Household borrowing constraints, fertility dynamics, and economic growth

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Journal of Economic Dynamics & Control 30 (2006) 27–54

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Household borrowing constraints, fertilitydynamics, and economic growth

Erasmo Papagni�

Dipartimento di Diritto ed Economia, Seconda Universita di Napoli, Corso del Gran Priorato di Malta,

81043 Capua, Italy

Received 7 May 2002; accepted 12 October 2004

Abstract

In this article we provide a model of growth with endogenous fertility in which multiple

steady states derive from the modelling of household liquidity constraints. We put forward an

innovative approach to the finance of higher education by assuming that youths can borrow

because their parents guarantee the loan repayment with their income. Young individuals can

renege on their debt and lenders provide them credit only up to an amount which is

commensurate to a collateral provided to children by their families. Parents care about

children’s education and choose a collateral which depends positively on family income and

negatively on family size. A stable trap of low-development is characterized by high-fertility

rates and low investment in human capital. On the other hand, in economies with a sufficiently

low starting rate of fertility borrowing constraints gradually vanish and the process of growth

reaches a steady state characterized by the optimality of fertility and schooling choices.

Government subsidies to education may reduce population growth and promote human

capital investment if fertility at steady states is lower than thresholds.

r 2004 Elsevier B.V. All rights reserved.

JEL classification: O41; O15; J13

Keywords: Development; Population; Borrowing constraints; Education subsidies

see front matter r 2004 Elsevier B.V. All rights reserved.

.jedc.2004.10.003

08 23272041.

dress: [email protected] (E. Papagni).

www.elsevier.com/locate/econbase

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E. Papagni / Journal of Economic Dynamics & Control 30 (2006) 27–5428

1. Introduction

An important tradition in the theoretical and applied analyses of economicdevelopment concentrates on the rate of population growth and human capital.Fertility and education are economic and demographic phenomena deeply rooted inthe family’s organization and behavior. From this point of view the seminal articlesby Neher (1971); Razin and Ben-Zion (1975); Barro and Becker (1989) provide co-ordinates for the analysis of economic growth with endogenous fertility.1 The recentliterature has focused on two important issues: conditions under which an economycan experience a demographic transition (e.g., Galor and Weil, 2000; Hansen andPrescott, 2002; Lucas, 2002); causes of persistence of differences in the levels andgrowth rates of per capita income across countries. While models of the first strandof the literature focus on the endogenous mechanisms which can be responsible forthe process of progressive reduction in fertility and economic development, modelsof non-convergent economic growth produce multiple steady states where povertytraps are characterized by high growth rates of population and low technicalprogress, and the opposite features describe growth equilibria.

In this article we provide a model of economic growth with endogenous fertility inwhich household borrowing is constrained and this market imperfection producesmultiple equilibria in the growth rate of the economy. The existing literature onendogenous fertility and poverty traps relies on different causes of non-convergencein economic growth. Becker et al. (1990) obtain multiple equilibria through theassumption that rates of return on investment in human capital do not decline withthe level of human capital. This assumption combined with the quantity–quality ofchildren trade-off brings about dynamics of fertility and education that drive theeconomy either toward a Malthusian steady state, where fertility is high andinvestment in human capital is null, or to the state of a growing economy whereparents have few children who acquire high education. A non-convexity in thetechnology of human capital production is also the cause of multiple equilibria inTamura (1996) and Morand (1999). A different approach to non-ergodic growth isthat of Kelemli-Ozcan (2002) who demonstrates that if mortality declines with thegrowth of per capita income, a Malthusian steady state, with high mortality and lowincome, can be locally stable and can co-exist with a steady state characterizingeconomic growth.

In all such models human capital is the engine of economic growth and the familyis deeply involved in the decision concerning investment in human capital. Theprevailing scheme in the literature is represented by parents who can borrow onperfect capital markets and finance their children’s education by bequests. Such amodelling strategy does not account for the variety of linkages among familymembers active in real economies and does not consider the joint involvement ofchildren and parents in financing education.

1Recent reviews of the literature on endogenous population and economic growth are Ehrlich and Lui

(1997) and Nerlove and Raut (1997).

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E. Papagni / Journal of Economic Dynamics & Control 30 (2006) 27–54 29

In this article we show how multiple steady states may derive from the modellingof household liquidity constraints. Actually, the provision of credit to households ishindered by several factors among which the most important are asymmetricinformation owned by lenders and households, and the low quality of human capitalas a collateral (Becker, 1993). We put forward an innovative approach to the fundingof higher education by assuming that youths can borrow because their parentsguarantee the loan repayment with their income. The market for student loans isimperfect because individuals can renege on their debt. In order to transfer all therisk of default to the parents, lenders provide credit to young individuals only up toan amount which is commensurate to a collateral provided to children by theirfamilies. Parents care about children’s education and this gives them a motive forproviding a guarantee in a loan contract. From adult life-cycle decisional problem itemerges that the amount of collateral in student loans depends positively on familyincome and negatively on family size. Indeed, the family’s income is a commonresource that each young agent shares with other members. As family size grows, theamount of credit available to each member shrinks. As in Banerjee and Newman(1993), this way of modelling imperfect credit markets produces pecuniary increasingreturns from which multiple steady states can derive. A novel feature of our model isthe involvement of fertility in this self-reinforcing mechanism. In fact, by choosing areduced number of children parents allow an increase in resources for investment ineducation which in turn will motivate a further reduction in fertility in the nextgeneration, while the opposite vicious circle may be at work under oppositeconditions.

The dynamics of fertility and education are linked through the financial system. Asa consequence of initial values of the rate of population growth, the model candescribe economic growth as a low-development trap in high-fertility economiescharacterized by severe household borrowing constraints and low investment inknowledge. Conversely, other economies may converge to a balanced growth pathwhere growth is strong because parents – generation after generation – have fewerchildren and devote a growing amount of income to their education.

In our article multiple equilibria concern also the model of the family in the tworegimes of economic growth. In fact an outcome of the model is the co-existence inlow growth equilibrium of a type of family where children depend on the parents tofund their education, opposite to a family in high growth equilibrium where youngindividuals are free to choose their future skills. This pattern ably fits the historicalexperience of industrialized countries in which economic development has beenintertwined with the weakening of economic linkages between parents and childrenafter adolescence.

It can be noted that our model does not produce a poverty trap with null economicgrowth but a low steady state in which growth is positive. Actually, cases of non-growing countries are important but not so frequent as to represent all theexperiences of underdevelopment around the world. Hence, this article provides amore general account of non-convergent economic growth. In fact, in manydeveloping countries tertiary schooling has a sizeable dimension and families canfinance the costs of education on credit markets, even if there are significant

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imperfections. This is confirmed by econometric evidence presented in De Gregorio(1996) and Benhabib and Spiegel (2000). Furthermore, empirical studies of theeconomics of the family (e.g., Behrman et al., 1989; Hanushek, 1992; Haveman andWolfe, 1995), and of the literature on economic development (e.g., Schultz, 1988;Parish and Willis, 1993) show that individual educational attainment depends onfamily resources and family size. This result was also obtained by Galor and Zang(1997) from cross-country regressions.

In almost all countries governments provide subsidies to education that partiallycompensate for limits to household borrowing. We extend the model with theconsideration of subsidies and flat-rate taxes and analyze the effects of these policyinstruments on fertility dynamics and growth. It could be noted how scant is theliterature on public policy in models of growth with endogenous fertility (exceptionsare: Zhang, 1997; de la Croix and Doepke, 2004). We derive non-trivial results as theeffects of education subsidies on fertility and growth can be positive or negativedepending on the state of the economy. If steady state fertility is higher than athreshold, then greater subsidies could produce higher fertility and be detrimental toeconomic growth.

Quite independently, Fernandez et al. (2001) have made the same hypothesis ofthis article that young agents can borrow with a collateral represented by parentalincome. However, they do not deal with economic growth but investigate sorting andinequality.

This paper is organized as follows. Section 2 contains the description of an OLGmodel of economic growth with endogenous fertility in which education can befinanced by loan contracts between young agents and lenders with the guarantee ofthe parent’s income. In Section 3, the general equilibrium dynamics of the model areanalyzed focusing on multiple steady states. In Section 4 we introduce publicsubsidies to education financed by flat rate taxes on adult consumption and income,and analyze the consequences of these policy instruments on equilibrium dynamics.Conclusions follow in Section 5.

2. The economic environment

This section puts forward a model of a small open economy in which the agent’slife is summarized in three ages: childhood, adulthood and old age. Agents of eachgeneration are all identical and in every period population is made up of threeoverlapping generations. Individuals consume during the three ages; becomeeducated when young; work, save and have children when adult; are retired whenold. Adult individuals care about the number and education of children. Youngagents acquire skills both from relations with their parents and attending school,which requires full time effort and physical resources. Students finance theireducation by borrowing on imperfect credit markets.

Adults have children and allocate their time endowment net of leisure –normalized to one – to work and child care. Savings are invested during adulthood

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and provide income for consumption in old age when individuals are retired.Individuals are endowed with perfect foresight.

2.1. Technology and preferences

Firms produce a single homogeneous good Y t – that can be used as consumptionand investment – under perfectly competitive conditions, with physical capital Kt

and labor in effective units as inputs. The labor force is composed by Nt adults, eachof whom is endowed with et units of education; hence the maximum amount ofefficiency units of labor is etNt: However, each adult has nt ¼ ðNtþ1Þ=Nt children,and child rearing takes t hours per child. Hence, the total amount of efficiency unitsof labor is Lt ¼ etNtð1� nttÞ; where ð1� nttÞ represents hours spent on the job. Awell-behaved concave production function with constant returns to scale describesproduction technology:

Y t ¼ F ðKt;LtÞ ¼ Ltyt;

where

yt ¼ f ðktÞ and kt ¼ Kt=Lt:

Hence, yt is output per unit of effective labor, and kt is the ratio of capital on laborinput measured in efficiency units. We assume that the capital stock depreciates fullyin one generation.

Human capital of young individuals born in period t – that we denote etþ1 – isproduced with their full time effort and two inputs: knowledge of the parents andphysical goods btþ1:

etþ1 ¼ Ae1�at ba

tþ1; a 2 ð0; 1Þ: (1)

The role of parents’ education in Eq. (1) is that of an intergenerational externality.Knowledge learning also requires the use of goods and services that must be drawnfrom other uses. What comes to mind straightaway is schooling, or less formalmeans for information and knowledge transmission which have some cost (booksand magazines, computers and TVs, travel, etc.). Students must finance theconsequent expenditures by borrowing in the credit market. The parameter A

represents the level of technology in human capital production. This technologyshows constant returns to scale, and the stock of knowledge does not depreciate.Hence, if enough resources are devoted to human capital accumulation it canproceed in the future without limit.

Concerning preferences of individuals over the life-cycle we make the simplifyingassumption that young agents do not derive any utility from consumption that isincluded in the cost of education btþ1: Hence we can represent preferences taking thepoint of view of an adult agent. Preferences of adults include their life-cycleconsumption, number and education of children. Such preferences are representedby a concave intertemporal utility function:

V ðct�1t ; ct�1

tþ1; etþ1; ntÞ ¼ logðct�1t Þ þ b logðct�1

tþ1Þ þ j logðetþ1Þ þUðntÞ; (2)

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where cij is consumption in period j of an individual born in period i. UðntÞ is a

continuous function with continuous first and second derivatives and

UnðnÞ40; UnnðnÞo0; UnnnðnÞo0; limn!0

UnðnÞ ¼ 1:

Number and average human capital of children enter the utility function of theparents because they appreciate children as consumption goods.

Parents have a direct influence on the human capital investment of their offspringwhen, as a consequence of credit market imperfections, borrowing for educationalfunding is constrained and the family provides collateral in student loan contracts.Otherwise, education can be financed without restrictions and parents enjoy theeducation of their sons without the need for any involvement.

2.2. Markets

In the model economy there are markets for goods, factors of production andloans. Perfect competition in all these markets implies that wage and rental rateequal respectively labor and capital productivity:

wt ¼ f ðktÞ � ktf kðktÞ;

rt ¼ f kðktÞ;

where wt is the wage rate per efficiency unit of labor and rt is the rental rate oncapital. The no arbitrage condition Rt ¼ rt ensures that the gross yield on loans Rt

equals the rental rate on capital rt thus agents are indifferent between accessing thecredit market and self financing.

Adult savings can be used to finance firms’ investment and the education of youngpeople. Entry in the market for loans is free. Firms can borrow from financialintermediaries. The latter collect funds as savers’ deposits and lend out. Lendersmake zero profits at an interest rate Rt because the market is perfectly competitive.The economy we consider is part of competitive international capital markets, hencedomestic capital accumulation is driven by the exogenous international rate ofinterest Rw

t ; and domestic savings are unrelated to investments of firms. Thishypothesis also implies that, at any time t, the marginal productivity of capital mustequal the international interest rate: Rt ¼ f kðktÞ ¼ Rw

t :We assume that the economyof the rest of the world is in steady state with constant Rw: From here onward wedrop the time subscript on the relevant variables k;R;w:

The demand for loans comes also from households who want to finance theeducation of their children by borrowing on a credit market that is not perfect.Human capital has a high degree of illiquidity, slavery is prohibited and humancapital cannot serve as collateral. Here we assume that individual debtors can defaulton their debt. In fact, in this model lenders have a technology which allows them tocollect from adult debtors a fraction of the loan, while they would get nothing fromyoung borrowers.

Competitive lenders collect deposits at the market interest rate R and issue loanswith zero profits. Hence, they are indifferent between lending to firms or to

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households whenever student loans yield R with certainty. The use of collateral inloan contracts may serve to reduce to zero the risk of default from lenders’ point ofview, and we assume that parents may provide collateral to debt contracts of theirchildren because they are altruistic and have tight control over children’s resources(see Banerjee and Newman, 1993).

Let us consider a child born at time t who applies for a loan Dt that she will spendon her educational project: Dt ¼ btþ1: Even if she will get a certain return W ðbtþ1Þ

from this investment, she cannot borrow because she cannot pledge her futureearnings and she has no resources to use as a collateral. The outcome of investmentsin human capital is not uncertain, but loan repayment can be under risk due to moralhazard since when students become adult they might default on their debt, forexample by changing domicile without leaving a trace. However, parents care aboutchild welfare, and may provide a collateral Mtþ1 out of their income. They also havea monitoring technology and can recover from children a fraction p of the loan.

In this context, child and parent can sign a loan contract in which the parent losesMtþ1 if the child does not repay her debt. Lenders have a technology which allowsthem to collect from the parents a fraction P of the collateral Mtþ1: Hence, whenchild and parent repay their debt, net returns are

W ðbtþ1Þ � btþ1Rtþ1;

while in the opposite choice they expect to earn:

W ðbtþ1Þ � PMtþ1Rtþ1:

Hence, the family repays the loan when

W ðbtþ1Þ � btþ1Rtþ1XW ðbtþ1Þ � PMtþ1Rtþ1;

which means that the incentive condition btþ1pPMtþ1 ensures debt repayment.Under the incentive condition, student loans signed by lenders, children and parentsare alternative forms of investment in the loan market perfectly substitutable withthe others.

It is in the interest of the family to take the maximum loan btþ1 ¼ PMtþ1: Fromthe point of view of the child we see that she never repays her debt because thefollowing inequality holds:

W ðbtþ1Þ � PpMtþ1Rtþ1XW ðbtþ1Þ � btþ1Rtþ1;

hence parents always repay loans and then get pbtþ1 from children. It can be notedthat the case p ¼ 1 means parents have perfect control over their children who havethe same incentives with respect to loan repayment.

A child will repay her debt when collateral is so high that she can take the optimalvalue of loan

botþ1pPpMtþ1;

meaning that parents’ altruism can provide to young agents full correction of creditmarket imperfections.

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2.3. Household choices over the life-cycle

Given the hypothesis of a small open economy, market equilibrium conditionsdepend on some crucial decisions made by households. For a complete analysis ofthe model, it will be useful to derive the economic implications of the case of perfectmarket for student loans. In fact, this case characterizes the economy when fertility islow and the finance of education is unconstrained.

2.3.1. Unconstrained investment in human capital

In the first part of their life agents choose the level of education they will beendowed with when adult. If young people have free access to the loan market, theywill choose investment in education to maximize life-cycle utility. As human capitalaffects the cost of having children, a youth’s decision is connected with her choice offertility. In this modelling set-up it is clear that the schooling decision has onlyeconomic consequences in the subsequent stages of life. Hence, we can formalize theproblem of a household from the point of view of adulthood.

The problem of a household is maximization of life-cycle utility Eq. (2) subject to abudget constraint. When adult she consumes and saves her labor income net ofrepayment of the debt she made when young to finance schooling. Her income is netof the wage she gives up to rear her children. Hence, we can specify the followingproblem:

Maxct�1

t ;ct�1tþ1;bt;nt

logðct�1t Þ þ b logðct�1

tþ1Þ þ j logðetþ1Þ þUðntÞ

s:t: ct�1t þ btRþ

ct�1tþ1

Rpetwð1� nttÞ;

et ¼ Ae1�at�1 ba

t :

(3)

Problem (3) highlights the main consequence of the assumption of free access tothe market for student loans on the behavior of households. Parents are not involvedin the finance of their children’s educational investment. The following are first-orderconditions for a maximum:

1

ct�1t

� lt ¼ 0; (4)

bct�1

tþ1

�lt

R¼ 0; (5)

jað1� aÞ

bt

� lt½R� Aae1�at�1 ba�1

t wð1� nttÞ� ¼ 0; (6)

UnðntÞ � ltetwt ¼ 0; (7)

where lt is a Lagrange multiplier.Conditions (4) and (5) are standard. Eq. (6) tells that the optimal choice of bt

requires the balance between marginal benefits and costs. A marginal increase of bt

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increases utility because of the effect on offspring education etþ1 and that on adultincome. Higher bt has a cost in terms of interests on loans. According to the lastf.o.c., Eq. (7), adults have children up to the number that equates marginal utility tothe time cost of a child. From the full set of f.o.c. we derive the following equationthat implicitly identifies the equilibrium rate of fertility no

t :

½1þ bþ jað1� aÞ�tUnðno

t Þ¼ ð1� aÞð1� no

t tÞ: (8)

It can be easily seen that the value of not is unique and it does not depend on human

capital. not decreases with both t;b and j: Parents have fewer children when rearing

is time expensive and when they have a particular preference for consumption andchild quality. Substitution of no

t in Eq. (6) provides the rule for the optimal choice ofbt:

bot ¼ et�1

½ð1þ bÞaþ jað1� aÞ�

½ð1þ bÞ þ jað1� aÞ�RAwð1� no

t tÞ� �1=ð1�aÞ

(9)

in which investment in education of an individual when young is reduced by thenumber of children she will have in adulthood, as less working time reduces thereturn on human capital investment.

2.3.2. Constrained investment in human capital

Let us imagine how child and parent make a joint decision on investment ineducation within the context of imperfect markets for student loans analyzed inSection 2.2. A young individual has a targeted expenditure for schooling given bybo

tþ1; but she can borrow only a lower amount under the guarantee that her parentswill repay debt in the case of default. Parents care about human capital of theiroffspring and underwrite a loan contract for the amount btþ1 ¼ PMtþ1 that theyalways pay back. Adults recover a fraction pbtþ1 of the loan and incur a cost ofð1� pÞbtþ1:

Given the rule btþ1 ¼ PMtþ1 that lenders follow to determine student loans, in thelife-cycle problem of an adult the choice of collateral Mtþ1 is equivalent to the choiceof btþ1; hence we make this simplifying change of variables:

Maxct�1

t ;ct�1tþ1;btþ1;nt



s:t: ct�1t þ pbtRþ

ct�1tþ1

Rþð1� pÞbtþ1nt

Rpetwð1� nttÞ;


tþ1:

(10)

In problem (10) the life-cycle budget constraint takes the point of view of an adultborn at time t� 1: As a consequence of credit market imperfections, she takes asgiven education that she acquired when young bt; and for that investment she has topay pbtR to her parents. The adult also knows that having nt children and gettingloans btþ1 from the bank, during old age she will incur a cost of ðð1� pÞbtþ1ntÞ=R:

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The novel feature of problem (10) that appears in the budget constraint is the costof children’s education that may cut old age consumption of the parent. This costincreases with the fertility rate and brings about a trade-off between number andquality of children. Problem (10) differs substantially from the standard case ofparents who provide their children with a bequest. In fact, here we consider the jointinvolvement of child and adult in funding education. It can be noted that problem(10) collapses to the case of bequests when p ¼ 0; hence such a framework seemsmore general. These innovations are embedded in the following first-orderconditions for a maximum:

1

ct�1t

� lt ¼ 0; (11)

bct�1

tþ1

�lt

R¼ 0; (12)

ja

btþ1� lt

ð1� pÞnt

R¼ 0; (13)

UnðntÞ � lt

ð1� pÞbtþ1

Rþ etwt

� �¼ 0: (14)

To analyze this case we make the following assumption on UðntÞ that ensures thatadults always choose a positive number of children:

Assumption A1.

1.
UnðnÞ þ nUnnðnÞ40: 2. limn!0 UnðnÞn ¼ y4ja: 3. limn!1UnðnÞn ¼ 1:
According to Eq. (13) parents choose the amount of child borrowing that equatesmarginal utility of btþ1 to the expected discounted marginal cost relative to childeducation. Substitution of Eq. (13) in (14) provides the following condition for therate of fertility:

UnðntÞ �ja

nt

� ltetwt ¼ 0; (15)

in which marginal utility of children equates the sum of the time cost and the cost ofone more child in terms of tighter borrowing constraints and lower education.Parents have children when UnðntÞnt � ja40; Assumption A1 is sufficient to ensurethat this condition holds. Eliminating l from the same equations we derive a trade-off between educational investment and fertility:

btþ1 ¼ et

wRtja

ð1� pÞ½UnðntÞnt � ja�: (16)

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Eq. (16) gives the maximum value of children’s borrowing as a function of theparent’s income and family size. Hence, in this model the effective extent of liquidityconstraint is an endogenous variable. Parents may choose a small number of childrenwho will be sustained with a large collateral as a guarantee for student loan.2Ifparents have tight control over their offspring (greater p) and can easily recover theloan then they invest more in child education. Rules for optimal rate of fertilityand investment in child education can be derived from the full set of f.o.c. toproblem (10):

etwtnt ¼ SnðntÞðetw� pRbtÞ;

where

SnðntÞ �UnðntÞnt � ja

1þ bþUnðntÞnt

;

ð1� pÞbtþ1nt

R¼ SbðntÞðetw� pRbtÞ;

where

SbðntÞ �ja

1þ bþUnðntÞnt

:

Parents decide the number of children by allocating to the time cost a share SnðntÞ

of their potential labor income net of debt repayment. In a similar way they choosetheir involvement in child education according to a share SbðntÞ: Both shares ofincome allocated to children are functions of adult preferences concerningconsumption and quantity and quality of children. The optimal value of fertilitycan be derived from the following equation:

½1þ bþUnðntÞnt�ntwtUnðntÞnt � ja

¼ w�pR

Abðnt�1Þ

1�a; (17)

2A reasonable question arises that concerns the educational investment. We would expect that the

unconstrained value b0t be greater than the constrained value bt for any given value of nt: The answer tellsus that this is the case when

½UnðnÞ � ja�UnðnÞ�

11�a4S;

where S is a positive constant:

S ¼ ð1�aÞRt½ð1þbÞaþjað1�aÞ�Aw

n o1=ð1�aÞwRtja1�p

;

the l.h.s. of this inequality is monotone increasing function of nt; starting from the origin. Since there is no

interest in values of nt in the neighborhood of zero, we can assume that UnðnÞ is so shaped that the

inequality is verified for realistic values of n:

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where

bðnt�1Þ �bt

et�1¼

wRtja

ð1� pÞ½Unðnt�1Þnt�1 � ja�(18)

is the investment in child education per unit of adult human capital. Eq. (17)highlights the dependence of the choice of fertility on the past rate nt�1: As inthe case of unconstrained finance of education, given nt�1; increasing preferencefor consumption reduces the fertility rate nt; and the same negative sign has the effectof preference for child quality. Equilibrium fertility also decreases when the time costof a child t increases and when the strength of parental control p is greater.

3. Equilibrium growth

Investment in education plays a crucial role in this model. Economic growth mayfollow multiple steady states according to the dynamics of fertility and the strengthof the accumulation of human capital. In the case of sub-optimal expenditure foreducation, the rate of growth of per capita income is low and decreasing such thatthe economy converges to a low state of development. The opposite regime ofsustained economic growth can be the outcome of dynamics driven by theunconstrained choices of fertility and education. The rate of fertility governs thischange of regime of economic growth. In this section we analyze the model dynamicsin the two regimes of growth starting from the case of a perfect loan market.

3.1. Unconstrained funding of education

The model describes an economy populated by individuals who, when young,choose the optimal amount of expenditure in human capital investment and find inthe credit market the relative amount of resources. The optimal value of the fertilityrate is given by Eq. (8). The optimal value of expenditure on education, bo

t can besubstituted in Eq. (1) to derive the gross rate of growth of human capital g:

gt �et

et�1¼ A

½ð1þ bÞaþ jað1� aÞ�

½ð1þ bÞ þ jað1� aÞ�RAwð1� no

t tÞ� �a=ð1�aÞ

: (19)

Knowledge evolves from generation to generation with a constant gross rate ofgrowth gt which is greater than one if – among other parameters – the technologicalparameter A and the ratio w=R are large enough.

Eqs. (8) and (19) describe the dynamics of the model economy in the case ofunconstrained borrowing of children. We can then characterize the competitiveequilibrium of the economy. Given the initial values N0; n0; e0; a dynamicequilibrium consists of sequences fkt; nt; etg

1t¼0 such that the capital market is in

equilibrium – kt ¼ k ¼ f 0�1ðRwÞ – and fertility and education are determined by

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Eqs. (8) and (19). A steady state equilibrium of this economy is defined by a pair ofstationary solutions ðn; gÞ to Eqs. (8) and (19).

Hence, at the initial period the economy instantly reaches an equilibrium, andsubsequently it grows along a unique steady state path. Households choose theoptimal rate of fertility which remains constant over time. When population growsalong this equilibrium trajectory, children can borrow on the credit market and canattain their planned level of human capital. Earned incomes grow with humancapital and the economy provides further finance for investment in education. Thisvirtuous process of economic growth unfolds into the future without limit.

The rate of growth of human capital is affected by the fundamentals of theeconomy: technology and preferences. As we saw in Section 2.3.1, the equilibriumrate of fertility depends negatively on t;b and j: It can be easily verified that theseeffects on nt become positive with respect to gt: As might be expected, if householdshave a particular preference for child quality they will reduce time for child rearingand increase investment in education. The gross rate of growth of output is given byg ¼ gþ n� 1; and the positive effects of t;b and j can be extended also to g if –among several parameters – the level of technology A is high enough.3

3.2. Constrained funding of education

A different regime of economic growth applies to the model economy when theoptimal choice of investment in human capital cannot be fully financed by loans. Thedesired amount of expenditure on education is greater than the borrowing limit thatlenders are willing to provide to each family member. In this context, a linkage arisesbetween children’s investment in human capital and the family’s resources and size.

Let us consider Eq. (17) and define two new functions:

DðntÞ �½1þ bþUnðntÞnt�ntwt

UnðntÞnt � ja;

Lðnt�1Þ � w�pR

Abðnt�1Þ

1�a:

Then the following dynamical system summarizes the evolution of the economy:

Fðnt; nt�1Þ � DðntÞ � Lðnt�1Þ ¼ 0;

gtþ1 ¼ AbðntÞa: ð20Þ

System (20) is recursive since the fertility rate nt determines gtþ1; but vice-versa doesnot hold. Given that fertility determines the growth rate of human capital, weconcentrate the analysis of dynamics on the function Fðnt; nt�1Þ ¼ 0: The dependenceof nt on the past size of the family derives from the mechanism of intergenerationallinkages arising from imperfections of the loan market. In order to analyze system(20) we need a further assumption on UðnÞ:

3The effect of parameters’ change on g is given by the sum of two effects with opposite sign: an effect on

n and another on g: If the level of technology A is high enough the second prevails on the first.

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Assumption A2.

n2UnnðnÞ þ jao0:

Assumptions A1 and A2 are sufficient to ensure that fertility dynamics aremonotone increasing. In fact, we can prove the following Lemma:

Lemma. Under the Assumptions A1, A2, the function Fðnt; nt�1Þ ¼ 0 for n 2 ½0;1Þ is

continuous, and it can be solved in the explicit form nt ¼ Cðnt�1Þ: Cðnt�1Þ is non-

negative, continuous, monotone increasing and concave for values of nt�1 2 ½n;1Þ;where at n we have LðnÞ ¼ 0; and

limnt�1!n

Cðnt�1Þ ¼ 0; limnt�1!1

Cðnt�1Þ ¼ n:

Proof. See the appendix.

This lemma provides a characterization of the function Cðnt�1Þ that describes thedynamics of the rate of fertility (see Fig. 1). From inspection of nt ¼ Cðnt�1Þ we seethat n is the value of nt�1; where Cðnt�1Þ crosses the horizontal axis, while n is theupper bound of Cðnt�1Þ: Hence, at first sight there is a corner steady state with nullfertility. However, if the economy moves toward zero fertility it meets a positivelower bound made by the optimal value no: Concavity of Cðnt�1Þ implies that it cancross the 451 line two times. More precisely, we can summarize the results concerningthe dynamics of fertility in the following proposition that can be easily provedrelying on the Lemma.

nt

no

nl nh nt-1

Ψ (nt-1)

45˚

Fig. 1. Multiple steady states in fertility dynamics.

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Proposition 1. Under the Assumptions A1, A2, any time path of nt that satisfies the

difference function nt ¼ Cðnt�1Þ is a monotone sequence bounded from below by no and

from above by n: There can be at most three steady states of time paths, no and two

other positive steady states, according to the cases:

1.
the function Cðnt�1Þ lies below and does not cross the 451 line; in this case no is a
globally stable steady state;
2. the function Cðnt�1Þ crosses the 451 line twice at nl and nh; with nlonh; and there are
two sub-cases:(a) no ¼ Cðnt�1Þ and no4C�1ðnoÞ; such that there is only one globally stable steady

state at nh;(b) no ¼ Cðnt�1Þ and nooC�1ðnoÞ; such that there are two locally stable steady

states no and nh and one unstable steady state nl:

3.3. General equilibrium dynamics

The study of general equilibrium economic dynamics is based on Proposition 1and the recursive system of difference equations:

nt ¼ Cðnt�1Þ

if nt4no;

gtþ1 ¼ AbðntÞa

8>><>>:nt ¼ no

t

if ntpno;

gtþ1 ¼ A bboðno

t Þa

8>><>>: ð21Þ

where bboðno

t Þ � ðboðno

t ÞÞ=et�1:The following definition of equilibrium provides a precise statement of this crucial

concept:

Definition. Given the initial values N0; n0; e0; a dynamic equilibrium consists ofsequences fkt; nt; etg

1t¼0 such that the capital market is in equilibrium – kt ¼ k ¼

f 0�1ðRwÞ – and fertility and education are determined by the dynamical system (21).Steady state equilibria of this economy are the pair of constant values ðn; gÞ which arethe stationary solution to system (21).

The case (a) of Proposition 1 describes the evolution of the economy when, givenany starting value of the fertility rate, family choices move in the direction of theoptimal unconstrained rate of fertility no: This case could be the consequence of tightcontrol of the parents over child behavior, that is high p. Under case (b) the first sub-case (i) characterizes an economy which is always trapped in a low-developmentbalanced growth path. In fact, when the economy converges to this stable dynamicequilibrium the rate of fertility nh remains high and consequently the constrainedinvestment in education is low. This case could be explained by low values of

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technology and preference parameters ðt;j;bÞ: In economies where rearing a childneeds few ours, or preference for consumption is low and parents do not care aboutchild education, the unconstrained rate of fertility no can be high and close to thesteady state rate nh; such that the basin of attraction of the high-fertility equilibriumis so wide that it can coincide with the positive part of the real line. During thetransition to nh; parents, generation after generation, reduce the share of net incomedevoted to child education, SbðntÞ; and increase the share SnðntÞ; which means theyreduce participation in the labor market to rear children. At the steady state nh

adults’ potential net income is allocated to quality and quantity of childrenaccording to constant shares Sbðn

hÞ; SnðnhÞ which reflect the fundamentals of the

economy.The model dynamics are even more interesting if Cðnt�1Þ crosses the 451 line twice,

as Fig. 1 shows. There is clear evidence of multiple steady states. The steady state inwhich fertility is high, nh; can be thought of as a low-development trap. WhenCðnt�1Þ crosses the 451 line from below, the steady state nl is unstable. On the left ofthis steady state, fertility moves towards the origin of the axes, but before reachingthat value it meets the threshold value that ensures escape from credit constraints. Atthat lower threshold, fertility takes the optimal value no and investment in educationis determined by the optimal choices of children.

A self-reinforcing mechanism concerning fertility choice drives nt towards no or tonh: Consider Eq. (15), the decisional rule on nt: Assumption A2 ensures that whenfertility is greater than threshold nl; increasing fertility reduces human capital and thetime cost of an additional child more than it increases the cost of a child in terms oflower education. The same assumption is sufficient for a negative trend in fertilitywhen it is lower than threshold nl:

Fig. 1 contains all important information on the different regimes of economicgrowth produced by the model economy. When the initial value of the fertility rate islow enough, noonl; a process of development can be described by the progressivereduction of fertility and the joint increase of investment in education by householdswho find an increasing amount of credit in the market. During this process ofeconomic development the model describes the evolution of the family’s organiza-tion. In fact, continuous reduction in fertility progressively increases the extent ofparticipation on the labor market of adult women and releases children frompecuniary linkages with their parents. The final outcome of this process is thetransition to a family where women depend less on husbands’ income and childrencan be free to decide their own lives. It is well known that these two different types offamilies characterize both different stages of development in the history ofindustrialized countries and, today, respectively the developing and the industria-lized countries. It should be noted that this result concerning young individualsderives from the model when preferences for children are the same in the tworegimes. Poor and rich parents love their children in the same way.

It is also worth noting that increasing values of parental control p shift the curveCðnt�1Þ downward and cause the shrinking of the basin of attraction of the low-development equilibrium which becomes closer to the high-growth fertility rate no:Effects with opposite sign on Cðnt�1Þ come from increases of parameters of the

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education production function. In fact, while a shifts the curve down, A, the level oftechnology, causes an upward shift due to the positive effect on adult net income perunit of adult human capital.

4. Government subsidies to education

In this section we extend the model with the introduction of the government.Subsidies to private education are an important feature of the modern system ofschooling around the world. Families are the main recipients of these policies whichaffect the behavior of the parents, in particular with respect to the choices involved inraising children. While the literature on economic growth and public policies towardhuman capital is well developed (e.g., Glomm and Ravikumar, 1992), the same issuein models of growth with endogenous fertility remains largely unexplored.4 Here wetackle the issue in the case of constraints to borrowing. This perspective is certainlyinteresting as financial markets’ imperfections are important reasons for public aidto students. The final effects of education subsidies on growth also depend onsubsidies which influence fertility. Parents should be able to make a better trade-offbetween quantity and quality of children when they share some of the costs ofeducation with the government.

We modify the model of the previous sections by introducing public subsidies toeducation financed by income and consumption flat-rate taxes. Given the total costof education btþ1 the government provides each child with a subsidy proportional tobtþ1; sbtþ1 with s40; such that the amount borrowed becomes btþ1ð1� sÞ: Thisexpenditure is financed by taxes on adult consumption and income with constantrates respectively tc; and ty: Hence, the problem of households becomes thefollowing:

Maxct�1

t ;ct�1tþ1;btþ1;nt



s:t: ct�1t ð1þ tcÞ þ pð1� sÞbtRþ

ct�1tþ1

Rþð1� pÞð1� sÞbtþ1nt

Rpetwð1� nttÞð1� tyÞ;


tþ1:

(22)

Solving problem (22) we derive a set of f.o.c. very close to those arising from problem(10). In particular the optimal amount of investment in schooling is

bgðntÞ ¼ et

wRtjað1� tyÞ

ð1� pÞð1� sÞ½UnðntÞnt � ja�:

This equation makes clear the opposite effects of income taxes and subsidies onbgðntÞ: In fact, income taxes reduce the net return to education, while subsidies

4An exception is Zhang (1997) where the effects of educational subsidies are considered financed by flat-

rate consumption and income taxes. The net effect on fertility is nil.

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increase it. Comparing this case with that of the previous sections we see thatbgðntÞ4bðntÞ if tyos:

Equilibrium dynamics of the economy can be investigated following the samesteps as the previous sections. However, choices of the representative agents must beconsistent with the government budget constraint, which is balanced in each period:

ct�1t tc þ etwð1� nttÞty � sbtþ1nt ¼ 0; (23)

where a bar over variables refers to the average in the population. In equilibriumindividual values of the variables are equal to the average, hence we can substitutethe government budget constraint in that of a household. Further substitution off.o.c. to problem (22) in the budget constraint provides the following equation forthe equilibrium dynamics of the rate of fertility:

½Gja� jaþUnðntÞnt�ntwtUnðntÞnt � ja

¼ w�pRð1� sÞ

Abbgðnt�1Þ

1�a; (24)

where

G ¼½1þ ðjaþ bÞð1þ tcÞ�ð1� tyÞ

jað1þ tcÞþ

sRð1� tyÞ

ð1� pÞð1� sÞ;

and

bbgðnt�1Þ ¼

bgðnt�1Þ

et

:

Similarity between Eqs. (24) and (17) is clear, such that it can be easily verified thatlemma and Proposition 1 also apply to Eq. (24), hence we can define nt ¼ Cgðnt�1Þ

the function derived from Eq. (24) that describes the dynamics of fertility. Thisfunction can be represented with a graph of the same type as that in Fig. 1, where itcrosses the 45� line at ngl and ngh: In Figs. 2 and 3 we show the two possible way inwhich the graph of Cgðnt�1Þ can differ from that of Cðnt�1Þ: In appendix we provethat Cgðnt�1Þ always starts from the horizontal axis at the left of Cðnt�1Þ: As nt�1

increases, Cgðnt�1Þ can either stay over Cðnt�1Þ or it can cross from above Cðnt�1Þ:The second case may result from low appreciation of children in adult utilityfunction.5

The interesting issue is investigation of the role of public policies in improvingborrowing constraints and in population dynamics. Subsidies to education shouldallow children to invest an amount of resources closer to their ideal target. However,this effect depends on the reaction of the parents to government policies as far asfertility is concerned. Fig. 2 tells us that one of the outcomes of the introduction of asystem of household taxation and subsidies may be an increase in the high-fertilitysteady state, ngh4nh; and a widening of the basin of attraction of this

5The precise condition shown in appendix is

1þ bþUnðnÞn4Gja� ð1þ bþ jaÞ

1� ð1� tyÞ1�að1� sÞa

:

.

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nt

nt-1

Ψg (nt-1)

Ψ (nt)

Fig. 3. The dynamics of fertility with and without education subsidies. The case of intersection between

CgðntÞ and CðntÞ:

nt

nt-1

Ψg (nt-1)

Ψ (nt)

Fig. 2. The dynamics of fertility with and without education subsidies. The case CgðntÞ4CðntÞ:


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low-development equilibrium. In this case, such a system may release parents fromsome of the cost of child education, with the consequence of allowing the choice of alarger family. This seems to be the case of economies in which parents derive highsatisfaction from the number of children. The introduction of subsidies to humancapital investment should imply that their positive effect is higher than the negativeeffect caused by greater fertility. Fig. 3 describes the more virtuous situation where asa consequence of the introduction of education subsidies, the low-developmentsteady state is lower and investment in education greater.

Taken for granted the existence of public subsidization of child education, theeffects of government policy on fertility dynamics are summarized in the followingproposition, that focuses on the steady states ngh; ngl:

Proposition 2. Under the Assumptions A1, A2, public policies have the following

effects on fertility dynamics described by equation nt ¼ Cgðnt�1Þ:

gh
1. (a) At steady state n ;
qn

qsw0 if gðnghÞ_pRð1� sÞ 1�

1� s

s

� �2

ð1� aÞ

" #;

while at steady state ngl

qn

qsw0 if gðnglÞwpRð1� sÞ 1�

1� s

s

� �2

ð1� aÞ

" #:

(b)
At steady state ngh;
qn

qty

w0 if gðnghÞ_A1

1�awð1� nghtÞapRð1� sÞ

� �a=ð1�aÞ

;

while at steady state ngl

qn

qty

w0 if gðnglÞwA1

1�awð1� ngltÞapRð1� sÞ

� �a=ð1�aÞ

:

(c)
qCgðnt�1; s; ty; tcÞ
qtc

40:

Proof. See the appendix.

According to Proposition 2 when the economy is trapped in a high-fertility
equilibrium path, the sign of the effects of subsidies and income taxes on fertilitydepends on the strength of growth. If the steady state rate ngh is low enough then apolicy that increases education subsidies can reduce equilibrium fertility and increasethe rate of growth of the economy, but when ngh is too high the reaction of theparents to higher subsidies can be increasing fertility which might worsen the

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dynamic capabilities of the economy. Since the effect of increasing subsidies oninvestment in human capital is the sum of two effects:

db

ds¼

qb

qsþ

qb

qn

qn

qswith

qb

qs40 and

qb

qno0;

this policy may be beneficial for economic growth if the consequences on fertility arenot too adverse on education. Such a policy may be promoted as it increases theunstable steady state ngl and it widens the basin of attraction of the high-growth andlow-fertility equilibrium.

A similar rationale emerges from the effects of income taxes on fertility. In thiscase, taxes reduce the return to education, but may also reduce fertility:

db

dty

¼qb

qty

þqb

qn

qn

qty

withqb

qty

o0;

hence, according to Proposition 2 at the stable steady state ngh an increase in incometax rates might reduce fertility if the value of ngh is low enough, and the converseapplies to the economy at ngl:

A policy that increases tax rates on adult consumption shifts the curve Cgðnt�1Þ

upward because consumption becomes more expensive with respect to raising andeducating children. Hence, the effect of tc on nt is positive and that on gðntÞ is alwaysnegative.

The set of results on policies to subsidize human capital accumulation should beappreciated since such effects are consistent with the trade-off in the decision onhaving and educating children which can change toward one or the other item of thetrade-off as a consequence of government intervention. Hence, these ambivalenteffects of government policies seem to suggest that policies towards the family mayhave complex effects on economic growth as a consequence of the reaction of parentsin environments characterized by imperfections in the credit market.

5. Conclusions

This paper has put forward a theoretical analysis of the family as a non-marketinstitution which, facing limited access to the credit market, provides their childrencollateral in loan contracts aiming to finance investments in education. From thiskind of intergenerational linkage multiple equilibria derive. A stable trap of low-development is characterized by high-fertility rates and low-investment in humancapital. On the other hand, economies with a lower than threshold starting rate offertility grow according to a process that may describe a demographic transition. Inthis case, borrowing constraints gradually vanish and the process of growth reachesa steady state characterized by optimal fertility and schooling choices. Thistransition brings young agents from a state of dependence on parents’ income to astate of self-sufficiency. When public policy is added to the analysis, it emerges thatgrowth might benefit from subsidies to education investment if parents do notsignificantly increase fertility.

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While the existing literature lacks deeper investigation of the relations betweenhouseholds and the financial markets, our model suggests a new direction for theanalysis of intergenerational linkages and growth. Further research could highlightways in which families choose to gather and redistribute finances on imperfect creditmarkets (Cigno, 1993; Behrman, 1997). This issue could be an important part ofmodels of economic growth and endogenous fertility. In this research program animportant role should be assigned to the study of public policies. Our analysis showshow important it is to consider endogenous fertility in order to derive soundtheoretical results.

Acknowledgements

I thank the editor, Wouter DenHaan and two referees for their comments andsuggestions. During this research I visited the Department of Economics of UCLAtwo times, which I thank for hospitality. I am grateful to Costas Azariadis for help-ful discussion and suggestions. I also thank seminar participants at the University ofNaples ‘‘Parthenope’’ and at CSEF-University of Salerno. The usual disclaimersapply. Financial support from the National Research Council of Italy is acknowl-edged.

Appendix A. Proof of the lemma

Continuity of Fðnt; nt�1Þ ¼ 0 derives from continuity of UðnÞ and Assumption A1.In order to apply the implicit function theorem to Fðnt; nt�1Þ ¼ 0; let us rewriteDðntÞ as

DðntÞ ¼ð1þ bþ jaÞð1� pÞ

jaRntbðntÞ þ nttw:

Then we have

DnðntÞ ¼ð1þ bþ jaÞð1� pÞ

jaR½bðntÞ þ nt

bbnðntÞ� þ tw:

But

bbnðntÞ ¼ �bðntÞUnnðntÞnt þUn

UnðntÞnt � jao0;

hence

DnðntÞ ¼ �ð1þ bþ jaÞð1� pÞ

jaRbðntÞ

n2UnnðnÞ þ ja

UnðntÞnt � ja

� �40

and it can be possible to solve DðntÞ � Lðnt�1Þ ¼ 0 in the function nt ¼ Cðnt�1Þ:

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Given Lnðnt�1Þ ¼ �pRð1� aÞbðnt�1Þ�abnðnt�1Þ40; we have

qnt

qnt�1¼

Lnðnt�1Þ

DnðntÞ40:

The proof of concavity of the function Cðnt�1Þ requires the evaluation of thesecond derivative:

qnt

qnt�1qnt�1¼

Lnnðnt�1ÞDnðntÞ � Lnðnt�1ÞDnnðntÞ

DnðntÞ2

:

From the first derivative we get

Lnnðnt�1Þ ¼ �pRð1� aÞbðnt�1Þ�a bnnðnt�1Þ � a

bnðnt�1Þ2

bðnt�1Þ

" #;

but

bnnðnt�1Þ ¼2bnðnt�1Þ

2

bðnt�1Þ� bðnt�1Þ

nt�1Unnnðnt�1Þ þ 2Unnðnt�1Þ

Unðnt�1Þnt�1 � ja40;

that can be substituted in Lnnðnt�1Þ to obtain

Lnnðnt�1Þ ¼ � pRð1� aÞbðnt�1Þ�a

� ð2� aÞbnðnt�1Þ

2

bðnt�1Þ� bðnt�1Þ

nt�1Unnnðnt�1Þ þ 2Unnðnt�1Þ

Unðnt�1Þnt�1 � ja

" #o0:

Let us consider the second derivative

DnnðntÞ ¼ð1þ bþ jaÞð1� pÞ

jaR½2 bbnðntÞ þ nt

cbnnðntÞ�;

substitution of bnnðntÞ provides


jaR

bbnðntÞ

nt

� 2bbnðntÞnt

bðntÞþ 2

bbnðntÞnt

bðntÞ

!2

� n2tUnnnðntÞnt

UnðntÞnt � jaþ

2UnnðntÞ

UnðntÞnt � ja

� �24 35:Let us denote with �b the elasticity of

bðnÞ : �b �bbnðntÞnt

bðntÞ;

then, applying Assumption A2 we have

�b ¼�UnnðntÞn

2t þUnðntÞnt

UnðntÞnt � ja4�ðUnðntÞnt � jaÞ

UnðntÞnt � ja¼ �1:

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We can then rewrite DnnðntÞ as


jaR

bbnðntÞ

nt

� 2�b þ 2�2b � n2t

UnnnðntÞnt

UnðntÞnt � jaþ

2UnnðntÞ

UnðntÞnt � ja

� �� 4ð1þ bþ jaÞð1� pÞ

jaR

bbnðntÞ2

nt

�b þ �2b � n2

t

UnnðntÞ

UnðntÞnt � ja

� �¼ð1þ bþ jaÞð1� pÞ

jaR

bbnðntÞ2

nt

2�b þ �2b þ

UnðntÞnt

UnðntÞnt � ja

� �;

since

�n2t

UnnðntÞ

UnðntÞnt � ja¼bbnðntÞnt

bðntÞþ

UnðntÞnt

UnðntÞnt � ja;

but

2�b þ �2b þ

UnðntÞnt

UnðntÞnt � ja42�b þ �

2b þ 140;

hence DnnðntÞ40: This results ensures that qnt=ðqnt�1qnt�1Þo0:As far as the extreme bounds of the function Cðnt�1Þ are concerned, we consider

the lower bound first. At n the function LðnÞ ¼ 0; and that implies bðnÞ ¼

ðw=pRÞ1=ð1�aÞ; hence n is positive. If LðnÞ ¼ 0 we have

CðnÞ ¼ nt 3 DðntÞ ¼ 0

and this value of the fertility rate is nt ¼ 0 since

limnt!0

ð1þ bþUnðntÞntÞntwtUnðntÞnt � ja

¼ð1þ bþ yÞwt

y� ja0 ¼ 0:

Concerning the upper bound of Cðnt�1Þ; it can be easily seen that limnt!1 bðnÞ ¼ 0and this implies: limnt�1!1Lðnt�1Þ ¼ w:

Since DðntÞ is monotone increasing and we can prove that it takes values in therange ½0;1�:

limnt!0

DðntÞ ¼ 0;

limnt!1


¼ limnt!1

ð1þ bþUnðntÞntÞwt

UnðntÞ �ja

nt

¼ 1;

then there exists a finite non-negative value of nt such that DðnÞ ¼ w; and

limnt�1!1

Cðnt�1Þ ¼ n:

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Appendix B. The functions Cðnt�1Þ and Cgðnt�1Þ on the plane ðnt�1; ntÞ

In this appendix we show how the introduction of subsidies and taxes in the modelof Section 2 modifies the function nt ¼ Cðnt�1Þ that characterizes the dynamics of theeconomy. Let us consider Eqs. (17) and (24) and write them as follows:

pR

Abbðnt�1Þ

1�a¼ w�


� BðntÞ;

pR

Abbðnt�1Þ

1�a¼

w

ð1� tyÞ1�að1� sÞa

�ðGjaþUnðntÞnt � jaÞntwt

UnðntÞnt � ja� BgðntÞ;

where both BðntÞ and BgðntÞ are monotone decreasing functions. Then we haveBgðntÞ4BðntÞ; if

1� ð1� tyÞ1�að1� sÞa4

½Gja� ð1þ bþ jaÞ�nttUnðntÞnt � ja

: (*)

This inequality always holds if Gja� ð1þ bþ jaÞo0; otherwise we know that

limnt!0

nt

UnðntÞnt � ja¼ 0;

hence the inequality (*) is satisfied when nt ! 0: Applying some results of the proofof Lemma, we derive

limnt!n

nttUnðntÞnt � ja

¼1

1þ bþUnðnÞn;

which means that at nt ¼ n the inequality (*) becomes

1� ð1� tyÞ1�að1� sÞa

Gja� ð1þ bþ jaÞ4

1

1þ bþUnðnÞn;

hence, if UnðnÞn is large enough, the inequality (*) holds, otherwise we haveBgðntÞoBðntÞ:

Given the above set of inequalities, we derive the following relations. Let us startby assuming that BgðntÞ4BðntÞ for nt 2 ½0; n� and set mt�1 � ðpR=AÞbðnt�1Þ

1�a: Then,monotonicity of BgðntÞ and BðntÞ implies

n0t ¼ Bg�1ðmt�1Þ4nt ¼ B�1ðmt�1Þ and Cgðnt�1Þ4Cðnt�1Þ:

This inequality is reversed if BgðntÞoBðntÞ; but we proved that this can be the caseonly for nt40:

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Appendix C. Proof of Proposition 2

Let us consider Eq. (24) and define the left side DgðntÞ and the right side Lgðnt�1Þ;then at steady states the effects of s; ty; tc on Cgðnt; s; ty; tcÞ derive from

qCgðn;xÞ

qx¼

Lgxðn; xÞ � Dg

xðn;xÞ

DgnðnÞ � Lg

nðnÞ; with x ¼ s; ty; tc:

We know that at the stable steady state ngh the curve Cgðnt; s; ty; tcÞ crosses the 451line from above while Dg

nðnghÞ � Lg

nðnghÞ40 and Dg

nðnghÞ � Lg

nðnghÞo0 at steady state

ngl: Given this result we prove parts (a) and (b) of Proposition 2.Part (a): Let us consider the derivatives Dg

s ðnÞ and Lgs ðnÞ: Considering that bbg

s ðnÞ ¼

ð bbgðnÞð1� sÞÞ=s2; we have

Dgs ðnÞ ¼

bbgðnÞ

1� s;

Lgs ðnÞ ¼

pR

AbbgðnÞ1�a

�pR

Að1� sÞð1� aÞ bbg

ðnÞ�a bbgs ðnÞ

¼pR

AbbgðnÞ1�a

�pR

A

ð1� sÞ

s

� �2

ð1� aÞ bbgðnÞ1�a:

Hence

Lgs ðnÞ � Dg

s ðnÞ ¼pR

AbbgðnÞ1�a 1�

1� s

s

� �2

ð1� aÞ

" #�bbgðnÞ

1� s;

but bbgðnÞa ¼ g=A; and we can derive

Lgs ðnÞ � Dg

s ðnÞw0; if g_pRð1� sÞ 1�1� s

s

� �2

ð1� aÞ

" #:

Part (b): Let us consider the derivatives:

LgtyðnÞ ¼

pRð1� sÞð1� aÞ

Að1� tyÞbbgðnÞ1�a40;

in which we use cbgtyðnÞ ¼ �

bbgðnÞ1�ty

and

DgtyðnÞ ¼

1

1� ty

½tntw� DgðnÞ�:

Using these results we obtain

LgtyðnÞ � Dg

tyðnÞ ¼

1

1� ty

pRð1� sÞð1� aÞ

AbbgðnÞ1�a

� tnwþ DgðnÞ

� �¼

1

1� ty

ð1� tnÞw� apRð1� sÞ

AbbgðnÞ1�a

� �w0;

if bbgðnÞ1�a

_Að1� tnÞw

apRð1� sÞ;

ARTICLE IN PRESS


but bbgðnÞ1�a

¼ ðg=AÞð1�aÞ=a; hence

LgtyðnÞ � Dg

tyðnÞw0 if g_A1=ð1�aÞ wð1� ntÞ

apRð1� sÞ

� �a=ð1�aÞ

:

Part(c): The effect of tc on the function Cgðnt�1; tcÞ derives from

qCgðnt�1; tcÞ

qtc

¼Lg

tcðnt�1Þ � Dg

tcðntÞ

DgnðntÞ

:

We already know that DgnðntÞ40: It is easy to verify that

LgtcðntÞ ¼ 0 and Dg

tcðntÞ ¼

�jatntwð1� tyÞ

½UnðntÞnt � ja�jað1þ tcÞ2o0;

which is enough for the proof of part (c) of Proposition 2.

References

Banerjee, A., Newman, A., 1993. Occupational choice and the process of development. Journal of Political

Economy 101, 274–298.

Barro, R.J., Becker, G.S., 1989. Fertility choice in a model of economic growth. Econometrica 57,

481–501.

Becker, G.S., 1993. Human Capital, third ed. University of Chicago Press, Chicago.

Becker, G.S., Murphy, K., Tamura, R., 1990. Human capital fertility and economic growth. Journal of

Political Economy 98, S12–S37.

Behrman, J.R., 1997. Intrahousehold distribution and the family. In: Rosenzweig, M.R., Stark, O. (Eds.),

Handbook of Population and Family Economics, vol. 1B. Elsevier Science, Amsterdam, pp. 125–187.

Behrman, J.R., Pollak, R.A., Taubman, P., 1989. Family resources, family size and access to financing for

college education. Journal of Political Economy 97, 398–419.

Benhabib, J., Spiegel, M., 2000. The role of financial development in growth and investment. Journal of

Economic Growth 5, 341–360.

Cigno, A., 1993. Intergenerational transfers without altruism: family, market and state. European Journal

of Political Economy 7, 505–518.

De Gregorio, J., 1996. Borrowing constraints, human capital accumulation, and growth. Journal of

Monetary Economics 37, 49–71.

de la Croix, D., Doepke, M., 2004. Public versus private education when differential fertility matters.

Journal of Development Economics 73, 607–629.

Ehrlich, I., Lui, F., 1997. The problem of population growth: a review of the literature from malthus to

contemporary models of endogenous population and endogenous growth. Journal of Economic

Dynamics and Control 21, 205–242.

Fernandez, R., Guner, N., Knowles, J., 2001. Love and money: a theoretical and empirical analysis of

household sorting and inequality. NBER Working Paper no. 8580.

Galor, O., Weil, D.N., 2000. Population, technology, and growth: from the malthusian regime to the

demographic transition and beyond. American Economic Review 90, 806–828.

Galor, O., Zang, H., 1997. Fertility, income distribution and economic growth: theory and cross-country

evidence. Japan and the World Economy 9, 197–229.

Glomm, G., Ravikumar, B., 1992. Public versus private investment in human capital: endogenous growth

and income inequality. Journal of Political Economy 100, 818–834.

Hansen, G.D., Prescott, E.C., 2002. Malthus to Solow. American Economic Review 92, 1205–1217.

Hanushek, E.A., 1992. The tradeoff between child quantity and quality. Journal of Political Economy 100,

84–117.

ARTICLE IN PRESS


Haveman, R., Wolfe, B.L., 1995. The determinants of children’s attainments: a review of methods and

findings. Journal of Economic Literature 33, 1829–1878.

Kelemli-Ozcan, S., 2002. Does the mortality decline promote economic growth? Journal of Economic

Growth 7, 411–439.

Lucas Jr., R.E., 2002. Lectures on Economic Growth. Harvard University Press, Cambridge MA.

Morand, O.F., 1999. Endogenous fertility, income distribution and growth. Journal of Economic Growth

4, 331–349.

Neher, P.A., 1971. Peasants, procreation and pensions. American Economic Review 61, 380–389.

Nerlove, M., Raut, L.K., 1997. Growth models with endogenous population: a general framework. In:

Rosenzweig, M.R., Stark, O. (Eds.), Handbook of Population and Family Economics, vol. 1B. Elsevier

Science, Amsterdam, pp. 1117–1176.

Parish, W.L., Willis, R.J., 1993. Daughters, education and family budgets. The Journal of Human

Resources 28, 863–898.

Razin, A., Ben-Zion, U., 1975. An intergenerational model of population growth. American Economic

Review 65, 923–933.

Schultz, T.P., 1988. Education investments and returns. In: Chenery, H., Srinivasan, T.N. (Eds.),

Handbook of Development Economics, vol. 1. Elsevier Science, Amsterdam, pp. 543–630.

Tamura, R., 1996. From decay to growth: a demographic transition to economic growth. Journal of

Economic Dynamics and Control 20, 1237–1261.

Zhang, J., 1997. Fertility, growth, and public investments in children. Canadian Journal of Economics 30,

835–843.

Household borrowing constraints, fertility dynamics, and economic growth

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