inequality and growth: why differential fertility...

Inequality and Growth:Why Differential Fertility Matters

By DAVID DE LA CROIX AND MATTHIAS DOEPKE*

We develop a new theoretical link between inequality and growth. In our model,fertility and education decisions are interdependent. Poor parents decide to havemany children and invest little in education. A mean-preserving spread in theincome distribution increases the fertility differential between the rich and the poor,which implies that more weight gets placed on families who provide little education.Consequently, an increase in inequality lowers average education and, therefore,growth. We find that this fertility-differential effect accounts for most of the empir-ical relationship between inequality and growth. (JEL J13, O40)

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How does the income distribution of a coutry affect its rate of economic growth? We argthat to answer this question, it is essentialaccount for the fertility differential between thrich and the poor. Our argument is simple: tfertility differential matters because it affecthe accumulation of human capital. Assumithat we identify human capital with educatiofuture human capital is a weighted averagethe education of today’s children from familiein different income groups, with the weighgiven by income-specific fertility rates. Poparents tend to have many children and provlittle education. If the fertility differential between the rich and the poor is large, moweight is put on children with little education

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* de la Croix: National Fund for Scientific Research (Bel-gium), IRES and CORE, Universite´ Catholique de Louvain,Place Montesquieu 3, B-1348 Louvain-la-Neuve, Belgium (email: [email protected]); Doepke: Department of Ecnomics, University of California, 405 Hilgard Avenue, LosAngeles, CA 90095 (e-mail: [email protected]). Wthank Costas Azariadis, Harold Cole, Roger Farmer, GaHansen, Lee Ohanian, Martin Schneider, and two anonymoreferees for helpful comments and suggestions. We also befited from discussions with seminar participants at the Stocholm School of Economics, the SED Meeting in StockholmUSC, GREQAM, the University of Namur, and the EuropeaUniversity Institute. de la Croix acknowledges financial support from the Belgian French-speaking community (Grant NoARC 99/04-235) and the Belgian Federal Government (GraNo. PAI P5/10); Doepke had support from the National Scence Foundation (Grant No. SES-0217051) and the UCLAcademic Senate.

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which lowers average education. The fertilitydifferential, in turn, is a function of the incomedistribution. If the differential increases withinequality, countries with higher inequality willaccumulate less human capital, and thereforgrow slower.

We develop a growth model which capturesthis channel from inequality to growth. Ourmodel is related to Gerhard Glomm and B.Ravikumar (1992), who analyze the effects ofpublic versus private education on growth in amodel with fixed fertility. We use a similaroverlapping-generations framework, but modeendogenous fertility decisions along the lines oGary S. Becker and Robert J. Barro (1988)Both fertility and education are thus chosenendogenously. Parents face a quality-quantittrade-off in their decision on children, and weshow that education increases with the incomof a family (richer families can afford moreeducation), while fertility decreases with in-come (the time cost of child rearing is high forrich parents). The aggregate behavior of themodel depends on the initial distribution of in-come. Other things being equal, we find thaeconomies with a less equitable income distribution have higher fertility differentials, accu-mulate less human capital, and have a lower ratof economic growth. A calibrated version of ourmodel with endogenous fertility choice showsthat this effect is quantitatively important andaccounts for most of the empirical relationshipbetween inequality and growth. In contrast, ifwe impose fertility to be constant across income

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1092 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2003

groups, the effects of inequality on human cap-ital and growth are small.

We also analyze the dynamic properties ofthe model. Since in our dynastic framework aperiod corresponds to one generation, the dy-namics of the model are to be interpreted aschanges which occur over a horizon of a centuryor more. Here we find that the predictions of themodel are similar to broad patterns of develop-ment observed in industrializing countries in thenineteenth and twentieth centuries. For realisticparameter values, the interaction of fertility andeducation decisions gives rise to nonmonotonebehavior in inequality and fertility: both vari-ables rise initially, and later start to fall. In otherwords, the model generates both a Kuznetscurve and a demographic transition. Thus, inaddition to accounting for the cross-sectionalrelationship between inequality and growth, themodel generates plausible implications for thedynamic interaction of inequality, fertility, andgrowth over long time horizons.

The relationships between inequality, differ-ential fertility, and growth postulated by themodel are supported by empirical results. Mi-chael Kremer and Daniel Chen (2000) examinethe relationship between inequality and differ-ential fertility. Using cross-country data, theyfind that more inequality tends to be associatedwith larger fertility differentials within a coun-try. This supports the first part of our hypothe-sis, linking inequality to differential fertility. Toexamine the second part of our hypothesis, thelink from differential fertility to growth, we adda differential-fertility variable to a standardgrowth regression and find large significant ef-fects of differential fertility on growth. In thesame regressions, the direct effect of inequalityas measured by Gini coefficients is insignifi-cant, once differential fertility is included.

The majority of the existing literature on in-equality and growth concentrates on channelswhere inequality affects growth through the ac-cumulation of physical capital (see Roland Ben-abou, 1996). To our knowledge, Paul G.Althaus (1980) is the only existing model thatanalyzes the effects of differential fertility ongrowth. However, in Althaus’ model fertilitydifferentials are exogenously given, and the roleof human capital is not considered. Endogenousfertility differentials arise in Momi Dahan andDaniel Tsiddon (1998), but since their model

does not allow for long-run growth, the analysisconcentrates on nonlinear dynamics during thetransition to the steady state. In Oded Galorand Hyoungsoo Zang (1997), inequality affectsgrowth through its effect on overall fertility andhuman capital. Financial market imperfectionsplay a crucial role in their analysis. Olivier F.Morand (1999) has a model of inequality andfertility in which the sole motive for fertility isold-age support. He concentrates on the possi-bility of poverty traps when the initial level ofhuman capital is too low. Our paper also relatesto empirical studies of the growth-inequalityrelationship such as Roberto Perotti (1996) andBarro (2000). Both Perotti and Barro find thatdemographic variables are important for under-standing the growth effects of the income distri-bution, but once again differential fertility is notconsidered directly.

In the following section, we introduce themodel. Section II presents theoretical results onthe quality-quantity trade-off and the long-rundynamics of the model. In Section III we cali-brate and simulate the model to assess the quan-titative importance of the differential-fertilitychannel, and to examine the implications for thedynamic evolution of inequality, fertility, andgrowth. Empirical evidence is discussed in Sec-tion IV, and Section V concludes.

I. The Model Economy

Consider an economy that is populated byoverlapping generations of people who live forthree periods: childhood, adulthood, and oldage. Time is discrete and goes from 0 to �. Alldecisions are made in the adult period of life.People care about adult consumption ct, old-ageconsumption dt � 1, their number of children nt,and the human capital of children ht � 1. Theutility function is given by:

ln�ct � � � ln�dt � 1 � � � ln�nt ht � 1 �.

The parameter � � 0 is the psychological dis-count factor and � � 0 is the altruism factor.The role of old-age consumption is to provide amotive for savings and therefore generate anendogenous supply of capital. Notice that par-ents care about both the quantity nt and thequality ht � 1 of their children. Raising one childtakes fraction � � (0, 1) of an adult’s time. An

1093VOL. 93 NO. 4 DE LA CROIX AND DOEPKE: INEQUALITY AND GROWTH

adult has to choose a consumption profile ct anddt � 1, savings for old age st, number of childrennt, and schooling time per child et. The budgetconstraint for an adult with human capital ht is:

(1) ct � st � et nt wth� t � wt ht �1 � �nt �,

where wt is the wage per unit of human capital.We assume that the average human capital ofteachers equals the average human capital in thepopulation h� t, so that education cost per child isgiven by etwth� t. The assumption that teachersinstead of parents provide education is crucialfor generating fertility differentials. It impliesthat the cost of education is fixed and does notdepend on the parent’s wage. Education istherefore relatively expensive for poor parents.1

In contrast, since raising each child takes a fixedamount of the parent’s time, having many chil-dren is more costly for parents who have highwages. Parents with high human capital andhigh wages therefore substitute child quality forchild quantity and decide to have less childrenwith more education.

The only friction in the model is that chil-dren cannot borrow to finance their own ed-ucation. Instead, education has to be paid forby the parents. This assumption is made inmost studies of the joint determination offertility and education. In the real world, chil-dren generally do not finance their own edu-cation (at least up to the secondary level).

The budget constraint for the old-age period is:

(2) dt � 1 � Rt � 1 st .

Rt � 1 is the interest factor. The human capital ofthe children ht � 1 depends on human capitalof the parents ht, average human capital h� t, andeducation et:

(3) ht � 1 � Bt �� et ��ht �

�h� t �.

Here the parameter � [0, 1] captures theintergenerational transmission of human capitalwithin the family, whereas � [0, 1 � ]

1 Alternatively, we could have assumed that education isprovided by parents, but that the parents’ teaching productivityincreases with their own human capital. In the model devel-oped by Omer Moav (2001) this assumption leads to povertytraps characterized by high fertility and low education.

represents externalities at the community orsociety level. Alternatively, can be inter-preted as measuring the effect of the quality ofschooling, since h� is the average human capitalof teachers. The efficiency parameter Bt in-creases deterministically at a constant rate:

(4) Bt � B�1 � ��1 � � �t.

The parameters satisfy B, � � 0, and � � (0,1). The presence of � guarantees that parentshave the option of not educating their children,because even with et � 0 future human capitalremains positive. As in Peter Rangazas (2000),equations (3)–(4) are compatible with endoge-nous growth for � 1 � , and with exogenousgrowth otherwise. We will later explore theimplications of exogenous versus endogenousgrowth for the long-run behavior of theeconomy.

Production of the consumption good is car-ried out by a single representative firm whichoperates the technology:

Yt � AKt�Lt

1 � �,

where Kt is aggregate capital, Lt is aggregatelabor supply, A � 0, and � � (0, 1). Physicalcapital completely depreciates in one period.The firm chooses inputs by maximizing profitsYt � wtLt � RtKt.

Human capital is distributed over the adultpopulation according to the distribution func-tion Ft(ht). Total population Pt evolves overtime according to:

(5) Pt � 1 � Pt �0

�

nt dFt �ht �,

and the distribution function of human capital,Ft(h) evolves according to:

(6) Ft � 1 �h� �Pt

Pt � 1�

0

�

nt I�ht � 1 h� dFt �ht�.

Here I� is an indicator function, and it isunderstood that the choice variables nt and ht � 1are functions of the individual state ht. Averagehuman capital h� t is given by:


(7) h� t � �0

�

ht dFt �ht �.

The market-clearing conditions for capital andlabor are:

(8) Kt � 1 � Pt �0

�

st dFt �ht �,

and:

(9) Lt � Pt� �0

�

ht �1 � �nt � dFt �ht �

� �0

�

et nth� t dFt �ht �� .

This last condition reflects the fact that the timedevoted to teaching is not available for goodsproduction. We are now ready to define anequilibrium for our economy.

Definition 1: Given an initial distribution ofhuman capital F0(h0), an initial stock of phys-ical capital K0, and an initial population size P0,an equilibrium consists of sequences of prices{wt, Rt}, aggregate quantities {Lt, Kt�1, h� t,Pt � 1}, distributions Ft � 1(ht � 1), and decisionrules {ct, dt � 1, st, nt, et, ht � 1} such that:

1. the households’ decision rules ctdt � 1, st, nt,et, ht � 1 maximize utility subject to the con-straints (1), (2), and (3);

2. the firm’s choices Lt and Kt maximizeprofits;

3. the prices wt and Rt are such that marketsclear, i.e., (8) and (9) hold;

4. the distribution of human capital evolves ac-cording to (6);

5. aggregate variables Pt and h� t are given by(4), (5), and (7).

II. Theoretical Results

We begin the analysis of the model bycharacterizing the quality-quantity trade-off

faced by individuals. We find that educationincreases and fertility decreases with in-come. The size of the differentials depends onthe initial dispersion of human capital. Toexamine the long-run implications of themodel, we characterize the balanced growthpath. Finally, we examine the dynamics ofindividual human capital as a function of theparameters.

A. The Trade-off Between the Quality andQuantity of Children

The key variable for decisions in our econ-omy is the human capital ht of a family relativeto the average human capital h� t of the popula-tion. We denote the relative human capital of ahousehold as:

xt �ht

h� t

.

For a household that has enough human capital

such that the condition xt ��

��holds, there is

an interior solution for the optimal educationlevel, and the first-order conditions imply:

(10) st ��

1 � � � �wt ht ,

(11) et ��xt � �

1 � �,

(12) nt ��1 � ��xt

��xt � ��1 � � � ��.

The second-order conditions for a maximum aresatisfied. Note that:

�et

�xt� 0 and

�nt

�xt� 0,

which reflects the well-documented fact thatskilled people invest relatively more in the qual-ity of their children than in their quantity. Thereason is that the cost of education is fixed,while the time cost of raising many childrenincreases with income.


The lowest possible fertility rate is given by:

limxt3�

nt ��1 � ��

��1 � � � ��.

For poorer households endowed with sufficiently

little human capital such that xt �

��holds, the

optimal choice for education et is zero. The first-order conditions imply equation (10) and:

(13) et � 0,

(14) nt ��

��1 � � � ��.

Once a household is at the corner solution and thechoice for education is zero, fertility no longerincreases as the human capital endowment falls.

Fertility as a function of human capital is plot-ted in Figure 1. The horizontal part of the rela-tionship corresponds to the range of human capitalwhich leads to a choice of zero for education et.Fertility depends negatively on human capital andmoves within a finite interval. The upper bound onthe fertility differential is given by:

limxt3 0 nt

limxt3� nt�

1

1 � �.

This relationship will turn out to be helpful tointerpret the role of the parameter � and tocalibrate its value.

The results derived so far reflect the maineffects of inequality on growth that we are in-

FIGURE 1. FERTILITY AS A FUNCTION OF HUMAN CAPITAL

terested in. Assuming that all dynasties choosepositive levels of education, equation (11)shows that education is a linear function ofrelative human capital. If the dispersion of hu-man capital increases for a given average levelof human capital, this linearity implies that theaverage education choice will still be the same.However, since the production function for humancapital is concave in education, future averagehuman capital will be lower if the distribution ofhuman capital is less equal. This would be trueeven if fertility were constant across families withdifferent human capital levels. The fact that fertil-ity is actually higher for people with low humancapital greatly amplifies the negative effect ofinequality on human capital accumulation.

B. The Balanced Growth Path

To analyze the dynamic behavior of the econ-omy, it is useful to rewrite the equilibrium condi-tions in terms of variables that are constant in thebalanced growth path. The capital–labor ratio kt,the growth rate of average human capital gt, thepopulation growth rate Nt, and the deflated level ofaverage human capital ht are defined by:

kt �Kt

Lt, gt �

h� t�1

h� t

, Nt �Pt�1

Pt, ht �

h� t

�1 � ��t .

We also need to define the distribution of therelative human capital levels:

Gt �xt � � Ft �xth� t �.

Rewriting equations (4), (5), (6), and (7) interms of the stationary variables leads to:

(15) h t � 1 �gt

1 � �h t ,

(16) Nt � �0

�

nt dGt �xt �,

(17) Gt � 1 �x� �1

Nt�

0

�

nt I�xt � 1 x� dGt �xt �,


(18) 1 � �0

�

xt dGt �xt �.

Prices follow from the competitive behavior offirms, which leads to equalization of marginalcosts and productivities:

(19) wt � A�1 � ��kt�,

Rt � A�kt� � 1.

Schooling and fertility decisions are given by(13) and (14) for xt � �/(��) and by (11) and(12) otherwise. The number of children for anadult with relative human capital xt is thusgiven by:

(20)

nt � min� �1 � ��xt

��xt � ��1 � � � ��,

�

��1 � � � �� .

From equation (3), the children’s human capitalis given by:

(21) xt � 1 �Bxt

gt� � � max�0,

��xt � �

1 � � ��

� �ht � � � 1.

From equation (9), labor input satisfies:

Lt

Pth� t� �

0

�/�� 1 � ��xt

1 � � � �dGt �xt �

� ��/��

� �1 � ��1 � ��xt � ��xt � ��

��xt � ��1 � � � �� xt dGt �xt �,

which leads to:

(22)Lt

Pth� t

�1 � �

1 � � � �.

Using (8), (10), (19), and (22), the capital stockevolves according to the following law of motion:

(23) kt � 1 ��

1 � �

1

gt NtA�1 � ��kt

�.

Given initial conditions k0, h0, and G0( x0),an equilibrium can be characterized by se-quences {ht � 1, gt, nt, Gt � 1( x), Nt, xt, kt � 1}such that (15), (16), (17), (18), (20), (21), and(23) hold at all dates.

This dynamic system is block recursive.Given the initial conditions, we can first use(20) to solve for nt. Then equations (18) and(21) determine xt � 1 and gt. Leading (18) byone period and replacing xt � 1 by its value from(21) yields an expression where gt can be com-puted as a function of past variables, xt and ht.The new distribution of relative human capitalis given by equation (17). The variable ht � 1 isobtained from (15), the aggregate populationgrowth rate Nt from (16), and the future capital–labor ratio kt � 1 from (23). This procedure canbe used to compute an equilibrium for any ini-tial conditions. The future distribution of humancapital is always well defined and nonnegative.Given any initial conditions, an equilibriumtherefore exists and is unique.

Concerning the long-run behavior of theeconomy, it follows from these equations thatthere is a balanced growth path in which every-one has the same human capital.

PROPOSITION 1: If �� , there is abalanced growth path characterized bydG(1) � 1 (i.e., the limiting distribution isdegenerate). The growth factor of output andhuman capital is:

g* � �B��

1 � � �� if � 1 �

�endogenous growth�,

1 � � otherwise�exogenous growth�,

and the growth factor of population is:

N* ��1 � ��

�� 1 � � � ��.

PROOF:See Appendix A.

Along this balanced growth path, there is nolonger any inequality among households. Thisholds because we have assumed that householdsdiffer only in their initial level of human capital.


If we had introduced ability shocks on top of anunequal initial distribution of human capital,inequality would persist along the balancedgrowth path. We abstract from idiosyncraticshocks in the presentation of the model, sincethey do not play a role in the channel frominequality to growth that we are interested in.However, shocks can influence the long-run dy-namic behavior of the model. Therefore weintroduce ability shocks as an extension in Sec-tion III, subsection C, below.

We will assume �� from here on. Wenow consider the dynamics of the human cap-ital of an individual dynasty (of mass zero)around an aggregate balanced growth path.This will be useful to understand the role ofthe parameter for the dynamic properties ofthe model.

C. The Dynamics of Individual HumanCapital

To study the dynamics of individual humancapital, we assume that the economy is on abalanced growth path, so that the growth rate ofaverage human capital is constant over time:gt � g*. We focus on the effect of the param-eter on the dynamics of individual humancapital. We consider the function xt � 1 � xt �( xt; ) (the change in relative human capitalxt as a function of xt and ), which is given by:

(24)

�x; � � � 1 � �

��

� x�� max�0,��x � �

1 � � ��

� x.

As shown in Appendix B, ( xt; ) is obtainedfrom equation (21) after replacing g* and h bytheir steady-state values. Note that in the endog-enous and exogenous growth cases the function( x; ) turns out to be the same.

A detailed study of ( xt; ) is performed inAppendix B. A complete characterization of thedynamics of xt as a function of the parameter is presented in the bifurcation diagram in Figure2. The steady states x are represented on thevertical axis as a function of . For small thereis only one steady state, x � 1, which is glo-

bally stable. Once reaches a threshold � (givenin the Appendix) two additional steady statesappear. The lower one is stable and the secondis unstable. This threshold arises at the pointwhere the cutoff value for an interior solution isa steady state of the individual dynamics.Moreover:

PROPOSITION 2: At the point:

� 1 ��

� � �

the dynamics of individual capital describedby xt � 1 � xt � ( xt; ) undergo a tran-scritical bifurcation. There are two steady-state equilibria, 1 and x� , near (1, ) for eachvalue of smaller or larger than . The equi-librium 1 (resp. x� ) is stable (resp. unstable)for � and unstable (resp. stable) for � .

PROOF:We check the five conditions that define such

a bifurcation in Stephen Wiggins (1990), p.365:

�1, � � 0, x �1, � � 0, �1, � � 0,

�xx �1, � � ��

�� 2 � 0,

�x �1, � � 1 � 0.

FIGURE 2. STEADY STATES AS A FUNCTION OF �


This bifurcation occurs when an unstable anda stable fixed point collide and exchange stabil-ity. That is, the unstable fixed point becomesstable and vice versa.2 When increases beyond, the high steady state increases and then van-ishes once � �. Thus, for individual dynamicsto be stable, it is essential that not be too high.In the next section, we calibrate the model pa-rameters to data and find that the stable regionfor is the empirically relevant case. The anal-ysis of the dynamics of xt at given aggregateconditions is helpful to understand the numeri-cal simulations carried out in the next section.

III. Computational Experiments

The theoretical results in the previous sectionhighlight two channels through which inequal-ity affects growth in this model. First, inequalityin human capital leads to inequality in educa-tion, and since the production function for hu-man capital is concave, inequality in educationlowers future average human capital. Second,people with lower human capital not onlychoose less education for their children, but alsoa higher number of children. This differential-fertility effect increases the weight in the pop-ulation on families with little education, whichalso lowers future human capital. The questionarises which effect is more important, and howlarge the effects are quantitatively. To answerthis question, we calibrate our model and pro-vide numerical simulations of the evolution offertility, inequality, human capital, and income.The main findings are that the effects of in-equality on human capital accumulation andgrowth are sizable, and that the differential-fertility effect is crucial for generating thisresult.

We also use the calibrated model to analyzethe dynamic implications of our theory. Here, akey finding is that for reasonable parameteriza-tions, the model generates a “hump shape” ininequality and population growth which firstincrease and then fall during development. Thisfeature enables the model to reproduce broadfeatures of the evolution of inequality, eco-

2 Note that beyond the bifurcation point the number offixed points does not change, whereas in a saddle-nodebifurcation two fixed points either appear or disappear.

nomic growth, and population growth in indus-trialized countries during development. Thisoutcome lends additional support to the rela-tionship between inequality, fertility, andgrowth postulated by the model.

A. Calibration

We choose the parameters of the model suchthat the balanced growth path resembles empir-ical features of the U.S. economy and popula-tion. The production function for human capitalis calibrated to match observed fertility differ-entials, as well as empirical estimates of theeffects of education on future earnings.

The model is calibrated under the assumptionthat one period (or generation) has a length of30 years. The parameter � is the capital share inthe consumption good sector and is set to 1⁄3 tomatch the empirical counterpart. The productiv-ity level A is a scale parameter and is set to A �1. The discount factor � mainly affects the ratioof human capital to physical capital in the bal-anced growth path. Since this ratio depends onthe choice of units, it does not provide a con-venient basis for calibrating �. Given that �does not influence qualitative features of themodel that we are interested in, we choose avalue that is standard in the real-business-cycleliterature, � � 0.99120 (i.e., 0.99 per quarter).The implied interest rate per year is 4.7 percent.The productivity growth rate � governs outputgrowth in the balanced growth path, and is set to1.0230 or 2 percent per year, which approxi-mates the average growth rate in the UnitedStates. With exogenous growth (i.e., � 1 � )as in our calibration, the overall productivity Bin the production function for human capital isa scale parameter and is set to B � 1.

The weight � of children in the utility func-tion governs the growth rate of population in thebalanced growth path. In the United States as inother industrialized countries, fertility rates areclose to the reproduction level. Accordingly, wechoose � such that the growth rate of populationin the balanced growth path is zero. This isachieved by choosing � � 0.271.3 The time-

3 Since convergence to the balanced growth path is slow,the model still allows for substantial population growth forlong time periods.


cost parameter � for having a child determinesthe overall opportunity cost of children. Evi-dence in Robert Haveman and Barbara Wolfe(1995) and John Knowles (1999) suggests thatthe opportunity cost of a child is equivalent toabout 15 percent of the parents’ time endow-ment. This cost only accrues as long as the childis living with the parents. If we assume thatchildren live with parents for 15 years and thatthe adult period lasts for 30 years, the overalltime cost should be 50 percent of the time costper year with the child present. Accordingly, wechoose � � 0.075. The parameter � also sets anupper limit on the number of children a personcan have. With our choice, a person spending alltime on raising children would have a littleabove 13 children. A family of two could havea little under 27 children.4

The parameter � influences the elasticity ofhuman capital with respect to education, as wellas the maximum fertility differential in theeconomy. Specifically, the maximum differen-tial written as a ratio is given by 1/(1 � �). Inthe data set that we use below, the highestfertility ratio between women at the lowest andthe highest education levels is 2.74 (Brazil).This differential is achieved by choosing � �0.635. Our choice of � guarantees that realizedfertility differentials in the model never exceedthe maximum differential observed in the data,which ensures that the role of the differential-fertility channel is not inflated. At the sametime, for evaluating the differential-fertilitychannel it is also important that the elasticity offuture human capital with respect to educationis calibrated realistically. In the model, this elas-ticity is determined jointly by � and �. Since �enters the education choice of parents, it deter-mines aggregate expenditures on education. Wechoose � such that in the balanced growth pathtotal education expenditure as a fraction of GDPmatches the corresponding value in U.S. data,which is 7.3 percent.5 The implied parametervalue is � � 0.0119. Our combined choices for� and � imply an elasticity of human capital

4 If we also modeled a goods cost of having children, theupper bound would be lower, and close to the maximumhuman fertility levels observed so far.

5 This figure (Digest of Education Statistics, 1998, U.S.Department of Education) does not include on-the-job train-ing, since it is not part of the parental investment in children.

with respect to education of 0.6 in the balancedgrowth path. This number is within the range ofestimates of the elasticity of earnings with re-spect to schooling. Specifically, estimates of thereturn to an additional year of schooling rangefrom 7.5 percent in Joshua D. Angrist and AlanB. Krueger (1991) to 12 to 16 percent in thestudy of twins by Orley Ashenfelter andKrueger (1994). The surveys by George Psacha-ropoulos (1994) and Krueger and Mikael Lin-dahl (2001) report estimates of the return toschooling in developed countries of 8–10 per-cent, with higher estimates for developing coun-tries and low levels of schooling. Assuming thatan additional year of schooling raises educationexpenditure by 20 percent, these returns trans-late into an earnings elasticity of schooling be-tween 0.4 and 0.8. The elasticity implied by ourparameter choices is exactly in the middle ofthis range.

The remaining parameters and do notinfluence individual decisions, but still have aneffect on growth rates. The elasticity of futurehuman capital with respect to average humancapital h� can be calibrated to evidence on theeffects of the quality of schooling. We interpretthe education choice et as the quantity ofschooling (corresponding to years of schoolingin the data) while h� measures the quality ofschooling, since it is the average human capitalof teachers. Compared to the quantity of school-ing, the quality of schooling (such as spendingper pupil at a given level of education) has beenshown to have smaller effects on earnings withan elasticity of around 0.1; see David Card andKrueger (1996) and Krueger and Lindahl(2001). In line with evidence on the effect of thequality of education, we set � 0.1. Alterna-tively, could also be interpreted as a measureof human capital externalities. Existing evi-dence [see Daron Acemoglu and Angrist (2001)and Krueger and Lindahl (2001)] suggests thatthese externalities are small as well, i.e., thesocial return to human capital accumulation isonly slightly larger than the private return, con-firming our low choice of . Our results arerobust with respect to the choice of this elastic-ity in the sense that matters only for thedetermination of the growth rate of averagehuman capital. Individual decisions and theevolution of inequality, fertility, and differentialfertility are independent of .


TABLE 1—INITIAL GROWTH WITH ENDOGENOUS AND EXOGENOUS FERTILITY

�2

Endogenous fertility Exogenous fertility

g0 N0 I0 D0 g0 N0 I0 D0

0.10 2.00 0.00 0.056 0.09 2.00 0 0.056 00.75 1.26 0.66 0.404 1.95 1.87 0 0.400 01.00 0.80 1.08 0.520 2.76 1.78 0 0.513 01.50 0.01 1.71 0.707 2.77 1.53 0 0.700 0

Notes: �2: Variance of income distribution. g0: Growth rate of human capital per worker. N0:Growth rate of population. I0: Income inequality (Gini coefficient). D0: Fertility differential.

variance of the underlying normal distribution.The parameter � is set such that ht is at itsbalanced growth level. We provide simulationsfor different variances of the distribution inorder to examine the effects of inequality. Theinitial level of physical capital K0 is chosensuch that the ratio of physical to human capitalis equal to its value in the balanced growthpath.6

B. Initial Inequality, Fertility, and Growth

As a first computational experiment, we ex-amine the effect of initial inequality on growthover the first period. Since a period is in fact ageneration, the growth rate should be inter-preted as a 30-year average. The main findingsare that inequality has a sizable effect ongrowth, and that most of this effect is accountedfor by the endogenous fertility differential.

Table 1 presents the initial annualized growthrates of human capital g0 and population N0 (inpercent), initial inequality I0, and the initialfertility differential D0 for different variances ofthe distribution of human capital. Inequality ismeasured by the Gini coefficient I0 computedon the earnings of the working population. Dif-ferential fertility is the difference between theaverage fertility of the top quintile and the bot-tom quintile; this quantity is then multiplied bytwo to yield a number per woman. To evaluatethe role of differential fertility in our model, wealso computed results under the assumption ofconstant, exogenous fertility.

The results in Table 1 show that inequalitylowers growth both with and without endoge-

6 The effect of inequality on growth is independent of �and K0; � and K0 only affect average growth rates.

The parameter determines the direct effectof parental human capital (or, equivalently, in-come) on the children’s human capital. Thus, captures the intergenerational transmission ofability, as well as human capital formationwithin the family that does not work throughformal schooling. Empirical studies detect sucheffects, but they are relatively small. Mark R.Rosenzweig and Kenneth I. Wolpin (1994) findthat an additional year of the mother’s educationat the high school level (roughly a 10-percentincrease in education) raises a child’s test scoresby 2.4 percent. Arleen Leibowitz (1974) findsthat even after controlling for schooling andeducation of the parents, parental income has asignificant effect on a child’s earnings. A 10-percent increase in parental income increases achild’s future earnings by up to 0.85 percent.Given that the long-run dynamics of the modelare sensitive to the choice of , we choose amoderate degree of intergenerational transmis-sion of human capital ( � 0.2) as the baselinecase, and provide a sensitivity analysis withrespect to alternative choices for . For individ-ual dynamics to be stable, must not exceed theupper bound from Proposition 2. Given ourchoices for the other parameters, the upper limitfor is 0.246, which is well above the calibratedvalue.

In addition to choosing parameters, we alsoneed to set the initial conditions for the simula-tions. The overall size of the population is ascale parameter which does not affect the re-sults, and is therefore set to one. Likewise, thedistribution of physical capital does not matter,since capital is owned by old people who havenothing left to decide. We therefore only spec-ify the aggregate value. The initial distributionof human capital follows a lognormal distribu-tion F(�, �2), where � and �2 are the mean and

7 The comparison to Barro’s result is complicated by thefact that Barro conditions his estimates on initial GDP,whereas in our simulations GDP is partly a function of theinitial income distribution. Thus, in principle, the simula-tions pick up the additional effect of changing initial GDP.In practice, however, this effect turns out to be negligible.Initial GDP varies only to the extent that child rearing timeis not included in GDP, and the resulting differences in GDPacross inequality levels are small (up to 0.3 percent). Wecomputed results with an additional adjustment in averagehuman capital that holds GDP per worker constant acrossinequality levels. The results are virtually indistinguishablefrom the ones reported in Figure 3.


nous fertility, but the effects are much largerwhen fertility is endogenous. When the varianceof the distribution of human capital is low (�2 �0.10), the difference between endogenous andexogenous fertility is small, and the growthrates are close to their values on the balancedgrowth path. When we increase the initial vari-ance to �2 � 0.75, substantial fertility differen-tials within the population begin to arise, andthe annual growth rate of human capital drops0.74 percent below the steady state. With con-stant exogenous fertility, the drop in the growthrate is six times smaller. Further increases in theinitial variance eventually lead to a negativegrowth rate (for �2 � 1.5) with endogenousfertility, while growth stays positive with exog-enous fertility.

The results are robust with respect to thechoice of . For example, with �2 � 0.75,initial growth with endogenous fertility is1.22 percent for � 0.05, and 1.32 percentfor � 0.3. With exogenous fertility, it is1.80 percent for � 0.05, and 1.94 percentfor � 0.3. We also carried out the samecomputations with a uniform instead of a log-normal distribution of initial human capital.We still found that growth declines muchfaster with inequality when fertility is endog-enous. For example, when the Gini index goesfrom 0 to 0.33, growth drops by 0.7 percentwith endogenous fertility and by 0.1 percentwith exogenous fertility.

The initial dispersion of human capital alsoinfluences the overall growth rate of population.When the variance of human capital rises, fer-tility of low-skilled households increases, whilehigh-skilled households decide to have fewerchildren. Because of the shape of the fertilityfunction (see Figure 1), the first effect domi-nates and aggregate fertility rises. This is in linewith empirical studies that report a high positivecorrelation between aggregate fertility rates andGini coefficients (see Barro, 2000).

Figure 3 depicts the growth rate of humancapital as a function of the Gini coefficient. Theslope is much steeper with endogenous fertilitythan with exogenous fertility. In the data, in-come Ginis for a country vary roughly in therange 0.2 to 0.65. In the model, raising theinitial Gini from 0.2 to 0.65 lowers the growthrate by only about 0.3 percent with exogenousfertility, but by 1.4 percent with endogenous

fertility. In a quantitative sense, fertility differ-entials within the population are essential forgenerating the relationship between inequalityand growth.

We have also represented the slope of theregression of economic growth on income Ginisrun by Barro (2000) when the fertility rate vari-able is omitted. In this regression, the Ginicoefficient captures the intrinsic effect of in-equality as well as the one going through fer-tility. Since actual Gini coefficients lie in theinterval [0.21, 0.64], the regression line hasbeen restricted to this interval. Our computa-tional experiment is consistent with the Barro(2000) finding: “A reduction of the Gini coef-ficient by 0.1 would be estimated to raise thegrowth rate on impact by 0.4 percent per year.”7

Perotti (1996) reports effects of similar magni-tude. Our calibrated model is therefore able toaccount for most of the empirical relationship

FIGURE 3. THE RELATIONSHIP OF INEQUALITY AND GROWTH

WITH ENDOGENOUS FERTILITY (SOLID LINE), EXOGENOUS

FERTILITY (DASHED LINES), AND IN BARRO’S REGRESSION

(DOTTED LINES)


between inequality and growth. Since the em-pirical estimates carry sizable standard errors,the finding does not rule out that other channelscould also play a role, but clearly the differential-fertility effect appears to be important.

C. The Dynamics of Inequality, DifferentialFertility, and Growth

We now turn to the dynamic implications ofour model. So far, we have only analyzed theeffects of inequality on growth during the initialperiod. Since in our dynastic model a period hasa length of 30 years, even the initial growtheffect extends over a long horizon, and conse-quently the dynamics of the model are to beinterpreted as changes which occur over a ho-rizon of a century or more. We therefore eval-uate the dynamic behavior of the model relativeto the evolution of income, fertility, and in-equality in industrializing countries in the last200 years. A central feature of the data for thisperiod is that the behavior of population growthand inequality is nonmonotone. As a benchmarkcase, consider England, the first country to in-dustrialize. Fertility rates increased until about1830 and started to decline rapidly only after1870 (Jean Claude Chesnais, 1992). Incomeinequality followed a similar pattern, with in-creasing inequality until about 1870 and a rapiddecline afterwards (Jeffrey G. Williamson,1985). The growth rate of income per capita, incontrast, does not display a hump shape.Growth rates were essentially zero before theindustrial revolution and then increased slowlythroughout the nineteenth century (Angus Mad-dison, 2001). Similar patterns can be observedfor Western Europe as a whole and, starting alittle later, in the United States.

To evaluate how our model performs relativeto these facts, we simulate the model with thebaseline calibration over a horizon of eight pe-riods, corresponding to 240 years. The mainfinding is that for plausible parameters, themodel can generate a hump shape in inequalityand the population growth rate that looks verysimilar to the data. This is a surprising finding,since unlike most existing theories the modelgenerates the pattern without requiring any ex-ogenous change to the economic environment.We also investigate the behavior of the model ifidiosyncratic shocks are added to the production

function for human capital, since for long-runanalysis the assumption that the only source ofinequality is the initial dispersion in humancapital is less attractive. We find that addingshocks slows growth, but leaves the qualitativebehavior of the model intact. Thus our modelturns out to be successful at accounting for boththe cross-sectional relationship between in-equality and growth, and the dynamic interac-tion between inequality, fertility, and growthover long time horizons.

Figure 4 shows the evolution of the growthrate of human capital, the population growthrate, inequality, and differential fertility for twodifferent values of . Growth rates are annual-ized. The initial distribution of human capital isassumed lognormal with �2 � 1, correspondingto an initial Gini coefficient of 0.5. With a low of 0.05, fertility, inequality, and differentialfertility converge monotonically to their steady-state values. Initially, inequality reduces growthbelow its balanced growth value for the reasonsexplained above, but subsequently the growthrate increases. Low initial growth implies thatthe capital stock increases more slowly than inthe balanced growth path. Since productivitygrowth is exogenous, the effective capital stockfalls, and the usual transitional dynamics inexogenous growth models result in higher sub-sequent growth. If is raised to 0.2, the modelgenerates the hump-shape patterns that charac-terize the data. Fertility and inequality first riseand then decrease, and the growth rate remainsbelow its balanced growth value for severalperiods as long as inequality remains high. Amoderate degree of intergenerational persis-tence in human capital is thus essential formatching the long-run evolution of inequalityand fertility to data.

The nonmonotone behavior of inequality andfertility is related to the corner solution foreducation. A fraction of the people in the firstperiod decides not to invest in education. Sincethis group has the highest fertility rates, theirchildren make up an even larger fraction of thepopulation in the next period. If there is a suf-ficient degree of intergenerational persistence,these children will be at the corner solution foreducation as well. This leads to an increase inpopulation growth in the first periods and slowgrowth of human capital, since investment ineducation is low. Because of exogenous tech-


FIGURE 4. GROWTH, FERTILITY, INEQUALITY, AND DIFFERENTIAL FERTILITY FOR DIFFERENT

small, i.e., we need some degree of intergen-erational persistence in human capital andearnings.8

The main disparity between the model andthe data is that in the data, growth rates wereslowly increasing throughout the nineteenthcentury, whereas the model produces a humpshape with first rapidly increasing and then fall-ing growth rates. This behavior of the model isgenerated by transitional dynamics due to theexogenous growth assumption. We thereforealso explore how the model performs if we

8 A unique equilibrium exists even if we increase above the bifurcation value of � 0.246, where we enterthe region of the parameter space in which the dynamics ofindividual human capital are no longer stable. The returns toparental human capital in the education function are notsufficiently decreasing to compensate the centrifugal forceof the quality-quantity trade-off. Computations of the dis-tribution of human capital after a large number of periodsindicate that an ever-decreasing share of the populationaccumulates an ever-increasing share of human capital. Themass of people with above-average human capital tends tozero, but the fraction of human capital accounted for bythem tends to one.

nological change and human capital externali-ties, however, after a few periods even thedynasties that initially did not invest in educa-tion find it optimal to start educating their chil-dren. From this point on, inequality andpopulation growth fall.

This nonmonotone behavior only occurs ifthe initial distribution of human capital issuch that some people choose not to invest ineducation. If everyone is above the thresholdinitially, convergence to the steady state ismonotone even for � 0.2. For parametersthat lead to an initial rise in fertility andinequality, the time paths for inequality,growth, and fertility look surprisingly similarto the patterns of development in WesternEurope between 1800 and 2000 described byGalor and David N. Weil (2000) and others.For example, let us assume that t � 1 in thegraphs corresponds to the period 1760 –1790in England. Then population growth peaksaround 1790 –1820, and inequality peaks 30years later. The timing of the subsequent fallis close to what is observed in the data. Forthe hump to occur, however, cannot be too

10 The ability shock is multiplicative and is drawn froma uniform distribution over [0.9, 1.1], which generates a


increase the human capital externality from �0.1 to � 1 � � 0.8 to allow for endogenousgrowth. As stressed above, individual decisionsand the evolution of inequality, fertility, anddifferential fertility are independent of the as-sumption on . However, the dynamic patternof the growth rate can be affected.9 Figure5 compares growth rates under the two differentregimes with U.K. data from Maddison (2001).While growth with � 0.1 increases quicklyand soon exceeds its balanced growth value,with � 1 � growth converges slowly andmonotonically to its long-run level. The patternunder the endogenous growth assumption iscloser to the observed pattern in England, wherethe average growth rate of 2 percent wasreached only towards the beginning of the twen-tieth century. Notice, however, that to generateendogenous growth we have to assume a muchhigher degree of externalities than suggested byempirical estimates in recent data. The resulttherefore indicates that either externalities playeda bigger role in the past, or that another endoge-nous growth mechanism was at work which gen-erated the slowly increasing growth rates.

So far, we have assumed that the only sourceof inequality is the initial dispersion in humancapital. As a consequence, inequality is transi-tory in the model and disappears in the balancedgrowth path. While this abstraction is of noconsequence for the initial growth effects, forlong-run dynamics idiosyncratic shocks wouldbe expected to be more important. To check therobustness of our results, we therefore extended

FIGURE 5. EXOGENOUS VERSUS ENDOGENOUS GROWTH

9 To calibrate the endogenous growth version of themodel, we keep at 0.2 and set � 0.8. The overallproductivity B in the production function for human capitalnow governs the growth rate of output per capita. We pickB � 0.367 which ensures a long-term growth rate of 2percent per year.

the model by introducing idiosyncratic abilityshocks in the production function for humancapital. With ability shocks, inequality persistseven in the long run.

The main outcome of our simulations with abil-ity shocks is that inequality decreases more slowlyand growth is reduced, which was to be expected,but that otherwise the qualitative features of thetransition remain intact. The quantitative impactof ability shocks is rather small as well, providedthat the shocks are calibrated to reproduce currentlevels of earnings inequality in the UnitedStates.10 Moreover, as can be seen in Figure 6, thedifference materializes only after several periods,since the ability shocks have little additional im-pact as long as the distribution of human capital iswidely dispersed.

In summary, we find that our relatively sim-ple model is surprisingly successful at reproduc-ing key features of the long-run evolution ofinequality, fertility, and growth. Under a mod-erate degree of intergenerational persistence inhuman capital (which could be generated bypersistence in innate ability), the model gener-ates a hump shape in inequality and fertility. Inorder to account for slowly increasing growthrates of income per capita, it is helpful to intro-duce an endogenous growth mechanism. Whilein our model endogenous growth is generatedby human capital externalities, we conjecturethat similar results could be obtained with othersources of endogenous growth.

Our results complement existing theories oflong-run growth. Relative to the literature, themain novelty is that we link the evolution ofgrowth and population to the income distri-bution. In contrast, the models developed byGalor and Weil (2000), Raouf Boucekkine etal. (2002), and Gary D. Hansen and EdwardC. Prescott (2002) abstract from distributionalissues.11 We find that allowing for inequality

long-term Gini coefficient of 0.35. We assume that theability shock is uncorrelated with parental human capital,since the parameter already captures intergenerationaltransmission of skills.

11 Doepke (2001) has a model of the industrial revolutionand the demographic transition which does allow for inequal-ity. However, since there are only two types of agents, theincome distribution has just two points. A hump in inequalityarises only if there are exogenous policy changes.


FIGURE 6. GROWTH AND INEQUALITY WITH AND WITHOUT ABILITY SHOCKS

evidence? The first part of our hypothesis,that income inequality leads to high fertilitydifferentials, has been analyzed by Kremerand Chen (2000). In line with our conjec-ture, they find that Gini coefficients have asignificant and sizable positive correlationwith fertility differentials. In this section weexamine the second part of our hypothe-sis, the link from fertility differentials togrowth. Our approach is to introduce adifferential-fertility variable into a standardgrowth regression. The analysis is designed tobe comparable to recent empirical studies ofinequality and growth. As predicted by ourmodel, we find that differential fertility has anegative effect on growth. Moreover, whenthe differential-fertility variable is present,the Gini variable is no longer significant inthe regression.

A. Data

Our sample contains 68 countries for whichdata on fertility differentials is available. Thedependent variable (GR) in all regressions is theaverage annual growth rate of GDP per capitaover the periods 1960 to 1976 or 1976 to 1992(the period depends on the availability of fertil-ity data). The GDP data is from the Penn WorldTables, and growth rates are continuously com-pounded and expressed as percentages.12 Sincewe are interested in long-run growth, we chose

12 For countries where data from 1960 and/or 1992 wasnot available, we computed growth rates over the closestavailable interval.

in human capital combined with endogenousfertility and education choice generates real-istic predictions for inequality, fertility, andgrowth in a simple and natural way. Galor andWeil (2000) generate a hump in fertility byintroducing a subsistence level of consump-tion, but the evolution of the income distri-bution is not explained. In Hansen andPrescott (2002) fertility is exogenous, so nei-ther the hump in fertility nor in inequality areaccounted for. A limitation of our approach isthat we take the initial conditions at the startof the industrial revolution as given. For a fullaccount of the evolution of the economy frompre-industrial stagnation to modern growthwe would have to add an element to the modelthat generates the initial stagnation phase. Wesuspect that this could be done along the linesof Galor and Weil (2000) or Hansen andPrescott (2002), but the extension is beyondthe scope of the current paper.

A main implication of our dynamic analysisfor the inequality-growth relationship is thatthe relationship can be modified by transi-tional dynamics. Since inequality first in-creases and then decreases during thetransition, inequality does not map one-to-oneinto growth rates. In the empirical analysis, itis therefore important to control for transi-tional dynamics to isolate the role of thedifferential-fertility channel.

IV. Empirical Evidence

Can the relationships between inequality,differential fertility, and growth postulatedby our model be supported by empirical


TABLE 2—DESCRIPTIVE STATISTICS

Sample Observations Variable MeanStandarddeviation Minimum Maximum

1960–1976 40 GR 1.95 3.65 �5.75 8.44GINI 44.32 11.14 23.38 68.00TFR 5.56 1.89 2.02 7.93DTFR 2.23 1.56 0.22 5.30

1976–1992 43 GR 0.39 1.89 �3.46 4.97GINI 45.91 9.56 28.90 69.00TFR 6.06 1.08 3.37 8.00DTFR 2.41 0.99 0.10 4.50

Total 83 GR 1.14 2.97 �5.75 8.44GINI 45.14 10.32 23.38 69.00TFR 5.82 1.54 2.02 8.00DTFR 2.32 1.29 0.10 5.30

Notes: GR: Annualized growth rate of income per capita in percent. GINI: Initial incomeinequality. TFR: Total fertility rate. DTFR: Fertility differentials. Detailed definitions and datasources are in the main text.

countries (AFR), and the initial total fertilityrate (TFR). I/GDP and G/GDP are from thePenn World Tables, the income Ginis are fromKlaus Deininger and Lyn Squire (1996), andtotal fertility rates and life expectancy (which isused as an instrument) are from the Barro-Leedata set.13 The inclusion of initial GDP and theinvestment ratio is important to control for tran-sitional dynamics.

One shortcoming of the data set is that theobservations on fertility differentials are closeto the end of the period over which we computegrowth rates. Since the fertility observations arefive-year averages and result from decisions andactions taken even earlier, the endogeneityproblem is not too severe. We correct for po-tential endogeneity of the differentials by usinginstrumental variables.

Table 2 provides descriptive statistics forthe main variables in our analysis. The twosubsamples are similar, except that the aver-age growth rate is much lower in the secondsample. Since we will allow for different con-

13 Where possible, the Gini coefficients are from theinitial year; otherwise we used the closest available year.For a few countries, no data is available in Deininger andSquire (1996). For Benin, Burundi, Central Africa, andNamibia, we relied on the Economic Report on Africa 1999:The Challenges of Poverty Reduction and Sustainability,United Nations (2000). For Haiti and Syria, only land Ginisfrom Idriss Jazairy et al. (1992) are available. We regressedthe available income Ginis on the land Ginis (correlation:0.61) and used the predicted values.

the longest subsample periods available in thePenn World Tables.

Following Kremer and Chen (2000), forfertility differentials we rely on informationfrom the World Fertility Survey and the De-mographic and Health Surveys on total fertil-ity rates by women’ s educational attainment[see Elise F. Jones (1982); United Nations(1987, 1995); Gora Mboup and Tulshi Saha(1998)]. For countries that participated in theWorld Fertility Survey, the independent vari-able is growth in GDP per capita in the firstperiod, and for countries that participated inthe Demographic and Health Surveys the left-hand-side variable is growth over the secondperiod. Our differential fertility variableDTFR is the difference in the total fertilityrate between women with the highest and thelowest education level. For some countries wehave two observations from the Demographicand Health Surveys, in which case we aver-aged the two resulting values. For 25 coun-tries we have only observations for the firstperiod, for 28 countries we have only obser-vations for the second period, and for 15countries there are observations from bothperiods.

The remaining independent variables are ini-tial GDP per capita (GDP), the average ratio ofinvestment to GDP (I/GDP), the average ratioof government expenditure to GDP (G/GDP),the initial Gini coefficient for the income distri-bution (GINI), a dummy variable for African


TABLE 3—GENERALIZED METHOD OF MOMENTS ESTIMATION

Independentvariable

Regression

(1) (2) (3) (4)

Constant A 12.35** (1.31) 12.79** (1.33) 15.30** (1.46) 13.92** (1.69)Constant B 10.41** (1.36) 10.98** (1.38) 13.40** (1.45) 12.18** (1.63)ln(GDP) �1.33** (0.17) �1.21** (0.16) �1.37** (0.15) �1.55** (0.20)I/GDP 0.14** (0.02) 0.13** (0.02) 0.07** (0.03) 0.08** (0.04)G/GDP �0.08** (0.03) �0.07** (0.03) �0.05* (0.03) �0.05* (0.03)AFR �1.75** (0.35) �1.80** (0.35) �1.95** (0.32) �2.41** (0.44)GINI �0.03** (0.01) 0.02 (0.03) 0.06 (0.05)ln(TFR) �1.84** (0.87) �1.01 (1.01)ln(DTFR) �1.22** (0.50)

J-test 17.71 [0.48] 17.11 [0.45] 16.79 [0.40] 9.58 [0.85]LR1 5.53 [0.01]LR2 2.08 [0.35]

Notes: The variables are described in more detail in the text. The dependent variable is the growth rate of real per capita GDP.Estimation by GMM. The instruments are: constant, log of initial GDP per capita, log of initial GDP per capita squared, initialinvestment/GDP ratio, initial government spending/GDP ratio, initial fertility, initial fertility squared, initial life expectancy,initial life expectancy squared, Africa dummy, and the tropics and access to the sea variables of Jeffrey Sachs and AndrewWarner (1997). Heteroskedasticity-consistent standard errors are reported in parentheses. J-test is the test for overidentifyingrestrictions of Lars P. Hansen (1982), asymptotically �2 distributed with n degrees of freedom, where n is the number ofoveridentifying restrictions. Corresponding p-values are reported in brackets. LR1 is a quasi-likelihood ratio test for theabsence of the differential fertility in the equation. LR2 is the test for the absence of both Gini and total fertility. The statisticsare computed as the normalized difference between the constrained and the unconstrained objective function, where theweighting matrix is provided by the unconstrained estimation (see Donald Gallant, 1987). The corresponding p-values arereported in brackets.

* Significant at the 10-percent level.** Significant at the 5-percent level.

reproduces standard results in the growth re-gression literature: the investment rate has apositive effect, whereas the government share,initial GDP, and the African dummy have neg-ative effects on growth. Regression (2) adds theGini coefficient. The estimated coefficient issignificantly negative, and its value is close toBarro’s estimate. The value of the parameterimplies that an increase in the Gini of 0.4(roughly the range of variation in the data)lowers growth by about 1.2 percent per year.However, when the total fertility rate is in-cluded [regression (3)], the coefficient on theGini coefficient changes sign and becomes in-significant, which is also in line with Barro(2000).

Regression (4) includes the differential-fertilityvariable. The coefficient on differential fertilityis significantly negative. The point estimate im-plies that an increase in the fertility differentialfrom one to two would lower growth by 0.8percent per year. With differential fertility in-cluded, the coefficients on both Gini and TFR

stant terms in the two subsamples, this differ-ence will not play a role in the results. Thesewill thus reflect cross-sectional differencesamong countries, as well as variation overtime within countries.

B. Estimation Results

Table 3 contains our estimation results. In allcases, the left-hand-side variable is the averagegrowth rate of GDP per capita. The regressionequation is estimated with the GeneralizedMethod of Moments (GMM). For countries thatare present in both sample periods, we allow theerror term to be correlated across the periods,and we use instrumental variables to correct forpossible endogeneity of I/GDP, G/GDP, GINI,and DTFR. We allow the constant to differacross the periods (Constant A for the earlyperiod and Constant B for the late period). Ourregressions are designed to be comparable toBarro (2000), but we include fewer variablebecause of the small sample size. Regression (1)


TABLE 4—GMM ESTIMATION WITH SQUARED GDP AND CROSS EFFECTS

Independentvariable

Regression

(1) (2) (3) (4)

Constant A 9.88 (10.1) �6.70 (17.5) �3.31 (18.8) 6.72 (19.9)Constant B 7.95 (10.1) �8.48 (17.4) �4.77 (18.6) 479 (19.7)ln(GDP) �0.65 (2.73) 4.65 (4.12) 2.37 (4.09) 1.39 (4.26)ln(GDP)2 �0.04 (0.18) �0.40 (0.24) �0.15 (0.20) �0.26 (0.22)I/GDP 0.14** (0.02) 0.10** (0.02) 0.07* (0.04) 0.08** (0.04)G/GDP �0.08** (0.03) �0.08** (0.03) �0.07** (0.04) �0.07** (0.03)AFR �1.73** (0.36) �1.64** (0.46) �2.23** (0.43) �2.10** (0.52)GINI �0.13 (0.22) 0.31 (0.27) �0.18 (0.37)ln(GDP)*GINI/100 �0.01 (0.03) �0.04 (0.04) 0.03 (0.05)ln(TFR) �2.20* (1.13) 0.14 (1.51)ln(DTFR) �1.56** (0.68)

J-test 17.63 [0.41] 13.87 [0.54] 16.58 [0.28] 8.38 [0.82]LR1 5.54 [0.02]LR2 0.52 [0.77]

Notes: The variables are described in more detail in the text. The dependent variable is the growth rate of real per capita GDP.Estimation by GMM. Instruments and test statistics as in Table 3. Heteroskedasticity-consistent standard errors are reportedin parentheses.

* Significant at the 10-percent level.** Significant at the 5-percent level.

are insignificant, and the point estimate on theGini is positive.

Based on the results in Section III, our modelpredicts that Gini, total fertility, and differentialfertility should all be equally negatively relatedto growth. It is therefore not clear why thecoefficients on the Gini and the total fertilityrate become insignificant once differential fer-tility is introduced. One possibility is that in-equality and total fertility are influenced byother factors which do not affect growth, whiledifferential fertility is observed with less noise.A second possibility is that total fertility andinequality have other effects on growth, whichare not present in our model and do not workthrough differential fertility. If some of theseeffects on growth are positive and thereforeoffset the negative effects, it would be plausiblethat the overall effect of total fertility and theGini becomes insignificant once the differential-fertility channel is controlled for.

Hansen’ s J-test measures how close theresiduals are to being orthogonal to the in-strument set. It can be seen as a global spec-ification test. The degrees of freedom equalthe number of restrictions imposed by theorthogonality conditions. These restrictionsare never rejected at the 5-percent level.

Moreover, there is a large improvement in thevalue of the test when differential fertility isintroduced. The significance of the differen-tial fertility variable is verified both by itst-statistic and by the quasi-likelihood ratiotest LR1. The test LR2 of joint insignificanceof GINI and TFR is not rejected.

Barro (2000) argues that there are importantnonlinear effects that relate the levels of devel-opment and inequality to growth rates. Heshows that unless these nonlinear effects areaddressed, the effects of inequality and fertilityon growth are difficult to distinguish. To checkwhether our findings are robust with respect tothe inclusion of nonlinear terms, we add asquared term for GDP per capita and an inter-action term involving GDP and inequality to theexplanatory variables. Table 4 presents the re-sults. The new terms are never significant anddo not affect the significance of the J-tests.Given our relatively small number of observa-tions, the inclusion of additional variables low-ers the significance of the other variables. Inparticular, the Gini coefficient is now neversignificant, but total fertility is significant inregression (3). Our main conclusion remains:when differential fertility is added in regression(4), it is significant, while total fertility is not.


Differential fertility again improves the value ofthe J-test.14

In summary, we find that standard growthregressions detect the effect of differentialfertility on growth postulated by our model.The effects implied by the regressions aresizable. At the same time, including differen-tial fertility leaves the direct effect of the Ginicoefficient insignificant, with a positive pointestimate.

V. Conclusion

Most of the theoretical literature on inequal-ity and growth has concentrated on channelswhere inequality affects growth through the ac-cumulation of physical capital. In this paper wepropose a different mechanism which links in-equality and growth through differential fertilityand the accumulation of human capital. In ourmodel, families with less human capital decideto have more children and invest less in educa-tion. When income inequality is high, largefertility differentials lower the growth rate ofaverage human capital, since poor families whoinvest little in education make up a large frac-tion of the population in the next generation. Acalibration exercise shows that these effects canbe fairly large. In the benchmark case, raisingthe Gini from 0.2 to 0.65 lowers the initialannual growth rate by 1.4 percent. We alsoexamine the role of differential fertility in thegrowth-regression framework used, among oth-ers, by Perotti (1996) and Barro (2000). In linewith the predictions of the theory, we find siz-able negative effects of differential fertility ongrowth. Both the empirical results and the quan-titative analysis of the model suggest that thedifferential-fertility channel is important for ac-

14 As an additional test of the robustness of our results,we also carried out regressions with initial life expectancyas an explanatory variable. Growth regressions generallyshow life expectancy to be more closely related to growththan other demographic variables such as the total fertilityrate. However, even if we add life expectancy to regression(4) in Table 3, differential fertility continues to have asignificantly negative effect on growth, albeit with a lowerpoint estimate of �0.79.

counting for the cross-sectional relationship be-tween inequality and growth.

We also examine the time-series implicationsof our model for the joint evolution of inequal-ity, fertility, and growth. Since in our overlap-ping generations model a period is onegeneration, the dynamics of the model are to beinterpreted as changes which occur over a ho-rizon of a century or more. Here we find that themodel is able to explain key features of theevolution of income, fertility, and inequality inindustrializing countries in the last 200 years.Specifically, the model generates an initial in-crease and ultimate decline in inequality andfertility. The same pattern has been observed inmany industrializing countries in the nineteenthand early twentieth centuries. In addition, if wespecify the model to allow for endogenousgrowth, the model generates steadily increasinggrowth rates throughout the transition, which isanother stylized feature of the data.

Our analysis provides a new perspective onthe link between economic growth and popula-tion growth. Existing studies have found littlecorrelation between the growth rates of popula-tion and output per capita (see Allen C. Kelleyand Robert M. Schmidt, 1999), which has ledsome researchers to conclude that populationdoes not matter for growth. The results in thispaper suggest that it is not overall populationgrowth, but the distribution of fertility withinthe population which is important. In otherwords, who is having the children matters morethan how many children there are overall.

A natural direction for further research con-cerns the policy implications of our model.Since differential fertility rather than inequalityper se is the main source of growth effects, it isnot clear that redistributional policies wouldincrease economic growth. Indeed, a typicaloutcome in models with endogenous fertility isthat income redistribution tends to increase fer-tility differentials (see Knowles, 1999), whichwould lower the growth rate. Here the policyimplications of our model are in stark contrastto other theories linking inequality and growth.Compared to income redistribution, policiesaimed at equalizing access to education wouldbe more effective.


APPENDIX A: PROOF OF PROPOSITION 1

Case � 1 � : The constant values gt � B(�(� � � )/(1 � �))�, Nt � nt � N*, xt � 1 �xt � 1, and:

kt � 1 � kt � �A��1 � � � ��1 � �� 1 � �

B��1 � ��1 � ��1 � � � 1/�1 � ��

solve equations (16), (17), (18), (20), (21), and (23).

Case 1 � : The constant values gt � 1 � �, Nt � nt � N*, xt � 1 � xt � 1,

h � � B

1 � � ��

1 � � � �� 1/�1 � � �

,

and:

kt � 1 � kt � �� 1 � � � ��A�1 � ��

�1 � ��1 � ��1 � �� 1/�1 � ��

solve equations (16), (17), (18), (20), (21), and (23).

APPENDIX B: ANALYSIS OF THE INDIVIDUAL TRANSITION FUNCTION

In order to understand the dynamics of the model, we examine the function xt � 1 � xt � ( xt;), i.e., the change in xt as a function of xt and :

�x; � �Bx

g* �� max�0,��x � �

1 � � ��

�h*� � � 1 � x.

Here g* and h* are the balanced growth path values. In the case � 1 � we replace g* byB(�(� � � )/(1 � �))�. In the case 1 � , we replace g* by 1 � � and h* by {[B/(1 ��)][�(� � � )/(1 � �)]�}1/(1 � � ). Both substitutions lead to the same expression, i.e., equation(24) of the main text. The expression can be rewritten as:

�x; � � x� ��1 � ��

�� max�0,

��x � �

��

� x.

We first consider the limits of this function. We have: (0; ) � 0 and x(0; ) � ��, and:

limx3�

�x; � � x � ��

� � ��

� x,

which implies:

limx3�

�x; � � �� if � � � 1 and limx3�

�x; � � �� otherwise.

Hence the function starts from (0, 0) with an infinite slope and goes either to �� or ��depending on parameter values.


By definition of the aggregate balanced growth path, x � 1 is a steady state and thus (1; ) �0. This steady state is locally stable if and only if x(1; ) � 0, i.e.,

��

� � �� 1 � 0.

At the point:

� 1 ��

� � �

the dynamics of individual capital described by: xt � 1 � xt � ( xt; ) undergo a transcriticalbifurcation, as proved in Proposition 2. There are thus two steady-state equilibria, 1 and x� , near (1,) for each value of smaller or larger than . The equilibrium 1 (resp. x� ) is stable (resp. unstable)for � and unstable (resp. stable) for � .

Another point of interest is xt � �

��. If, at this point, the function is negative, it crosses

the horizontal axes between 0 and �

��. The existence of this steady state results from the infinite slope

of at 0 and from its continuity; uniqueness results from the concavity of the function in the interval(0, �

��). We evaluate:

� �

��, � � � ��1 � ��

��

��

��

��.

This is negative if is above a threshold �:

� ��1 � ��ln��/�� ln � � � ln��1 � ��/��

ln � � ln��.

We are now able to fully characterize the dynamics of x as a function of the parameter . Thebifurcation diagram is presented in Figure 2.

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