Abstract
In the so-called Rapport Sauvy (1962), the French demographer Alfred Sauvy argued that Wallonia’s fertility rate was socially suboptimal, and recommended a 20 % rise of fertility, on the grounds that a society with too low a fertility leads to a low-productive economy composed of old workers having old ideas. This paper examines how Sauvy’s intuition can be incorporated in the Samuelsonian optimal fertility model (Samuelson, Int Econ Rev 16:531–538, 1975). We build a four-period OLG model with physical capital and with two generations of workers (young and old), the skills of the latter being subject to some form of decay. We characterize the optimal fertility rate and show that this equalizes, at the margin, the sum of the capital dilution effect (Solow effect) and the labour age-composition effect (Sauvy effect) with the intergenerational redistribution effect (Samuelson effect). Numerical simulations show that it is hard, from a quantitative perspective, to reconcile Sauvy’s recommendation with facts. This leads us to examine other potential determinants of optimal fertility, by introducing technological progress and a more general social welfare function.
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Notes
From a formal perspective, one may be tempted to interpret the parameter α differently. For instance, one may interpret it as reflecting the health of older workers, or as reflecting their participation rate to the labour force. Note, however, that our model supposes that α does not affect individual utility directly, whereas health or participation would have such a direct influence. Therefore, we prefer here to keep our interpretation of α as a technical parameter reflecting the decay in human skills.
Note that this objective is here similar to maximizing the average ex post (i.e. realized) lifetime well-being among agents living at the steady-state (some agents being short-lived, whereas others are long-lived).
The necessity to raise fertility when longevity increases raise the number of inactive persons is also emphasized in Sauvy (1956).
More formally, there exists a particular level of α, denoted \(\hat {\alpha }\), for which \(\alpha F_{L}\left (k^{\ast },1+\frac {\alpha }{n^{\ast }}\right ) \) takes its maximum, that is, for which the marginal loss in product per young worker due to a rise in fertility is the largest. This level is defined by the equality:
$$F_{L}\left( k^{\ast},1+\frac{\hat{\alpha}}{n^{\ast}}\right) =-\frac {\hat{\alpha}}{n^{\ast}}F_{LL}\left( k^{\ast},1+\frac{\hat{\alpha}}{n^{\ast} }\right) $$When the first influence dominates the second, a larger decay of old workers’ skills (i.e. a lower α) contributes to reduce the negative effect induced by a higher n, leading, ceteris paribus, to a higher optimal fertility n ∗. But when the second influence dominates the first, a larger decay raises the negative effect induced by a higher n, leading, ceteris paribus, to a lower optimal fertility n ∗.
Indeed, from the resource constraint, we have:
$$\begin{array}{@{}rcl@{}} F\left( k,1+\frac{\alpha}{n}\right) & =&\left[ n\beta d^{-\frac{1}{\sigma} }\right]^{-\sigma}+\frac{d}{n}+ \frac{\left[\frac{d_{t}^{-\frac{1}{\sigma}}}{\beta n}\right]^{-\sigma}\pi}{n^{2}}+nk\\ & \iff& d=\frac{F\left( k,1+\frac{\alpha}{n}\right) -nk}{\left[ \left[ n\beta\right]^{-\sigma}+\frac{1}{n} +\frac{\left[\frac{1}{\beta n}\right]^{-\sigma}\pi}{n^{2}}\right] } \end{array} $$Given that, at time t, there are four generations born at times t−3, t−2, t−1 and t, we start the aggregation of well-being by considering the cohort born at t = −3. Since that cohort start consuming at t = −2, we consider a time horizon starting at t = −2.
Note that, since the time horizon of the social planner is infinite, an upper bound requires that ρn ϕ<1 (see Stelter 2016).
We consider here the baseline production function without labour-augmenting technological progress.
Stelter (2016) derives a similar generalized golden rule taking ethical judgements on numbers into account, but when there is only one cohort of workers (instead of two here).
That is, u(c t+1) + βu(d t+2) + β 2 πu(b t+3)>0.
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Acknowledgments
The authors would like to thank Alessandro Cigno, David de la Croix, Thomas Baudin, and Pierre Mohnen, as well as two anonymous referees for their comments. We are also grateful to the participants of the XXIemeCongrès des Economistes Belges de Langue Française in Liege.
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Pestieau, P., Ponthiere, G. Optimal fertility under age-dependent labour productivity. J Popul Econ 30, 621–646 (2017). https://doi.org/10.1007/s00148-016-0627-7
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DOI: https://doi.org/10.1007/s00148-016-0627-7