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

F_k - δ = r*.                           (20c)

²⁰The partial derivatives of the pseudo-welfare function are

W_c = U_c[1 + Ψ(1 + ε_c)], and W_n = U_n [1 + Ψ(1 + ε_n)]; ε_c and ε_n are general equilibrium
elasticities defined as

Ucc   (1-T) [Unc - (b + k)Ucc]

^εc = (^{c -} q) -Ц- ^- —,--U_c------;

_, U^jn∙ ɪ (1 - T )T"  n. ɪʌɪ

^εn = ^{(c -} q) un   + τ,2      ^{(b + k)} un [^{1 +}

19

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