Abstract
We investigate the effects of pay-as-you-go pension and child allowances on fertility, labor supply of the old, and welfare. For this purpose, we analyze a small open overlapping-generations model in which fertility and an old agent’s labor supply (retirement time) are endogenized with pay-as-you-go pension and child allowances. We find that how the pay-as-you-go pension tax rate affects the fertility rate depends on whether an old agent retires. When an old agent fully retires, then the size of the interest rate and fertility rate determine the effect of the pay-as-you-go pension tax rate on the fertility rate. When an old agent works, the pay-as-you-go pension tax rate certainly reduces the fertility rate. In addition, how child allowances affect the fertility rate depends on whether an old agent works. If an old agent retires fully, then an increase in the child allowance tax rate increases the fertility rate. When an old agent works, this is not necessarily true, which suggests that an old agent’s labor status should be taken into account when we evaluate the effects of the social security system on economic variables. In addition, we examine the effect of the social security tax rates on welfare and provide numerical examples.
Acknowledgement
We thank the Managing Editor, Tiago Cavalcanti, and an anonymous referee for their excellent comments and suggestions. Chen gratefully acknowledges financial support from the Program for Globalization Studies at the Institute for Advanced Studies in Humanities at National Taiwan University (grant number: NTU-ERP-103R890502). Miyazaki gratefully acknowledges financial support from the Japan Society for the Promotion of Science KAKENHI (grant number: 16K17086).
A Appendix
A.1 Proof of Proposition 1
Proof.
(i) Taking the derivative of
The sign of
(ii) From Equation (13),
⊡
A.2 Proof of Proposition 2
Proof.
Let
Then, Equation (20) is written as nt+1 = G(nt). Note that when Equation (16) holds,
To prove the local stability, first, we show that
where we use the fact that
Since G(n) and n are continuous in n, L(n) is also continuous in n. From the above argument, L(n) < 0 for all
where we use the fact that G is strictly decreasing and strictly convex in n, and
where
A.3 Proof of Proposition 3
Proof.
(i) By differentiating both sides in Equation (21) with respect to τ, we obtain
(ii) Taking the derivative of
Since η < 1, the sign of
and
Rearranging this equation, we obtain
Since
Thus,
From Equation (28),
where
From Equation (21), equilibrium
Note that this equation has one negative solution and one positive solution when
for all τ ∈ [0, 1 − η). Thus,
for all τ ∈ [0, 1 − η). ⊡
A.4 Proof of Corollary 1
Proof.
(i) If τ ≥ (1 − η)2, then the left-hand side (LHS) of Equation (24) is less than or equal to 0, while the RHS is positive. Thus,
(ii) For any given η ∈ [0, 1), if τ moves to 0, then the LHS of Equation (24) moves to
where
Since the LHS of Equation (24) is continuous in τ, for sufficiently small τ ∈ [0, 1), the inequality in Equation (24) is reversed, which implies
(iii) Differentiating
where we use the fact that
Then,
Thus,
if
A.5 Proof of Proposition 5
Proof.
Define
If
A.6 Proof of Proposition 6
Proof.
First, at (τ, 0),
Taking the derivative of W with respect to τ and evaluating it at (τ, 0), we obtain
The numerator of
where
Notice that
V(0) < 0 if and only if
Hence, if θ <
Suppose θ >
A.7 Proof of Lemma 2
Proof.
When τ = 0,
Recall that
Taking the derivative of W with respect to η and evaluating it at (0, η), we obtain
From Equation (23), the second term is
By applying Equation (30) to the denominator of Equation (31), we obtain
Then,
Since the denominators of both terms are positive, the sign of
This is less than 0 if
References
Bradshaw, Jonathan, and Naomi Finch. 2002. A Comparison of Child Benefit Packages in 22 Countries. Research Report No.174. Department for Work and Pensions, University of York.Search in Google Scholar
Chen, Hung-Ju, and I-Hsiang Fang. 2013. “Migration, Social Security, and Economic Growth.” Economic Modelling 32: 386–399.10.1016/j.econmod.2013.02.026Search in Google Scholar
de la Croix, David, and Matthias Doepke. 2003. “Inequality and Growth: Why Differential Fertility Matters.” American Economic Review 93: 1091–1113.10.1257/000282803769206214Search in Google Scholar
Fenge, Robert, and Volker Meier. 2005. “Pensions and Fertility Incentive.” Canadian Journal of Economics 38: 28–48.10.1111/j.0008-4085.2005.00267.xSearch in Google Scholar
Hu, Sheng Cheng. 1979. “Social Security, the Supply of Labor, and Capital Accumulation.” American Economic Review 69: 274–283.Search in Google Scholar
Kitao, Sagiri. 2014. “Sustainable Social Security: Four Options.” Review of Economic Dynamics 17: 756–779.10.1016/j.red.2013.11.004Search in Google Scholar
Michel, Philippe, and Pierre Pestieau. 2013. “Social Security and Early Retirement in an Overlapping-Generations Growth Model.” Annals of Economics & Finance 14: 705–719.Search in Google Scholar
Miyazaki, Koichi. 2017. “Optimal Pay-As-You-Go Social Security with Endogenous Retirement.” Macroeconomic Dynamics DOI:10.1017/S1365100517000062. Forthcoming.10.1017/S1365100517000062Search in Google Scholar
Mizuno, Masakatsu, and Akira Yakita. 2013. “Elderly Labor Supply and Fertility Decisions in Aging-Population Economies.” Economics Letters 121: 395–399.10.1016/j.econlet.2013.09.022Search in Google Scholar
Stokey, Nancy L., Robert E. Rucas, and Edward C. Prescott. 1989. Recursive Methods in Economic Dynamics. Cambridge, MA: Harvard University Press.10.2307/j.ctvjnrt76Search in Google Scholar
van Groezen, Bas, and Lex Meijdam. 2008. “Growing Old and Staying Young: Population Policy in an Aging Closed Economy.” Journal of Population Economics 21: 573–588.10.1007/s00148-006-0067-xSearch in Google Scholar
van Groezen, Bas, Theo Leers, and Lex Meijdam. 2003. “Social Security and Endogenous Fertility: Pensions and Child Allowances as Siamese Twins.” Journal of Public Economics 87: 233–251.10.1016/S0047-2727(01)00134-7Search in Google Scholar
Wigger, Berthold U. 1999. “Pay-as-You-Go Financed Public Pensions in a Model of Endogenous Growth and Fertility.” Journal of Population Economics 12: 625–640.10.1007/s001480050117Search in Google Scholar
Zhang, Jie, and Junsen Zhang. 2003. “Long-Run Effects of Unfunded Social Security with Earnings-Dependent Benefits.” Journal of Economic Dynamics and Control 28: 617–641.10.1016/S0165-1889(03)00037-XSearch in Google Scholar
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