J Econ Growth (2007) 12:351–387DOI 10.1007/s10887-007-9022-2

Escaping high mortality

Javier A. Birchenall

Published online: 15 November 2007© Springer Science+Business Media, LLC 2007

Abstract This paper develops an economic analysis of mortality to account for themortality decline during the demographic transition. We propose a unified growth modelin which there is intra- and intergenerational health transmission: parental health affectschild survival and investments in early life improve health over the entire life-cycle. Basedon data from England and Wales between 1640 and 2000, we show that the role of economicchanges in mortality is larger than previously estimated. As in current estimates, contem-porary income has a minor impact on mortality change. Most of the economic influences,however, are unaccounted for by contemporary relationships.

Keywords Mortality · Population growth · Economic development

JEL Classifications I12 · O11 · O33

1 Introduction

This paper studies the role of economic changes in the mortality decline observed during thedemographic transition and argues that economic changes are the most important influencein the mortality decline of developed countries. We propose a dynamic model of healthtransmission and argue that, despite a weak association between contemporaneous incomeand mortality, changes in economic conditions have improved health beyond the short-termconsiderations commonly studied in the literature. We show that economic changes have amuch larger effect on mortality than what can be predicted based on contemporary relation-ships because improvements in early-life lower infant and child mortality but also improveadult health and lower adult mortality. This secondary improvement would be mislabeledas an ‘exogenous’ improvement in mortality if only contemporaneous effects are taken intoaccount.

J. A. Birchenall (B)Department of Economics, 2127 North Hall University of California at Santa Barbara,Santa Barbara, CA 93106, USA

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In the paper, we develop a simple theoretical model of health transmission based ontraditional models of infectious disease mortality. The model consists of two main parts.First, we assume that parents have preferences over the number of surviving children perfamily and that child survival depends on child quality investments and the health capital ofparents. Because we relate parental health to a cohort-specific infectious disease component,a high level of child mortality can be seen as the outcome of low investments in childquality and/or as a consequence of unhealthy disease environments. A second part of themodel studies the evolution of health capital over the life cycle and the intergenerationallinks between the health status of parents and children. In particular, we assume that thereturn from child quality investments extends beyond increasing child survival as higherchild quality also improves adult health and consequently lowers adult mortality. One wayto interpret our second assumption is to consider that the health of children who survivechildhood infections is permanently ‘scarred’ as they mature to be unhealthy adults and todie prematurely. Following the epidemiological literature, we define this secondary effect asa ‘cohort’ effect.1

Because changes in child quality investments have a dual effect on survival, we arguethat economic changes induce long-lasting effects on mortality not previously accountedfor in the literature. For instance, economic changes are often assumed to be secondaryfor the secular mortality decline because income is only weakly correlated with mortalityrates (Preston 1975, 1980, 1985). If only contemporary or ‘period’ effects take place, anincrease in parental income (or in parental health) reduces child mortality but has no long-term consequences for the population as the health of the survivors will not be affected bychanges in early life. This amounts to assuming that economic changes have only transitoryeffects on health.

The dynamic aspects considered in our analysis of cohort effects suggest that income-driven improvements in child quality lower child mortality but also improve adult healthin general. For example, nutritional gains have reduced stunting of height in developedcountries as shown in Fig. 1. Since the improvements in height and in adult health arealso in part a consequence of changes in early-life, it is misleading to associate changesin economic conditions with contemporaneous effects exclusively. Moreover, since babiesborn to undernourished mothers are more likely to die prematurely and to suffer from lowbirthweight, changes in the health of one generation have permanent or long-lasting effectson future generations. Those secondary effects are also ignored in the current discussionson the role of economic changes in the mortality decline. For example, although we mostlyfocus on the early stages of the demographic transition, our interpretation of mortality changesuggests that the decline in old-age mortality observed in developed countries after 1960,often attributed to improvements in medical technology, could in part be the result of thebetter health conditions for the cohorts born during the first half of the 20th century whenpublic health eliminated the urban penalty in mortality.

Despite the importance of the mortality decline for the demographic transition and econo-mic development in general, theoretical analyses of the secular decline in mortality are much

1 The importance of conditions in early life for health and mortality at later ages was originally stressed byBarker (1994). Fogel (1994) also highlighted the role of early childhood in determining adult health outcomes.Almond (2006) recently evaluated Barker’s hypothesis using the increase in prenatal exposure to diseases dueto the influenza pandemic of 1918. Almond (2006) shows that disease exposure had a large and negativecausal effect on adult educational and health outcomes. Bengtsson and Lindstrom (2000) have also stressedthe importance of early-life conditions relative to the contemporary conditions for the outcome of airborneinfectious diseases (see also Elo and Preston 1992 for a survey on the effect of early-life conditions on adultand old-age mortality).

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155

160

165

170

175

180

1650 1700 1750 1800 1850 1900 1950 2000

Year of birth

Height (cm)

England US (free) Sweden France

Fig. 1 Height in selected countries. Data from Komlos (1996), Komlos and Baten (2004), Komlos andCinnirella (2005), and Fogel (1994)

less prevalent in the economics literature than studies of fertility change. It is often suggestedthat public health and exogenous changes in mortality are the main factors responsible for themortality decline mainly because of the spectacular mortality decline experienced by poorcountries during the first half of the 20th century. Nonetheless, in developed countries mostpublic health changes took place initially in cities at a time when the mortality decline wasalready under way. As indicated by the presence of an urban mortality penalty, cities weredeadlier than rural areas until well into the 20th century (Birchenall 2004).

McKeown (1976) downplayed exogenous changes and instead argued for a ‘Malthusian’channel that focused on contemporaneous improvements in income. The decline of tubercu-losis and airborne diseases in general, argued McKeown (1976), can only be explained byeconomic changes because no other intervention could have effectively contributed to theirdecline as no therapeutical measures existed prior to the 1950s. McKeown’s (1976) thesis,however, is inadequate since income is only partially responsible for changes in nutritionalstatus. All infections, regardless of the agent, increase metabolic demands, reduce nutrientabsorption, and lower overall health. By this synergism, income alone cannot be used tocharacterize the health status of the population or their mortality risks.2 In fact, inferencesbased exclusively on income generate very ‘puzzling’ results. For example, the decades prior

2 Technical treatments of the synergism between nutrition and infection include Scrimshaw and SanGiovanni(1997) and Lunn (1991). Harris (2004) revisits McKeown’s (1976) thesis and provides the needed qualifica-tions. Additional studies of the mortality decline include Bengtsson (1998), Fogel (1992), Fridlizius (1984),Livi-Bacci (1991), and Mercer (1990). Birchenall (2004) presents a brief overview that includes a quantitativeanalysis for modern countries and some stylized facts for mortality change.

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to the American Civil War are one of the periods of faster economic growth in the UnitedStates, but during that period adult life expectancy and height markedly declined (see Fig. 1and Fogel et al. 1978).3

The two distinctive features of the paper, the emphasis on a concept of health capital ornutritional status jointly determined by income and disease loads and the dynamic effects ofincome changes on mortality, can be traced to Fogel’s (1994) hypothesis for the mortalitydecline. Using measures of complete height (a net indicator of nutritional status), Fogel(1994) has shown that improvements in economic conditions are able to account for 90%of the mortality decline before 1870 and for 50% after 1870. The findings in our paper areobtained by studying the evolution of age-specific mortality rates and not by indirect measuresof nutritional status. We employ historical estimates of mortality for England and Wales from1640 to 2000 from Wrigley et al. (1997) and present linear estimates for the role of early-lifeconditions on adult mortality rates. Using a calibrated version of our model economy, wealso generate several counterfactuals to measure the contribution of contemporaneous effects,cohort (or intra- and intergenerational) effects, and changes in disease transmission for thesecular mortality decline. Overall, we show that intra- and intergenerational changes playedan important role in the secular mortality decline.

Because we argue that parental investments generate long-lasting effects on health, theideas in the paper can be naturally related to alternative models of human capital accumulationsuch as education. (We discuss the related research in the next section.) The paper also seesparental investments through Becker and Lewis’s (1973) quantity–quality model with childquality associated with health or nutrition. The use of health as a measure of child quality hasappeal for pre-modern economies because most of the mortality change was concentratedat early ages when no schooling decisions are made. Associating child quality with healthor nutrition also helps to understand the inverse U-shaped relationship between populationgrowth and income characteristic of all demographic transitions. Since a decline in mortalityacts as a reduction in the price of survivors (Fernandez-Villaverde 2003; Doepke 2005), fasterpopulation growth is the initial optimal response to an improvement in child nutrition or inpublic health. However, if parents value child quality beyond survival, population growth willalso decline in the later stages of the demographic transition, producing the characteristicinverse U-shaped relationship between income and population growth initially considered inGalor and Weil’s (2000) unified growth model.

The rest of the paper proceeds as follows: Sect. 2 places the paper in the literature ongrowth and population change. Section 3 describes the baseline model. Following the currentliterature, we associate pre-transitional conditions with a Malthusian economy. It is possibleto argue that the Malthusian setting is a reasonable characterization of pre-modern Englandand Wales because historically wages varied inversely with population, as the Malthusianmodel would predict. For example, Fig. 2 depicts wages and population since 1250 forEngland and Wales. Prior to 1650, the cross products between population growth and realwage growth are always negative. In contrast, the century following 1650 experienced positivepopulation and wage growth suggesting important technological changes (see Overton 1996;Kogel and Prskawetz 2001). Section 4 considers the conditions that produce an escape fromhigh mortality. To focus on the evolution of health along the transition, we study the responseto exogenous technological change as in Hansen and Prescott (2002). Section 5 presents a

3 The causes of the increase in mortality, the antebellum puzzle, are still debated, but it seems that the worseningof the disease environment played a major role in the decline in health associated with industrialization. Otherinfluences include changes in food prices, market integration (due to the development of railroads, canals, andsteamboats), the widening of income inequality, and the allocation of nutritious foods to the market rather thanto household consumption (see Komlos 1996; Komlos and Baten 2004 for detailed anthropometric analyses).

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3.0

4.0

5.0

6.0

7.0

0.5 1.5 2.5 3.5 4.5

Ln(population)

Ln(wages)

1650

1450

1340

Fig. 2 Population and wages in England and Wales, 1250–2000. Population for 1149, 1230, 1262, 1292, and1317 is based on Hallam (1988, p.536). For the years of the Black Death, estimates are based on Hatcher(1977, Fig. 1). Afterwards, we use the values of Wrigley et al. (1997, p.614) and Mitchell (1988). Wagescorrespond to Clark (2005)

quantitative analysis of the model and several counterfactual exercises. Section 6 concludesthis paper and offers some remarks for less developed countries.

2 Related research

Mortality change: The role of mortality change has been amply recognized in the currentliterature on the macroeconomics of population growth, but most of the emphasis has beenplaced on the consequences of mortality change rather than on the reasons for why mortalitydeclined or the behavioral aspects of child mortality. It is commonly assumed that mortalitydeclined exogenously or that the changes took place in response to contemporaneous incomebut were driven as an external effect not recognized by parents.

Some studies that focus on the consequences of exogenous child mortality change forfertility include Boldrin and Jones (2002), Ehrlich and Lui (1991) (where old-age securityis studied), and Doepke (2005) and Sah (1991) (where risk considerations on fertility arestudied). The effect of changes in adult mortality on the accumulation of human capital hasalso been studied by Soares (2005), where a quantity–quality trade-off arises between thequantity of children and the quality of parents (in a broad sense); as adult mortality declines,the quality of parents increases and the quantity of children declines (see also Boucekkineet al. 2003; Cervellati and Sunde 2005; Hazan and Zoabi 2006 for analyses of longevity).

Kalemli-Ozcan (2002) provides an example in which mortality varies with income as anexternal effect. Thus, fertility and education respond to changes in child mortality (in turn

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driven by contemporaneous income changes) but parental decisions have no direct influenceon mortality rates. 4 In Kalemli-Ozcan (2002), lower child mortality creates an initial incentivefor faster population growth due to the lower cost of survivors but afterwards a decline inmortality lowers a ‘precautionary’ demand for children (as in Sah 1991).

An example in which mortality plays a central role is Lagerlöf (2003). Lagerlöf (2003)considered a model with epidemics as the cause of death but assumed that mortality does notrespond to parental decisions. In Lagerlöf (2003), child mortality varies exogenously witheducation, population density, and epidemic shocks, but fertility and education decisionsare not directly affected by any of the parameters that govern mortality.5 In fact, in theMalthusian stage, parents set a target number of births independent of the child mortalityrate, see Lagerlöf (2003, Eq. 3.3).6 Epidemic shocks serve only to translate the number ofbirths into survivors and, through scale effects of population, to modify schooling decisions.

It is important to note that epidemic diseases and mortality crises in general are secondaryfor explaining the secular decline in mortality because the contribution of crises, at least inEngland and Wales, does not seem to be especially large.7 Instead of focusing on diseasesthat fluctuate in epidemic waves, we consider an equilibrium in which diseases becomeendemic rather than epidemic. Epidemic diseases need to be re-introduced systematically tohave a permanent effect on mortality while endemic diseases are constantly present in thepopulation so they can be seen as a factor responsible for the widespread presence of chronicmalnutrition in the past (see Fogel 1994).

As we assume that parental decisions influence mortality, our paper is closely related tothe work of Cigno (1998), Fernandez-Villaverde (2003), Galor and Moav (2004); and Jones(2001). In those papers, child mortality results from decisions undertaken by parents to offsetthe negative effects of additional inputs in the production of child health often associated withinfectious diseases (see also Blackburn and Cipriani 1998; Chakraborty 2004 for endogenoustreatments of survival). In the previous models, an increase in parental resources reduces childmortality but leaves health and morbidity unchanged once children reach adulthood. Since weexplicitly consider nutrition and parental care as health investments whose returns are spreadalong the life cycle, we study intra- and intergenerational aspects not previously consideredin the literature.

Support for the role of parental decisions in affecting child mortality comes from variedsources because parental resources could compensate for the disadvantages of some childrenor reinforce the advantage of a given child. Chu’s (1991) analysis suggest that primogeniture

4 As we show in the quantitative analysis, the correlation between income and mortality is rather weak (seealso Wrigley and Schofield 1981, Fig. 10.5). However, a contemporaneous channel fails to take into account thecomplete effects of economic changes. Also, as we noted in the introduction, income is not always sufficientto describe mortality in pre-modern populations (hence the puzzling aspect of the antebellum mortality).5 Tamura (2006) studies child mortality risk and also assumes externalities from human capital in childmortality. In general, the external effects of human capital are assumed to be positive but this is not alwaysthe case because schools are a common source of infectious diseases (see, for example, Miguel and Kremer2004).6 As Lagerlöf (2003, p.762) notes, since “[child mortality] depends on things that are taken as given [byparents],” under a logarithmic utility function the decisions of parents with respect to the number of survivorsare equivalent to the decisions with the number of births. Also note that epidemics are not produced by theeffect of disease vectors or by contact between infected agents.7 Before the national-level estimates of mortality of Wrigley and Schofield (1981), mortality crises receiveda disproportionate attention as the study of specific regions suggested a large role for epidemics. Since crisestended to average out spatially, the contribution to national mortality was much lower than the contributionat each specific location. And while the volatility of crude death rates declined during the secular mortalitydecline, the analysis by Fogel (1992) suggests that the elimination of crises mortality only accounts for 10%of the overall decline.

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is a parental response designed to avoid extinction in high mortality environments. Infanticideand neglect, as well as the favoring of one sex over the other, also indicate deliberate parentalinfluences on survival (Scrimshaw 1978). The close birth spacing associated with high fertilityalso elevates child mortality risk (Cleland and Sathar 1984; Curtis 1993).8 Finally, patternsof food consumption also support an endogenous link between the quantity of children andtheir health capital. Deaton and Paxon (1998, p.903) have shown that in both rich and poorcountries “there is a large negative association between food consumption per person andhousehold size.” Since diets are an important input in health production, a reduction in foodconsumption per capita would most likely translate into lower health. The previous claim issupported by the fact that children with many siblings are shorter in stature, Tanner (1978,p.89). In fact, as Weir (1993) has shown, the early decline in total fertility rates in France hada beneficial influence on heights (see also Schneider 1996).

Quantity–quality in Malthusian worlds: Following the lead of Becker et al. (1990) andGalor and Weil (2000), recent research emphasizes investment in human capital as animportant factor for the evolution of income per capita and population beginning in a longpre-transitional period and extending to the demographic transition and beyond. In both ofthe previous papers, pre-transitional conditions are seen as the outcome of a Malthusianequilibrium (or epoch) characterized by low investments in human capital, low income percapita, and low population growth. Both models also associate a modern regime with a steadygrowth in income per capita and with a negative relationship between fertility and educationarising from a quantity-quality trade-off.9

In the literature that follows Becker et al. (1990) and Galor and Weil (2000) (as in Galor andMoav 2002; Fernandez-Villaverde 2003; Lagerlöf 2003; Lucas 2002; Soares 2005, and mostof the models surveyed recently in Galor 2005), the demographic transition takes place byeconomic changes that lead parents to favor education instead of a large quantity of children.While related to the previous research, our view in this paper employs a complementary formof human capital given by health capital and measured by morbidity rates or the prevalenceof infection in a given period. In our paper, a decline in fertility improves overall health andsurvival.

A model of health capital differs from a model of education because an increase in healthinvestments lowers child mortality and hence makes children less expensive (i.e., raises thedemand for survivors). In models of education, an increase in human capital always makes

8 In close-birth intervals, protein-calorie malnutrition commonly affects the elder of two children since mothersfavor breast-feeding a younger sibling, Scrimshaw (1978, p.390). Similar effects on health exist for differentialsin birth order and for variations in the quantity of children. For example, firstborns are usually taller and livelonger possibly due to the absence of siblings during pregnancy and in early life (as siblings are a commonsource of infectious diseases). As Scrimshaw (1978, p.391) notes, the effects of fertility on child mortalityextend beyond ‘maternal depletion’ because if the first child of a closely spaced pair dies, the mortality riskof the second child remains unaltered.9 There are important differences between both approaches. Becker et al. (1990) model features multipleequilibria and requires an exogenous regime change to induce a demographic transition. Their Malthusianequilibrium and the transition to modern growth also induce a series of counterfactual predictions, see Galor(2005, p.235). In Galor and Weil’s (2000) model, there is no multiplicity of equilibria, and the transition isa feature of the equilibrium growth path. In their model, technology improves and increases the return toinvestments in human capital. Technological progress initially induces parents to spend more resources onraising children, and later on, it induces a reallocation of these increased resources toward education. Overall,the level of output and the growth rate of population feature an inverse U-shaped relationship along thetransition from Malthusian stagnation to modern growth. Lagerlöf (2006) presents a quantitative analysis ofGalor and Weil’s (2000) model. See also Bar and Leukhina (2007), Boldrin and Jones (2002), Doepke (2005),and Hansen and Prescott (2002) for quantitative analyses of the British case.

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children more expensive.10 To make the connection with models of education assume thatparents make decisions about the number of births b and that χ represents the probability ofsurvival to adulthood. The number of children who survive is defined by n ≡ bχ , and so thenumber of children who die is b[1 − χ]. The cost of children who do not survive is κ , andas in quantity–quality models, all survivors enjoy the same level of child quality c. Spendingon children is thus given by nc + bκ [1 − χ]. If q denotes income per adult and x parentalconsumption, the budget constraint is:

q = x + n

{c + κ

[1 − χ

χ

]}. (1)

Note that a costly child mortality induces a ‘wedge’ in the cost of the quantity of children;i.e., if κ = 0 or χ = 1, the marginal cost of child quality will be proportional to c (as inthe standard case of education). Since we assume that survival varies with c, investments inchild quality reduce the total cost of surviving children. As a consequence, we would observequantity and quality moving in the same direction in environments of high mortality and anegative relationship between both variables during the the late stages of the demographictransition.

3 A Malthusian economy

The environment: Consider an economy populated by a sequence of overlapping generations.The economy is described by the size of the adult population N and the size of the adultinfected population Y ≤ N . The state of the economy is (N, y) with y ≡ Y/N as the adultinfection rate. Infection rates translate into mortality by the case-fatality rate but we let yalso describe the adult mortality rate. All adults die at the end of the second period.

Income derives from agriculture under a Cobb-Douglas technology N1−βT β with landT and population N as inputs. We assume that health affects productivity because income islower for those adults who are infected. Income per adult is:

q (N, y) = h (y)

(N

T

)−β, (2)

with 0 ≤ h (y) ≤ 1 as a decreasing function of the adult infection rate.The function h(y) is not essential for any of the results of the paper, but makes it pos-

sible to consider the effect of infectious diseases on labor productivity as in efficiencywage models (Schultz and Tansel 1997). Obvious alternatives in which income per adultis h(y)1−β (N/T )−β would generate the same economic predictions:

Lemma 1 Income per adult, q (N, y), is decreasing in both arguments.

Some children die from infectious diseases during the first period of their lives. A fraction zof the children of any generation become infected. Of these, a fraction δ die and a fraction 1−δsurvives. The child survival probability is then χ = (1 − δz). Children become susceptibleand acquire infectious diseases by contact with an infected adult. (We ignore contagion from

10 In models of education, the quantity–quality interaction in the budget constraint often takes the followingmultiplicative form: nh′ with n as the number of survivors and h′ as their average education. Since a marginalchange in human capital h′ must be applied to all children n, the marginal cost of education is proportional ton, and, similarly, the marginal cost of child quantity is proportional to h′. Then, an exogenous increase in h′raises the cost of children and prompts a reduction in fertility rates and population growth.

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other children as a simplification but also because adults are an important source of infectionfor young children.)

The child infection rate z is a function of child quality and previously infected (infectious)adults, z = ψ (c, y). The functionψ (c, y) represents a ‘matching’ technology that describeshow infective and healthy individuals meet.11 For simplicity we assume that the mixingpattern of the population is random, by which we mean that ψ (c, y) = ψ (c)× y. In otherwords, the probability of child infections is proportional to the fraction of infected adults.

In models of mathematical epidemiology the functionψ(c) is called the (pairwise) contactrate or the rate at which a susceptible agent that contacts an infective source becomes infected.

In the interest of parsimony we take the contact rate as ψ (c) =(θ

c

)with θ > 0 as

measure of public health effectiveness, knowledge, and other exogenous influences (i.e.,parental education). Since public health affects infectivity, it is possible to think of publichealth changes as innovations that make an infective individual less able to infect healthypopulations. In general, c and θ can be thought as different forms of ‘technical change’ in theway diseases are produced. (Anderson and May 1991; Daley and Gani 1999 provide detailedanalyses of mathematical models in epidemiology.)

As Eq. 1 suggests, the odds of child mortality or the likelihood of a child death, i.e.,1 − χ(c, y)

χ(c, y)≡ δψ(c)y

1 − δψ(c)y, is the relevant concept for all the decisions in the model.

Assumption 1 The likelihood of a child death is well-represented by a function:

δψ(c)y

1 − δψ(c)y� δψ(c)y = δ

(θ

c

)y. (3)

Assumption 1 holds as an approximation but the qualitative properties of the economywill be maintained if we study the child mortality rate 1 − χ(c, y) instead of the likelihood

of a child death. For instance, if one assumes that δψ(c, y) = θ(yc

)/[1 + θ

(yc

)], the

approximation of the relative child mortality risk in (3) would be exact.Assumption 1 implies, as in other forms of ‘household production,’ that infected adults

are required to spread the disease. Also, as child quality increases, the odds of mortalitydecrease toward zero, and as child quality decreases, the odds of child mortality increase.Finally, there is a synergism between child quality and exposure to infection because themarginal effect of child quality on survival is higher in unhealthier environments.

Parents are responsible for all the decisions in the household. They care about theirconsumption x, the per capita quality of their children c, and the number of surviving childrenn. They maximize the utility function:

u(x, c, n) = (1 − γ ) ln(x)+ γ [ln(n− n)+ v ln(c + c)] . (4)

Assumption 2 In (4), n and c are strictly positive with 0 < γ < 1 and v > 1.

11 The view in which a healthy individual randomly meets an infected one has been previously studiedin the economic literature to understand the dynamics of infection in the transmission of HIV/AIDS byKremer (1996), and to evaluate the effectiveness of vaccinations by Philipson (2000). See also Gersovitzand Hammer (2004). Since the model offers a general view of diffusion, mathematical models of infectiousdisease epidemiology have also been used to study the diffusion of technology and expectations (see Aghionand Howitt 1999; Carroll 2005).

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The parameters n and c are non-standard, but they allow for deviations from the homo-theticity of logarithmic utility functions. As in Becker et al. (1990), c can be defined as anendowment of child quality. The role of the lower bound on the desired number of survivorsn will be evident in the analysis of modern growth because n > 0 is required for a constantpopulation growth. A positive c is needed to induce a negative relationship between quantityand quality in modern regimes.12 The assumption of v > 1, or that parents have a higherpreference for child quality, is important for interior solutions since this allows an equilibriumin which parents shift toward child quality at the expense of a constant asymptotic fertilityrate. We provide an example later on in this section.

Household decisions: Optimal allocations can be characterized by n (N, y) and c (N, y),continuous functions of the state of the economy. For all c and n, define the indifference andisocost curves by U ≡ {(c, n) : ln(n− n)+ v ln(c + c) = u}, and Q ≡ {(c, n) :n

(c + κ

[1 − χ(c, y)

χ(c, y)

])= q

}. The first order condition for the allocation between the

quantity and quality of children is:

− dn

dc

∣∣∣∣U : v (n− n)(c + c)

= �(n, c, y)

P (c, y): − dn

dc

∣∣∣∣Q . (5)

�(n, c, y) and P (c, y) define the ‘endogenous’ cost of child quality and the quantityof children (Becker and Lewis 1973; Blomquist 1989). In traditional consumer theory thesecosts are constant but in quantity–quality models the costs are themselves a function ofparental decisions. Because�(n, c, y) andP (c, y) change as mortality declines and incomeincreases, price effects will produce an initial increase in fertility followed by a fertilitydecline during in the later stages of the demographic transition. (The price effects due tolow mortality reinforce a positive income effect often considered as the cause of the initialincrease in population growth.)

From the budget constraint (1), these marginal costs are P (c, y) ≡ c+ κ

[1 − χ(c, y)

χ (c, y)

]

and �(n, c, y) ≡ nPc (c, y) or:

P(c, y) = c{

1 + κδθ( yc2

)}, and �(n, c, y) = n

{1 − κδθ

( yc2

)}. (6)

The next Lemma characterizes the marginal costs faced by parents.

Lemma 2 The marginal cost of survivors, P (c, y), is increasing in both arguments, andthe marginal cost of child consumption,�(n, c, y), is increasing in n and c, and decreasingin y.

To interpret the Lemma, note that the marginal cost of the quantity (quality) of childrenis increasing in the quality (quantity) of children as in standard quantity–quality models(i.e., Becker and Lewis 1973). This implies that quality investments increase the cost of thequantity of children and lead parents to substitute away from a large number of children. Underendogenous survival,�(n, c, y) is also an increasing function of c (i.e., the cost of quality isnPc(c, y)). This implies that there are increasing marginal costs in quality investments andso parents are limited in the amount of resources they can use to achieve low mortality. If

12 A baseline quality is also featured in Lagerlöf (2003) and in Greenwood et al. (2005) where c is interpretedas a level of child quality produced with household technologies. The role of c and n in providing deviationsfrom homotheticity is also related to the non-balanced growth model of Kongsamut et al. (2001).

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mortality is high, the investments needed to reduce child mortality are too costly and parentswould opt for increases in the quantity of children in pre-modern environments.13

It is illustrative to consider a simplifying example that will later describe modern regimes.Recall that κ introduces a ‘wedge’ in the cost of children and that if κδθy = 0, the modelbecomes identical to Becker and Lewis (1973). In that case, which we see as part of moderngrowth regimes since non-survival becomes negligible, the cost structure is �(n) = n and

P(c) = c. The first order condition (5) generates an allocation given by: v(

1 − nn

)= 1+ c

c.

Along such an allocation, the quantity and quality of children ‘move’ in opposite directions,

i.e.,dn

n= −1

v

( cn

) (nc

) dcc

. Asymptotically, as c tends to infinity, parents set a target

number of children given by vn/(v − 1); hence the need of v > 1 and n > 0.The allocation of resources when child mortality is costly is considered next:

Proposition 1 Let Assumptions 1 and 2 hold. Then, if non-survival is costly, n (N, y) andc (N, y) are decreasing inN . Moreover, if the effects of adult infection on income are small,n (N, y) is decreasing in y while c (N, y) is increasing in y.

By the endogeneity of prices, demands cannot be characterized by the response to incomeand prices as relative prices also change. Yet, a convenient way to interpret the previous resultis to consider demands that respond as if prices were not subject to a non-linear interaction.For example, higher population reduces income per adult and lowers child quality and thenumber of survivors through income effects. If h(y) = 1 in Eq. 2, a change in adult infectionrates affects demands by price effects alone. From Lemma 2, lower infection reduces the costof survivors and increases the cost of child quality (i.e.,�y (n, c, y) < 0 and Py (c, y) > 0).As a consequence, the number of survivors will increase and child quality will decline wheny declines.

Population and infection dynamics: Since the child infection rate is ψ(c)y and the case-fatality rate is δ, the fraction of children who die is given by δψ(c)y and the fraction ofinfected children who survive is (1 − δ)ψ(c)y. The fraction of survivors (or the relativecohort size) is given by χ(c, y) = 1 − δψ(c)y which equals the fraction of non-infectedchildren plus the fraction of infected children who survive, 1 − ψ(c)y + (1 − δ)ψ(c)y .

The dynamics of infection are generally very complex, but as in most endemic diseases,we assume that infected children who survive create a reservoir for the disease or becomethe only remaining source of infection (i.e., survivors become disease carriers):

Assumption 3 The fraction of infected adults in the next period y′ is:

y′ = (1 − δ)ψ(c(N, y))y

χ(c(N, y), y)� (1 − δ) θ

[y

c(N, y)

]. (7)

13 Since the budget set is not necessarily convex by the quantity–quality interaction, the necessary first orderconditions might not be sufficient. To ensure optimality of the interior solution, the slope of the indifferencecurve must diminish faster than the slope of the isocost curve as c increases. Indifference curves and isocostsare downward sloping and satisfy:

∂

∂c

(dn

dc

∣∣∣∣U)

= −[

1 + v

c + c

]{dn

dc

∣∣∣∣U}> 0,

∂

∂c

(dn

dc

∣∣∣∣Q)

=[�c (n, c, y)

� (n, c, y)− 2

(Pc (c, y)

P (c, y)

)] {dn

dc

∣∣∣∣Q}

.

That is, indifference curves between c and n are always convex but not necessarily more convex than isocostcurves. Since Pc (c, y) and�c (n, c, y) are positive, to ensure that second order conditions hold globally, it issufficient to strengthen the concavity of P(c, y).

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362 J Econ Growth (2007) 12:351–387

An explanation of Assumption 3 is as follows. Recall that child quality lowers childinfection rates. Thus, a direct effect of child quality is to reduce child mortality. This effectis represented by δψ(c)y. Because child quality also affects the fraction of survivors or theirhealth, (1 − δ)ψ(c)y, Assumption 3 guarantees that adult health is an increasing function ofchild quality.14 In other words, investments in early life generate the two fold effect consideredin the introduction: higher child quality increases child survival and also increases adult healthand survival (i.e., child quality reduces the adult prevalence of infectious diseases).

A generalization of Assumption 3 would allow non-infected children 1 − ψ(c)y to alsoacquire infections by contact with infected children ψ(c)y not just with adults, as we haveconsidered so far. For example, under a Cobb-Douglas mixing function, the prevalence ofadult infection in the next period can be seen as:15

y′ � [ψ(c)y]1−α [1 − ψ(c)y]α − δψ(c)y. (8)

Notice that the evolution of future infection in (8) depends very non-linearly on childquality and infection so the tractability of the model is drastically reduced under additionalinfectivity (we will consider this case in the quantitative section though). There are alsoseveral diseases that remain undetected for life and create persistent infectivity of which themost notable is tuberculosis.16 Tuberculosis represents by far the greatest decline of a singlecondition in the mortality decline in England and Wales (Harris 2004) and one of the mostimportant diseases in the mortality decline of less developed countries (Birchenall 2004).

For a given (N, y), the Malthusian economy can be described by a pair of differenceequations in the adult population and the adult infection rate:

N ′ = n (N, y)N , (9)

y′ = (1 − δ) θ

(y

c (N, y)

). (10)

Since we considered a simplified version of disease transmission, the dynamics of healthoutcomes in (10) relate cohort effects to early-life conditions expressed in terms of childquality investments, c(N, y), and the disease environments during the growing years, y.

Proposition 2 Let Assumptions 1–3 hold. Then, the adult infection rate, y′ is increasing inN and y.

Proposition 2 describes an intuitive pattern of intergenerational health transmission. ByAssumption 3, adult infection decreases with child quality or increases with population

14 Notice that y′ can be written as the number of infected survivors divided by total cohort size. Also, theapproximation in Eq. 7 is the same as in Assumption 1 because if survivors become carriers:

y′ =(

1 − δ

δ

)(1 − χ

χ

)�

(1 − δ

δ

)δψ(c)y,

in which the odds of a child death are also the fundamental concept.15 Assumption 3 corresponds to α = 0 in Eq. 8. In a general case, there is a non-monotonic transmission ofinfection. This non-monotonicity characterizes epidemic outbreaks in which the infection rate first increasesfrom very few cases and then declines as the potential population at risk gets reduced. See Anderson and May(1991) and Daley and Gani (1999).16 Individuals infected with tuberculosis remain so for life unless treated, while typhoid fever patients carrythe bacilli in the gallbladder and spread the disease continuously. Fenner (1982) presents additional examplesof infectious diseases with human-to-human transmission and recurrent infectivity. See also Anderson andMay (1991, Sect. 10.3).

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J Econ Growth (2007) 12:351–387 363

(by Proposition 1). Child quality also increases with infection but the change is less thanproportional so there is intergenerational persistence in disease prevalence. This intergene-rational persistence in health outcomes is analogous to other forms of human capital trans-mission such as education because health status is transmitted by parental investments inchild quality and influenced by parents’ health endowment. As in models of education, forexample, Proposition 2 implies that economic changes have long-lasting effects on healthoutcomes.17

Equilibrium and stability: Since land is fixed and production features decreasing returnsto population, the previous economy achieves a steady state equilibrium with a constantpopulation, a constant mortality rate, and zero income growth.

Definition 1 A steady state Malthusian equilibrium is described by an adult populationlevelN∗ and an adult infection rate y∗ such that fertility and child mortality offset each other(i.e., n(N∗, y∗) = 1), adult infection remains constant across generations, and householdsmaximize utility. If y∗ is strictly positive, the Malthusian equilibrium is endemic; otherwise,it is a disease-free equilibrium.

To characterize the equilibrium, define N≡ { (N, y) : n (N, y) = 1} and Y≡{(N, y) :y = (1 − δ) θy/c (N, y)}. Equilibrium corresponds to the intersection of N and Y .Therefore, an endemic equilibrium can be summarized as consisting of two equations,n (N∗, y∗)= 1 and c (N∗, y∗) = (1 − δ) θ , and two unknowns, N∗ and y∗. A disease-free equilibrium is given by a pair N̄ and c(N̄, 0) in which child quality is determined whendemands are evaluated at n

(N̄, 0

) = 1.

Lemma 3 A disease-free Malthusian equilibrium always exists. In addition, there is a uniqueendemic Malthusian equilibrium.

As is traditional in infectious disease modeling, a disease-free equilibrium always existssince no infection generates no contagion in future generations, see Eq. 10. In addition,there is an endemic equilibrium in which infectious diseases have a constant and positiveprevalence rate and, consequently, there is a constant infectious-disease adult mortality ratey∗ and a constant child mortality rate:

δψ(c(N∗, y∗))y∗ =(

δ

1 − δ

)y∗. (11)

There is no population growth in any of the equilibria as the model is purely Malthusian.However, in the endemic equilibrium child quality is determined entirely by the diseaseenvironment represented by θ and δ, i.e., c(N∗, y∗) = (1 − δ)θ . A constant child quality asthe one described by the endemic equilibrium can be seen as an micro-founded ‘subsistenceconsumption.’ Subsistence would obviously vary with the disease environment and willnot be “located at the edge of a nutritional cliff, beyond which lies demographic disaster.”The level of child quality c(N∗, y∗) will ensure that health is perfectly persistent betweengenerations although conditions that facilitate the spread of diseases will obviously affect the

17 In models of education, the return to parental investments is given by higher future income, while in ourmodel investments in child quality lower child mortality and through changes in child mortality, also the adultmortality rate. A referee has pointed out that a dynastic representation with valuations for future adult incomewould generate a different investment policy. The main features of the model, however, will not affected (i.e.,a dynastic model would still exhibit a dynamic system as the one in the model). See Lucas (2002, pp.125–134)for a comparison of different arrangements and parental valuations in Malthusian settings. See also Bar andLeukhina (2007, Proposition 3).

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364 J Econ Growth (2007) 12:351–387

N

y

N*

y*

N

1=)y,N(n

-1(=)y,N(c δ)θ

Fig. 3 Malthusian steady states and equilibrium dynamics. The point (N̄, 0) represents the disease-freeequilibrium and (N∗, y∗) the endemic equilibrium. The properties of the trajectories that lead to (N∗, y∗)are studied in detail the Appendix where an example of the stability conditions for a parametrized example ispresented

level of ‘subsistence’ as well as the equilibrium level of infection and mortality (as we willlater show).18

The phase diagram of the system is in Fig. 3. In the figure, N̄ is the population in thedisease-free equilibrium and (N∗, y∗) the endemic equilibrium. By Proposition 1, along

the N locus, population and infection are negatively related,dy

dN

∣∣∣∣N = −nN (N, y)ny (N, y)

<

0, because a stable mortality schedule must exactly offset fertility to produce a stationarypopulation (i.e., higher infection rates must be compensated for by higher child qualityor by lower population levels). Along Y , population and infection are positively related,dy

dN

∣∣∣∣Y = −cN (N, y)cy (N, y)

> 0.

Although linear approximations to non-linear dynamical systems are subject to the usualcaveats, we present the analysis of local stability in Theorem 1.

Theorem 1 Let (N, y) ∈ R++ × [0, 1] be any economy that satisfies Assumptions 1–3, andlet J(N, y) be the Jacobian matrix of (9) and (10). Then, the disease-free equilibrium isunstable. If det

[J(N∗, y∗)

]< 1, the endemic Malthusian equilibrium is stable.

The disease-free equilibrium can be ruled unstable because a small perturbation initiatesa disease outbreak with dynamics that diverge toward the endemic equilibrium. By theMalthusian nature of the economy, the introduction of a ‘new disease’ will lower popu-lation and hence increase income per adult. Income gains, however, are not enough to fully

18 The previous quote was taken from Fogel (1994, p.377). It is followed by “rather than one level of subsis-tence, there are numerous levels at which population and food supply can be in equilibrium. However, somelevels will have smaller people and higher” normal “(non-crisis) mortality than others.” The standard approachin the literature has been to assume a fixed subsistence level of consumption, Galor and Weil (2000) and Galorand Moav (2002).

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J Econ Growth (2007) 12:351–387 365

compensate for the increase in infectivity, and a new equilibrium will be achieved with higherincome but also higher mortality.

The disease-free equilibrium is an interesting feature of the model because the introductionof new diseases plays a fundamental role in multiple ‘Malthusian’ instances. Ignoring therole of additional causes of death, the spread of HIV/AIDS in Africa can be seen as atransition from a disease-free to an endemic equilibrium. Young (2004) has argued that theHIV/AIDS epidemic will most likely lead to income gains for future African generationsdue to the presence of diminishing returns to scale in production. Since the infectivity ofthe population will also change, mortality will permanently increase. Also, as we showed inFig. 1, Americans had a health advantage over European populations because Americans werewell fed by pre-modern standards. Yet, the health advantage vanished by the introduction ofepidemic and endemic diseases into American cities (Fogel et al. 1978). As in the AfricanHIV/AIDS case, income and health gains in the antebellum varied in opposite directions.19

The stability of the endemic equilibrium is simple indeed since sources of instability canonly take limited forms. Assume that the economy begins in a situation with low populationand low infection rates. From Lemma 1 and Proposition 1, the low population levels corres-pond to high income per adult, high fertility, and low mortality. The previous changes inducefaster population growth and, as a consequence of population growth, a decline in incomeper adult. Population will increase and income will decline until the economy reaches thestable equilibrium.

The disease transmission reinforces the role of income changes in population growthbecause health capital is persistent in the model. For instance, as child quality improvesin response to income changes, mortality declines, and this gives a further incentive forfaster population growth. These reinforcing effects, nonetheless, could generate oscillationsin mortality and population. For example, fertility will increase due to income changes (asin the standard Malthusian case) but also due to changes in the cost of survivors (by theeffect of infectivity on the cost of survivors). As an excessively high fertility will lowerincome and increase infection, the next generation will have incentives for a rapid reductionin fertility. Their fertility decline will increase income and lower infection once more creatingthe oscillatory behavior described in Fig. 7 in the Appendix.

Land availability and public health: Health capital will slowly change in response toimprovements in public health or land availability but the long-term effects are entirelyMalthusian. By constant returns to scale in production, equilibrium can be written asn(N∗/T , y∗) = 1, and c(N∗/T , y∗) = (1 − δ)θ . Then, higher land availability increasespopulation (proportionally) and leaves mortality from infectious diseases unaffected. This isthe case because both Y and N shift outwards in the same proportion leaving equilibriuminfection rates and income per adult constant. Any other transitory change in technology willhave the same effect because all the gains will be fully devoted to more population but notto higher income per capita.

19 A referee has suggested an alternative view in which the dynamical system is subject to ‘shocks’ responsiblefor mortality fluctuations. Although a view based on shocks offers an interesting perspective on why mortalityfluctuates, shocks are not very important per se for the mortality decline. Also, a view based on shocks suffersfrom important shortcomings. For example, it is known that the Black Death was an economic boom for thesurvivors of the epidemic (i.e., Clark 2005; Young 2004). Since the vector responsible for the disease retreatedquickly, new shocks (not based on the epidemic) would have to be introduced frequently in order to explainwhy mortality remained very high (i.e., Hatcher 1986). Because we treat health as a form of human capital, aone-time shock such as the Black Death would generate persistent effects on health and mortality.

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366 J Econ Growth (2007) 12:351–387

Public health, represented by a reduction in the contact rate θ , reduces equilibrium infec-tivity:

dy∗

dθ= −(1 − δ)nN(N

∗, y∗)ny(N∗, y∗)cN(N∗, y∗)− nN(N∗, y∗)cy(N∗, y∗)

> 0, (12)

with the denominator positive as Y is steeper than N . Child mortality will also declineas θ declines, see Eq. 11. Because of the mortality reduction, population will increase as

θ declines, i.e.,dN∗

dθ= − ny(N

∗, y∗)nN(N∗, y∗)

dy∗

dθ< 0. Since technological conditions remain

unaltered, improvements in public health lower mortality, increase population, but lowerincome per adult. In fact, exogenous changes in mortality actually lower an already stagnantstandard of living.

4 The escape from high mortality

By the nature of the equilibrium, the only possibility for escaping from (N∗, y∗) is thatwages no longer decline as the population increases. Several possibilities, including endoge-nous or exogenous technological change, are likely to eliminate the Malthusian restriction onlong-run economic growth. However, it is possible to show that any form of technologicalchange that produces positive long-run income growth is consistent with a demographic tran-sition in our model, because the Malthusian economy described above behaves asymptoticallyas economies originally devised by Becker and Lewis (1973), economies in which higherincome is associated with higher investments in child quality and a decline in populationgrowth.

Technological change: Consider, as in Hansen and Prescott (2002), a modern technologywith constant returns to scale given by ANh (y) s, in which A represents labor productivityand s the share of employment. Income per adult is:

q (A,N, y) = maxs

{(1 − s)1−β(N/T )−β + As

}h(y). (13)

If (1−β)(N/T )−β > A, s = 0 and the economy reverts to the one in the previous section.Otherwise, s is given by: s(A,N) = 1 − (1 − β)1/βA−1/β(N/T ), an increasing functionof A and a decreasing function of N . When the modern technology is in use, q (A,N, y) isalso increasing in A, and decreasing in N and y just as in Lemma 1.

In order to keep the model simple, we next assume exogenous technological change:

Assumption 4 Productivity in the modern technology satisfies: A′ = gA, with g > 1.

Theorem 2 Let (A,N, y) ∈ R2++ × [0, 1] be any economy that satisfies Assumptions 1–4.

Then, the economy described by (A,N, y) escapes from the endemic Malthusian equilibriuminto a balanced growth disease-free equilibrium in which population grows at a rate b̂ =n̂ = vn/(v − 1) and income per capita grows at a rate g.

Assumption 4 is not crucial to Theorem 2. Modern productivity may grow by learning-by-doing that depends on how fast adoption is taking place, by scale effects on population,or by alternative mechanisms. Assumption 4, however, influences the way the transitioninto modern growth is studied and the interpretation of the Malthusian equilibrium. Forexample, a transition from a Malthusian economy in Becker et al. (1990) (and Lucas 2002)requires an exogenous change in the return to education while in Galor and Weil (2000) (as in

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J Econ Growth (2007) 12:351–387 367

Hansen and Prescott 2002; Galor and Moav 2002) the transition is part of the equilibriumpath of the economy.

Under Assumption 4, the introduction of a modern technology makes a transition fromstagnation inevitable. Trajectories move away from the Malthusian equilibrium into a ba-lanced growth path in which infection and mortality are zero, and fertility reaches a targetn̂ = vn/(v − 1). Nonetheless, when A is below a threshold given by (1 − β)(N/T )−β , theeconomy remains in a Malthusian regime because s = 0 in Eq. 13. Hence, while the onlyequilibrium is the disease-free equilibrium in Theorem 2, the economy behaves as predictedby the Malthusian model until the threshold is reached.20

Finally, a transition also generates the correct pattern for demographic variables or ademographic transition:

Corollary 1 Under the conditions of Theorems 1 and 2, an escape from the Malthusianequilibrium generates a decline in adult and child mortality rates, and an inverse U-shapedrelationship between the quantity and quality of children.

In pre-transitional environments, higher income increases child quality and the quantity ofchildren, while in modern environments higher income increases child quality but reduces thequantity of children. This transition features two aspects that differ from related research onunified growth models (see the survey by Galor 2005). First, we have related human capitalto the health of children and adults, and hence quality investments yield returns in the formof higher survival. Second, we considered a ‘goods cost’ of non-survival rather than a timecost. Then, as mortality declines, the economic costs of non-survival and the incentives fora higher quantity of children decline so the quantity and quality of children end up movingin opposite directions.

5 Quantitative findings

Data: We next present a quantitative analysis of the model and the main proposed hypotheses:the role of early-life conditions on adult mortality as highlighted in Assumption 3, and therelationship between child and adult mortality rates as a mechanism for health transmission assuggested by Assumption 1. We employ long-term data for mortality in pre-transition Englandfrom the family-reconstruction of 26 parishes (Wrigley et al. 1997). The overall pattern ofmortality in the family-reconstruction data is similar to the pattern using back-projection byWrigley and Schofield (1981), but reconstructions have generated age-specific death rates notavailable before.21 Having access to age-specific mortality is important because we assumethat adult mortality varies with economic conditions and the disease environment in earlylife.

To examine the determinants of adult mortality rates, Table 1 presents OLS regressionsbetween the adult mortality rate for ages between 25 and 65 (40q25) and the contemporaneous

20 A qualitative change in dynamical systems due to threshold externalities is a feature studied by Azariadisand Drazen (1990). Previous versions of this paper assumed that productivity growth in modern technologiesdepended on the rate of adoption, g(s) with g(0) = 1. A transition under this alternative assumption is notinevitable but it only requires the eventual adoption of modern technologies and that income does not declinefollowing adoption.21 Wrigley and Schofield (1981) presented adult mortality estimates, but those were “not based on directobservation in the same sense as the reconstitution-based rates,” Wrigley et al. (1997, p.286). The estimates foradult mortality represent mortality for individuals aged 25–65 because there is no information for individualsaged 15–25. Mortality beyond age 65 is less reliable by reduced sample sizes, Wrigley et al. (1997, p.289).

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368 J Econ Growth (2007) 12:351–387

real wage, ln (waget ). Table 1 includes the years between 1640 and 1870 under the assumptionthat no major public health change took place before that time.22 Table 1 also includesmeasures of the wage and the child mortality rate during ‘growing years’ representing pasteconomic and mortality conditions. We considered lags for 3 – 5 decades as adult mortalityrepresents mortality for individuals whose average age of death was 45. Table 2 is similarto Table 1 but extends the sample until the year 2000 and includes a linear time trend after1870 to capture public health changes. Both samples, however, generate similar results.Contemporaneous or period wages have a negative effect on adult mortality rates if consideredalone (or jointly with a time trend), but when past wages are included, the period effect ofincome is no longer statistically significant.

The effect of lagged wages is negative, statistically significant, and larger than the effectof contemporary or period income. While the effect of contemporary wages is about −0.16,past wages have an influence as large as −0.54. Moreover, while period effects alone onlyexplain about 35% of mortality change in pre-transitional times, including past wages almostdoubles the explanatory power of economic conditions, Table 1.

To the extent that past wages capture economic conditions in early life, we conclude fromTables 1 and 2 that cohort effects are more important for adult mortality than the period effectof economic changes. The lagged value of child mortality is also important and significant inthe complete sample. Since the estimated coefficient is about 0.6, Tables 1 and 2 also suggestthat improvements in child mortality during growing years have persistent effects on adultmortality, as Assumption 3 suggests.

Since the model also assumes that adult health and economic variables affect child mor-tality, Table 3 studies the relationship between wages and mortality before age 15 (mortalityin the first year of life is also studied separately in Table 3). As before, wages have a negativeeffect on mortality rates if considered jointly with a time trend. However, the period effectof wages becomes statistically non-significant when adult mortality rates are included. Aswe showed that adult mortality varied with past economic conditions, income actually has a‘multiplier’ effect on child and adult mortality, as suggested by the dynamic links discussedin the paper. Note also that the estimated effect of exogenous changes, while still significant,is reduced once adult mortality is included because part of the assumed exogenous changesare captured by adult mortality.

Calibration/estimation: The results from Tables 1 to 3, while supportive of the role ofearly-life conditions and intergenerational health transmission, only consider the mortalityaspects of the model. We next calibrate a complete version of our model to study mortalitychange and the demographic transition.

The first aspect we study is the production function. Because the effect of adult disease onlabor productivity tends to be quantitatively small, we assume h(y) = 1.23 Table 4 presentsan estimate of β in Eq. (2) based on the time series relationship between population andwages (see also Fig. 2). The estimates in Table 4 suggest that β is near 0.65 in pre-modern

22 Of the major breakthroughs in disease control, only vaccinations against smallpox took place before themid-19th century but the effects are not uniform or sufficiently big to be a major contribution; see Bengts-son (1998). Also, the continuing decline in the annual death rate from smallpox in England was disruptedimmediately after smallpox vaccination laws were enforced, McKeown (1976).23 Evidence from potential income losses due to illness and days disabled in very poor countries (modernCote d’Ivoire and Ghana) suggests that most of the effects of disease are on earnings, see Schultz and Tansel(1997). An extra day ill reduces log-earnings by about −0.011 and −0.0036 in Schultz and Tansel (1997,Table 5). Because the effects of disease are small, they are and unlikely to play a major role in the outcomeof the model. Schultz and Tansel (1997) provide larger estimates based on instrumental variable although theestimates are likely to be biased upwards. Fogel (1994) also argues that a large fraction of the population inpre-modern economies lacked the energy required for any substantial work.

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J Econ Growth (2007) 12:351–387 369

Tabl

e1

Peri

odan

dco

hort

effe

cts

onad

ultm

orta

lity

inE

ngla

ndan

dW

ales

Pre-

mod

ern

sam

ple,

1640

–187

0

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Con

tem

pora

ryec

onom

icco

ndit

ions

Ln(

wag

e t)

−0.1

66∗ (

0.04

)−0.0

0(0

.08)

−0.0

1(0

.08)

−0.0

1(0

.06)

0.03

6(0

.11)

0.07

3(0

.11)

0.06

9(0

.08)

Past

econ

omic

cond

itio

ns

Ln(

wag

e t−3

0)

−0.2

88∗ (

0.11

)−0.3

34∗ (

0.14)

Ln(

wag

e t−4

0)

−0.3

27∗ (

0.13

)−0.4

43∗ (

0.18

)L

n(w

age t

−50)

−0.4

17∗ (

0.12

)−0.5

42∗ (

0.14

)

Past

mor

tali

tyco

ndit

ions

CM

Rt−

300.

245

(0.4

1)C

MRt−

400.

616

(0.6

8)C

MRt−

500.

656

(0.4

0)R

2(N

.Obs

=24

)0.

340.

450.

500.

620.

460.

530.

65

Not

e:In

pare

nthe

ses,

we

repo

rtro

bust

stan

dard

erro

rs.∗

Indi

cate

sth

atth

eco

effic

ient

isdi

ffer

entf

rom

zero

atth

e5%

confi

denc

ele

vel.

Inal

lreg

ress

ions

,the

depe

nden

tvar

iabl

eis

the

adul

tmor

talit

yra

teA

MR

( 40q

25).

The

wag

ese

ries

isfr

omC

lark

(200

5,an

dth

eA

MR

was

com

pute

dfr

omW

rigl

eyet

al.(

1997

,Tab

le6.

19)

from

1640

to18

00an

dfr

omth

eH

uman

Mor

talit

yD

atab

ase

(200

7)af

ter

1840

(we

fille

dth

ega

pby

extr

apol

atio

nbu

tthe

excl

usio

nof

the

3po

ints

has

noef

fect

onth

ere

gres

sion

s)

123

370 J Econ Growth (2007) 12:351–387

Tabl

e2

Peri

odan

dco

hort

effe

cts

onad

ultm

orta

lity

inE

ngla

ndan

dW

ales

Com

plet

esa

mpl

e,16

40–2

000

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Con

tem

pora

ryec

onom

icco

ndit

ions

Ln(

wag

e t)

−0.1

11∗ (

0.02

)−0.0

27(0

.04)

−0.0

0(0

.04)

0.01

3(0

.04)

0.00

1(0

.04)

0.05

3(0

.04)

0.06

6(0

.04)

Lin

ear

tren

d t:t>

1870

−0.0

13∗ (

0.00

)−0.0

0(0

.00)

−0.0

01(0

.00)

0.00

1(0

.00)

0.00

1(0

.00)

0.01

9(0

.01)

0.02

7∗(0

.01)

Past

econ

omic

cond

itio

ns

Ln(

wag

e t−3

0)

−0.1

49∗ (

0.06

)−0.2

00∗ (

0.07

)L

n(w

age t

−40)

−0.2

28∗ (

0.08

)−0.3

52∗ (

0.10

)L

n(w

age t

−50)

−0.3

38∗ (

0.09

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300.

378

(0.2

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400.

641∗

∗ (0.

33)

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574∗

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=37

)0.

890.

900.

910.

930.

900.

920.

94

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123

J Econ Growth (2007) 12:351–387 371

Tabl

e3

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erm

inan

tsof

child

and

infa

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orta

lity

inE

ngla

ndan

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Chi

ldm

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rate

Pre-

mod

ern

sam

ple

Com

plet

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mpl

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–187

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870

1580

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1580

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016

40–2

000

Con

tem

pora

ryec

onom

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ndit

ions

ln(w

age t

)−0.0

16(0

.01)

0.01

2(0

.02)

−0.0

22∗ (

0.11

)0.

001

(0.0

0)−0.0

15∗ (

0.00

)0.

007

(0.0

0)L

inea

rtr

end t

:t>18

70

−0.0

21∗ (

0.00

)−0.0

14∗ (

0.00

)−0.0

11∗ (

0.00

)−0.0

07∗ (

0.00

)

Con

tem

pora

rym

orta

lity

cond

itio

ns

AM

Rt

0.32

5∗(0

.05)

0.36

3∗(0

.04)

0.23

4∗(0

.02)

R2

(N.O

bs)

0.02

(30)

0.54

(24)

0.92

(43)

0.95

(37)

0.91

(43)

0.95

(37)

Not

e:D

efini

tions

asin

Tabl

e1.

∗In

dica

tes

that

the

coef

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ble

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).T

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mor

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te( 1q

0)

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,Tab

le6.

6)fr

om15

80to

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and

from

the

Hum

anM

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Dat

abas

e(2

007)

ther

eaft

er

123

372 J Econ Growth (2007) 12:351–387

Table 4 Population and wages in England and Wales

Pre-modern sample Complete sample

1200–1650 1200–1650 1200–2000 1200–2000

Ln(populationt ) −0.651∗ (0.05) −0.696∗ (0.05) 0.636∗ (0.07) −0.643∗ (0.05)Linear trendt :t ≥ 1200 −0.002 (0.00)Linear trendt :t > 1650 −0.045∗ (0.00)Linear trendt :t > 1650×Ln(populationt ) 0.043∗ (0.00)R2 (N. Obs) 0.71 (46) 0.73 (46) 0.63 (81) 0.96 (81)

Note: Data as in Fig. 2. ∗ Indicates that the coefficient is different from zero at the 5% confidence level. Inall regressions, the dependent variable is Ln(waget ). Linear trendt :t ≥ 1200 is a time trend beginning in 1200,and Time trendt :t > 1650 a trend beginning in 1650. Linear trendt :t > 1650×Ln(populationt ) is a post 1650interaction between time and population. The coefficient for Ln(populationt ) corresponds to −β in Eq. (2)

samples and that the inclusion of a time trend has no effect on the relationship of populationto wages (i.e., there was no sustained technical change before 1650). In the complete sample,the association between wages and population is positive (with a coefficient of 0.63), butincluding an interaction between time and population generates an estimate of β consistentwith the evidence in the pre-modern sample. Table 4 thus suggests that the positive associationbetween wages and population in modern times is due to post-1650 technical change and anaccelerating effect well captured by population (see also Fig. 2).

To calibrate preference and disease parameters we follow a simple approach. We employthe observed wage series as input in the demand system obtained from the model. Thosehousehold demands are then used to generate net reproduction rates and measures of childand adult mortality rates. Those measures are contrasted with the actual time series forEngland and Wales using the parameters {v, γ,n, c, κ, δ, θ} to minimize the distance betweenobserved moments and the moments obtained from the model (as in a standard method ofmoments). For child and adult mortality we employ the distance in the first moment, and forfertility the distance between the predicted asymptotic level of fertility and a replacementfertility level which represents modern conditions (population growth is non-linear in thedata so its average is not informative). We assume that the effect of early-life conditions onadult mortality has a lag of four decades. The solution is numerical and the parameters areupdated until convergence (see Fernandez-Villaverde 2003 for a similar approach).

The values of the parameters that provide the best fit to observed moments are reportedin Table 5. The parameters are estimated in three samples. The first sample ends in 1800because child mortality increased between 1800 and 1860 due to changes in urbanizationnot accounted for by the model. (Assuming that disease environments remained unaffecteddespite rapid changes in urbanization is clearly inadequate, see Fig. 1.) The second sampleincludes the years until 1870, and the complete sample includes a declining time trend incontact rates after 1870 as a proxy for public health changes. Overall, the parameters arestable between samples although the estimated share of child spending and the fatality rateslightly change.

One way to interpret the mortality parameters in Table 5 is the following. Assume thatthe adult infection rate equals one and that there is no investment in child quality beyondthe endowment c. Under these circumstances, the estimated contact rate θ predicts a childmortality rate (i.e., Eq. 3) equal to 0.4 × 21.6 × 1/14.2 = 0.60, or 60%. This estimate fitswell a notion of an upper bound on child mortality in pre-modern times. A case-fatality rateof 40% is in the range of observed fatality rates for respiratory diseases such as tuberculosis(Birchenall 2004); see also Anderson and May (1991) for similar estimates of fatality rates.

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J Econ Growth (2007) 12:351–387 373

Table 5 Estimated parameter values

Parameter description Pre-modern sample Complete sample

1640–1800 1640–1870 1640–2000

(1) (2) (3)

Utility function and children’s costv Preference for child quality versus quantity 2.86 2.86 2.86γ Share of child spending 0.12 0.11 0.09n Bound on survivors per couple 0.64 0.64 0.65c Endowed child quality 14.20 14.20 14.20κ Cost of non-surviving children 28.05 28.05 28.05Adult and child mortalityδ Case-fatality rate for child infection 0.40 0.42 0.43θ Pairwise contact rate for infection 21.62 21.62 21.61gθ Exogenous decline in contact rates (post-1870) −0.15

Note: The parameters are explained in detail in the text. The calibration minimizes the distance betweenobserved and predicted moments using observed wages as inputs in the demand for the quantity and qualityof children. Sample period refers to the sample used to compute the data moments and parameters

Now consider preference and costs parameters. To interpret n, note that the lowest netreproduction rate in England and Wales in the complete sample was 0.75 in 1930. Theestimated n is close to the previous figure. Since the asymptotic level of fertility is n̂ ≡vn/(v−1), which the calibration sets to one, the estimated value of child quality is v = 2.86.This coefficient suggests that parents value child quality over twice as much as the quantity ofchildren. There is no accurate measure of the share of life-time income spent on children, but,on average, the fraction of time in child rearing before the age of five was 15%; see Reher(1995, Table 4). Because we estimate, γ = 0.12, the parametrized model suggests lowerspending although the spending share is not exactly equal to γ because of c and n. There isalso no comparison in terms of κ , but in historical populations, an additional child became apositive asset after the age of 12. Before that age, there were substantial costs associated witheach child. Lindert (1980, Table 1.8) estimates that an extra child lowered earnings by about4.7% before the age of one and about 10% before the age of 12. Our estimates are larger thanLindert’s (1980, Table 1.8) because income changed little in the long 18th century and themodel relies heavily on price effects to induce population growth (Proposition 1) (The costκ only reflect non-survival and so it does not take into account the cost of having unhealthychildren.).

Using the parameters in the first column of Table 5, Figs. 4–6 compare the observedvalues of adult and child mortality, and population growth (measured by the net reproductionrate) with the predicted values obtained from the model. As in the OLS regressions, themodel provides a good fit to mortality rates. The model also generates an inverse U-shapedrelationship between population and income but the predicted levels of fertility are belowthe observed values for England and Wales. With regards to fertility, the main quantitativedifficulty of the model is not the decline in population growth at the later stages of thedemographic transition or the weak association between income and fertility at high incomelevels but the rapid increase in fertility during the 18th century. Since the initial child mortalitywas small, the reduction in the cost of survivors is not sufficiently strong to induce a largeincrease in population growth.

The main use of our model is to measure the role of different factors in explaining mortalitychange. Using the model, we next consider the following counterfactual scenarios for the

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374 J Econ Growth (2007) 12:351–387

100

250

400

550

700

850

1600 1660 1720 1780 1840 1900 1960 2020

Adult mortality rate (per 1000)

Data Model (estimated)

Model (predicted)

Fig. 4 Observed, estimated, and predicted adult mortality rates (40q25) in England and Wales, 1640–2000.Data sources as in Table 1. The estimated values are obtained using the parameters in Table 5(1). The predictedvalues assume wage growth as in the data

decline of child mortality rates: In (a) we assume that economic changes take place asobserved in the data. (Scenario (a) is also depicted in Figs. 4–6.) In (b) we estimate periodeffects or assume that y′ = y at all times. That is, all cohorts are endowed with the samehealth capital or exposure to infectious diseases in early life. In (c), adult infectivity is allowedas in Eq. 8 with α = 0.5. Finally, in scenario (d), we estimate child mortality in the absenceof productivity gains as an indication of mortality levels under Malthusian dynamics. Sincethe scenarios mainly eliminate influences that affect child mortality through changes in adultmortality rates, it is uninformative to describe the dynamics of adult mortality.

The results are in Table 6. As the table shows, the model underpredicts mortality rates inthe 19th century because income gains failed to translate into lower adult and child mortality.(See Figs. 4 and 5. Notice also that the model predicts a monotone path for mortality if θis constant, see Corollary 1.) The speed of the decline in child mortality in the post-1900period is also slower in scenario (a) than in the data because we have ignored modern changesin public health in θ and in δ. To account for the increase in mortality in the 19th centuryand the subsequent role of public health changes, one can assume a transitory increase inthe contact rate θ due to urbanization. Alternatively, the predicted series in scenario (a) canbe shifted forward so that the mortality conditions predicted in 1800–1840 apply to the1850–1890 period.24 This case suggests that the predicted child mortality in the 1950–2000

24 Both approaches yield similar results because wage growth has been very stable in modern times. Formally,one way to include the adverse effects of urbanization is to consider that θ , the contact rate between healthyand unhealthy individuals, is increasing in s until the late-19th-century advent of the germ theory of disease.

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J Econ Growth (2007) 12:351–387 375

0

75

150

225

300

375

450

1600 1660 1720 1780 1840 1900 1960 2020

Child mortality rate (per 1000)

Data Model (estimated)

Model (predicted) Period effects only

Fig. 5 Observed, estimated, and predicted child mortality rates (15q0) in England and Wales, 1640–2000.Data sources as in Table 3. The estimated values are obtained using the parameters in Table 5(1). The ‘periodeffects only’ represents a counterfactual based on the contemporary relationship between child mortality andincome. The post-1800 path for the ‘period effects only’ scenario is described in Table 6(b). The predictedvalues assume wage growth as in the data, Table 6(a)

period would be 65.7 per thousand. Based on period conditions only, the predicted mortalityis twice as large, 125 per thousand.

Scenario (c), in which adult infectivity is generalized, reveals that additional infectivityoperates mainly to increase adult mortality. Since adults acquire new infections, adult morta-lity declines at a slower rate. By the association between adult and child mortality, additionalinfectivity reduces the speed of the decline in child mortality rates. As scenario (d) shows,the absence of productivity change in Malthusian settings generates a constant child morta-lity rate that would be similar to the levels in the 17th century. The adjustment path in theMalthusian model is described in the Appendix.

Overall, scenarios (b) and (c) have special difficulties with the initial decline in childmortality between 1640 and 1840 since most health improvements in those years were mate-rialized through a decline in adult mortality rates (see Fig. 4). For instance, scenarios in whichadult mortality remains high or constant underpredict the child mortality decline between1640 and 1840. As scenario (b) shows, a period effect predicts an increase in child mortalitybetween 1640 and 1840. The estimated period effect in scenario (b) obviously raises objec-tions about the role of economic conditions in child mortality, see Fig. 5. Note, however, as

Footnote 24 continuedChanges in the case-fatality rate δ have also been important in the post-1940 mortality decline in developedcountries and in less developed countries.

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376 J Econ Growth (2007) 12:351–387

0.65

0.95

1.25

1.55

1.85

1600 1660 1720 1780 1840 1900 1960 2020

Fertility rate (per adult)

Data Model (estimated)

Model (predicted)

Fig. 6 Observed, estimated, and predicted fertility rates in England and Wales, 1640–2000. The measure offertility is the net reproduction rate produced by the generalized inverse projection in Wrigley et al. (1997,Table A9.1) and Mitchell (1988) thereafter. The predicted values assume wage growth as in the data. Theasymptotic level of fertility implied by the parameters in Table 5 is n̂ = 1

Table 6 Counterfactual scenarios for child mortality in England and Wales

Time Data Estimated Counterfactual scenariosPeriod model

Complete Period Adult Malthusianmodel effects infectivity dynamics

(a) (b) (c) (d)

1640–1800 311.9 (30.71) 311.9 (20.76)1800–1840 271.3 290.2 346.0 324.7 337.01850–1890 270.1 157.0 211.3 196.3 331.01900–1940 129.7 65.7 125.0 116.9 326.91950–2000 17.3 21.3 51.7 50.8 324.5

Note: The estimated model uses the parameters from Table 5(1). In parentheses we report standard deviations.The complete model, scenario (a), takes wages as observed in the data for the post-1800 period. The constantvalue of adult infectivity for scenario (b) is the observed adult mortality in 1640. Scenario (c) treats adultinfection as in Eq. 8 with α = 0.5. The Malthusian scenario (d) assumes that wages and population arenegatively related as suggested by Table 4. The Appendix contains a complete analysis of the Malthusianmodel

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J Econ Growth (2007) 12:351–387 377

we have stressed throughout this paper, that period effects represent only one influence ofeconomic conditions on mortality.

Since income changes were positive but small before 1850 (see Clark 2005), the quantita-tive analysis of the model suggests that economic changes have only a small direct effect onchild mortality. Because lagged income has important effects on adult mortality, economicinfluences in early life also contributed to the secular mortality decline by improving overallhealth and lowering adult mortality. For example, the post−1700 gains in adult mortalitycan be linked to improvements in economic conditions between 1640 and 1700 when thepost−1700 adults were at growing ages. In terms of the timing of the mortality transition,adult mortality declined first because of economic influences when those adults were at gro-wing ages. These improvements in adult mortality later lowered child mortality up until theadverse effects of industrialization started to take place.

6 Concluding remarks

Summary: The secular decline in mortality is commonly associated with exogenous publichealth changes and/or with improvements in the standard of living, with exogenous changesreceiving relatively more support as a result of a weak association between period economicconditions and mortality. That point is not contradicted by our analysis since period changesare quantitatively small to account for a significant change in mortality. Our paper, however,argues that the influence of period conditions is only one of the multiple channels that relateeconomic conditions to mortality change and that inferences based exclusively on periodeffects are misleading.

We developed a theoretical model of health transmission in which we employed a conceptof health capital determined by income and disease prevalence, and we allowed for a broaderrole of economic conditions on mortality. A separate role for income and disease in mortalityis important because despite rapid income gains, antebellum mortality initially stagnated oreven increased due to higher disease exposure and mortality in cities. (Other instances whenthe dichotomy mattered were discussed in the paper.) We have also shown that period effectsassign a very narrow role to economic aspects in mortality because in a period interpretationhealth capital remains unaltered as economies develop.

Since empirical evidence suggests that income gains and health changes in early lifelower child mortality but also improve overall health for the survivors, inferences based onperiod effects alone are misleading. As we consider health as a form of capital, economicchanges generate persistent effects on health analogous to the intergenerational transmissionof education or human capital in general.25 Thus, as economic conditions improve, childmortality declines, and the health of survivors improves. As health improves, economicconditions generate additional intra- and intergenerational effects not accounted for in aperiod analysis.

Using long-term data for mortality in England and Wales, we documented important effectsof past economic conditions on adult mortality. We also documented a positive relationship

25 Perhaps an example in a standard life-cycle model will indicate the dimensions not correctly accountedfor by period effects. In a life-cycle model of savings, an increase in adult income is often used for higheradult consumption but also for higher savings destined to increase old-age consumption in the future. Sincethe increase in consumption in old age is also ‘caused’ by higher adult income, a weak period relationshipbetween adult consumption and adult income will incorrectly suggest that economic conditions have a limitedeffect on consumption. The case we have considered in the paper shares the same implications because westudied a life-cycle model of health accumulation.

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378 J Econ Growth (2007) 12:351–387

Table 7 Life expectancy and income in world samples

1930 Sample 1960 Sample

Period effects Period effects Dynamic effects Dynamic effects

Ln(income1930) 0.213∗ (0.01) −0.024 (0.02) 0.497∗ (0.20)Ln(income1960) 0.119∗ (0.01) 0.099∗ (0.03) 0.101∗ (0.02)

Ln(e01930) 0.192∗∗ (0.10) 0.791∗ (0.28)

Ln(e01930) × Ln(income1930) −0.130∗ (0.04)

R2 (N. Obs) 0.81 (38) 0.77 (34) 0.80 (34) 0.85 (34)

Note: In all regressions, the dependent variable is log-life expectancy. In parentheses, we report robust standarderrors. ∗ and ∗∗ indicate that the coefficient is different from zero at the 5% and 10% confidence level,respectively. ln(incomet ) represents log-income per capita in 1930 and 1960 (both from Preston 1975). e0

1930Is life expectancy at birth in 1930, and Ln(e0

1930)× Ln(income1930) an interaction between income andmortality in 1930. Life expectancy is also from Preston (1975)

between child and adult mortality rates as a mechanism for health transmission. Both channelssuggest that the role of economic changes in mortality is larger than previously estimated.Although we limited our study to northwestern Europe, the demographic experience of En-gland and Wales has not been unique. The reductions in mortality initiated in northwesternEurope have spread throughout the world, and by the end of the 20th century, no countryremained at the first stage of the demographic transition with high crude death and birthrates. The experience of poor countries differs in important respects from the experienceof northwest Europe (see, for example, Birchenall 2004), but in poor countries, the role ofeconomic conditions also seems to extend beyond period changes.

Less developed countries: A weak association between contemporary income and morta-lity has been used repeatedly to argue that mortality in less developed countries is not relatedto changes in economic development and that most gains in mortality are unaccounted forby improvement in economic conditions. It is unquestionable that technological change hasplayed a very important role in the mortality transition of poor countries, but as we have notedin the paper, using a period link gives a very narrow role to economic conditions becauseperiod effects assume that economic development lowers mortality but has no long-termconsequences for the health of survivors.

To illustrate how period effects provide a limited account of the role of economic condi-tions, Table 7 presents OLS regressions between income per capita and life expectancy in1930 and in 1960 for a world sample of 38 countries from Preston (1975). We first estimatethe relationship between mortality and income in terms of period effects for both samples.As first noted by Preston (1975), the period response to income declines markedly between1930 and 1960 as the estimated income elasticity declines from 0.21 to half of such value in1960.

Note also that on average income in 1930 has no effect on the mortality conditions in 1960(actually the average effect is −0.024 but not different from zero statistically), but when aninteraction between income in 1930 and the health conditions in those years is included inthe estimation, income in 1930 has important effects on the mortality conditions in the 1960s.Moreover, the effect of health and economic conditions in 1930 does not alter the role ofperiod effects in 1960 since the estimated elasticity is almost identical in all cases. That is,past economic and health conditions reduce the unexplained part of mortality differences andnot the period effect of income. Since the unexplained part is assumed to be due to exogenous

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J Econ Growth (2007) 12:351–387 379

changes not related to income, the estimates based exclusively on period effects provide onlya partial view of the role of economic conditions in mortality change.

A complete account of the role of cohort influences in mortality for poor countries isbeyond the scope of the present paper, but as in the experience of developed countries, theeffect of economic and health conditions on mortality extends beyond a period reduction inmortality. Consider the following example. A 1% permanent increase in income increaseslog-life expectancy in 1930 by 0.21 (Table 7). Such 1% increase also generates the followingindirect effect on log-life expectancy in 1960: 0.49 − 0.13 × ln(e0

1930). In poor countries, apermanent 1% increase in income starting in 1930 will thus improve life expectancy in 1960by: 0.10 + 0.49 − 0.13 × 3.2 = 0.174, where the first term represents the period effect in1960 and 3.2 represents the logarithm of life expectancy at birth in the countries with thehighest mortality rates in 1930 (e0

1930 = 24.5). Hence, while the period effect declines inhalf, the total effect of permanent income changes in poor countries is only slightly lowerbetween the two periods.26

Period effects also leave aside important long-term effects of public health improvements.A permanent change in health conditions due to exogenous factors in 1930 would also reducethe unexplained differentials in mortality during the 1960s. An exogenous 1% increase in lifeexpectancy in 1930 increases life expectancy in 1960 by: 0.79−0.13× ln(income1930) or by0.1929 using the average estimate of Table 7. This additional gain is also unaccounted for byperiod changes and would also have to be incorrectly attributed to exogenous changes in 1960.Actually, economic and health improvements induce long-lasting changes in mortality thatwould be masked as unexplained gains in a period interpretation. Since changes in economicconditions have both intra- and intergenerational influences on health and human capital,the relationship between mortality and period income provides an inadequate measure forunderstanding the role of economic changes in the escape from high mortality.

Acknowledgements This paper is based on my dissertation research. The article has improved thanks tothe detailed suggestions of the Editor and three referees. I would like to thank Robert W. Fogel, Robert E.Lucas Jr., and Nancy L. Stokey for their advice. I benefited from comments by Gary S. Becker, TommyBengtsson, Gregory Clark, Guy David, Stanley Engerman, Joe Gani, Bernard Harris, Patrick Heuveline, PaulHuck, Susan Jones, D. Gale Johnson, Finn Kydland, Ethan Lieber, Claudia Linares, David Meltzer, TomasPhilipson, Robert Pollak, Peter Rupert, Raaj Sah, Nevin Scrimshaw, Beatriz Urdinola, researchers at the Centerfor Population Economics, University of Chicago, and participants of seminars at the University of Chicago,University of California Santa Barbara, University of Pittsburgh, University of Cambridge, University ofCalifronia Davis, and Universidad de los Andes. Financial support from Banco de la República Colombia isgratefully acknowledged.

7 Appendix

7.1 Proofs

Proof of Lemma 2 The proof follows directly from Eq. 6. Recall thatnPc (c, y) = �(n, c, y)

> 0 so nPc(c, y) is also a cost and hencePc(c, y) is positive. Also,�c (n, c, y) = nPcc (c, y)

= 2nκδθ( yc3

)> 0. From (6), Py (c, y) > 0 and �y (n, c, y) < 0. Note that if we assume

26 The interaction term in Table 7 indicates larger effects of income in unhealthy countries (as in ourAssumption 1). Table 7 also shows that life expectancy in poor countries grew at faster rates from the transferof public health technologies after 1930. The exogenous health improvements were particularly important forthe period between 1930 and 1960, see Preston (1975). Additional estimates for the role of period incomeare available in Pritchett and Summers (1996). Estimates for the role of period economic conditions in poorcountries cluster at around 30–40%, Birchenall (2004).

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380 J Econ Growth (2007) 12:351–387

0

150

300

450

600

750

900

0.5 1.5 2.5 3.5 4.5

Ln(population)

Adult mortality rate (per 1000)

Data Malthusian model Endemic steady-state

Fig. 7 Trajectories in the parametrized Malthusian model in response to a decline in adult population and anincrease in adult infection rates. Parameters as in Table 4 and Table 5(1). Land available is T = 26, 8. Thefigure describes total population

that κ represents the cost per births rather than the cost of non-surviving children, we willobtain the same properties for the cost functions. Under this alternative view, P (c, y) ≡c + κ/χ (c, y) = c + κ + κδθ(y/c), but marginal costs are identical since the differencebetween both interpretations is a fixed component κ . �

Proof of Proposition 1 The demand system is: x = (1 − γ )[q − nP(c)],

n− n = γ

(q − nP(c)P (c)

), and c + c = γ v

(q − nP(c)�(n, c)

).

Under the assumptions in the text, it is possible to re-write the demands for n and c as:

n− (1 − γ )n = γ(qc

)(c2

c2 + κδθy

), and (14)

c + c = γ v(qn

)(n− n

n− (1 − γ )n

) (c2

c2 − κδθy

). (15)

The constants n and c only create minor differences with the interpretation of demandsso we will later approximate the demands around n = c = 0. Note also that when non-surviving children are costly, κδθy > 0, the demand system includes a ‘wedge’ into anotherwise standard quantity–quality trade-off (i.e., Becker and Lewis 1973).

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J Econ Growth (2007) 12:351–387 381

Substitution of n into the demand for child quality is possible and generates (after somesimple but tedious algebra):

c + c = γ vc

(n

(c2 + κδθy

) + cq

(1 − γ )n(c2 + κδθy

) + γ cq

) (c2 + κδθy

c2 − κδθy

), (16)

which implicitly defines the optimal allocation of child quality.27

Totally differentiating the previous equation and approximating the result around n = c= 0 gives:28

(4c2κδθy[

c2 − κδθy] [c2 + κδθy

] − 1

)dc

c�

(1−γγ

)dq

q+

(2c2κδθy[

c2 − κδθy] [c2 + κδθy

])dy

y.

(17)

The previous expression shows that if non-survival is costly, i.e., 4[c2κδθy+ (κδθy)2] >c4, child quality will be an increasing function of income per adult. By Lemma 1, c will bedecreasing in N . Child quality will also be increasing in y if the effects of adult infection onincome are small. Similarly, totally differentiating the demand for n gives:

dn

n� dq

q+

(κδθy

c[c2 + κδθy] − 1

)dc

c− y

[c2 + κδθy]dy

y.

Once the feedback effects are taken into account, the number of survivors will also beincreasing in income when non-survival is very costly. On the other hand, nwill be decreasingin y because the positive effect of y on child quality is small relative to the negative directprice effect of y on n. �Proof of Proposition 2 The effect ofN on y′ is mediated through child quality. Because c isdecreasing in N and y′ is decreasing in child quality (by Proposition 1), y′ is increasing inN . For y note

dy′

dy= (1 − δ)θ

c

[1 − dc

c

/dy

y

],

which according to the proof of Proposition 1:

dc

c

/dy

y� 2c2κδθy

4c2κδθy − c4 + (κδθy)2.

The previous expression is most likely positive and less than one so despite the parentalefforts to compensate children when the disease environment worsens, there is intergenera-tional persistence in health status. �27 For example, if the cost of non-survival and the endowed quality are small relatively to total child quality,κ/c � 0 and c/c � 0, the demand Eq. 16 is linear in income:

c =[

γ (v − 1)

n(1 − γ (v + 1))

]q.

Note again that v > 1 is needed for child quality to be an increasing function of income. The shareof spending in children must also be smaller than (1 + v)−1.28 The approximation is done over the total differential because if n and c are set to zero, the demand functionswill be indeterminate. That is, the allocation will determine total child spending but not how to divide resourcesbetween quantity and quality. The total differential is available upon request. The proof of Corollary 1 containsan alternative derivation of the positive income effects.

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382 J Econ Growth (2007) 12:351–387

Proof of Lemma 3 It follows, by the mechanics of disease transmission, that (N, 0) ∈ Y forallN > 0 since no infection produces no contagion to begin with. A disease-free equilibriumexists as in standard Malthusian models. From Eqs. 14 and 15 a disease-free equilibrium isa pair {N̄, c(N̄, 0)} which solves:

1 − (1 − γ )n = γ

(h(0)

c(N̄, 0)

)(N̄

T

)−β, and

c(N̄, 0)+ c = γ vh(0)

(N̄

T

)−β (1 − n

1 − (1 − γ )n

).

As we noted, child quality in the disease-free steady-state is not affected by any of themortality parameters. The endemic equilibrium follows from Fig. 3 because Proposition 1ensures that N and Y cross at least once as Y is steeper than N everywhere (i.e., Y positivelysloped). The endemic equilibrium is a pair {N∗, y∗} that solves:

1 − (1 − γ )n = γ

(h(y∗)(1 − δ)θ

)(N∗

T

)−β ( [(1 − δ)θ ]2

[(1 − δ)θ ]2 + κδθy∗

), and

(1 − δ)θ + c = γ vh(y∗)(N∗

T

)−β (1 − n

1 − (1 − γ )n

) ( [(1 − δ)θ ]2

[(1 − δ)θ ]2 − κδθy∗

).

�Proof of Theorem 1 The Jacobian matrix of the Malthusian system is

J(N∗, y∗) ≡⎡⎣ 1 + nN (N

∗, y∗)N∗ ny (N∗, y∗)N∗

−y∗ cN (N∗, y∗)c(N∗, y∗)

1 − y∗ cy (N∗, y∗)

c(N∗, y∗)

⎤⎦,

and the characteristic equation of J(N∗, y∗) is P (λ) = λ2 − tr[J(N∗, y∗)

]λ + det[J(N∗,

y∗)] = 0.A set of conditions for the stability of the disease-free equilibrium

(N̄, 0

), i.e., Bernussou

(1977, p. 5), is given by:

P (1) = 1 − tr[J(N̄, 0)

] + det[J(N̄, 0)

]> 0,

P (−1) = 1 + tr[J(N̄, 0)

] + det[J(N̄, 0)

]> 0,

P (0) = det[J(N̄, 0)

]< 1.

Transforming the inequalities into equations, the previous conditions define the saddle-node, period-doubling, and Neimark-Hopf bifurcation loci.

Notice that det[J(N̄, 0)

] ≡ 1 + nN(N̄, 0

)N̄ is related to the stability of unidimensional

Malthusian models. Notice also that because tr[J(N̄, 0)

] = 1 + det[J(N̄, 0)

], P (1) = 0, a

small perturbation near y = 0 sets off an infectious disease outbreak that moves the economyaway from that equilibrium pointing out its instability.

The trace at the endemic equilibrium can be calculated as:

tr[J(N∗, y∗)

] = 2 + nN (N∗, y∗)N∗

n(N∗, y∗)− y∗cy (N∗, y∗)

c(N∗, y∗).

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J Econ Growth (2007) 12:351–387 383

One can verify that at the endemic equilibrium tr[J (N∗, y∗)

] = 1 + det[J (N∗, y∗)

] +X (N∗, y∗), where

X(N∗, y∗) ≡ y∗N∗ [

cy(N∗, y∗)nN (N∗, y∗)− ny (N

∗, y∗) cN (N∗, y∗)]

c(N∗, y∗)n(N∗, y∗).

The stability conditions correspond to X (N∗, y∗) < 0, 2 tr[J (N∗, y∗)

]−X (N∗, y∗) > 0,and det

[J (N∗, y∗)

]< 1. X (N∗, y∗) < 0 if and only if Y is steeper than N near the

steady state. The second condition corresponds to 4+2(nNN

∗ − y∗cy/c)−y∗cynNN∗/c+

y∗nycNN∗/c > 0 whose only negative term is the one in parentheses (which is small as wehave shown in the proof of Proposition 2). For the third condition, det

[J (N∗, y∗)

]< 1,

note that there are not conditions that would guarantee the stability of the economy hencethe statement in the Theorem.29 If det

[J(N∗, y∗)

] = 1, the economy loses stability througha Neimark-Hopf bifurcation, while if det

[J(N∗, y∗)

]> 1, the endemic Malthusian equili-

brium is unstable. Instability will arise if a decline in population produces higher income anda sufficiently high fertility to overcompensate for the initial population decline. �Proof of Theorem 2 and Corollary 1 Assume that the introduction of A takes place at anymoment along the transition toward (N∗, y∗). Notice that by the increasing productivity, theMalthusian economy will eventually adopt the modern technology. Without loss of generality,assume that s(A,N) > 0, when A is introduced but assume to the contrary that (N, y) →(N∗, y∗).

By Assumption 4, A′ > A, and by Proposition 1, N ′ > N and y′ < y; thus, s(A′, N ′) >

s (A,N) > 0. However, as Eq. 13 shows, income is decreasing in N so the introductionof the modern technology might even produce an initial income decline if it induces a largepopulation increase. Assume that income remains constant after the introduction ofA; that is,q

(A′, N ′, y′) = q (A,N, y). Then, since y′ < y, price effects on fertility (Proposition 1) will

further increase population,N ′′ > N ′, and reduce infection y′′ < y′ (Proposition 2). The newlevels of population further encourage adoption so s

(A′′, N ′′) > s

(A′, N ′) > s (A,N) > 0.

If there is no income gain after two periods price effects still produce higher population, lowerinfection, and higher adoption. Therefore, even if income remains constant indefinitely, priceeffects ensure that N → ∞ and y → 0, contradicting the initial assumption.

In general, the first order condition in Eq. 5 can be alternatively written as:

v(

1 − nn

)=

(1 + c

c

) [c2 − κδθy

c2 + κδθy

]. (18)

Totally differentiating the first order condition (18) gives:

v(nn

) dnn

= −(cc

) [c2 − κδθy

c2 + κδθy

]dc

c+

(1 + c

c

)(4cκδθy

[c2 + κδθy]2

)dc

c.

Substituting the first order condition in the previous expression generates

(nn

) dnn

= −(

1 − nn

) [c

c + c− 4cκδθy

[c2 + κδθy][c2 − κδθy]]dc

c.

29 If stable, the invariant curves take a parabolic form with a common axis related to the eigenvectors of theJacobian. Thus, convergence to the stable Malthusian endemic equilibrium, i.e.,

∥∥(N, y)− (N∗, y∗)∥∥, can

take several forms. Convergence is either monotone (node of type I), or it oscillates in one (node of type II) orin two variables (node of type III). Convergence could also exhibits spiral oscillations, see Bernussou (1977,Fig. 1.1).

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384 J Econ Growth (2007) 12:351–387

The previous expression, very much related to the proof of Proposition 1, see Eq. 17,shows that there is a non-linear relationship between n and c. If the endowed child quality cis small relatively to the cost of non-survival κδθy, quantity and quality will move in the samedirection. As mortality declines, the positive association gives way to a negative relationshipbetween n and c. �7.2 The Malthusian model

The analysis of the Malthusian model only needs to determine land available,T , so we employthe assumed relationship between observed wages and population in 1640 (T = 26.80). Wealso assume h(y) = 1. The stability of the endemic Malthusian equilibrium is describedin Fig. 7 where we estimate total population as (1 + b)N with b as the fertility rate. Adultmortality equals y so Fig. 7 can be seen as analogous to the phase diagram in Fig. 3. The onlydifference is that Fig. 7 describes total population while the phase diagram refers to adultpopulation N . The difference is unimportant for the representation of the model.

The adult population in the endemic steady-state is N∗ = 0.6 million while total popu-lation is 3.7 million. Adult mortality in the endemic steady-state is y∗ = 0.395 and childmortality is 0.186. The adjustment path in Fig. 7 shows oscillations in both variables as anode of type III in Bernussou (1977, Fig. 1.1). These endogenous cycles, and their disap-pearance in the transition from Malthusian stagnation are also a quantitative feature of Galorand Weil’s (2000) model, see Lagerlöf (2006).

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