Population, Technology, and Growth: From Malthusian ...cwyuen/seminar/papers/Galor-Weil (AER00...Method in Macroeconomics Conference, Paris; and the National Bureau of Economic Research

Population, Technology, and Growth: From MalthusianStagnation to the Demographic Transition and Beyond

By ODED GALOR AND DAVID N. WEIL*

This paper develops a unified growth model that captures the historical evolution ofpopulation, technology, and output. It encompasses the endogenous transition betweenthree regimes that have characterized economic development. The economy evolvesfrom a Malthusian regime, where technological progress is slow and population growthprevents any sustained rise in income per capita, into a Post-Malthusian regime, wheretechnological progress rises and population growth absorbs only part of output growth.Ultimately, a demographic transition reverses the positive relationship between incomeand population growth, and the economy enters a Modern Growth regime with reducedpopulation growth and sustained income growth.(JEL J13, O11, O33, O40)

This paper analyzes the historical evolutionof the relationship between population growth,technological change, and the standard of liv-ing. It develops a unified model that encom-passes the transition between three distinctregimes that have characterized the process ofeconomic development: the “Malthusian Re-gime,” the “Post-Malthusian Regime,” and the“Modern Growth Regime.” We view the unified

modeling of this long transition process, fromthousands of years of Malthusian stagnationthrough the demographic transition to moderngrowth, as one of the most significant researchchallenges facing economists interested ingrowth and development.

The analysis focuses on the two most impor-tant differences between these regimes from amacroeconomic viewpoint: first, in the behaviorof income per capita; and second, in the rela-tionship between the level of income per capitaand the growth rate of population.

The Modern Growth Regime is characterizedby steady growth in both income per capita andthe level of technology. In this regime there is anegative relationship between the level of out-put and the growth rate of population: the high-est rates of population growth are found in thepoorest countries, and many rich countries havepopulation growth rates near zero.

At the other end of the spectrum is theMalthusian Regime in which technologicalprogress and population growth were glacial bymodern standards, and income per capita wasroughly constant. Further, the relationship be-tween income per capita and population growthwas the opposite of that which exists in theModern Growth Regime: “The most decisivemark of the prosperity of any country,” ob-served Adam Smith (1776), “is the increase inthe number of its inhabitants.”

The Post-Malthusian Regime, which oc-curred between the Malthusian and Modern

* Galor: Department of Economics, Brown University,Providence, RI 02912, and Department of Economics, HebrewUniversity, Jerusalem 91905, Israel (e-mail: [email protected]); Weil: Department of Economics, Brown Univer-sity, Providence, RI 02912 (e-mail: [email protected]).The authors thank the referees and seminar participants atBar-Ilan University; Boston College; Boston University; theBoard of Governors of the Federal Reserve; Brandeis, Brown,Columbia, Cambridge, Harvard, and Hebrew Universities; theInstitute of International Economic Studies, Stockholm Uni-versity; the London School of Economics; the University ofMichigan; Michigan State University; the Federal ReserveBank of Minneapolis; the Massachusetts Institute of Technol-ogy; Pennsylvania State University; the Universities of Roch-ester and Rome; Tel Aviv University; the University ofToulouse; University College, London; Uppsala and Yale Uni-versities; the 1997 European Science Foundation Growth con-ference, Castelvecchio Pascoli, Italy; the 1998 Aiyagari’sMemorial Conference, Haifa, Israel; the 1999 Technology,Human Capital, and Growth conference, Hebrew University,Jerusalem; the 1999 Dynamics, International Trade, andGrowth conference, Tilburg University; the 2000 Theories andMethod in Macroeconomics Conference, Paris; and theNational Bureau of Economic Research Summer Institute,Cambridge, 1999, for useful comments. Galor’s research issupported by National Science Foundation Grant No. SBR-9709941.

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Growth Regimes, shared one characteristic witheach of them. Income per capita grew duringthis period, although not as rapidly as it wouldduring the Modern Growth Regime. At thesame time, the Malthusian relationship betweenincome per capita and population growth wasstill in place. Rising income was reflected inrising population growth rates.

The most basic description of the relation be-tween population growth and income was pro-posed by Thomas R. Malthus (1798). TheMalthusian model has two key components. Thefirst is the existence of some factor of production,such as land, which is in fixed supply, implyingdecreasing returns to scale for all other factors.The second is a positive effect of the standard ofliving on the growth rate of population.

According to Malthus, when population sizeis small, the standard of living will be high, andpopulation will grow as a natural result of pas-sion between the sexes. When population size islarge, the standard of living will be low, andpopulation will be reduced by either the “pre-ventive check” (intentional reduction of fertil-ity) or by the “positive check” (malnutrition,disease, and famine).

The Malthusian model implies that, in theabsence of changes in technology or in theavailability of land, the size of the populationwill be self-equilibrating. Further, increases inavailable resources will, in the long run, beoffset by increases in the size of the population.Countries with superior technology will havedenser populations, but the standard of livingwill not be related to the level of technology,either over time or across countries.

The Malthusian model’s predictions are con-sistent with the evolution of technology, popu-lation, and output per capita for most of humanhistory. For thousand of years, the standard ofliving was roughly constant and did not differgreatly across countries. As depicted in Figure1, Angus Maddison (1982) estimates that thegrowth rate of GDP per capita in Europe be-tween 500 and 1500 was zero. Furthermore,Ronald D. Lee (1980) reports that the real wagein England was roughly the same in 1800 as ithad been in 1300. According to Kang Chao’s(1986) analysis, real wages in China were lowerat the end of the eighteenth century than theyhad been at the beginning of the first century.Joel Mokyr (1990), Lant Pritchett (1997), and

Robert E. Lucas, Jr. (1999) argue that even inthe richest countries, the phenomenon of sus-tained growth in living standards is only a fewcenturies old.

Similarly, the pattern of population growth isconsistent with the predictions of the Malthu-sian model. Population growth was nearly zero,reflecting the slow pace of technologicalprogress. As depicted in Figure 1, the rate ofpopulation growth in Europe between the years500 and 1500 was 0.1 percent per year. Further-more, Massimo Livi-Bacci (1997) estimates thegrowth rate of world population from the year 1to 1750 at 0.064 percent per year.

Fluctuations in population and wages alsobear out the predictions of the Malthusianmodel. Lee (1997) reports positive income elas-ticities of fertility and negative income elastic-ities of mortality from studies examining a widerange of preindustrial countries. Similarly, Ed-ward A. Wrigley and Roger S. Schofield (1981)find that there was a strong positive correlationbetween real wages and marriage rates in En-gland over the period 1551–1801. Negativeshocks to population, such as the Black Death,were reflected in higher real wages and fasterpopulation growth (Livi-Bacci, 1997).

Finally, the prediction of the Malthusian modelthat differences in technology should be reflectedin population density but not in standards of livingis also borne out. As argued by Richard Easterlin(1981), Pritchett (1997), and Lucas (1999), priorto 1800 differences in standards of living amongcountries were quite small by today’s standards;yet there did exist wide differences in technology.China’s sophisticated agricultural technologies,for example, allowed high per-acre yields, butfailed to raise the standard of living above subsis-tence. Similarly in Ireland a new productivetechnology—the potato—allowed a large increasein population over the century prior to the GreatFamine without any improvement in standards ofliving (Livi-Bacci, 1997). Using this interpreta-tion, Michael Kremer (1993) argues that changesin the size of population can be taken as a directmeasure of technological improvement.

Ironically, it was only shortly before the timethat Malthus wrote that humanity began toemerge from the trap that he described. Asis apparent from Figure 1 the process ofemergence from the Malthusian trap was a slowone. The figure shows the growth rate of total

807VOL. 90 NO. 4 GALOR AND WEIL: POPULATION, TECHNOLOGY, AND GROWTH

output in Western Europe between the years500 and 1990, as well as the breakdown be-tween growth of output per capita and growth ofpopulation. The growth rate of total output inEurope was 0.3 percent per year between 1500and 1700, and 0.6 percent per year between1700 and 1820. In both periods, two-thirds ofthe increase in total output was matched byincreased population growth, so that the growthof income per capita was only 0.1 percent peryear in the earlier period and 0.2 percent in thelater one. In the United Kingdom, where growthwas the fastest, the same rough division be-tween total output growth and populationgrowth can be observed: total output grew at anannual rate of 1.1 percent in the 120 years after1700, whereas population grew at an annual rateof 0.7 percent over that period.

Thus the initial effect of faster income growthin Europe was to increase population. Incomeper capita rose much more slowly than did totaloutput, and as income per capita rose, popula-tion grew ever more quickly. Only the fact thatoutput growth accelerated allowed income per

capita to continue rising. During this Post-Malthusian Regime, the Malthusian mechanismlinking higher income to higher populationgrowth continued to function, but the effect ofhigher population on diluting resources per cap-ita, and thus lowering income per capita, wascounteracted by technological progress, whichallowed income to keep rising.

Both population and income per capita con-tinued to grow after 1820, but increasingly thegrowth of total output was expressed as growthof income per capita. Indeed, whereas the rateof total output growth increased, the rate ofgrowth of population peaked in the nineteenthcentury and then began to fall. Populationgrowth was 40 percent as large as total outputgrowth over the period 1820–1870, but only 20percent as large as total output growth over theperiod 1929–1990. Over the next several de-cades much of Western Europe is forecast tohave negative population growth.

The dynamics of population growth reflectedboth changes in constraints and qualitativechanges in household behavior induced by the

FIGURE 1. OUTPUT GROWTH IN WESTERN EUROPE, 500–1990

Notes:Data from 500–1820 are from Angus Maddison (1982) and apply to Europe as a whole. Data for 1820–1990 are fromMaddison (1995), Table G, and apply to Western Europe.

808 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2000

economic environment. The Malthusian demo-graphic regime had been characterized by highlevels of both fertility and mortality. As livingstandards rose, mortality fell. Between the1740’s and the 1840’s, life expectancy at birthrose from 33 to 40 in England and from 25 to 40in France (Livi-Bacci, 1997). Robert Fogel(1997) estimates that 85 percent of the declinein mortality in France between 1785 and 1870was simply the result of better nutrition. Mor-tality reductions led to growth of the populationboth because more children reached breedingage and because each person lived for a greaternumber of years.

The initial effect of higher income was also toraise fertility directly, primarily by raising thepropensity to marry. Fertility rates increased inmost of Western Europe until the second half ofthe nineteenth century, peaking in England andWales in 1871 and in Germany in 1875 (TimDyson and Mike Murphy, 1985; Ansley J.Coale and Roy Treadway, 1986). Thus, inMalthusian terms, the positive check was beingweakened and the preventive check was beingless assiduously enforced. But as income con-tinued to rise, population growth fell furtherbelow the maximum rate that could be sustainedgiven the mortality regime. The reduction infertility was most rapid in Europe around theturn of the twentieth century. In England, forexample, live births per 1,000 women aged15–44 fell from 153.6 in 1871–1880 to 109.0 in1901–1910 (Wrigley, 1969). Notably, the rever-sal of the Malthusian relation between incomeand population growth corresponded to an in-crease in the level of resources invested in eachchild. For example, the average number of yearsof schooling in England and Wales rose from2.3 for the cohort born between 1801 and 1805to 5.2 for the cohort born 1852–1856 and 9.1 forthe cohort born 1897–1906 (Robert C. O. Mat-thews et al., 1982).

This historical evidence suggests that the keyevent that separates the Malthusian and Post-Malthusian Regimes is the acceleration in thepace of technological progress, whereas theevent that separates the Post-Malthusian andModern Growth eras is the demographic transi-tion that followed the industrial revolution. Theemergence from the Malthusian trap and theonset of the demographic transition raise in-triguing questions. How was the link between

income per capita and population growth, whichhad for so long been a constant of humanexistence, so dramatically severed? How doesone account for the sudden spurt in growthrates? Is there a unified framework of analysisthat can account for this intricate evolution ofpopulation, technology, and growth throughouthuman history?

Neoclassical growth models with exogenouspopulation clearly are unable to capture this intri-cate transition process. Further, the existing liter-ature on the relation between population growthand output has tended to focus on only one of theregimes described earlier. The majority of theliterature has been oriented toward the modernregime, trying to explain the negative relationbetween income and population growth, eithercross-sectionally or within a single country overtime (e.g., Robert J. Barro and Gary S. Becker,1989). Among the mechanisms highlighted in thisliterature are that higher returns to child quality indeveloped economies induce a substitution ofquality for quantity (Becker et al., 1990); thatdeveloped economies pay higher relative wages ofwomen, thus raising the opportunity cost of chil-dren (Galor and Weil, 1996); and that the net flowof transfers from parents to children grows (andpossibly switches from negative to positive) ascountries develop (John W. Caldwell, 1976).1 Thenegative effect of high income on fertility is oftenexamined in conjunction with a model in whichhigh fertility has a negative effect on income as aresult of capital dilution. Recent papers that areconcerned with the Malthusian Regime are Kre-mer (1993) and Lucas (1999). Lucas presents aMalthusian model in which households optimizeover fertility and consumption, whereas in Kremer(1993) a feedback loop between technology andpopulation generates a transition from the proxim-ity of a Malthusian equilibrium to the Post-Malthusian Regime.2

1 See Isaac Ehrlich and Fracis Lui (1997), James A.Robinson and T. N. Srinivasan (1997), and T. Paul Schultz(1997) for surveys of the literature in this area, and RichardR. Nelson (1956) and Momi Dahan and Daniel Tsiddon(1998) for an alternative mechanism.

2 To generate a demographic transition, Kremer assumesthat population growth increases with income at low levelsof income and then decreases with income at high levels ofincome. Another strand of literature (Marvin Goodfriendand John McDermott, 1995; Daron Acemoglu and FabrizioZilibotti, 1997) has attempted to model the acceleration of


This paper accounts for the transition from theMalthusian Regime, through the Post-MalthusianRegime and the demographic transition, to theModern Growth Regime in a unified model. Atthe heart of our model is a novel explanation forthe reduction in fertility that has allowed incomeper capita to rise so far above subsistence. Moststudies of the demographic transition focus on theeffect of a highlevelof income in inducing parentsto switch to having fewer, higher-quality children.In our model, parents also switch out of quantityand into quality, but do so not in response to thelevel of income but rather in response to techno-logical progress. The “disequilibrium” broughtabout by technological change raises the rate ofreturn to human capital, and thus induces the sub-stitution of quality for quantity.

The argument that technological progress itselfraises the return to human capital was most clearlystated by Theodore W. Schultz (1964). Examiningagriculture, Schultz argued that when productivetechnology has been constant for a long period oftime, farmers will have learned to use their re-sources efficiently. Children will acquire knowl-edge of how to deal with this environment directlyfrom observing their parents, and formal school-ing will have little economic value. But whentechnology is changing rapidly, the knowledgegained from observing the previous generationwill be less valuable and the trial-and-error pro-cess, which led to a high degree of efficiencyunder static conditions, will not have had time tofunction. New technology will create a demandfor the ability to analyze and evaluate new pro-duction possibilities, which will raise the return toeducation.3 Such an effect would be a naturalexplanation for the dramatic rise in schooling inEurope over the course of the nineteenth century.

The effect of technology on the return tohuman capital in which we are most interestedis the short-run impact of a new technology. In

the long run, technologies may be “skill biased”or “skill saving.” But we would argue that theintroduction of new technologies is mostly skillbiased.4 If technological changes are skill bi-ased in the long run, then the effect on which wefocus will be enhanced, whereas if technologyis skill saving it will be diluted.

The second piece of the model is morestraightforward: the choice of parents regardingthe education level of their children affects thespeed of technological progress. Children withhigh levels of human capital are, in turn, morelikely to advance the technological frontier or toadopt advanced technologies.5

The third piece of the model links the size ofthe population to the rate of technologicalprogress and to the take-off from the MalthusianRegime. Holding the level of education con-stant, the speed of technological progress is alsoa positive function of the overall size of thepopulation. For a given level of education,higher population generates a larger supply,larger demand, and more rapid diffusion of newideas.

The final piece of the model is the mostClassical. The economy is characterized by theexistence of a fixed factor of production, land,and a subsistence level of consumption belowwhich individuals cannot survive. If technolog-ical progress permits output per worker to ex-ceed the subsistence level of consumption,population rises, the land-labor ratio falls, and,in the absence of further technological progress,wages fall back to the subsistence level of con-sumption. Income per capita is therefore self-equilibrating. Sustained technological progress,however, can overcome the offsetting effect ofpopulation growth, allowing sustained incomegrowth.

The model produces a Malthusian “pseudosteady state” that is stable over long periods oftime, but vanishes endogenously in the long run.In this Malthusian regime output per capita isstationary. Technology progresses only slowly,and is reflected in proportional increases in output

output growth at the time of the Industrial Revolution with-out considering the determinants of population growth. Seealso Assaf Razin and Uri Ben-Zion (1975), Zvi Eckstein etal. (1988), John Komlos and Mark Artzrouni (1990) andLakshmi K. Raut and T. N. Srinivasan (1994).

3 Schultz (1975) cites a wide range of evidence in sup-port of this theory. Similarly, Andrew D. Foster and MarkR. Rosenzweig (1996) find that technological change duringthe green revolution in India raised the return to schooling,and that school enrollment rates responded positively to thishigher return.

4 See Galor and Tsiddon (1997) and Claudia Goldin andLawrence F. Katz (1998).

5 This link between education and technological changewas first proposed by Nelson and Edmund S. Phelps (1966).For supportive evidence see Easterlin (1981) and MarkDoms et al. (1997).


and population. Shocks to the land to labor ratiowill induce temporary changes in the real wageand fertility, which will in turn drive income percapita back to its stationary equilibrium level. Be-cause technological progress is slow, the return tohuman capital is low, and parents have little in-centive to substitute child quality for quantity. TheMalthusian pseudo steady state vanishes in thelong run because of the impact of populationsize on the rate of technological progress. At asufficiently high level of population, the rate ofpopulation-induced technological progress is highenough that parents find it optimal to provide theirchildren with some human capital. At this point, avirtuous circle develops: higher human capitalraises technological progress, which in turn raisesthe value of human capital.

Increased technological progress initially hastwo effects on population growth. On the onehand, improved technology eases households’budget constraints, allowing them to spend moreresources on raising children. On the other hand, itinduces a reallocation of these increased resourcestoward child quality. In the Post-Malthusian Re-gime, the former effect dominates, and so popu-lation growth rises. Eventually, however, morerapid technological progress resulting from theincrease in the level of human capital triggers ademographic transition: wages and the return tochild quality continue to rise, the shift away fromchild quantity becomes more significant, and pop-ulation growth declines. In the Modern GrowthRegime, technology and output per capita increaserapidly, whereas population growth is moderate.

The rest of this paper is organized as follows.Section I formalizes the assumptions about thedeterminants of fertility and relative wages pre-sented earlier, and incorporates them into anoverlapping generations model. Section II de-rives the dynamical system implied by themodel, and analyzes the evolution of the econ-omy along transitions to the steady state. Sec-tion III concludes.

I. The Basic Structure of the Model

Consider an overlapping-generations econ-omy in which activity extends over infinite dis-crete time. In every period the economyproduces a single homogeneous good, usingland and efficiency units of labor as inputs. The

supply of land is exogenous and fixed over time.The number of efficiency units of labor is de-termined by households’ decisions in the pre-ceding period regarding the number and level ofhuman capital of their children.

A. Production of Final Output

Production occurs according to a constant-returns-to-scale technology that is subject toendogenous technological progress. The outputproduced at timet, Yt, is

(1) Yt 5 Hta~At X!1 2 a,

whereX and Ht are the quantities of land andefficiency units of labor employed in productionat timet, a [ (0, 1), andAt . 0 represents theendogenously determined technological level attime t. The multiplicative form in which tech-nology (At) and land (X) appear in the produc-tion function implies that the relevant factor forthe output produced is the product of the two,which we define as “effective resources.”

Output per worker produced at timet, yt, is

(2) yt 5 htaxt

~1 2 a! ; y~ht , xt !,

whereyh(ht, xt) . 0 andyx(ht, xt) . 0 @(ht, xt) @0, ht [ Ht/Lt is the number of efficiency units oflabor per worker andxt [ (AtX)/Lt is the amountof effective resources per worker at timet.

Suppose that there are no property rights overland. The return to land is therefore zero, andthe wage per efficiency unit of labor is thereforeequal to its average product:

(3) wt 5 ~xt /ht !1 2 a ; w~ht , xt !,

where wh(ht, xt) , 0 and wx(ht, xt) . 0,@(ht, xt) @ 0.

We base the modeling of the production side ontwo simplifying assumptions. First, capital is notan input in the production function; second, thereturn to land is zero. Alternatively we could haveassumed that the economy is small and open to aworld capital market in which the interest rate isconstant. In this case, the quantity of capital willbe set to equalize its marginal product to theinterest rate, whereas the price of land will follow


a path such that the total return on land (rent plusnet price appreciation) is also equal to the interestrate. This is the case presented in Galor and Weil(1998). Capital, however, has no role in the mech-anism that we examine, and the qualitative resultswould not be affected if the supply of capital wereendogenously determined.6 Allowing for capitalaccumulation and property rights over land wouldcomplicate the model to the point of intractability.

B. Preferences and Budget Constraints

In each periodt a generation that consists ofLt identical individuals joins the labor force.Each individual has a single parent. Members ofgenerationt live for two periods. In the firstperiod of life (childhood),t 2 1, individualsconsume a fraction of their parent’s time. Therequired time increases with children’s quality.In the second period of life (parenthood),t,individuals are endowed with one unit of time,which they allocate between child-rearing andlabor force participation. They choose the opti-mal mixture of quantity and quality of childrenand supply their remaining time in the labormarket, consuming their wages.

The preferences of members of generationtare defined over consumption above a subsis-tence levelc . 0, as well as over the potentialaggregate income of their children. They arerepresented by the utility function7

(4) ut 5 ~ct !~1 2 g!~wt 1 1nt ht 1 1!g,

wherent is the number of children of individualt, ht 1 1 is the level of human capital of each

child, andwt 1 1 is the wage per efficiency unitof labor at timet 1 1.

The utility function is strictly monotonicallyincreasing and strictly quasi-concave, satisfyingthe conventional boundary conditions that en-sure that, for sufficiently high income, thereexists an interior solution for the utility maxi-mization problem. However, for a sufficientlylow level of income the subsistence consump-tion constraint is binding and there is a cornersolution with respect to the consumption level.8

Following the standard model of householdfertility behavior (Becker, 1960) the householdchooses the number of children and their qualityin the face of a constraint on the total amount oftime that can be devoted to child-raising andlabor-market activities. We further assume thatthe only input required to produce both childquantity and child quality is time.9 Since allmembers of a generation are identical in theirendowments, the budget constraint is not af-fected if child quality is produced by profes-sional educators rather than by parents.

Let tq 1 teet 1 1 be the time cost for amember of generationt of raising a child with alevel of education (quality)et 1 1. That is,tq isthe fraction of the individual’s unit time endow-ment that is required to raise a child, regardlessof quality, andte is the fraction of the individ-ual’s unit time endowment that is required foreach unit of education for each child.

Consider members of generationt who areendowed withht efficiency units of labor attime t. Define potential incomezt as the amountthat they would earn if they devoted their entiretime endowment to labor-force participation:zt [ wtht. Potential income is divided betweenexpenditure on child-rearing (quantity as well

6 An alternative mechanism to deal with land in themodel would be to assume that land is owned by a smallfraction of the population, which consumes the rents that itreceives and which has a negligible impact on the evolutionof population.

7 The second component of the utility function mayrepresent either intergenerational altruism or implicit con-cern about potential support from children in old age. Theinterpretation that emphasizes intergenerational altruism re-flects an implicit bounded rationality on the part of theparent. Alternative formulations, according to which indi-viduals generate utility from the utility of their children orfrom theactualaggregate income of their offspring, wouldrequire parental predictions about fertility choices of theirdynasty. These approaches would greatly complicate themodel and we conjecture that they would not affect thequalitative results.

8 As will become clear below, the presence of a subsis-tence consumption constraint provides the Malthusian pieceof our model. The formulation that we use implicitlystresses a “demand” explanation for the positive incomeelasticity of population growth at low-income levels, sincehigher income will allow individuals to afford more chil-dren. However, one could also cite “supply” factors, such asdeclining infant mortality and increased natural fertility, toexplain the same phenomenon. See Nancy Birdsall (1988)and Randall J. Olsen (1994).

9 If both time and goods are required to produce childquality, the process we describe would be intensified. As theeconomy develops and wages increase, the relative cost ofa quality child will diminish and individuals will substitutequality for quantity of children.


as quality), at an opportunity cost ofwtht[tq 1teet 1 1] per child, and consumptionct.

Hence, in the second period of life (parent-hood), the individual faces the budget constraint

(5) wt ht nt ~tq 1 teet 1 1! 1 ct # wt ht .

C. The Production of Human Capital

An individual’s level of human capital is deter-mined by his/her quality (education) as well as bythe technological environment. Incorporating theinsight of Schultz (1964), discussed earlier, tech-nological progress is assumed to raise the value ofeducation in producing human capital. The levelof human capital of children of members of gen-erationt, ht11, is an increasing function of theireducationet11, and a decreasing function of therate of progress in the state of technology fromperiodt to periodt 1 1, gt11 [ (At11 2 At)/At.The higher the children’s quality, the smaller theadverse effect of technological progress.

(6) ht 1 1 5 h~et 1 1 , gt 1 1!,

whereh(et 1 1, gt 1 1) . 0, he(et 1 1, gt 1 1) .0, hee(et 1 1, gt 1 1) , 0, hg(et 1 1, gt 1 1) , 0,hgg(et 1 1, gt 1 1) . 0, andheg(et 1 1, gt 1 1) .0 @(et 1 1, gt 1 1) $ 0.

Hence, the individual’s level of human capitalis an increasing, strictly concave function of edu-cation, and a decreasing strictly convex functionof the rate of technological progress.10 Further-more, education lessens the adverse effect oftechnological progress. That is, technology com-plements skills in the production of human capital.

Moreover, although the number of efficiencyunits of labor per worker is diminished duringthe transition from one technological state toanother—the “erosion effect”—theeffectivenumber of the efficiency units of labor perworker, which is the product of the workers’level of human capital and the economy’s tech-nological state (reflected in the wage per effi-ciency unit of labor), is assumed to be higher as

a result of technological progress. That is, y(ht, xt)/gt . 0.

D. Optimization

Members of generationt choose the numberand quality of their children, and therefore theirown consumption, so as to maximize their in-tertemporal utility function. Substituting (5) and(6) into (4), the optimization problem of a mem-ber of generationt is

(7) $nt , et 1 1%

5 argmax$wt ht @1 2 nt ~tq 1 teet 1 1!#%1 2 g

3 $~wt 1 1nt h~et 1 1 , gt 1 1!%g

subject to

wt ht @1 2 nt ~tq 1 teet 1 1!# $ c;

~nt , et 1 1! $ 0.

The optimization with respect tont implies that, aslong as potential income at timet is sufficientlyhigh so as to ensure thatct . c, the time spent byindividual t raising children isg, whereas 12 g isdevoted for labor-force participation. However,for low levels of potential income, the subsistenceconstraint binds. The individual consumes thesubsistence levelc, and uses the rest of the timeendowment for child-rearing.

Let z be the level of potential income at whichthe subsistence constraint is just binding; that is,z [ c/(1 2 g)). It follows that forzt $ c

(8) nt @tq 1 teet 1 1#

5 H g if zt $ z1 2 @c/wt ht # if zt # z.

As long as the potential income of a memberof generationt, zt [ wtht, is below z, thenthe fraction of time necessary to ensure subsis-tence consumptionc is larger than 12 g, andthe fraction of time devoted for child-rearingis therefore belowg. As the wage per effi-ciency unit of labor increases, the individual can

10 Strict convexity with respect togt11 is not essential. It isdesigned to ensure that the level of human capital will notbecome zero at high rates of technological progress. Alterna-tive assumptions will not affect the qualitative analysis.


generate the subsistence consumption withsmaller labor force participation and the fractionof time devoted to child-rearing increases.11

Figure 2 shows the effect of an increase inpotential incomezt on the individual’s choice oftotal time spent on children and consumption.The income expansion path is vertical until thelevel of income passes the critical level thatpermits consumption to exceed the subsistencelevel. Thereafter, the income expansion pathbecomes horizontal at a levelg in terms of timedevoted for child-rearing.12

Regardless of whether potential income isabove or belowz, increases in wages will notchange the division of child-rearing time be-tween quality and quantity. Whatdoesaffect thedivision between time spent on quality and timespent on quantity is the rate of technologicalprogress, which changes the return to education.

Specifically, using (8), the optimization withrespect toet 1 1 implies that, independently ofthe subsistence consumption constraint, the im-plicit functional relationship betweenet 1 1 andgt 1 1, as depicted in Figures 3–5 and derived inLemma 1, is given by

(9) G~et 1 1 , gt 1 1!

; ~tq 1 teet 1 1!he~et 1 1 , gt 1 1!

2 teh~et 1 1 , gt 1 1!

H5 0 if et 1 1 . 0# 0 if et 1 1 5 0,

where Ge(et 1 1, gt 1 1) , 0 and Gg(et 1 1,gt 1 1) . 0 @gt 1 1 $ 0 and@et 1 1 . 0.

To ensure the existence of a positive level ofgt 1 1 such that the chosen level of education is0, it is assumed that

(A1) G~0, 0! 5 tqhe~0, 0!

2 teh~0, 0! , 0.

LEMMA 1: If (A1) is satisfied, then the level ofeducation chosen by members of generation t fortheir children is a nondecreasing function of gt11.

et 1 1 5 e~gt 1 1!H5 0 if gt 1 1 # g. 0 if gt 1 1 . g,

where, g . 0, and

e9~gt 1 1! . 0 ; gt 1 1 . g.

PROOF:As follows from (6) and (9),G(0, gt 1 1) is

monotonically increasing ingt 1 1. Furthermore,(6) implies that limgt 1 13` G(0, gt 1 1) . 0,whereas (A1) implies thatG(0, 0) , 0. Hence,there existsg . 0 such thatG(0, g) 5 0, andtherefore, as follows from (9)et 1 1 5 0 forgt 1 1 # g. Furthermore, it follows from (9) thatet 1 1 is a single-valued function ofgt 1 1, wheree9t 1 1( gt 1 1) 5 2Gg(et 1 1, gt 1 1)/Ge(et 1 1,gt 1 1) . 0.

As is apparent from (9),e0( gt 1 1) depends onthe third derivatives of the production functionof human capital. A concave reaction of thelevel of education to the rate of technologicalprogress appears plausible economically, henceit is assumed that13

(A2) e0~gt 1 1! , 0 ; gt 1 1 . g.

11 John D. Durand (1975) and Goldin (1994) report that,across a large sample of countries, the relationship betweenwomen’s labor-force participation and income isU-shaped.The model presented here explains the negative effect of in-come on labor-force participation for poor countries, and fur-ther predicts that this effect should no longer be operative oncepotential income has risen sufficiently high; it does not, how-ever, explain the positive effect of income on participation forricher countries. See, however, Galor and Weil (1996) for amodel that does explain this phenomenon.

12 An alternative way of generating a qualitatively sim-ilar result would be to assume a Stone-Geary utility functionof the formut 5 (ct 2 c)(12 g)(wt 1 1ntht 1 1)g. In this casethe income expansion path would be nearly vertical for lowlevels of potential income and asymptotically horizontal forhigh levels of potential income. Adopting this formulationwould raise the dimensionality of the system, however.

13 Alternatively, if e(gt11) is strictly convex we may as-sume that for physiological or other reasons, the maximumamount of education that a child can receive is bounded fromabove. In the model we ignore integer constraints on thenumber of children, so that absent a constraint on the qualityper child, parents might choose to have an infinitesimally smallnumber of children with infinitely high quality. Thus the ex-istence of integer constraints may be taken as one justificationfor an upper bound on level of education.


Furthermore, substitutinget 1 1 5 e( gt 1 1)into (8), it follows thatnt is

(10) nt 5 5g

tq 1 tee~gt 1 1!; nb~gt 1 1! if zt $ z

1 2 @c/zt #

tq 1 tee~gt 1 1!; na~gt 1 1 , zt ! if zt # z.

As follows from (3), (6), and the definition ofzt,

(11) zt 5 wt ht 5 htaxt

~1 2 a! ; z~et , gt , xt !,

whereze(et, gt, xt) . 0, zx(et, gt, xt) . 0, andzg(et, gt, xt) , 0 @(et, gt, xt) @ 0.14

The following proposition summarizesthe properties of the functionse( gt 1 1), na( zt,gt 1 1), andnb( gt 1 1) and their significance forthe evolution in the substitution of quality forquantity in the process of development.

PROPOSITION 1:Under (A1)–(A2)

(a) Technological progress that is expected tooccur between the first and second periodsof children’s lives results in a decline in theparents’ chosen number of children and anincrease in their quality, i.e.,

nt /gt 1 1 # 0 and et 1 1/gt 1 1 $ 0.

(b) If parental potential income is below z˜ (i.e.,if the subsistence consumption constraint isbinding), an increase in parental potentialincome raises the number of children, buthas no effect on their quality, i.e.,

nt/zt . 0 and et11/zt 5 0 if zt , z.

(c) If parental potential income is above z˜, anincrease in parental potential income doesnot change the number of children or theirquality, i.e.,

nt / zt 5 et 1 1/ zt 5 0 if zt . z.

PROOF:Follows directly from Lemma 1, (8)–(10),

and assumptions (A1)–(A2).

14 It should be noted that, whereas the partial derivativeof zt with respect togt is negative (holdingxt and thusAt

constant), the total derivative ofzt with respect togt (hold-ing At 2 1 constant) is positive.

FIGURE 2. PREFERENCES, CONSTRAINTS, AND INCOME EXPANSION PATH

Notes:The figure depicts the household’s indifference curves, budget constraints, as well as the subsistence consumptionconstraintc $ c. The income expansion path, as derived in Proposition 1, is vertical as long as the subsistence consumptionconstraint is binding and horizontal at a levelg once the subsistence consumption constraint is not binding.


It follows from Proposition 1 that if the sub-sistence consumption constraint is binding, anincrease in the effective resources per workerraises the number of children, but has no effecton their quality, whereas if the subsistence con-straint is not binding, an increase in the effec-tive resources per worker does not change thenumber of children or their quality.

E. Technological Progress

Suppose that technological progressgt 1 1,which takes place between periodst andt 1 1,depends on the education per capita among theworking generation in periodt, et, and thepopulation size of the working generation inperiod t, Lt.

15

(12) gt 1 1 ;At 1 1 2 At

At5 g~et , Lt !,

where forLt @ 0 andet $ 0, g(0, Lt) . 0,gi(et, Lt) . 0, andgii (et, Lt) , 0, i 5 et, Lt.

16

Hence, for a sufficiently large populationsize, the rate of technological progress betweentime t and t 1 1 is a positive, increasing,strictly concave function of the size and level ofeducation of the working generation at timet.Furthermore, the rate of technological progressis positive even if labor quality is zero.

As will become apparent, the dynamical sys-tem of the described economy is rather com-plex; hence, to simplify the exposition, thedynamical system is analyzedinitially under theassumption that an increase in population sizehas no effect on technological progress. In par-ticular, it is initially assumed that

(A3) gL ~et , Lt ! 5 0 ; Lt . 0.

In later stages of the analysis the effect of thesize of population on the relationship betweentechnological progress and the level of educa-tion as specified in (12) is fully incorporatedinto the analysis.

F. The Evolution of Population, Technology,and Effective Resources

The size of the working population at timet 1 1, Lt 1 1, is

(13) Lt 1 1 5 nt Lt ,

whereLt is the size of the working population attime t, nt is the number of children per person,andnt 2 1 is the rate of population growth. Thesize of the working population at time 0 ishistorically given at a levelL0.

The state of technology at timet 1 1, At 1 1,as derived from (12), is

(14) At 1 1 5 ~1 1 gt 1 1!At ,

where the state of technology at time 0 is his-torically given at a levelA0.

The evolution of effective resources perworker, xt [ ( AtX)/Lt, depends on the evolu-tion in the technological level and the rate ofpopulation growth:

(15) xt 1 1 51 1 gt 1 1

ntxt ,

wherex0 [ A0X/L0 is historically given.Substituting (11) and (12) into (10), and (10)

and (12) into (15),

(16) xt 1 1

5 5@1 1 g~et, Lt !#@tq 1 tee~g~et, Lt !!#

gxt

; fb~et, Lt !xt if zt $ z

@1 1 g~et, Lt !#@tq 1 tee~g~et, Lt !!#

1 2 @c/z~et , gt , xt !#xt

; fa~et , gt , xt, Lt !xt if zt # z,

15 We consider a modification of equation (12) along thelines suggested by Jones (1995) in Section II.D.

16 It should be noted that we assume that for a suffi-ciently small population the rate of technological progress isstrictly positive only every several periods. That is, for asufficiently smallLt . 0, g(0, Lt) $ 0, gi(et, Lt) $ 0, forall t, and g(0, Lt) . 0, gi(et, Lt) . 0, for some t.Furthermore, the number of periods that pass between twoepisodes of technological improvement declines with thesize of population. These assumptions ensure that in earlystages of development the economy is indeed in a Malthu-sian steady state. Clearly, if technological progress occurredin every time period at a pace that increased with the size ofpopulation, the growth rate of output per capita wouldalways be positive, despite the adjustment in the size ofpopulation.


where, as follows from Lemma 1, (11), and (12),fe

b(et, Lt) . 0, andfxa(et, gt, xt, Lt) , 0 @et $ 0.

II. The Dynamical System

The development of the economy is charac-terized by the evolution of output per worker,population, technological level, education perworker, human capital per worker, and effectiveresources per worker. The evolution of theeconomy, is fully determined by a sequence {et,gt, xt, Lt} t 5 0

` that satisfies (12)–(16) andLemma 1 in every periodt.

The dynamical system is characterized bytwo regimes. In the first regime the subsistenceconsumption constraint is binding and for agiven population sizeL, the evolution of theeconomy is governed by athree-dimensionalnonlinear first-order autonomous system:17

(17) H xt 1 1 5 fa~et , gt , xt; L !xt

et 1 1 5 e~g~et; L !!gt 1 1 5 g~et; L !

if zt # z

where the initial conditionse0, g0, andx0 arehistorically given. In the second regime thesubsistence consumption constraint is notbinding and, for a given population sizeL, theevolution of the economy is governed by atwo-dimensional system:

(18) H xt 1 1 5 fb~et , xt; L !xt

et 1 1 5 e~g~et; L !!. if zt $ z

In both regimes, however, the analysis of thedynamical system is greatly simplified by thefact that, as follows from Lemma 1, (12), and(A3), the joint evolution ofet and gt is deter-mined independently of thext. Furthermore, theevolution ofet andgt is independent of whetherthe subsistence constraint is binding, and istherefore independent of the regime in whichthe economy is located. The education level ofworkers in periodt 1 1 depends only on thelevel of technological progress expected be-

tween periodt and period t 1 1, whereastechnological progress between periodst andt 1 1 depends only on the level of education ofworkers in periodt. Thus for a given populationsizeL, we can analyze the dynamics of technol-ogy and education independently of the evolu-tion of resources per capita.

A. The Evolution of Technologyand Education

The evolution of technology and education,given (A3), is characterized by the sequence{ gt, et} t 5 0

` that satisfies in every periodt theequations gt 1 1 5 g(et; L) and et 1 1 5e( gt 1 1). Although this dynamical subsystemconsists of two independent one dimensional,nonlinear first-order difference equations, it ismore revealing to analyze them jointly.

In light of the properties of the functionse( gt 1 1) andg(et; L) given in Lemma 1, (A2),(A3), and (12), it follows that if population sizedoesplay a role in technological progress, thisdynamical subsystem is characterized by threequalitatively different configurations, which aredepicted in Figures 3–5. The economy shiftsendogenously from one configuration to anotheras population increases and the curveg(et; L)shifts upward to account for the effect of anincrease in population.

In Figure 3, for a range of small populationsizes, the dynamical system is characterized byglobally stable steady state equilibria. For agiven population size in this range, the steady-state equilibrium is (e, g) 5 (0, gl). As impliedby (12), the rate of technological change in atemporary steady state increases monotonicallywith the size of population, whereas the level ofeducation remains unchanged.

In Figure 4, for a range of moderate populationsizes, the dynamical system is characterized bythree steady-state equilibria. For a given popula-tion size in this range, there exist two locallystable steady-state equilibria: (e, g) 5 (0, gl) and(e, g) 5 (eh, gh), and an interior unstable steadystate (e, g) [ (eu, gu). The steady-state equilibria(eh, gh) andgl increase monotonically with the sizeof population.

Finally, in Figure 5, for a range of large popu-lation sizes, the dynamical system is characterizedby globally stable steady state equilibria. For agiven population size in this range, there exists a

17 For a given population, the entire dynamical systemcan be represented by the sequence (gt, xt)t50

` . However,sincee( gt) is not invertible, the sequence (et, xt)t50

` doesnot represent the dynamical system, and a dynamical systemthat incorporates the evolution ofet is necessarily three-dimensional in the first regime.


unique globally stable steady-state equilibrium:(e, g) 5 (eh, gh). These temporary steady-statelevels increase with population.

B. Global Dynamics

This section analyzes the evolution of theeconomy from the Malthusian Regime,through the Post-Malthusian Regime, to thedemographic transition and Modern Growth.The global analysis is based on a sequence ofphase diagrams that describe the evolution ofthe system within each regime and the tran-sition between the different regimes in theplain (et, xt). The phase diagrams, depicted inFigures 6 – 8 contain three elements: theMalthusian Frontier, which separates the re-gions in which the subsistence constraint isbinding from those where it is not; theXXlocus, which denotes the set of all pairs (et,xt) for which effective resources per workerare constant; and theEE locus, which denotesthe set of all pairs for which the level ofeducation per worker is constant.

The Malthusian Frontier.—As was estab-lished in (17) and (18) the economy exitsfrom the subsistence consumption regimewhen potential incomezt exceeds the criticallevel z. This switch of regime changes thedimensionality of the dynamical system fromthree to two.

Let theMalthusian Frontierbe the set of alltriplets of (et, xt, gt) for which individuals’incomes equalz.18 Using the definitions ofztand z, it follows from (6) and (11) that theMalthusian FrontierMM is

(19) MM ; $~et , xt , gt ! : xt~1 2 a!h~et , gt !

a

5 c/~1 2 g!%.

18 As was shown in Proposition 1, below the MalthusianFrontier, the effect of income on fertility will be positive,whereas above the frontier there will be no effect of incomeon fertility. Thus the Malthusian Frontier separates theMalthusian and Post-Malthusian Regimes, on the one hand,from the Modern Growth Regime, on the other.

FIGURE 3. THE EVOLUTION OF TECHNOLOGY AND EDUCATION FOR A SMALL POPULATION

Notes: The figure describes the evolution of educationet and the rate of technological changegt for a constant smallpopulationLl. The curve labeledgt 1 1 5 g(et; Ll) shows the effect of education on the growth rate of technology as presentedin equation (12). The curve labeledet 1 1 5 e( gt 1 1) shows the effect of expected technological change on optimal educationchoices derived in Lemma 1. The point of intersection between the two curves is the globally stable steady-state equilibrium(0, gl). In early stages of development, the economy is in the vicinity of this steady state in which education is zero and therate of technological progress is slow.


Let the Conditional Malthusian Frontierbethe set of all pairs (et, xt) for which, conditionalon a given technological levelgt, individuals’incomes equalz. Following the definitions ofztand z, equations (6) and (11) imply that theConditional Malthusian FrontierMM ugt

, as de-picted in Figures 6–8, is

(20) MM ugt; $~et , xt ! : xt

~1 2 a!h~et , gt !a

5 c/~1 2 g!ugt%.

LEMMA 2: If (et, xt) [ MM ugtthen xt is a

decreasing strictly convex function of et.

PROOF:The lemma follows from (6) and (20).

Hence, the Conditional Malthusian Frontier, asdepicted in Figures 6–8, is a strictly convex,downward sloping, curve in the (et, xt) space.Furthermore, it intersects thext axis and asymp-totically approaches theet axis asxt approaches

infinity. The frontier shifts upward asgt increasesin the transition to a Modern Growth regime.

The XX Locus.—Let XX be the locus of alltriplets (et, gt, xt), such that for a given popu-lation size the effective resources per worker,xt, are in a steady state:

XX ; $~et , xt , gt ! : xt 1 1 5 xt%.

Along theXX locus the growth rates of populationand technology are equal. Above the MalthusianFrontier, the fraction of time devoted to child-rearing is not dependent on the level of effectiveresources per worker. In this case, the growth rateof population will just be a negative function ofthe growth rate of technology, since for highertechnology growth, parents will spend more oftheir resources on child quality and thus less onchild quantity. Thus there will be a particular levelof technological progress that induces an equalrate of population growth. Since the growth rate oftechnology is, in turn, a positive function of thelevel of education, this rate of technology growth

FIGURE 4. THE EVOLUTION OF TECHNOLOGY AND EDUCATION FOR A MODERATE POPULATION

Notes:The figure describes the evolution of educationet and the rate of technological changegt once the size of the population hasgrown to reach a moderate size,Lm. The system is characterized by multiple steady state equilibria. The steady- state equilibria (0,gl) and (eh, gh) are locally stable, whereas (eu, gu) is unstable. Given the initial conditions, in the absence of large shocks theeconomy remains in the vicinity of the low steady-state equilibrium (0,gl), in which education is still zero but the rate oftechnological progress is moderate.


will correspond to a particular level of education,denotede. Below the Malthusian Frontier, thegrowth rate of population depends on the level ofeffective resources per capitax, as well as on thegrowth rate of technology. The lower the value ofx, the smaller the fraction of the time endowmentdevoted to child-rearing, and so the lower thepopulation growth. Thus, below the MalthusianFrontier, a lower value of effective resources percapita will mean that lower values of technologygrowth (and thus education) will be consistentwith population growth being equal to technologygrowth. Thus, as drawn in Figures 6–8, lowervalues ofx will be consistent with lower values ofe on the part of theXX locus that is below theMalthusian Frontier.

Lemmas 3, 4, and 5, derive the properties ofthis locus. To simplify the exposition withoutaffecting the qualitative nature of the dynamicalsystem, the parameters of the model are re-stricted so as to ensure that theXX locus isnonempty whenzt $ z; that is,

(A4) g , ~g/tq! 2 1 , g~eh~L0!, L0!.

LEMMA 3: If (A1)–(A4) are satisfied, then forzt $ z, there exists a unique value0 , e , eh,such that xt [ XX. Furthermore, for zt $ z

xt 1 1 2 xt H. 0 if et . e5 0 if et 5 e, 0 if et , e.

PROOF:For zt $ z, it follows from (16) that

xt 1 1 5 xt if and only if fb~et; L !

; @1 1 g~et; L !#[tq 1 tee~g~et; L !!]/

g 5 1.

Since fb(et; L) is strictly monotonically in-creasing inet and since (A4) implies that for allLt . 0, fb(0; L) , 1 and fb(eh; L) . 1,there exists a unique value 0, e , eH, suchthat fb(e; L) 5 1 and hencext [ XX. Fur-thermore, sincefb(et; L) is strictly monotoni-cally increasing inet, it follows from (16) thatxt 1 1 . xt if and only if fb(et; L) . 1 and

FIGURE 5. THE EVOLUTION OF TECHNOLOGY AND EDUCATION FOR A LARGE POPULATION

Notes: The figure describes the evolution of educationet and the rate of technological changegt once the size of thepopulation grows to a high level,Lh. The system is characterized by a unique globally stable steady-state equilibrium (eh,gh). In mature stages of development, the economy converges monotonically to this steady state with high levels of educationand technological progress.


henceet . e, whereasxt11 , xt if and only iffb(et; L) , 1 and henceet , e.

Hence, theXX locus, as depicted in Figures6 – 8 in the space (et, xt), is a vertical lineabove the Conditional Malthusian Frontier ata levele. This critical level decreases with thesize of the population.

Lemma 3 holds as long as consumption isabove subsistence. In the case where the sub-sistence constraint is binding, the evolution ofxt, as determined by equation (16), is based onthe rate of technological changegt, the effectiveresources per workerxt, as well as the quality ofthe labor forceet.

Let XXugtbe the locus of all pairs (et, xt), such

thatxt11 5 xt for a given level ofgt; that is,

XXugt; $~et , xt ! : xt 1 1 5 xtugt%.

LEMMA 4: If (A1)–(A4) are satisfied, then forzt # z and for 0 # et # e, there exists a single-valued function xt 5 x(et), such that(x(et), et) [XXugt

. Furthermore, for zt # z,

xt 1 1 2 xt5,0 if ~et , xt ! . ~et , x~et !!

and 0 # et # e,

50 if xt 5 x~et !

and 0 # et # e,

.0 if @~et , xt ! , ~et , x~et !!

and 0 # et # e#, or @et . e#.

PROOF:As follows from (16),xt11 5 xt if and only if

fa~et , gt , xt ! 5 @1 1 g~et; L !#

3 @tq 1 tee~g~et; L !!#

4 $1 2 @c/z~et , gt , xt !#%

5 1.

Sincefa(et, gt, xt; L) is strictly monotonicallydecreasingin xt, there exists a single-valuedfunctionxt 5 x(et), such thatfa(et, xtugt) 5 1and therefore (et, x(et)) [ XXugt

. Moreover,

sincefea(et, gt, xt; L) is not necessarily mono-

tonic, x9(et) is not necessarily monotonic aswell. Furthermore, sincefa(et, xtugt) is strictlymonotonically decreasing inxt, it follows from(16) that for 0# et # e and forzt # z

xt 1 1 . xt

if and only if xt , max@x~et !, xtM#,

where~et, xtM! [ MM ugt

and

xt 1 1 , xt if and only if xt . x~et !.

Hence, without loss of generality, the locusXXugt

is depicted in Figures 6–8, as an upward-sloping curve in the space (et, xt), defined foret # e. XXugt

is strictly below the ConditionalMalthusian Frontier for the value ofet , e, andthe two coincide ate.

LEMMA 5: Let (e, x) [ MMugt. If (A4) is

satisfied, then

~e, x! 5 XXugtù MM ugt

ù XX.

PROOF:Let (e, x) [ MM ugt

. It follows from thedefinition of MM ugt

that z(e, xugt) 5 z.Hence, Lemma 2 implies that (e, x) [ XX.Furthermore, since Lemmas 2 and 3 are bothvalid for zt 5 z, it follows that x(e) 5 x andhence (e, x) [ XXugt

.

Hence, the Conditional Malthusian Frontier, theXX locus, and theXXugt

locus, as depicted in Fig-ures 6–8 in the (et, xt) space, coincide at (e, x).

The EE Locus.—Let EE be the locus of alltriplets (et, gt, xt), such that the quality of laboret is in a steady state:

EE ; $~et , xt , gt ! : et 1 1 5 et%.

As follows from the analysis in Section II, sub-section A for a given population size, thesteady-state values ofet are independent of thevalues of xt and gt. The locus EE evolvesthrough three phases in the process of develop-ment, corresponding to the three phases that


describe the evolution of education and technol-ogy depicted in Figures 3, 4, and 5.

In the early stages of development, when pop-ulation size is sufficiently small, the joint evolu-tion of education and technology is characterizedby a globally stable temporary steady-stateequilibrium, (e, g) 5 (0, gl), as depicted in Fig-ure 3. The correspondingEE locus, depicted in thespace (et, xt) in Figure 6,is vertical at the levele50, for a range of small population sizes. Further-more, for this range, the global dynamics ofet inthis configuration are given by

(21) et 1 1 2 et H5 0 if et 5 0, 0 if et . 0.

In later stages of development, as populationsize increases sufficiently, the joint evolution ofeducation and technology is characterized bymultiple locally stable temporary steady-stateequilibria, as depicted in Figure 4. The corre-spondingEE locus, depicted in the space (et,

xt) in Figure 7,consists of three vertical linescorresponding to the three steady-state equilib-ria for the value ofet—that is,e 5 0, e 5 eu,ande 5 eh. The vertical linese 5 eu ande 5eh shift rightward as population size increases.Furthermore, the global dynamics ofet in thisconfiguration are given by

(22) et 1 1 2 et5, 0 if 0 , et , eu or et . eh

5 0 if et [ $0, eu, eh%

. 0 if eu , et , eh.

In mature stages of development, when popula-tion size is sufficiently large, the joint evolutionof education and technology is characterized byglobally stable steady-state equilibrium at thepoint (e, g) 5 (eh, gh), as depicted in Figure5. The correspondingEE locus, as depicted inFigure 8 in the space (et, xt), is vertical at thelevel e 5 eh. This vertical line shifts rightwardas population size increases. Furthermore, the

FIGURE 6. THE CONDITIONAL DYNAMICAL SYSTEM FOR A SMALL POPULATION

Notes:This figure describes the evolution of educationet and effective resource per workerxt for a constant small populationLl. The curveet 1 1 5 et is the set of all pairs (et, xt), for which education is constant over time. The curvext 1 1 5 xt isthe set of all pairs (et, xt), givengt, for which effective resource per worker is constant over time (Lemmas 3, 4, and 5). Thepoint of intersection between the two curves is the unique globally stable steady-state equilibrium. In early stages ofdevelopment, the system is in the vicinity of this conditional Malthusian steady-state equilibrium. The Conditional MalthusianFrontier as defined in equation (20) is the set of all pairs (et, xt), givengt, below which the subsistence consumption constraintis binding.


global dynamics ofet in this configuration aregiven by

(23) et 1 1 2 etH. 0 if 0 # et , eh

5 0 if et 5 eh., 0 if et . eh.

C. Conditional Steady-State Equilibria

In early stages of development, when popu-lation size is sufficiently small, the dynamicalsystem, as depicted in Figure 6 in the space(et, xt), is characterized by a unique andglobally stable conditional steady-state equi-librium.19 It is given by a point of intersection

between theEE locus and theXX locus. Thatis, conditional on a given rate of technologi-cal progressgt and a given population size,the Malthusian steady state (0,x( gt)) isglobally stable.20 In later stages of develop-ment, as population size increases suffi-ciently, the dynamical system as depicted inFigure 7 is characterized by two conditionalsteady-state equilibria. The Malthusian con-ditional steady-state equilibrium is locallystable, whereas the conditional steady-stateequilibrium (eu, xu) is a saddlepoint.21 In addi-tion, for education levels aboveeu the systemconverges to a stationary level of educationeh

and possibly to a steady-stategrowth rateof xt,

19 Since the dynamical system is discrete, the trajectoriesimplied by the phase diagrams do not necessarily approxi-mate the actual dynamic path, unless the state variablesevolve monotonically over time. As shown in Section II,subsection A, the evolution ofet is monotonic, whereas theevolution and convergence ofxt may be oscillatory. Non-monotonicity may arise only ife , e. Nonmonotonicity inthe evolution ofxt does not affect the qualitative descriptionof the system. Furthermore, iffx

a(et, gt, xt; L)xt . 21 theconditional dynamical system is locally nonoscillatory. The

phase diagrams in Figures 6–8 are drawn under the assump-tions that ensure that there are no oscillations.

20 The local stability of the steady-state equilibrium (0,x( gt)) can be derived formally. The eigenvalues of theJacobian matrix of the conditional dynamical system eval-uated at the conditional steady-state equilibrium are bothsmaller than 1 (in absolute value) under (A1)–(A3).

21 Convergence to the saddlepoint takes place only if thelevel of education iseu. That is, the saddlepath is the entirevertical line that corresponds toet 5 eu.

FIGURE 7. THE CONDITIONAL DYNAMICAL SYSTEM FOR A MODERATE POPULATION

Notes:This figure describes the evolution of educationet and effective resource per workerxt, once the size of the population hasgrown to reach a moderate size,Lm. The system is characterized by multiple steady state equilibria. Given the initial conditions, inthe absence of large shocks, the economy remains in the vicinity of the conditional Malthusian steady state equilibrium.


given the population size. In mature stages ofdevelopment when population size is sufficientlylarge, the system converges globally to an educa-tional level eh and possibly to a steady-stategrowth rateof xt, given the population size.

D. Analysis

The transition from the Malthusian regimethrough the Post-Malthusian regime to the demo-graphic transition and a Modern Growth regimeemerges from Proposition 1 and Figures 2–8.Consider an economy in the early stages of devel-opment. Population is low enough that the impliedrate of technological change is very small, andparents have no incentive to provide education totheir children. As depicted in Figure 3 in the space(et, gt), the economy is characterized by a singletemporary steady-state equilibrium in which tech-nological progress is very slow and children’slevel of education is zero. This temporary steady-state equilibrium corresponds to a globally stableconditional Malthusian steady-state equilibrium,drawn in Figure 6 in the space (et, xt). For a givenrate of technological progress, effective resourcesper capita, as well as the level of education, areconstant and hence, as follows from (2) and (6),output per capita is constant as well. Moreover,

shocks to population or resources will be undonein a classic Malthusian fashion. Population will begrowing slowly, in parallel with technology.

As long as the size of the population is suf-ficiently small, no qualitative changes occur inthe dynamical system described in Figures 3and 6. The temporary steady-state equilibriumdepicted in Figure 3 gradually shifts verticallyupward, reflecting small increments in the rateof technological progress as the size of thepopulation increases, while the level of educa-tion remains constant at zero. Similarly, theconditional Malthusian steady-state equilib-rium, drawn in Figure 6 for a constant rate oftechnological progress, shifts upward vertically.However, output per capita remains constant atthe subsistence level.

Over time, the slow growth in population thattakes place in the Malthusian regime will raise therate of technological progress and shift theg(et11;Ll) locus in Figure 3 upward so that it has theconfiguration shown in Figure 4. At this point, thedynamical system of education and technologywill be characterized by multiple, history-dependent steady states. One of these steady stateswill be Malthusian, characterized by constantresources per capita, slow technological progress,and no education. The other will be characterized

FIGURE 8. THE CONDITIONAL DYNAMICAL SYSTEM FOR A LARGE POPULATION

Notes: The figure describes the evolution of educationet and the rate of technological changext, once the size of thepopulation has reached a high level,Lh. The dynamical system changes qualitatively and the conditional Malthusian steadystate vanishes. The economy evolves through a Post-Malthusian Regime until it crosses the Conditional Malthusian Frontierand enters the Modern Growth Regime.


by a high level of education, rapid technologicalprogress, growing income per capita, and moder-ate population growth.

For the deterministic description of a take-offfrom the Malthusian equilibrium that is empha-sized in this paper, however, the existence ofmultiple steady states turns out not to be rele-vant. Since the economy starts out in theMalthusian steady state, it will remain there atthis intermediate stage. If we were to allow forstochastic shocks to education or technologicalprogress, it would be possible for an economyin the Malthusian steady state of Figure 4 tojump to the Modern Growth steady state, but wedo not pursue this possibility.

Figure 5 shows that the increasing size of thepopulation continues to raise the rate of techno-logical progress, reflected in a further upwardshift of theg(et 1 1; Lt) locus. At a certain levelof population, the steady-state Malthusian van-ishes, and the economy transitions out of theMalthusian Regime. Increases in the rate oftechnological progress and the level of educa-tion feed back on each other until the economyconverges to the unique, stable steady state.

Although both the evolution of education andtechnological progress traced in Figure 5 aremonotonic once the Malthusian steady state hasbeen left behind, the evolution of populationgrowth and the standard of living, which can beseen in Figure 8, are more complicated. The rea-son for this complication is that technologicalprogress has two effects on the evolution of pop-ulation, as shown in Proposition 1. First, by in-ducing parents to give their children moreeducation, technological progress will ceteris pa-ribus lower the rate of population growth. But,second, by raising potential income, technologicalprogress will increase the fraction of their timethat parents can afford to devote to raising chil-dren. Initially, while the economy is in theMalthusian region of Figure 8, the effect of tech-nology on the parent’s budget constraint will dom-inate, and so the growth rate of the population willincrease. This is the Post-Malthusian Regime.22

The positive income effect of technologicalprogress on fertility functions only in the Malthu-sian region of Figure 8, however; as the figureshows, the economy eventually crosses theMalthusian frontier. Once this has happened, fur-ther improvements in technology no longer havethe effect of changing the amount of time devotedto child-rearing, whereas faster technologicalchange will continue to raise the quantity of edu-cation that parents give each child. Thus once theeconomy has crossed the Malthusian Frontier,population growth will fall as education and tech-nological progress rise.

In the Modern Growth Regime, resources percapita will rise, as technological progress outstripspopulation growth. Figure 5 shows that the levelsof education and technological progress will beconstant in the steady state, provided that popula-tion size is constant (i.e., population growth iszero). This implies that the growth rate of re-sources per capita, and thus the growth rate ofoutput per capita, will also be constant. However,if population growth is positive in the ModernGrowth Regime and if its effect on technologicalprogress remains positive, then education andtechnological progress will continue to rise, and,similarly, if population growth is negative theywill fall. In fact, the model makes no firm predic-tion about what the growth rate of population willbe in the Modern Growth Regime, other than thatpopulation growth will fall once the economyexits from the Malthusian region. It may be thecase that population growth will be zero, inwhich case the Modern Growth Regime wouldconstitute a global steady state, in whiche andg were constant. Alternatively, populationgrowth could be either positive or negative inthe Modern Growth Regime, withe andg be-having accordingly if the effect of populationsize on the rate of technological progress re-mains positive.23

22 Literally, income per capita does not change during thePost-Malthusian Regime. It remains fixed at the subsistencelevel. This is an artifact of the assumption that the only inputinto child quality is parental time, and that this time input doesnot produce measured output. If child-rearing, especially theproduction of quality, requires goods or time supplied through

a market (e.g., schooling), the shift toward higher child qualitythat takes place during the Post-Malthusian Regime would bereflected in higher market expenditures (as opposed to parentaltime expenditures) and rising measured income.

23 Jones (1995) has argued for a model of technologycreation in which the steady-state growth rate of technology isrelated to the growth rate of population, rather than to its level.Under such a specification, our model would have a steady-state modern growth regime, in which the growth rates ofpopulation and technology would be constant. Further, such a


III. Concluding Remarks

This paper develops a unified endogenousgrowth model in which the evolution of popula-tion, technology, and output growth is largelyconsistent with the process of development in thelast millennia. The model generates an endoge-nous takeoff from a Malthusian Regime, througha Post-Malthusian Regime, to a demographictransition and a Modern Growth Regime. Inearly stages of development—the MalthusianRegime—the economy remains in the proximity of aMalthusian trap, where output per capita is nearlystationary and episodes of technological changebring about proportional increases in output and pop-ulation. In the intermediate stages of development—the Post-Malthusian Regime—the intensified pace oftechnological change that is caused by the increase inthe size of population during the Malthusian Regimepermits the economy to take off. Production takesplace under a state of technological disequilibrium inwhich the relative return to skills rises, inducing thehousehold to shift its spending on children towardquality and away from quantity. Output per capitaincreases along with an increase in the rate of pop-ulation growth and human-capital accumulation.Eventually, rapid technological progress, which re-sults from high human-capital accumulation, triggersa demographic transition in which fertility rates per-manently decline.

One of the significant components of the modelis the effect of technological change on the returnto education. Specifically, technological transi-tions, in and of themselves, are assumed to raisethe return to education. An alternative assumptionthat would produce many of the same results isthat the return to education rises with theleveloftechnology, so that, for example, a technologicallystagnant economy with a high level of technologywould have a higher return to education than asimilarly stagnant economy with a low level ofeducation. A model incorporating this assumptionwould produce a technological takeoff that was notrelated to the size of population: even if populationwere constant, technological progress would eventu-ally raise the rate of return to education sufficiently toinduce parents to give their children more schooling,

and this would in turn feed back to raise the rate oftechnological progress. Making this assumption,however, would be equivalent to assuming thatchanges in technology were skill biased throughouthuman history. Although, on average, technologicalchange may have been skilled biased, our mech-anism allows us to consider those periods in whichtechnological change was unskilled biased in thelong run (most notably, elements of the industrialrevolution).

The model abstracts from several factors thatare relevant for economic growth. Differences be-tween countries in the determination of populationgrowth or in the process of technological change(as a result of institutions and cultural factors, forexample) would be reflected in their ability toescape the Malthusian trap and in the speed oftheir takeoff. Similarly, differences in policies,such as the public provision of education, wouldchange the dynamics of the model. One interest-ing possibility that the model suggests is that theinflow of grain and other commodities as well asthe outflow of migrants during the nineteenth cen-tury may have played a crucial role in Europe’sdevelopment. By easing the land constraint at acrucial point—when income per capita had begunto rise rapidly, but before the demographic transitionhad gotten under way—the “ghost acres” of the NewWorld provided a window of time, which allowedEurope to pull decisively away from the Malthusianequilibrium (Kenneth Pomeranz, 1999).

Even though the model presents a unified de-scription of the development process followed byEurope and its offshoots, it is clearly not fullyapplicable to countries that are developing today.For currently developing countries, a large stockof preexisting technology is available for import,and so the relationship between population sizeand technology growth, which helped trigger thedemographic transition in Europe, is no longerrelevant. Similarly, the relationship between in-come and population growth has changed dramat-ically, resulting from the import of healthtechnologies. Countries that are poor, even by thestandards of nineteenth-century Europe, are expe-riencing growth rates of population far higher thanthose ever experienced in Europe.

We end by stressing the importance of the con-struction of unified models of population and de-velopment that encompass the endogenoustransition between the three fundamental regimesthat have characterized the process of develop-

steady state would be stable: if population growth fell, the rateof technological progress would also fall, inducing a rise infertility.


ment.24 Imposing the constraint that a singlemodel account for the entire process of developmentis a discipline that would improve the understandingof the underlying phenomena and generate superiortestable predictions and more accurate analysis of theeffects of policy interventions.

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