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Page 1: Demographic Transition, Income Distribution,

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30 MOMI DAHAN AND DANIEL TSIDDON

Figure 1. Fertility in various countries.

becomes more equally distributed, and output per capita takes off. Since the poor have a

growing weight in the economy in the first phase and a declining weight in second phase,

the economywide average rate of fertility first increases and subsequently declines.

The article considers two sources for the difference in the net  return to investment on

human capital between rich (educated) and poor (uneducated) individuals. One source is

capital market imperfections. We assume that borrowing funds for investment in education

is more costly than self-finance due to the nonacceptability of human capital as a collateral.A second, companion, source of the difference in net returns on education between the rich

and the poor is the informal education imparted by parents to their children; we show that

both channels generate qualitatively similar results.

The relationship between fertility and economic growth described in this article differs

from previous research in two major respects: (1) these variables are linked via income

distribution, and (2) the initial surge in the rate of fertility is the cause of the subsequent

fertility decline. Furthermore, onceembeddedin a conventionalgrowthmodel, the evolution

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 31

of the economy is consistent with the takeoff of sustained economic growth that succeeded

the decline in the rate of fertility.

The article is organized as follows. The rest of the introduction surveys the relevant

literature briefly. Section 2 develops the basic argument in its simplest form. Section 3

expands on the linkages between income distribution and demographic transition, as well as

on theeffect of an exogenous decline in child mortality rate on this linkage.1 Section 4 brings

the idea a step closer to the data and explores the links between demographic transition and

human-capital-based economic growth. Section 5 demonstrates the role of parental effect

on child capacity to acquire education in this context, and Section 6 concludes.

1.1. Evidence About Income Distribution, Education, and Fertility

The connection between income distribution, differential fertility rates, and economic

growth is discussed by Simon Kuznets (1973, pp. 46–48; originally 1967):

In many societiesand over long periods, fertilityand therate of natural increase have

been greater for the poorer and lower social status groups than for the richer and

higher social status groups. . . . This negative correlation between birth rates and

rates of natural increase, on one hand, and economic status and per capita economic

performance, on the other hand, raises problems with respect to the economic

advance of the poor and generally less favored groups within any society—not

only in keeping economic and social inequality from widening, . . . but also [raises

problems] in providing a sufficient upward economic flow of potential human talent

from the surplus at the low economic levels.

In a way, our research formalizes a similar idea. Unlike Kuznets, however, we show that

the well-described deterioration in the status of the poor, which is partly due to “excess

fertility,” is indeed the source of the later phase of demographic transition. This later phase

coincides with the shiny side of the “Kuznets curve” and with an increase in the growth rate

of income per capita.

Along with Kuznets many authors have documented various aspects of demographic

transition and the relation between income distribution and fertility. Dyson and Murphy

(1985) used cross-country and cross-continent data over a very long period of time to

document the fact that the rate of  fertility (not only that of population growth) rises before

it declines. Moreover, they show that while fertility rates do fluctuate, they tend to peak 

 just before starting a prolonged decline. Dyson and Murphy (1985, p. 423) conclude that

a fertilityrise hasto [be] viewedas an integral part of theopening phase of transition.Indeed, the first sign of an impeding decline is a rise, which often starts many years

before the predecline peak. . . . Paradoxically a better appreciation of the changes

that trigger such rises may enhance our understanding of the causes and timing of 

subsequent declines.

Our work indeed shows that the seeds of the decline in fertility are shown in the previous

increase.

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32 MOMI DAHAN AND DANIEL TSIDDON

Another well-documented piece of evidence is that the transition to low rates of fertility

is tied in with an increase in investment in human capital. For example, Caldwell (1980,

p. 225) suggests, “It is argued that the primary determinant of the timing of the onset of 

the fertility transition is the effect of mass education on the family economy.” Birdsall

(1983), discussing the negative correlation between wages and fertility, also emphasizes

the role of education in bringing down fertility rates. These studies, and many others, have

emphasized the role of mass education beyond and above the role of human capital per se

in the explanation of fertility decline. Since mass education coincides with a decline in the

private costs of education, this evidence is consistent with the spirit of our model. 2

The relation between infant mortality decline and fertility decline is a conventional expla-

nation of the demographic transition. The evidence, however, is not unambiguous. Francine

van de Walle (1986) documented the fact that fertility decline in Europe took place first

in cities and only later in the countryside, although infant mortality was higher in the city

during this period. Furthermore, inspecting a cross-section of European countries in 1870,

1900, and 1930, she shows that the data reveal no reduction in infant mortality prior to the

beginning of the demographic transition. Reenforcing the view that infant mortality did

not lead to the decline of fertility, she adds that fertility decline started with very different

infant mortality rates across countries and regions in Europe.

Information about family planning seems not to be the major factor in this process as

well. Delayed marriage was used to reduce fertility throughout Europe for centuries before

changes in fertility occurred. Also, documentation of women’s behavior by their doctors

reveals that during the first half of the nineteenth century Europeans knew how to control

the number of children. On that matter Holbing wrote (Coale and Watkins, 1986, p. 16)

that “in most Hungarian villages of the county the young wives consider it a shame to bear

in the first four or even ten years of their marriage, and even the healthiest and strongestwomen bear not more than two children.” Furthermore, Timothy W. Guinnane, Barbara

S. Okun, and James Trussell (1994) show that fertility decline in Europe did not happen

as fast as demographers used to think. They interpret this finding as evidence against an

important role for the diffusion of information in this context.

Last but not least, demographers have long argued about whether an increase in income

equality yields a decline in the rate of fertility. Part of this debate appears in Boulier

(1982), who claims it does not, and Repetto (1982), who claims it does. Perotti (1996),

in a cross-country study, shows that higher income inequality is linked with a higher the

rate of fertility and that higher rate of fertility is linked with a lower secondary education

attainment. His evidence is of the cross-section type and our work analyzes the dynamics

of a single economy, but the evidence are largely consistent with our explanation.

1.2. Related Theories

Research on the interaction between income growth and fertility is not new in economics.3

Razin and Ben-Zion (1975) found a negative relationship between economic growth and

fertility, while Eckstein and Wolpin (1985) discuss the optimality of fertility choice in

the context of a growth model. Based on the work of Becker and Barro (1988), Becker,

Murphy, and Tamura (1990) show that in a world characterized by multiplicity of equilibria,

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 33

one equilibrium is characterized by a low level of output and a high rate of fertility, while

another is characterized by a high level of output and a low rate of fertility. Dahan (1996)

investigated the role of capital market imperfections in the codetermination of population

growth and economic growth, while Kremer (1993), Goodfriend and McDermott (1995),

and Lucas (1996) focus on the long-run correlation between population and income growth.

A second, more closely related, strand of literature deals explicitly with the relation

between income growth and demographic transition. Recent papers that explicitly discuss

demographic transition offer diverse explanations. Azariadis and Drazen (1991) show that

demographic transition may occur due to an increase in the bargaining power of children as

production becomes more urban oriented. Ehrlich and Lui (1991) show that the interaction

between “companionship” and altruism may lead to demographic transition. Galor and

Weil (1996) show that when capital complements women’s wages more than that of men’s,

fertility decisions are altered by capital accumulation, and demographic transition is a by-

product of development. All these explanations can serve in conjunction with the one we

provide below. We think that yet another explanation is called for since most of the facts

documented in Section 1.1 are not explained with these theories.

2. The Structure of the Economy4

Consider a small, open, overlapping-generations economy in which agents live for two

periods and capital flows freely at a fixed world interest rate r . In the first period of life,

agents are children: each child consumes a fixed quantity of his parents’ time. Children

can either perform simple tasks (unskilled work) or invest in human capital. In the second

period of life they either benefit from higher income if they invest in human capital or

work as unskilled workers for a low pay. In either case, they decide on the number of 

their offspring, become parents, and spend time bringing up their children. For simplicity,

assume that agents consume only in the second period of life.

 2.1. Production

Production occurs in one sector with constant returns to scale. There are three factors

of production—skilled workers, unskilled workers, and physical capital. The production

function is

Y t  = F (K t , L u,t , Ls,t ) = AK d t  Le

s,t  L1−d −eu,t  , (1)

where Y t  is output in period t , A is the level of technology and is constant, L u,t  is the number

of unskilled workers in period t , and L s,t  is the number of skilled workers in period t . The

economy is open, the international interest rate r  is exogenous, and competition is perfect.

The pay to factors of production is therefore

r  = F K ,t  = d A

 L u,t 

K t 

1−d  Ls,t 

 Lu,t 

e

, (2)

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34 MOMI DAHAN AND DANIEL TSIDDON

ws,t  = F  Ls,t = e A

 Lu,t 

K t 

−d  Lu,t 

 Ls,t 

1−e

= e A

d A

d 1−d 

 L u,t 

 Ls,t 

1−d −e1−d 

, (3)

wu,t  = F  L u,t 

= (1 − d − e) A

 L u,t 

K t 

−d  L u,t 

 L s,t 

−e

= (1 − d − e) A

d A

d 1−d 

 L u,t 

 L s,t 

−e1−d 

. (4)

In each period the ratio of returns to each type of labor is therefore a linear function of the

ratio of unskilled to skilled labor:

ws,t 

wu,t 

=

e

1 − d − e

L u,t 

 L s,t 

. (5)

 2.2. Utility Maximization

Agents derive utility from consumption in the second period, from leaving bequests behind

them (“a direct bequest motive”) and from the number of their children. There is no

uncertainty. The utility function of an individual born at time t  is5

U t  = α ln(C t +1) + β ln( N t +1) + γ  ln( Bt +1), α + β + γ  = 1, (6)

where C t +1 is second-period consumption, N t +1 is the number of children, and Bt +1 is

the total estate bequeathed.6 This formalization of the intergenerational link, the “joy of 

giving,” has recently gained direct empirical support in Wilhelm (1996).

An individual’s lifetime income, in terms of his second-period consumption I t +1, is spent

on consumption, child rearing, and bequests. The cost of rearing children is measured in

terms of work time forgone, at δ per child:

C  j,t +1 + δ N  j,t +1w j,t +1 +  B j,t +1 = I  j,t +1 where j = u, s, (7)

where C  j,t +1 is (second-period) consumption of individual of type j , the second term is the

cost of rearing children, wu,t +1 is the second period wage for unskilled unit of work, ws,t +1,

is the (second-period) wage for a skilled unit of work, N  j,t +1 is the corresponding number of 

children of type j parent, and δ is a constant. B j,t +1

is the bequest type j leaves, and I  j,t +1

is type j lifetime income in terms of the second-period ( I  j,t +1 is specified in Section 2.11).

Each individual maximizes his utility subject to his budget constraint. He has three

decision variables—consumption, number of children, and bequest. For each generation t ,

the optimal level of each choice variable is

C t +1 = α I t +1, N t +1 =β

δw j,t +1

 I t +1, Bt +1 = γ I t +1, for j = u, s. (8)

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 35

Using (2.8), the (indirect) utility at the optimum is

U  j,t  = ln( I t +1) + ε j,t +1 j = u, s (9)

where

εu,t +1 = α ln(α) + β ln(β) + γ  ln(γ ) − β ln(δwu,t +1),

εs,t +1 = α ln(α) + β ln(β) + γ  ln(γ ) − β ln(δws,t +1).

Assuming parents divide bequest equally among heirs, one immediately observes that

bequest per child is a function only of second-period income only and not of wealth:7

( B/ N )t  ≡ bt  = (γ/β)δw j,t  where j = u, s. (10)

Thus, once the choice is set to include the number of offspring, the bequest is not linked

across periods, and the system becomes block recursive (wealth per dynasty is not  a state

variable anymore).

 2.3. Investment in Human Capital 8

Each individual has one unit of time in each period of life. At the end of the first period

the individual receives his share of his parent’s bequest. In the first period the individual

decides whether or not to invest in human capital.9 An individual who chooses to invest

in human capital spends all his working time in the first period of life at school, investing

an amount h that is due at the end of the period (when he receives his bequest). In the

second period he works as a skilled worker, earning ws . An individual who does not investin human capital engages in unskilled labor in both periods of his life and earns wu each

period.10

We proceed in two steps. In this section we analyze the static choice of an individual

born on an arbitrary date with some arbitrary level of bequest. We relegate the dynamics

of investment in human capital to Section 2.5 (once fertility and bequests are known). To

focus on the novelty of this work we make the following assumptions:

A1. Inheritance within the skilled dynasty is always larger than investment in human

capital.11

A2. The direct costs of investment in human capital are a constant fraction of the skilled 

wage h = θ ws .

The second assumption is based on the fact that, in general, human capital is acquired viaschooling and that teachers are skilled individuals.12

At the end of the first period, an individual who chooses to be educated must pay for that

education. Borrowing for investment in human capital is, however, expensive. It is assumed

that an individual cannot borrow at the world rate of interest to invest in human capital.

In a more general formulation, this may result from costly monitoring or uncollaterizable

human capital. For the sake of brevity, this work assumes that the rate of interest to lenders

r  is less than that to borrowers i .

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36 MOMI DAHAN AND DANIEL TSIDDON

Since the borrowing rate exceeds the lending rate, an individual who chooses to invest in

human capital borrows only the amount bt −1 − θ ws,t . For someone born at time t , using

(2.10) and A2 lifetime income in period two prices is one of the following three forms:

 I u,t +1 = [wu,t (1 + γδ/β)(1 + r ) + wu,t +1] iff  the individual remains (11a)

unskilled,

 I us ,t +1 = [ws,t +1 + ((γ δ/β)wu,t  − θ ws,t )(1 + i )] iff  the unskilled borrows (11b)

and invests in human capital,

 I t +1 = [ws,t +1 + ((γδ/β)w j,t  − θ ws,t )(1 + r )] iff  ALL investment in (11c)

human capital is from personalendowment.

According to equation (2.9), indirect utility is affected by two factors: one is lifetime

income, and the other, ε j , reflects differences in the cost of rearing children. The human-

capital investment decision determines these two factors. It is therefore easy to infer who

will choose to invest by comparing the indirect utility in two states of nature.

The utility of a poor individual who is born in period t  and does not invest in human

capital is

U u,t  = ln[wu,t (1 + γδ/β)(1 + r ) + wu,t +1] + εu,t +1, (12)

while for a poor individual who is born at time t  and does borrow to invest in human capital,

the utility is

U us,t  = ln{[(γ δ/β)wu,t  − θ ws,t ](1 + i) + ws,t +1} + εs,t +1, (13)

where i is the interest rate for borrowers and εs is defined in (2.9).

Since the wealthy always have enough to invest in human capital (A1), their utility is

U s,t  = ln{ws,t [(γ δ/β) − θ ](1 + r ) + ws,t +1} + εs,t +1. (14)

While members of a skilled dynasty always invest in human capital, the poor invest in

human capital if and only if  the following condition holds:

ws,t +1 + [(γ δ/β)wu,t  + θ ws,t ](1 + i)

> [wu,t (1 + γ δ/β)(1 + r ) + wu,t +1](ws,t +1/wu,t +1)β . (15)

Assuming momentarily constant wages, from equation (2.15) one can derive the minimal

level of bequest necessary for investment in human capital f :

 f  =

wu (2 + r ) + ws [(1 + i )θ − 1]

wu

ws

β

(1 + i )

wu

ws

β

− (1 + r )

iff  R H S  > 0,

0 iff  R H S ≤ 0.

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 37

Consider for a moment the case of exogenous fertility (β = 0). In this simplified case, the

factors that determine the minimal size of a bequest that suffices to guarantee investment

in education are13

• Borrowing and lending rates Aggravating capital market imperfection increases the

critical value f  and prevents more people from investing in human capital.

• Skilled and unskilled wages A rise in unskilled wages drives f  up because it reduces

the return to investment on human capital while an increase in the skilled wage acts in

opposite direction.

• The cost of education For similar reasons, the cost of human capital acts to increase

 f .

In the general case of endogenous fertility (β > 0), the qualitative effects of  ws and

wu on f  are not changed (the algebra is omitted). However, a rise in ws has two effects:

it increases the return to human capital through the same channel as before, but at the

same time it reduces the return to human capital because of the time allocated to rearing

more children (children are a normal good). Given the log utility, the former effect always

dominates the latter. The case of  wu is similar, and the effects of i , r , θ  remain unchanged.

 2.4. Fertility Choice

From equations (2.8) and (2.10), one can solve for the number of children per parent for

each type of parent. Since the bequest per child depends only on second-period wage and

not on total income (2.10), there are potentially only three types of individuals: (1) u, theunskilled, (2) s, the skilled offspring of skilled parents, and (3) us , skilled offspring of 

unskilled  parents.14 Denoting by N  j,t  the number of offspring of a parent born in period

t − 1, where j = u, s, us , these numbers are

 N u,t  =β

δ

wu,t −1

wu,t 

γ δ

β+ 1

(1 + r ) + 1

, (16)

 N s,t  =β

δ

ws,t −1

ws,t 

γ δ

β− θ 

(1 + r ) + 1

, (17)

 N us,t  = βδ

wu,t −1

ws,t 

γ δβ

− θ ws,t −1

ws,t 

(1 + i ) + 1

. (18)

Note that for those who do not switch from unskilled to skilled status (equations (2.16)

and (2.17)), only the ratio of previous to current wage shows up as a determinant in the

decision about fertility. The previous wage appears because it determines current wealth,

both through previous work in the labor market (for the unskilled) and through the bequest

generation t − 1 received (for both skilled and unskilled); the current wage represents the

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38 MOMI DAHAN AND DANIEL TSIDDON

price of raising children. While the previous wage has a positive effect on this number of 

offspring (an income effect), the current wage has a negative effect (dominancy of the price

effect). The effects of all other variables are similar to their effects in the steady state and

are discussed below.

We show below that a steady-state wage rate indeed exists. Rewriting (2.16) to (2.18),

the expression for the steady state are

 N u =β

δ

γ δ

β+ 1

(1 + r ) + 1

, (16)

 N s =

β

δγ δ

β − θ 

(1 + r ) + 1

, (17

)

 N us =β

δ

wu

ws

γ δ

β− θ 

(1 + i ) + 1

. (18)

The gap between the number of an unskilled parent’s offspring and the number of a skilled

parent’s offspring at the steady state is a simple combination of the parameters:

 N u −  N s = (β/δ)(1 + r )(1 − θ ). (19)

Equation (2.19) immediately yields the following lemma:

Lemma 1: At the steady state,

(i) Unskilled individuals have more children than skilled individuals.

(ii) Among skilled individuals, the number of offspring per parent whose own parents were

unskilled is smaller than the number of offspring per parent whose own parents were

skilled.

Proof. Equations (2.16) to (2.18). Note that being “us” is a choice, so N us is either larger

than zero or meaningless.

A steady-state comparison between the number of children born to the poor and to the

rich shows that a number of factors account for the observed gap: (1) the cost of acquiring

human capital θ , (2) the intensity of the desire to have kids β, and (3) forgone earnings δ.A decline in the cost of acquiring human capital θ  will contribute to an increase the number

of children among the rich, but will not affect poor people directly, thereby reducing the

gap by creating a positive income effect (for the rich) without altering the relative price of 

raising children. Note, however, that the role of  θ  in the model is also of an entry barrier, so

that the implications of a decline in θ  are not fully captured by this partial assessment. An

increase in β also increases the gap between fertility rates: while β increases both N u and

 N s , it increases fertility among the rich by less, since the increase in the number of children

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 39

is a decrease in the net return to investment in human capital. This price effect moderates

the effect of β on the rich and increases the gap in fertility rates. A decrease in δ increases

the fertility gap (we return to this effect in our discussion of child mortality in Section 5).

Capital-market imperfections thus split population into three groups. The population that

invests in human capital grows more slowly than the population that does not. The inverse

relation between the level of human capital and the rate of population growth has been

documented in many empirical studies (Caldwell, 1980; Rosenzweig, 1990; Perotti, 1996),

and the fact that the rate of population growth is positively correlated with the proportion

of poor in the economy is also supported in many papers (Kuznets, 1973; Repetto, 1978;

Perotti, 1996). The fact that the division of skills leads to a division in rates of reproduction

is the cornerstone of the analysis in this work.

 2.5. General Equilibrium Dynamics

To examine the dynamic behavior of the economy, we must proceed in steps. We first

characterize the steady state and show its uniqueness. Next, we examine one possible

dynamic path. In the third step, we analyze the bifurcation point of this path and show that

this is the only way the system can reach a unique steady state. Finally, we show that this

path is indeed the only path, thus completing the characterization.

2.5.1. Step 1: The Steady State Consider first the decision of the skilled. Given as-

sumption A1, they always invest in human capital. As for the unskilled, as long as the

following inequality holds, they do not invest in human capital:

ws,t  +

γ δ

βwu,t −1 − θ ws,t −1

(1 + i ) <

wu,t −1

1 +

γ δ

β

(1 + r ) + wu,t 

ws,t 

wu,t 

β

.

(20)

Define a stationary equilibrium to be an equilibrium with a constant population compo-

sition (in terms of the proportion of skilled to unskilled workers). Equation (5) implies that

this constant population composition corresponds to a constant wage ratio or a constant

education premium. Thus, together with equation (19) it implies that a steady state without

transition cannot exist. Population composition can therefore remain stable if and only if 

some of the poor choose to acquire education and becomeskilled. Since allpoor individuals

are alike, a necessary condition for such a transition to occur is that the utility of those poor

who remain unskilled needs to be the same as the utility of the poor that opt for education.15

Thus, in steady state the following condition must hold:

(1 + i )(γδ/β) + [1 − θ (1 + i )](ws /wu ) =

1 +

γ δ

β

(1 + r ) + 1

(ws /wu )β .

(21)

Figure 2 characterizes the steady state. The right-hand side (RHS) of equation (21) is

a concave curve that starts at the origin of the positive quadrant where the left-hand side

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40 MOMI DAHAN AND DANIEL TSIDDON

Figure 2. The composition of the labor force and the incentive to invent.

(LHS) of equation (21) is a straight line with a positive intercept in the same quadrant.Since, if intersecting, the line intersects the curve twice, one can use (20) to show that the

only relevant intersection is the rightmost one. Two other issues should be stated:

• A negative slope would have implied that the marginal cost of investment in human

capital for the poor in the steady state increases faster with ws than the marginal benefit

and that in this case the poor never invest in human capital. We exclude this possibility

by assumption—that is, i θ < (1 − θ ).

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 41

• For any possible borrowing rate, i < (1 − θ)/θ , two intersections exist.16

Since every wage ratio corresponds to a single composition of the labor force (equation

(5)), the fourth quadrant of Figure 2 shows the unique composition of the workforce Lu / L s ,

which is consistent with thesteady state. These two—the uniquewage ratio that is consistent

with the fact that in the stationary equilibrium some of the poor acquire education, and the

unique corresponding labor-force composition—fully characterize the unique candidate for

a steady state with transition from unskilled to skilled (steady-state values are denoted by

asterisks).

Given the steady-state differences in fertility rates (19), the nature of the production

function, and the unique relationship between the skill and unskilled wage ratio and the

composition of the labor force (5), one immediately observes that this economy obtains no

steady state without transition from unskilled to skilled. If there is a steady state, therefore,

it must be one with transition. At such a stationary equilibrium, skilled parents have skilled

children, and the unskilled, who have a lower utility than the skilled, are indifferent between

remainingpoor (and having many children)or going to school(and a having fewerchildren).

Once in the steady state, in every generation some of the young unskilled do attend school,

and the stationary level of population growth coexists with a transition from unskilled to

skilled. In the long run, therefore, income and utility have a nontrivial distribution within

each generation at the stationary state.

2.5.2. Step 2: The Dynamic Path with No Transition from Unskilled to Skilled  Step 1

showed that there exists at most one wage ratio (labor-force composition) that corresponds

to a stationary state. Assume for a moment that the economy is not yet in this steady

state and therefore no children of unskilled workers choose to acquire education. Since the

rate of change of wages shows up in both equations (16) and (17), it would seem that the

difference N u,t  −  N s,t  cannot be signed. As shown in Step 4, however, this is not true: this

difference is signable as long as A1 holds. To understand the dynamics before transition

from poor to rich takes place, it is therefore enough to note two facts:

• When wages of both skilled and unskilled workers do not change from one period to

the next, the number of children of an unskilled worker N u,t , is necessarily higher for

all t  ((16) and (17)).

• Given the production function (2.1) and the (temporary) assumption that no child of an

unskilled worker chooses to acquire education, wu,t  < wu,t −1, and ws,t  > ws,t −1 (since

 L u / Ls increases).

Thus, as long as no transition occurs from being unskilled to being educated, the fertility

gap between poor and rich at any period t  is necessarily positive and higher than it would

have been with constant wages; ( N u,t / N s,t ) > ( N u / N s ) > 1 for all t .

When the initial distribution of wealth is unequal, but not unequal enough to induce

some of the poor to immediately invest in human capital, the proportion of poor people

in the economy increases and their wages decline. The ratio L u / L s becomes larger and

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42 MOMI DAHAN AND DANIEL TSIDDON

larger. Figure 3 shows the dynamics of population composition. The dynamic path that

corresponds to this relative expansion of the unskilled labor force must start above the 45

degree line and, since N u is larger than N s , the line draws further away from the 45 degree

line (the L L curve in Figure 3). The economy moves to the right along the L L line as long

as the skill premium is not large enough to induce investment in human capital by some

of the poor. It can be shown that along the L L curve the economy-average rate of fertility

increases as well. As long as the economy continues along this path, it never reaches a

steady state.

2.5.3. Step 3: Bifurcation—The PoorFind It Profitable to Invest in Human Capital17 At

thecurrent production structure (equation (1)), as the number of unskilled workers increasesrelative to the number of skilled workers, their wage declines, while the wage of skilled

workers rises. As shown in Figure 2, at some point it must  become lucrative for some of 

the poor to invest in human capital. At this point the system bifurcates, whereupon the

numbers N u and N s no longer suffice to characterize population dynamics. Once some of 

the poor choose to acquire education, the population of those who remain poor increases by

less than N u . By the same token, the population of the skilled grows by more than N s ; the

actual number of skilled workers in the next generation is N s + N us while the actual number

of unskilled workers from this generation is N u − N us . According to the utility function

we use, the bequest per child is a constant fraction of the second period wage, so that once

a person moves into the skilled group his dynasty remains skilled. In the steady state,

therefore, there is always a group of unskilled children receiving education and becoming

skilled; the net rate of reproduction of the unskilled is equal to the net rate of reproduction

of the skilled by definition. The point where the skill premium is high enough—the pointwhere the system bifurcates and settles on the steady state—is ( Lu / Ls )∗ in Figure 3.

2.5.4. Step 4. Uniqueness of the Dynamic Path Thus far, the described dynamic path

shows only one possibility out of many. This step rules out all other possibilities as being

inconsistent with the set of assumptions made.

Consider Figure 3. We first show that the model exhibits no dynamics to the right of the

steady state. When the number of poor is sufficiently large (so that the economy is to the

right of the steady state), the incentive to become educated is necessarily strong enough to

make some of the poor immediately invest in human capital (Figure 2 and the discussion

in Step 1). Therefore, the economy necessarily jumps to the steady state, and there are no

dynamics to its right.We now rule out imploding paths. Suppose the economy lies to the left of the steady state

and that the ratio ws,t −1/ws,t  is sufficiently large relative to wu,t −1/wu,t  to make N s > N u .

In this case, Lu / Ls must grow smaller over time. As the ratio of unskilled to skilled falls,

one of two results must ensue: the dynamic path may either intersect the 45 degree line,

or it may go toward the origin. The former case implies that there is a second steady state.

This is inconsistent both with the fact that there is only one steady state with transition

from poor to rich (Step 1) and with the fact that a steady state with no transition cannot

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 43

Figure 3. The evolution of the compostion of the labor force.

exist (19). If the dynamic path approaches the origin, it must eventually cross the point at

which education is not worthwhile for the rich—contrary to assumption A1. 18

Thus, there exists no other dynamic path but the one described in Steps 1 to 3. The

economy may be on this path or start at the steady state. If not at the steady state, the

economy, therefore, must  undergo demographic transition.19 In the first phase, fertility

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44 MOMI DAHAN AND DANIEL TSIDDON

increases, and income becomes less equally distributed. It is only when the skill premium

becomes high enough that unskilled fertility goes down, income becomes more equally

distributed, and the rate of accumulation of human capital peaks. Clearly, at this point

outputalso rises (a “level effect”). If thestock of human capital hasa strong enough external

effect on the production process, sustained economic growth takes off on the transition into

low fertility rates. We return to this last point in Section 4.

3. Further Aspects of Demographic Transition

This section details two issues: (1) the relationship between demographic transition and the

dynamicsof income inequality and (2) the relationshipbetween child mortality demographic

transition and income inequality.

 3.1. Demographic Transition and the Distribution of Income

The previous section linked demographic transition to the accumulation of human capi-

tal. Clearly, linking investment in human capital to changes in fertility rates immediately

links these changes to the distribution of income across individuals. In fact, in this work,

demographic trends are strongly associated with trends in the distribution of income.

Since demography is not of a monotone trend, neither is the distribution of income. The

process of demographic transition coexists with the well-known Kuznets curve. In the

first epoch, as long as poor people do not find it productive to invest and become skilled,

their numbers increase, and their wage (per worker) declines. During this phase, the rich

population grows more slowly than does the poor population, and the wages of the rich

increase. An ever-diminishing proportion of the population grows richer and richer. Thus,

throughout this phase income distribution becomes less and less equal. When the day of 

reckoning arrives, when the poor change their decision about investment in skill and some

of the poor choose to invest in human capital, there is an immediate decline in the supply of 

the unskilled and an increase in the supply of the skilled; the wages of the unskilled increase

and the wages of the skilled decrease. Demographic transition therefore coincides with the

point in time at which the trend of inequality is reversed: when fertility declines, so does

inequality.20

 3.2. Child Mortality and Demographic Transition

The term demographic transition is sometimes taken to imply a decline in fertility that lags

behindthe decline in mortality. While Francine van de Waile (1986)(as well as many others)

show that this is imprecise, there is no doubt that decreasing mortality rates contribute to

a secular increase in population size. This section shows that lower mortality rates indeed

contribute to the transition from high to low rates of fertility, without referring to partial

information or “near rationality” arguments that, in this context, seems counterfactual for

many. This section focuses only on child mortality.

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 45

To discuss mortality we add one (very plausible) assumption to our model. We assume

that parents care for living offspring only—that is, it is not the number of births that yields

utility but the number of children who survive childhood that enters the utility function.21

With this change in the utility function, an exogenous decrease in child mortality is modeled

here as a decrease in the cost of raising a child to adulthood, and a decrease in mortality is a

decrease in δ. As shown earlier (equation (2.19)), the fertility gap between poor and rich is a

positive function of δ. With lower mortality rate, δ is lower, and thegap in fertility higher . A

decrease in the time needed for raising surviving children increases fertility among the poor

by more than it increases fertility among the rich and therefore contributes to an increase

in the relative supply of poor people. Originated at the lower death rate, this increase in the

relative supply of the unskilled drives the relative wage of the unskilled further down and

hastens the onset of demographic transition. Thus, a decline in child mortality rates can

contribute in an important way to demographic transition without relying on implausibly

long information lags.

4. Demographic Transition and Economic Growth

The close association between the decline in fertility and the reversal of trends in inequality

leaves at least one important issue open. In the phase of rising average fertility, the model

of Section 3 predicts no correlation between the growth rate of income per capita and

demographic trends. In fact, with different parametrization of the model, one can get either

a positive correlation—that is, that income growth is positively correlated with fertility

(Kremer, 1993; Lucas, 1996)—or a negative correlation. It is only on the decline in the rate

of fertility that per capita output is positively correlated with this decline.

Keeping the basic mechanism intact, this section, in accordance with much of the endoge-

nous growth theory, modifies the model slightly to explicitly account for the contribution of 

human capital to economic growth. This modification highlights and qualifies the correla-

tion between demographic transition, on one hand, and the accumulation of human capital

and economic growth, on the other hand. It is therefore a crucial extension in terms of 

bringing the mechanism we suggested in Section 2 closer to the well-documented facts

(Kremer, 1993; Lucas, 1996). We use a two-sector version of the previous model to show

that technological progress in the high-skill sector generates a phase of growth of income

per capita that coincides with population growth before the economy settles down to a lower

rate of reproduction and a faster rate of growth.

Accounting for endogenous growth, demographic transition occurs in two stages. The

first stage is one of economic growth in which average fertility increases, and in the second

stage (once the incentive to invest in human capital gains sufficient momentum) fertilitydecreases and remains low and constant thereafter. This second stage is characterized by an

increase in the rate of growth of income per capita, sustained by two sources: the decline

in the rate of fertility and the increase in the rate of accumulation of human capital.22

As a by-product, the analysis shows that the situation of the poor need not deteriorate

before transition takes place; it is only their relative position, or the skill premium, that

matters.

To streamline the exposition we change the production structure to its simplest version.

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46 MOMI DAHAN AND DANIEL TSIDDON

Everything not specified is kept intact. Assume that production of the same aggregate

output is performed in two sectors: the unskilled produce using a linear technology and no

capital Y u , while skilled work is complemented with capital in a Cobb-Douglas form Y s :

Y u = wu L u , (22)

Y s = At K at  L 1−ast  where At  is the technological parameter of time t . (23)

In a perfectly competitive world, where capital is free to flow at the rate of interest r , the

wage of an unskilled worker is fixed at wu ; the wage of skilled workers ws is constant and

equals

ws = At (1 − α)

α At 

α1−α

. (24)

Wages of both skilled and unskilled workers are independent of the skill composition;

given the free flow of capital they do not change with population size as well. Changes

in the wage differential between the skilled and the unskilled may result from changes in

parameters or from technological progress.

Suppose technological progress is a function of a past societywide stock of human capital.

A simple way to capture this effect is to assume A is a function of the aggregate level of 

human capital in the economy in the previous period. Since human capital per educated

person is fixed by construction in this model, an aggregate change comes out of an increase

in the population of educated persons only, At  = A( N s,t −1), which is a Kremer-type

assumption; A( ) > 0, A > 0, A < 0. If we repeat our previous assumption so that

 N s,t  > N s,t −1 wages of skilled workers grow, while wages earned by unskilled workers

do not change; technological progress is skilled biased. The increase in skilled workers’

wages increases the return to education and therefore drives down the threshold level of 

bequest necessary to induce investment in human capital f . However, it may take time

before f  decreases sufficiently to induce the poor to invest in human capital. Until the

threshold is reached, both output and the average rate of fertility grow, while output per

capita grows if and only if  the fruits of growth are not diluted by the growing numbers of 

the poor. As long as bequest remains below the critical level f , the population of the poor

grows larger relative to the population of the rich. Thus, income polarization is the rule, and

average rate of fertility increases.23 The second phase of the demographic transition begins

when the poor, too, benefit from investing in human capital. The transition to low fertilityoccurs immediately and permanently once the net  incentive to invest in human capital is

sufficiently high.24

The decrease in theoverallrate of fertility coincideswith an increase in the level of income

per capita and with an increase in the rate of growth of income per capita. Two factors

cause the level to change: (1) since the transition occurs when everybody invests in human

capital, and since the return to human capital is higher than the return to any other form of 

investment, output grows and so does income per capita; and (2) this higher level of output

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 47

is now distributed among a small number of individuals since fertility has declined (this

effect is sometimes referred to as the dilution effect ).25 The fact that after the transition all

individuals are skilled implies that the rate of growthof income per capita is unambiguously

increasing as well.

Accounting for human capital as the engine of growth shows that indeed output and

output per capita can be positively correlated with fertility for a long period of time. It

is only once skill becomes sufficiently important that this relationship is reversed and the

takeoff in growth rates is associated with a decline in fertility. Note also that with skilled

biased technological progress, the state of the poor need not deteriorate before choosing

education. Its relative position does indeed deteriorate in all formulations of the mechanism

we propose and is the source of the qualitative change in the behavior of the poor.

Portraying economic growth this way clarifies another issue. Different countries, regions,

or classes have experienced very high levels of income long before demographic transition

have occurred. The current formalization helps us to understand what is the exact relation-

ship between income and fertility. It is not income per se, but it is the return to education,

which is just one component of income, that brings income and fertility together.

5. Demographic Transition with No Capital Market Imperfections

In the previous sections we demonstrated that demographic transition is the outcome of 

rational choice when investmentin human capital is morecostly for the poor. The conclusion

might be to focus government policy on a tax-transfer system in poor and rapidly growing

populations. The issue at handis, however, moresubtle. While capital-market imperfections

are an important factor in determining human capital, they are clearly not the sole source

of difference between rich and poor. More generally, as long as there is a wedge in the net 

return to investment in human capital demographic transition occurs in a nonmonotone way

and the relationship between income inequality and growth is of an inverted U-shape. This

section outlines this argument. We use the minimal model to show that the crucial factor

behind the previously described dynamics is indeed the net return and not  capital market

imperfections.

We assume that the skill premium for a child of a skilled parent is higher than the skill

premium for the child of the unskilled parent. More specifically, we assume that the child

of a skilled parent obtains more units of efficiency from the same investment in human

capital: investing one period of time and θ ws in real resources, the child of the skilled

parent obtains E s efficiency units, while the skilled child of an unskilled parent obtains

only E us units of efficiency ( E s > E us ). We assume there is no comparative advantage inbeing unskilled. There are many explanations for this lead: informal education, cultural

aspects, the time spent searching for a job or the quality of the match. Whatever the reasons,

evidence supporting such a connection abounds (Galor and Tsiddon, 1997a, 1997b).

Production in the unskilled sector is the same as in equation (22); production in the

skilled sector uses two factors of production—capital and efficiency units of labor. The

total number of efficiency units E  in this sector is a weighted average of  E s and E us , where

the weights are the numbers of  s-type individuals (skilled children of skilled parents) and

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48 MOMI DAHAN AND DANIEL TSIDDON

of us-type individuals (skilled children of unskilled parents), respectively. The production

function is

Y s,t  = A( E t −1)K αt  E 1−αt  , where E t  = N t ,us E us +  N t ,u E u .

Since the parental effect exists in human capital, an operative bequest motive is not

necessary in this context. While the rest of the details are relegated into the appendix, let

us note that the model yields the following birth rates:

 N s =β

δ 1 −

θ (1 + r )

 E s , N us =

β

δ 1 −

θ (1 + r )

 E us , N u =

β

δ(2 + r ).

One immediately observes that just like the previous section birth rates of the unskilled

 N u are higher than the birth rates of the skilled N s , which are higher than those of the

“switchers” from unskilled to skilled N us . Also, since birth rates of the skilled have a spill-

over effect through technology advancement, the wage per efficiency unit increases, and

eventually it becomes profitable for the unskilled to invest in human capital. The premium

that induces demographic transition, a rise in equality, and a rise in the rate of growth is

derived in the appendix to be

ws

wu

=

(2 + r ) E 

βus

 E us − θ (1 + r )

11−β

.

Demographic transition will occur provided the net  skill premium of the child of an

unskilled worker is lower than the net  skill premium of the child of a skilled worker and

provided the cost of child rearing is higher for the skilled. In the first phase of this transition,

the average fertility rate increases, and income becomes less equally distributed; in the

second stage, fertility decreases, income becomes more equally distributed, and a phase of 

persistent growth starts.

6. Conclusion

This article uses a growth model with endogenous fertility to show that the demographic

transition along with a Kuznets-type dynamics of income distribution are the foundations

for economic growth based on human capital accumulation. We show that in the first phaseof development the average rate of fertility increases and income becomes less equally

distributed. In the second phase fertility declines, and income becomes more equally

distributed. At this stage the economy accumulates human capital more rapidly. The

demographic transition and the U-shaped dynamics of equality are therefore necessary for

knowledge-based growth.

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 49

Appendix: The Details of the Model with Parental Lead

The production function in the skilled sector was detailed in the text to be

Y s = AK a E 1−a, where E = N us E us +  N u E u . (25)

Since educated parents provide a lead, no bequest motive is needed to link parents and

children. We therefore simplify the utility function to

u = (1 − β) ln(C ) + β ln( N ). (26)

The budget constraint for each type is

 I s = C + δ N E s ws , I us = C + δ N E us ws , I u = C + δ N wu , (27)

where ws is the wage per one unit of efficiency in the skilled sector.

Following the same steps as in Section 2, one can show that the optimal number of 

offspring for each type is

 N s =β

δ

1 −

θ (1 + r )

 E s

, N us =

β

δ

1 −

θ (1 + r )

 E us

, N u =

β

δ(2 + r ). (28)

Since E s > E us , N u > N s > N us always. The number of children of an unskilled worker

is always higher than the number of children of the skilled worker, which, in turn, is always

higher than the number of children of a person born to unskilled parents who acquired

education.

We proceed further as in Section 2. The utility function of each type can be rewritten asa function of net income and wage only:

us = ln( I s ) + ε − ln(δ E s ws ), uus = ln( I us ) + ε − ln(δ E us ws ), (29)

uu = ln( I u ) + ε − ln(δ E s ws ),

where

 I s = E s ws − θ ws (1 + r ), I us = E us ws − θ ws (1 + r ), I u = wu (2 + r ). (30)

Of course, for the model to come to life the skill premium must be high enough so that

U s > U u . As long as U u > U us , children of unskilled parents decide to remain unskilled.

Once this inequality is reversed (or turns into an equality), children of unskilled parents

choose to be skilled. The child of an unskilled parent chooses to acquire education if and

only if 

 E us ws − θ ws (1 + r ) ≥ wu (2 + r )

 E us ws

wu

β

. (31)

Equation (31) describes the relationship between the utility of thoseborn to unskilled parents

who choose to acquire education and those born to unskilled parents who choose to remain

unskilled. Thus, after dividing through by wu one can use Figure 2 without alterations. The

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50 MOMI DAHAN AND DANIEL TSIDDON

concave curve that starts at the origin represents the right-hand side (RHS) of equation (31),

while the upward sloping line represents the left hand side (LHS). As long as RHS is above

LHS, the child of an unskilled parent chooses to remain unskilled. If LHS is greater than or

equal to RHS—the child of an unskilled parent chooses education over no education. The

switch happens when the wage ratio is

ws

wu

=

(2 + r ) E 

βus

 E us − θ (1 + r )

11−β

. (32)

Acknowledgments

This article was presented in seminars in the Bank of Israel, Bar Ilan University, the Hebrew

University, Tel-AvivUniversity, as well as in the ESF conference July1997 and NBER group

on income distribution July 1997. Comments of participants are gratefully acknowledged.

Notes

1. It is commonly argued that demographic transition occurs due to an initial decline in infants’ death rate, to

which birth rates respond with a lag. The present study suggests another link from the decline in mortality to

demographic transition. The link here does not lean against lack of information or near-rationality arguments.

2. Rosenzweig (1990) uses micro data to find direct support for the claim that an increase in the return to human

capital simultaneously increases investment in human capital and decreases the rate of fertility.

3. Fertility choice was analyzed by economists for generations before it was merged into economic growth. For

surveys see Razin and Sadka (1995), Easterlin (1987), and Birdsall (1988).4. We assume throughout that the reproduction rate of both unskilled and skilled is larger than one—that is,

population growth is positive in each group. Analyzing other cases is possible but yields no new insight.

5. Abstracting from fertility choice, this model borrows elements from Galor and Zeira (1993).

6. Results do not change if the individual benefits from the bequest per child (the coefficient of ln( N ) becomes

β − γ ). Assuming consumption in the second period only makes no difference but simplifies the notation.

7. We assume here that parents divide the bequest equally among their children. This is always the case when

the utility is from bequest per child. This work ignores the effects of both ex-ante or ex-post heterogeneity.

8. We omit the time subscript when inessential throughout this section.

9. The model could be easily modified to make the parent take the education decision of the child.

10. Investment in human capital indivisible; it is composed of both direct investment and forgone earnings.

11. Assumption A1 restricts the set of possible initial conditions of the composition of the labor force not to

have too many educated at the initial period. Doing otherwise, yields no insights. Also one could use a less

restrictive assumption at the cost of a bit more complex dynamics.

12. The model is simple to solve under many other formulations (including fixed costs, and so on). Some

indivisibility of investment in human capital is, however, needed. The choice of formulation is based on thedesire to ignore “level effects” and make the dynamics as simple as possible.

13. In this case the critical value f  = {wu (2 + r ) + [θ (1 + i ) − 1]ws}/(i − r ).

14. Strictly speaking, this is true only from the second period of the world on.

15. Since their net total income is higher, the utility of the born-rich is higher than that of the “switchers” and thus

from that of the poor who choose to remain poor.

16. To see that an intersection indeed exists, consider the limit of  i = (1− θ)/θ . Since the LHS is horizontal and

the RHS is increasing, the two must intersect. From that one can show that for every relevant borrowing rate

the curve and the line intersect twice. We later show that only the right intersection matters.

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DEMOGRAPHIC TRANSITION, INCOME DISTRIBUTION, AND ECONOMIC GROWTH 51

17. In this section the transition to steady state occurs after the wage of the unskilled declines. As we show in

Section 4, all that matters is the skill premium.

18. Arithmetically, this happens when (2 + r )wu > ws [1 − (1 + r )θ ].

19. Strictly speaking, for overall average fertility to decline one needs an additional technical assumption. At the

intuitive level this assumption means that the share of the unskilled in total population size is large enough.

This assumption is necessary since once in the steady state, the fertility of the skilled increases relative to the

dynamic path (compare (2.17) to (2.17)) while fertility of the remain to be unskilled declines (compare (2.16)

to (2.16)). The source for this effect is the following. In the steady state skilled-wages stop growing. Since

one raises children in the second period of life, the costs of raising children relative to permanent income

(bequest + wage) go down, and therefore fertility among the skilled goes up. On the other hand, besides the

attenuating effect of “us” type fertility, the fertility of unskilled that remains unskilled decreases in steady

state relative to the pre-steady-state level due to similar though opposite effect of the wage decline in the

pre-steady-state period.

20. The change in fertility rates as well as the change in inequality occurs here instantaneously—a natural result of 

ignoring exogenous heterogeneity (such as ability). As long as heterogeneity remains moderate, in the sensethat it does not overwhelm the wage-occupation nexus between parent and child, the correlation between the

trend of inequality and fertility rate remains.

21. See also Ehrlich and Lui (1991). If parents derive disutility from a child’s death, it strengthens the results.

22. Since capital is free to flow in any amount, this model does not include a “dilution effect” of fertility.

23. Since the details are similar to those derived in Section 2, they are omitted here.

24. The term net  means net of both the direct cost of capital and the reduction in utility caused by having fewer

children.

25. Reproduction rate is always positive. Population size is smaller than it would have been with no transition.

References

Azariadis, C., and Alan D. (1991). “Demographic Transition in a dual Economy.” Mimeo.

Becker, G. S., and R. J. Barro. (1988). “A Reformulation of the Economic Theory of Fertility.” Quarterly Journalof Economics 103, 1–26.

Becker, G. S., Kevin M. Murphy, and Robert Tamura. (1990). “Human Capital, Fertility and Economic Growth.”

 Journal of Political Economy 98, 12–37.

Birdsall,N. (1983). “Fertility andEconomic Change inEighteenth-and Nineteenth-CenturyEurope: A Comment.”

Population and Development Review 9, 111–135.

Birdsall,N. (1988). “EconomicApproaches to Population Growth.” In Hollis Chenery and T. N. Srinivasan (eds.),

The Handbook of Development Economics. Amsterdam: North Holland.

Boulier, B. L. (1982). “Income Redistribution and Fertility Decline: A Skeptical View.” Population and Devel-

opment Review (supp.), 159–173.

Caldwell, J. C. (1980). “Mass Education as a Determinant of the Timing of Fertility Decline.” Population and 

 Development Review 6, 225–255.

Coale, A. J., and S. Cotts Watkins (eds.). (1986). The Decline of Fertility in Europe. Princeton, NJ: Princeton

University Press.

Dahan, M. (1996). “Income Distribution and Economic Growth.” Ph.D. dissertation, Hebrew University of 

Jerusalem, Jerusalem.

Dyson, T., and M. Murphy. (1985). “The Onset of Fertility Transition.” Population and Development Review 11,

399–440.

Easterlin, R. A. (1987). “Fertility.” In JohnEatwell, Murray Milgate, and PeterNewman(eds.),The New Palgrave:

 A Dictionary of Economics (vol. 2, pp. 302–308). London: Macmillan.

Eckstein, Z., and K. I. Wolpin. (1985). “Endogenous Fertility and Optimal Population Size.” Journal of Public

 Economics 27, 93–106.

Ehrlich,I., and F. T.Lui. (1991). “Intergenerational Trade, Longevity and Economic Growth.” Journal of Political

 Economy 99, 1029–1059.

Galor, O., and D. Tsiddon. (1997a). “The Distribution of Human Capital and Economic Growth.” Journal of 

 Economic Growth 2, 93–124.

Page 24: Demographic Transition, Income Distribution,

7/30/2019 Demographic Transition, Income Distribution,

http://slidepdf.com/reader/full/demographic-transition-income-distribution 24/24

52 MOMI DAHAN AND DANIEL TSIDDON

Galor, O., and D. Tsiddon. (1997b). “Technological Progress, Mobility, and Economic Growth.” American

 Economic Review 87, 363–382.

Galor, O., and J. Zeira. (1993). “Income Distribution and Macroeconomics.” Review of Economic Studies 60,

35–52.

Galor, O., and D. N. Weil. (1996). “The Gender Gap, Fertility, and Growth.” American Economic Review 86,

374–387.

Goodfriend, M., and J. McDermott. (1995). “Early Development.” American Economic Review 85, 116–133.

Guinnane,T. W., B. S. Okun, and J. Trussell. (1994). “WhatDo We Know About the Timingof Fertility Transition

in Europe?” Demography 31(1), 1–20.

Kremer, M. (1993). “Population Growth and Technological Change: One Million B.C. to 1990.” Quarterly

 Journal of Economics 108, 681–716.

Kuznets, S. (1967). “Population and Economic Growth.” Proceedings of the American Philosophical Society 2(3)

(June), 170–193.

Kuznets, S. (1973). Population, Capital and Growth. New York: Norton.

Livi-Bacci, M. (1986). “Social Group Forerunners of Fertility Control in Europe.” in A. J. Coale and S. CottsWatkins (eds.), The Decline of Fertility in Europe (pp. 182–200). Princeton, NJ: Princeton University Press.

Lucas, R. (1996). “Ricardian Equilibrium: A Neoclassical Exposition.” Israel Institute of Technology Economics

Workshop Series.

Perotti, R. (1996). “Income Distribution, Democracy and Growth: An Empirical Investigation.” Journal of 

 Economic Growth 1, 149–187.

Razin, A., and U. Ben-Zion. (1975). “An Intergenerational Model of Population Growth.” American Economic

 Review 66, 923–933.

Razin, A., and E. Sadka. (1995). Population Economics. Cambridge, MA: MIT Press.

Repetto,R. (1978). “The Interaction of Fertilityand SizeDistribution of Income.” Journal of Development Studies

14, 22–39.

Repetto, R. (1982). “A Reply.” Population and Development Review (supp.), 174–178.

Rosenzweig, M. R. (1990). “Population Growth and HumanCapital Investments: Theory and Evidence.” Journal

of Political Economy 98, S38–S70.

Sharlin, A. (1986). “Urban-Rural Differences in Fertility in Europe During the Demographic Transition.” in

A. J. Coald and S. Cotts Watkins (eds.), The Decline of Fertility in Europe (pp. 234–260). Princeton, NJ:

Princeton University Press.

van de Waile, F. (1986). “Infant Mortality and European Demographic Transition.” in A. J. Coale and S. Cotts

Watkins (eds.), The Decline of Fertility in Europe (pp. 201–303). Princeton, NJ: Princeton University Press.

Wilhelm, O. M. (1996). “Bequest Behavior and the Effect of Heirs’ Earnings: Testing the Altruistic Model of 

Bequest.” American Economic Review 86, 874–892.


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