2.1. A Brief History of Fertility in Europe and the US: 1930–2000

As already mentioned, we are interested in understanding how much of the following two facts, depicted in Figures 1 and 2 below, can be accounted for by the difference in the national Social Security systems.

Figure 1. TFR in the USA: 1800–1990.

Figure 2. TFR’s in Europe: 1900–1990.

Fact 1: Both in Europe and in the US fertility rates, as measured by the TFR, have decreased constantly over most of the 20th century. The total variation over the 50-year period 1950–2000 is about 1.3 children per woman in Europe, where it has fallen from about 2.8 to about 1.5, and about 1.0 in the US, where it has fallen from about 3.0 to about 2.0.

Fact 2: While in 1920 the average TFRs in Europe and the US were roughly equal, in 2000 they are about 0.4–0.8 children apart (depending on country); the TFR in the US is at 2.0 children per woman, while in Europe it is between 1.2 and 1.6 children per woman.

There are several other relevant facts to keep in mind when interpreting these differences in the historical patterns of fertility in the US and Europe. In the demographic literature, two factors are usually treated as the main driving forces behind long run movements in fertility: reductions in infant mortality rates (IMR) and increases in Female Labor Force Participation Rates (FLFPR).

While IMR might be reasonably thought of as exogenous in individual fertility decisions, labor force participation is clearly endogenous to household decisions. As such, any explanation of variations in TFR based on variations in FLFPR only begs for the common factor(s) affecting both. Leaving this objection aside, it is also clear from the data that the facts cannot be accounted for on the basis of the correlation between TFR and FLFPR. While it is true that FLFPRs have increased over time in both Europe and the US, this has occurred at very different rates. Moreover, over the 1980s and 1990s the cross-country correlation between TFR and FLFPR has turned positive instead of negative (Adsera, Reference Adsera2004). In particular, current LFPRs are higher in the U.S. than in Europe, while TFRs are lower in Europe. Thus, while the time series changes in TFRs in each individual country are consistent with an increase in FLFPR and a negative correlation between TFR and FLFPR this explanation alone cannot account for the cross-sectional evidence. Indeed, the cross-sectional evidence would require the opposite correlation.

Similar, even if less extreme, problems arise with the IMR. The separate time series behavior of TFRs in Europe and the US are consistent with the observed drop in IMR; the respective drops in IMR were from 37/1000 for the U.S. to about 7/1000 (from 1950 to 2000) and, from values between 22/1000 and 60/1000 (depending on the country) to values between 4/1000 and 7/1000 in Europe. Taking 0.030 as our point estimate of the correlation between IMR and TFR (which is halfway between the two estimates of Regressions II and IV given below), the observed time series variations in country by country IMR can account for a drop in fertility that ranges from 0.5 (Sweden) to 1.6 (Spain) children per woman. But an elasticity of 0.030 cannot possibly account for the current differences in TFR between the US and Europe, neither now nor fifty years ago. Mortality rates among infants are basically identical on the two sides of the Atlantic these days, and were higher, not lower, in Europe than in the US in the 1950s. Hence, while a reduction in IMRs has certainly played a role, along the lines of, e.g., Boldrin and Jones (Reference Boldrin and Jones2002), in the fertility decline of both the US and Europe, this explanation also has difficulty with the observed cross-sectional differences over this period.

Similar problems arise with other putative explanations, e.g., increases in income per capita, female education levels, or in the degree of urbanization.

Thus, to account coherently for both facts on the basis of changes in factors that are usually associated to long run movements in fertility, appears difficult.

In contrast, the size and timing of the growth in government pension systems correlates well with both the time series and cross-sectional observations: Beginning shortly after WWII the size and relevance of Social Security were roughly the same as in the US and in European countries. Since then, Social Security has grown everywhere, but this increase has been much more dramatic in Europe than in the US. When the system was first introduced in the US, it was quite small—there were about 50 thousand beneficiaries in 1937, and only two hundred thousand in 1940; it is only right after WWII that the system takes off, and in 1950 the number of beneficiaries reached 3.5 million. Thus, as an approximation, the size of the pension system was 0% of labor income in 1935;Footnote ⁶ currently, tax receipts and payments are approximately 10% of labor income. In Europe, the payments of the systems were also approximately 0% of labor income in 1935, but the growth has been much more dramatic, in some countries pension payments stand as high as 20–25% of labor income. The history of the U.K system lies someplace in between; for details compare the historical section of the chapters in Gruber and Wise (Reference Gruber and Wise1999) dedicated to European countries.

We would be remiss if we did not point out the anomalous behavior of fertility rates during the 1920–1950 period both in Europe and in the US (where the changes are larger). In both, measured TFR, which had been steadily decreasing since 1800 in parallel with the decrease in IMR and the increase in urbanization, took a sharp swing downward around 1920, reaching particularly low levels during the 1930–1940 decade. Fertility snapped back to much higher levels (about 50% higher, in fact) during the “baby boom” period—1940–1960—after which it decreased again to the current low levels.Footnote ⁷ Both of these movements are hard to account for on the basis of movements in the standard variables used by demographers to track long fun movements in fertility (IMR, urbanization, female education, and the other, assorted socioeconomic variables used in empirical studies). Thus, although explaining the whole 1920–1960 fertility ” swing” is a fascinating and challenging task, it will not be taken up here.Footnote ⁸

2.2. Cross-Sectional Data

The loose, but suggestive, discussion of the relative sizes and timing of changes in government pension systems in Europe and the US and their relationship to observed changes in fertility given above is further strengthened by an examination of cross-sectional evidence. We examine a cross section of 104 countries taken from 1997. The raw data are shown in Figure 3.

Figure 3. Cross-country correlation, social security tax and TFR.

Although one must be careful about causal interpretations, the data in cross section show a strong negative relationship between the TFR in a country and the size of its Social Security and pension system. This plots TFR for the country in 1997 versus Social Security expenditures as a fraction of GDP in 1997, denoted SST, for these countries. Since this second variable is a measure of the average tax rate for the Social Security system as a whole, we identify it with the SST in what follows. Although the relationship is far from perfect, as can be seen, there is a strong negative relationship between these two variables. Most notably, there are only four countries for which SST is at least 6% and TFR is above 2 (children per woman).Footnote ⁹

In contrast to this, in those countries where TFR is above three, none has an SST above 4%. This is suggestive of the overall relationship between these two variables. Regression results from this data set confirm and quantify the visual impression, as summarized in the table below.Footnote ¹⁰ For cross-section regressions, the dependent variable is TFR, SST is the Social Security tax rate estimated as total expenditures on the Social Security system as a fraction of GDP (in 1997), GDP is per capita GDP in 1995 (in USD 1,000 ), IMR is the Infant Mortality Rate, estimated as the number of deaths per 1,000 live births (in 1997).

As can be seen, the coefficient on SST is negative and highly statistically significant. It is also economically significant. Most LDCs (Lesser Developed Countries) have either no Social Security system or a very small one. In contrast, SST is between 7% and 16% for most developed countries, but only European countries have ratios above 10%. Thus, the relevant range for calculations is in changes in SST from 0% (0.00) to 10% (0.10). Our regressions imply that, everything else the same, an increase in SST of this size (i.e., from 0% to 10%) is associated with a reduction in the number of children per woman of between 0.7 and 1.6. In Regression II, we include two other variables that might either give alternative explanations for the results in column I or allow for a sharper estimation of the conditional correlation between SST and TFR. They are per capita GDP and IMR. Although the size and significance of SST do fall somewhat, it remains substantially negative and statistically significant, while the coefficient on GDP is not significant; the coefficient on IMR has the expected positive sign and is highly significant, which is consistent with the quantitative theoretical predictions of Boldrin and Jones (Reference Boldrin and Jones2002). We also did regressions including education variables from the Barro–Lee data set as additional predictors. The addition of these variables left the coefficient estimates on SST and IMR virtually unchanged and still highly significant. The addition of these variables, while not significant themselves, did increase the size of the GDP coefficient and made it statistically significant.Footnote ¹¹

2.3. A Small Panel Study

We find similar results when we look at panel data. Here, we look at a panel data set of TFRs and SST’s in eight developed countries over the period from 1960 to 2000.Footnote ¹² The eight countries are: Austria, Belgium, Denmark, Finland, France, Ireland, Norway, and Spain. A summary of the data are shown in Figure 4 and the regression results are given in Table 1. The columns labeled Regression III and IV show the results of two simple regressions for this panel data set. (Uncorrected for autocorrelation and/or heteroscedasticity.) The variables here have the following meaning: TFR is still TFR in that country/year, SST is the Social Security tax rate measured as Social Security expenditure over labor earnings, IMR is as before, and 65% is the share of the population aged 65 years or older; per capita GDP has been omitted as it is never significant.

Table 1. Fertility and social security, cross section and country panel

Figure 4. Social security tax and TFR in eight European countries.

The results from this panel regression are qualitatively similar to what we saw above in the cross section—viz., an increase in SST leads to a reduction in TFR, even after controlling for IMR and for the share of elderly people in the population. Quantitative comparisons are more delicate, as the measure for SST adopted here differs from the previous one. Still, if one takes the rough, but overall accurate, approximation that labor earnings are 2/3 of GDP, then an increase in the Social Security expenditure over GDP from 5% to 15% is associated also in the panel data with a fall in TFR of between 1.0 and 1.8 children per woman, similar to the estimates in the cross-section data.

These findings are subject to the same cautions which always accompany regression studies, but they are highly suggestive that SST may indeed have an effect on fertility decisions, that this effect is to reduce the number of children that people have, and that this effect is fairly large in size: an increase of the Social Security system on the order of 10% of GDP is associated with a reduction in TFR of between 0.7 and 1.6 children per woman.

These results are of considerable interest but also must be interpreted with care. In many countries, the Social Security system not only provides old-age insurance (i.e., an annuity) financed with an ad-hoc tax on labor income, but also has an element of forced savings. That is, the benefits paid out to an individual are dependent, to varying degrees in different countries, on the contributions made over the working lifetime of the payee. Because of this, the exact relationship between SST in these regressions and the SST rate in subsequent sections is imperfect. That is, in the models, we will assume that SST is financed through a labor income tax and is paid out lump sum. Thus, from the point of view of testing the model predictions, we would ideally like to have data on that part of SST that most closely mirrors our lump-sum payment mechanism. Data limitations prevent us from attempting this, however. Thus, the effective change in the SST that is relevant for the models is probably smaller than what we have found in the previous regressions.

3.1. Exogenous Growth and BGPs

We assume that there is exogenous labor augmenting technological change, A_t = γ ^t _A A ₀. As it is well known, for there to be balanced growth it must also be that a_t = γ ^t _A a ₀, b_t = b, and τ _t = τ. Accordingly we define the de-trended variables in the standard way. That is, $\hat{c}_{t}^{o}=c_{t}^{o}/\gamma _{A}^{t}$ , $\hat{c} _{t}^{m}=c_{t}^{m}/\gamma _{A}^{t}$ , $\hat{d}_{t}=d_{t}/\gamma _{A}^{t}$ , $\hat{s}_{t}=s_{t}/\gamma _{A}^{t}$ , $\hat{x}_{t}=x_{t}/\gamma _{A}^{t}$ , and $\hat{T}_{t}^{o}=T_{t}^{o}/\gamma _{A}^{t}$ . Finally, we denote n_t /n _{t − 1} = γ _nt . Under our assumptions, if $\hat{x}_{t}$ , $\hat{s }_{t}$ , and γ _nt converge to constants then, so do $\hat{w}_{t}$ , R_t , and $\hat{\theta }_{t}$ and, consequently, the equilibrium quantities. The balanced growth equations that these must satisfy are given by:

(4) \begin{equation} \hat{c}^{o}=\zeta ^{1/\sigma }\hat{c}^{m} \end{equation}

(5) \begin{equation} \hat{c}^{o}=\frac{\beta ^{1/\sigma }}{\gamma _{A}}\hat{c}^{m}\left[ \frac{ \partial c^{o}}{\partial s}\right] ^{1/\sigma } \end{equation}

(6) \begin{equation} \hat{c}^{o}=\left[ \frac{\beta }{\hat{\theta }\gamma _{A}^{(\sigma -1)}} \right] ^{1/\sigma }\hat{c}^{m}\left[ \frac{\partial c^{o}}{\partial n} \right] ^{1/\sigma } \end{equation}

(7) \begin{equation} \frac{\partial c^{o}}{\partial s}=\frac{\zeta ^{1/\sigma }(1-\xi )R}{\zeta ^{1/\sigma }+\gamma _{n}} \end{equation}

(8) \begin{equation} \frac{\partial c^{o}}{\partial n}=\frac{\zeta ^{1/\sigma }}{(\zeta ^{1/\sigma }+\gamma _{n})^{2}}\left[ \zeta ^{1/\sigma }\left( (1-\tau )\hat{w }-\hat{s}-\hat{\theta }\gamma _{n}\right) -(1-\xi )R\hat{x}-\hat{T}^{o}\right] \end{equation}

(9) \begin{equation} \hat{c}^{m}=(1-\tau )\hat{w}(1-b\gamma _{n})-\hat{a}\gamma _{n}-\hat{d}-\hat{ s} \end{equation}

(10) \begin{equation} \hat{c}^{o}=\gamma _{n}\hat{d}+(1-\xi )R\hat{x}+\hat{T}^{o} \end{equation}

(11) \begin{equation} \hat{x}=\frac{\xi R\hat{x}}{\gamma _{A}\gamma _{n}}+\frac{\hat{s}}{\gamma _{A}} \end{equation}

(12) \begin{equation} \hat{w} =F_{\ell }(\hat{x},A_{0}\gamma _{n}(1-b\gamma _{n}))\text{,} \end{equation}

(13) \begin{equation} R =(1-\delta )\;+F_{k}(\hat{x},A_{0}\gamma _{n}(1-b\gamma _{n}))\text{,} \end{equation}

(14) \begin{equation} \hat{\theta } =\hat{a}+(1-\tau )b\hat{w}\text{,} \end{equation}

(15) \begin{equation} \hat{T}^{o} =\gamma _{n}\tau \hat{w}(1-b\gamma _{n})\text{.} \end{equation}

Simple manipulations give the following expression for the growth rate of population:

\begin{equation*} \gamma _{n}=\zeta ^{1/\sigma }\left( \frac{\beta (1-\xi )R}{\gamma _{A}^{\sigma }\zeta }-1\right). \end{equation*}

From the above equation it is clear that steady state fertility only depends on the preference parameters ζ, β, and σ, the exogenous rate of growth of technological progress γ _A , the equilibrium interest rate, R, and the degree of capital market imperfection ξ. This implies that the other parameters, such as the costs of having children or the size of the Social Security system, impact steady state fertility only indirectly, through general equilibrium effects embedded in the interest rate. Therefore, in small closed economies, or in economies with a linear technology and fixed prices, there would be no such effects. Most notably, fertility would be invariant to both the size of the Social Security system and the costs of having children. The Barro and Becker model of fertility, as shown in the Appendix, displays a similar feature. In both models, the effects of Social Security on fertility come from general equilibrium effects.

Increasing ξ corresponds to forcing the old to pass on more of their savings to their children and thus represents reducing access to capital markets. This has a direct effect on the growth rate of population as can be seen. Surprisingly, holding R constant and increasing ξ causes γ _n to fall, the opposite of what one would expect. There is also an indirect effect of a change in ξ on R. A careful examination of the rate of return condition shows that the indirect effect goes in the opposite direction. In fact, due to the general equilibrium equalization of the rate of return on saving with the rate of return on fertility, an increase in ξ leads to lower investment in physical capital and, hence, a higher value of R in equilibrium. Because of these offsetting effects, the overall impact of more efficient capital markets on the value of (1 − ξ)R and, hence, on the growth rate of population depends on parameters. In Section 6, below, we find that the overall effect is negative as would be expected.

The detailed analysis of Social Security in the Barro and Becker model is presented in the Appendix. As with the Caldwell model, it turns out that any effects on steady state fertility from changes in the size of a PAYGO Social Security system must come through indirect effects working off changes in the equilibrium interest rate.

In sum then, neither of the two models delivers an explicit and unambiguous prediction about the direction of the effect of the introduction of a PAYGO Social Security system on fertility and the growth rate of population. Thus, any effect can only be identified through a more thorough, quantitative exercise. This is what we turn to next.

4.1. Functional Forms

Utility

Recall from Section 3 that for the Caldwell model, the period utility function is assumed to be given by:

\begin{equation*} u(c_{t}^{m},c_{t}^{o},c_{t+1}^{o})=\frac{({c_{t}^{m}})^{1-\sigma }}{1-\sigma }+\zeta \frac{({c_{t}^{o}})^{1-\sigma }}{1-\sigma }+\beta \frac{({c_{t+1}^{o} })^{1-\sigma }}{1-\sigma }. \end{equation*}

Production

We assume that the production function is CRS with constant depreciation, and is given by:

\begin{equation*} (1-\delta )K+F(K,L)=(1-\delta )K+AK^{\alpha }L^{1-\alpha }. \end{equation*}

Inputs and output markets are assumed competitive.

4.2. Facts to Match

Setting ξ = 0, there are a total of nine parameters in the Caldwell model.

A number of these parameters are used in macroeconomic models of growth and the business cycle, hence, in calibrating them we follow the existing literature for as many as we can. Accordingly, we normalize A to 1, we set annual depreciation to 8%, and we fix the share of income that goes to capital to either 0.22 or 0.33.Footnote ¹⁵ We have set the parameter γ _A equal to 1.25% on a yearly basis following Oliveira Peires and Garcia (Reference Oliveira Pires and Garcia2012) estimation for developed countries over the 1970–2000 period, and Dennison’s calculations for the 20th century US.

Additionally, we have made the choice to set the relative weights on the flow utility from current consumption of the old (ζ) to be one for both models. While it makes our life easier, this choice implies, obviously, that on a per capita basis consumption of old parents is equal to that of middle age children. This contradicts the evidence from the empirical life time consumption literature that suggests a drop in all measures of per capita consumption after retirement; estimates of the ratio between average consumption while working and while retired yield values of about 0.70–0.80 . For this ratio to be obtained by c^o /c^m we need to set ζ < 1.0. The impact of this different calibration is also considered in the section on sensitivity analysis, below.

Given these choices, we still need to determine the values of the four parameters β, σ, a, and b. To make our results as clear as possible, for each model we consider two extreme cases: one in which all of the costs of raising children are in terms of goods (b = 0), the other in which they are completely in terms of time (a = 0). This implies calibrating three parameters at a time. The model makes either explicit or implicit predictions about a large number of potentially measurable variables that could be used to help in the calibration: the real rate of return on safe investments, donations as a share of income or consumption, the TFR and the growth rate of the population, the amount of time and/or resources devoted to rearing children, the composition of the population by age group, etcetera. As we must pin down only three parameters we need three independent observations.

To do this, the first step is to choose the country and the historical period the calibrated model is anchored to. Several alternatives are possible, the most obvious choices would be to use data from either the US or Europe at some point in time before government pension programs took off. The US Social Security Administration was created in 1935, thus it would seem natural to calibrate to the US in 1935. However, the period 1930–1950 is also characterized by two anomalous events—the second World War and the Great Depression. In principle both events might have had a major impact on fertility rates, and they certainly had large impacts on the capital–output ratio, measured TFP, and the rate of return on capital; the latter are all relevant macro variables we are taking into consideration to calibrate our model. For these reasons, we calibrate the model to observations from 2000. Because the US is much more homogeneous than Europe, and because we have already set a number of model’s parameters on the basis of US observations, our calibration benchmark is the US in year 2000.

The independent observations we aim at matching are the TFR, the capital–output ratio, and the childbearing costs. In the US the TFR was at 1.75 in 1980, at 2.03 in 1990, and it is around 2.06 currently. Thus, we will take a TFR of 2.00 to be the current “steady-state” level. From Maddison (Reference Maddison1995a, b) we take the capital to output ratio to be between 2.4 and 2.5. We also need to have an estimate of the cost of raising a child. Focus first on the case in which this cost is entirely in time, i.e., a = 0, and b > 0. For this, we set b to be 3% of the available family time, which corresponds to roughly 6% of the mother’s time per child. When total fertility is about 2.0 children per woman this number is consistent with the estimates on time-use data reported by Juster and Stafford (Reference Juster and Stafford1991), with the one estimated by Echevarria and Merlo (Reference Echevarria and Merlo1999) using data fitted to an international cross section, and also with the estimates reported by Moe (Reference Moe1998) based on Peruvian micro data. This number (b = 3%) may seem surprisingly low, in fact the opposite is true. In our context, the fraction b is applied to the total time available for work during the whole working life, while the 6% of mother’s time per child reported in the quoted studies refers only to the infancy-childhood years, which are generally substantially fewer than the active years of a mother. From this point of view, then, a value of b between 2% and 2.5% may be more appropriate; again, we refer to the sensitivity analysis section for this case.

Finally, the parameters describing the Social Security system must be chosen for the model. The exact form of the US Social Security system is much more complex than what we allow for here. Payments received depend, to some extent, on what was paid in and are therefore not exactly lump-sum. Figure 5 shows the time paths of both receipts and expenditures of the Social Security system from 1937 to date. These figures include both Social Security and Medicare, but omit Social Security Disability Insurance since this is not restricted to the elderly. As can be seen these are approximately 7% of GDP over the last two decades of the 20th century. Since labor’s share in income is 67% in the model, this corresponds to an average labor income tax rate of approximately 10%, and this is the value we used in the calibration.

Figure 5. Social security receipts and expenditures/GDP: 1937–2004.

Given this discussion, we will adopt the following three target values for our calibration for the year 2000, when τ = 10%:

1. a. capital–output ratio: 2.4 (annual basis),

2. b. the TFR: 2.0 children per woman, and,

3. c. the amount of time allocated to rearing children: 3% of family time per child.

The model has trouble matching these targets perfectly.Footnote ¹⁶ When ζ = 1.0, the elasticity of intertemporal substitution in consumption plays a very secondary role. Our choice of σ = 0.95 and β = 0.99 (yearly) yields a TFR of 1.82 (lower than the targeted value of 2.00) and an annual capital–output ratio of about 2.4 when τ = 0.10 . These two choices together imply an interest rate of about 2.9% per year, perhaps a bit on the low side, when α = 0.33.Footnote ¹⁷ For the case in which the time cost (b) is zero, we keep all other parameters the same and we set the good cost of rasing children (a) so that the resulting good cost of raising children as a fraction of per-capita output turns out to be 4.5%. This is a value for which we have a hard time finding real-world estimated counterparts, so we picked it only because it was consistent with observed TFR, capital/output ratios and interest rates at τ = 0.10 when all other calibrated parameters remained the same as above.

The parameter values used in the baseline calibration are summarized in Table 2.

Table 2. Model parameters

5.1. Basic Steady State Calculations

Each of the three questions raised above is addressed by comparing steady state calculations of fertility changing only the labor income tax rate used to finance the pension system (with a corresponding, period-by-period balanced budget change in lump sum transfers.) For this reason, we begin by presenting and discussing the basic calculations of comparative steady states that the model implies at our calibrated parameter values.

We begin by examining the case in which there are only time costs of having children. The figures graph different BGP values for a given variable as a function of the SST rate. Figures 6–10 plot, in order, the values of γ _n , K/Y, c^m /y and c^o /y Footnote ¹⁸ , s/y and d and nd corresponding to the values of τ on the horizontal axis.

Figure 6. Fertility and the SS tax, Caldwell model.

Figure 7. Capital output ratio and the SS tax, Caldwell model.

Figure 8. Consumption of O’s and M’s and the social security tax, Caldwell model.

Figure 9. Savings and the social security tax, Caldwell model.

Figure 10. Old age support and the SS tax, Caldwell model.

As we can see, in this framework when the SST moves from 0% to about 10%, the number of children decreases from about 1.15 to about 0.91 (0.9 if there are only goods costs to raising children), the capital–output ratio increases from about 2.2 to 2.4, and there is a sizeable decline in consumption of about 3.0% for both middle aged and old. Finally, donations (both total and per-child) and savings also decrease. The drop in output caused by the introduction of Social Security is large, roughly a 10% deviation from the undistorted BGP level. This drop is larger than that for savings, generating an increase in the capital–output ratios. The drop in fertility is also large as it is equivalent to 0.48 children per woman. When the SST is moved further to about 20–25%, the number of children decreases further to about 0.62–0.65, the capital–output ratio increases to 2.7–2.8 and per-capita consumption also decreases further.

5.2. Comparisons to the Data

Comparisons between Europe and the US, and across time

Comparing this to US and European data reported in Sections 1 and 2, we see that the drop predicted by the model is equal to 50% of the observed total drop in TFR between 1950 and 2000; the latter was about equal to one child per woman in the US and 1.3 children per woman in Europe.

Recall the basic facts that we want to examine. These are that in the US the TFR was about 3.0, and in Europe approximately 2.6, in 1950. At this time, the SST rate was approximately τ = 1% in both regions. By 2000, the SST rate in the US had climbed to around 10% while TFR fell to approximately 2.0. In Europe, both τ and TFR depend on the country, but the relevant range for τ is from around 20% (e.g., France or Germany) to 25% (Italy).

The model predictions for these quantities are contained in Tables 3 and 4.

Table 3. Model and Data, US 1950, and 2000

Table 4. Model and data, Europe in 2000

As can be seen in Table 3, the predicted value for TFR for the US is slightly low; 1.82 at τ = 10%, versus the targeted value of 2.0. This was discussed in the section on calibration, and is something that is true for all of the calculated values of TFR from the model. The model predicts that in 1950 fertility should have been 2.2 in both the US and in Europe, substantially lower than the actual value of 3.0 in the US and 2.6 in Europe. But, the predicted change in TFR is 0.38 children per woman or about 40% of the actual difference seen in the US data.

The relevant comparisons for countries like France and Germany with SST rates of τ = 20% are 1.44 for 2000, and 2.2 in 1950. (Here we use the value τ = 1% for 1950.) Again, the model predictions are systematically too low but as can be seen the predicted change in fertility is 2.2 − 1.44 = 0.76 children per woman. This is 50–60% of the observed drop in fertility, depending on the country. Further increasing τ to 25%, the value for Italy, we can see that the model predicts TFR to be 1.30, just slightly above the actual value, and about 75% of the observed change over the 1950 to 2000 period.

As far as comparisons between the US and Europe are concerned, the relevant comparison is between τ = 10% and τ ∈ [20%, 25%]. As can be seen, this implies a difference in TFRs of 1.92 − 1.37 = 0.55 children, comparable to the differences actually seen.

Comparison to the Regression Results of Section 2

Finally, using the cross section of countries studied in Section 2 we constructed two subgroups of countries, one with “large” Social Security systems, one with small.Footnote ¹⁹

From these groups, see Table 5, we can see that the changes predicted by the model are roughly in line with what is seen in the data. Indeed, the size of fertility difference predicted from the model when moving from the low SST group to the high SST group is 0.73 children per woman, while that in the data is 0.87 children per woman. With respect to the cross-section regressions presented earlier on, notice that the low SST group has, roughly, the same IMR rate as the high SST group but much lower values for the 65% variable: the range is 4.6–11.5, averaging at 7.9%, versus a range of 13.5–17.5 averaging at 15.8% for the high SST group. We should, however, compare our results also to what we found in our econometric estimates; there, once we control for infant mortality and the fraction of the population over 65, a 20% increase in the SST is associated with a drop in TFR of between 1.3 and 2.4 children per woman. Thus, our model accounts for between 30% and 55% of the observed differences in fertility in the overall cross section.

Table 5. Model and data for three groups of countries

In the Caldwell-type framework, the quantitative effects of changes in the size of the Social Security system are similar for the two alternative cost structures (time costs and goods costs). This is because in this framework the key mechanism governing fertility is how fertility translates into transfers to parents, and how sensitive these are to changes in the number of children. The introduction of a Social Security system reduces per-child donations, and hence fertility. The difference between the two is in the distortionary effect of taxation on the child-rearing versus market activities. If the costs of children are solely in terms of goods, in this framework with inelastic labor supply, there is no offsetting substitution effect when τ is increased. Thus, the effects on fertility are larger, if only slightly, in this case.

As an additional dimension along which the two models’ predictions should be compared, we note that the Caldwell-type model predicts an increase in the capital–output ratio, while the Barro and Becker model predicts a decrease of the capital–output ratio as Social Security increases. In the data, the US capital–output ratio has either remained constant or increased since early in the 20th century; also, the capital–output ratio is substantially higher among the European countries, relative to the US, and the European countries have, with the sole exception of the UK, a substantially higher SST than the US. This lends further empirical support to Caldwell-type models of fertility as an alternative to dynastic models.

6.1. Parameters of Preferences and Technology

The long and the short of the sensitivity analysis results is: varying preference parameters within reasonable intervals does not change the qualitative predictions of the two models, nor the magnitude of Δγ _n /Δτ as a percentage of the initial value of γ _n . It is still and uniformly true that increasing τ from about 0% to 10% decreases TFR by between 20% and 25% in a Caldwell-type model (the corresponding change is slightly less than 1% in the Barro–Becker model—see the Appendix). Similarly, pushing τ from about 10% to 25% decreases TFR by roughly 30% (it is about 5% in the Barro–Becker model).

What varies substantially, and sometimes dramatically, with the preference parameters are the levels of both fertility and the capital–output ratio, and this sensitivity in levels is common to both models.

As illustrated earlier on, at the baseline parameter values the implied TFR is slightly below the current value of 2.06 in the US for the BJ model; for the Barro and Becker model, as shown in the Appendix, the values for b (resp. a) needed to match observations are much larger than the estimated 3% of time. This seems to point to a lack of richness of the models overall. Clearly, however, a model with features of both would do much better. Since the aim of this paper is partially to compare the two models, this was not attempted.

Our findings for changes in the parameters governing technology are similar to those for preferences: small changes in either α, γ _A , a, b, or δ bring about changes in fertility and in the capital–output ratio that are sometimes substantial. However, they leave the comparative static results basically unaltered when it comes to fertility. Indeed, in the BJ model, reducing the time cost of children from the b = 3% value adopted in the baseline case to values slightly higher than b = 2% suffices to make the predicted level of fertility to match current averages in the US, i.e. about 2.06 per woman. This choice may be justified by the fact that in our model the effective time cost of having children is artificially increased by the assumption that, with only three periods, the length of working life is equal to that of the retirement period. As explained above, this is a gross distortion of the real world, where the number of years spent working is roughly twice the number of years spent in retirement. Because of this fact, one may argue that b = 2% is a preferable baseline calibration for the BJ-type setting; should this choice be made, our model can easily match current US fertility levels when τ = 10% and the remaining parameters are as in Table 2, without affecting any of the comparative statics results.

One experiment that is of particular interest is the effects of changes in the growth rate of productivity. Our value of 1.012 is fairly low and is based on Dennison’s work, which makes substantial adjustments for the observed changes in labor quality. We also performed our baseline experiments on the effects of changes in τ on γ _n for values of γ _A up to and including 1.02. These gave rise to very similar results: when the size of the SST increases from 0% to 10% , fertility drops of almost half a child per woman.

Another alteration that is particularly relevant concerns changing α . In the household production literature (which treats the stock of housing and durables as inputs into the production of home goods, and removes the housing service component from GNP) an estimate of α = 0.22 has been found in McGrattan, Rogerson and Wright (Reference McGrattan, Rogerson and Wright1997). Recalibrating the BJ model to this target does not change the overall effects of changes in Social Security on fertility, but it does greatly enhance our ability to hit the targets set out in the previous section. In particular, with α = 0.22, γ _n = 1, is attainable even with ζ = 0.65. When this alternative calibration is adopted for the dynastic model, increasing the SST rate still increases fertility, and still only very marginally. In the BJ-type model, increasing the Social Security tax rate reduces fertility of more or less the same percentage as in the base line model.

6.2. The Role of Financial Markets Imperfections

In our version of the BJ model the parameter ξ ∈ (0, 1) measures the extent to which financial market imperfections prevent middle age individuals from using private saving as a means of financing late age consumption. In the baseline model we assumed ξ = 0, so that financial markets are functioning perfectly. As reported in the introduction, a number of empirical studies have found evidence that different measures of the ability to save for retirement are strongly correlated with fertility decisions. In fact, a study by Cigno and Rosati using Italian and German micro data have estimated that the impact of financial market accessibility on fertility is comparable to that of public pensions: the easier it is to save for retirement, the lower is fertility. In the BJ model the intuition for this result is simple: in the equation for the equilibrium donations (see Section 3) the terms (1 − ξ)R_tx_t and T^o _t are interchangeable—a variation in ξ has the same effect as a change in the public pension transfer. The more imperfect capital markets are, the less valuable physical capital is for financing consumption in late age and, therefore, the more valuable children are in this regard. One would expect, then, that when ξ > 0 fertility would be higher than in the baseline case; the question is: how much higher?

The answer, reflected in Figure 11, is: a lot higher. Figure 11 plots the mapping from the pair (τ, ξ) ∈ [0.0, 0.25] × [0.0, 0.20] into the equilibrium values of γ _n , while Figure 12 has K/Y on the vertical axis and the same two parameters on the horizontal plane. As the reader can verify, even small changes in the efficiency of financial markets make children a very valuable form of investment. This in turn pushes fertility to levels similar to those observed in the earlier part of the 20th century. In quantitative terms, we find that, even in the presence of a Social Security system of roughly the same magnitude as the current one, a reduction in the rate of return on capital of about 20% (ξ = 0.2) would increase fertility 30%, or 0.66 more children per woman in our setting. Equally important, the same degree of financial market inefficiency leads to a substantial decrease in aggregate savings resulting in a K/Y ratio which is almost 50% lower than in the baseline case. These are large effects by historical standards.

Figure 11. Fertility, social security tax and $\xi$ , Caldwell model.

Figure 12. K/Y, social security tax and $\xi$ , Caldwell model.

Financial instruments through which one can reliably save for retirement are limited both historically in the developed countries and currently in the developing countries. It is difficult to know if changing ξ from 0 to 0.2 corresponds to an interesting quantitative exercise without further data work. But, the fact that the effects that we find are so large makes this an interesting possibility to explore further.