The authors gratefully acknowledge two anonymous referees for their insightful comments and suggestions. This paper was read at the 13th Annual Conference of the Association for Public Economic Theory (PET12), held at the Academia Sinica in Taiwan. The authors thank David de la Croix, Fabio Mariani, and the session participants for their insightful comments. They are also grateful to Kazutoshi Miyazawa, Kazumasa Oguro, Shinji Yamashige and the seminar participants at the Demographic Transition and Public Finance Workshop (Hitotsubashi University) and the Nagoya Macroeconomics Workshop (Nagoya City University) for their helpful comments on earlier versions. The financial support from the Japan Security Scholarship Foundation (Nihon Shoken Shogaku Zaidan) is gratefully acknowledged. Email: yakita@econ.nagoya-cu.ac.jp; mkt.hirazawa@gmail.com; kitaura@hosei.ac.jp

Original Article

# Fertility, Intra-Generational Redistribution, and Social Security Sustainability

Article first published online: 27 JAN 2014

DOI: 10.1111/caje.12069

© Canadian Economics Association

Issue

## Canadian Journal of Economics/Revue canadienne d'économique

Volume 47, Issue 1, pages 98–114, February/février 2014

Additional Information

#### How to Cite

Hirazawa, M., Kitaura, K. and Yakita, A. (2014), Fertility, Intra-Generational Redistribution, and Social Security Sustainability. Canadian Journal of Economics/Revue canadienne d'économique, 47: 98–114. doi: 10.1111/caje.12069

#### Publication History

- Issue published online: 27 JAN 2014
- Article first published online: 27 JAN 2014

- Abstract
- Article
- References
- Cited By

### Keywords:

- H55;
- J13;
- J14;
- J18

### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

Incorporating heterogeneity in preference to having children into an overlapping generations model of a small open economy, we examine the effects of changes in the size of pay-as-you-go (PAYG) social security on fertility choices of individuals and population growth of the economy. It is shown that PAYG social security will raise population growth by increasing the number of individuals who have children and the number of children parents have if the system involves redistribution between retirees with different contributions, whereas, if it has no redistribution, PAYG social security does not affect the fertility decisions of individuals.

*Fécondité, redistribution intra-générationnelle, et durabilité de la sécurité sociale*. Incorporant l'hétérogénéité des préférences pour avoir des enfants dans un modèle d'une petite économie ouverte avec des générations qui se chevauchent, on examine les effets de changements dans l'importance de la sécurité sociale par répartition sur les choix de fécondité des individus et sur la croissance de la population dans l’économie. On montre que cette forme de sécurité sociale augmente la croissance de la population en accroissant à la fois le nombre de personnes qui ont des enfants et le nombre d'enfants que les parents choisissent d'avoir si le système implique une redistribution entre les gens à la retraite qui ont fait des contributions différentes, alors que, si le régime n'implique aucune redistribution, le régime de sécurité sociale par répartition n'affecte pas les décisions de fécondité des individus.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

Declining fertility rates put pressure on the financing of pay-as-you-go (PAYG) social security systems, especially, in western countries. The fertility decline may also endanger the sustainability of society itself in the sense that the population may approach zero (e.g., Cigno 1993; Sinn 2004). The cost of rearing children must be shouldered by each household, while the size of a person's pension benefits depends on everyone else's fertility decisions, giving some individuals the incentive for free-riding by receiving benefits without paying the cost (e.g., Folbre 1994). Such free-riding is considered to exert negative effects on the fertility decisions of individuals. Thus, social security reforms have been proposed to resolve the problem by, for example, conversion of social security benefits to a parental dividend (Bental 1989; Burggraf 1993), to a (voluntary) self-financing social security program that promises a return equal to the individual fertility rate (Eckstein and Wolpin 1985) or to a PAYG social security cum child allowance system (van Groezen et al. 2003). As a matter of fact, the social security systems employed in most developed countries involve some degree of such intra-generational redistribution, although the works cited above assumed homogeneous individuals. For example, a flat-rate benefit system partly financed by consumption taxes will be introduced in Japan (e.g., the 2004 revision).1 The flat-rate benefit scheme involves intra-generational redistribution when individuals are heterogeneous, while consumption taxes will be proportional to the respective consumption levels of individuals.

Our purpose in the present study is to examine the intra-generational redistribution effects of PAYG social security on the sustainability of the social security system through changes in fertility decisions of individuals, that is, decisions as to whether or not to have children and how many, while focusing on the intra-generational redistribution through pension benefits between individuals with and without children. The longer the child-rearing time, the shorter the working time, and hence the lower the contribution to PAYG social security. Most of the literature has not explicitly taken into account the effects of the intra-generational redistribution through PAYG social security benefits on fertility decisions of individuals in considering reforms of PAYG social security, although there are many works assuming heterogeneous agents.2 The paper closest to our own is that of Cremer, Gavari, and Pestieau (2008), who showed that, assuming both endogenous fertility and heterogeneity in the ability to raise children, the optimal PAYG schemes require a marginal subsidy on fertility to correct for the externality under perfect information and additional subsidy, depending on whether the redistribution is geared more to people with more children. However, they did not consider the forgone income of child rearing.3 The present paper takes into account the trade-off in time allocation between market work and child rearing, which is important in the present context if the child-rearing cost consists of units of parents’ time, that is, forgone income of parents and units of goods (Barro and Becker 1989, 486). This is our contribution to the literature. The present study assumes heterogeneity among individuals in the degree of their preference for having children.4

We show first that for a given social security contribution rate, the number of individuals who have children is greater when the benefit level is not linked to the contribution, that is, under the Beveridgean benefit scheme, than when the benefit level is proportional to the contribution, that is, under the Bismarckian benefit scheme. Second, it is shown that a rise in the contribution rate increases the fertility rate through increases in the number of individuals who have children and the number of children they have under the Beveridgean scheme. In this case, therefore, it enhances the sustainability of the social security system in the sense that the supporters of both will increase in the future. In contrast, under the Bismarckian benefit scheme without intra-generational redistribution, an increase in the contribution rate does not affect the population growth rate, that is, the rate of return to PAYG social security contributions.

This paper is organized as follows. The next two sections introduce the model of a small open economy, and Section 'Changes in the Contribution Rate' examines the effect of a PAYG social security system on the population growth rate of the economy and then the welfare effects of a change in the contribution rate. We assume a defined-contribution system in the present study. Our analysis concentrates on steady states in order to examine the long-term effects of the scheme change. Section 'Concluding Remarks' concludes the paper.

### 2. Model

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

We consider a small open economy facing the world interest rate, *r*, which is assumed to remain constant over time. Assuming a neoclassical constant-returns-to-scale production function and perfectly competitive factor markets, the wage rate, *w*, is also constant.

The economy is populated by overlapping generations of people who live for three periods. Each individual is reared by his parent in the first period of life, works and possibly rears children in the second, and retires in the third. Individuals in each generation differ only in their preference for having children. The degree of the preference of an individual is represented by the (marginal) utility weight, α, of having children relative to material consumption.5 We assume that α is distributed over according to the cumulative distribution function , where is the density function and .6 The distribution is assumed to be the same for every generation, though the population size may change over time.

#### 2.1. Individuals

Normalizing the time endowment during the working period to one, and assuming that the rearing time per child ε is constant, the budget constraints of an individual of generation *t*, who works in period *t*, in the second and third periods are given, respectively, as

- (1a)

- (1b)

where τ is the social security contribution rate in each period, is the number of children he rears, and are consumption in the second and third periods, is savings, is the gross rate of interest, and is the social security benefits paid to him in period . denotes the social security benefits paid to the individuals who choose to have no children, that is, those who contribute most to the social security system, and the parameter σ reflects the extent of intra-generational redistribution among retirees or, in other words, from the viewpoint of individuals, the degree of actuarial fairness of public pensions. The case of corresponds to full redistribution within the generation (i.e., a flat-rate benefit scheme or the Beveridgean scheme), while the case of involves no redistribution (i.e., a contribution-proportional benefit scheme or the Bismarckian scheme).

In order to solve the optimization problem of individuals and to allow them to have no child at the optimum, we assume the utility function of an individual with α to be log-linear:

- (2)

where ρ is the subjective discount factor.7 The problem for the individual is to choose consumption during two periods and the number of children.8 By solving the optimization problem, we may have the optimal plans of the individual: , , and , which gives the labour supply .9 The explicit solutions under the Beveridgean and Bismarckian schemes will be given in the respective sections to follow.

#### 2.2. Labour Supply and Population Growth

Since the numbers of children individuals have depend on the degree of their preferences for children, the evolution of the total population of this economy is

- (3)

where and is the population of the working generation in period *t*.10

The total labour supply of this economy in period *t* is given as

- (4)

#### 2.3. Social Security Schemes

The authority operating an unfunded social security scheme determines the benefit levels so as to balance the budget constraint for a given contribution rate in each period:

- (5)

It should be noted that the parameter σ may depend on the contribution of the individual, , which in turn rests on the degree of preference for having children, α. In this study we consider two schemes: the Beveridgean scheme and the Bismarckian scheme.

##### 2.3.1. The Beveridgean Scheme

The Beveridgean scheme is characterized by a flat rate of benefits: . The budget constraint (5) can be rewritten as

- (5′)

The social security benefit is the same for all retirees regardless of their contribution .

##### 2.3.2. The Bismarckian Scheme

Under the Bismarckian scheme, social security does not involve any income redistribution among individuals with different contributions; that is, the benefits are proportional to the contribution. Therefore, we have the following condition:

- (5′′)

where is the rate of replacement. The social security benefit for a retiree is proportional to his contribution .11

### 3. Fertility Choices of Individuals

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

We examine the optimization problem of individuals under the two schemes.

#### 3.1. Under the Beveridgean Scheme

The optimization problem of individuals is to choose consumption and the number of children so as to maximize the lifetime utility (2) with respect to the budget constraints (1) and the non-negative constraint on the number of children:

- (6)

- (7)

Denoting the Lagrangean multipliers attached to the constraints (6) and (7) by and , respectively, the first-order conditions are

- (8a)

- (8b)

- (8c)

- (8d)

Condition (8d) implies that (i) and or (ii) and . In case (i), the conditions (8a)–(8c) give the solution , while in case (ii), the conditions determine . The optimum plans can be obtained as follows:

- (9)

- (10)

- (11a)

- (11b)

- (11c)

- (12)

- (13)

where

- (14)

Defining as the degree of preference for having children α such that in (13), we have

- (15)

This is the cut-off degree of preference for having children. Individuals with a degree of preference less than will not have children. Rather, they prefer to reduce their child-rearing time further, increasing the labour supply and their consumption, if possible. Individuals with lower α want to do so all the more. But, of course, it is not possible for them to decrease the child-rearing time below zero. That is, they are constrained in their fertility choices. In contrast, individuals with a stronger preference for children than will have children and, from (11a), the stronger the preference for children, the greater the number of children they have.

#### 3.2. Under the Bismarckian Scheme

Under the Bismarckian scheme, individuals are expected to know that the social security benefits are proportional to the contribution and therefore that changes in the contribution, that is, the working hours, bring about proportional changes in the benefits. In this scheme the authority balances its budget by adjusting the ratio of the total benefit payments to the total contribution, while the relative benefit level of a retiree is kept proportional to his relative contribution during his working period. Thus, the replacement ratio is equal to the population growth rate under this scheme. We assume here the perfect foresight of individuals with respect to the replacement rate. Therefore, we have 12

Taking (5″) into account, the budget constraint of an individual of generation *t* can be rewritten as

- (16)

Thus, the maximization problem of an individual can be written as

s.t. (16)

- (7)

Denoting the Lagrangean multipliers attached to the constraints (16) and (7) by and , respectively, we obtain

- (17a)

- (17b)

- (17c)

- (17d)

As in the previous case, we have two cases: (i) and and (ii) and. We obtain the optimal plan in each case as follows:

- (18)

- (19)

where

- (20)

- (21)

- (22)

- (23)

Defining as the degree of preference for having children α such that , we have

- (24)

This is the cut-off degree of preference for having children. Individuals with a degree of preference less than will not have children. As in the Beveridgean case, they rather wish to further reduce their child-rearing time, increasing the labour supply and their consumption if possible. Individuals with lower α want to do so all the more, but they cannot. Thus, individuals with are constrained in their fertility choices. In contrast, individuals with a greater preference for children than will have children and, from (20), the stronger the preference for children, the greater the number of children they have.

#### 3.3. The Beveridgean Scheme versus the Bismarckian Scheme

If there are individuals with sufficiently little preference for having children, some individuals will not have children under both schemes, as shown in (15) and (24). Since the social security benefits are positive, we have

- (25)

That is, for the same contribution rate, the cut-off degree of preference for having children under the Beveridgean scheme is lower than that under the Bismarckian scheme. There are individuals who would have children under the Beveridgean scheme but not under the Bismarckian scheme. Comparing (10) with (19), we can see that for an individual with a degree of preference for children, if he has children, the number of children preferred under the Beveridgean scheme is greater than that under the Bismarckian scheme. Thus, we have the following result:13

Proposition 1. Other things being equal, both the number of individuals who have children and the number of children they have are greater under the Beveridgean scheme than under the Bismarckian scheme. Therefore, the rate of population growth will be higher under the Beveridgean scheme.

### 4. Changes in the Contribution Rate

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

In this section we examine the steady-state effects of changes in the contribution rate, focusing on the long-term effect of the policy changes. As can be readily seen, under the Bismarckian scheme, changes in the contribution rate do not affect either the cut-off degree of preference for children (and therefore the number of individuals who have children) or the number of children they have (see (19) and (24)); that is, .14

On the other hand, under the Beveridgean scheme, from (3), (4), and (5) and making use of the result in Section 'Under the Beveridgean Scheme', we obtain

- (26)

- (27)

where is used. (In this section, notation is used instead of when it is not confusing.) Inserting (27) into (26), we have

- (28)

where, assuming , we have15

Differentiating in (15) with respect to τ, we have

- (29)

From (28) and (29) we have

- (30)

From (30) we obtain

- (31)

- (32)

where

We assume here that .16 Therefore, it follows that

- (33)

and . An increase in the contribution rate lowers the cut-off degree of preference for children and therefore increases the number of individuals who have children under the Beveridgean scheme. Together with (11b) and (11c), the increased contribution rate raises the fertility rate in the economy as a whole. Thus, we have the following proposition.17

Proposition 2.

- Under the Beveridgean scheme, an expansion of social security reduces the cut-off degree, thereby raising the population growth rate of the economy through increasing both the number of individuals who have children and the number of children they have.
- Under the Bismarckian scheme, an increase in the social security contribution does not affect either the number of individuals who have children or the number of children they have.

The intuition behind the results can be explained as follows. When the proportional contribution rate increases, the cost of having children in terms of forgone income becomes lower than the marginal utility. Although the lower child-rearing cost tends to increase the number of children, the increased child-rearing time reduces the labour supply and thereby the social security contribution. Under the Beveridgean scheme, since the level of social security benefits does not depend on the number of children, these changes in the cost and benefits of having children make the benefit greater than the cost for individuals with . These changes are also true for individuals with a higher degree of preference. Thus, the increase in the contribution rate increases both the number of individuals who have children and the number of children parents have. In contrast, under the Bismarckian scheme, when the replacement rate is perfectly foreseen, individuals decide the number of children by reckoning the effects on the costs and benefits of social security. A change in contribution will be reflected in the benefits straightforwardly, and with a perfect capital market they are perfectly offset.

At this stage, it should be noted that, in contrast to the previous studies (e.g., Zhang, Zhang, and Lee 2001; Yakita 2001), an increase in the after-tax wage rate may reduce the fertility rate of the society, even though raising children brings about positive utility to the parent in our model.18 Boldrin, de Nardi, and Jones (2005) and Ehrlich and Kim (2007), among others, also recently documented a negative relationship between fertility and the size of the social security system using panel data sets. Although they suggest that the mechanism in the present study may be qualitatively less important, they do not necessarily show that the mechanism is null in the real world. In fact, the link between contributions and benefits varies in the social security systems from country to country.19

Next we briefly examine the utility effects. Since the behaviours of individuals do not change with the tax changes under the Bismarckian scheme, we analyze here the effects under the Beveridgean scheme.

The lifetime utility of an individual is given as

- (34)

where and are given by (9) and (10). Differentiating (34), we obtain

- (35)

From (9) and (10) we obtain

- (36)

- (37)

If (> 0) is sufficiently great (small), we have *d**c*_{t}^{1}/*d*τ > (<) 0 in (36).

Therefore, we have the following two cases. If is sufficiently great, (35) is positive. In this case, the utility effect of an increase in the contribution rate is positive for all degrees of preference for children.20 On the other hand, if is small, the first term on the right-hand side of (35) is negative. Therefore, for individuals whose degree of preference for children is too small to have children, the utility effect is negative, since the second term on the right-hand side of (35) is absent. For individuals with a sufficiently high degree of preference for children, the positive second term will dominate the negative first, making the utility effect positive. For individuals with intermediate preference, the utility effect is ambiguous. When an increase in the contribution rate does not increase the benefit level sufficiently, the redistribution from individuals without children to those with children will improve the welfare of the latter individuals at the cost of the former.

Since the capital labour ratio of the economy as a whole remains constant, the decreased total labour supply is associated with a reduction in total capital per worker, , where is the total capital. Under the constant-returns-to-scale technology, total output per worker also decreases. However, since a higher fertility rate increases the number of workers, whether the total capital decreases or not is ambiguous. On the other hand, savings of an individual is given as , from which we have

- (38)

From (36) and (37), we can show that, if is great enough to hold , we have for all workers, that is, total savings decreases, and that if , the effect on the total savings is not clear. Therefore, whether capital flows into or out of the economy is ambiguous in general. It is noted that, even if is sufficiently great and hence the lifetime utility increases for all degrees of preference, increases in the contribution rate do not necessarily increase capital flows into the economy.

### 5. Concluding Remarks

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

Assuming heterogeneity of individuals with respect to the preference for having children in an overlapping-generations model of a small open economy, we examined the effect of a change in the size of the PAYG social security system on fertility through redistribution among beneficiaries with different contributions due to different child-rearing time. The feature of this model is the existence of the cut-off degree of preference: that is, individuals with a lower degree of preference than the cut-off degree do not have children. Redistribution from individuals without children and with a high contribution to those with children and low contributions raises the fertility rate of the economy under the Beveridgean scheme. If social security benefits do not involve redistribution among retirees, that is, under the Bismarckian scheme, the size of the social security system does not affect fertility. Thus, the social security scheme with flat-rate, rather than contribution-proportional, benefits will enhance the sustainability of the social security system in the sense that the fertility rate is higher for a given contribution rate. This result may indicate a promising direction of social security reforms for sustainability.

We have assumed away other public expenditures in order to focus on social security. Long-term care, for example, may expand life expectancy, requiring in turn more pension benefit payment. We have also assumed a small open economy. If the wage rate is affected by the labour supply, as in a closed economy, the effect of an increase in the contribution rate on the benefit level may be negative when the wage decrease is too great. In that case, the fertility rate may be lowered by the raised contribution rate. Finally, technical progress may raise the wage rate, which may in turn decrease the number of individuals having children. These issues are subjects for future research.

- 1
It is often said that the trend of social security reform in the world is the switch from defined-benefit to defined-contribution systems. In contrast, the reform in Japan can be said to maintain the property of “collective annuities” à la Cremer, Lozachmeur, and Pestieau (2010) through a defined-benefit system.

- 2
For example, assuming both wage inequality and longevity difference among individuals, Cremer, Lozachmeur, and Pestieau (2010) showed that when the former dominates, a flat rate benefit (Beveridgean) system is more welfare improving than a contribution (Bismarckian) system. Cremer, Lozachmeur, and Pestieau (2004) also showed that, assuming heterogeneous individuals in the levels of productivity and health status, redistribution through social security may impose an implicit tax on postponed retirement, thus inducing early retirement for some individuals. In the present study, we assume inequality in contributions, owing to differences in the lengths of working time rather than wage inequality.

- 3
Their focus is, in contrast, the second-best policies rather than a positive analysis of social security reforms.

- 4
Zhang and Zhang (1998) emphasized the parental motive to have children in modelling endogenous fertility. However, unlike ours, they considered two basic utility configurations: altruistic and non-altruistic.

- 5
Alternatively, we may assume that α denotes the probability of having children. The parameter should be interpreted cautiously, since the number of children is not necessarily determined by parents’ preference in reality. There are many couples who do not have children even though they want them. We exclude this case in our study.

- 6
The upper bound is assumed to be sufficiently great.

- 7
Letting be the utility from having children, we have , , and . We can instead assume a more general utility function such as where without altering our conclusion. The logarithmic utility function is a special case of .

- 8
Considering labour-leisure choices of individuals does not essentially affect the results. The effect of an increase in the social contribution rate on leisure is qualitatively the same as that on fertility.

- 9
The upper bound of the number of children is given by, with which the wage income approaches zero. We also assume that even if individuals desire to borrow against their future pension benefits in order to finance current spending, they are constrained to have less than the upper bound of children since there is no mechanism through which they can obligate the future pension benefits.

- 10
Assuming unisex individuals without infant mortality, the present study supposes that the sustainable growth rate of population is 1. Since not all the individuals have children, the population growth rate () temporarily can be less than 1.

- 11
Conceptually, the Bismarckian scheme may be split up into contribution-related and fertility-related components, as shown by Fenge and Meier (2005, 2009). Fenge and Meier (2005) also showed that fertility-related pensions are equivalent to child allowances in achieving the optimum allocation. See also Fenge and Weizsäcker (2010), who obtained a Corlett-Hague result regarding the optimal mix of the two components. Our focus, however, is a comparison between the Bismarckian and the Beveridgean schemes.

- 12
In the present setting, although the per worker wage rate remains constant, the aggregate income grows at the rate of population growth.

- 13
The social security systems of Ireland and New Zealand are the flat-rate schemes, that is, the Beveridgean scheme, while those in Italy and the Slovak Republic have a close link between contributions and benefits, that is, nearly the Bismarckian scheme (see OECD 2005); Ireland and New Zealand had total fertility rates of 1.90 and 1.97 during 1995–2000, respectively, higher than those in Italy and Slovak Republic of 1.21 and 1.40, respectively (UN 2003). See Figure 1.

- 14
In contrast, Zhang and Zhang (2007) showed that social security has a larger negative effect on fertility in the earning-dependent benefit scheme than in the earning-independent scheme in an overlapping-generations model with operative bequest. The difference between us stems from the fact that the bequest costs tend to reduce the number of children in their model, while there are no such costs in our model.

- 15
Since, if , as in de la Croix and Doepke (2003), we have , the condition holds plausibly.

- 16
Otherwise, if , we have ; that is, decreases in the contribution rate raise the level of social security benefits, implausible as it seems.

- 17
Our results crucially depend on the assumption of the payroll tax for social security. However, it is not implausible, and is even common in the literature, to assume non-lump-sum contributions. See, for example, Sinn (2004) and Zhang, Zhang, and Lee (2001), although Bental (1989) and van Groezen, Leers, and Meijdam (2003) assumed lump-sum taxes. Samuelson (1975) showed the effect of PAYG social security on the dynamic resource allocation assuming a lump-sum contribution by identical individuals. In the real world, for instance, Sweden introduced a proportional tax in 1999, while Japan has adopted different schemes with lump-sum and proportional contributions.

- 18
Galor and Weil (1996) emphasized the effect of an increase in women's relative wages in lowering fertility, taking into account the differences between men and women. The negative effect of after-tax wage on fertility is also well recognized in Beckerian models such as that in Ehrlich and Lui (1991).

- 19
See footnote 13 and Figure 1.

- 20
Under plausible values of parameters of τ and ε (as in fn15), if population growth rate is low relative to the interest rate, for example, when the economy is in the dynamic efficient range, we will have .

### References

- Top of page
- Abstract
- 1. Introduction
- 2. Model
- 3. Fertility Choices of Individuals
- 4. Changes in the Contribution Rate
- 5. Concluding Remarks
- References

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