The name is absent



From the Hamiltonian first order conditions for this problem,

and


λ i(t) = 2cιx(t) + λi(t)


2x(t)
(x
2(t) + 1)2


λ j (t) = 2c2x(t) + λj (t)


b+ρ-


2x(t)
(x
2(t) + 1)2


(24)

(25)


The optimal steady-state tax rate is found by setting each of λ(t), λi(t) and
λ
j (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

1 + 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




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