxE {ΠG}
m
∑n= 1 mj
∑i=11 li + ∑i= j mj
ln aH τ - c2 x2H τ
(34)
∑ ∑j 1 mj Ал 2λ
( ∑i= ι li + ∑ ι mj ) Vn aL τ c2xL '
- mj
where
τ The tax rate applied to phosphorous loading.
L τ denotes the low tax rate and H τ denotes the high tax rate.
l The lobbying effort of farmers in favour of a low tax rate.
m The lobbying effort of greens in favour of a high tax rate.
To find the optimum lobbying efforts l* and m*, one finds the values of li
and mj for which the first derivative is equal to zero, i.e. dE {ΠF} /dli = 0 and
dE {ΠG} /dmj = 0. (These values will maximize the expected pay-offs provided
the profit functions are concave, i.e. if their second derivatives are negative.)
Therefore we begin by solving:
dE {Πf}
~~dli
d_ ∑η= 1 li Λ 2 ʌ
dl ∑ηι l l ∑η rn (ɪn αLτ c1xLτ)
dli i=1 li + j=1 mj
1-
∑η= 1 li Ал 2 ∖ ,'
=0
∑nι / -J-∑η m Vn aHτ - c1xH τ) - li
i=11 li + j=2 1 mj
dE {Πf}
dli
d
dli
∑ηι l-
i=1 li
(∑η= 1 li + ∑η= 1 mj∙)
lnaLτ - c1x2L τ
(35)
∑n= 1 mj
(∑ηi1 li + ∑η= 1 mj )
lnaHτ - c1xH τ2 -li
=0
For simplicity, we assume that all the communities are approximately the
same size, that is, they contribute an equal amount of lobbying effort, so that
∑n= 1 li = n1l and ∑n== 1 mj = n2m. Equation (35) then becomes
dE{ΠF} d
dli
n1l
dl (n1l + n2m)
(ɪn aL τ - c1χV) + (ηιl'+m12m) (In αHτ
- c1 x2H τ - l
⇔ n1n2m lnaLτ - c1x2Lτ - ln aH τ + c1x2Hτ = (n1l + n2m)2
which expanded is
(n1l)2 + 2n2mn1l + (n2m)2 - n1n2m ln aLτ - c1x2Lτ - ln aHτ + c1x2Hτ = 0
(36)
19