We recognise this as a second-order polynomial of the form dX2 + eX + f = 0,
where
X = n1l
d= 1,
e = 2n2m and
f = (n2m)2 - n1n2m lnaL τ - c1x2L τ - lnaH τ + c1x2H τ ,
which has two roots X of the form
X=
-e - √e2 - 4df
2d
Substituting for d, e and f , and keeping only the positive root of the polynomial,
we obtain the farmers’ optimal lobbying effort:
l=
-n2m
nɪ
1
H—
n1
nn∙2m,m (lnaLτ - cιxLτ - lnант + cιx2Hτ)
(37)
Similarly, from equation (34) we obtain:
n2n1l lnaHτ - c2x2Hτ - lnaLτ + x2Lτ = (n1l + n2m)2 ,
(38)
of which the positive real root gives us:
m=
-n1 l
П2
1
+--
n2
nП2П11 (ln ан τ - C2x2H τ - ln aLτ + c2xL τ)
(39)
We note that l andm are dependent on each other. By noticing, from equations
(36) and (38), that
l ln ан τ
- c2χH τ
- ln aLτ + c2x2Lτ^ = m (ln alτ
- cixL τ - ln ан τ + cixH τ
and by substituting back into equations (37) and (39), we can express the
respective optmimal lobbying effort of farmers and ‘greens’, l* and m* as
l* = nɪn (lnaLτ - cιx2Lτ - lnaHτ + cιx∏τ)2 (lnahτ - C2X2Hτ - lnaLτ + C2X2Lτ)
[nɪ (ln aLτ - cιX2Lτ - ln aHτ + cιX2Hτ) + n (lnaHτ - C2X2H τ - ln aLτ + c2xLτ)]2
(40)
and
m* = nɪn (ln aLτ - C1X2Lτ - ln aHτ + C1X2lτ) (ln aHτ - C2X21 τ - ln aLτ + C2X2Lτ)2
[nɪ (ln aLτ - CiX2Lτ - lnaHτ + cιx∏τ) + n2 (ln ahτ - c2x2Hτ - lnaLτ + c2xLτ)]2
(41)
In the following section, we use these results to study the impact of the
lobbying efforts on the optimal tax policy derived in section 3.6 and depicted
in section 3.8.
20