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Evolution, Income-dependent Relative Concerns, and Modern

Fertility Transition

Junji Kageyama∗

Revised on April 30, 2013


The aim of this study is to examine how the income-dependency of relative concerns affects the general pattern of modernization. Incorporating this type of preferences into an economic model, this study provides an explanation for the relationship between modern economic growth and modern fertility transition.

JEL: A12, B41, D01, J13, O11

Department of Economics, Meikai University, Akemi 1, Urayasu, Chiba 279-8550, Japan (e-mail: I thank Joshua R. Goldstein for valuable discussions and suggestions on the relationship between

fertility and social interaction, and Michael Kuhn for providing me helpful comments. Any remaining errors are of

course my own. This study is financially supported by Grant-in-Aid for Challenging Exploratory Research from the




An increasing number of studies in economics search for biological bases of preferences. The

bi-ological basis of time preferences, for example, has been examined by Rogers (1994), Sozou and

Seymour (2003), Robson and Samuelson (2009), Kageyama (2011, 2012b), and Chowdhry (2011).

The origin of relative concerns, likewise, has been investigated by Cole, Mailath and Postlewaite

(1992, 1995), Robson (1996, 2001), Samuelson (2004), N¨oldeke and Samuelson (2005), Rayo and

Becker (2007), De Fraja (2009), and Kageyama (2012a).

Yet, the aim of these studies is often limited to providing explanations for the preferences that

have already been discovered in empirical and experimental studies, such as the age-trajectory of

time preference, hyperbolic discounting, and the income-happiness paradox (Easterlin, 1974). An

attempt to make novel predictions on preferences purely from biological theories is still rare.

The contribution of such an attempt is not only to fulfill theoretical interests. More importantly,

particularly in social science, its significance lies in its application to provide explanations for

economic and social phenomena.

Along this line, this study focuses on the finding in Kageyama (2012a) that, for low income levels,

people care about absolute, but not relative, standing such as the absolute level of consumption,

but, when income reaches a certain level, people start to care about both absolute and relative

standing, suggesting that the degree of concerns for relative standing is at least partially increasing

in income.

The rationale for this finding comes from behavioral ecology studies that, in pre-modern

hu-man population, social standing affects lifetime reproductive success through reproduction, which

depends on both fertility and the survival of offspring, rather than through one’s own survival.

In-corporating this effect into a biological model, Kageyama (2012a) shows that, as resources increase,

reproduction and, thus, relative standing become more decisive to lifetime reproductive success,

leading to the spread of preferences that induce greater concerns for relative standing when income


Introducing this result into an economic model, the present study examines how the

income-dependency of relative concerns affects reproductive behavior and economic performance. This


and Tsoukis (2008) that incorporate status concerns in examining fertility and economic growth. In

particular, Tournemaine (2008) and Tournemaine and Tsoukis (2008) show that a greater concern

for status results in having fewer but higher quality children. In this respect, the novelty of the

present analysis is to include the income dependency in status concerns.

The rest of the paper is organized as follows. The next section provides a theoretical model,

and Section 3 discusses its implications. The results show that, once the economy escapes from

the Malthusian trap, this type of preferences induces endogenous economic growth and the fertility

transition characterized by the initial increase in fertility and the reduction in fertility thereafter.

Section 4 concludes.



I assume the following properties to hold. First, individuals are homogenous. Second, individuals

live for two periods, the first period we refer to as childhood, and the second period, adulthood.

Third, individuals make all decisions in adulthood. Forth, at period t, income, yt, is allocated to

consumption,ct, the number of children,mt, and educational input,st. Thus, the budget constraint

is given by

yt≥ct+ (ψ+σst)mt (1)

where ψ and σ respectively denote the cost of raising a child and the price of educational input,

both of which are time-invariant. Fifth, income depends on human capital,ht, and human capital

depends on educational input in the previous period. Specifically, they are respectively given by

yt=ωht and ht=sβt−1 where ω is wage rate andβ <1 is an efficiency parameter.

Sixth, individuals are endowed with income-dependent relative preferences, meaning that

in-dividuals are concerned with their own relative standing and that the degree of such concerns is

increasing in income. Seventh, parents care about the well-being of their children as well as their

own. However, as parents cannot directly observe their children’s utility, I assume that parents

perceive children’s utility through empathy. Empathy helps the parents understand the feelings of

children but through the parents’ own scale of utility.1



This has two implications. First, parents use the same income-dependent relative preferences to

measure their children’s well-being. Second, since parents obtain their own utility from consumption

in adulthood, they care about their children’s consumption in their adulthood. To capture this

latter aspect, I assume that parents are concerned with children’s growth that signals the children’s

consumption in adulthood and measure children’s growth with educational output.

By putting these conditions together and by following Weber’s law that suggests that we measure

utility in logarithmic form, the utility function of the adult at period tcan be written as

Ut=γln(ct−cˆt) + (1−γ) ln




t −sˆβt



where γ is the importance of one’s own consumption relative to reproduction, and ˆct and sˆβt are

the reference levels for comparison.

Here, ˆct and sˆβt depend on their social averages, ¯ct and s¯βt, such that ˆct = ztct¯ and sˆβt = zts¯βt

whereztdenotes the degree of concerns for the social averages. Asztincreases, the individual cares

more about the social averages, and ifzt= 0, the individual cares only about absolute levels. Note

that, given that zmax is the upper limit ofzt, I assume for technical simplicity that the condition

zmax<1−β holds. To capture the income-dependency of relative concerns, I further assume that

zt=z(yt) and, for anyz(yt)≤zmax,z′(yt)>0 hold.2

2.1 Intratemporal equilibrium

With these specifications, the Lagrangian for the adult at periodt becomes

L(ct, mt, st, λt) = γlog(ct−ˆct) + (1−γ) log




t −sˆβt


+λt[yt−ct−(ψ+σst)mt]. (3)

whereλtis the Lagrangian multiplier. Note thatytis not controllable for the adult at periodtas it

is determined by educational input in the previous period. Subsequently, the first-order conditions

scale of utility to measure the feelings of others.


I assume that parents have positional concerns on the education, but not on the number, of children, considering

that positional concern on the number of children is significantly less, if not non-existent, as compared to those on

education. Namely, parents don’t gain utility from having more children but from higher-educated children. However,

I acknowledge the possibility that a greater number of children may result in enhancing actual social status in some


are given by

∂L ∂c =γ


c−ˆc−λ= 0, (4)


∂m = (1−γ) 1

m −λ(ψ+σs) = 0, (5) ∂L

∂s = (1−γ)


sˆβ −λσm= 0, (6)



∂λ =y−c−(ψ+σs)m= 0 (7)

where period subscripts are suppressed whenever possible.

Next, assume that the population is sufficiently large that the changes incandsat the individual

level have a negligible impact on the social averages. Then, by applying the condition that adults

are homogeneous and thus choose the same levels of cand s, ¯c and ¯sβ are respectively given byc

and sβ. As a result, consumption, fertility, and education become

c= γ

γ+ [1−z(y, α)](1−γ)y, (8)

m= [1−z(y, α)−β](1−γ) γ+ [1−z(y, α)](1−γ)


ψ, (9)


s= ψ σ


1−z(y, α)−β. (10)

With these results, we can examine howc,s, andmrespond to a change iny. By differentiating

c,s, and mwith respect to y, we have

∂c ∂y =


γ+ [1−z(y)] (1−γ) +


{γ+ [1−z(y)] (1−γ)}2y >0, (11)

∂s ∂y =

ψ σ


[1−z(y)−β]2 >0, (12)


∂m ∂y =

[1−z(y)−β] (1−γ) γ+ [1−z(y)] (1−γ)

1 ψ−

[γ+β(1−γ)] (1−γ)z′(y) {γ+ [1−z(y)] (1−γ)}2


ψ. (13)

Equations (11), (12), and (13) show that the impact of income on the resource allocation consists

of two effects: the income effect, i.e., the direct effect of income, and the relative concern effect,

i.e., the change in concerns for relative standing. Consumption increases with income because


positive while the income effect is nil. By contrast, the impact of income on fertility is indeterminate.

Equation (13) shows that the relative concern effect is negative while the income effect is positive.

To further examine this point, consider the case, which I call the benchmark scenario, where

z(y) is linear in y such that z(y) =ηy forz(y)≤zmax. In this case, equation (13) becomes

∂m ∂y =


[γ+ (1−ηy)(1−γ)]2 1 ψ




forz(y)≤zmax. Here, the term in the final bracket, (1−ηy)2


−β, is positive ifyis sufficiently

small, and is negative if y is sufficiently large. The income effect dominates the relative concern

effect when income is low, and vice versa when income is high. As a result, in the present model,

the relationships between education and income, and fertility and income can be summarized as:

Proposition 1. Given the positive effect of income on relative concerns, education increases with

income. Moreover, under the benchmark scenario, fertility increases with income when income is

sufficiently low, and decreases with income when income is sufficiently high.

2.2 Intertemporal equilibria

At the same time, income depends on reproductive decision. Reproductive decision intertemporally

affects income through the level of human capital. This intertemporal effect is summarized in the

dynamics of human capital. By inserting equation (10) andyt=ωht intoht+1=sβt,ht+1 becomes

ht+1=j(ht) =



β 1−z(ωht)−β


for z(ωht)≤zmax. Equation (15) shows that j(ht) is positive at z(ωht) = 0 and increasing in ht.

Furthermore, under the benchmark scenario wherez(yt) =ηyt,j(ht) is convex inht. This leads to

the following proposition.

Proposition 2. Under the benchmark scenario:

(1) If j(0) and j′(h

t) are sufficiently small, there exist at most two steady states that satisfy

ht+1 = ht where the one with the lower ht is stable and the one with the higher ht is unstable.


satisfiesht+1 =j(hmax) wherehmax= zmaxηω , corresponding to the corner solution at zmax.

(2) Ifj(0) andj′(h

t) are sufficiently large, there exists no steady state that satisfies ht+1 =ht, but

exists a stable steady state that satisfiesht+1 =j(hmax).

Figure 1 presents these two cases. The convex curve at the bottom represents the first case,

and the one at the top represents the second case. In the second case, human capital continues to

grow until it reaches its upper limit where ht+1=j(hmax) holds.

Place Figure 1 around here



The present analysis yields the following implications. First, it provides an explanation for the

big-push and the subsequent economic growth in modern history. To see this, suppose that the

economy is originally trapped at its stable steady state presented by the pointE1 in Figure 2, and

that educational efficiency eventually increases. This lowers the price of education, σ, and shifts

j(ht) upwards. If the shift is sufficiently large, j(ht) no longer intersects with ht+1 =ht, and, as

shown in Figure 2, human capital and income jump to the new levels and grow steadily until the

economy reaches the new steady state E2.

Place Figure 2 around here

In this growth process, fertility increases initially but decreases thereafter. The concave curve

at the bottom of Figure 2 presents this dynamics. This is consistent with the empirical evidence

on the modern fertility transition. As presented by Dyson and Murphy (1985), most countries,

both developing and developed, had undergone a rise in fertility in the initial phase of the modern

economic growth before they experienced a reduction in fertility.


Galor and Weil (2000) and Tournemaine and Tsoukis (2008). Galor and Weil (2000) show that

technological progress that increases the return to education can explain this phenomenon.

Tourne-maine and Tsoukis (2008), on the other hand, argue that an exogenous shock in status concerns

is the cause of the fertility transition. Relating to this point, the present study suggests that the

change in status concerns is indeed endogenous.3

Second, the present study shows that the time cost of raising children is not a prerequisite for

the quality-quantity trade-off, pointing to a possibility that fertility may decline even when the

opportunity cost of raising children remains constant.4

This result is consistent with the pattern of fertility decline in Japan in the third quarter of

the 20th century where marital fertility declined dramatically while being a housewife remained a

common practice. In this period, although women’s participation in the labor force did not rise,

the fertility rate declined from 3.65 to 1.91 mainly due to the change in marital fertility (National

Institute of Population and Social Security Research of Japan, 2008, Table 4.15).5

This indicates

that the cause of the fertility decline during this period is not the increase in working opportunities

for women. Instead, recalling that Japan experienced a rapid economic growth in this period, we

can attribute it to the relative concern effect.


Concluding Remarks

This study incorporates the finding in Kageyama (2012a) that the degree of relative concerns is

income-dependent, and shows that, once the economy escapes from the Malthusian trap, this type

of preferences induces endogenous economic growth and the fertility transition characterized by

the initial increase in fertility and a reduction thereafter. By incorporating biologically-predicted

preferences into economics, it provides an explanation for the general pattern of modernization.


Applying these results, the present analysis also provides an explanation for the transition from a Malthusian

Regime, through a Post-Malthusian Regime, to a Modern Growth Regime discussed in Galor and Weil (2000).


See Jones, Schoonbroodt and Tertilt (2008) for the importance of the time cost for generating the quality-quantity



Women’s labor force participation rate was 48.6% in 1950, 50.9% in 1960, 50.9% in 1970, and 46.1% in 1975

(Statistics Bureau of Japan, 2011b). Focusing on married women, the rate declined from 51% in 1962 to 45% in 1975


These results suggest that limiting research to a narrow and safe definition of preferences would

incur costs. The income-dependency of relative concerns, being taken as an example, potentially

accounts for various economic and social phenomena. A novel prediction on preferences inserts a

unique perspective into the study of sociality.


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ht hmax

•• ht+1 = ht j(ht)


ht+1 = ht

ht hmax

mt E1

E2 j(h

t) with high σ j(ht) with low σ

Figure 1: Steady states and dynamic paths.

Figure 1:

Steady states and dynamic paths. p.12
Figure 2: Modern economic growth and modern fertility transition.

Figure 2:

Modern economic growth and modern fertility transition. p.13