- Email: [email protected]
Family regulation as a moving target in the demographic transition
Mathematical Social Sciences 59 (2010) 239–248
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Family regulation as a moving target in the demographic transition Noël Bonneuil ∗ Institut national d’ études démographiques, France École des hautes études en sciences sociales, France
article
info
Article history: Received 18 December 2008 Received in revised form 29 June 2009 Accepted 10 July 2009 Available online 13 August 2009 JEL classification: C02 J11 J13 J16 N33
abstract The consideration of mortality changes equilibrium with unilateral gifts in the economics of family regulation. Expressions for each spouse’s allocation to the common good are obtained that are consistent with the pre-eminent role of mortality in the fertility decline. The other determinant of equilibrium is the husband–wife productivity ratio. Furthermore, in a context of demographic transition, the adjustment of allocations to their equilibrium values may be not instantaneous, and the transition resembles the pursuit of a moving target rather than a shift in equilibrium. One consequence is the lack of correlation between the fertility decline and economic variables. © 2009 Elsevier B.V. All rights reserved.
Keywords: Demographic transition Unilateral gift equilibrium Family economy Target pursuit
1. Introduction The demographic transition is the decline of fertility and mortality from high to low levels. Beginning in France in three regions (Normandy, Garonne Valley, and Champagne) in the mideighteenth century, it spread to other regions during the nineteenth century (Bonneuil, 1997). By the late-nineteenth century it was under way throughout Western Europe. Before focusing on the explanation Bergstrom (2007) gives for the demographic transition in terms of family economics, let
∗
Corresponding address: Institut national d’ études démographiques, 133, bld Davout, 75980, Paris cedex 20, France. E-mail address: [email protected].
0165-4896/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2009.07.005
240
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
me, at the editor’s request, briefly situate the demographic transition in the context of recent economic literature. Galor and Weil (2000) believe that the demographic system before the transition was a ‘‘Malthusian regime in which technological progress and population growth were glacial by modern standards and income per capita was roughly constant’’ (:806), and they build a scenario of the demographic transition centered on technological progress. This is however only one scenario among many possible, and it is barely consistent with most data-documented demographic literature. Thus for the old demographic regime, somewhat hastily designated as Malthusian, an investigation based on the French parish register data shows that fertility and mortality in fact varied irregularly and widely in the seventeenth century, responding to the sharp effects of epidemics and wars of the Little Ice Age (Bonneuil, 1990). Beside high levels of fertility and mortality, the other salient feature of the pre-transition regime was large fluctuations in these levels, exactly the opposite of ‘‘glacial’’, in fact. The same strong irregularity of price time series indicates that income also fluctuated widely. With regard to the transition itself, their model, centered on England where transition did not begin until the 1870s, ignores and thus fails to explain why it began in France as early as the mid-eighteen century, where by comparison technological progress had yet to take off. For example, in 1806, the fertility index is estimated at 0.25 in Calvados (France) (Bonneuil, 1997) and ≈0.45 in England (Wrigley and Schofield, 1981). Similarly, their model also fails to explain why, compared with Morocco, Egypt, which had a higher proportion of literate people (51% versus 44% in 1995), was richer ($3,829 versus $3,477 in product per head) and had a more modern economy, began its transition later and more slowly, in the early 1980s versus in the early 1960s (Bonneuil and Dassouki, 2006, 2007). The Malthusian framework is merely a theoretical interpretation; historical data teach us that epidemics, not food shortages, were the main obstacle to population growth (Lee, 1990). A Malthusian homeostatic regulation of population by food prices was the subject of a celebrated demonstration by Wrigley and Schofield (1981), on which Galor and Weil rely heavily. One criticism is that Wrigley and Schofield claim to prove the presence of Malthusian checks from an analysis of short-term fluctuations, whereas the Malthusian equilibrium is long term (Bonneuil, 1994b). A fatal flaw is that all their population data rely on a back projection which has been proved erroneous: Lee (1985) showed that the model contains T more unknowns than equations, where T is the span of the reconstruction period. The published solution depends heavily on the initialization of the algorithm, which produces as many different solutions as there are different starting values (Bonneuil, 1992). The generalized inverse projection (Oeppen, 1993) designed to remedy the lack of identifiability of the back projection method, was discredited by Lee (1993, 11) who rightly objected to the characterization ‘‘that the incomplete cohorts arose from a stable population’’ (Oeppen cited in Lee, 1993: 11). A stable population corresponds to a situation where demographic forces are constant over time, and the chance of observing such a constraint is remote in a real-world context of time-varying demographic forces where the initial population was not stable. The problem is that any error in the initial age structure has rapidly uncontrollable effects in the estimation of demographic forces. Beside income or technological progress, the adoption of the ‘‘Solow technology’’ (Hansen and Prescott, 2002), or various ‘‘endogenous’’ feedback loops between the concerned variables in an optimal acquisition of human capital (Cervellati and Sunde, 2005) have been invoked to explain the demographic transition. As Watkins (1987b, 29) put it, ‘‘anything that distinguishes traditional from modern societies has been considered relevant to the explanation of fertility decline’’. Is there still a place for non-economic factors? Carlsson (1969) put economic and non-economic explanations in competition, by suggesting two mechanisms of the transition: did couples have less children because they adapted to better economic conditions or because they experienced family life differently? Examination of transitions in the world gives advantage to the latter interpretation, and economic conditions do not interfere directly. The upset down of cultural marks seems a key determinant, while the attention given to children turned into attention given to oneself. In nineteenth century France, Bonneuil (1997) showed that fertility at a high level was sensitive to mortality and to environment, but, after the onset of the decline and this decline made possible, this link vanished and the decline responded to an overall diffusion of changing behaviors, from person to person, from group to group, from region to region. In tropical Latin America, after it was launched, the fertility decline spread quickly to the neighboring countries independently of the economic or technological
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
241
level. The rise of education, urbanization, and non-agricultural salaried work increased the receptivity of young couples to the idea of fewer pregnancies, which are made possible by the infant and juvenile mortality decline. The substantial role played by family renders the model of Bergstrom (2007) of prior interest, and that is why I suggest to revisit it here. This author addressed the question of ‘‘why is fertility not positively related with income’’ in modern economies, a result confirmed by most empirical studies ((Ariès, 1971, 1948); Carlsson, 1969; Cleland and Wilson, 1987; Watkins, 1987a; Bonneuil, 1997; Bonneuil and Dassouki, 2006, 2007). He drew on the micro-economics of the family ((Ruskin, 2008, 1864); Lundberg and Pollak, 1993; Knibiehler, 2000; Young, 2005) to show that if the income earning productivity of both spouses increases while the mother’s productivity in child breeding does not change, there would be no change in the time and resources allocated to children. Apparently, the perpetuation of this productivity balance would neutralize the effect of the accumulation of human capital on fertility through economic growth (Boucekkine and de la Croix, 2002; Boucekkine et al., 2004). I shall extend his analysis and show firstly that equilibrium with unilateral gifts is driven primarily by mortality, a result that all observations confirm, but which is ignored by Bergstrom. Secondly, Bergstrom emphasizes the possible influence of the male–female wage–earning ratio on fertility, but I will show that empirical data do not testify for any significant relationship between these two variables during the French demographic transition, one of the best documented case study. Thirdly, I show that situating family regulation in a dynamic process towards an equilibrium that appears as a moving target explains how the demographic transition can be driven by economic factors even while fertility shows no significant correlation with economic variables such as productivity or the male–female productivity ratio. 2. Food and fire: A model of family partnerships Bergstrom (2007) employs an arboreal allegory to represent the choices confronted by a man and a woman whose offspring share their genes. ‘‘Bob and Alice live in a forest [...] Alice divides her time between collecting food and gathering wood [...] Bob spends all his time gathering food’’, and ‘‘every night huddles beside Alice’s fire [...] Bob leaves morsels of food by the fire for Alice. Warmth and food are both ‘normal goods’ for Alice. The extra food Bob leaves induces her to increase her food consumption, but not by the total amount Bob leaves her. She uses some of her time saved by Bob’s gifts to gather more firewood’’. Bergstrom focuses on the case of unilateral gifts, though he also discusses common interests, and Pareto efficiency and conditional payments. With unilateral gifts, Alice maximizes her utility function U (cA , y) = cA y where ca is the amount of food she eats, and y the amount of wood on the fire. ‘‘She has T hours to allocate between collecting food and wood. In an hour, she collects one unit of wood or πA units of food. If Bob leaves g units of food, she maximizes her utility by choosing: y=
1 2
T+
g
πA
and cA =
1 2
(πA T + g ).
(1)
Bob’s utility function is U (cb , y) = cB y, where cB is his food consumption, and y is still the amount of wood on Alice’s fire. Bob is useless at collecting wood, but gathers πB units of food per hour for T hours. If 3πA ≥ πB ≥ πA , Bob maximizes his utility by choosing: y=
T πA + πB 4
πA
and cB =
T 2
(πA + πB ).
(2)
He achieves this outcome by giving g = T2 (πB − πA ) units of food to Alice. In this case, Alice spends the fraction (3πA − πB )/4 of time gathering food and the rest collecting wood. If 3πB ≤ πA , Bob gives Alice no food and Alice divides her time equally between collecting food and wood’’. The fire is a public good for Alice and Bob, and if ‘‘Alice is not spending all her time gathering wood, and if the sum of Alice’s and Bob’s marginal rates of substitution exceeds pA , then both are better off
242
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
if Alice gathers more wood. At a Pareto optimal allocation’’ y=T
πA + πB 2πA
if πB ≤ πA
(3)
and y=T
if πB ≥ πA .
(4)
The third possibility discussed by Bergstrom is common interest, when ‘‘the only thing Alice and Bob really care about is the size of the fire [...] Each maximizes utility by choosing consumption to maximize his or her net output’’. The micro-economics of the family was not developed specifically to explore the demographic transition, rather to analyze household consumption behavior (Lundberg and Pollak, 1993; Browning et al., 1994). Bergstrom extends the scope of the model to explain the possible role of productivity in the demographic transition. Data on gender differentials in productivity during the demographic transition are hard to come by. ‘‘Even the most comprehensive micro-data sets lack good information on potentially important productivity determinants such as physical strength, energy, and individual work habits’’ (Cox and Nye, 1989: 904). For the French textile industry, these authors publish male–female wage ratios of around 0.50 for cotton and wool spinning or weaving in 1839–45 and 1860–65. With aggregate data, those for England taken from Wrigley and Schofield (1981) must be handled with great care, as I mentioned in the introduction. For France, fertility was reconstructed in the 89 départements over the period 1806–1906 (Bonneuil, 1997), and schooling data for boys and girls aged 5–15 were corrected by Luc (1985, 1986) and Diebolt (1999) for 1837, 1850, 1867 and 1876, corresponding to the key period of the French demographic transition. The ratio of the schooling rate of boys to the schooling rate of girls in generation G is a proxy for the lifetime productivity ratio between men and women of generation G. For each French département, I compared the fertility index in 1850 with the schooling ratio for 18–28 year olds – measured on those aged 5–15 in 1837 –, the fertility index in 1867 with the schooling ratio for 22–32 year olds – measured on those aged 5–15 in 1850 – and the fertility index in 1876 with the schooling ratio of 16–26 year olds – measured on those aged 5–15 in 1867. Regression of differenced fertility (to eliminate temporal correlations) on differenced mortality, on differenced schooling ratio and on region shows no clear significant relationship whereby fertility would vary with the schooling ratio, but the coefficient of differenced mortality is always significant in the early phase of the transition. Although research consistently shows that mortality and fertility had little or no correlation to real wages in historical demographic regimes (Lee, 1990), as Bergstrom correctly observes, it equally consistently shows the decline in mortality as preceding that in fertility. Fertility decline followed mortality decline, either closely as in France, or only after mid-nineteenth century in the rest of Europe, such as Sweden after 1860 or England and Wales after 1880 (Coale and Watkins, 1986). The family partnership interpretation of the demographic transition thus requires the inclusion of mortality in the utility function. My view is that on average people behaved as if they were aware of the mortality level, and were lucid enough to realize that they must have at least yA eδ children, where e−δ represents survivorship, and yA is a target level of the number of children expected to be alive at their own age of reproduction. To return to the food and fire parabola, the rain falling on Alice’s fire renders part of the wood definitively unfit for burning and part of food unfit for eating (mortality), and the fire is enough to warm Alice and Bob once a certain size (target parity) has been attained. Indeed, in Bergstrom’s model where the marginal utility from the fire is simply the time y devoted to the fire, if Alice and Bob were not limited by their time resources, the larger the fire, the greater their happiness. In my interpretation, Alice and Bob need a fire of a sufficient size if they are to maintain health and combat cold, but a fire that is too large is too hot. I understand Bergstrom’s metaphor to be predicated on the belief that evolution favors those with the largest fire. However, if evolution favors the strategies that maximize reproductive success, individuals do not favor evolution, but simply their capacity to stay in the game. This theme has been developed in biology: evolution certainly favors the most prolific species, but, more tautologically, the species that can be observed today are merely those which have managed to perpetuate themselves
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
243
so far. Biologists are well aware of the question of survival and co-existence. For example, GarcíaMartinez et al. (1998) pointed out the role played by a time-varying natural selection in maintaining the co-existence of two mitochondrial haplotypes called I and II, whereas cage experiments invariably lead to the fixation of haplotype II. Constancy of conditions favors the fittest haplotype, but fluctuations of the environment allow polymorphism, because the fittest is not always the same. The concept of ‘‘fittest’’ must be replaced by that of ‘‘staying in the game’’. Other examples: Giesel (1976, 75) concludes from studies of passerine birds that variability in fertility is an ‘‘important adaptation to account for environmentally induced unpredictability of the major selective forces’’. Barker et al. (1986), studying temporal variation in allozyme frequencies at six loci from monthly collections over four years in one population of the cactophilic species Drosophila buzzatii, show significant variation over time in allele frequencies, and deny the presence of any clear cyclical or seasonal pattern. The mathematics of evolution are concerned at least as much with maintenance as with maximization (Pianka, 1978; Vincent and Brown, 1986; Bonneuil, 1994a,b, 2008; Bonneuil and Müllers, 1997; Bonneuil and Saint-Pierre, 2000, 2002, 2005, 2008). If parents have a target parity, the time devoted to child care increases their marginal utility when this time is less than the target, and diminishes it thereafter. Instead of a term y as in Bergstrom, I suggest a marginal utility of logistic form, starting from a positive level y0 representing the utility of being alive – while in Bergstrom, utility is nil in the absence of fire – and upper-bounded by a term δ
depending on mortality, such as y0 (1 + e−β yA e ), such that the inflexion point is yA eδ . Alice’s utility is then δ
UA (cA , y, δ) = cA
y0 (1 + e−β yA e ) δ
1 + eβ(y−yA e ) where β is a negative parameter, and where consumption cA is: cA = πA (T − y) + g .
(5)
(6)
Similarly, Bob’s marginal utility can be taken logistic: δ
UB (cB , y, δ) = cB
y0 (1 + e−β yA e )
(7)
δ 1 + eβ(y−yA e )
with cB = πB T − g
(8)
assuming, for the sake of simplicity, that Bob and Alice have the same target yA eδ and the same childless marginal utility y0 . Alice maximizes her utility at g given to obtain y(g ), and Bob maximizes his utility by selecting g = gmax with y(g ). The calculus is less simple than for the case of linear marginal utility of children: Proposition 2.1. If utilities of both spouses depend on mortality in a logistic manner, according to Eqs. (5) and (7), the solution of the unilateral gift maximization program under constraints of Eqs. (6) and (8) is: y(gmax ) ≈
2
β
yA eδ
+
2
−
1 2β
1/2 πB β 2 y2A e2δ + 4β yA eδ + 8 − 4β T 1 + πA
(9)
Proof. Alice maximizes her utility in Eq. (5) at g given to obtain: y(g ) = T +
g
πA
+
1
β
+
1
β
PLog
−1−β πg +T −yA eδ A e
(10)
where PLog(z) is the dominant root of the equation w ew = z. Expressing exp(y(g )) using Eq. (10) yields the expression of exp(w), which, multiplied by w , whose expression is taken from Eq. (10), leads to w exp(w), and after simplification, to the inverse formula: g = πA
y(g ) −
1
β
−T −
1 −1−β e
β
g
πA +T −yA e
δ
.
(11)
244
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
Bob maximizes his utility in Eq. (7) by selecting g = gmax such that
∂ UB (gmax ) = 0 ∂g
(12)
given y(g ). Differentiating Eq. (11) with respect to g yields:
∂ y(g ) 1 1 = ∂g πA 1 + e−β(y(g )−yA )
(13)
which helps solving the maximization of UB in Eq. (12):
∂ UB ∂ y(g ) = −y(g ) + (πB T − g ) = 0. ∂g ∂g
(14)
Numerical simulations show that y(gmax ) is greater than yA exp δ(t ), but close to it, a property consistent with the empirical observation that fertility followed mortality. Hence, a first-order development in the neighborhood of yA exp δ is possible. The positive solution (with t omitted) is then given by Eq. (9). π
The final result is the analytical expression of y(gmax ) as a function of the productivity ratio πB and A of yA eδ . It is close to a linear function of yA eδ always above yA eδ . So Bergstrom’s result that y depends only on the productivity ratio between the man and woman, and not on the productivity of one spouse alone, still holds. The second result is the dependence of y(gmax ) on mortality, varying linearly and with a positive coefficient with yA eδ . 3. The demographic transition: Moving equilibrium or moving to a moving target? To present the demographic transition as the shifting of an economic equilibrium, whether unilateral, Pareto, or common interest, is not devoid of paradox. In this view, agents would be conveyed through the demographic transition astride a shifting equilibrium. There would be no restriction on modifying secular habits governing the division of labor between spouses. Yet in both France and Great Britain, at least until the 1940s, women were ‘‘locked up’’ in familial dependence. In Great Britain, the marriage bar imposed from the mid-nineteenth century onwards kept married women out of salaried work (Hakim, 1987). The result was the low activity rate of women in England until the 1930s (10% against 50% in France in 1901 (Hantrais, 1990: 109)). In France, women’s choice between the home and wage–earning was limited by the role they were expected to play, with their work outside the home being perceived as a danger to the family (Daune-Richard, 2004). So economic equilibria did not shift without encountering resistance rooted in customs and habits. After all, demographic behavior is not a mechanistic device such that the decline in mortality would automatically lead to a fall in fertility. The decision to limit family size may be made possible by improved living conditions, but it remains the product of couples’ willingness to have or to avoid having children. Taking into account the time necessary for people to alter their habits changes the formulation of the problem from the configuration of preferences at a static equilibrium to the dynamic of preferences on a path to a moving target – moving because mortality continues its decline. Weir (1984, 28), for example, after pointing out ‘‘the a-historical character of any equilibrium analysis’’, concluded, from empirical observations, that trend shifts may have dominated equilibrium adjustments. Let us focus, like Bergstrom, on economic regulation with unilateral gifts. This model is relevant for describing France and Great Britain during their demographic transition. In spite of differences notably in activity rates or productivity (Maddison, 1987), the family was organized according to the male breadwinner model under which the husband was a salaried worker while the wife was in charge of running the home (Scott and Tilly, 1987; Daune-Richard, 2004).
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
245
)
0
80
Fig. 1. Simulated transition: the equilibrium-target follows mortality, the state follows with delay, confusing its relationship with mortality. The variable representing mortality, δ , is normalized for comparison with the time allocated to child care y. π yA = 2, β = −0.2, T = 12, πB = 2. A
In the case of unilateral gift, letting the argument δ(t ) appear, Alice and Bob start from (y, g ) = (y(gmax (δ(0)), δ(0)), gmax (δ(0))) and move in direction of the target at time t
(y(gmax (δ(t )), δ(t )), gmax (δ(t ))),
(15)
which is itself moving under the influence of mortality and the male–female productivity ratio. The dynamics is given:
y0 (t ) = dA (y(t ), g (t ), vA (t )) g 0 (t ) = dB (y(t ), g (t ), vB (t ))
(16)
where dA and dB specify a dynamic, and vA (t ) and vB (t ) are controls admissible in closed sets VA = [ [ ] [ [ ] [vA , vA ] and VB = [vB , vB ], with vA < 0, vB < 0 (to make fertility decline possible). VA represents Alice’s capacity for action and VB Bob’s. VA and VB are closed because Alice and Bob are limited in their capacity for change, and because the transition takes time to unfold. One particular case is when Alice and Bob can change their own variables independently of other variables:
y0 (t ) = vA (t ) g 0 (t ) = vB (t ).
(17)
There is no reason why y0 (t ), the change of the time y devoted to children, and y0 (gmax ) = ∂ y(gmax ) 0 ) πB 0 δ (t )+ ∂ y(gπmax ( π ) the change of the target, which is the equilibrium at date t, should produce B ∂δ ∂π
A
A
a significant relationship in an econometric study (with discrete time). The relationship of fertility to mortality and to the ratio of productivity between spouses is confused in the short term by the fact that the system is not in equilibrium but pursuing a target, which is itself driven by mortality and economic change. For an example, Fig. 1 shows a simulation where mortality δ is decreasing in a logistic pattern, consistent with what is observed in France and England during their respective π transitions. For πB fixed, the target y(gmax(δ) , δ)(t ) follows mortality. In this simulation, vA is taken as A null until time 40, which is the onset of the transition, negative thereafter, so that the state y(t ) of the system follows this characteristic curve of the fertility decline, delayed with respect to the mortality decline. A uniform noise is added to these three variables, and time series yn , y(gmax )n , δn are drawn from means over each time interval [nh, (n + 1)h) with h representing a duration of one year. The
246
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
regression of the target y(gmax )n on δn yields a significant coefficient, while the regression of the state π yn on δn is not significant. The same can be done with the ratio of productivity πB . A The pace of the transition results from a competition between the velocity at which the target moves and the velocity at which the system pursues the target. This is exactly what is revealed by the co-integration analysis of the demographic transition in France based on a finely detailed spatial reconstruction for each of the 89 (in 1861) départements. The French transition unfolded in two stages: at the onset of transition, the improvement in mortality provided an opportunity to undergo fewer pregnancies, but fertility quickly became insensitive to economics and to mortality decline, with the response to the environment fading away as harsh conditions disappeared. Further change was increasingly dependent on culture, habits and attitudes (Bonneuil, 1997). Co-integration and analysis of short-term fluctuations show no significant relationship between fertility and education, mortality, productivity ratio between spouses, or urbanization, but the overall pattern of fertility catching up with these variables is clear to see, and is consistent with a target pursuit model (rather than with an error correction model or homeostatic model). An important parameter of the transition is its duration. The presentation of the demographic transition in terms of target pursuit allows its expression to be given for simple cases. With time 0 corresponding to the onset of the transition, one minimal time from onset to target at t is obtained by solving: t
Z min
dt
(18)
0
[
[
under the dynamic of Eq. (17). If uA ≥ uB , the maximization of the associated Hamiltonian shows that [
one optimal trajectory is obtained with uA = uA , so that the minimal time to horizon H, considering that the target is moving with δ(t ) is: tmin (H ) =
H
Z 0
∂ y(gmax (δ(t ))) dt . ∂δ uA (t ) 1
(19)
[
[
For uA constant in time, as simulations and the approximation of Eq. (9) show, y(gmax ) is close to a linear function of yA eδ (t ) remaining above yA eδ (t ), so that tmin (H ) ≈
y(gmax (δ(H ))) − y(gmax (δ(0))) [
uA [
.
(20)
[
Similarly, when uA ≤ uB , tmin (H ) ≈
gmax (δ(H )) − gmax (δ(0)) [
uB
.
(21)
4. Conclusion Bergstrom (2007) showed that equilibrium with unilateral gifts in family regulation incorporates only the productivity ratio between spouses, which explains why the fertility decline was not correlated with the variations in female productivity alone. However, all historical transitions persistently identify the precedence of the mortality decline (a fact not contested by Bergstrom but that he ignores for the sake of his demonstration). Continuing the analysis, I introduced mortality into the spouses’ utilities through a logistic marginal utility of children to show that, at equilibrium, fertility follows the mortality decline. The maximization of utilities is tractable, and I gave analytical expressions. However, while empirical cases illustrate the pattern of mortality precedence, the short-term covariation of fertility with mortality, or with socio-economic variables disappears after the onset of the transition, at least in nineteenth century France, which is among the most fully documented
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
247
historical examples at a fine spatial scale. An account of the transition as a dynamic system in which the time devoted to children moves in the direction of the equilibrium allows resistance to change to be incorporated. It explains the lack of short-term correlation between fertility, mortality, and economic variables, but fits the pattern of fertility catching up with these latter variables in the long run. It also allows for the existence of varied responses to the fall in fertility and the improvement in productivity, in conformity with historical observations, notably the conflicting behavior of the British, who retained high fertility long after the onset of the mortality decline, and the French, who adjusted to it immediately. References Ariès, P., 1971. Histoire des populations françaises et leurs attitudes devant la vie. Self, Paris, (1948), Le Seuil, Paris. Barker, J.S.F., East, P.D., Weir, B.S., 1986. Temporal and microgeographic variation in allozyme frequencies in a natural population of Drosophila buzzatii. Genetics 112, 577–611. Bergstrom, T.C., 2007. Some evolutionary economics of family partnerships. American Economic Review 97 (2), 482–486. Bonneuil, N., 1990. Turbulent Dynamics in a 17th century population. Mathematical Population Studies 2, 289–311. Bonneuil, N., 1992. Cohérence démographique et non-identifiabilité de la rétro-projection. in: Blanchet, D., Blum, A., Bonneuil, N. (Eds), Méthodes de la démographie historique. INED, Congrès et Colloques 11, Paris, pp. 99–108. Bonneuil, N., 1994a. Capital accumulation, inertia of consumption and norms of reproduction. Journal of Population Economics 7, 49–62. Bonneuil, N., 1994b. Malthus, Boserup and population viability. Mathematical Population Studies 4, 107–119. Bonneuil, N., 1997. Transformation of the French Demographic Landscape, 1806–1906. Clarendon Press, Oxford. Bonneuil, N., 2008. The mathematics of maintenance and acquisition. In: Chen, J., Guo, C. (Eds.), Ecosystem Ecology Research Trends. Nova Science Publishers, New York, pp. 153–175. Bonneuil, N., Dassouki, C., 2006. Women’s education and diffusion of the fertility transition: The case of Egypt 1960–1996 in 4905 administrative subdivisions. Journal of Population Research 23 (1), 27–39. Bonneuil, N., Dassouki, C., 2007. Economics, Geography, Family Planning and Rapidity of change in the demographic transition: The case of the Egyptian muhafazas 1960–1996. Journal of Developing Areas 40 (2), 185–210. Bonneuil, N., Müllers, K., 1997. Viable Populations in a prey–predator system. Journal of Mathematical Biology 35, 261–293. Bonneuil, N., Saint-Pierre, P., 2008. Beyond optimality: Managing children, assets, and consumption over the life cycle. Journal of Mathematical Economics 44 (3–4), 227–241. Bonneuil, N., Saint-Pierre, P., 2000. Protected polymorphism in the two-locus haploid model with unpredictable fitnesses. Journal of Mathematical Biology 40 (3), 251–277. Bonneuil, N., Saint-Pierre, P., 2002. Minimal number of generations out of Polymorphism in the one-locus two-allele model with unpredictable fertilities. Journal of Mathematical Biology 44 (6), 503–522. Bonneuil, N., Saint-Pierre, P., 2005. Population viability in three trophic-level food chains. Applied Mathematics and Computation 169–2, 1086–1105. Boucekkine, R., de la Croix, D., 2002. Vintage human capital, demographic trends, and endogenous growth. Journal of Economic Theory 104, 340–375. Boucekkine, R., de la Croix, D., Licandro, O., 2004. Modelling vintage structures with DDEs: Principles and applications. Mathematical Population Studies 11 (3–4), 151–180. Browning, M., Bourguignon, F., Chiappori, P.-A., Lechene, V., 1994. Income and outcomes: A structural model of intrahousehold allocation. Journal of Political Economy 102 (6), 1067–1096. Coale, A., Watkins, S.C. (Eds.), 1986. The Decline of Fertility in Europe: The Revised Proceedings of a Conference on the Princeton European Fertility Project. Princeton University Press, Princeton, NJ. Carlsson, G., 1969. The decline of fertility: Innovation or adjustment process. Population Studies 20, 149–174. Cervellati, M., Sunde, U., 2005. Human capital formation, life expectancy, and the process of development. American Economic Review 95 (5), 1653–1672. Cleland, J., Wilson, C., 1987. Demand theories of the fertility transition: An iconoclastic view. Population Studies XLI (2), 5–30. Cox, D., Nye, J.V., 1989. Male–Female wage discrimination in nineteenth century France. Journal of Economic History 49 (4), 903–920. Daune-Richard, A.-M., 2004. Les femmes et la société salariale: France, Royaume-Uni, Suède. Travail et Emploi. Travail et Emploi 100, 69–84. Diebolt, C., 1999. Les effectifs scolarisés en France: XIXe et XXe siècles. International Review of Education 45, 197–213. Galor, O., Weil, D.N., 2000. Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond. The American Economic Review 90 (4), 806–828. García-Martinez, J., Castro, J.A., Ramón, M., Latorre, A., Moya, A., 1998. Mitochondrial DNA haplotype frequencies in natural and experimental populations of Drosophila subobscura. Genetics 149, 1377–1382. Giesel, J.T., 1976. Reproductive strategies as adaptations to life in temporally heterogeneous environments. Annual Review Ecological Systems 7, 57–79. Hakim, C., 1987. Trends in flexible workforce. Employment Gazette (November), 549–555. Hansen, G.D., Prescott, E.C., 2002. Malthus to Solow. Amercican Economic Review 92 (4), 1205–1217. Hantrais, L., 1990. Managing Professional and Family Life. Darmouth Publishing Company, Hants and Vermont. Knibiehler, Y., 2000. Histoire des mères et de la maternité en Occident. PUF, Que sais-je ?, Paris. Lee, R.D., 1985. Inverse projection and back projection: A critical appraisal, and comparative results for England, 1539 to 1871. Population Studies 39, 233–248.
248
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
Lee, R.D., 1990. The demographic response to economic crisis in historical and contemporary populations. Population Bulletin of the United Nations 29, 1–15. Lee, R.D., 1993. In: Reher, D., Schofield, R.S. (Eds.), Inverse Projection and Demographic Fluctuations, in Old and New Methods in Historical Demography. Oxford University Press, Oxford, pp. 7–28. Luc, J.-N., 1985. La statistique de l’enseignement primaire XIXe-XXe siècles. Politique et mode d’emploi, Inrp/Cnrs, Paris. Luc, J.-N., 1986. L’illusion statistique. Annales E.S.C. 41, 887–911. Lundberg, S., Pollak, R.A., 1993. Separate spheres bargaining and the marriage market. Journal of Political Economy 101 (6), 988–1010. Maddison, A., 1987. Growth and slowdown in advanced capitalist economies: Techniques of quantitative assessment. Journal of Economic Literature 25, 649–698. Oeppen, J., 1993. Back projection and inverse projection: Two members of a wider class of constrained projection models. Population Studies 47, 245–267. Pianka, E.R., 1978. Evolutionary Ecology. Harper and Row Publishers. Scott, J.-W., Tilly, L., 1987. Les femmes, le travail et la famille. Rivages, Paris. Ruskin, J., 2008. Sesame and Lilies. Bibliolife. Powell’s Books, Portland OR, 1864. Vincent, T.L., Brown, J.S., 1986. Evolution under nonequilibrium dynamics. Mathematical Modelling in Science and Technology 29 (31–8), 766–771. Watkins, S.C., 1987a. The fertility transition: Europe and the Third World compared. Sociological Forum 2 (4), 645–673. Watkins, S.C., 1987b. In: Stycos, J.M. (Ed.), Demography as an Inter-Discipline. Transaction Publishers, pp. 27–55. Weir, D., 1984. Life under pressure: France and England, 1670–1870. The Journal of Economic History 44 (1), 27–47. Wrigley, E.A., Schofield, R.S., 1981. The Population History of England 1541–1871, A Reconstruction. Edward Arnold, London. Young, A., 2005. The gift of the dying: The tragedy of Aids and the welfare of future African generations. The Quaterly Journal of Economics 2, 423–466.
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Family regulation as a moving target in the demographic transition Noël Bonneuil ∗ Institut national d’ études démographiques, France École des hautes études en sciences sociales, France
article
info
Article history: Received 18 December 2008 Received in revised form 29 June 2009 Accepted 10 July 2009 Available online 13 August 2009 JEL classification: C02 J11 J13 J16 N33
abstract The consideration of mortality changes equilibrium with unilateral gifts in the economics of family regulation. Expressions for each spouse’s allocation to the common good are obtained that are consistent with the pre-eminent role of mortality in the fertility decline. The other determinant of equilibrium is the husband–wife productivity ratio. Furthermore, in a context of demographic transition, the adjustment of allocations to their equilibrium values may be not instantaneous, and the transition resembles the pursuit of a moving target rather than a shift in equilibrium. One consequence is the lack of correlation between the fertility decline and economic variables. © 2009 Elsevier B.V. All rights reserved.
Keywords: Demographic transition Unilateral gift equilibrium Family economy Target pursuit
1. Introduction The demographic transition is the decline of fertility and mortality from high to low levels. Beginning in France in three regions (Normandy, Garonne Valley, and Champagne) in the mideighteenth century, it spread to other regions during the nineteenth century (Bonneuil, 1997). By the late-nineteenth century it was under way throughout Western Europe. Before focusing on the explanation Bergstrom (2007) gives for the demographic transition in terms of family economics, let
∗
Corresponding address: Institut national d’ études démographiques, 133, bld Davout, 75980, Paris cedex 20, France. E-mail address: [email protected].
0165-4896/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2009.07.005
240
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
me, at the editor’s request, briefly situate the demographic transition in the context of recent economic literature. Galor and Weil (2000) believe that the demographic system before the transition was a ‘‘Malthusian regime in which technological progress and population growth were glacial by modern standards and income per capita was roughly constant’’ (:806), and they build a scenario of the demographic transition centered on technological progress. This is however only one scenario among many possible, and it is barely consistent with most data-documented demographic literature. Thus for the old demographic regime, somewhat hastily designated as Malthusian, an investigation based on the French parish register data shows that fertility and mortality in fact varied irregularly and widely in the seventeenth century, responding to the sharp effects of epidemics and wars of the Little Ice Age (Bonneuil, 1990). Beside high levels of fertility and mortality, the other salient feature of the pre-transition regime was large fluctuations in these levels, exactly the opposite of ‘‘glacial’’, in fact. The same strong irregularity of price time series indicates that income also fluctuated widely. With regard to the transition itself, their model, centered on England where transition did not begin until the 1870s, ignores and thus fails to explain why it began in France as early as the mid-eighteen century, where by comparison technological progress had yet to take off. For example, in 1806, the fertility index is estimated at 0.25 in Calvados (France) (Bonneuil, 1997) and ≈0.45 in England (Wrigley and Schofield, 1981). Similarly, their model also fails to explain why, compared with Morocco, Egypt, which had a higher proportion of literate people (51% versus 44% in 1995), was richer ($3,829 versus $3,477 in product per head) and had a more modern economy, began its transition later and more slowly, in the early 1980s versus in the early 1960s (Bonneuil and Dassouki, 2006, 2007). The Malthusian framework is merely a theoretical interpretation; historical data teach us that epidemics, not food shortages, were the main obstacle to population growth (Lee, 1990). A Malthusian homeostatic regulation of population by food prices was the subject of a celebrated demonstration by Wrigley and Schofield (1981), on which Galor and Weil rely heavily. One criticism is that Wrigley and Schofield claim to prove the presence of Malthusian checks from an analysis of short-term fluctuations, whereas the Malthusian equilibrium is long term (Bonneuil, 1994b). A fatal flaw is that all their population data rely on a back projection which has been proved erroneous: Lee (1985) showed that the model contains T more unknowns than equations, where T is the span of the reconstruction period. The published solution depends heavily on the initialization of the algorithm, which produces as many different solutions as there are different starting values (Bonneuil, 1992). The generalized inverse projection (Oeppen, 1993) designed to remedy the lack of identifiability of the back projection method, was discredited by Lee (1993, 11) who rightly objected to the characterization ‘‘that the incomplete cohorts arose from a stable population’’ (Oeppen cited in Lee, 1993: 11). A stable population corresponds to a situation where demographic forces are constant over time, and the chance of observing such a constraint is remote in a real-world context of time-varying demographic forces where the initial population was not stable. The problem is that any error in the initial age structure has rapidly uncontrollable effects in the estimation of demographic forces. Beside income or technological progress, the adoption of the ‘‘Solow technology’’ (Hansen and Prescott, 2002), or various ‘‘endogenous’’ feedback loops between the concerned variables in an optimal acquisition of human capital (Cervellati and Sunde, 2005) have been invoked to explain the demographic transition. As Watkins (1987b, 29) put it, ‘‘anything that distinguishes traditional from modern societies has been considered relevant to the explanation of fertility decline’’. Is there still a place for non-economic factors? Carlsson (1969) put economic and non-economic explanations in competition, by suggesting two mechanisms of the transition: did couples have less children because they adapted to better economic conditions or because they experienced family life differently? Examination of transitions in the world gives advantage to the latter interpretation, and economic conditions do not interfere directly. The upset down of cultural marks seems a key determinant, while the attention given to children turned into attention given to oneself. In nineteenth century France, Bonneuil (1997) showed that fertility at a high level was sensitive to mortality and to environment, but, after the onset of the decline and this decline made possible, this link vanished and the decline responded to an overall diffusion of changing behaviors, from person to person, from group to group, from region to region. In tropical Latin America, after it was launched, the fertility decline spread quickly to the neighboring countries independently of the economic or technological
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
241
level. The rise of education, urbanization, and non-agricultural salaried work increased the receptivity of young couples to the idea of fewer pregnancies, which are made possible by the infant and juvenile mortality decline. The substantial role played by family renders the model of Bergstrom (2007) of prior interest, and that is why I suggest to revisit it here. This author addressed the question of ‘‘why is fertility not positively related with income’’ in modern economies, a result confirmed by most empirical studies ((Ariès, 1971, 1948); Carlsson, 1969; Cleland and Wilson, 1987; Watkins, 1987a; Bonneuil, 1997; Bonneuil and Dassouki, 2006, 2007). He drew on the micro-economics of the family ((Ruskin, 2008, 1864); Lundberg and Pollak, 1993; Knibiehler, 2000; Young, 2005) to show that if the income earning productivity of both spouses increases while the mother’s productivity in child breeding does not change, there would be no change in the time and resources allocated to children. Apparently, the perpetuation of this productivity balance would neutralize the effect of the accumulation of human capital on fertility through economic growth (Boucekkine and de la Croix, 2002; Boucekkine et al., 2004). I shall extend his analysis and show firstly that equilibrium with unilateral gifts is driven primarily by mortality, a result that all observations confirm, but which is ignored by Bergstrom. Secondly, Bergstrom emphasizes the possible influence of the male–female wage–earning ratio on fertility, but I will show that empirical data do not testify for any significant relationship between these two variables during the French demographic transition, one of the best documented case study. Thirdly, I show that situating family regulation in a dynamic process towards an equilibrium that appears as a moving target explains how the demographic transition can be driven by economic factors even while fertility shows no significant correlation with economic variables such as productivity or the male–female productivity ratio. 2. Food and fire: A model of family partnerships Bergstrom (2007) employs an arboreal allegory to represent the choices confronted by a man and a woman whose offspring share their genes. ‘‘Bob and Alice live in a forest [...] Alice divides her time between collecting food and gathering wood [...] Bob spends all his time gathering food’’, and ‘‘every night huddles beside Alice’s fire [...] Bob leaves morsels of food by the fire for Alice. Warmth and food are both ‘normal goods’ for Alice. The extra food Bob leaves induces her to increase her food consumption, but not by the total amount Bob leaves her. She uses some of her time saved by Bob’s gifts to gather more firewood’’. Bergstrom focuses on the case of unilateral gifts, though he also discusses common interests, and Pareto efficiency and conditional payments. With unilateral gifts, Alice maximizes her utility function U (cA , y) = cA y where ca is the amount of food she eats, and y the amount of wood on the fire. ‘‘She has T hours to allocate between collecting food and wood. In an hour, she collects one unit of wood or πA units of food. If Bob leaves g units of food, she maximizes her utility by choosing: y=
1 2
T+
g
πA
and cA =
1 2
(πA T + g ).
(1)
Bob’s utility function is U (cb , y) = cB y, where cB is his food consumption, and y is still the amount of wood on Alice’s fire. Bob is useless at collecting wood, but gathers πB units of food per hour for T hours. If 3πA ≥ πB ≥ πA , Bob maximizes his utility by choosing: y=
T πA + πB 4
πA
and cB =
T 2
(πA + πB ).
(2)
He achieves this outcome by giving g = T2 (πB − πA ) units of food to Alice. In this case, Alice spends the fraction (3πA − πB )/4 of time gathering food and the rest collecting wood. If 3πB ≤ πA , Bob gives Alice no food and Alice divides her time equally between collecting food and wood’’. The fire is a public good for Alice and Bob, and if ‘‘Alice is not spending all her time gathering wood, and if the sum of Alice’s and Bob’s marginal rates of substitution exceeds pA , then both are better off
242
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
if Alice gathers more wood. At a Pareto optimal allocation’’ y=T
πA + πB 2πA
if πB ≤ πA
(3)
and y=T
if πB ≥ πA .
(4)
The third possibility discussed by Bergstrom is common interest, when ‘‘the only thing Alice and Bob really care about is the size of the fire [...] Each maximizes utility by choosing consumption to maximize his or her net output’’. The micro-economics of the family was not developed specifically to explore the demographic transition, rather to analyze household consumption behavior (Lundberg and Pollak, 1993; Browning et al., 1994). Bergstrom extends the scope of the model to explain the possible role of productivity in the demographic transition. Data on gender differentials in productivity during the demographic transition are hard to come by. ‘‘Even the most comprehensive micro-data sets lack good information on potentially important productivity determinants such as physical strength, energy, and individual work habits’’ (Cox and Nye, 1989: 904). For the French textile industry, these authors publish male–female wage ratios of around 0.50 for cotton and wool spinning or weaving in 1839–45 and 1860–65. With aggregate data, those for England taken from Wrigley and Schofield (1981) must be handled with great care, as I mentioned in the introduction. For France, fertility was reconstructed in the 89 départements over the period 1806–1906 (Bonneuil, 1997), and schooling data for boys and girls aged 5–15 were corrected by Luc (1985, 1986) and Diebolt (1999) for 1837, 1850, 1867 and 1876, corresponding to the key period of the French demographic transition. The ratio of the schooling rate of boys to the schooling rate of girls in generation G is a proxy for the lifetime productivity ratio between men and women of generation G. For each French département, I compared the fertility index in 1850 with the schooling ratio for 18–28 year olds – measured on those aged 5–15 in 1837 –, the fertility index in 1867 with the schooling ratio for 22–32 year olds – measured on those aged 5–15 in 1850 – and the fertility index in 1876 with the schooling ratio of 16–26 year olds – measured on those aged 5–15 in 1867. Regression of differenced fertility (to eliminate temporal correlations) on differenced mortality, on differenced schooling ratio and on region shows no clear significant relationship whereby fertility would vary with the schooling ratio, but the coefficient of differenced mortality is always significant in the early phase of the transition. Although research consistently shows that mortality and fertility had little or no correlation to real wages in historical demographic regimes (Lee, 1990), as Bergstrom correctly observes, it equally consistently shows the decline in mortality as preceding that in fertility. Fertility decline followed mortality decline, either closely as in France, or only after mid-nineteenth century in the rest of Europe, such as Sweden after 1860 or England and Wales after 1880 (Coale and Watkins, 1986). The family partnership interpretation of the demographic transition thus requires the inclusion of mortality in the utility function. My view is that on average people behaved as if they were aware of the mortality level, and were lucid enough to realize that they must have at least yA eδ children, where e−δ represents survivorship, and yA is a target level of the number of children expected to be alive at their own age of reproduction. To return to the food and fire parabola, the rain falling on Alice’s fire renders part of the wood definitively unfit for burning and part of food unfit for eating (mortality), and the fire is enough to warm Alice and Bob once a certain size (target parity) has been attained. Indeed, in Bergstrom’s model where the marginal utility from the fire is simply the time y devoted to the fire, if Alice and Bob were not limited by their time resources, the larger the fire, the greater their happiness. In my interpretation, Alice and Bob need a fire of a sufficient size if they are to maintain health and combat cold, but a fire that is too large is too hot. I understand Bergstrom’s metaphor to be predicated on the belief that evolution favors those with the largest fire. However, if evolution favors the strategies that maximize reproductive success, individuals do not favor evolution, but simply their capacity to stay in the game. This theme has been developed in biology: evolution certainly favors the most prolific species, but, more tautologically, the species that can be observed today are merely those which have managed to perpetuate themselves
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
243
so far. Biologists are well aware of the question of survival and co-existence. For example, GarcíaMartinez et al. (1998) pointed out the role played by a time-varying natural selection in maintaining the co-existence of two mitochondrial haplotypes called I and II, whereas cage experiments invariably lead to the fixation of haplotype II. Constancy of conditions favors the fittest haplotype, but fluctuations of the environment allow polymorphism, because the fittest is not always the same. The concept of ‘‘fittest’’ must be replaced by that of ‘‘staying in the game’’. Other examples: Giesel (1976, 75) concludes from studies of passerine birds that variability in fertility is an ‘‘important adaptation to account for environmentally induced unpredictability of the major selective forces’’. Barker et al. (1986), studying temporal variation in allozyme frequencies at six loci from monthly collections over four years in one population of the cactophilic species Drosophila buzzatii, show significant variation over time in allele frequencies, and deny the presence of any clear cyclical or seasonal pattern. The mathematics of evolution are concerned at least as much with maintenance as with maximization (Pianka, 1978; Vincent and Brown, 1986; Bonneuil, 1994a,b, 2008; Bonneuil and Müllers, 1997; Bonneuil and Saint-Pierre, 2000, 2002, 2005, 2008). If parents have a target parity, the time devoted to child care increases their marginal utility when this time is less than the target, and diminishes it thereafter. Instead of a term y as in Bergstrom, I suggest a marginal utility of logistic form, starting from a positive level y0 representing the utility of being alive – while in Bergstrom, utility is nil in the absence of fire – and upper-bounded by a term δ
depending on mortality, such as y0 (1 + e−β yA e ), such that the inflexion point is yA eδ . Alice’s utility is then δ
UA (cA , y, δ) = cA
y0 (1 + e−β yA e ) δ
1 + eβ(y−yA e ) where β is a negative parameter, and where consumption cA is: cA = πA (T − y) + g .
(5)
(6)
Similarly, Bob’s marginal utility can be taken logistic: δ
UB (cB , y, δ) = cB
y0 (1 + e−β yA e )
(7)
δ 1 + eβ(y−yA e )
with cB = πB T − g
(8)
assuming, for the sake of simplicity, that Bob and Alice have the same target yA eδ and the same childless marginal utility y0 . Alice maximizes her utility at g given to obtain y(g ), and Bob maximizes his utility by selecting g = gmax with y(g ). The calculus is less simple than for the case of linear marginal utility of children: Proposition 2.1. If utilities of both spouses depend on mortality in a logistic manner, according to Eqs. (5) and (7), the solution of the unilateral gift maximization program under constraints of Eqs. (6) and (8) is: y(gmax ) ≈
2
β
yA eδ
+
2
−
1 2β
1/2 πB β 2 y2A e2δ + 4β yA eδ + 8 − 4β T 1 + πA
(9)
Proof. Alice maximizes her utility in Eq. (5) at g given to obtain: y(g ) = T +
g
πA
+
1
β
+
1
β
PLog
−1−β πg +T −yA eδ A e
(10)
where PLog(z) is the dominant root of the equation w ew = z. Expressing exp(y(g )) using Eq. (10) yields the expression of exp(w), which, multiplied by w , whose expression is taken from Eq. (10), leads to w exp(w), and after simplification, to the inverse formula: g = πA
y(g ) −
1
β
−T −
1 −1−β e
β
g
πA +T −yA e
δ
.
(11)
244
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
Bob maximizes his utility in Eq. (7) by selecting g = gmax such that
∂ UB (gmax ) = 0 ∂g
(12)
given y(g ). Differentiating Eq. (11) with respect to g yields:
∂ y(g ) 1 1 = ∂g πA 1 + e−β(y(g )−yA )
(13)
which helps solving the maximization of UB in Eq. (12):
∂ UB ∂ y(g ) = −y(g ) + (πB T − g ) = 0. ∂g ∂g
(14)
Numerical simulations show that y(gmax ) is greater than yA exp δ(t ), but close to it, a property consistent with the empirical observation that fertility followed mortality. Hence, a first-order development in the neighborhood of yA exp δ is possible. The positive solution (with t omitted) is then given by Eq. (9). π
The final result is the analytical expression of y(gmax ) as a function of the productivity ratio πB and A of yA eδ . It is close to a linear function of yA eδ always above yA eδ . So Bergstrom’s result that y depends only on the productivity ratio between the man and woman, and not on the productivity of one spouse alone, still holds. The second result is the dependence of y(gmax ) on mortality, varying linearly and with a positive coefficient with yA eδ . 3. The demographic transition: Moving equilibrium or moving to a moving target? To present the demographic transition as the shifting of an economic equilibrium, whether unilateral, Pareto, or common interest, is not devoid of paradox. In this view, agents would be conveyed through the demographic transition astride a shifting equilibrium. There would be no restriction on modifying secular habits governing the division of labor between spouses. Yet in both France and Great Britain, at least until the 1940s, women were ‘‘locked up’’ in familial dependence. In Great Britain, the marriage bar imposed from the mid-nineteenth century onwards kept married women out of salaried work (Hakim, 1987). The result was the low activity rate of women in England until the 1930s (10% against 50% in France in 1901 (Hantrais, 1990: 109)). In France, women’s choice between the home and wage–earning was limited by the role they were expected to play, with their work outside the home being perceived as a danger to the family (Daune-Richard, 2004). So economic equilibria did not shift without encountering resistance rooted in customs and habits. After all, demographic behavior is not a mechanistic device such that the decline in mortality would automatically lead to a fall in fertility. The decision to limit family size may be made possible by improved living conditions, but it remains the product of couples’ willingness to have or to avoid having children. Taking into account the time necessary for people to alter their habits changes the formulation of the problem from the configuration of preferences at a static equilibrium to the dynamic of preferences on a path to a moving target – moving because mortality continues its decline. Weir (1984, 28), for example, after pointing out ‘‘the a-historical character of any equilibrium analysis’’, concluded, from empirical observations, that trend shifts may have dominated equilibrium adjustments. Let us focus, like Bergstrom, on economic regulation with unilateral gifts. This model is relevant for describing France and Great Britain during their demographic transition. In spite of differences notably in activity rates or productivity (Maddison, 1987), the family was organized according to the male breadwinner model under which the husband was a salaried worker while the wife was in charge of running the home (Scott and Tilly, 1987; Daune-Richard, 2004).
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
245
)
0
80
Fig. 1. Simulated transition: the equilibrium-target follows mortality, the state follows with delay, confusing its relationship with mortality. The variable representing mortality, δ , is normalized for comparison with the time allocated to child care y. π yA = 2, β = −0.2, T = 12, πB = 2. A
In the case of unilateral gift, letting the argument δ(t ) appear, Alice and Bob start from (y, g ) = (y(gmax (δ(0)), δ(0)), gmax (δ(0))) and move in direction of the target at time t
(y(gmax (δ(t )), δ(t )), gmax (δ(t ))),
(15)
which is itself moving under the influence of mortality and the male–female productivity ratio. The dynamics is given:
y0 (t ) = dA (y(t ), g (t ), vA (t )) g 0 (t ) = dB (y(t ), g (t ), vB (t ))
(16)
where dA and dB specify a dynamic, and vA (t ) and vB (t ) are controls admissible in closed sets VA = [ [ ] [ [ ] [vA , vA ] and VB = [vB , vB ], with vA < 0, vB < 0 (to make fertility decline possible). VA represents Alice’s capacity for action and VB Bob’s. VA and VB are closed because Alice and Bob are limited in their capacity for change, and because the transition takes time to unfold. One particular case is when Alice and Bob can change their own variables independently of other variables:
y0 (t ) = vA (t ) g 0 (t ) = vB (t ).
(17)
There is no reason why y0 (t ), the change of the time y devoted to children, and y0 (gmax ) = ∂ y(gmax ) 0 ) πB 0 δ (t )+ ∂ y(gπmax ( π ) the change of the target, which is the equilibrium at date t, should produce B ∂δ ∂π
A
A
a significant relationship in an econometric study (with discrete time). The relationship of fertility to mortality and to the ratio of productivity between spouses is confused in the short term by the fact that the system is not in equilibrium but pursuing a target, which is itself driven by mortality and economic change. For an example, Fig. 1 shows a simulation where mortality δ is decreasing in a logistic pattern, consistent with what is observed in France and England during their respective π transitions. For πB fixed, the target y(gmax(δ) , δ)(t ) follows mortality. In this simulation, vA is taken as A null until time 40, which is the onset of the transition, negative thereafter, so that the state y(t ) of the system follows this characteristic curve of the fertility decline, delayed with respect to the mortality decline. A uniform noise is added to these three variables, and time series yn , y(gmax )n , δn are drawn from means over each time interval [nh, (n + 1)h) with h representing a duration of one year. The
246
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
regression of the target y(gmax )n on δn yields a significant coefficient, while the regression of the state π yn on δn is not significant. The same can be done with the ratio of productivity πB . A The pace of the transition results from a competition between the velocity at which the target moves and the velocity at which the system pursues the target. This is exactly what is revealed by the co-integration analysis of the demographic transition in France based on a finely detailed spatial reconstruction for each of the 89 (in 1861) départements. The French transition unfolded in two stages: at the onset of transition, the improvement in mortality provided an opportunity to undergo fewer pregnancies, but fertility quickly became insensitive to economics and to mortality decline, with the response to the environment fading away as harsh conditions disappeared. Further change was increasingly dependent on culture, habits and attitudes (Bonneuil, 1997). Co-integration and analysis of short-term fluctuations show no significant relationship between fertility and education, mortality, productivity ratio between spouses, or urbanization, but the overall pattern of fertility catching up with these variables is clear to see, and is consistent with a target pursuit model (rather than with an error correction model or homeostatic model). An important parameter of the transition is its duration. The presentation of the demographic transition in terms of target pursuit allows its expression to be given for simple cases. With time 0 corresponding to the onset of the transition, one minimal time from onset to target at t is obtained by solving: t
Z min
dt
(18)
0
[
[
under the dynamic of Eq. (17). If uA ≥ uB , the maximization of the associated Hamiltonian shows that [
one optimal trajectory is obtained with uA = uA , so that the minimal time to horizon H, considering that the target is moving with δ(t ) is: tmin (H ) =
H
Z 0
∂ y(gmax (δ(t ))) dt . ∂δ uA (t ) 1
(19)
[
[
For uA constant in time, as simulations and the approximation of Eq. (9) show, y(gmax ) is close to a linear function of yA eδ (t ) remaining above yA eδ (t ), so that tmin (H ) ≈
y(gmax (δ(H ))) − y(gmax (δ(0))) [
uA [
.
(20)
[
Similarly, when uA ≤ uB , tmin (H ) ≈
gmax (δ(H )) − gmax (δ(0)) [
uB
.
(21)
4. Conclusion Bergstrom (2007) showed that equilibrium with unilateral gifts in family regulation incorporates only the productivity ratio between spouses, which explains why the fertility decline was not correlated with the variations in female productivity alone. However, all historical transitions persistently identify the precedence of the mortality decline (a fact not contested by Bergstrom but that he ignores for the sake of his demonstration). Continuing the analysis, I introduced mortality into the spouses’ utilities through a logistic marginal utility of children to show that, at equilibrium, fertility follows the mortality decline. The maximization of utilities is tractable, and I gave analytical expressions. However, while empirical cases illustrate the pattern of mortality precedence, the short-term covariation of fertility with mortality, or with socio-economic variables disappears after the onset of the transition, at least in nineteenth century France, which is among the most fully documented
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
247
historical examples at a fine spatial scale. An account of the transition as a dynamic system in which the time devoted to children moves in the direction of the equilibrium allows resistance to change to be incorporated. It explains the lack of short-term correlation between fertility, mortality, and economic variables, but fits the pattern of fertility catching up with these latter variables in the long run. It also allows for the existence of varied responses to the fall in fertility and the improvement in productivity, in conformity with historical observations, notably the conflicting behavior of the British, who retained high fertility long after the onset of the mortality decline, and the French, who adjusted to it immediately. References Ariès, P., 1971. Histoire des populations françaises et leurs attitudes devant la vie. Self, Paris, (1948), Le Seuil, Paris. Barker, J.S.F., East, P.D., Weir, B.S., 1986. Temporal and microgeographic variation in allozyme frequencies in a natural population of Drosophila buzzatii. Genetics 112, 577–611. Bergstrom, T.C., 2007. Some evolutionary economics of family partnerships. American Economic Review 97 (2), 482–486. Bonneuil, N., 1990. Turbulent Dynamics in a 17th century population. Mathematical Population Studies 2, 289–311. Bonneuil, N., 1992. Cohérence démographique et non-identifiabilité de la rétro-projection. in: Blanchet, D., Blum, A., Bonneuil, N. (Eds), Méthodes de la démographie historique. INED, Congrès et Colloques 11, Paris, pp. 99–108. Bonneuil, N., 1994a. Capital accumulation, inertia of consumption and norms of reproduction. Journal of Population Economics 7, 49–62. Bonneuil, N., 1994b. Malthus, Boserup and population viability. Mathematical Population Studies 4, 107–119. Bonneuil, N., 1997. Transformation of the French Demographic Landscape, 1806–1906. Clarendon Press, Oxford. Bonneuil, N., 2008. The mathematics of maintenance and acquisition. In: Chen, J., Guo, C. (Eds.), Ecosystem Ecology Research Trends. Nova Science Publishers, New York, pp. 153–175. Bonneuil, N., Dassouki, C., 2006. Women’s education and diffusion of the fertility transition: The case of Egypt 1960–1996 in 4905 administrative subdivisions. Journal of Population Research 23 (1), 27–39. Bonneuil, N., Dassouki, C., 2007. Economics, Geography, Family Planning and Rapidity of change in the demographic transition: The case of the Egyptian muhafazas 1960–1996. Journal of Developing Areas 40 (2), 185–210. Bonneuil, N., Müllers, K., 1997. Viable Populations in a prey–predator system. Journal of Mathematical Biology 35, 261–293. Bonneuil, N., Saint-Pierre, P., 2008. Beyond optimality: Managing children, assets, and consumption over the life cycle. Journal of Mathematical Economics 44 (3–4), 227–241. Bonneuil, N., Saint-Pierre, P., 2000. Protected polymorphism in the two-locus haploid model with unpredictable fitnesses. Journal of Mathematical Biology 40 (3), 251–277. Bonneuil, N., Saint-Pierre, P., 2002. Minimal number of generations out of Polymorphism in the one-locus two-allele model with unpredictable fertilities. Journal of Mathematical Biology 44 (6), 503–522. Bonneuil, N., Saint-Pierre, P., 2005. Population viability in three trophic-level food chains. Applied Mathematics and Computation 169–2, 1086–1105. Boucekkine, R., de la Croix, D., 2002. Vintage human capital, demographic trends, and endogenous growth. Journal of Economic Theory 104, 340–375. Boucekkine, R., de la Croix, D., Licandro, O., 2004. Modelling vintage structures with DDEs: Principles and applications. Mathematical Population Studies 11 (3–4), 151–180. Browning, M., Bourguignon, F., Chiappori, P.-A., Lechene, V., 1994. Income and outcomes: A structural model of intrahousehold allocation. Journal of Political Economy 102 (6), 1067–1096. Coale, A., Watkins, S.C. (Eds.), 1986. The Decline of Fertility in Europe: The Revised Proceedings of a Conference on the Princeton European Fertility Project. Princeton University Press, Princeton, NJ. Carlsson, G., 1969. The decline of fertility: Innovation or adjustment process. Population Studies 20, 149–174. Cervellati, M., Sunde, U., 2005. Human capital formation, life expectancy, and the process of development. American Economic Review 95 (5), 1653–1672. Cleland, J., Wilson, C., 1987. Demand theories of the fertility transition: An iconoclastic view. Population Studies XLI (2), 5–30. Cox, D., Nye, J.V., 1989. Male–Female wage discrimination in nineteenth century France. Journal of Economic History 49 (4), 903–920. Daune-Richard, A.-M., 2004. Les femmes et la société salariale: France, Royaume-Uni, Suède. Travail et Emploi. Travail et Emploi 100, 69–84. Diebolt, C., 1999. Les effectifs scolarisés en France: XIXe et XXe siècles. International Review of Education 45, 197–213. Galor, O., Weil, D.N., 2000. Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond. The American Economic Review 90 (4), 806–828. García-Martinez, J., Castro, J.A., Ramón, M., Latorre, A., Moya, A., 1998. Mitochondrial DNA haplotype frequencies in natural and experimental populations of Drosophila subobscura. Genetics 149, 1377–1382. Giesel, J.T., 1976. Reproductive strategies as adaptations to life in temporally heterogeneous environments. Annual Review Ecological Systems 7, 57–79. Hakim, C., 1987. Trends in flexible workforce. Employment Gazette (November), 549–555. Hansen, G.D., Prescott, E.C., 2002. Malthus to Solow. Amercican Economic Review 92 (4), 1205–1217. Hantrais, L., 1990. Managing Professional and Family Life. Darmouth Publishing Company, Hants and Vermont. Knibiehler, Y., 2000. Histoire des mères et de la maternité en Occident. PUF, Que sais-je ?, Paris. Lee, R.D., 1985. Inverse projection and back projection: A critical appraisal, and comparative results for England, 1539 to 1871. Population Studies 39, 233–248.
248
N. Bonneuil / Mathematical Social Sciences 59 (2010) 239–248
Lee, R.D., 1990. The demographic response to economic crisis in historical and contemporary populations. Population Bulletin of the United Nations 29, 1–15. Lee, R.D., 1993. In: Reher, D., Schofield, R.S. (Eds.), Inverse Projection and Demographic Fluctuations, in Old and New Methods in Historical Demography. Oxford University Press, Oxford, pp. 7–28. Luc, J.-N., 1985. La statistique de l’enseignement primaire XIXe-XXe siècles. Politique et mode d’emploi, Inrp/Cnrs, Paris. Luc, J.-N., 1986. L’illusion statistique. Annales E.S.C. 41, 887–911. Lundberg, S., Pollak, R.A., 1993. Separate spheres bargaining and the marriage market. Journal of Political Economy 101 (6), 988–1010. Maddison, A., 1987. Growth and slowdown in advanced capitalist economies: Techniques of quantitative assessment. Journal of Economic Literature 25, 649–698. Oeppen, J., 1993. Back projection and inverse projection: Two members of a wider class of constrained projection models. Population Studies 47, 245–267. Pianka, E.R., 1978. Evolutionary Ecology. Harper and Row Publishers. Scott, J.-W., Tilly, L., 1987. Les femmes, le travail et la famille. Rivages, Paris. Ruskin, J., 2008. Sesame and Lilies. Bibliolife. Powell’s Books, Portland OR, 1864. Vincent, T.L., Brown, J.S., 1986. Evolution under nonequilibrium dynamics. Mathematical Modelling in Science and Technology 29 (31–8), 766–771. Watkins, S.C., 1987a. The fertility transition: Europe and the Third World compared. Sociological Forum 2 (4), 645–673. Watkins, S.C., 1987b. In: Stycos, J.M. (Ed.), Demography as an Inter-Discipline. Transaction Publishers, pp. 27–55. Weir, D., 1984. Life under pressure: France and England, 1670–1870. The Journal of Economic History 44 (1), 27–47. Wrigley, E.A., Schofield, R.S., 1981. The Population History of England 1541–1871, A Reconstruction. Edward Arnold, London. Young, A., 2005. The gift of the dying: The tragedy of Aids and the welfare of future African generations. The Quaterly Journal of Economics 2, 423–466.