while the pseudo-welfare function of the social planner becomes
(1 - T)
W(c,n,k + b, Ψ) = U(c,n) + Ψ{(c - q)Uc--tt~^ [Un - (k + b)Uc]},
where Ψ is the positive Lagrange multiplier on (19).
The optimal tax structure can be obtained by maximizing the present
discounted value of the pseudo-welfare function subject to the balance of
payments equation (16), once the production function and the time allocation
constraint (3) are taken into account.
We can state that:
Proposition 2 In an immortal small open economy that operates under per-
fect capital mobility and exhibits endogenous population growth, a residence-
based system of taxation implies that the optimal tax rate on wealth is nega-
tive in the steady state. Therefore, labor should bear the burden of taxation
necessary to finance all government outlays.
Proof. The optimal social planner problem entails20
Wn
Wc
= Fl T, + k + b,
(20a)
- di ɪn Wc = Wa + Fk
dt Wc
- ρ - δ - n,
(20b)
Fk - δ = r*. (20c)
20The partial derivatives of the pseudo-welfare function are
Wc = Uc[1 + Ψ(1 + εc)], and Wn = Un [1 + Ψ(1 + εn)]; εc and εn are general equilibrium
elasticities defined as
Ucc (1-T) [Unc - (b + k)Ucc]
εc = (c - q) -Ц- - —,--Uc------;
(1 - T)T,,
T ,2 ]
(1 - T) [Unn - (b + k)Unc]
T, Un
_, Ujn∙ ɪ (1 - T )T" n. ɪʌɪ
εn = (c - q) un + τ,2 (b + k) un [1 +
19
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