PAYG Pensions and Economic Cycles

Abstract

In this article we compare the dynamics and long-run outcomes of an overlapping generations closed economy under exogenous and endogenous fertility. Individuals have myopic foresight, and pay-as-you-go (PAYG) public pensions exist. Although large PAYG transfers may cause endogenous fluctuations in both contexts, cyclical instability and deterministic chaos more likely occur when fertility is an economic decision variable. We find that the existence of endogenous fertility and PAYG pensions can explain the occurrence of demographic oscillations that mimic the baby boom and baby bust, in contrast with the unrealistic case of constant population, and also economic cycles. We also show that an increase in the social security contribution rate under endogenous fertility, often advocated as a remedy against the crisis of public pension budgets, may prolong both the phases and size of demographic (economic) cycles around a lower (higher) long-run level of population growth (income per worker) than under exogenous fertility.

Keywords

demoeconomic fluctuations myopic foresight PAYG pensions stability OLG model

Unfunded pay-as-you-go (PAYG) social security is a controversial issue in the economic literature. Some authors highlight the effects of pensions in reducing either saving (Feldstein 1974) or welfare if the capital stock is not too much larger (Diamond 1965), while others find that the crowding out effect of pensions on voluntary saving does not occur if individuals take care of their future infinite progeny via private intergenerational transfers (Hansson and Stuart 1989).

Although the existence of the PAYG system has been justified on equity grounds in a political equilibrium model because of an intergenerational redistribution effect as in Tabellini (1990), it has also been criticised due to the perils of unviable budgets, especially in countries facing demographic changes because of population ageing (e.g., Italy, Japan, and Spain). This has raised debates on the role PAYG pensions can play in overlapping generations (OLG) growth models when fertility changes (e.g., van Groezen, Leers, and Meijdam 2003). While in this class of models, the issue of PAYG pensions has usually been studied under exogenous fertility (Samuelson 1975a), a recent body of theoretical literature, pioneered by the seminal article by Becker (1960), argues that fertility should actually be considered as a decision variable influenced by economic incentives and constraints (Becker and Barro 1988; Barro and Becker 1989; Becker, Murphy, and Tamura 1990) rather than being left out of the economic sphere.

However, to the best of our knowledge, most studies have not yet explored the dynamic features of an economy with endogenous fertility in comparison with those of the standard case of exogenous fertility when PAYG pensions are in existence. In particular, less attention has been paid to the role of PAYG pensions and endogenous demographic changes on long-run demoeconomic outcomes and economic and demographic stability. The aim of this article, therefore, is to fill this gap by reconsidering the issue of unfunded social security in the neoclassical OLG growth model à la Diamond (1965) under exogenous and endogenous fertility.

In particular, as in Eckstein and Wolpin (1985) and Galor and Weil (1996), the choice of fertility is assumed to be motivated by the so-called weak form of altruism of parents (Zhang and Zhang 1998), in which individuals draw utility from the number of descendants they have and choose fertility by comparing the benefits and costs of children. Moreover, since childbearing can reasonably be considered either a time-based or income-based activity irrespective of whether fertility is endogenous or exogenous (due, for instance, to religion belief, customs, unchecked sexuality, and the like), in this article we assume that raising children is costly even under exogenous fertility.

Given the difficulty of studying the dynamics of OLG models with endogenous fertility, we postulate a lognormal utility function and a Cobb–Douglas production function, which permit closed form analytical solutions as well as economic interpretations of the dynamical outcomes, otherwise prevented by the use of other more general functional forms.^1,2 These assumptions are standard, and, beyond the analytical simplification, they have much empirical support.³

Our results reveal that PAYG pensions strongly matter for both the steady-state and dynamical events depending on whether fertility is endogenous or exogenous. In particular, in the absence of public pensions, fertility still remains constant even when it is an economic decision variable, and thus the model is not suited to explain the observed population dynamics in such a case. Indeed, the working of pensions generates endogenous population dynamics that may mimic the observed evolution of fertility, especially in Western countries. Moreover, PAYG pensions are crucial to determine the long-run level of population growth as well as the appearance of permanent and chaotic phases of baby booms and baby busts.

Therefore, the interaction between PAYG pensions and endogenous fertility significantly matters for stability, and the destabilizing effect of public pensions is higher than when fertility is exogenous. The economic reasons for this result are several.

First, regardless of whether fertility is exogenous or endogenous, the existence of a PAYG transfer crowds out voluntary savings because of the need to sustain consumption when old becomes lower than in the absence of it. As a consequence, when individuals are shortsighted, raising pensions beyond a certain critical level may determine a sharp reduction in savings and capital accumulation as well as the destabilization of the equilibrium point. Second, the negative crowding out effect of public pensions on private savings is lower, the higher is the subjective discount factor because, ceteris paribus, the relative weight of old-age consumption increases as the intergenerational discount factor becomes larger. Third, in the case of endogenous fertility, the individual discount factor also reduces the weight of the positive effect of pensions on the demand for children. (Note that the positive relationship between fertility and PAYG pension exists because individuals save less than when intergenerational transfers are absent, and thus the disposable income for childbearing increases in such a case.) Moreover, the weight of the subjective discount factor on saving and fertility through public pensions is exactly the same. Finally, since capital accumulation per worker is determined as the ratio between savings and fertility, the subjective discount factor does not affect the public pension component on capital accumulation when fertility is endogenous, while affecting in a negative way saving and the public pension component on capital accumulation when fertility is exogenous.

The present article, therefore, contributes to three strands of literature: the neoclassical OLG growth literature with public pensions, the theory of endogenous population dynamics (Manfredi and Fanti 2006), and endogenous economic cycles framed in the OLG context (Grandmont 1985). In this regard, the article studies an issue so far not explored, such as dynamic and steady-state demoeconomic outcomes with PAYG pensions, by contrasting models with exogenous and endogenous fertility, while also focusing on the effects of increasing the contribution rate (Cigno 2007; Liikanen 2007). Although it is well known that OLG economies with myopic expectations may typically show cyclical (and even chaotic) dynamics when the elasticity of substitution in production (Farmer 1986; Reichlin 1986) and in utility functions (Michel and de la Croix 2000; Fanti and Spataro 2008) are fairly low and high, respectively, it is remarkable that cyclical instability and deterministic chaos can occur in the standard double Cobb–Douglas economy because of the financing of PAYG pensions.

Our findings can have interesting policy implications. In particular, to the extent that fertility is an economic decision variable, a rise in the contribution rate, often invoked as a remedy against the crisis of public pension budgets, may reduce the desired number of children of individuals and trigger baby booms and baby busts around the long-run level of fertility.⁴

The rest of the article is organized as follows: the section Exogenous Fertility analyzes the steady-state and local stability properties of an OLG economy with exogenous fertility and is followed by a section on Endogenous Fertility. The section Exogenous Versus Endogenous Fertility: Some Analytical and Numerical Results compares the dynamic and steady-state outcomes under exogenous and endogenous fertility and is followed by the Conclusion.

Exogenous Fertility

Firms are identical and markets are competitive. Aggregate production at time $t$ ( $Y_{t}$ ) takes place by combining capital ( $K_{t}$ ) and labor ( $L_{t} = N_{t}$ in equilibrium, where $N_{t}$ is the number of young individuals at $t$ ) through a Cobb–Douglas technology $Y_{t} = A K_{t}^{α} L_{t}^{1 - α}$ , where $A > 0$ and $0 < α < 1$ . Assuming that output is sold at unit price and capital fully depreciates at the end of every period, profit maximization implies that production factors are paid their marginal products, that is:

r_{t} = α A k_{t}^{α - 1} - 1,

w_{t} = (1 - α) A k_{t}^{α},

where

k_{t} = K_{t} / N_{t}

is capital per worker,

r_{t}

is the interest rate, and

w_{t}

is the wage rate.

At time $t$ young population of measure $N_{t}$ grows at the constant rate $\frac{N_{t}}{N_{t - 1}} - 1 = \bar{n} - 1$ , where $\bar{n}$ is the exogenously given number of children per each individual (by assuming, as usual, the existence of a single parent only). The government redistributes from the young to the old with unfunded public pensions. Therefore, the pension expenditure at $t$ $(P_{t} = p_{t} N_{t - 1})$ is constrained by the amount of tax receipts $θ w_{t} N_{t}$ , where $0 < θ < 1$ is the (wage) contribution rate to the PAYG system. The (per pensioner) budget constraint of the government can then be written as

p_{t} = θ w_{t} \bar{n} .

Individuals are identical and live two periods in an OLG closed economy (Diamond 1965): youth (working period) and old age (retirement period). Agents of generation t draw utility (U_t ) from young- and old-age consumptions, c _1t and c _2,t+1, respectively. Young individuals are endowed with one unit of labor inelastically supplied to firms and receive wage income

w_{t}

. Starting from the seminal articles by Diamond (1965), Samuelson (1975b), and Deardorff (1976), the economic literature did not consider children costs in models with exogenous fertility. However, whatever the reasons for a positive fertility rate, it is natural to conjecture that the bearing of children is costly even when fertility is not an economic decision variable (Fanti and Gori, 2011). In particular, we assume that the amount of resources needed to care for a child is

q w_{t}

, where

0 < q < 1

is the percentage of child rearing cost on working income, as in Boldrin and Jones (2002). Therefore, the budget constraint of the young at

t

reads as

c_{1, t} + s_{t} + q w_{t} \bar{n} = w_{t} (1 - θ) .

that is, the disposable income is used to consume, save (

s_{t}

), and take care of

\bar{n}

descendants. Old individuals retire and live with the amount of resources saved when young plus the expected interest accrued from

t

t + 1

at the rate

r_{t + 1}^{e}

and the expected pension benefit

p_{t + 1}^{e}

. The budget constraint of the old born at time t is therefore

c_{2, t + 1} = (1 + r_{t + 1}^{e}) s_{t} + p_{t + 1}^{e} .

The typical agent of generation t chooses how much to save out of his or her disposable income to maximize the utility function:

U_{t} = ln (c_{1, t}) + β ln (c_{2, t + 1}),

subject to equations (4) and (5), where

0 < β < 1

is the subjective discount factor.⁵ The saving function therefore is

s_{t} = \frac{β w_{t} (1 - θ - q \bar{n})}{1 + β} - \frac{p_{t + 1}^{e}}{(1 + β) (1 + r_{t + 1}^{e})} .

where

q < (1 - θ) / \bar{n}

. Now, inserting the one-period forward pension accounting rule equation (3) into equation (7) to substitute for

p_{t + 1}^{e}

yields

s_{t} = \frac{β w_{t} (1 - θ - q \bar{n})}{1 + β} - \frac{\bar{n} θ}{1 + β} \cdot \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}} .

From equation (7), we see that saving is the result of two components: a positive private component (the first term on the right-hand side of equation (7)), which depends on the propensity to save out of wage income (net of taxes and children costs);⁶ and a negative public pension component (the second term on the right-hand side of equation (7)) because of the existence of a pension transfer to sustain consumption when old, which induces the young to save less than in the absence of it.

Note that the negative effect of future pensions on current savings depends on the relative weight between the subjective discount rate (ρ) and the market interest rate (r). The higher the relative importance of the former, the higher the negative effect of pensions on savings.

Given the government budget equation (3) and knowing that $N_{t + 1} = \bar{n} N_{t}$ , the equilibrium condition on the capital market can be expressed as

\bar{n} k_{t + 1} = s_{t} .

From equations (8) and (9), we get

k_{t + 1} = \frac{β w_{t} (1 - θ - q \bar{n})}{(1 + β) \bar{n}} - \frac{θ}{1 + β} \cdot \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}} .

In an OLG economy, the dynamic evolution of capital is different depending on whether expectations about factor prices are either rational or myopic (Michel and de la Croix, 2000; de la Croix and Michel, 2002; Chen, Li, and Lin 2008).

While our economy does not exhibit any interesting dynamic events (i.e., the steady state is locally stable and the dynamic path is always monotonic) if expectations are rational, nonmonotonic dynamics and cyclical instability may occur if expectations are myopic. Therefore, in the following section we concentrate on the case of shortsighted agents to study the complex dynamic properties of this stylized economy.

Exogenous Fertility: Steady-State and Local Stability Analyses with Myopic Foresight

With myopic expectations, both the expected interest and wage rates depend on the time t stock of capital per worker, or

\{\begin{matrix} 1 + r_{t + 1}^{e} = α A k_{t}^{α - 1} \\ w_{t + 1}^{e} = (1 - α) A k_{t}^{α} \end{matrix} .

Therefore, exploiting equations (1), (2), (10), and (11) the dynamic path of capital accumulation is

k_{t + 1} = \frac{1}{1 + β} [\frac{β (1 - θ - q \bar{n}) (1 - α) A}{\bar{n}} k_{t}^{α} - θ \frac{1 - α}{α} k_{t}] .

Steady states of the phase map equation (12) are defined as

k_{t + 1} = k_{t} = k^{*}

. Therefore, the unique steady state is

k^{*} = {\{\frac{β α (1 - α) A (1 - θ - q \bar{n})}{\bar{n} [α (1 + β) + θ (1 - α)]}\}}^{\frac{1}{1 - α}} .

As expected, the equilibrium stock of capital negatively depends on the pension contribution rate because saving reduces for a twofold reason when

θ

raises: the disposable income becomes lower because a larger fraction of wages is used for the payment of pensions; and the substitution of private savings for public transfers to sustain consumption when old becomes more attractive.

From equations (12) and (13), therefore, the following proposition holds:

Proposition 1: [Dynamics under exogenous fertility]. (1) Let $0 < α < α_{4}$ hold. Then $θ_{1} < θ_{2} <$ 1, and: (1.1) if $0 < θ < θ_{1}$ , the dynamics of capital is monotonic and convergent to $k^{*}$ ; (1.2) if $θ_{1} < θ < θ_{2}$ , the dynamics of capital is oscillatory and convergent to $k^{*}$ ; (1.3) if $θ = θ_{2}$ , a flip bifurcation emerges;⁷ (1.4) if $θ_{2} < θ < 1$ , the dynamics of capital is oscillatory and divergent to $k^{*}$ . (2) Let $α_{4} < α < α_{2}$ hold. Then $θ_{1} <$ 1, $θ_{2} >$ 1, and (2.1) if $0 < θ < θ_{1}$ , the dynamics of capital is monotonic and convergent to $k^{*}$ ; (2.2) if $θ_{1} < θ <$ 1, the dynamics of capital is oscillatory and convergent to $k^{*}$ . (3) Let $α_{2} < α <$ 1 hold. Then $θ_{2} > θ_{1} >$ 1 and the dynamics of capital is monotonic and convergent to $k^{*}$ for any $0 < θ <$ 1, where

θ_{1} = θ_{1} (α, β) = \frac{α^{2} (1 + β)}{{(1 - α)}^{2}},

θ_{2} = θ_{2} (α, β) = θ_{1} \frac{1 + α}{α},

α_{2} = α_{2} (β) = \frac{- 1 + \sqrt{1 + β}}{β} > 0, 0 < α_{2} < 1 / 2,

α_{4} = α_{4} (β) = \frac{- (3 + β) + \sqrt{β^{2} + 10 β + 9}}{2 β} > 0, 0 < α_{4} < 1 / 3.

Proof: See appendix A.

Indeed, a rise in the capital stock causes a twofold effect. It increases saving because the current wage raises, while also decreasing it because of the rise in the present value of next period transfers. Moreover, the lower the capital share in production and the subjective discount factor (or, alternatively, the higher the relative weight of both the labor income in production and the public pension component in capital accumulation), the more likely in $θ$ causes cyclical instability because the relative weight of the public pension component increases.

Endogenous Fertility

In this section, we assume that individuals enjoy to have children and choose the number of descendants (e.g., the so-called weak form of altruism, as in Zhang and Zhang [1998]) by comparing between benefits and costs of upbringing.

The public pension budget and the individual budget constraints when young and old are still determined by equations (3),(4), and (5), respectively, with the only difference that the number of children is now an economic decision variable, and, as can easily be ascertained below (see equation (19)), it depends on both the wage and interest rates, which in turn depend on the dynamic variable $k_{t}$ . As a consequence, the number of children is no longer constant.

Individuals have preferences toward material consumption over the life cycle and the number of children, as in Eckstein and Wolpin (1985) and Galor and Weil (1996). The typical agent that enters the working period at t chooses fertility and saving to maximize the utility function:

U_{t} = ln (c_{1, t}) + β ln (c_{2, t + 1}) + ϕ ln (n_{t}),

subject to equations (4) and (5), where

φ > 0

captures the parents' taste for children. The constrained maximization of equation (18) gives⁸

n_{t} = \frac{1}{1 + β + φ} [\frac{φ (1 - θ)}{q} + \frac{φ p_{t + 1}^{e}}{q w_{t} (1 + r_{t + 1}^{e})}],

s_{t} = \frac{1}{1 + β + φ} [β w_{t} (1 - θ) - \frac{(1 + φ) p_{t + 1}^{e}}{1 + r_{t + 1}^{e}}] .

Upon substitution of the one-period forward pension accounting rule equation (3) for

p_{t + 1}^{e}

into equations (19) and (20), fertility and saving are given, respectively, by

n_{t} = \frac{ϕ w_{t} (1 - θ)}{(1 + β + ϕ) q w_{t} - ϕ θ \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}}},

19'

s_{t} = \frac{w_{t} (1 - θ)}{(1 + β + φ) q w_{t} - φ θ \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}}} (β q w_{t} - φ θ \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}}) .

20'

It is important to note that the young choose fertility by taking the future level of pensions as given; that is, they consider their own number of children to be too small to affect the size of the benefit they will receive when old. In other words, each individual does not internalize the government pension budget when choosing fertility and saving. In contrast, if the pension system were of the so-called fertility-related type (Kolmar 1997; Fenge and Meier 2005), so that pension transfers are contingent on the individual number of children, then each young parent would take into account the influence of the government budget on saving and fertility decisions by internalizing it.

From equation (19′), we note that fertility is constant at $n_{t} = n = \frac{φ}{q (1 + β + φ)}$ if $θ = 0$ , while becoming a time-dependent variable that positively depends on capital accumulation if $0 < θ < 1$ .

Using equations (9), (19′), and (20′), the equilibrium in the capital market is:

k_{t + 1} = \frac{β}{φ} q w_{t} - θ \frac{w_{t + 1}^{e}}{1 + r_{t + 1}^{e}} .

Endogenous Fertility: Steady-State and Local Stability Analyses with Myopic Foresight

Exploiting equations (1), (2), (11), and (21), the dynamic path of capital accumulation becomes

k_{t + 1} = \frac{β}{φ} q (1 - α) A k_{t}^{α} - θ \frac{1 - α}{α} k_{t} .

The steady state is now given by

k^{*} = {\{\frac{q β α (1 - α) A}{φ [α + θ (1 - α)]}\}}^{\frac{1}{1 - α}} .

Though equation (23) reveals that a rise in

θ

negatively affects capital accumulation even under of endogenous fertility, the intensity of such a negative effect is different depending on whether fertility is exogenous or endogenous. Comparison of equations (13) and (23) gives the following remark:

Remark 1: Ceteris paribus, the negative effect of a rise in the pension contribution rate on the long-run economic growth is smaller under endogenous fertility.

Indeed, a rise in the contribution rate causes a negative income effect that reduces the disposable income irrespective of whether fertility is exogenous or endogenous. However, while in the former case such a negative effect only reduces savings, in the latter case it causes the same identical reduction in both savings and fertility, and then the negative income effect on capital accumulation is sterilized in such a case. An illustration of this result is postponed to the section on Change in PAYG Pensions: Policy Experiment.

We now proceed with the steady-state analysis of the relationship between fertility and the pension contribution rate. Using equations (1), (2), (19′), (20′), and (23), the long-run fertility rate can be written, after some straightforward algebra, as

n^{*} (θ) = \frac{φ (1 - θ) [α + θ (1 - α)]}{q \{α β + (1 + φ) [α + θ (1 - α)]\}} .

We now analyze how the long-run fertility rate reacts to a change⁹ in

θ

. First, we define the following threshold value of the output elasticity of capital:

\hat{α} = \frac{β}{1 + 2 β + φ} < \frac{1}{4},

for any

0 < φ < 1

. Then the following proposition holds.

Proposition 2: (1) Let $0 < α < \hat{α}$ hold. Then $n^{*} (θ)$ is inverted U-shaped with $θ = \hat{\hat{θ}}$ being the fertility-maximising contribution rate. (2) Let $\hat{α} < α <$ 1 hold. Then a rise in $θ$ always causes the fall of fertility in the long run.

Proof: The proof uses the following derivative:

\frac{\partial n^{*} (θ)}{\partial θ} = \frac{φ {- {(1 - α)}^{2} (1 + φ) θ^{2} - 2 α (1 - α) (1 + β + φ) θ - α [α (1 + 2 β + φ) - β]}}{q {α β + (1 + φ) [α + θ (1 - α)]}^{2}} .

Solving the numerator of equation (26) with respect to

θ

gives

θ = \hat{θ} = - \frac{α (1 + β + φ) + \sqrt{α β (1 + α β + φ)}}{(1 - α) (1 + φ)} < 0,

θ = \hat{\hat{θ}} = \frac{- α (1 + β + φ) + \sqrt{α β (1 + α β + φ)}}{(1 - α) (1 + φ)},

with

\hat{\hat{θ}} > \hat{θ}

. Then, by applying the Descartes' rule of sign, we find that if

0 < α < \hat{α}

, then

\hat{θ} < 0

and

, so that equation (27) can be ruled out. Therefore,

\frac{\partial n^{*} (θ)}{\partial θ}

\frac{>}{<} 0

if and only if

with

being an interior global maximum. This proves Point 1. If instead

\hat{α} < α < 1

, then

\hat{θ} < 0

and

. Therefore,

\frac{\partial n^{*} (θ)}{\partial θ} < 0

for any

0 < θ < 1

. This proves point 2 and proposition 2 follows. Q.E.D.

Note that the output elasticity of capital $\hat{α}$ is found to be below 0.2 for a wide range of values of both the subjective discount factor $β$ and taste for children $ϕ$ . Thus the restriction given by equation (25) is not satisfied for some realistic values of the output elasticity of capital recently observed in several developed countries, which is generally not lower than 0.25 (Gollin 2002). Therefore, we may reasonably conclude raising PAYG transfers in actual several economies reduces fertility.¹⁰

As regard local stability, the analyses of equations (22) and (23) give the following proposition:

Proposition 3: [Dynamics under endogenous fertility]. (1) Let $0 < α < 1 / 3$ hold. Then $θ_{3} < θ_{4} <$ 1, and (1.1) if $0 < θ < θ_{3}$ , the dynamics of capital is monotonic and convergent to $k^{*}$ ; (1.2) if $θ_{3} < θ < θ_{4}$ , the dynamics of capital is oscillatory and convergent to $k^{*}$ ; (1.3) if $θ = θ_{4}$ , a flip bifurcation emerges; (1.4) if $θ_{4} < θ < 1$ , the dynamics of capital is oscillatory and divergent to $k^{*}$ . (2) Let $1 / 3 < α < 1 / 2$ hold. Then $θ_{3} < 1$ , $θ_{4} > 1$ , and (2.1) if $0 < θ < θ_{3}$ , the dynamics of capital is monotonic and convergent to $k^{*}$ ; (2.2) if $θ_{3} < θ < 1$ , the dynamics of capital is oscillatory and convergent to $k^{*}$ . (3) Let $1 / 2 < α < 1$ hold. Then $θ_{4} > θ_{3} > 1$ , and the dynamics of capital is monotonic and convergent to $k^{*}$ for any $0 < θ < 1$ , where

θ_{3} = θ_{3} (α) = \frac{α^{2}}{{(1 - α)}^{2}},

θ_{4} = θ_{4} (α) = θ_{3} \frac{1 + α}{α} .

Proof: See appendix B.

Similar to the case of exogenous fertility, large PAYG transfers may destabilize the economy when the output elasticity of capital is fairly low. However, we note the subjective discount rate now plays no role on stability. Indeed, comparison of propositions 1 and 3 reveals that the stability properties dramatically change depending on whether fertility is endogenous or exogenous, as shown in the next section.

Exogenous versus Endogenous Fertility: Some Analytical and Numerical Results

In order to compare exogenous and endogenous fertility economies focusing specifically on dynamical features, we now assume that the same steady-state number of children is raised in both contexts. This amounts to saying that the same unique steady-state capital stock is achieved.¹¹ First of all, it is important to remark that

Remark 2: When PAYG pensions are absent ( $θ = 0$ ) and the number of children is the same under exogenous and endogenous fertility, the dynamic paths of capital accumulation and the steady states coincide.

In contrast, when PAYG pensions exist

(0 < θ < 1)

, we find that the same positive steady state is achieved but with two different dynamic adjustment processes. Moreover, the following proposition holds:

Proposition 4: The risk of cyclical instability caused by PAYG transfers is higher under endogenous fertility. Moreover, under endogenous (exogenous) fertility the subjective discount factor is neutral for stability (acts as an economic stabilizer).

Proof: Under exogenous (endogenous) fertility cyclical instability emerges only when $0 < α < α_{4}$ and $θ_{2} < θ < 1$ ( $0 < α < 1 / 3$ and $θ_{4} < θ < 1$ ). As $α_{4} < 1 / 3$ , then $θ_{4} < θ_{2} = θ_{4} (1 + β)$ . Moreover, $\frac{\partial θ_{2}}{\partial β} = θ_{4} > 0$ and $\frac{\partial θ_{4}}{\partial β} = 0$ . Q.E.D.

Indeed, when fertility is exogenous, a rise in

β

reduces the relative weight of the public pension component on savings, and this in turn tends to mitigate the negative effect of a higher contribution rate on capital accumulation (see equation (12), especially). In contrast, when fertility is endogenous, the subjective discount factor does not alter capital accumulation through the public pension component (see equation (22)) because it negatively affects in the same way both savings and the demand for children. Therefore, the cyclical unstable region in the parameter space

(α, θ)

is larger when fertility is endogenous.

Figure 1 illustrates the content of proposition 4 and displays the different size of the cyclically unstable regions under exogenous and endogenous fertility.

Figure 1.

Stability-instability regions in the space $(α, θ)$

In the following two sections (Change in PAYG Pensions When the Number of Children is the Same Under Exogenous and Endogenous Fertility and Change in PAYG Pensions: Policy Experiment), we compare the dynamic evolution of demoeconomic variables and long-run outcomes in both economies when $θ$ changes.

Change in PAYG Pensions When the Number of Children Is the Same Under Exogenous and Endogenous Fertility

While the previous section has presented some analytical results, the quantitative implications will be better seen through numerical simulations, which can help to reveal other effects of social security when ambiguity exists in the analysis (e.g., the effects of the pension contribution rate on the population growth).

To illustrate the different dynamic adjustment processes under exogenous and endogenous fertility, we take the following parameter set: $A = 10$ , $α = 0.2$ , and $β = 0.4$ for both economies. Moreover, without losing generality, we assume that $\bar{n} = 1$ , (i.e., stationary population). Then we choose $ϕ = 0.3$ and calibrate $q$ (for different values of $θ$ ) to obtain $n^{*} = 1$ under endogenous fertility. In particular, we assume $θ = 0.05$ ( $0.35$ ) [ $0.46$ ] to obtain $q = 0.1745$ ( $0.133$ ) [ $0.1124$ ]. The simulations results are summarized in result 1 and Figures 2 through 4.

Result 1: Given the same initial condition, $k_{0}$ , economic and demographic variables under exogenous and endogenous fertility show the following three distinct dynamic behaviors, depending on the relative size of the PAYG system.

Monotonic and convergent dynamics (the pension contribution rate is fairly low). Figure 2A illustrates the phase map under exogenous and endogenous fertility when θ = 0.05 and q = 0.1745. Figure 2B represents the corresponding time plot for n_t (k ₀ = 0.01 and n ₀ = 0.7), showing that under endogenous fertility population increases in the first stages of development and then approaches the stationary state.

Nonmonotonic dynamics, which may result in (i) nonmonotonic convergence toward the stationary state in capital and fertility (for intermediate values of the contribution rate). Figure 3A displays the phase map of capital accumulation, which is unimodal in both cases, when θ = 0.35 and q = 0.133. The steady state is approached with dampened oscillations. (Given the different slopes of the phase maps at the equilibrium point, fluctuations in demoeconomic variables under endogenous fertility are larger than under exogenous fertility, as shown in the corresponding time plot for both n _t and k _t; see Figure 3B through D, where k ₀ = 0.01 and n ₀ = 0.7); (ii) irregular fluctuations in capital and fertility (e.g., the pension contribution rate is fairly high). Figure 4A displays the phase map when θ = 0.46 and q = 0.1124. The corresponding time plots for n _t and k _t reveal that oscillations are permanent also in the very long run when fertility is endogenous, while converging toward the stationary state when fertility is exogenous (see Figure 4B–D, where $k_{0} = 0.01$ and $n_{0} = 0.7$ ).

The economic intuition is the following. In the first stages of development saving grows faster than fertility, the latter being univocally of the Malthusian type (endogenous fertility); that is, the higher income, the higher the number of children. When economies develop, however, the reduction in saving is on average larger than that of fertility, and fluctuations are larger under endogenous fertility. It is important to note that when

θ

becomes larger and families rationally compares benefits and costs of having children, we can observe stages of development in which fertility subsequently increases and decreases. Moreover, the intensity of these fluctuations either shrinks as the steady state is approached or indefinitely persists. In the latter case (when the contribution rate is fairly high), our model predicts economic cycles as well as persistent and marked baby booms and baby busts, which may be of irregular (bounded) amplitude and length (see Figure 4B and D). This behavior accords with the observed income fluctuations over time, the baby boom and baby bust observed in the last century, especially in Europe and the United States (Greenwood, Seshadri, and Vandenbroucke 2005), and the persistent and steady reduction in fertility experienced in the Western world (Murphy, Simon, and Tamura 2008). Notice that, if fertility were exogenous, then ceteris paribus not only baby booms and baby busts but also economic cycles would be prevented (see Figure 4C).

Figure 2.

A, Phase map ( $θ = 0.05$ and $q = 0.1745$ ). B, Time plot for $n_{t}$ ( $n^{*} = n = 1$ ).

Figure 3.

A, Phase map ( $θ = 0.35$ and $q = 0.133$ ). B, Time plot for $n_{t}$ ( $n^{*} = n = 1$ ), C, Exogenous fertility: time plot for $k_{t}$ . D, Endogenous fertility: Time plot for $k_{t}$ .

Figure 4.

A, Phase map ( $θ = 0.46$ and $q = 0.1124$ ). B, Time plot for $n_{t}$ ( $n^{*} = n = 1$ ). C, Exogenous fertility: time plot for $k_{t}$ . D, Endogenous fertility: Time plot for $k_{t}$ .

Therefore, the existence of large PAYG transfers is sufficient to generate realistic predictions about demoeconomic outcomes in an OLG model with endogenous fertility, such as the observed phase of fertility drop, which has otherwise been explained, for instance, by the quantity–quality theory of fertility (Becker and Lewis 1973; Becker and Tomes 1976; Barro and Becker 1989) or by the theory of female labor participation (Mincer 1962).

Change in PAYG Pensions: Policy Experiment

In the current political debate in several developed countries, there is wide consensus to increase pension contributions with the aim to balance public pension budgets. What are the effects of an increase in pension contributions on the time evolution of capital accumulation and fertility?

Taking into account some forecasts on the social security contribution rate, such as Liikanen (2007), we conduct a policy experiment using the same values of technological and preference parameters as in the section Change in PAYG Pensions When the Number of Children is the Same Under Exogenous and Endogenous Fertility, in order to compare the short- and long-run outcomes in fertility and capital corresponding to a change from $θ = 0.1$ to $θ = 0.3$ at a certain point in time ( $t = 2$ ) in models with exogenous and endogenous fertility. We start by assuming $\bar{n} = n^{*} = 1$ and $q = 0.1702$ in both economies. Note that before the change in the social security contribution rate, the steady-state stock of capital under exogenous and endogenous fertility is the same (see Figure 5B and C).

Figure 5.

A, Endogenous fertility: change from $θ = 0.1$ to $θ = 0.3$ at $t = 2$ (Time plot for $n_{t}$ ). B, Exogenous fertility: change from $θ = 0.1$ to $θ = 0.3$ at $t = 2$ (Time plot for $k_{t}$ ). C, Endogenous fertility: change from $θ = 0.1$ to $θ = 0.3$ at $t = 2$ (Time plot for $k_{t}$ ).

Fertility

Figure 5A represents the time plot for n_t portrayed from t = 2 (the time when θ changes) onward. Note that the upper curve (θ = 0.1) started out at t = 0 (not shown in the figure), with k ₀ = 0.01 and n ₀ = 0.7 as initial conditions. Then we assume k ₂ = 1.42 (see Figure 5C) as the initial condition for k at the time of the change in θ. Case θ = 0.1: after a few oscillations, fertility approaches the stationary state (n = 1). Case θ = 0.3: the rise in θ causes an 18 percent reduction in fertility (i.e., almost 2 children per couple when θ = 0.1 against 1.65 children per couple when θ = 0.3) followed by phases of baby booms and baby busts, dampened over a dozen generations, until the new and lower long-run fertility rate is approached.

Capital

Figure 5B and C show that the capital stock installed at t = 2 when θ = 0.1 is k ₂ = 1.35 (k ₂ = 1.42) under exogenous (endogenous) fertility. In the short run (t = 2), the rise in θ causes a 48 (24) percent reduction in the capital stock under exogenous (endogenous) fertility. This is because the reduction in saving is mitigated by the reduced number of children when fertility is endogenous. In the long run, there are phases of small and fast (wide and slow) oscillations under exogenous (endogenous) fertility until the new low (high) steady state is achieved around the seventh (thirtieth) generations.

Conclusion

This article dealt with the study of the dynamical and steady-state outcomes of a double Cobb–Douglas OLG economy with public PAYG pensions and shortsighted individuals under exogenous and endogenous fertility. We showed, ceteris paribus, that an economy is likely to be exposed to endogenous fluctuations when fertility is an economic decision variable. Indeed, in such a case our model generates a population dynamics that mimic the historical evolution of population with an increasing trend with increasing income (i.e., the Malthusian-type fertility observed in the first stages of development) and phases of baby booms and baby busts before either approaching the steady-state level of fertility or permanently oscillating around it.¹² Indeed, such oscillations may be consistent, in a broad sense, with the baby boom and the subsequent fertility drop observed in some actual developed economies in the last sixty years (Greenwood, Seshadri, and Vandenbroucke 2005). Moreover, the existence of an endogenous population dynamics allows for a novel realistic explanation of regular and even chaotic demoeconomic cycles. Therefore, to the extent that fertility choices are based on a comparison between benefits and costs of children, the effects of PAYG pensions may be markedly different from those expected in the case in which fertility is determined out of the economic sphere.

Since the rational choice of fertility may be considered going hand to hand with the stages of economic development, and since social security reforms are currently high on the political agenda in several developed economies, our findings constitute policy advice about the economic and demographic effects of raising pension transfers when fertility is endogenous.¹³

Footnotes

Appendix A

Appendix B

Appendix C

Notes

We gratefully acknowledge the editor Prof. James Alm and two anonymous referees for valuable comments on an earlier draft. The usual disclaimer applies.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Barro

Robert J.

Becker

Gary S

. 1989. “Fertility Choice in a Model of Economic Growth.” Econometrica 57:481–501.

Becker

Gary S.

1960. “An Economic Analysis of Fertility.” In Demographic and Economic Change in Developing Countries, edited by National Bureau Committee for Economic Research, 225–56. Princeton, NJ: Princeton University Press.

Becker

Gary S.

Barro

Robert J

. 1988. “A Reformulation of the Economic Theory of Fertility.” Quarterly Journal of Economics 103:1–25.

Becker

Gary S.

Gregg Lewis

. 1973. “On the Interaction between the Quantity and Quality of Children.” Journal of Political Economy 81:S279–88.

Becker

Gary S.

Tomes

Nigel

. 1976. “Child Endowments and the Quantity and Quality of Children.” Journal of Political Economy 84:S143–62.

Becker

Gary S.

Murphy

Kevin M.

Tamura

Robert

. 1990. “Human Capital, Fertility and Economic Growth.” Journal of Political Economy 98:S12–37.

Boldrin

Michele

Jones

Larry E

. 2002. “Mortality, Fertility and Saving in a Malthusian Economy.” Review of Economic Dynamics 5:775–814.

Chen

Hung-Ju

Ming-Chia

Lin

Yung-Ju

. 2008. “Chaotic Dynamics in an Overlapping Generations Model with Myopic and Adaptive Expectations.” Journal of Economic Behavior & Organization 67:48–56.

Cigno

Alessandro

. 2007. “Low Fertility in Europe: Is the Pension System the Victim or the Culprit?” Europe and the Demographic Challenge. CESifo Forum 8:37–41.

10.

Deardorff

Alan V.

1976. “The Optimum Growth Rate for Population: Comment.” International Economic Review 17:510–15.

11.

de la Croix

David

Michel

Philippe

. 2002. A Theory of Economic Growth. Dynamics and Policy in Overlapping Generations. Cambridge: Cambridge University Press.

12.

Diamond

Peter A.

1965. “National Debt in a Neoclassical Growth Model.” American Economic Review 55:1126–50.

13.

Eckstein

Zvi

Wolpin

Kenneth I

. 1985. “Endogenous Fertility and Optimal Population Size.” Journal of Public Economics 27:93–106.

14.

Epstein

Larry G.

Zin

Stanley E

. 1991. “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis. Journal of Political Economy 99:263–86.

15.

Farmer

Roger E. A.

1986. “Deficits and Cycles.” Journal of Economic Theory 40:77–86.

16.

Fanti

Luciano

Gori

Luca

. 2011. “Fertility and PAYG pensions in the overlapping generations model.” Journal of Population Economics. (Forthcoming). doi:10.1007/s00148-011-0359-7.

17.

Fanti

Luciano

Spataro

Luca

. 2008. “Poverty Traps and Intergenerational Transfers.” International Tax and Public Finance 15:693–711.

18.

Feldstein

Martin

. 1974. Social Security, Induced Retirement, and Aggregate Capital Accumulation. Journal of Political Economy 82:905–26.

19.

Fenge

Robert

Meier

Volker

. 2005. “Pensions and Fertility Incentives.” Canadian Journal of Economics 38:28–48.

20.

Galor

Oded

Weil

David N

. 1996. “The Gender Gap, Fertility, and Growth.” American Economic Review 86:374–87.

21.

Gollin

Douglas

. 2002. “Getting Income Shares Right.” Journal of Political Economy 110:458–74.

22.

Grandmont

Jean-Michel

. 1985. “On Endogenous Competitive Business Cycles.” Econometrica 53:995–1045.

23.

Greenwood

Jeremy

Seshadri

Ananth

Vandenbroucke

Guillaume

. 2005. “The Baby Boom and Baby Bust.” American Economic Review 95:183–207.

24.

Hansson

Ingemar

Stuart

Charles

. 1989. “Social Security as Trade Among Living Generations.” American Economic Review 79:1182–95.

25.

Jones

Charles I.

2005. “The Shape of Production Functions and the Direction of Technical Change.” Quarterly Journal of Economics 120:517–49.

26.

Kolmar

Martin

. 1997. “Intergenerational Redistribution in a Small Open Economy with Endogenous Fertility.” Journal of Population Economics 10:335–56.

27.

Tien-Yien

Yorke

James A

. 1975. “Period Three Implies Chaos.” American Mathematical Monthly 82:985–92.

28.

Liikanen

Erkki

. 2007. “Population Ageing, Pension Savings and the Financial Markets.” Bank for International Settlements Review 53:1–6.

29.

Manfredi

Piero

Fanti

Luciano

. 2006. “The Complex Effects of Demographic Heterogeneity on the Interaction between the Economy and Population.” Structural Change and Economic Dynamics 17:148–73.

30.

Mankiw

N. Gregory

Romer

David

Weil

David N

. 1992. “A Contribution to the Empirics of Economic Growth.” Quarterly Journal of Economics 107:407–37.

31.

Michel

Philippe

Croix

David de la

. 2000. “Myopic and Perfect Foresight in the OLG Model. Economics Letters 67: 53–60.

32.

Michel

Philippe

Pestieau

Pierre

. 1999. “Social Security and Early Retirement in an Overlapping-Generations Growth Model.” CORE Discussion Paper No. 9951.

33.

Mincer

Jacob

. 1962. “Labor Force Participation of Married Women: A Study of Labor Supply. In Aspects of Labor Economics, edited by Gregg Lewis

, 63–105. Princeton, NJ: Princeton University Press.

34.

Murphy

Kevin

Simon

Curtis

Tamura

Robert

. 2008. “Fertility Decline, Baby Boom, and Economic Growth.” Journal of Human Capital 2:262–302.

35.

Reichlin

Pietro

. 1986. “Equilibrium Cycles in an Overlapping Generations Economy with Production.” Journal of Economic Theory 40:89–102.

36.

Samuelson

Paul A.

1975a. “Optimum Social Security in a Life Cycle Growth Model.” International Economic Review 16:539–44.

37.

Samuelson

Paul A.

1975b. “The Optimum Growth Rate of Population.” International Economic Review 16:531–38.

38.

Tabellini

Guido

. 1990. “A Positive Theory of Social Security.” National Bureau of Economic Research Working Paper No. 3272.

39.

van Groezen

Bas

Leers

Theo

Meijdam

Lex

. 2003. “Social Security and Endogenous Fertility: Pensions and Child Allowances as Siamese Twins.” Journal of Public Economics 87: 233–51.

40.

Zhang

Junxi

. 1999. “Environmental Sustainability, Nonlinear Dynamics and Chaos.” Economic Theory 14:489–500.

41.

Zhang

Junsen

Zhang

Junxi

. 1998. “Social Security, Intergenerational Transfers, and Endogenous Growth.” Canadian Journal of Economics 31:1225–41.