From the Hamiltonian first order conditions for this problem,
and
λ i(t) = 2cιx(t) + λi(t)
2x(t)
(x2(t) + 1)2
λ j (t) = 2c2x(t) + λj (t)
b+ρ-
2x(t)
(x2(t) + 1)2
(24)
(25)
The optimal steady-state tax rate is found by setting each of λ(t), λi(t) and
λj (t) equal to zero and solving to find λ*, λ* and λ*. From optimal management
equation (14):
(nι + П2)
a(t)
λ(t) =
This holds for all t, therefore this is also true for steady-state λ* and a*, i.e.
λ*
(nι + n2)
a*
To find λ*, one solves
λ i(t)
⇒ λ*
2cιx(t) + λi(t)
—2c1x(t)
b+ρ-
b+ρ-
2x(t)
(x2(t) + 1)2
2x(t)
(x2(t) + 1)2
Similarly, one finds that
λj =
— 2C2X(t)
2x(t)
(x2(t) + 1)2
(26)
Substituting back into (22), gives the optimal constant tax:
(nι + П2) + П1
a* (nι + n2)
/
—2c1x(t)
П2
(П1 + П2 )
/ ∖
— 2C2X(t)
2x(t)
(x2(t) + 1)2
∖
b + ρ —
2x(t)
(x2(t) + 1)2
)
|
/ (n1 + n2) 1 |
∖ 2x(t)(n1c1 + n2c2) |
|
a* (n1 + n2) ∖ |
b+ρ— ]) |
(nɪ + n2) _ ɪ
a* a*
14