Lothar Grall. March 15, 2011

Transcription

1 Climate Change, Somatic Capital, and the Timing of the Neolithic Transition Lothar Grall March 15, 2011 Abstract This research proposes somatic capital as a hitherto neglected variable in the discussion of factors impacting the timing of the Neolithic transition. The paper develops a model of long-run growth in population and technology that builds on the trade-off between quantity and quality of offspring. It suggests that harsh environmental conditions during the ice age altered the evolutionary optimal allocation of resources towards offspring quality. Higher somatic investment in offspring increased research productivity of individuals and accelerated the rate of technical progress. Thus, the adaptive response triggered within human populations living in cold and harsh climate for thousands of years had a significant impact on the timing of the Neolithic transition. The theory further suggests that differential somatic investment can be identified as deep-rooted determinant of comparative economic development. Keywords: Out-of-Africa hypothesis, Neolithic Revolution, Somatic Capital, Climate Change, Ice Age Conditions, Technological Growth, Population Growth JEL Classification Numbers: J10, O10, O30 Department of Economics, Justus-Liebig-University Giessen, Licher Str. 66, Giessen, Germany. Lothar.Grall@wirtschaft.uni-giessen.de.

2 1 Introduction The Neolithic transition to agriculture was a remarkable episode in human history. The shift to plant cultivation and animal husbandry from hunting and gathering significantly changed the long-run economic outcome of mankind. Agriculture began about 12,000 years ago (12 kya) in the fertile crescent in the Middle East, and not long after that in North China and in Europe. For the first time human societies were capable of supporting a non-food producing class of specialized craftsmen and knowledge creators. Agriculture is therefore a necessary condition for modern civilization itself. Modern research on long-run growth attempts to identify deep-rooted determinants of comparative economic development. The aims are to explain the transition from a period of Malthusian stagnation to one of sustained economic growth and to identify factors that generated the differential pattern of development across the world (Galor 2005, 2011; Galor and Moav 2002; Galor and Weil 2000). The timing of the Neolithic transition is a key event in this respect. Per-capita incomes, life expectancies, and population densities all seem to be correlated with the region-specific timing of the transition (Ashraf and Galor 2010; Galor and Moav 2007; Olsson and Hibbs 2005). Specifically, Diamond (1997) argues that favorable biogeographic conditions expedited the transition from foraging to farming in well endowed areas. Olsson and Hibbs (2005) provide empirical support for the Diamond hypothesis and show that current variations in economic prosperity still embody the effects of the timing of the Neolithic transition. Similarly, Putterman (2008) finds that the year of the transition significantly effects incomes today. Apparently, it is important to identify factors that influenced the timing of the transition. Here I want to add a novel twist to the discussion. According to this research it is not only biogeographic factors (Diamond 1997; Olsson and Hibbs 2005; Putterman 2008) or climatic fluctuations (Ashraf and Michalopoulos 2010; Dow, Reed, and Olewiler 2009) that had an impact on the timing, but also climate induced variation in humans itself. The theory builds on Robson (2010) who develops a model of preferences over the quality and quantity of children. Following Robson, it is a key assumption of this paper that individuals make an evolutionary optimal decision of somatic investment in offspring, maximizing the long-run number of descendants. 1 The term somatic capital or embodied capital is, in a physical sense, organized somatic tissue. In a functional sense, somatic capital includes physical stature and strength, but also factors like immune function, coordination, skill, and knowledge, all of which affect the profitability of human activities like resource acquisition (Kaplan 1996; Kaplan et al. 1 The idea of a trade-off between offspring quantity and offspring quality is not new. Fitness-maximizing trade-offs have long been studied within the context of life history theory in biology (Charnov 1993; Lessels 1997; Roff 1992, 2002; Stearns 1992) and anthropology (Hawkes and Paine 2006; Hill 1993; Lummaa 2007). In economics, Becker was the first to introduce both a qualitative and a quantitative dimension to the demand for children (Becker 1960; Becker and Lewis 1973; Willis 1973) - 1 -

3 1995). Concerning anatomically modern humans it is also reasonable to treat somatic capital as brain size or intelligence. It is therefore a second key assumption of this paper that higher somatic investment increases the innovation capability of individuals, leading to higher per-capita research productivity. In this paper I argue that the migration of early modern humans out of Africa into Western Asia and Eastern Europe between 40 and 45 kya and then into Europe between 40 and 36 kya triggered an evolutionary adaptation to the cold and harsh climate in Eurasia during the ice age. Lower hunter-gatherer productivity in climatically challenging environments was overcompensated by increased somatic investment in offspring, reallocating parental investment from child quantity towards child quality. A higher research productivity of individuals accelerated the rate of technical progress in human populations living in cold areas of the ice age, eventually leading to an earlier Neolithic transition. The theory differs from Robson (2010) in several important respects. First, I add climate as an exogenous variable that influences the optimal allocation between offspringquantity and offspring-quality. Robson s analysis focusses on an exogenous change in the production mode instead. Second, I consider the long time span between the appearance of anatomically modern humans more than 160 kya and the Neolithic Revolution. I do not restrict the analysis on the transition itself, but investigate the run-up to the transition. Third, I introduce technological growth and therefore endogenize the timing of the transition. In this context, I investigate the interaction between climate, optimal somatic investment, population growth, and technological growth. The aim of this research is twofold. First, it adds somatic capital as a hitherto neglected variable to the discussion of factors impacting the timing of the Neolithic transition. According to the theory it is not only biogeographic factors or climatic fluctuations but also climate induced variation in human somatic capital that influenced the timing of transition. This variable is interesting because it potentially fits as an omitted variable causing both an early switch to agriculture and rapid subsequent growth, possibly leading to spurious regressions in the studies of Olsson and Hibbs (2005) and Putterman (2008). Second, differential somatic investment and its link to research productivity is identified as a potentially deep-rooted determinant of comparative economic development. The main argument of this paper is grounded in anthropological, archaeological, and climatological evidence. It is helpful to summarize the respective findings before developing the model. Thus, the paper is organized as follows. Section 2 outlines the relevant literature on modern human origins, shortly discusses the climate of the past, and gathers evidence for differential somatic investment. Section 3 formalizes the assumptions about somatic investment and research productivity, and incorporates them into a simple model. Section 4 characterizes the dynamical system and analyzes the economy with respect to the timing of the Neolithic transition. Section 5 concludes

4 2 Modern human origins and evidence on differential somatic investment 2.1 The Out-of-Africa hypothesis and the climate of the past The Out-of-Africa hypothesis for modern human origins is widely accepted today: living humans exhibit remarkably little genetic diversity compared to nearly all other mammals, and the implication is that all living people share a recent common African ancestor (Klein 2009, p. 615). However, mounting evidence points to a lag between the emergence of modern human anatomy and the emergence of modern human behavior. Recent studies suggest modern anatomy evolved at least kya (White et al. 2003). By contrast, early evidence of symboling behavior in the form of personal adornment dates to only 77 kya (d Errico and Vanhaeren 2007). Even more, the total package of modern human behaviors language, art, personal ornamentation, specialized tools, complex social organization, extensive trade networks that indicates a capacity for abstract thought, was not in place until about kya (Nowell 2010). Although it is impossible to show that this capacity did not exist earlier, it was at this time when human populations in Africa gained a significant fitness advantage that finally led to the Out-of-Africa expansion. The spread of African populations to western Asia and eastern Europe is currently dated to between 45 and 40 kya, whereas western Europe was only reached between 40 and 36 kya (Klein 2009, p. 743). However, the timing of the human dispersal is still a fundamental issue and earlier southern routes of expansion are actively debated (e. g. Armitage et al. 2011). Those earlier expansions are of limited interest here, since they appear to have been non-permanent and lack signs of full modern behavior. When anatomically and behaviorally modern humans migrated into Eurasia around 45 kya, they had to cope with challenging climatic conditions. Proxy data such as ocean sediments and ice cores show that the climate of the past 100,000 years (100 kyr) was significantly colder than today. The average temperature in the northern hemisphere at the height of the last ice age was around 12 to 14 C lower than current values, whereas the global average cooling was at least 5 C. This is not true for the tropics, where average temperatures seem to have been much closer to current figures (Burroughs 2005, p. 41). Overall, the Earth cooled significantly at mid-to-high latitudes during the ice age, but surprisingly little everywhere else. The climate record for the northern hemisphere in figure 1 draws a detailed picture of the climatic challenges faced by African migrants in Eurasia after 45 kya. While the initial migration itself probably took place during an interstadial, a period of relative warmth, ice age conditions returned and reached a peak during the Last Glacial Maximum (LGM) between ca. 24 and 21 kya. The ice sheets reached their greatest extent around 21 kya and covered most of North America, all of Scandinavia, the northern half of the British Isles, and the Urals (Burroughs 2005, p. 40). Portions of northern Europe, the central East European Plain, and much of Siberia seem to have been abandoned at this time. People - 3 -

5 #$ -./ $56278.$9:5;$ %$!#$!&%$!&#$!"%$!"#$ %$ &%$ "%$ '%$ (%$ #%$ )%$ *%$ +%$,%$ &%%$ <8.$9=>2;$ Figure 1: Temperature change in the northern hemisphere of the past 100,000 years. The time series uses GISP2 ice core data with average values for every 200 years (World Data Center for Palaeoclimatology, Boulder, Colorado, USA). continued to occupy southwest Europe, although there is evidence on population stress (Hoffecker 2005, p. 192). The end of the last ice age was initiated by a profound warming in 14.5 kya. However, this interstadial was relatively short-lived and around 12.9 kya the climate dropped back into near glacial conditions. The temperature rose to modern values only around 10 kya. This date constitutes the beginning of the Holocene and marks the break from ice age conditions that characterized Eurasia before (Burroughs 2005, p. 47). 2.2 The Neolithic transition to agriculture It is generally accepted that the beginning of the Holocene created climatic conditions that allowed farming in certain parts of the world. In the ensuing millennia, the transition to agriculture independently occurred in several regions of the world. However, the timing of the transition differed significantly between the regions. The rough dates range between 11.5 kya in the Middle East from where it spread to Europe, followed by about 9.5 kya in North China, 8 kya in South China, 5.5 kya in Central Mexico, and 4.5 kya in Sub- Saharan-Africa. Apparently, there is a rough ecogeographic patterning in the timing of the Neolithic transition: societies from over 30 N latitude experienced on average an earlier transition than societies from under 30 N latitude. The literature has identified several important factors that influenced the timing of the Neolithic transition. First of all, favorable biogeographic conditions have been identified as key drivers. In his seminal book, Diamond (1997) tries to explain the dominance of Eurasian societies. He argues that geographic aspects like the major axes of continents and biogeographic aspects like the availability of domesticable plants and species generated favorable conditions for agriculture in Eurasia. Olsson and Hibbs (2005) provide - 4 -

6 empirical support for the Diamond hypothesis and find evidence for a link between biogeographic endowments and subsequent economic development. Beyond that, climatic fluctuations themselves have been identified as an important factor. Ashraf and Michalopoulos (2010) highlight the role of climatic sequences as a determinant of technological sophistication in a hunter-gatherer society. They find evidence for the idea that climatic volatile environments forced foragers to improve their productive endowments faster. Similarly, Dow, Reed, and Olewiler (2009) propose a model in which population and technology respond endogenously to climate. They argue that a negative climate shock that increases local population densities above a critical level is the trigger for the transition to farming. 2.3 Evidence on differential somatic investment Living with the ice age had profound evolutionary consequences for human populations outside Africa. As modern humans spread out across the globe, their physique and physical size changed in response to environmental and climatic challenges. Ruff (1994) demonstrates an ecogeographic patterning in body mass and body height in living humans, with populations from higher latitudes being larger on average than those from lower latitudes. Ruff, Trinkaus, and Holliday (1997) use the breadth of the head of the femur to calculate the body mass of early modern humans. They show that the ecogeographic patterning can be confirmed for them as well: specimens from over 30 N latitude were significantly larger than those from under 30 N latitude. They calculate an average weight of 67 kg for early modern humans dating from the period 35 to 21 kya, while those from the period 21 to 10 kya averaged 63 kg. Thus, body mass of early modern humans declined over the respective period. Ruff, Trinkaus, and Holliday (1997) estimate this decline to be 16% within higher-latitude males between 50 and 10 kya. Comparing early modern humans from the end of the Pleistocene (around 10 kya) to living humans, they show that the body mass of high-latitude males and females was not significantly different from that of high-latitude specimens today, whereas lower-latitude males and females were about 10% larger than living lower-latitude samples. Apparently, an ecogeographic patterning can be demonstrated for both living and prehistoric humans: people from more northerly latitudes are usually taller, broader, and heavier than people living near the equator. These differences in body proportions directly translate to differences in brain size. A number of studies show a robust correlation between body dimensions and brain volume. For example, Witelson, Beresh, and Kigar (2006) calculate body height to account for 1 to 4% of the variation in cerebral volume within each sex. There is also a robust empirical relationship between brain volume and intelligence. In a meta-analysis of 37 samples across 1530 people, McDaniel (2005) calculates a correlation of brain volume and intelligence of Thus, about 10% of the differences in IQ test scores can be attributed to differences in brain size. If these corre

7 lations are valid for early modern humans, people from more northerly latitudes were, on average, not only larger, but had also bigger brains and were slightly smarter than people living near the equator. This can be considered as indirect support for the proposed link between somatic investment and individual research productivity. Note that it is impossible to directly test the link between body size and intelligence for early modern humans, since skeletal remains are quiet about cognitive skills. However, the frequency of certain brain regulating genes in living populations might unexpectedly support this hypothesis. Microcephalin, a gene regulating brain size, is currently discussed in this respect (Evans et al. 2005; Mekel-Bobrov et al. 2007). A key element of the proposed theory is a relationship between geographic latitude (i. e. environmental harshness) and somatic investment on the one hand and somatic investment and technological progress on the other hand. Interestingly, the archaeological record supports this link, as a significant correlation has been found between technological complexity and geographic latitude (Hoffecker 2005, p. 191). Although it seems logical to assume that declining temperatures directly acted as a stimulus to technological change, Hoffecker (2005, p. 190) notes that, first, many early inventions probably were not critical to survival in these latitudes and, second, a series of important innovations took place after climates in northern Eurasia began to warm (i. e. following the coldest phase of the LGM). Thus, archaeological findings support a more indirect link between climate and technological progress, as proposed in this paper. 3 The basic structure of the model Consider an overlapping-generations economy in which activity extends over infinite discrete time. Individuals are identical and live for either one period (childhood) or two periods (childhood and adulthood). Children face a mortality risk that may prevent them from reaching adulthood. In their childhood, individuals consume part of their parental income. Those that survive, reach adulthood, work as hunter-gatherers, and produce a single homogenous good. Adults allocate their income between quantity and quality of offspring, that is, they choose the number of children, n t, and the amount of somatic capital, k t+1, invested in each of these. The model abstracts from adult consumption. Somatic capital is embodied energy investment and reflects factors like body size, strength, immune function, coordination, skill, or knowledge. It can be defined as a stock of attributes embodied in the soma of an organism which can be used for fitness-enhancing activities such as production (Kaplan 1996; Kaplan et al. 1995). Somatic capital is certainly influenced by genetic factors which might change just slowly over time. However, somatic capital can also change rapidly, since it is directly affected by nutrition and parental care during child growth (Robson 2010, p. 285)

8 3.1 Production Consider a population of p t adult hunter-gatherers that support their progeny on a fixed amount X of land or resources. In every period t, the economy produces output Y t with somatic capital k t, land X, and labor p t as inputs. The production function exhibits constant returns to scale in land and labor. Production is influenced by environmental harshness e t. A high degree of environmental harshness, that is cold and harsh climate, reduces production. In the process of development, the hunter-gatherer technology level A t is subject to endogenous growth. Since the amount of resources X is fixed it can be normalized to one (X = 1). Per-capita output of an adult hunter-gatherer can thus be written as (1) y t = f (k t, p t, A t, e t ). The following restrictions are imposed on the per-capita production function: Assumption 1. (i) As a function of k, adult output is zero if somatic capital falls short of a minimum amount k, and is positive with diminishing marginal productivity for k > k. More precisely, for each (p, A, e) > 0, there exists k > 0 such that f (k, p, A, e) = 0, for k [0, k]; f k (k, p, A, e) > 0 and f kk (k, p, A, e) < 0, for all k > k; and f k (k, p, A, e) 0, as k. (ii) An increase in environmental harshness is disadvantageous and reduces output. That means f e (k, p, A, e) < 0, for all k > k, (p, A, e) > 0. (iii) An increase in the population size, p, decreases per-capita output, since there is a fixed amount of land. Thus, f p (k, p, A, e) < 0, for all k > k, (p, A, e) > 0. (iv) An increase in the level of technology, A, is advantageous, in that f A (k, p, A, e) > 0, for all k > k, (p, A, e) > Preferences regarding quantity and quality of offspring It is a general accepted view among cultural anthropologists that culture has liberated humans from most constraints of biological evolution. While this may be true for contemporary humans, it is certainly not for prehistoric hunter-gatherer societies. Especially human fertility should always have been under relatively strong selection, at least until very recent times. The human system of fertility regulation seems well organized to respond adaptively to variable environments (Kaplan 1996, p ). Following Robson (2010), the fundamental trade-off between quantity and quality of offspring and its relationship to environmental variables is formalized in a simple way. Let k t be somatic capital of an adult hunter-gatherer at time t. An individual s level of somatic capital is determined by parental investment during childhood. Hence, the adult - 7 -

9 hunter-gatherer decides on somatic investment per child, k t+1. Since the model abstracts from adult consumption, the number of children is (2) n t = f (k t, p t, A t, e t ) k t+1, where f (k t, p t, A t, e t ) is the adult s income from hunting-gathering activities. Following the insight of Kaplan (1996), it is a key assumption of this paper that the human fertility system is naturally selected to generate preferences regarding quantity and quality of offspring that are evolutionary optimal. Individuals do not make particular numerical choices of quantity and quality. Rather, preferences are assumed to maximize reproductive fitness, which is measured by a dynastic setup in terms of the number of descendants at a distant future. 2 Reproductive fitness, W t, of an adult at time t is given by (3) W t = ω φ(p t+ j )n t+ j, j=0 where n t+ j is the number of children in generation j at time t + j, and φ(p t+ j ) is their probability of survival to adulthood. The entire product is equal to the expected number of living descendants after ω generations have passed, with ω 0. Assumption 2. The probability of survival to adulthood, φ(p) (0, 1), decreases with population size, φ (p) < 0, for all p > 0. This reflects increased prevalence of diseases in a denser population. It is possible to extend the model and to introduce survival probabilities that depend both on population and somatic capital. Somatic capital would then additionally reflect investments in immune function that improve survivability of individuals. 3.3 Optimization Members of generation t choose the number and quality of their children so as to maximize their reproductive fitness. Substituting (2) into (3) and taking the logarithm of both sides, the optimization problem of a member of generation t is (4) k t+1 = arg max k t+1 subject to k t+1 > k. { ω ( ) } ln φ(pt+ j ) + ln f (k t+ j, p t+ j, A t+ j, e t+ j ) ln k t+1+ j j=0 The first order condition of this optimization problem depends on expected values for population, p e t+1, environmental harshness, ee t+1, and technology, Ae t+1. That means, a hy- 2 Reproductive fitness is typically measured in terms of a quantity, r, called intrinsic rate of increase of a population. r depends on factors like age-specific birth rates, age-specific death rates, and the age distribution, which are not representable in an overlapping generations setup (Kaplan 1996, p. 97)

10 f k k t+1 k Figure 2: Optimal somatic investment. pothesis on expectation formation is needed. Since the growth rate of both population and technology was less than 0.01% per generation until 10 kya (Kremer 1993, p. 683) and hence not noticeable for prehistoric hunter-gatherers, static expectations are assumed: Assumption 3. Adults use static expectations to predict next-generation values of population, technology, and environmental harshness: p e t+1 = p t, A e t+1 = A t, and e e t+1 = e t. Following assumption 3, optimization with respect to k t+1 leads to the condition (5) f k (k t+1, p t, A t, e t ) = f (k t+1, p t, A t, e t ) k t+1, as depicted in figure 2. Somatic investment is optimal if the elasticity of the production function is one with respect to somatic capital. Note that (5) also follows from maximizing the term f (k t+1, p t, A t, e t )/k t+1 with respect to k t+1. This leads to the following proposition: Proposition 1. For each (p t, A t, e t ) > 0, there exists a unique amount of somatic capital k t+1 > k, maximizing f (k t+1, p t, A t, e t )/k t+1. An adult that invests k t+1 units of output in each child and bears f (k t, p t, A t, e t )/k t+1 children maximizes reproductive fitness W t. Proof. Follows immediately from assumptions 1 and 3. Condition (5) implies that members of generation t invest in somatic capital of their children according to an implicit functional relationship between k t+1, p t, A t, and e t, (6) K(k t+1, p t, A t, e t ) f k (k t+1, p t, A t, e t ) f (k t+1, p t, A t, e t ) k t+1 = 0. The derivatives of (6) can be positive or negative in general. However, several assumptions appear economically plausible. First, it is reasonable that hunting and gathering under ice age conditions in Eurasia was somatically more demanding than hunting and gathering under milder African conditions. This is the case if travel distances to prey - 9 -

11 animals increased and edible plants were harder to find. Hence, the marginal product of somatic capital increases with environmental harshness, f ke > 0. Second, it appears plausible that food acquisition became somatically less demanding with growing population size, for example by increased cooperation possibilities between hunter-gatherers. Thus, the marginal product of somatic capital decreases with population size, f kp < 0, but less than the average product, f p k < f kp, so that there is a net gain in somatic capital to offset the negative impact of population size on per-capita output. Third, it is reasonable that hunting and gathering became somatically more demanding with growing technological level, for example by increased cognitive demands to handle sophisticated hunting technology. Hence, the marginal product of somatic capital is assumed to increase with technology, f ka > 0, but less so than the average product, f A k This implies a net loss in somatic capital with increasing technological level. > f ka. Assumption 4. Let for all k t > k and (p t, A t, e t ) > 0 the marginal product of somatic capital increase with climate and technology, f ke > 0 and f ka > 0, but decrease with population, f kp < 0. The increase with technology and the decrease with population are smaller than the one of average product of somatic capital: f ka < f A k and f kp > f p k. Further assume f k k + f kk > 0 which simply assures that the factor demand for k is elastic. With assumption 4, the derivatives of (6) are readily given as K k < 0 K p > 0, K A < 0, and K e > 0 for all k t+1 > k and (p t, A t, e t ) > 0. This leads to the following proposition: Proposition 2. Optimal somatic investment is a single-valued function of e t, p t, and A t, (7) k t+1 k(p t, A t, e t ), where k e (p t, A t, e t ) > 0, k p (p t, A t, e t ) > 0, and k A (p t, A t, e t ) < 0. Proof. The proposition follows from (6) and assumption 4. The second derivatives of (7) depend on the third derivatives of the production function. A concave reaction of optimal somatic investment to population size appears plausible economically, since it is increasingly difficult to offset a negative impact on per-capita output by raising somatic investment. There might also be a genetic limit to body size. Assumption 5. For all (p t, A t, e t ) > 0, the function of optimal somatic capital is assumed to be concave in population size, k pp (p t, A t, e t ) < Research productivity and the evolution of technology Although Palaeolithic societies did not support a class of specialized knowledge creators, the term research is used in this paper to indicate improvements of the level of technology, A t. Technological innovations in the late Palaeolithic between 50 and 10 kya have been

12 things like bone awls, bone points, stone lamps, heated shelters, eyed needles, tailored clothing, storage pits, domesticated dogs, fishhooks, bow and arrow, and so on. Klein (2009, p ) provides a brief overview on late Palaeolithic technology. With the emergence of anatomically modern humans between kya technological innovations increasingly occurred, although very infrequently at first. With the total package of modern human behaviors being in place around kya, the rate of technological change significantly accelerated. People became far more inventive and made technological innovations at an unprecedented rate (Klein 2009, p. 672). As discussed in section 2, there is a significant correlation in living humans between body dimensions and brain size on the one hand and brain size and intelligence on the other hand. Larger people have on average bigger brains and are slightly smarter than people with smaller body proportions. To reflect this, it appears reasonable to treat somatic capital as brain size or intelligence which raise an individual s ability to innovate. Hence, it is a second key assumption of this paper that the research productivity of an adult hunter-gatherer increases with somatic capital k t. Assumption 6. Let g(k t ) > 0 be the research productivity of an adult hunter-gatherer with somatic capital k t. It is assumed that g(k t ) increases with k t, g (k t ) > 0. Following Kremer (1993, p. 686), I use a simplified formulation of the growth rate of technology. This formulation assumes that population affects the growth rate of technology linearly. That means a larger population produces proportionally more ideas. Thus, if the per-capita research productivity g(k t ) is independent of population size and if the level of technology affects research output linearly, the evolution of the technology parameter A t over time is given by (8) A t+1 = g(k t )p t A t, where g(k t )p t > 1 for p t 0 and k t > k. The initial level of technology and somatic capital at time 0 is historically given at A 0 and k 0. For a sufficiently large population size the rate of technological progress between time t and t +1 is positive, Â t = g(k t )p t 1 > 0. This research equation captures the fact that it is both quantity p t and quality k t of people that influence technological growth The evolution of population and climate The size of the population of adult hunter-gatherers at time t + 1, p t+1, is given by (9) p t+1 = f (k t, p t, A t, e t ) φ(p t )p t, k(p t, A t, e t ) 3 It is possible to extend the model and to introduce a generalized research equation following the lines of Kremer (1993, p ). However, for the considered period of time, a generalized research equation generates qualitatively similar results (Kremer 1993, p. 692)

13 where p t is the size of the population at time t, k(p t, A t, e t ) is optimal somatic investment per child, n t = f (k t, p t, A t, e t )/k(p t, A t, e t ) is the number of children per adult, and φ(p t ) is the survival probability of progeny. The growth rate of population is then ˆp t = n t φ(p t ) 1. The size of the population at time 0 is historically given at a level p 0. The evolution of environmental harshness, e t, over time is exogenously given. The exact time series is population-specific. It depends on the original climate in Africa before the migration out of Africa, on the timing of the migration itself, on the destination climate in Eurasia if a migration occurred, and on the development of world climate over time. Figure 1 gives an impression of climate change in the northern hemisphere during the considered period of time. Of course, it is impossible to incorporate a complex and exogenously given time series into the theoretical analysis of a dynamical system. This can only be done in a simulation. Hence, the dynamical system is analyzed under the assumption that the level of environmental harshness is fixed at an initially low level, e l. Later, the dispersal of anatomically and behaviorally modern humans out of Africa around 45 kya into Eurasia is modeled by an increase in environmental harshness from a low to a high level, e h, with e h > e l. 3.6 The transition to agriculture This paper promotes the idea that a region-specific rise in somatic investment has significantly accelerated pre-transition growth rates and therefore expedited the transition to agriculture in areas with harsh climatic conditions during the ice age. To show this formally, a certain level of technology, Ā, is assumed to be necessary for the Neolithic transition. A hunter-gatherer society that exhibits higher growth rates will reach this level of technology earlier in time and therefore observes an earlier transition to agriculture. Assumption 7. Let a technology level of Ā > A 0 > 0 be necessary for the Neolithic transition. A society that reaches this level of technology changes the mode of production from foraging to farming. Note that this assumption is very simplistic. It is illusory to expect to identify some precise moment in history when foragers decided to take their first steps as farmers. According to Barker (2006, p. 31), the transition from hunting to farming must be understood in terms of gradually evolving relationships between people, animals, and plants. Nevertheless, this oversimplification is helpful to clearly state the key issue of this paper. 4 The dynamical system The development of the economy is characterized by the evolution of population, technology, somatic capital, and climate. It is determined by a sequence {p t, A t, k t, e t } t=0 that

14 satisfies a three-dimensional nonlinear first-order autonomous system in every period t: (10) k t+1 = k(p t, A t, e t ) p t+1 = f (k t,p t,a t,e t ) k(p t,a t,e t ) φ(p t )p t A t+1 = g(k t )p t A t, where the evolution of climate {e t } t=0 is exogenously given. The dynamical system is analyzed in four steps. First, I assume a fixed level of technology and climate and characterize the evolution of somatic capital and population towards a conditional steady state. Second, the comparative static of the conditional steady state is described. Third, I relax the assumption of fixed technology and study global dynamics for a constant climate. Finally, I explore the effect of an increase in environmental harshness on global dynamics and derive the consequence for the timing of the Neolithic transition. 4.1 The evolution of somatic capital and population Initially, suppose a fixed level of technology, A t, and a constant climate, e. The conditional evolution of population and somatic capital is characterized by a sequence {p t, k t } t=0 that satisfies the following two-dimensional system: (11) k t+1 = k(p t, A t, e) p t+1 = f (k t,p t,a t,e) k(p t,a t,e) φ(p t )p t. This dynamical subsystem is characterized by a globally stable steady state equilibrium ( p, k). To see this, consider the phase diagram depicted in figure 3. The phase diagram contains a Conditional KK locus, which denotes the set of all pairs (p t, k t ) for which, conditional on a given technological level A t and climate e, somatic capital is constant, (12) KK At {(p t, k t ) : k t = k(p t, A t, e) A t }, and a Conditional PP locus, which denotes the set of all paris (p t, k t ) for which, conditional on a given technological level A t and climate e, the population size is constant, (13) PP At {(p t, k t ) : f (k t, p t, A t, e)φ(p t ) = k(p t, A t, e) A t }. Obviously, the locus KK At is a strictly concave and monotonically increasing curve in the (p t, k t ) space. It intersects the k t axis somewhere above k. This follows immediately from proposition 2 and assumption 5. Somatic capital is increasing below the locus KK At and decreasing above. The locus shifts upward as e increases. It shifts downward as A t increases in the process of technological growth

15 k t p t+1 = p t k k t+1 = k t k p pt Figure 3: The conditional dynamical system for fixed technology and climate. Proposition 3. The locus PP At is a single-valued function of p t, A t, and e t, (14) k t p(p t, A t, e t ), where p p (p t, A t, e t ) > 0, p e (p t, A t, e t ) > 0, and p A (p t, A t, e t ) < 0. The locus can be convex or concave in general. However, for all (A t, e t ) > 0 there exists p > 0 such that p p (p t, A t, e t ) > k p (p t, A t, e t ) for all p t > p. It intersects the k t axis at k > 0. Proof. See the appendix. The locus PP At is a strictly upward sloping curve in the (p t, k t ) space. It intersects the k t axis below the locus KK At. Furthermore, it intersects the locus KK At at a point ( p, k), as depicted in figure 3. Population increases above the locus PP At and decreases below. The locus shifts upward as e increases. It shifts downward as A t increases in the process of technological growth. Point ( p, k) is the conditional steady state equilibrium. Proposition 4. Let (A t, e) > 0 be arbitrary, but fixed. There exists a unique steady state equilibrium of the conditional dynamic system (11). The conditional steady state equilibrium values p > 0 and k > k satisfy the equations (15) φ( p) f k ( k, p, A t, e) = 1 and φ( p) f ( k, p, A t, e) = k. Proof. Follows immediately from (5), assumption 5, and propositions 2 and 3. Interestingly, the steady state level of population, p, has the property that it also yields the maximum possible population size. To see this, maximize population p t by choosing somatic capital k t+1 subject to the steady state constraint φ(p t ) f (k t+1, p t, A t, e) = k t

16 Then, reproduction is socially optimal if the surviving marginal product of somatic capital is one, φ(p t ) f k (k t+1, p t, A t, e) = 1, which is just the steady state condition of optimal individual reproduction (15). It follows that the optimal social reproduction rate and the optimal individual reproduction rate are identical. Proposition 5. Let (A t, e) > 0 be arbitrary, but fixed. The unique equilibrium steady state ( p, k) is also the unique solution to max k t+1 {p t : φ(p t ) f (k t+1, p t, A t, e) = k t+1 }. Proof. See the appendix. 4.2 Comparative statics of the conditional steady state This section analyzes the impact of an increase in environmental harshness and technology on the conditional steady state. Initially, suppose a fixed level of technology, A t, and a low level of environmental harshness, e l. As follows from proposition 4, there exists a conditional steady state equilibrium ( p, k) = (p l, k l ). Consider now an exogenous increase in environmental harshness from e l to e h, as experienced by a population of hunter-gatherers migrating out of Africa into Eurasia around 45 kya. The following lemma derives appropriate comparative static results: Lemma 1. Let A T > 0 be arbitrary, but fixed. A rise in environmental harshness, e, increases the conditional steady state level of somatic capital, k, but decreases the one of population, p. It also follows that the conditional steady state values of survivability φ( p) and per-capita output f ( k, p, A t, e) rise, whereas fertility f ( k, p, A t, e)/ k = 1/φ( p) falls. Proof. See the appendix. Hence, the cold and harsh climate in Eurasia triggered an evolutionary adaptation to a new conditional steady state, ( p, k) = (p h, k h ), where somatic capital has increased, k h > k l, but population has decreased, p h < p l. Survivability has increased due to the smaller population size. As fertility is the inverse of survivability in the conditional steady state, it is obvious that fertility has declined. The effect on productivity f ( k, p, A t, e) is unambiguously positive. This is interesting, as the model exhibits much richer behavior than a simple version of a Malthusian model. In particular, there is no biologically predetermined subsistence level here. Rather, the steady state equilibrium level of per-capita income reflects influences of environment and technology. A close look at survivability helps to get an intuition for the comparative static results. It is clear from the perspective of the whole population that fertility falls if survivability increases because a lower fertility is needed to keep population at a constant level. This is, however, not clear from the perspective of an individual because the fertility decline is the consequence of a free decision to substitute offspring-quantity for offspring-quality. A rise in survivability increases both the marginal benefit of quantity and the marginal

17 benefit of quality. Contrary to quantity, however, quality is related to the potential number of grandchildren, which are twice-discounted by survivability and doubly benefit of an increase. Hence, a rise in survivability increases the marginal benefit of quality more than it increases the marginal benefit of quantity. So, adult hunter-gatherers improve reproductive fitness if they substitute offspring-quantity for offspring-quality. Now, suppose a fixed level of environmental harshness, e, and a low level of technology A l. The conditional steady state is (p l, k l ). The following lemma derives comparative static results for an exogenous increase in the level of technology from A l to A h : Lemma 2. Let e > 0 be arbitrary, but fixed. A rise in the technological level, A t, decreases the conditional steady state level of somatic capital, k, but increases the one of population, p. It also follows that the conditional steady state values of survivability φ( p) and percapita output f ( k, p, A t, e) fall, whereas fertility f ( k, p, A t, e)/ k = 1/φ( p) rises. Proof. Analog to the proof of lemma 1. Hence, an exogenous rise in the level of technology from A l to A h leads to a new conditional steady state equilibrium, ( p, k) = (p h, k h ), where somatic capital has decreased, k h < k l, but population has increased, p h > p l. Interestingly, productivity f ( k, p, A t, e) unambiguously falls. These comparative static results simplify the analysis of global dynamics in the next section. 4.3 Global dynamics In this section the assumption of a fixed level of technology is relaxed and the evolution of the technology parameter A t over time according to (8) is fully incorporated into the analysis. The assumption of a constant climate e is maintained for the time being. Hence, the development of the economy is characterized by a sequence {p t, A t, k t } t=0 that satisfies the three-dimensional system (10) in every period t: (16) k t+1 = k(p t, A t, e) p t+1 = f (k t,p t,a t,e) k(p t,a t,e) φ(p t )p t A t+1 = g(k t )p t A t. This dynamical system is dominated by hyperbolic growth in population p t and technology A t for a population size that is significantly greater than zero, p t 0. To see this, note that the growth rate of technology increases with the level of population and that the growth rate of population increases with the level of technology, since f A (k t, p t, A t, e) > 0 and k A (p t, A t, e) < 0. Thus, there is a simple feedback loop between population and technology that leads to increasing growth rates over time. Hyperbolic growth implies that the growth rate is proportional to the level. It is thus faster than exponential growth, where the

18 A t A t+1 = A t p t+1 = p t à p = g(k t ) 1 pt Figure 4: The conditional dynamical system for fixed somatic capital and climate. growth rate is just constant (Korotayev, Malkov, and Khaltourina 2006, p ). This prediction of the model is entirely in line with empirical data about the prehistoric growth of the world population (Kremer 1993, p. 683). 4 The global dynamics of the system is illustrated by a phase diagram, depicted in figure 4, that describes the evolution of technology and population in the plain (p t, A t ). The phase diagram contains a Conditional AA locus, which denotes all pairs (p t, A t ), such that for a given amount of somatic capital k t the level of technology is constant, (17) AA kt {(p t, A t ) : g(k t )p t = 1 k t }, and a Conditional PP locus, which denotes all pairs (p t, A t ), such that for a given amount of somatic capital k t and environmental harshness e the population size is constant, (18) PP kt {(p t, A t ) : f (k t, p t, A t, e)φ(p t ) = k(p t, A t, e) k t }. Proposition 6. For each k t > k there exists a unique p = g(k t ) 1 such that A t AA kt and > 0 if p t > ˆp (19) A t+1 A t = 0 if p t = ˆp < 0 if p t < ˆp. Proof. Follows immediately from the definition of locus AA kt in (17). 4 Interestingly, the model approximately behaves like a set of two difference equations that Korotayev, Malkov, and Khaltourina (2006, p. 24) propose for the simulation of world population growth. See the proof of theorem 1 for details

19 Hence, the AA kt locus, as depicted in figure 4, is a vertical line at a level ˆp = g(k t ) 1. This critical level decreases with the amount of somatic capital, k t. Furthermore, it is independent from environmental harshness, e. Proposition 7. The locus PP kt is a single-valued function of p t, k t, and e t, (20) A t h(p t, k t, e t ), where h p (p t, k t, e t ) > 0, h k (p t, k t, e t ) < 0, and h e (p t, k t, e t ) > 0. The locus can be convex or concave in general. It intersects the A t axis at a value à > 0. Proof. See the appendix. Hence, the PP kt locus is a monotonically increasing curve in the (p t, A t ) space. It intersects the A t axis at a point (0, Ã) and shifts downward as somatic capital increases, but shifts upward with a rise in environmental harshness. As arrows indicate, it is apparent from figure 4 that the economy either enters the upper right area and winds up to a situation of increasing population and technology, or enters the lower left area and collapses to the origin. Which situation occurs, depends on the initial values of k 0, p 0, and A 0. If for a given A 0 > 0 and k 0 > k the initial population size is large enough, p 0 0, the economy approximately follows a hyperbolic growth pattern. What happens to somatic capital if population and technology both grow hyperbolically? The growth rate of somatic capital between time t and t + 1 is given by (21) ˆk t = k(p t, A t, e t ) k t 1. It is clear from lemma 2 that the conditional steady state value of somatic capital decreases whereas the one of population increases with the level of technology. Hence, ongoing technological progress leads to decreasing somatic capital. It follows that the growth rate of somatic capital in (21) is negative. This prediction of the model is well in line with the findings of Ruff, Trinkaus, and Holliday (1997) summarized in subsection 2.3, which estimate a decline in body mass of early modern humans of 16% within higher-latitude males between 50 and 10 kya. The conditional steady state values of somatic capital and population change continually, because the growth rate of technology is strictly positive in every period t. Hence, the steady state values are never reached. 5 Rather, it is clear from figure 3 that somatic capital jumps into the lower left area of the phase diagram with an increase in technology. From there it would monotonically adjust to the new steady state unless another rise in 5 It is possible to change this aspect of the model, if the growth rate of technology is assumed to be positive only every several periods. If the time between episodes of growth declines with population size, the long-run behavior of the model would be the same. In the short-run, however, the conditional steady state of somatic capital and population would indeed be reached (Galor and Weil 2000, p. 816)

20 technology occurred. Hence, ongoing technological progress keeps somatic capital in the lower left area of the phase diagram. It is clear from (21) and proposition 2 that the growth rate of somatic capital increases with the size of population, k p > 0, but decreases with the level of technology, k A < 0. Since both variables grow hyperbolically, the net effect is ambiguous. Since there are no signs of an accelerating decline in body mass of early modern humans, a negative but increasing growth rate of somatic capital seems more plausible. Theorem 1. Suppose k p + k A > 0. Let e > 0 be arbitrary, but fixed, and the initial values k 0, p 0, and A 0 large enough so that the economy enters the upper right area of the phase diagram in figure 4. There, the growth rates of p t and A t are positive and approximately follow a hyperbolic growth path, ˆp t p t and  t A t. The growth rate of k t is negative, ˆk t < 0, but increasing, ˆk t / t > 0. Proof. See the appendix. 4.4 The impact of an increase in environmental harshness on the timing of the Neolithic transition This section analyzes the impact of an increase in environmental harshness on global dynamics. Initially suppose a low level of environmental harshness, e l. The aim is to show that an exogenous increase from e l to e h, as experienced by a population of huntergatherers migrating into Eurasia around 45 kya, temporarily increases the growth rate of technology. In a hyperbolic growth setting this implies under plausible assumptions a permanently higher growth rate of technology and an earlier Neolithic transition. The analysis is divided into two parts. First, I assume a fixed level of technology, A, and analyze the short-run dynamics of an increase in environmental harshness. Second, I relax the assumption of fixed technology and incorporate the short-run dynamics into the analysis of long-run dynamics from theorem 1. Suppose a fixed level of technology, A. The impact of an increase in environmental harshness is derived in the following theorem: Theorem 2. Let (A, e l ) > 0 be arbitrary, but fixed, and (k l, p l ) the initial steady state. If e increases to e h, there exists a new steady state (k h, p h ) with k l < k h and p l > p h. There can be no jump in population, but p t decreases to p t+1 > p h. There is a jump up in somatic capital to k t+1 > k h. After the initial jump, population and somatic capital decrease monotonically to the new steady state equilibrium (k h, p h ). Proof. See the appendix. The phase diagram in figure 3 helps to to get an intuition for the result. It is clear from propositions 2 and 3 that both the Conditional KK locus and the Conditional PP locus