Pay-as-You-Go Social Security and Educational Subsidy in an Overlapping Generations Model with Endogenous Fertility and Endogenous Retirement

Hung-Ju Chen; Koichi Miyazaki

doi:10.1515/bejm-2021-0046

Article

Pay-as-You-Go Social Security and Educational Subsidy in an Overlapping Generations Model with Endogenous Fertility and Endogenous Retirement

Hung-Ju Chen and Koichi Miyazaki

Published/Copyright: December 3, 2021

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From the journal The B.E. Journal of Macroeconomics Volume 22 Issue 2

Abstract

This study analytically investigates the effects of pay-as-you-go social security and educational subsidies on the fertility rate, retirement age, and GDP per capita growth rate in an overlapping generations model, where parents invest resources toward their children’s human capital. We find that an old agent retires fully when his or her labor productivity is low and retires later when the labor productivity is high. Under the unique balanced-growth-path (BGP) equilibrium, when an old agent is still engaged in work, tax rates are neutral to the fertility rate, higher tax rates encourage him or her to retire earlier, a higher social security tax rate depresses the GDP per capita growth rate, and a higher tax rate for educational subsidies can accelerate growth. However, when an old agent fully retires, higher tax rates increase the fertility rate, a higher social security tax rate lowers the GDP per capita growth rate, and a higher tax rate for educational subsidies boosts growth. Additionally, if an old agent’s labor productivity increases, the fertility rate also increases. We also conduct numerical simulations and analyze how an old agent’s labor productivity affects the retirement age, fertility rate, and GDP per capita growth rate under the BGP equilibrium.

Keywords: pay-as-you-go social security; educational subsidy; fertility; endogenous retirement; GDP per capita growth rate

JEL Classification: H55; J26; J13; I25

Corresponding author: Koichi Miyazaki, School of Economics, Hiroshima University, 1-2-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 7398525, Japan, E-mail: komiya@hiroshima-u.ac.jp

Acknowledgement

We thank the editor and two anonymous referees of this journal for their helpful comments and suggestions, which helped us to considerably improve this paper. Chen acknowledges financial support provided by the Ministry of Science and Technology of Taiwan (grant number: MOST-110-2410-H-002-197-MY3). Miyazaki acknowledges financial support by JSPS KAKENHI Grant Number 19K01631.

Appendix A

A.1 Proof of Proposition 3.2

Proof

First, we show there exists a unique ϕ ̂ * ∈ ( 0,1 ) that satisfies Eq. (19) with ϕ ̂ * = ϕ t − 1 = ϕ t .

Let LHS ̂ ( ϕ t ) and RHS ̂ ( ϕ t − 1 ) be the left- and right-hand sides of Eq. (19), respectively. Then, LHS ̂ ( ϕ t ) is strictly increasing in ϕ _t, LHS ̂ ( 0 ) = 0 , and lim ϕ t → 1 LHS ̂ ( ϕ t ) = + ∞ . RHS ̂ ( ϕ t − 1 ) is strictly decreasing in ϕ _t−1, because χ > γ ( 1 + r ̄ ) 1 + σ from Eq. (22). Additionally, RHS ̂ ( 1 ) = τ 2 1 − τ 2 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ ≥ 0 and is finite. This implies that RHS ̂ ( 0 ) > 0 . These facts show there is a unique ϕ ̂ * ∈ ( 0,1 ) so that LHS ̂ ( ϕ * ) = RHS ̂ ( ϕ * ) .

Given that ϕ ̂ * , l ̂ o * is uniquely determined by Eq. (21). Therefore, there exists a unique BGP equilibrium where an old agent supplies strictly positive amount of labor. □

A.2 Proof of Proposition 3.3

Proof

Taking the derivative of l ̂ o * with respect to τ ₁, we have:

∂ l ̂ o * ∂ τ 1 = − 1 1 + σ + γ 1 1 − τ 2 1 + σ − γ ( 1 + r ̄ ) χ − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ 1 1 − τ 2 q η ( 1 − τ 1 − τ 2 ) ( 1 − ϕ ̂ * ) ( 1 − η ) w ̄ η

(36) × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ + ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 1 − τ 2 × q η w ̄ 1 − η η 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ × η 1 − τ 1 − τ 2 1 − ϕ ̂ * η − 1 − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 1 .

To know the sign of ∂ l ̂ o * ∂ τ 1 , we need to know ∂ ϕ ̂ * ∂ τ 1 .

Since ϕ ̂ * satisfies:

(37) ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) ( 1 − τ 1 − τ 2 ) n ̂ * = τ 2 ( 1 − q n ̂ * ) + 1 n ̂ * τ 2 1 − τ 2 1 − τ 1 − τ 2 1 + σ + γ × χ ( 1 + σ ) − γ ( 1 + r ̄ ) ( 1 − ϕ ̂ * ) η θ q η ( 1 − τ 1 − τ 2 ) 1 − η w ̄ η − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ ,

∂ ϕ ̂ * ∂ τ 1 = ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ × χ ( 1 + σ ) − γ ( 1 + r ̄ ) ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ q η 1 − η ( 1 − τ 1 − τ 2 ) n ̂ * ( 1 − ϕ ̂ * ) 2 + 1 n ̂ * τ 2 1 − τ 2 × ( 1 − τ 1 − τ 2 ) 1 − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η − 1 θ q η 1 − η w ̄ η .

From this:

(38) 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 1 = 1 1 − ϕ ̂ * ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ × χ ( 1 + σ ) − γ ( 1 + r ̄ ) ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ q η 1 − η n ̂ * 1 − ϕ ̂ * + 1 n ̂ * τ 2 1 − τ 2 ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η < 1 1 − ϕ ̂ * ,

where we use:

ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ × χ ( 1 + σ ) − γ ( 1 + r ̄ ) ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * τ 2 1 − τ 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ < ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * < q η 1 − η n ̂ * 1 − ϕ ̂ * + 1 n ̂ * τ 2 1 − τ 2 ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η .

Therefore, the last term in Eq. (36) is − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 1 < 0 , which implies ∂ l ̂ o * ∂ τ 1 < 0 .

Next, taking the derivative of l ̂ o * with respect to τ ₂, we have:

(39) ∂ l ̂ o * ∂ τ 2 = − τ 1 ( 1 − τ 2 ) 2 1 1 + σ + γ χ ( 1 + σ ) − γ ( 1 + r ̄ ) χ − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ × τ 1 ( 1 − τ 2 ) 2 q η ( 1 − τ 1 − τ 2 ) ( 1 − ϕ ̂ * ) ( 1 − η ) w ̄ η 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 1 − τ 2 q η w ̄ 1 − η η × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ × η 1 − τ 1 − τ 2 1 − ϕ ̂ * η − 1 − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 2 .

From Eq. (37):

(40) ∂ ϕ ̂ * ∂ τ 2 = ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * + ( 1 − q n ̂ * ) + 1 n ̂ * ( 1 − τ 1 − τ 2 ) − η [ 1 − τ 1 − τ 2 − τ 2 ( 1 − τ 2 ) ( 1 − η ) ] ( 1 − τ 2 ) 2 χ ( 1 + σ ) − γ ( 1 + r ̄ ) 1 + σ + γ ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ q η 1 − η ( 1 − τ 1 − τ 2 ) n ̂ * ( 1 − ϕ ̂ * ) 2 + 1 n ̂ * τ 2 1 − τ 2 × ( 1 − τ 1 − τ 2 ) 1 − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η − 1 θ q η 1 − η w ̄ η .

Note that the denominator of Eq. (40) is strictly positive. Therefore, ∂ ϕ ̂ * ∂ τ 2 > 0 if and only if the numerator of Eq. (40) is strictly positive. Including n ̂ * into ( 1 − q n ̂ * ) − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ , we have:

1 − τ 1 ( 1 − τ 2 ) 2 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ .

Since 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ > 0 , ( 1 − q n ̂ * ) − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ > 0 if and only if τ 1 < ( 1 − τ 2 ) 2 . If τ 1 < ( 1 − τ 2 ) 2 , the numerator of Eq. (40) is strictly positive, which implies ∂ ϕ ̂ * ∂ τ 2 > 0 . Given ∂ ϕ ̂ * ∂ τ 2 > 0 , consider the last two terms in Eq. (39), that is:

− ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 ( 1 − τ 2 ) 2 q η ( 1 − τ 1 − τ 2 ) ( 1 − ϕ ̂ * ) ( 1 − η ) w ̄ η × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 1 − τ 2 q η w ̄ 1 − η η 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ × η 1 − τ 1 − τ 2 1 − ϕ ̂ * η − 1 − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 2 = − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 1 − τ 2 q η ( 1 − ϕ ̂ * ) ( 1 − η ) w ̄ η ( 1 − τ 1 − τ 2 ) η − 1 × 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ 1 − τ 1 − τ 2 1 − τ 2 − η − ( 1 − η ) σ θ ( 1 + σ + γ ) q 1 χ + 1 1 + r ̄ τ 1 1 − τ 2 q η w ̄ 1 − η η 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ η × 1 − τ 1 − τ 2 1 − ϕ ̂ * η − 1 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 2 .

These are strictly negative if 1 − τ 1 − τ 2 1 − τ 2 > η or τ ₁ < (1 − η)(1 − τ ₂), given ∂ ϕ ̂ * ∂ τ 2 > 0 . Since the first term in Eq. (39) is strictly negative, if 0 < τ 1 < min ( 1 − τ 2 ) 2 , ( 1 − η ) ( 1 − τ 2 ) , then ∂ l ̂ o * ∂ τ 2 < 0 . Inserting τ ₁ = 0 into Eq. (39), ∂ l ̂ o * ∂ τ 2 = 0 . □

A.3 Proof of Proposition 3.4

Proof

To investigate the effects of τ ₁ and τ ₂ on g ̂ * , we focus on:

G ̂ * ≔ 1 − τ 1 − τ 2 1 − ϕ ̂ * .

Taking the derivative of G ̂ * with respect to τ ₁, we have:

∂ G ̂ * ∂ τ 1 = − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 1 < 0 ,

which was shown for Eq. (38).

Taking the derivative of G ̂ * with respect to τ ₂, we have:

∂ G ̂ * ∂ τ 2 = − 1 1 − ϕ ̂ * + 1 − τ 1 − τ 2 ( 1 − ϕ ̂ * ) 2 ∂ ϕ ̂ * ∂ τ 2 = 1 1 − ϕ ̂ * × − 1 + ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * + ( 1 − q n ̂ * ) + 1 n ̂ * ( 1 − τ 1 − τ 2 ) − η [ 1 − τ 1 − τ 2 − τ 2 ( 1 − τ 2 ) ( 1 − η ) ] ( 1 − τ 2 ) 2 × χ ( 1 + σ ) − γ ( 1 + r ̄ ) 1 + σ + γ ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q × 1 + χ 1 + r ̄ τ 1 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ q η 1 − η n ̂ * 1 − ϕ ̂ * + 1 n ̂ * τ 2 1 − τ 2 × ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η .

Recall that, if τ 1 ≥ ( 1 − τ 2 ) 2 , ( 1 − q n ̂ * ) − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ ≤ 0 . Furthermore, if τ 1 ≥ ( 1 − τ 2 ) 2 :

1 n ̂ * ( 1 − τ 1 − τ 2 ) − η [ 1 − τ 1 − τ 2 − τ 2 ( 1 − τ 2 ) ( 1 − η ) ] ( 1 − τ 2 ) 2 χ ( 1 + σ ) − γ ( 1 + r ̄ ) 1 + σ + γ × ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η ≤ 1 n ̂ * τ 2 1 − τ 2 ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η .

Therefore, since ϕ ̂ * < 1 :

ϕ ̂ * q η ( 1 − ϕ ̂ * ) ( 1 − η ) n ̂ * + ( 1 − q n ̂ * ) + 1 n ̂ * ( 1 − τ 1 − τ 2 ) − η [ 1 − τ 1 − τ 2 − τ 2 ( 1 − τ 2 ) ( 1 − η ) ] ( 1 − τ 2 ) 2 × χ ( 1 + σ ) − γ ( 1 + r ̄ ) 1 + σ + γ ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η − 1 n ̂ * 1 ( 1 − τ 2 ) 2 ( 1 − η ) σ ( 1 + σ + γ ) q 1 + χ 1 + r ̄ τ 1 1 − ( 1 − η ) σ 1 + σ + γ 1 + χ 1 + r ̄ q η 1 − η n ̂ * 1 − ϕ ̂ * + 1 n ̂ * τ 2 1 − τ 2 ( 1 − τ 1 − τ 2 ) − η 1 + σ + γ [ χ ( 1 + σ ) − γ ( 1 + r ̄ ) ] η ( 1 − ϕ ̂ * ) η θ q η 1 − η w ̄ η < 1 ,

which leads to ∂ G ̂ * ∂ τ 2 < 0 . Therefore, if a slight increase in τ ₂ accelerates the growth rate under the BGP equilibrium, then τ 1 < ( 1 − τ 2 ) 2 must be satisfied. □

A.4 Proof of Lemma 3.1

Proof

From Eq. (31):

n t + 1 n t = ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ n t + 1 ( 1 − q n t + 2 ) 1 + r ̄ h t + 2 h t + 1 ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ n t ( 1 − q n t + 1 ) 1 + r ̄ h t + 1 h t

holds. Let k ̃ = n t + 1 n t and g ̃ = h t + 1 h t under the BGP equilibrium. Then, from the above equation:

n t + 1 n t = ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ k ̃ n t 1 − q k ̃ 2 n t 1 + r ̄ g ̃ ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ n t ( 1 − q k ̃ n t ) 1 + r ̄ g ̃ = k ̃ , ⇒ ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ k ̃ n t 1 − q k ̃ 2 n t 1 + r ̄ g ̃ = k ̃ ( 1 − τ 1 − τ 2 ) w ̄ + τ 1 w ̄ n t ( 1 − q k ̃ n t ) 1 + r ̄ g ̃ , ⇒ ( 1 − k ̃ ) ( 1 − τ 1 − τ 2 ) w ̄ = τ 1 w ̄ k ̃ n t 1 + r ̄ g ̃ q n t k ̃ ( k ̃ − 1 )

must hold. Since 1 − τ ₁ − τ ₂ > 0 and all parameters are strictly positive, this equation holds if and only if k ̃ = 1 . This completes the proof. □

A.5 Proof of Lemma 3.2

Proof

Since at equilibrium, n _t > 0 for all t, dividing both sides of Eq. (31) by n _t, we have:

1 − σ ( 1 − η ) q ( 1 + σ ) 1 n t = σ ( 1 − η ) 1 + σ τ 1 θ ( 1 − q n t + 1 ) ( 1 + r ̄ ) ( 1 − τ 1 − τ 2 ) q × η ( 1 − τ 1 − τ 2 ) q n t + ( 1 − η ) τ 2 ( 1 − q n t ) n t w ̄ 1 − η η .

Moreover, dividing both sides by the last term in the right-hand side of the above equation, we obtain:

1 − σ ( 1 − η ) q ( 1 + σ ) 1 n t η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n t ) n t η = σ ( 1 − η ) 1 + σ τ 1 θ ( 1 − q n t + 1 ) ( 1 + r ̄ ) ( 1 − τ 1 − τ 2 ) q w ̄ 1 − η η .

Let

LHS ̃ ( n t ) ≔ 1 − σ ( 1 − η ) q ( 1 + σ ) 1 n t η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n t ) n t η

and

RHS ̃ ( n t + 1 ) ≔ σ ( 1 − η ) 1 + σ τ 1 θ ( 1 − q n t + 1 ) ( 1 + r ̄ ) ( 1 − τ 1 − τ 2 ) q w ̄ 1 − η η .

Note that LHS ̃ ( σ ( 1 − η ) q ( 1 + σ ) ) = 0 and that LHS ̃ ( 1 q ) = 1 + σ η 1 + σ [ η ( 1 − τ 1 − τ 2 ) q ] η > 0 . Since the numerator of LHS ̃ ( n t ) is strictly increasing in n _t and the denominator is strictly decreasing in n _t, LHS ̃ ( n t ) is strictly increasing in n _t on ( 0 , 1 q ) . However, RHS ̃ ( n t + 1 ) is strictly decreasing in n _t+1 on ( 0 , 1 q ) , RHS ̃ ( σ ( 1 − η ) q ( 1 + σ ) ) = σ ( 1 − η ) 1 + σ τ 1 θ 1 + σ η 1 + σ ( 1 + r ̄ ) ( 1 − τ 1 − τ 2 ) q w ̄ 1 − η η > 0 , and RHS ̃ ( 1 q ) = 0 . Thus, there is a unique n ̃ * ∈ ( σ ( 1 − η ) q ( 1 + σ ) , 1 q ) such that LHS ̃ ( n ̃ * ) = RHS ̃ ( n ̃ * ) . □

A.6 Proof of Proposition 3.6

Proof

From Eq. (32), we obtain:

∂ n ̃ * ∂ τ 1 = Z n ̃ * ( 1 − q n ̃ * ) 1 − τ 1 − τ 2 η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η − 1 × η ( 1 − η τ 1 − τ 2 ) q + ( 1 − η ) ( 1 − τ 2 ) τ 2 ( 1 − q n ̃ * ) ( 1 − τ 1 − τ 2 ) n ̃ * 1 − Z τ 1 1 − τ 1 − τ 2 η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η − 1 × η ( 1 − 2 q n ̃ * ) ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) ( 1 − η − 2 q n ̃ * ) n ̃ *

and

∂ n ̃ * ∂ τ 2 = Z τ 1 ( 1 − q n ̃ * ) ( 1 − η ) 1 − τ 1 − τ 2 η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η − 1 × η + τ 2 ( 1 − q n ̃ * ) 1 − τ 1 − τ 2 1 − Z τ 1 1 − τ 1 − τ 2 η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η − 1 × η ( 1 − 2 q n ̃ * ) ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) ( 1 − η − 2 q n ̃ * ) n ̃ * ,

where Z ≔ σ ( 1 − η ) 1 + σ θ ( 1 + r ̄ ) q w ̄ 1 − η η . Since the numerators of both equations are positive, how the tax rates affect the fertility rate is determined by the signs of the denominators of both equations. That is, ∂ n ̃ * ∂ τ i > 0 for i = 1, 2 if and only if:

(41) 1 > Z τ 1 1 − τ 1 − τ 2 η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η × η ( 1 − 2 q n ̃ * ) ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) ( 1 − η − 2 q n ̃ * ) n ̃ * η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * .

If τ ₁ is sufficiently close to 0, the right-hand side of Eq. (41) is sufficiently close to 0. Since it is continuous in τ ₁, as long as τ ₁ is sufficiently close to 0, Eq. (41) holds, which implies that ∂ n ̃ * ∂ τ i > 0 for all i = 1, 2.

Specifically, when τ ₁ = 0, the numerator of ∂ n ̃ * ∂ τ 2 = 0 for all τ ₂, which implies that ∂ n ̃ * ∂ τ 2 τ 1 = 0 = 0 . □

A.7 Proof of Proposition 3.7

Proof

Since g ̃ * in Eq. (33) is:

g ̃ * = θ w ̄ 1 − η η η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ * η − 1 ,

we focus on how

G ̃ * ≔ η ( 1 − τ 1 − τ 2 ) q + ( 1 − η ) τ 2 ( 1 − q n ̃ * ) n ̃ *

will change as tax rates change, instead of g ̃ * .

Taking the derivative of G ̃ * with respect to τ ₁, we have:

∂ G ̃ * ∂ τ 1 = − η q − ( 1 − η ) τ 2 1 ( n ̃ * ) 2 ∂ n ̃ * ∂ τ 1 .

When the condition in Proposition 3.6 is satisfied, ∂ n ̃ * ∂ τ 1 > 0 . Therefore, ∂ G ̃ * ∂ τ 1 < 0 .

For τ ₂, it is not as straightforward as the case of τ ₁, because an increase in τ ₂ can increase the parental education subsidies, which encourages an agent to invest in his or her children. Taking the derivative of G ̃ * with respect to τ ₂, we obtain:

∂ G ̃ * ∂ τ 2 = − η q + ( 1 − η ) 1 − q n ̃ * n ̃ * − ( 1 − η ) τ 2 1 ( n ̃ * ) 2 ∂ n ̃ * ∂ τ 2 = − q + 1 − η n ̃ * 1 − τ 2 n ̃ * ∂ n ̃ * ∂ τ 2 .

We consider a special case here. Assume τ ₁ = 0 and χ ≤ γ ( 1 + r ̄ ) 1 + σ . Then, n ̃ * = σ ( 1 − η ) q ( 1 + σ ) and ∂ n ̃ * ∂ τ 2 = 0 . Therefore, at τ ₁ = 0:

∂ G ̃ * ∂ τ 2 = − q + 1 − η n ̃ * = 1 n ̃ * 1 − η 1 + σ > 0 .

Since ∂ G ̃ * ∂ τ 2 is continuous in τ ₁, when τ ₁ is sufficiently close to 0, ∂ G ̃ * ∂ τ 2 > 0 . □

A.8 Proof of Lemma 4.1

Proof

Let

A ( χ ) ≔ χ

and

B ( χ ) ≔ γ ( 1 + r ̄ ) 1 + σ + τ 1 1 − τ 1 − τ 2 1 + σ 1 + σ + γ n ̂ * ( 1 − q n ̂ * ) ( 1 + g ̂ * ) .

First, since n ̂ * is continuous in χ,

lim χ → χ ̄ n ̂ * = 1 q .

Therefore, lim χ → χ ̄ ( 1 − q n ̂ * ) = 0 . Next, we consider lim χ → χ ̄ ϕ ̂ * . The first term on the right-hand side of Eq. (20) goes to 0 as χ → χ ̄ . The left-hand side of Eq. (20) and the second term on its right-hand side goes to some positive value as χ → χ ̄ . This implies that lim χ → χ ̄ ϕ ̂ * < 1 . Hence, lim χ → χ ̄ ( 1 + g ̂ * ) < + ∞ . Therefore, lim χ → χ ̄ B ( χ ) = γ ( 1 + r ̄ ) 1 + σ . It is not difficult to verify χ ̄ ( = ( 1 + r ̄ ) 1 + γ + η σ ( 1 − η ) σ ) > γ ( 1 + r ̄ ) 1 + σ . Therefore, for a χ sufficiently close to χ ̄ , A(χ) > B(χ). Since for χ < γ ( 1 + r ̄ ) 1 + σ , A(χ) < B(χ), A(χ), and B(χ) are continuous in χ, there exists χ * ∈ ( 0 , χ ̄ ) so that A(χ*) = B(χ*). From this argument, for all χ > χ*, A(χ) > B(χ) holds. □

A.9 Proof of Proposition 4.1

Proof

For 1, by differentiating n ̂ * with respect to χ, we have: ∂ n ̂ * ∂ χ = ( 1 − η ) σ q ( 1 + σ + γ ) 1 1 + r ̄ > 0 , which completes the proof.

For 2, when χ = χ*, Eq. (34) holds with equality. This means that, when χ = χ*, under the BGP equilibrium, an old agent retires fully and the fertility rate and GDP per capita growth rate under the BGP equilibrium are n ̃ * and g ̃ * , respectively. Therefore:

(42) χ * = γ ( 1 + r ̄ ) 1 + σ + τ 1 1 − τ 1 − τ 2 1 + σ 1 + σ + γ n ̃ * ( 1 − q n ̃ * ) ( 1 + g ̃ * )

holds. Then:

n ̂ * = ( 1 − η ) σ q ( 1 + σ + γ ) 1 + χ 1 + r ̄ > ( 1 − η ) σ q ( 1 + σ + γ ) 1 + χ * 1 + r ̄ = ( 1 − η ) σ q ( 1 + σ + γ ) 1 + γ 1 + σ + 1 1 + r ̄ τ 1 1 − τ 1 − τ 2 1 + σ + γ 1 + σ n ̃ * ( 1 − q n ̃ * ) ( 1 + g ̃ * ) = ( 1 − η ) σ q ( 1 + σ ) 1 + 1 1 + r ̄ τ 1 1 − τ 1 − τ 2 n ̃ * ( 1 − q n ̃ * ) ( 1 + g ̃ * ) = n ̃ * ,

where the last equality is derived from Eqs. (32) and (33). □

References

Becker, G. S., K. M. Murphy, and R. Tamura. 1990. “Human Capital, Fertility, and Economic Growth.” Journal of Political Economy 98 (5): S12–37. https://doi.org/10.1086/261723.Search in Google Scholar

Bovenberg, A. L., and B. Jacobs. 2005. “Redistribution and Education Subsidies are Siamese Twins.” Journal of Public Economics 89 (11–12): 2005–35. https://doi.org/10.1016/j.jpubeco.2004.12.004.Search in Google Scholar

Chen, H.-J. 2015. “Child Allowances, Educational Subsidies and Occupational Choice.” Journal of Macroeconomics 44: 327–42. https://doi.org/10.1016/j.jmacro.2015.03.004.Search in Google Scholar

Chen, H.-J. 2018. “Fertility, Retirement Age, and Pay-as-You-Go Pensions.” Journal of Public Economic Theory 20 (6): 944–61. https://doi.org/10.1111/jpet.12343.Search in Google Scholar

Chen, H.-J., and K. Miyazaki. 2018. “Fertility and Labor Supply of the Old with Pay-as-You-Go Pension and Child Allowances.” The B.E. Journal of Macroeconomics 18: https://doi.org/10.1515/bejm-2016-0182. 20160182.Search in Google Scholar

Cigno, A., and M. Werding. 2007. Children and Pensions. Cambridge: The MIT Press.10.7551/mitpress/7513.001.0001Search in Google Scholar

Cipriani, G. P. 2018. “Aging, Retirement, and Pay-as-You-Go Pensions.” Macroeconomic Dynamics 22 (5): 1173–83. https://doi.org/10.1017/s1365100516000651.Search in Google Scholar

Cipriani, G. P., and T. Fioroni. 2021. “Endogenous Demographic Change, Retirement and Social Security.” Macroeconomic Dynamics 25 (3): 609–31. https://doi.org/10.1017/s1365100519000269.Search in Google Scholar

de la Croix, D., and M. Doepke. 2003. “Inequality and Growth: Why Differential Fertility Matters.” The American Economic Review 93 (4): 1091–113. https://doi.org/10.1257/000282803769206214.Search in Google Scholar

Fenge, R., and P. Pestieau. 2005. Social Security and Early Retirement. Cambridge: The MIT Press.Search in Google Scholar

Hu, S. C. 1979. “Social Security, the Supply of Labor, and Capital Accumulation.” The American Economic Review 69 (3): 274–83.Search in Google Scholar

Kaganovich, M., and I. Zilcha. 1999. “Education, Social Security, and Growth.” Journal of Public Economics 71 (2): 289–309. https://doi.org/10.1016/s0047-2727(98)00073-5.Search in Google Scholar

Kitao, S. 2014. “Sustainable Social Security: Four Options.” Review of Economic Dynamics 17 (4): 756–79. https://doi.org/10.1016/j.red.2013.11.004.Search in Google Scholar

Lucas, R. E. 1988. “On the Mechanics of Economic Development.” Journal of Monetary Economics 22 (1): 3–42. https://doi.org/10.1016/0304-3932(88)90168-7.Search in Google Scholar

Michel, P., and P. Pestieau. 2013. “Social Security and Early Retirement in an Overlapping-Generations Growth Model.” Annals of Economics and Finance 14 (2): 705–19.Search in Google Scholar

Miyazaki, K. 2013. “Pay-as-You-Go Social Security and Endogenous Fertility in a Neoclassical Growth Model.” Journal of Population Economics 26 (3): 1233–50. https://doi.org/10.1007/s00148-012-0451-7.Search in Google Scholar

Miyazaki, K. 2019. “Optimal Pay-as-You-Go Social Security with Endogenous Retirement.” Macroeconomic Dynamics 23 (2): 870–87. https://doi.org/10.1017/s1365100517000062.Search in Google Scholar

Momota, A. 2003. “A Retirement Decision in the Presence of a Social Security System.” Journal of Macroeconomics 25 (1): 73–86. https://doi.org/10.1016/s0164-0704(03)00007-7.Search in Google Scholar

Wigger, B. U. 1999. “Pay-as-You-Go Financed Public Pensions in a Model of Endogenous Growth and Fertility.” Journal of Population Economics 12 (4): 625–40. https://doi.org/10.1007/s001480050117.Search in Google Scholar

Zhang, J. 1997. “Fertility, Growth, and Public Investments in Children.” Canadian Journal of Economics 30 (4a): 835–43. https://doi.org/10.2307/136272.Search in Google Scholar

Zhang, J., and R. Casagrande. 1998. “Fertility, Growth, and Flat-Rate Taxation for Education Subsidies.” Economics Letters 60 (2): 209–16. https://doi.org/10.1016/s0165-1765(98)00097-4.Search in Google Scholar

Received: 2021-02-15

Revised: 2021-11-15

Accepted: 2021-11-16

Published Online: 2021-12-03

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Articles in the same Issue

https://doi.org/10.1515/bejm-2021-0046

Keywords for this article

pay-as-you-go social security; educational subsidy; fertility; endogenous retirement; GDP per capita growth rate