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Childcare Allowances and Public Pensions: Welfare and Demographic Effects in an Aging Japan

Akira Okamoto

Abstract

In this study, reforms on childcare allowances and public pensions are examined in an extended lifecycle simulation model with endogenous fertility. A slight increase in family policies such as childcare allowances leads to increases in the total population but the magnitude of change is not large. As childcare allowances increase, however, the total population is cumulatively and progressively augmented, resulting in substantial growth in the total population and national income in the long run. Furthermore, from a long-term perspective, increases in childcare subsidies or decreases in public pension benefits are potentially Pareto-improving.

JEL Classification: H30; C68

Acknowledgements

I am grateful for the insightful comments and suggestions of Professors Alan J. Auerbach, Ronald D. Lee and Emmanuel Saez (The University of California, Berkeley). In addition, I want to thank Associate Professor Yoshinari Nomura (Okayama University), who helped me in writing the computer program used for the simulations. Professors Toshihiro Ihori (GRIPS), Yasushi Iwamoto (The University of Tokyo), Akira Yakita (Nagoya City University), Chuanchuan Zhang (The Central University of Finance and Economics), and Hyun-Hoon Lee (Kangwon National University) kindly offered many useful comments. I also wish to express my appreciation to the Editor of this journal, Professor Johann Brunner (Johannes Kepler University Linz), and to the two anonymous referees, for providing numerous insightful and thoughtful comments. Finally, I wish to acknowledge the financial support of the Ministry of Education, Culture, Sports, Science and Technology in Japan (Grant-in-Aid for Scientific Research (C) No. 15K03514).

Appendix

A Utility Maximization Problem

The utility maximization problem over time for each household in Section 2 is regarded as the maximization of Utin eq. (1) subject to eqs. (2) and (8). Let the Lagrange function be

L t = U t + s = 21 85 λ s t [ A s + 1 t + { 1 + r t + s ( 1 τ r ) } A s t + [ 1 τ w τ t + s p ] w t + s e s ( 1 l s t ) + b s t ( { l u t } u = 21 R E )

(32) ( 1 + τ t c ) C s t ( 1 + τ t c ) Φ s t ] + s = 21 R E η s t ( 1 l s t ) ,

where λst and ηst represent the Lagrange multiplier for eqs. (2) and (8), respectively.

The first-order conditions on the number of children nst, consumption Cst, leisure lst, and assets As+1t for s=21,22,,85 can be expressed by

(33) α ( 1 + δ ) ( s 21 ) k 1 1 γ ( n s t ) 1 γ = λ s t { ( 1 + τ t c ) g = 0 20 Ω s , g t ξ t ( 1 ρ ) } ,

where Ωs,0t = 1 for g=0, Ωs,gt=(k=1g{1+rt+s1+k(1τr)})1for g=1,2,,20,

(34) ( 1 α ) ( 1 + δ ) ( s 21 ) { ( C s t ) ϕ ( l s t ) 1 ϕ } 1 γ ϕ ( C s t ) ϕ 1 ( l s t ) 1 ϕ = λ s t ( 1 + τ t c ) ,

( 1 α ) ( 1 + δ ) ( s 21 ) { ( C s t ) ϕ ( l s t ) 1 ϕ } 1 γ ( 1 ϕ ) ( C s t ) ϕ ( l s t ) ϕ

(35) = λ s t { ( 1 τ w τ t + s p ) w t + s e s } + k = S T 85 λ k t θ w t + s e s R E 20 η s t ( s R E ) ,

(36) λ s t = { 1 + r t + s ( 1 τ r ) } λ s + 1 t ,

(37) η s t ( 1 l s t ) = 0 ( s R E ) ,

(38) 1 l s t = 0 ( s > R E ) ,

(39) η s t 0.

The combination of eqs. (33) and (36) produces eqs. (9) and (10). If the initial value, n21t, is given, the initial value, W21t, can be derived from eq. (10). If the value, W21t, is specified, the value of each age, Wst, can be derived from eq. (9), which generates the value of each age, nst. If the value, nst, is specified, the childrearing cost for lifetime is calculated, which gives the lifetime budget constraint represented by eq. (42).

The combination of eqs. (34) and (36) produces eqs. (11) and (12). If the initial value, V21t, is specified, the value of each age, Vst, can be derived from eq. (11). If Vstis specified, the values of consumption, Cst, and leisure, lst, at each age are obtained in the method that follows.

For s=21,22,,RE, the combination of eqs. (34) and (35) yields the following expression:

(40) C s t = [ ϕ { ( 1 τ w τ t + s p ) w t + s e s + k = S T 85 λ k t λ s t θ w t + s e s R E 20 + η s t λ s t } ( 1 ϕ ) ( 1 + τ t c ) ] l s t .

If the value of lst is given under ηst=0, the value of Cst can be obtained using a numerical method, and then the value of Vst can be derived from eq. (12). The value of lst is chosen so that the value of Vst obtained in the simulation is the closest to that calculated by evolution from V21t through eq. (11). If the value of lst chosen is unity or higher, the value of Cst is obtained from eq. (12) under lst=1. If it is less than unity, the value of Cst is derived from eq. (40).

For s=RE+1,RE+2,,85, the condition of lst=1 leads to the following equation:

(41) V s t = ( 1 α ) ϕ ( C s t ) ϕ γ + ϕ 1 1 + τ t c .

The value of Cst is chosen to satisfy this equation.

From eq. (2) and the terminal condition A21t=A86t=0, the lifetime budget constraint for an individual (=NWt) is derived:

s = 21 R E Ψ s t [ 1 τ w τ t + s p ] w t + s e s ( 1 l s t ) + s = S T 85 Ψ s t b s t ( { l u t } u = 21 R E )

(42) = s = 21 85 Ψ s t ( 1 + τ t c ) C s t + s = 21 40 k = 21 s Ψ s t ( 1 + τ t c ) ξ t ( 1 ρ ) n k t + s = 41 60 k = s 20 40 Ψ s t ( 1 + τ t c ) ξ t ( 1 ρ ) n k t

where Ψ21t = 1 for s=21, Ψst=(u=22s{1+rt+u(1τr)})1for s=22,23,,85.

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Published Online: 2020-01-11

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