Optimal Taxation of Capital Income in Models with Endogenous Fertility



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)- - —,--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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