6.1 Introduction

Beginning with this chapter we shift from a focus on technological developments to a focus on institutions. Chapters 6–8 explore the origins of inequality and warfare in a pre-state setting. Chapters 9–11 study the origins of cities and states.

These chapters go beyond the food production technologies of Chapters 3–5. In the rest of the book food production will still matter, but we will also have to consider technologies for coercion; that is, the technologies used by one social group against another social group. Chapter 6 introduces a simple technology through which organized insiders can exclude unorganized outsiders from a production site. We show that under suitable conditions, such a technology leads to endogenous group property rights and inequality. Chapters 7–8 consider technologies for conflict between organized groups, leading to an exploration of the conditions for war and peace. Chapters 9–11 study technologies of confiscation, where an organized elite can seize goods, land, or labor from unorganized commoners. This will lead to a theory of taxation and the state.

Economists and archaeologists use terms like “equal” and “egalitarian” in various ways, so we need to be careful about definitions. First, it should be noted that inequality could be social, political, or economic in nature. These types of inequality are positively but imperfectly correlated. Our concern here is only with economic inequality. Second, economic inequality could be defined in terms of wealth, income, or consumption. The theoretical framework of this chapter focuses on consumption.

We use the term “equality” in a modeling context where all of the individuals in a group or region consume the same amount of food, which is the only consumption good in the formal model of this chapter. When inequality arises in our model it is a matter of degree, and we will explain later how inequality can be quantified (see the discussion of Gini coefficients in Section 6.3). Of course, food varies both in quality and quantity, and people value additional consumption goods such as housing and clothing. For modeling purposes we will ignore these complications.

We use the term “egalitarian” (following archaeologists and anthropologists) when referring to situations where agents have equal access to subsistence resources, but some may consume more food than others due to special skills, hard work, or other forms of personal achievement (see Mattison et al., Reference Mattison, Smith, Shenk and Cochrane2016, 185). We are not concerned with such inequalities because we want to explain inequalities across sites or across the economic classes at a site, not among the individuals who make up these groups. When reporting on empirical evidence from archaeology or anthropology we interpret “highly egalitarian” to mean something approaching equal consumption among the members of a group.

We use the term “inegalitarian” when differences in food consumption arise from structural differences in access to subsistence resources such as the exclusion of outsiders from a site or control over land by an elite. This is consistent with common usage among archaeologists. For example, Mattison et al. (Reference Mattison, Smith, Shenk and Cochrane2016) call a society “inegalitarian” when access to subsistence resources is unequal due to inheritance or ascribed roles, so that inequality is persistent and institutionalized.

Archaeologists agree that almost every society was egalitarian before 10,000 BP, and that most societies have become inegalitarian in the last 10,000 years (Kohler and Smith, Reference Kohler and Smith2018). There are exceptions (for example, southwest Asia shows some signs of inequality well before 10,000 BP, as we will discuss in Section 6.11), but they are rare. This is consistent with evidence from ethnography, which indicates that mobile foraging groups are highly egalitarian. Such societies have been studied in the Kalahari Desert, Australia, Southeast Asia, Amazonia, and the Arctic (Kelly, Reference Kelly2013a). In these societies strong social norms support food sharing and oppose self-aggrandizement (see Boix, Reference Boix2015, 46–51, and the sources cited there). There may be some differences in work tasks associated with age, sex, and ability, but food consumption is evened out by sharing, and gender-based inequality is frequently muted (Leacock, Reference Leacock1992; Endicott, Reference Endicott, Lee and Daly1999). Hereditary class distinctions are absent.

Several reasons are often given for why mobile foraging groups tend to be highly egalitarian: (a) production technology is simple and available to everyone; (b) individuals have access to similar natural resources; (c) mobility limits the accumulation of material assets; (d) food storage technology is frequently minimal; (e) the technology for the use of violence is simple and widely available; (f) hunting often involves teamwork; and (g) there are large gains from risk sharing. Anthropologists particularly emphasize the value of food sharing as a form of insurance against bad luck, illness, or injury (Kelly, Reference Kelly2013a).

Sedentary foraging societies are more diverse. Some are relatively egalitarian, but others display considerable inequality both across and within communities. As we discussed in Section 4.1, group sizes for sedentary foragers are larger than for mobile foragers (Kelly, Reference Kelly2013a, 171–172). Compared to mobile foragers, sedentary foragers also have higher populations relative to natural productivity, more food storage, and greater inequality (Rowley-Conwy, Reference Rowley-Conwy, Panter-Brick, Layton and Rowley-Conwy2001, 40–44). Some societies of this kind have hereditary elites, as along the northwest coast of North America (Ames and Maschner, Reference Ames and Maschner1999).

More generally, sedentary foraging is correlated with “social hierarchies and hereditary leadership, political dominance, gender inequality, and unequal access to resources” (Kelly, Reference Kelly2013a, 104). To illustrate Kelly’s point about gender inequality, we refer again to Abu Hureyra (see Sections 4.3 and 5.3). This village along the Euphrates River was based on sedentary foraging during 13,500–13,000 BP and began to cultivate cereals in subsequent centuries. Molleson (Reference Molleson, Moore, Hillman and Legge2000, 311–317) reports that women suffered from severe skeletal deformation due to repetitive dehusking and grinding tasks. The specialization of male and female activities was clearly causing systematic and lifelong physical injuries to women.

Ethnography suggests that in agricultural economies all forms of inequality tend to become more pronounced, and class structures are often based upon elite control over land (Johnson and Earle, Reference Johnson and Earle2000). However, as with sedentary foraging, there is diversity among agricultural societies. Bogaard et al. (Reference Bogaard, Fochesato and Bowles2019) argue that labor-constrained (and chronologically earlier) forms of agriculture tended to exhibit less inequality than land-constrained (and chronologically later) forms. We return to these issues in Section 6.3.

We argued in Chapter 5 that cultivation triggered a long process of learning by doing that raised productivity. Over the course of one or two millennia, cultivation often led to domestication, with additional productivity gains. Here we treat this productivity growth as exogenous and explore its implications for population, property rights, and inequality, which are endogenous. Specifically, we explain the associations among (a) rising productivity, (b) rising population, and (c) rising inequality. The link between (a) and (b) results from Malthusian dynamics while the link between (b) and (c) results from endogenous property rights.

Our starting point is a region in which many identical agents have free mobility across many food acquisition sites. These sites vary in quality due to water availability, soil fertility, and other factors. In a world of open access, food consumption is equal across sites (and individuals) because if it were not, agents would move from sites with low consumption to sites with high consumption. As in Chapter 5, equilibrium implies that local populations must be highest at the best sites.

The new element in this chapter is a technology of exclusion, which underpins the formation of group property rights over production sites. When the number of agents at a site exceeds a critical mass, as it eventually will due to Malthusian population growth, the insiders can drive away or kill outsiders who try to enter. This prevents any further entry and converts open-access land into an exclusive resource.

The insiders at a closed site continue to share food equally. They are better off than outsiders who remain in the commons, and insiders at higher-quality sites are better off than insiders at lower-quality sites. We call this situation insider–outsider inequality. Additional growth in productivity and population leads to the enclosure of lower-quality sites. Regional inequality increases in the long run, and the agents still in the commons become increasingly impoverished.

When productivity and population are sufficiently high, stratification emerges at the best sites and then spreads across the region. At stratified sites, insiders control land and employ outside labor at a wage equal to food per person in the commons, generating inequalities within sites rather than just across sites. The marginal product of labor in the stratified sector is equal to its average product in the open sector. Members of the elite at a stratified site receive both land rent and an implicit wage. We call this elite–commoner inequality. Our theory predicts that insider–outsider inequality will arise first, with elite–commoner inequality following later if regional population density and settlement sizes become large enough.

Although in our formal modeling we assume that elites hire commoners by paying wages, one can easily reformulate the model to have commoners pay land rents to elites. Our conclusions are not affected by the institutional distinction between a labor market and a land market. Elites and commoners could also adopt sharecropping arrangements. In any of these cases we assume commoners have freedom of movement. This includes movement among open sites in the commons, as well as between open and closed sites (assuming a willingness to work at the terms offered by the insiders at a closed site).

For brevity we omit consideration of slave societies. We think of slavery as the type of commoner labor that occurs when it is easy to monitor and enforce work effort, and to prevent exit from a site. Prehistoric slavery tended to arise through the taking of prisoners in raiding or other forms of warfare. When the technology of coercion makes slavery viable, it is not hard to understand how inequality can emerge. However, in this chapter we will show that inequality can develop even without direct control over labor.

We need to be clear about the distinction between the absolute population at a site and the population density at the site. We assume that members of an insider group help each other repel intruders, and their ability to do so is a function of the physical distances among the insiders. There are no specialized guards or warriors. A site becomes closed when an organized group of insiders becomes sufficiently numerous relative to the land area of the site it is defending. Thus there is a minimum local population density that is necessary in order for insider–outsider inequality to occur.

In the formal model of this chapter, we assume all sites have the same land area. Thus the distinction between local population levels and local population densities will not be important. But in later chapters we sometimes use formal models where the sites have differing land areas and in such cases the proper theoretical concept is local density. We generally simplify by assuming that the required insider density is a parameter that is invariant across sites and over time. But in practice this threshold could depend upon the terrain, the efficiency of the technology used to repel outsiders, the value of the resources at the site, or external events that may encourage entry attempts by outsiders.

A central simplifying assumption is that insiders are organized while outsiders are not. This means that intruders only need to be deterred or repelled one at a time. We will study conflicts over land between organized groups in Chapters 7 and 8, which deal with warfare when the contending groups have internal equality and inequality, respectively.

Another simplifying assumption is that the number of children for a given adult is proportional to that adult’s food income, and that all adults convert food into children at an identical rate. This implies that aggregate population growth or decline depends only on aggregate food output, not the way in which food income is distributed among agents. Redistributing income would simply lower the number of children for some agents while raising the number for others by an equal amount, without affecting regional population.

Our theory has implications for upward and downward economic mobility. In long-run equilibrium, aggregate population is stationary, so in each generation parents have just enough children to replace themselves. However, elite agents at stratified sites have more children than are needed for replacement, and the same is true for the insiders at high-quality closed sites without stratification. Conversely, commoners have fewer children than are needed for replacement, and the same is true for the insiders at lower-quality closed sites. To maintain a stationary class structure over time, there must be a mechanism to ensure some downward mobility among children of rich parents. There must also be a process for replenishing the membership of poor insider groups at low-quality closed sites, as well as the membership of the commoner class.

We handle these issues by assuming that in each generation, the children of the previous insiders at a site form a new insider group based, for example, on birth order. After the critical mass needed for exclusion is reached, other offspring are expelled from the site. As a result, all elite agents at a stratified site have elite parents, so elite status is hereditary. However, not all children of elite parents stay in the elite. The same is true for insider groups at high-quality closed sites. On the other hand, when insider parents have too few children to maintain exclusion at a site, all of their children retain insider status. In addition, some commoners will enter until exclusion is restored.

Our theory focuses on the development of corporate landowning groups based on kinship or common descent, rather than property rights at the level of individuals, nuclear families, or households. Agents inherit land by inheriting membership in these corporate groups. We do not rule out more individualistic types of landownership after corporate groups have formed. But the central concept is that insider groups operate as collective actors with respect to the appropriation of land, and we think of them as being the early arrivers at a site. We assume individual members of corporate landowning groups share resources and food equally. In general, we are only concerned with inequalities emerging across classes of insiders, outsiders, elites, and commoners, not with inequalities internal to such classes.

This point requires emphasis because a number of researchers instead focus on inequality across individuals or households (see Sections 6.2 and 6.3). Our perspective leads us to expect bimodal distributions of consumption, income, or wealth at stratified sites, rather than continuous distributions like those of a normal or log normal type. This expectation carries over to diet, health, stature, housing, grave goods, life expectancy, and similar measures of well-being. We do not deny that ranking or inequality occurs within elites or other social classes, but our theory deals with the structural economic inequality across classes.

The reader may recall from Section 1.2 that our “main sequence” went from mobile foraging with open access to sedentary foraging with open access, and then to agriculture with open access, closed access, and stratification. In Chapters 3–5 this sequence enabled us to model the technological trajectory leading to agriculture without the complication of distinguishing between open and closed production sites. However, in practice sedentary foragers can sometimes reach population densities sufficiently high to yield closed access and possibly stratification, which gives an alternative sequence in Figure 1.2. The formal model in this chapter applies equally well to foraging and agricultural societies.

We pause for a moment to consider possible objections from non-economists. It should be clear from the preceding discussion that our theory is highly stylized. To take just one example: the motivations of the agents are extraordinarily simple. We assume they care only about food consumption, and are attracted to food sources like moths to a flame. Readers may ask why our agents do not sacrifice some food income for the sake of proximity to their families and friends, the enjoyment of beautiful landscapes, or an emotional attachment to their places of birth.

As we discussed in Chapter 1, we do not claim to account for the full richness of human psychology. We only claim that simple models can yield interesting insights. It would be impossible to understand the interactions among our variables without adopting various strategic simplifications. The test is not whether our theory is true in the sense of being descriptively accurate, but whether it does a better job of explaining a certain set of empirical facts than other currently available theories.

The rest of the chapter is organized as follows. Section 6.2 discusses conceptual issues and surveys theoretical ideas from archaeologists, economists, and others about the origins of prehistoric inequality. Section 6.3 describes several empirical generalizations from archaeology and anthropology. In Section 6.4 we deviate briefly from prehistory to discuss the California gold rush. This example suggests how group property rights over valuable production sites can be established in the absence of a state.

Section 6.5 develops a short-run model in which a given regional population is distributed across sites, and property rights are endogenously determined at each site. Section 6.6 introduces an aggregate production function for the region as a whole and identifies the distortions associated with property rights issues. Section 6.7 combines the aggregate production function with Malthusian population dynamics to define long-run equilibrium and characterize the adjustment path leading to equilibrium.

Sections 6.8, 6.9, and 6.10 give formal results regarding poverty, inequality, and demography. These sections show that when some sites are closed, higher productivity makes commoners poorer and increases inequality for the region. When stratification exists, some elite children move down into the commoner class in each generation.

Section 6.11 presents archaeologically-based narratives describing the emergence of inequality in four regions: Southwest Asia, Europe, Polynesia, and the Channel Islands off the coast of California. For each of these cases we argue that the sequence of events is consistent with our theory. Section 6.12 summarizes our results and Section 6.13 is a postscript. Proofs of all formal propositions are available at cambridge.org/economicprehistory.

6.2 Theories of Inequality

In one perspective, the egalitarianism of mobile foraging groups represents the natural human condition, and inequality is an anomaly that requires an explanation. In another perspective (common among modern anthropologists and archaeologists), most egalitarian societies harbor “aggrandizers” or “aspiring elites,” and the capacity of these societies to restrain incipient inequality is what requires an explanation.

We assume that individuals and small groups routinely engage in self-interested behavior. However, we are also impressed by the fact that for most of human existence, egalitarianism prevailed. At a general level our goal is to explain variations across time and space in the degree of inequality, and we want to identify causal mechanisms linking inequality with the natural environment, technology, and population. But the key item on our agenda is to explain how societies became inegalitarian.

We begin with a conceptual point. Economic inequality (e.g., with respect to food consumption) can involve differences across communities or differences within communities. The archaeological and anthropological literature focuses heavily on the emergence of elite and commoner classes within communities. Much less attention has been paid to inequality across sites, settlements, or communities, each of which may be internally egalitarian. A complete theory of early inequality needs to explain inequality both across the sites within a region (insider–outsider inequality) and within individual sites (elite–commoner inequality).

Inequality across social groups is likely to arise whenever groups are sedentary, control natural resources of unequal value, are physically able to prevent outsiders from accessing their resources, and can overcome the free-rider problems associated with the defense of a site or territory. There are strong incentives for local groups to solve such problems in order to ensure that entry by outsiders does not reduce food per capita for those already present. When they succeed in creating property rights of this kind, local groups have permanently better standards of living than outsiders who must obtain their food in the remaining open locations. This disables the mechanism that brought about equal food per capita across sites in Chapters 4–5, because now individual agents cannot migrate from sites with low food per capita to sites with high food per capita. However, it does not necessarily imply any departure from egalitarianism within groups.

Another conceptual issue involves the role of biology. Many writers assume that inequality emerges when technology becomes productive enough to provide more food than the biological minimum needed by a society. This generates a surplus that can be distributed unequally, in some cases to support a non-food-producing elite. Such ideas are widespread in the literatures of archaeology and anthropology, and are occasionally adopted by economists as well (for example, Milanovic et al., Reference Milanovic, Lindert and Williamson2011).

We do not believe commoners’ living standards are fixed biologically, for several reasons. First, nutrition, health, and life expectancy are continuous variables and respond to food intake in a continuous way. Second, ethnographic data indicate that under normal conditions, foragers do not devote extraordinary labor time to food acquisition, contrary to what might be expected if they were on the brink of starvation (see table 1.1 in Kelly, Reference Kelly1995). Third, living standards within egalitarian societies evidently declined during the transition from mobile to sedentary foraging as discussed in Section 4.10, and during the transition from sedentary foraging to agriculture as discussed in Section 5.9. Fourth, if commoners in a stratified society have insufficient food to replace themselves demographically, they can be replaced through downward mobility from the elite, as we argued above and will show in Section 6.10. And finally, archaeological evidence suggests that standards of living for commoners in stratified societies have varied across time and space. For example, there is substantial debate over whether early states made commoners better off or worse off, a topic to be discussed in Chapters 9–11. These points support the view that the standard of living among commoners is an economic variable, not a biological constant, even when one adopts a Malthusian perspective.

We do not treat “surplus” as a concept defined by technical productivity in relation to subsistence requirements, and we do not say that rising productivity directly increases “surplus” in some mechanical way. In our theory, the gap in food consumption between elite and commoner agents is traceable to the rent an elite appropriates from the land it controls. Rising productivity in food technology does indeed lead to inequality, but this occurs indirectly by causing regional population density to rise for Malthusian reasons. As a result local populations also rise, and insiders at the most valuable sites eventually become numerous enough to exclude outsiders. The endogenous emergence of property rights enables insiders to capture land rent, and can lead eventually to stratification. As this process unfolds, the rich get richer and the poor get poorer. This contrasts with the prediction that living standards for the poor remain constant at a biological minimum.

In the rest of this section, we survey a range of theoretical perspectives. We start with authors who emphasize natural resources, technology, and population. Next, we go to authors who assign a larger role to coercion. This is followed by a review of ideas on property rights, and finally some comments on culture. As will no doubt become clear, these categories are artificial, and individual authors often cite multiple causal factors. We omit a history of archaeological thought on the subject, but see Price and Feinman (Reference Price and Feinman1995, Reference Price and Feinman2012) for references to earlier literature.

Mattison et al. (Reference Mattison, Smith, Shenk and Cochrane2016) have proposed a synthesis of archaeological ideas about the transition to inequality (see also the review by Kohler and Smith, Reference Kohler and Smith2018, ch. 1). Mattison et al. argue that the Holocene shift to a more stable climate was a necessary condition for the emergence of persistent institutionalized inequality, because the mobile foraging strategies associated with climate instability in the Pleistocene were incompatible with wealth accumulation. However, this climate change was not sufficient (many foraging societies remained egalitarian in the Holocene). A second necessary condition was the presence of dense and predictable resource patches where the benefits of group defense exceeded the costs. A third necessary condition was a role for material assets, such as land or livestock, whose ownership could be readily passed from parents to children. Together these three conditions were sufficient for the emergence of systematic and persistent economic inequality.

Mattison et al. are skeptical about the role of population as a causal factor behind the growth of inequality. They observe that according to some archaeologists, inequality arises when resources are scarce relative to population, while according to others it arises when resources are abundant relative to population. Mattison et al. also tend to see early inequality as emerging from prestige and resulting in broad social benefits. They grant a significant role to coercion only in larger-scale chiefdoms and states.

We agree with Mattison et al. about the importance of high-value sites that can be readily defended. In our model, we assume that regional geography generates a resource gradient from low to high quality sites. We adopt a simple technology for site defense: if the current insider group is too small, it cannot stop further entry, but if it is large enough, it can. The valuable resource in our framework is land. Rights to land can be transmitted from parents to children but only through the inheritance of membership in a social group that collectively controls access to a site.

We do not see population pressure (people relative to resources) as an important factor. Instead, what matters in our theory is population density (people relative to land). It is only at high population densities that an insider group will achieve the size needed to exclude outsiders from a site or territory. In this sense our view is closer to Kelly (Reference Kelly2007, ch. 8), who believes population density is the key proximate cause of inequality among hunter-gatherers. In our framework the ultimate causes are nature and technology, which determine population density in the long run through Malthusian dynamics. When nature becomes more benign or technology improves, population rises and inequality increases.

Kennett et al. (Reference Kennett, Winterhalder, Bartruff, Erlandson and Shennan2009) offer an explanation for the emergence of stratification that stresses the role of heterogeneous production sites, demographic change, constraints on mobility, and competition over scarce resources. Kennett et al. see increasing population density as a source of social stress that allows dominant individuals to shape institutions in ways favorable to stratification. They argue that this is more likely when productivity differences across sites are large and geographic or social barriers inhibit migration. We agree about the importance of heterogeneous sites and demographic factors. However, by contrast with Kennett et al., we emphasize the endogenous nature of population, the causal dependence of mobility constraints on endogenous property rights, and the idea that insider–outsider inequality is a necessary precursor to stratification.

Johnson and Earle (Reference Johnson and Earle2000) develop a broad theory of human social evolution from foraging bands to local groups, chiefdoms, and agrarian states. Using many ethnographic studies, they argue that population growth and technological innovation led to subsistence intensification, which generated a series of problems involving risk, conflict, investment, and trade. The solutions to these problems typically implied greater political integration and social stratification.

In the Johnson and Earle framework, population and technology serve as prime movers. Other writers endogenize these variables. For example, Kirch (Reference Kirch2010, 190–201) combines an agricultural production function with Malthusian population dynamics to explain the evolution of inequality in Hawaii. His focus on resources, technology, and population is very much in the spirit of our approach.

Henrich and Boyd (Reference Henrich and Boyd2008) endogenize technology. They assume that people acquire economic strategies through imitation, learning, and teaching, and that cultural variants spread if they enhance economic success. There are benefits from specialization and exchange across groups, with unequal productivity gains over time through learning by doing. Poorer groups cannot easily imitate the techniques used by richer groups, so the resulting inequalities become permanent. Henrich and Boyd argue that their model can explain stratification without coercion, deception, or exogenous group differences.

We agree that cultural learning tends to make food technology more productive over time (see Chapters 3–5), especially in the context of cultivation and domestication. However, our model differs by treating technological innovations as common knowledge throughout a region. Another difference is that we require a technology of exclusion by which group property rights over land can be enforced. To this extent, we do include a role for coercion, but only insofar as insiders threaten outsiders with violence in order to deter unauthorized entry at a site. The relationships between elite and commoner agents within stratified sites are voluntary in the sense that the commoners receive wages equal to the food income they could obtain by exiting to one of the remaining open sites.

We turn next to authors who give a larger role to coercion. Gilman (Reference Gilman1981) argues that stratification in Bronze-Age Europe resulted from site-specific capital investments, high exit costs, and the use of force by “protectors” who exacted tribute from other agents at the site. Webster (Reference Webster1990) maintains instead that stratification in prehistoric Europe was caused by incipient control over labor through patron–client relationships, rather than by control over non-human wealth or natural resources such as land. Arnold (Reference Arnold1993, Reference Arnold, Douglas Price and Feinman1995) surveys a wide range of anthropological explanations for the emergence of inequality in hunter-gatherer societies. Her primary argument is that aspiring elites gain control over household labor in times of environmental or social stress, but she provides few details.

Jared Diamond (Reference Diamond1997) has proposed a theory about the origin of inequality that combines increasing productivity with a political economy constraint involving social cohesion. According to Diamond, early foraging bands were kinship-based groups of no more than a few dozen people. With the development of a sedentary lifestyle, tribes of perhaps a few hundred people arose. Diamond sees tribes as egalitarian units within which social cohesion was maintained through kinship ties. He argues that inequality emerged with the formation of chiefdoms whose populations were in the thousands. At high population densities, conflicts between individuals and groups could no longer be resolved through kinship ties alone. The continued existence of such societies required the transfer of coercive power to a “chief” and this concentrated power led to inequality.

The political economist Boix (Reference Boix2015) argues that among simple foragers, social norms prevent predatory behavior. As technology advances, inequality begins to develop because some people are better innovators and some control resources complementary to the innovations. Growing inequality eventually destabilizes the cooperative equilibrium and causes agents to specialize in production or predation according to their comparative advantage. Two possible trajectories can occur: one where coalitions of producers try to keep order and another where predators become rulers. The former path usually leads to less inequality than the latter.

Borgerhoff Mulder et al. (Reference Borgerhoff Mulder2009) have provided an influential view of inequality in 21 small-scale societies studied by ethnographers. We discuss their empirical findings in Section 6.3, but it is useful to discuss their theoretical perspective here. They define four types of societies based on food technology: hunter-gatherer, horticultural, agricultural, and pastoral. In this classification, horticultural societies have domesticated plants but not plows. They are constrained by labor but not land, and have no land markets.

Borgerhoff Mulder et al. present an equation, applying to all four types of society, in which the wealth of an individual household is a weighted average of wealth inherited from parents and the average wealth of the society, plus an exogenous shock associated with accidents, illnesses, theft, and other random events that are independent across households. They show that in a steady state, the variance of wealth for the society as a whole depends positively on the variance of shocks and also positively on the transmissibility of wealth from parent to child. While random shocks to households will create some initial inequality, the inequality can become permanent over time through inheritance.

There are three general kinds of assets that children can inherit from their parents. The first is embodied wealth (skill, strength, health, intelligence, personality traits). The second is material wealth (land, cattle, tools, houses, resource locations). The third is relational wealth (friendships, kinship connections, positions in social networks). The authors argue that material assets are most easily transmitted from parent to child, while embodied and relational assets are less easily transmitted.

Different types of society tend to have different levels of inequality because their food technologies depend more or less heavily on the various asset types. As we discuss in Section 6.3 below, Borgerhoff Mulder et al. find that hunter-gatherer and horticultural societies tend to have relatively low inequality, while agricultural and pastoral societies tend to have higher levels of inequality. They attribute this to the reliance of the former on embodied and relational assets with low levels of heritability, and the reliance of the latter on material assets such as land or animal herds with greater heritability.

This approach has a number of limitations. First, it focuses heavily on random shocks at the level of the household, rather than structural features of the economy as a whole. Second, it does not explain how property rights over material assets such as land were created and instead simply assumes the inheritability of these assets. Third, it does not provide a theory about the origins of inequality except in the sense that technological developments caused food producers to make greater use of material assets. The model facilitates cross-sectional comparisons among societies of different types, but says little about the causal mechanisms that increased inequality over time within a given society.

Because our story for the origin of inequality stresses the creation of endogenous property rights over land, it is useful to contrast our analysis with that of other economists who have theorized about the evolution of property rights to land in prehistoric societies. North and Thomas (Reference North and Thomas1977), for example, argue that foragers had open access to natural resources that were subject to depletion by over-harvesting. Population growth made this problem more severe and motivated the development of communal property rights over land, which in turn made agriculture attractive.

De Meza and Gould (Reference De Meza and Gould1992) assume that agents choose whether or not to enforce private ownership claims on resource sites. Our model resembles theirs in having many resource sites with costless mobility among sites. However, they assume that there is a fixed cost of enclosing a site, while we assume that group property rights are a costless byproduct of food acquisition when insiders are sufficiently numerous.

Baker (Reference Baker2003) seeks to explain why foraging societies have varying land tenure institutions. The explanatory variables include resource density and predictability, as well as production and conflict technology. The model involves strategic interactions between insiders and outsiders, where groups must decide how much of their endowed territory to defend and whether they will intrude on territories defended by others.

Rowthorn and Seabright (Reference Rowthorn and Seabright2010) develop a model of property rights that applies to agriculturalists during the Neolithic transition. They cite evidence showing that increased productivity in early farming was associated with lower levels of nutrition and health. In their model this is explained by an increase in resources allocated to the defense of crops and land from outside groups.

Bowles and Choi (Reference Bowles and Choi2019) study the co-evolution of agriculture and property rights. In their view, mobile foraging bands had communal control over land. With the arrival of the Holocene, a few sedentary groups with unusually productive and easily defended resources developed private property institutions. A subset of these groups then adopted agriculture. This led to a positive feedback loop in which agriculture reinforced private property rights, and private property rights encouraged agriculture (see Section 5.2).

Our theory differs from North and Thomas (Reference North and Thomas1977), de Meza and Gould (Reference De Meza and Gould1992), Baker (Reference Baker2003), Rowthorn and Seabright (Reference Rowthorn and Seabright2010), and Bowles and Choi (Reference Bowles and Choi2019) in several ways. First, these authors study the creation of group or individual property rights over land but not the emergence of inequality across or within social groups. Second, Baker (Reference Baker2003) limits attention to foraging societies, North and Thomas (Reference North and Thomas1977) and Bowles and Choi (Reference Bowles and Choi2019) study the co-evolution of property rights and agriculture, and Rowthorn and Seabright (Reference Rowthorn and Seabright2010) assume an agricultural transition. Our theory applies to both foraging and agricultural societies, while explaining why inequality increases as agricultural societies develop. Finally, unlike these authors we treat population as endogenous.

We end this section with some remarks on the book The Creation of Inequality by Flannery and Marcus (Reference Flannery and Marcus2012), who assign a central role to culture. Their book provides a vast array of case studies, ranging from mobile foragers to kingdoms and empires. One key transition is from societies where inequality is based on achievement to those where it is based on hereditary rank. Other transitions involve stratification and monarchy.

Flannery and Marcus often provide rich and insightful cultural descriptions, but their causal stories about transitions from one social structure to another are sometimes frustratingly vague. The following statement about causality is representative.

Our problem with this approach is that in virtually every society, some people seek to be treated as superior, and seek to manipulate prevailing social norms in their favor. If this were sufficient to create inequality, then inequality would be universal. It is essential to identify the objective conditions under which such strategies are likely to succeed or fail. The relevant conditions almost certainly involve population, technology, and the natural environment. Unfortunately, Flannery and Marcus pay little attention to these variables, and often dismiss them or minimize their importance, although they do concede near the end of the book that population growth, intensive agriculture, and climatic improvement “could create a favorable environment for inequality” (553).

In our theory culture does not instigate major economic transitions, but it is an enabling factor that operates in the background to facilitate collective decision-making and learning by doing. Our foreground factors for explaining the origins of inequality, as in previous chapters, are the natural environment, technology, and population.

6.3 Evidence on Inequality

This section surveys broad empirical generalizations about prehistoric inequality in order to provide a backdrop for the formal model we will develop in Sections 6.5–6.10. The ideal data for testing our theory would involve evidence on the emergence of insider–outsider and elite–commoner inequality within a geographic region over time. A number of archaeological narratives of this sort will be presented in Section 6.11. Here we focus mainly on cross-sectional data at the level of individuals or households, although a few time-series findings will also be discussed.

Archaeologists have several sources of information on inequality in prehistoric societies, including house sizes, artifact distributions, burials, and health status inferred from bones and teeth. Each source has its advantages and disadvantages. For example, house size may vary with family size over the life cycle rather than reflecting permanent inequalities in wealth. Grave goods may reflect prestige rather than the tangible wealth owned by a person in life. Nevertheless, evidence of each kind offers valuable insights.

Archaeological measures of inequality typically involve stocks of wealth such as housing or jewelry. On the other hand, our formal models refer to flows per unit of time (food income and food consumption). An individual with permanently high food income is likely to have high wealth because food resources can be exchanged for labor services such as house construction, or for the jewelry placed in burials. Thus in practice there is likely to be a strong association between stock and flow measures of inequality.

The most commonly used quantitative index of inequality within a population is the Gini coefficient. The construction of this index will be explained using Figure 6.1. One first ranks the individual members of a population on the dimension of interest (e.g., house size). One then creates a graph where the horizontal axis indicates the poorest 10% of the population, the poorest 20%, and so on, up to the poorest 100%. By definition the latter is the entire population. On the vertical axis, one plots the percentage of the total wealth (e.g., total housing space) owned by the corresponding subset of the population. For example, the poorest 10% might own 3% of the total housing space, the poorest 20% might own 8%, and so on, up to the poorest 100%, who necessarily own 100% of the total housing space.

Figure 6.1. Lorenz curve and Gini coefficient

The resulting curve is called a Lorenz curve. In a society with perfect equality the Lorenz curve would be the straight diagonal line from the lower left to the upper right in Figure 6.1 (the “bottom” 10% would have 10% of the total housing space, and so on). The curve bows outward below this diagonal line if there is some inequality. In the extreme case where one person owns all of the wealth in question, the Lorenz curve is identical to the horizontal axis except for a jump to 100% along the right side of the graph because the richest person is included in the poorest 100% of the population.

The Gini coefficient is defined to be twice the area between the diagonal line of perfect equality and the actual Lorenz curve. This yields a numerical value between zero (complete equality) and one (extreme inequality). The same methods can be applied to other forms of wealth, such as inequality in grave goods. Assuming the availability of data, a Gini coefficient can also be used as a summary measure for inequality in income or consumption. Moreover, the units of observation need not be individuals; they could be households or other subunits within a larger social group.

The Gini coefficient has limitations. For example, suppose a site is stratified and there is a bimodal distribution where a small elite is quite wealthy and a large majority is very poor. Although the resulting Gini value will be high, it will not reveal the existence of the underlying class structure. For this, one needs the entire Lorenz curve. The shape of the Lorenz curve predicted by our theory of stratification will be discussed in Section 6.9. Partly for these reasons, archaeological narratives like those in Section 6.11 provide an important supplement to Gini coefficients and other numerical indexes of inequality.

More broadly, archaeological applications of the Gini coefficient frequently have problems of non-comparability, bias, and lack of precision. Fochesato et al. (Reference Fochesato, Bogaard and Bowles2019) find that the statistical adjustments needed to deal with grave goods, slaves, and exclusion of households without property are often substantial. On the other hand, the adjustments to deal with differing populations and sample sizes are generally minor.

Kohler et al. (Reference Kohler and Smith2018) offer the most comprehensive overview of prehistoric Gini coefficients with respect to housing inequality. The authors draw upon archaeological data from 62 societies running the gamut from foraging groups to state-level societies such as Old Babylonia and Pompeii. Foraging societies generally have the lowest inequality, with Gini coefficients that vary within a narrow band below 0.2. Horticultural societies, which have domesticated plants, move frequently, face minimal land constraints, and use some foraged resources, typically have higher Gini coefficients for housing than foragers, as well as a greater range of variation. Agriculturalists have the highest levels of housing inequality and also the greatest range of variation, with some overlap in the distributions for agriculturalists and horticulturalists.

The same authors explore relationships between settlement sizes and inequality. They define a village as a settlement of fewer than a thousand residents lacking a central feature; a town as a settlement of more than a thousand residents without a central feature or more than two hundred with a small or moderately sized central feature; and a city as a settlement with more than a thousand residents and a large central feature. They find that Gini coefficients tend to be larger as one moves from villages to towns to cities, although there is substantial overlap in the distributions of Ginis for each category.

Kohler et al. find no relationship between regional population size and the Gini coefficient for housing. This is perhaps not surprising, given that regions can differ in their geographic scale. A more relevant measure is population density. For the 11 Old World cases where data exist, they find a strong positive relationship between population density and inequality. No such relationship is found for the 28 New World cases. They also find a positive relationship between population at a local site and inequality. This relationship is stronger in the Old World than in the New World.

In both hemispheres, Ginis start low and rise as cultivation begins. The authors associate this with rising productivity, greater residential stability, and increasing size of social groups (309). For the first two millennia after cultigens are observed, the Ginis are similar in the two hemispheres. The Ginis then flatten out in the New World. But in the Old World there is a pause about three millennia after the appearance of cultigens, which is followed by resumption in the growth of Ginis. Fochesato et al. (Reference Fochesato, Bogaard and Bowles2019, 867) comment that “no plausible adjustments … would change the result that post-Neolithic wealth inequality in Eurasia tended to be higher than inequality in the Western hemisphere.”

Kohler et al. consider various explanations for this difference. One possibility is a difference in the nature of state institutions. However, the difference in Gini coefficients across hemispheres becomes evident prior to state formation (by about 6000 BP). Their preferred explanation is that Old World societies had greater access to large domesticated animals. This could have increased inequality through several causal channels: enhanced agricultural productivity through plowing, inequality in the ownership and inheritance of animals, the use of horses and camels for military conquest, and so on. For more on this topic, see Kohler et al. (Reference Kohler2017).

Bogaard et al. (Reference Bogaard, Fochesato and Bowles2019) distinguish between labor-limited and land-limited farming technologies. The ratio of the marginal product of labor to the marginal product of land is higher in societies of the former type than societies of the latter type. Intuitively, when agriculture is labor-limited the society treats labor as scarce and land as abundant. When agriculture is land-limited the society instead treats labor as abundant and land as scarce.

Bogaard et al. draw on evidence from western Eurasia to argue that Neolithic agriculture was labor-limited and had levels of inequality similar to foraging societies. Over time, land scarcity tended to increase due to changes in the underlying production technology, such as the use of oxen for plowing. This was associated with rising Gini coefficients. The authors attribute increasing inequality partly to opportunities for the unequal ownership and inheritance of valuable land (as well as animals). Some of their data comes from societies with states and written texts but the general pattern is clearly visible among pre-state farming societies.

Prehistoric inequality can also be assessed using grave sizes or grave goods. We will provide some qualitative information of this sort in Section 6.11 when we present our regional narratives. Yu et al. (Reference Yu, Chen and Fang2019) employ quantitative methods to study two Neolithic Chinese settlements, each of which had strong class stratification.

The Dawenkou archaeological site was a central settlement in North China. A total of 133 graves were excavated. Most were small (less than 5 m²), but a few were huge (around 14 m²). The latter were filled with ceramic vessels along with jade, bone, and antler implements. The early phase of occupation (6000–5500 BP) gives a Gini for grave areas of 0.23; the middle phase (5500–5000 BP) a Gini of 0.34; and the late phase (5000–4500 BP) a Gini of 0.65. The differences are strongly statistically significant and reveal “gradually increasing social inequality” (4953). Yu et al. suggest that this may help to explain an escalation of violent conflict and social turmoil in the late phase.

Liangzhu was a central settlement in East China during the late Neolithic period. A total of 80 graves were excavated, again with a clear class distinction. Most graves were less than 3 m² and commonly contained pottery. A few elite graves were 5–7 m² and had jade, silk, ivory, and lacquer artifacts. In the early phase (5500–5000 BP), the Gini for grave sizes is estimated to be 0.43. During the middle phase (5000–4500 BP), this falls to 0.19, and in the late phase (4500–4000 BP) it is 0.20. The decline is strongly significant, indicating that elite–commoner inequality can sometimes drop substantially.

Another way to assess prehistoric inequality is through variation in the heights of individuals inferred from skeletal data (Boix and Rosenbluth, Reference Boix and Rosenbluth2014; Boix, Reference Boix2015). While much of this variation (perhaps 80%) is due to genetic factors, some (perhaps 20%) is due to environmental conditions. In particular, short stature tends to be associated with a lack of food during childhood and adolescent growth spurts. There are methodological issues surrounding such inferences (see Boix and Rosenbluth, Reference Boix and Rosenbluth2014, 4–6), but skeletal remains do provide an interesting additional perspective on inequality.

Boix and Rosenbluth describe intriguing time-series findings for Japan. The early Jomon, a largely sedentary hunter-gatherer population (see Section 4.3), had a coefficient of variation for male height of 2.70. By contrast, the agricultural Yayoi who supplanted the Jomon had a male coefficient of variation of 3.56. The authors say this is “suggestive but not conclusive evidence” that agriculture resulted in more male inequality (14). They argue that male height dispersion increased further, again indicating greater inequality, as agriculture became more labor intensive and warfare became more widespread.

Ethnography offers further information about inequality in small-scale societies. As usual, one should be cautious about extrapolating back from recent societies to those of the distant past. However, some empirical generalizations seem very robust and it is difficult to imagine that they would not also have applied in prehistory.

As we observed in Section 4.1 and reiterated in Section 6.1, mobile and sedentary foragers tend to differ along several dimensions. Sedentary foraging societies generally have highly predictable natural resources, large settlement sizes, perimeter defense, and tightly controlled resource ownership. There is often, although not always, a hierarchical class structure based upon descent (Kelly, Reference Kelly2007, ch. 8). In some situations, technological innovation and climate change have led to transitions from mobile to sedentary strategies, resulting in the development of closed settlements and stratification (see the discussions of northwest Alaska and the northwest coast of North America by Boix, Reference Boix2015, 96–101). Thus it can be misleading to lump all hunter-gatherers together. However, in the work of Borgerhoff Mulder et al. discussed below, the sedentary hunter-gatherer societies tend to be near the egalitarian end of the spectrum, which mitigates this problem.

Borgerhoff Mulder et al. (Reference Borgerhoff Mulder2009) have estimated Gini coefficients for 21 societies studied by ethnographers, classified into four broad types: hunter-gatherer, horticultural, agricultural, and pastoral. We discussed the theoretical framework used by these authors in Section 6.2. In their model, the welfare of a household is determined by its embodied, material, and relational capital, with weights on these three kinds of capital that can vary across societies. These weights come from estimates by experts who have studied each society and indicate the percentage increase in welfare that would occur in response to a 1% increase in each type of wealth. The authors estimate heritability measures for each type of capital in each society using regression techniques.

The central results can be summarized as follows. First, material wealth (such as land or animals) is highly heritable for agricultural and pastoral societies. It is not highly heritable for hunter-gatherers or horticulturalists, probably because material wealth in the latter cases tends to take a different form (for example, in a hunter-gatherer society it may include tools made from non-durable materials). Second, material wealth is much more important as a determinant of household welfare in agricultural and pastoral societies by comparison with hunter-gatherer and horticultural societies. Third, the societies in which wealth is more hereditable have more inequality measured by Gini coefficients. Hunter-gatherers have the lowest Ginis while horticulturalists have somewhat higher Ginis. The Ginis for agriculturalists and pastoralists are much higher, with pastoralists at the top.

For hunter-gatherer and horticultural societies, parents in the top wealth decile are three times more likely to have children in the top wealth decile as compared to parents in the bottom decile. For agricultural societies, this becomes eleven times more likely, and for pastoral societies, it is twenty times more likely. If we aggregate hunter-gatherer and horticultural societies into one group and agriculturalists and pastoralists into another group, 45% of the difference in inequality between the two groups is due to technology (differences in the relative importance of various assets) and 55% is due to institutions (differences in the hereditability of assets).

In our view, the general similarity of hunter-gatherer and horticultural societies suggests that it is not the availability of domesticated plants that matters for inequality. Rather, it is the degree to which good land is a scarce resource relative to population or labor supply. For details, see Gurven et al. (Reference Gurven2010). Related articles appear in a special issue of Current Anthropology (Symposium, 2010).

Boix (Reference Boix2015, 37–44) uses data on 1,100 foraging and non-foraging societies from the Ethnographic Atlas. He finds a strong association between production technologies and settlement sizes. Among simple foragers, only 6.2% of societies have settlements of more than 200 people. For complex foragers (often those having aquatic food resources), this figure is 7.4%. Among societies with extensive or shifting agriculture (often called horticulturalists), the fraction rises to 44.1%, and for intensive agriculturalists it is 78.7%. No foraging societies in the sample had settlements larger than 1,000 residents, while 10% of horticulturalists and 51.5% of agriculturalists had such settlements.

For the same four categories defined by production technology, the percentage of societies having some inheritance rules about land is 8.3% among simple foragers, 24.7% for complex foragers, 87.1% for horticulturalists, and 96.5% for intensive agriculturalists. Importantly for us, the percentage of societies having an elite or other class structure is 1.9% among simple foragers, 25.6% for complex foragers, 28.7% for horticulturalists, and 55.4% for intensive agriculturalists. Boix carries out ordered probit regressions and finds that these differences are statistically significant at conventional levels (see also the results for settlement patterns and stratification in table 5.1 of Boix, Reference Boix2015, 182–184).

Taking the evidence from this section as a whole, it seems indisputable that there are strong correlations between food production technology and inequality, with farming and pastoralism leading to greater differences in housing, stronger inheritance processes, and more stratification into elite and commoner classes. These patterns are visible from both prehistory and ethnography. There are also suggestions that the link between food production technology and economic inequality might be mediated by variables such as regional population density, settlement size, or both. We address these connections later in the chapter. First, however, we need another piece of the puzzle: the creation of group property rights over valuable production sites. For an illuminating recent case we turn to the California gold rush of the nineteenth century.

6.4 The California Gold Rush

When the California gold rush started in 1848, mineral rights were held by the US government (all information is from Umbeck, Reference Umbeck1981). However, enforcement proved impossible due to high rates of desertion from the military. From 1848 to 1866 open access prevailed, except at sites where property rights were created endogenously.

Miners foraged for gold using simple technologies involving little human capital. The exclusion technology was based only on numbers of miners. Miners would wait a short period after a discovery at a site had been made public before allocating gold land. At that point, a majority vote would be held and early arrivers would be allowed to stay, while late arrivers would be compelled to leave. The values of the claims of the insiders were then equalized.

To maintain their exclusive mining rights, miners were required to be present at the mining site and to work their claims a specified number of days per week. Miners were collectively required to enforce each other’s mining rights. Failure to comply with these rules led to expulsion from the site. Although all miners were well armed, there is little evidence of violence in the gold fields. There is also no evidence that specialized gunfighters were hired, or that unusual abilities in the use of violence were important.

The population of miners was perhaps 800 in May 1848 and had grown to over 100,000 one year later. Early in this process, there is no evidence that any sites were closed. Individuals or small teams worked under open access conditions. Population densities were higher at better sites (often those closest to water supplies).

By 1849, contracts between miners show that exclusive mining rights had arisen at some sites. These contracts describe explicit responsibilities for group enforcement of property rights. The inequality in the gold fields through 1866 was insider–outsider, in the sense that some groups of miners had more valuable sites and thus higher per capita incomes. It never involved stratification among miners at an individual site, except in a few instances where miners hired local Native Americans to search for gold. In 1866 the US Congress passed legislation that ended the need for self-enforcement by the miners.

This narrative captures several features of our formal model, including (a) many sites of heterogeneous quality within a region; (b) initial conditions of open access; (c) creation of group property rights when local population densities became high enough; (d) a low cost of property right enforcement after a critical mass of agents was reached; (e) little difficulty with free-rider problems among insiders at a site; (f) no specialization with respect to violence; and (g) roughly equal endowments of human and physical capital among the agents. The main difference from our framework is that we treat population growth as a Malthusian response to rising productivity, while in the gold fields population grew rapidly through in-migration.

Putting the latter point aside, we believe that small groups of hunter-gatherers or agriculturalists would most likely have cooperated to defend territories using generally available weapons (clubs, spears, bows and arrows) much as the gold miners did using rifles, and that the number of insiders would have been the key variable in deterring potential intruders. While our model does not deal with coalitions of intruders (that is, warfare), it should be noted that coalitions of gold miners did not engage in warfare over mining sites.

6.5 Short-Run Equilibrium

At an individual production site, food output (in calories) is θsL^α where θ > 0 reflects region-wide climate, resources, and technology; s > 0 is the quality of the site; L ≥ 0 is labor used for food production; and 0 < α < 1. The input of land is normalized at unity. Variations in site quality reflect local geographic factors such as terrain, soil, or availability of fresh water. Labor has diminishing returns due to the fixed land input.

Each individual agent is negligible relative to the number of agents at the site as a whole. An agent is endowed with a unit of time, which is used for food production. We ignore leisure. Any agent at the site can migrate to another site and obtain a quantity of food w. There is an infinitely elastic supply of outsiders who will enter the site if they are not excluded and can obtain more than w by doing so.

A group of $\mathrm{d}>0$ or more food producers can cooperate to prevent outsiders from appropriating land at the site. This is an automatic by-product of food production and does not have an opportunity cost (exclusion involves deterrence rather than building a fence or patrolling a perimeter). The weapons used to defend a site are readily available to all. Deterrence fails whenever there are fewer than d agents at the site, and in this case outsiders will enter if they can enhance their food consumption by doing so.

Let n be the number of agents born at the site. There are two ways in which a group of size d may arise. If $\mathrm{n}\ge \mathrm{d}$, a subset of size d is determined by birth order. This core group, called insiders, appropriates all of the land at the site. The other n-d agents become outsiders and have no land endowment. Alternatively, if $\mathrm{n}<\mathrm{d}$, those born at the site fall short of the number needed for exclusion. But if enough outsiders are attracted to the site, exclusion occurs after the first d-n agents have arrived from other sites.

Next consider a region with a continuum of production sites. Physical mobility among sites is costless but the agents cannot leave the region due to natural barriers such as deserts, mountains, or ocean. Site qualities are distributed uniformly on the interval $\mathrm{s}\in \left[0,1\right]$. Let n(s) be the number of people born at a site of quality s. Define the short run to be a period in which the total regional population $\mathrm{N}\equiv {\int }_0^1\mathrm{n}\left(\mathrm{s}\right)$ ds does not change. We interpret this as one human generation.

In the short run, labor is allocated across sites as follows.

1. (a) If $\mathrm{L}\left(\mathrm{s}\right)<\mathrm{d}$ then $\mathrm{w}=\mathrm{\theta sL}{\left(\mathrm{s}\right)}^{\mathrm{\alpha }-1}$ (open sites)

2. (b) If $\mathrm{L}\left(\mathrm{s}\right)\ge \mathrm{d}$ then (closed sites)

   1. (i) L(s) maximizes ${\mathrm{\theta sL}}^{\mathrm{\alpha }}-\mathrm{w}\left(\mathrm{L}-\mathrm{d}\right)$ subject to $\mathrm{L}\ge \mathrm{d}$, and

   2. (ii) $\mathrm{r}\left(\mathrm{s}\right)\equiv \mathrm{\theta sL}{\left(\mathrm{s}\right)}^{\mathrm{\alpha }}-\mathrm{wL}\left(\mathrm{s}\right)\ge 0.$

Part (a) states that all sites where exclusion is impossible must have the same food per capita w, which is equal to the average product of labor. Because mobility is costless these sites must be equally attractive in equilibrium. We call such sites open and refer to the set of all open sites as the commons.

Part (b) describes sites where insiders can exclude outsiders. We call such sites closed. At these sites the d insider agents choose some number of outsiders $\mathrm{L}-\mathrm{d}\ge 0$ who are admitted to the site and allowed to produce food there. We assume that the methods used to exclude outsiders can also be used to prevent them from appropriating land after they are admitted. Any outsiders allowed to produce food at the site receive the wage w.

When $\mathrm{L}=\mathrm{d}$ so that no outsiders are admitted, we say that the site is closed but unstratified. When $\mathrm{L}>\mathrm{d}$, the site is both closed and stratified. In the latter situation, we distinguish between the elite (the insiders who control access to land) and the commoners (the hired outsiders who have no land claim and receive the wage w). Commoner labor does not contribute to entry deterrence.

Condition (b)(i) in the definition of SRE requires that when a site is closed, the number of hired agents is chosen to maximize the net food income of the insider group. The resulting food is shared equally among the d landowners, so that each receives $\mathrm{w}+\left[\mathrm{\theta sL}{\left(\mathrm{s}\right)}^{\mathrm{\alpha }}-\mathrm{wL}\left(\mathrm{s}\right)\right]/\mathrm{d}$. Condition (b)(ii) requires that the land rent $\mathrm{r}\left(\mathrm{s}\right)\equiv \mathrm{\theta sL}{\left(\mathrm{s}\right)}^{\mathrm{\alpha }}-\mathrm{wL}\left(\mathrm{s}\right)$ obtained at a closed site be non-negative. Otherwise the insiders would be better off abandoning the site and moving to the commons themselves.

Landowners in the stratified sector maximize profit, and thus at every site in this sector the marginal product of labor is equal to the wage. The wage, in turn, is equal to the average product of labor in the commons. At a site that is closed but unstratified, the marginal product of labor at the site is below the average product in the commons, so it is unprofitable for insiders to employ outsiders.

We prove the existence and uniqueness of SRE in three steps. First, we consider an arbitrary wage w and characterize the property rights and labor inputs that must occur at each site in an SRE. This is done in Lemma 6.1 below. The main result is that there are quality bounds s^a and s^b with $0<{\mathrm{s}}^{\mathrm{a}}<{\mathrm{s}}^{\mathrm{b}}$ such that sites with qualities below s^a are open, sites of intermediate quality are closed but unstratified, and sites with qualities above s^b are both closed and stratified. Lemma 6.2 establishes that if a wage and a labor allocation satisfy the conditions of Lemma 6.1 and also clear the labor market, then a SRE occurs.

Finally, Proposition 6.1 shows that equating labor demand with the labor supply N yields a unique equilibrium wage. This wage determines a unique set of property rights through Lemma 6.1. Because the definition of SRE only involves the ratio $\mathrm{x}\equiv \mathrm{w}/\mathrm{\theta }$ rather than w and $\mathrm{\theta }$ separately, in this section we fix θ and work with the normalized wage x.

Lemma 6.1

Fix the normalized wage $\mathrm{x}\equiv \mathrm{w}/\mathrm{\theta }>0$. Let L(⋅, x) be a density function that satisfies (a) and (b) in D6.1 for the given x. Define ${\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)\equiv {\mathrm{xd}}^{1-\mathrm{\alpha }}$ and ${\mathrm{s}}^{\mathrm{b}}\left(\mathrm{x}\right)\equiv \left(\mathrm{x}/\mathrm{\alpha }\right){\mathrm{d}}^{1-\mathrm{\alpha }}$.

(a)	If $\mathrm{s}<{\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)$	then	$\mathrm{L}\left(\mathrm{s},\mathrm{x}\right)={\left(\mathrm{s}/\mathrm{x}\right)}^{1/\left(1-\mathrm{\alpha }\right)}<\mathrm{d}$	(open sites)
(b)	If ${\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)\le \mathrm{s}\le {\mathrm{s}}^{\mathrm{b}}\left(\mathrm{x}\right)$	then	$\mathrm{L}\left(\mathrm{s},\mathrm{x}\right)=\mathrm{d}$	(unstratified sites)
(c)	If ${\mathrm{s}}^{\mathrm{b}}\left(\mathrm{x}\right)<\mathrm{s}$	then	$\mathrm{L}\left(\mathrm{s},\mathrm{x}\right)={\left(\mathrm{\alpha s}/\mathrm{x}\right)}^{1/\left(1-\mathrm{\alpha }\right)}>\mathrm{d}$	(stratified sites)

Lemma 6.1 shows that for any wage level, a commons must exist. If x is high enough that $1<{\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)$, all sites are in the commons. At intermediate wage levels we have $0<{\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)\le 1\le {\mathrm{s}}^{\mathrm{b}}\left(\mathrm{x}\right)$, where at least one of the latter inequalities is strict. Inferior sites are open and superior sites are closed, but no sites are stratified. If the wage is low enough that $0<{\mathrm{s}}^{\mathrm{a}}\left(\mathrm{x}\right)<{\mathrm{s}}^{\mathrm{b}}\left(\mathrm{x}\right)<1$, all three sectors exist, with the best sites both closed and stratified.

The latter case is illustrated in Figure 6.2. The density L(⋅, x) showing labor input as a function of site quality is rising in the open sector, flat on the interval where insiders do not use hired labor, and rising again in the stratified sector. The areas D_O(x), D_I(x), D_E(x), and D_C(x) under the density curve show the total labor input (i) at open access sites; (ii) by insiders at unstratified sites; (iii) by elites at stratified sites; and (iv) by commoners at stratified sites, respectively.

Figure 6.2. Labor input as a function of site quality for a fixed normalized wage

Lemma 6.2 shows that an equilibrium can be found by associating each normalized wage with the corresponding density function from Lemma 6.1, computing the area under this density as in Figure 6.2, and then varying the wage until the total area under the density curve is equal to the regional population N. This procedure gives the following results.

Proposition 6.1

(short-run equilibrium).

Define $\mathrm{Q}\equiv \left(1-\mathrm{\alpha }\right)/\left(2-\mathrm{\alpha }\right)$, ${\mathrm{N}}^{\mathrm{a}}\equiv \mathrm{Qd}$, ${\mathrm{N}}^{\mathrm{b}}\equiv 2\mathrm{Qd}$, ${\mathrm{x}}^{\mathrm{a}}\equiv {\mathrm{d}}^{\mathrm{\alpha }-1}$, and ${\mathrm{x}}^{\mathrm{b}}\equiv {\mathrm{\alpha d}}^{\mathrm{\alpha }-1}$. Let the labor demand function be $\mathrm{D}\left(\mathrm{x}\right)\equiv {\int }_0^1\mathrm{L}\left(\mathrm{s},\mathrm{x}\right)$ ds where L(s, x) is obtained from Lemma 6.1.

(a)	$\mathrm{N}<{\mathrm{N}}^{\mathrm{a}}$	implies ${\mathrm{x}}^{\mathrm{a}}<\mathrm{x}$	and $\mathrm{D}\left(\mathrm{x}\right)={\mathrm{Qx}}^{-1/\left(1–\mathrm{\alpha }\right)}$
(b)	${\mathrm{N}}^{\mathrm{a}}\le \mathrm{N}\le {\mathrm{N}}^{\mathrm{b}}$	implies ${\mathrm{x}}^{\mathrm{b}}\le \mathrm{x}\le {\mathrm{x}}^{\mathrm{a}}$	and $\mathrm{D}\left(\mathrm{x}\right)=\mathrm{d}-{\mathrm{xd}}^{2-\mathrm{\alpha }}/\left(2–\mathrm{\alpha }\right)$
(c)	${\mathrm{N}}^{\mathrm{b}}<\mathrm{N}$	implies $\mathrm{x}<{\mathrm{x}}^{\mathrm{b}}$	and $\mathrm{D}\left(\mathrm{x}\right)=\mathrm{Q}\left[{\mathrm{xd}}^{2-\mathrm{\alpha }}/\mathrm{\alpha }+{\left(\mathrm{\alpha }/\mathrm{x}\right)}^{1/\left(1-\mathrm{\alpha }\right)}\right]$
(d)	The labor demand function $\mathrm{D}\left(\mathrm{x}\right)$ is continuously differentiable with ${\mathrm{D}}^{\prime }\left(\mathrm{x}\right)<0$ for all $\mathrm{x}>0$. Also, ${lim}_{\mathrm{x}\to 0}\mathrm{D}\left(\mathrm{x}\right)=\infty$ and ${lim}_{\mathrm{x}\to \infty }\mathrm{D}\left(\mathrm{x}\right)=0$.
(e)	For each $\mathrm{N}>0$, there is a unique $\mathrm{x}>0$ such that $\mathrm{D}\left(\mathrm{x}\right)=\mathrm{N}$. The combination of this normalized wage and the associated labor allocation L(⋅, x) from Lemma 6.1 form a unique SRE for the given N. The equilibrium wage x(N) is continuously differentiable with ${\mathrm{x}}^{\prime }\left(\mathrm{N}\right)<0$, ${lim}_{\mathrm{N}\to 0}\mathrm{x}\left(\mathrm{N}\right)=\infty$, and ${lim}_{\mathrm{N}\to \infty }\mathrm{x}\left(\mathrm{N}\right)=0$.

Figure 6.3 shows the relationships among the regional population N, the wage, and property rights. The downward sloping labor demand curve D(x) is obtained from Proposition 6.1, and the regional population N gives a vertical supply curve for labor. At low population levels, the wage is high and all sites are open, because even the best sites fail to reach the threshold d. Population growth leads to a falling wage. Beyond N^a, the best sites become closed. Further population growth leads to more enclosures in order of decreasing site quality. Beyond N^b, the best sites are both closed and stratified.

Figure 6.3. Normalized wage and property rights as a function of regional population

6.6 The Aggregate Production Function

Before considering long-run equilibrium, we need to develop some ideas about aggregate output. First, suppose that all sites are open. Due to the constant elasticity of output, equalizing the average product of labor across sites is the same as equalizing the marginal product of labor (MPL = αAPL). Therefore labor is allocated efficiently. But if some sites are closed, a distortion occurs because labor input is constant on [s^a(x), s^b(x)] while site qualities are not, so marginal products are unequal. If some sites are stratified, a second distortion arises because the marginal product of labor at such sites is equal to the average product at open sites. Here we study the implications for regional output.

Total food output for the region is

$\mathrm{Y}={{\int }_0}^1\mathrm{\theta sL}{\left(\mathrm{s},\mathrm{x}\right)}^{\mathrm{\alpha }}\mathrm{ds}$(6.1)

where the labor allocation L(⋅, x) satisfies Lemma 6.1 and the equilibrium wage is derived from D(x) = N in Proposition 6.1. Thus the aggregate production function becomes

$\mathrm{Y}\left(\mathrm{N}\right)={{\int }_0}^1\mathrm{\theta sL}{\left[\mathrm{s},\mathrm{x}\left(\mathrm{N}\right)\right]}^{\mathrm{\alpha }}\mathrm{ds}$(6.2)

or

$\mathrm{Y}\left(\mathrm{N}\right)=\mathrm{\varphi }\left[\mathrm{x}\left(\mathrm{N}\right)\right]\text{where}\mathrm{\varphi }\left(\mathrm{x}\right)={{\int }_0}^1\mathrm{\theta sL}{\left(\mathrm{s},\mathrm{x}\right)}^{\mathrm{\alpha }}\mathrm{ds}$(6.3)

Figure 6.4 shows the production function Y(N). This function is increasing with a continuous marginal product, but its second derivative is discontinuous at N^a and N^b. In the population interval N < N^a, all sites are open and there are no distortions. Output has a Cobb–Douglas form as in part (a) of the Corollary, where Q can be interpreted as the endowment of quality-adjusted land for the region as a whole. In the range N^a < N < N^b, higher-quality sites are closed and labor input is constant at d across these sites. In this interval Y(N) is below the dashed continuation of the Cobb–Douglas function from (a).

Figure 6.4. Aggregate production function

When N^b < N, the highest-quality sites become stratified. This creates a further distortion because the marginal product of labor is not equalized between the open and stratified sectors. As shown in Figure 6.4, for population levels near N^b the production function is convex. The rising marginal product results from the shifting property rights boundaries between the open and closed sectors as well as the unstratified and stratified sectors. Y(N) has an inflection point at N^c and then returns to concavity. At very high population levels almost all sites are stratified, marginal products are equal at almost all sites, and Y(N) approaches the dashed Cobb–Douglas curve in Figure 6.4.

Aggregate per capita food Y(N)/N is globally decreasing in total population N, as indicated in part (d) of the Corollary. This occurs because the usual tendency for average product to decline at each individual site outweighs the property rights effects responsible for the non-concave portion of Y(N).

6.7 Long-Run Equilibrium

Population is endogenous in the long run. Each adult in period t has children who survive to become adults in period t+1. Adults from period t die at the end of that period. The number of children for an individual is equal to the parent’s food income multiplied by a constant ρ > 0. This relationship arises because fertility is an increasing function of food, childhood mortality is a decreasing function of food, or both.

We assume that ρ is identical for all adults, so the aggregate number of new adults in period t+1 is ρY^t where Y^t is regional food output in period t. Let N^t be the population of adults in period t for the region as a whole. The dynamics for the region are given by N^t+1 = ρY(N^t; θ) where Y(N; θ) is the aggregate production function in Section 6.6 with the productivity parameter θ included as an explicit argument. In a long-run equilibrium we always have Y(N; θ)/N = 1/ρ. This is the Malthusian feature of the model: as long as the demographic parameter ρ is constant, every long-run equilibrium yields the same food per capita at the regional level. A permanent increase in productivity (θ) is thus absorbed in the long run through higher population rather than higher food per person.

The results in Proposition 6.3 are shown in Figures 6.5(a), 6.5(b), and 6.5(c). The ray from the origin with slope $1/\mathrm{\rho }$ is identical in all three graphs. The only distinction involves the productivity parameter θ, which is low in Figure 6.5(a), intermediate in Figure 6.5(b), and high in Figure 6.5(c). The values of N^a and N^b are independent of θ. In each case the LRE is unique because the aggregate average product of labor Y(N; θ)/N is globally decreasing in N (see part (d) of the Corollary to Proposition 6.2).

Figure 6.5.(a) Long-run equilibrium with low productivity

Figure 6.5.(b) Long-run equilibrium with intermediate productivity

Figure 6.5.(c) Long-run equilibrium with high productivity

Figure 6.5(a) shows the case where $0<\mathrm{\theta }<{\mathrm{\theta }}^{\mathrm{a}}$ and $0<\mathrm{N}\left(\mathrm{\theta }\right)<{\mathrm{N}}^{\mathrm{a}}$. The ray from the origin with slope $1/\mathrm{\rho }$ intersects the production function $\mathrm{Y}\left(\mathrm{N};\mathrm{\theta }\right)$ at a population less than N^a. All sites are open because $\mathrm{N}\left(\mathrm{\theta }\right)<{\mathrm{N}}^{\mathrm{a}}$. Figure 6.5(b) shows the case where ${\mathrm{\theta }}^{\mathrm{a}}<\mathrm{\theta }<{\mathrm{\theta }}^{\mathrm{b}}$ and ${\mathrm{N}}^{\mathrm{a}}<\mathrm{N}\left(\mathrm{\theta }\right)<{\mathrm{N}}^{\mathrm{b}}$. Because the production function $\mathrm{Y}\left(\mathrm{N};\mathrm{\theta }\right)$ has now shifted up, the ray from the origin and $\mathrm{Y}\left(\mathrm{N};\mathrm{\theta }\right)$ intersect at a higher value of N. Due to the larger population the best sites are closed, although none are stratified. Finally, Figure 6.5(c) shows the case where ${\mathrm{\theta }}^{\mathrm{b}}<\mathrm{\theta }$ and ${\mathrm{N}}^{\mathrm{b}}<\mathrm{N}\left(\mathrm{\theta }\right)$. Here the productivity θ is high enough to support a long-run population above N^b and accordingly the best sites are stratified.

We conclude this section with a brief description of the process through which the system converges to long-run equilibrium for a fixed θ. This is shown in Figure 6.6. We start from an initial population N⁰ with a corresponding output Y⁰. Suppose N⁰ is below the long-run population N*. The output Y⁰ determines the adult population N¹ in the next generation through ${\mathrm{N}}^1={\mathrm{\rho Y}}^0$. This implies that the point (N¹, Y⁰) lies on the ray from the origin with slope $1/\mathrm{\rho }$. N¹ yields the output Y¹ through the production function, and so on. The arrows show the process by which the system converges monotonically to the point (N*, Y*). The same dynamics apply in reverse starting from a population above N*.

Figure 6.6. Monotonic convergence to a long-run equilibrium

In later sections two distinct thought experiments will be considered. In the first case we start from an LRE with a low productivity level θ* and a low population N*. For example N* might be associated with an open access LRE as shown in Figure 6.5(a). We then consider a discrete permanent jump in productivity to ${\mathrm{\theta }}^{\ast \ast }>{\mathrm{\theta }}^{\ast }$. Depending on θ** the new LRE may generate insider–outsider inequality without stratification as in Figure 6.5(b) or it may involve stratification as in Figure 6.5(c). In either case the dynamics of the convergence process are qualitatively the same as in Figure 6.6.

The second thought experiment involves continuous productivity growth at a rate that is slow compared with the rate at which population adjusts. In this case, it would be a reasonable approximation to assume that population is always near (though not exactly equal to) the LRE level N* associated with current productivity θ*. We can then use the results from Proposition 6.3 to study the comparative static effects of θ on N and w.

6.8 Poverty

The least well-off agents are those receiving the wage w, either in the commons or at stratified sites (if any). The dynamics of poverty can be investigated using the thought experiments described at the end of Section 6.7. We start in each case from a LRE with the productivity θ*, population ${\mathrm{N}}^{\ast }=\mathrm{N}\left({\mathrm{\theta }}^{\ast }\right)$, and non-normalized wage ${\mathrm{w}}^{\ast }=\mathrm{w}\left({\mathrm{\theta }}^{\ast }\right)$.

First consider a permanent productivity increase to ${\mathrm{\theta }}^{\ast \ast }>{\mathrm{\theta }}^{\ast }$ in period 0. We want to examine the dynamics of the wage along the adjustment path leading to the new LRE (θ**, N**). The immediate effect of the productivity shock is to raise the short-run equilibrium wage w⁰ to a level such that ${\mathrm{w}}^0/{\mathrm{\theta }}^{\ast \ast }={\mathrm{w}}^{\ast }/{\mathrm{\theta }}^{\ast }$. This follows from the fact that the ratio $\mathrm{x}\equiv \mathrm{w}/\mathrm{\theta }$ is determined solely by N in the short run, and the initial population N* remains in place. The result is ${\mathrm{w}}^0>{\mathrm{w}}^{\ast }$, so the short run effect of a positive productivity shock is always to alleviate poverty by making commoners better off.

Along the adjustment path, we have an increasing population $\left({\mathrm{N}}^0,,,{\mathrm{N}}^1,,,.,,.\right)$ where ${\mathrm{N}}^0={\mathrm{N}}^{\ast }$ and the sequence converges to N**. Because population is rising, the ratio $\mathrm{x}=\mathrm{w}/{\mathrm{\theta }}^{\ast \ast }$ is falling due to the downward sloping labor demand curve. The new productivity level θ** is constant throughout the adjustment process and so the non-normalized wage sequence $\left({\mathrm{w}}^0,{\mathrm{w}}^1,..\right)$ is decreasing.

Whether the final wage w** in the new LRE is at or below the initial wage w* depends on θ**. If the new productivity level is consistent with universal open access $\left({\mathrm{\theta }}^{\ast \ast }<{\mathrm{\theta }}^{\mathrm{a}}\right)$, then ${\mathrm{w}}^{\ast }={\mathrm{w}}^{\ast \ast }=1/\mathrm{\rho }$ due to Proposition 6.3(d). The productivity shock makes everyone better off along the adjustment path with eventual convergence back to the original wage. On the other hand, if ${\mathrm{\theta }}^{\ast \ast }>{\mathrm{\theta }}^{\mathrm{a}}$ so that some sites are closed in the new LRE, Proposition 6.3(d) implies ${\mathrm{w}}^{\ast }>{\mathrm{w}}^{\ast \ast }$. In the long run the poor become worse off through the closure of more sites and the contraction of the commons.

The second thought experiment involves a gradual improvement in productivity from θ* to θ**, where we ignore short-run adjustments and focus on the long-run wage w(θ). Again Proposition 6.3(d) gives a straightforward verdict. As long as θ is low enough to maintain universal open access, the wage remains constant at w(θ) = 1/ρ. However, once θ > θ^a so that some sites are closed, further productivity growth reduces the wage and leads to worsening absolute poverty.

One implication of these results involves the distinction drawn by Bogaard et al. (Reference Bogaard, Fochesato and Bowles2019) between labor-limited and land-limited agriculture. As explained in Section 6.3, these concepts are defined by the ratio of the marginal products for labor and land. Our food technology in Section 6.5 does not explicitly include land but in Section 8.4 we will extend it to include both inputs. With this extension, and the assumption that each site has one unit of land, it is easy to show that as the wage (w) declines, the stratified sites become relatively less labor-limited and more land-limited. This occurs even though in our model productivity growth is neutral between labor and land (the result stems from endogenous property rights to land). In the long run this strengthens incentives for elites to adopt innovations that enable them to substitute abundant labor for scarce land.

6.9 Inequality

Here we address two separate issues: Inequality within a given site, and inequality for the region as a whole. First consider an individual stratified site. The total number of agents is L, where L - d are commoners and d are members of the elite. Each commoner receives the wage w and each elite agent receives w + r(s)/d where r(s) > 0 is land rent at a site of quality s. The Lorenz curve consists of two line segments, one for each class of agents (see the discussion of Lorenz curves and Gini coefficients from Section 6.3). The Gini coefficient for an individual stratified site is G = (1 − α)[1 − d/L(s, x)] where L(s, x) is the optimal labor input from Lemma 6.1(c) and α and d are fixed parameters.

In a short-run equilibrium, one can compare the Gini coefficients across stratified sites. Because L(s, x) is an increasing function of s, the better sites have more inequality. One can also compare the same site across short-run equilibria with different values of x. In the SRE where x is lower, or equivalently population N is higher, a given site has more inequality because the elite hires more commoners. As N approaches infinity and x goes to zero, L(s, x) goes to infinity for any fixed site quality s. Therefore the Gini at the site approaches 1 − α. This is the share of food output captured by landowners in a perfectly competitive equilibrium, as we will explain in connection with Proposition 6.4 below.

We now turn to the effect of population on regional inequality. When N < N^a so that all sites are open, there is no inequality. In what follows, we assume N > N^a so that some sites are closed. A Lorenz curve y(z) for this case is presented in Figure 6.7. The horizontal axis z indicates the fraction of the regional population (ordered from poorest to richest) whose food income falls below some given level, and the vertical axis y indicates the fraction of regional food income received by this subset of agents.

Figure 6.7. Lorenz curve y(z) for a fixed population when some sites are closed

The Lorenz curve has two segments. First there is a linear portion involving the worst-off agents, who receive the wage w either in the commons or through employment at stratified sites. This group makes up a fraction z^a ≡ (D_O + D_C)/N of the population and determines the Lorenz curve on the interval [0, z^a]. The slope of this segment is wN/Y, which is direct plus imputed labor cost as a fraction of total food output. The non-linear segment of the Lorenz curve beyond z^a involves agents who receive land rent. This part of the curve is derived as follows.

An insider or elite agent at a site of quality s receives the income w + r(s)/d where w is the imputed value of the agent’s time, r(s) is land rent, and d is the number of insider agents. For any income level above w, there is some fraction of the population z > z^a receiving that income or less. This group includes all commoners, as well as insiders who occupy sites at or below a quality cutoff s(z) given by

$\mathrm{s}\left(\mathrm{z}\right)=1-\left(1-\mathrm{z}\right)\mathrm{N}/\mathrm{d}\mathrm{for}{\mathrm{z}}^{\mathrm{a}}\le \mathrm{z}\le 1.$(6.4)

The total rent R(z) appropriated by the poorest z of the population is

$\begin{array}{ll}\mathrm{R}\left(\mathrm{z}\right)\equiv 0& \mathrm{for}0\le \mathrm{z}\le {\mathrm{z}}^{\mathrm{a}}\mathrm{and}\\ \mathrm{R}\left(\mathrm{z}\right)={{\int }_0}^{\mathrm{s}\left(\mathrm{z}\right)}\mathrm{r}\left(\mathrm{s}\right)\mathrm{ds}& \mathrm{for}{\mathrm{z}}^{\mathrm{a}}\le \mathrm{z}\le 1\end{array}$(6.5)

where r(s) ≡ 0 for the open sites s ∈ [0, s^a) and r(s) = θsL(s)^α − wL(s) ≥ 0 for the closed sites s ∈ [s^a, 1]. In the latter case L(s) is the optimal labor input from Section 6.5. This is equal to d for unstratified sites, with L(s) > d for stratified sites (if any).

For a fixed N > 0, the Lorenz curve y(z) is given by

$\mathrm{y}\left(\mathrm{z}\right)=\left(1/\mathrm{Y}\right)\left[\mathrm{wNz}+\mathrm{R}\left(\mathrm{z}\right)\right]$(6.6)

The Gini coefficient is twice the area between the 45° line and the Lorenz curve, or

$\mathrm{G}=2\left[1/2-{{\int }_0}^1\mathrm{y}\left(\mathrm{z}\right)\mathrm{dz}\right]$(6.7)

For z = 1, the right-hand vertical axis in Figure 6.7 shows the division of income across classes of agents. The fraction of total food going to commoners is w(D_O + D_C)/Y, the fraction going to insiders and elites as imputed wages is w(D_I + D_E)/Y, and the fraction going to the latter two groups as land rent is 1 − wN/Y.

Part (a) shows that starting from insider–outsider inequality, population growth always leads to a higher Gini. Indeed we have Lorenz curve dominance: when N rises, the fraction of income y(z) for the worst-off z of the population falls at every 0 < z < 1. This is true whether the society remains in an insider–outsider regime or shifts to an elite–commoner regime. Although the Gini is not necessarily increasing in N throughout the elite–commoner range, it must increase within the insider–outsider range.

The absolute productivity θ drops out of the Gini coefficient and is irrelevant for Proposition 6.4. The only parameters that influence inequality are the elasticity α and the exclusion threshold d. Part (b) shows that any elite–commoner equilibrium has greater inequality than any insider–outsider equilibrium. The boundary value G(N^b) is inversely related to α because when food output is more responsive to labor, commoners are paid more and land rents become less significant as a source of inequality.

The limit result for G(∞) follows from the fact that when the population is large, almost all agents are commoners, and almost all commoners are employed at stratified sites. The property rights distortions discussed in Section 6.6 become negligible, and the aggregate production function is approximately described by part (a) of the Corollary to Proposition 6.2. As a result, the slope of the linear part of the Lorenz curve in Figure 6.7 is approximately the labor share α arising in a competitive economy. Because the boundary z^a in Figure 6.7 approaches one, the area under the Lorenz curve approaches α/2 and (6.7) gives a Gini of 1 − α. Hence in a large elite–commoner society, the Gini coefficient is the output share that would go to landowners in a perfectly competitive economy.

No assumptions about the long run are used in Proposition 6.4. However, suppose a society starts from long-run equilibrium with insider–outsider inequality. If better climate or improved food production technology stimulates population growth, this must increase inequality both along the adjustment path and in the new LRE. Higher productivity does not directly cause higher inequality; the two variables are linked only through the channel of Malthusian population growth and endogenous property rights.

6.10 Demography

The definition of long-run equilibrium in Section 6.7 ensures that at the regional level, each generation produces exactly enough children to replace itself. But agents with high incomes produce more children than are needed to replace themselves, while agents with low incomes produce fewer children than are needed. The following proposition relates site quality to reproduction.

Whenever there is inequality, the commoners have an income too low to permit demographic replacement. Insiders at sites with quality near s^a have land rents near zero and must also fail to replace themselves. On the other hand, insiders at the best sites (s = 1) must more than replace themselves in order to maintain a steady-state population for the economy as a whole. Since land rent is an increasing function of site quality, there is an intermediate value s^r at which the insiders are in demographic equilibrium.

Recall from Section 6.5 that when the number of agents born at a site (n) exceeds the number needed to close the site (d), the latter group is selected by birth order while the remaining n-d agents are excluded. Proposition 6.5 shows that all stratified sites have downward mobility in the sense that some children of the elite are excluded from elite status. These agents go to an open site or are hired at a stratified site.

There is also downward mobility at high-quality unstratified sites (those with s^r < s ≤ s^b) because some children of insiders do not become insiders themselves. However, at lower-quality unstratified sites (those with s^a ≤ s < s^r) the number of children n born to insiders is less than d. As discussed in Section 6.5, in this case d-n commoners enter the site, and at that point the insiders become sufficiently numerous to block further entry.

6.11 Regional Cases

During the Upper Paleolithic, foraging groups were small, mobile, and exploited large territories at low population densities (see Chapter 3). Our theory predicts minimal inequality under such conditions because almost all sites would have been open.

In some parts of the world foragers became sedentary during the recovery from the Last Glacial Maximum or shortly after the onset of the Holocene. We believe these transitions were caused by positive climate change and led to higher population densities (see Chapter 4). Our theory predicts that this process could have led to insider–outsider inequality and perhaps even elite–commoner inequality at the best sites.

Shortly before the Holocene a negative climate shock (the Younger Dryas) led to initial cultivation in southwest Asia and perhaps elsewhere. The effect of the Younger Dryas on sedentism is unclear (see Section 5.8), but learning by doing and the domestication of plants and animals raised productivity. This and the benign Holocene climate resulted in population growth along the trajectory leading to full agriculture, often resulting in settlements much larger than those of sedentary foragers (see Chapter 5). Our theory predicts that these trends would have intensified both insider–outsider and elite–commoner inequality.

This broad portrait is consistent with the empirical generalizations from Section 6.3. The evidence reviewed there indicates that inequality has tended to rise with the transitions from mobile to sedentary foraging and from sedentary foraging to agriculture. These trends are visible in archaeological data on housing and burials. Ethnography also supports these generalizations, and strongly suggests that population plays an important mediating role between productivity and inequality. It also suggests that class structures and inheritance are more common in agricultural societies than those based on foraging.

Here we present a series of regional case histories for Southwest Asia, Europe, Polynesia, and the Channel Islands of California. These examples are chosen mainly because they have been intensively studied by archaeologists, and reflect a variety of food acquisition strategies including foraging, fishing, and farming.

6.12 Conclusion

We have developed a theory that links rising technical productivity with rising population density and inequality. Our approach attributes inequality to the creation of group property rights over land. Rising productivity does not lead to growing inequality directly, but rather indirectly via Malthusian population growth and endogenous property rights. The model applies equally well to foragers and farmers. It accounts both for the emergence of inequality within a society and variation of inequality across societies. In Section 6.11 we argued that several regional narratives from archaeology are supportive.

Our theory highlights the formation of corporate groups that collectively control access to valuable sites or territories. We do not examine landownership by individuals or households, but this could develop once sites are closed. In long-run equilibrium with stratification, elite agents inherit membership in these corporate landowning groups from their parents, although some offspring of the elite will suffer downward mobility into the commoner class. The same is true for insiders at high-quality unstratified sites. Insiders at lower-quality unstratified sites must accept some new outsiders in each generation to preserve their property rights. Commoners do not replace themselves demographically, so this class is maintained partly by downward mobility from elite and insider groups.

The aggregate production function shows how property rights affect allocative efficiency as population rises. Due to our Cobb–Douglas functional form, open access yields an efficient allocation of labor. When some sites are closed, output is below its theoretical maximum because marginal products of labor are not equated across closed sites. When population crosses into the elite–commoner range, another distortion arises because the marginal product of labor at stratified sites is equal to the average product in the commons. At very high population levels, almost all sites are stratified and almost all labor is paid its marginal product. As a result, aggregate output approaches its theoretical upper bound, and the Gini coefficient approaches the output share that would go to landowners in a perfectly competitive economy.

Our theory has numerous testable implications. Assume that climate and/or food production technology improves over time, and that regional population therefore grows over time. In this setting our predictions include the following.

1. (a) Insider–outsider inequality will arise first at the best sites in the region and then spread to sites of lower quality.

2. (b) Elite–commoner inequality will arise after insider–outsider inequality. It will also begin at the best sites and then spread to sites of lower quality.

3. (c) The commons will shrink and the average quality of open sites will decline.

4. (d) The distribution of food consumption within stratified sites will be bimodal.

5. (e) Inequality within stratified sites will be greater at the higher-quality sites.

6. (f) Once there is insider–outsider inequality, commoners will become increasingly impoverished over time.

7. (g) Once there is insider–outsider inequality, the regional Gini coefficient will rise over time (although not necessarily after stratification has begun to emerge).

8. (h) All elite individuals at a stratified site will have elite parents, but some children of elite individuals will become commoners.

Future archaeological research could refute these (or other) implications of our theory. In that case we would need to modify the model or look for alternative explanations.

6.13 Postscript

This chapter is based on our article “The origins of inequality: Insiders, outsiders, elites, and commoners” published in the Journal of Political Economy (Dow and Reed, Reference Dow and Reed2013). Leanna Mitchell provided exceptional research assistance for the original article. We are grateful for comments from colleagues at Simon Fraser University, participants at the Workshop on the Emergence of Hierarchy and Inequality hosted by the Santa Fe Institute in February 2009, and participants at the SFU Conference on Early Economic Developments in July 2009. We also thank two anonymous referees and two editors at the Journal of Political Economy for detailed comments on earlier drafts. The Human Evolutionary Studies Program at SFU and the Social Sciences and Humanities Research Council of Canada provided funding. We received valuable feedback on an earlier draft of this chapter from Samuel Bowles, Richard Lipsey, John Chant, Jon Kesselman, Craig Riddell, Herbert Grubel, and Richard Harris. All opinions are those of the authors.

Sections 6.1 and 6.2 have been extended and updated for this volume, and Section 6.3 is entirely new. The descriptions of the gold rush in Section 6.4 and the formal model in Sections 6.5–6.10 are lightly edited versions of their counterparts from the JPE article. All formal propositions are identical. New material has been added at the start of Section 6.11 and we have rewritten the archaeological narrative for southwest Asia, but the other regional narratives are essentially unchanged. Section 6.12 has been rewritten. When the formal model from Sections 6.5–6.10 was constructed, we were not aware of the empirical findings reported in Section 6.3 with publication dates of 2014 or later.

7.1 Introduction

Chapter 6 addressed the transition to inequality. In this chapter and the next, we address the transition to organized warfare over land. We regard these as the two most significant institutional developments before cities and states. This chapter explores the incidence of warfare among foragers and early farmers in a setting where the rival groups are egalitarian. This will be followed in Chapter 8 by an examination of warfare among elites in stratified societies.

In our theoretical framework inequality and warfare both involve technologies of coercion. We showed in Chapter 6 that inequality can emerge through a technology of exclusion, where a group of organized insiders prevents entry to a site by unorganized outsiders. For the purposes of Chapters 7 and 8 we introduce a technology of combat, where one organized group fights another organized group for control of a site.

Warfare has been defined in a variety of ways. For example, LeBlanc (Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020, 40) takes it to mean “collective action by one group against another, without there being a larger overarching political entity, the membership of which includes both groups.” In Chapters 7 and 8 we assume the absence of an overarching political entity and simply define warfare to be lethal conflict between organized groups.

Our focus is specifically on conflict between groups over land. This differs from other forms of lethal violence motivated by revenge, feuding, status competition, or theft. Violence has numerous proximate causes but it will be convenient to distinguish raiding, displacement, and conquest. Raiding means the theft of moveable wealth or kidnapping of humans. It does not involve a transfer of land from one group to another. We are not proposing a theory of raiding. Whenever we use the term “war” without qualification, we mean “warfare over land.”

Displacement means permanently taking over the territory of an opposing group and driving away or killing the previous inhabitants, with the intention of using the new territory as a source of food. Conquest means seizure of territory from a rival elite while keeping commoners in place to produce food and pay land rent. This chapter will focus on displacement and Chapter 8 will focus on conquest.

Our goal is not to resolve empirical disputes about the prevalence of warfare in prehistory. There has been intense debate over this issue, as will be seen in Section 7.2. Instead, we study a theoretical question: under what conditions are groups more likely or less likely to engage in warfare over land? However, we will argue that the evidence is consistent with the formal model we develop.

The central message of this chapter is that when groups are internally egalitarian and individual agents can migrate easily between groups, warfare over land is unlikely. The reason is that both individual migration and Malthusian population dynamics tend to generate a positive correlation between site quality and group size. This prevents war. A small group with a poor site might like to take over land controlled by a large group with a rich site, but a military victory over the larger group is unlikely. Hence the smaller and poorer group refrains from attacking. Conversely the larger and richer group has a high probability of military success, but gains little by taking over the site of the poorer group. We do not entirely rule out warfare among egalitarian groups but this occurs only under narrow conditions involving shocks from nature or technology. The story will be very different in Chapter 8, which will show that elites in stratified societies routinely engage in warfare with other elites, or use threats of force to bully their opponents into fleeing.

In both chapters groups consist of individuals who maximize their expected food consumption. There is a production technology where food is obtained from labor and land, and a military technology where the probability of a successful attack on another group depends on the relative size of the two groups. There are two sites or territories where food can be obtained and their productivities may differ.

In this chapter we assume that groups are internally egalitarian in the sense that they share food equally and reach unanimous collective decisions. Attacks are motivated by a group’s attempt to increase the quantity and/or quality of land it controls. In order to focus on situations in which wars are actually fought, we assume a group defends its land when it is attacked, because fleeing to another location is even less attractive. Chapter 8 will instead consider three strategies: attack, defend, and flee.

Time periods are the length of a human generation. At the beginning of a period each site has a population inherited from the past. Individual agents can move between sites, although this is generally costly. Once the final group sizes are determined, each group decides whether to attack the other. Finally production takes place, agents have children, and fertility-driven Malthusian dynamics generate a new initial population for each site in the next period. The process then repeats.

As mentioned above, in this framework war is generally deterred by the positive correlation between site qualities and group sizes resulting from individual migration and Malthusian dynamics. There are two necessary conditions for warfare among egalitarian groups: an exogenous productivity shock from nature or technology to serve as a trigger, and costly individual mobility. Exogenous shocks that change the relative productivities of sites or territories can create a temporary imbalance between site qualities and group sizes while costly individual mobility makes it difficult to undo these imbalances through migration. Under our assumptions warfare tends to be self-limiting because it reallocates population toward better sites. Therefore a series of wars requires a series of shocks.

Our analysis suggests that warfare over land would have been rare among mobile foraging bands with exogamous marriage, where agents could respond to negative local shocks by exploiting kinship networks to join more prosperous groups. The analysis also suggests that warfare over land would have become more common after the emergence of sedentary foraging or farming groups with endogamous marriage and rigid membership criteria. Even in this case, exogenous shocks that changed the relative productivities of sites or territories would also have been necessary.

We make numerous simplifying assumptions in our formal model. For example, we employ a military technology where the winners suffer no casualties while the losers are exterminated. We also ignore potential advantages accruing to the role of defender, such as the ability to build fortifications. Some readers may be uncomfortable with these assumptions. Such readers should note that our assumptions about military technology usually stack the deck in favor of aggression. This only strengthens our conclusion that early warfare over land was rare, because more reasonable assumptions would make war even less likely than we claim. This issue will be discussed in more detail in Section 7.8.

In the formal model of this chapter we assume there are two sites or territories, with a group of agents at each location. We ignore the larger regional setting of Chapter 6 with many sites that are open, closed, or stratified. The model here can be interpreted as a simple region with only two sites rather than a continuum of sites. Accordingly, we do not utilize a technology of exclusion involving a critical mass of insiders who prevent entry by unorganized outsiders (there is no equivalent to the parameter “d” used in Chapter 6). Instead, we consider a technology of combat where insiders try to prevent entry by an organized coalition of outsiders. Section 7.9 discusses relationships between the models of Chapters 6 and 7.

Section 7.2 reviews the literature on the subject, and Section 7.3 surveys some archaeological and anthropological evidence about warfare in small-scale foraging and farming societies. Sections 7.4, 7.5, and 7.6 develop the formal model through a process of backward induction. Section 7.4 treats group sizes and site qualities as exogenous and derives conditions where war and peace occur. Section 7.5 moves back to the individual migration stage and characterizes locational equilibrium where no individual agent wants to change sites. Section 7.6 introduces our Malthusian fertility assumption and presents necessary and sufficient conditions for warfare. Non-economists may want to skip over the formal modeling in Sections 7.4–7.6. However, we recommend reading the opening paragraphs in these sections, which provide non-technical summaries of the results and create a pathway through the theory for those who prefer a verbal presentation.

Section 7.7 discusses how our theory can account for the evidence from Section 7.3. Section 7.8 examines a number of modeling assumptions that could be altered, and Section 7.9 links the models from Chapters 6 and 7. Section 7.10 is a postscript. Proofs of the formal propositions are available at cambridge.org/economicprehistory.

7.2 Theories of Early Warfare

There is a consensus that warfare is common among elites in stratified societies (see Chapter 8), but the frequency and intensity of warfare among internally egalitarian groups is more variable. Here we seek to shed light on the question of whether warfare over land had an origin, and more generally, how one might explain variation across small-scale foraging and farming societies in the incidence of warfare.

A number of scholars (Keeley, Reference Keeley1996; LeBlanc, Reference LeBlanc1999, Reference LeBlanc2007, Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020; LeBlanc and Register, Reference LeBlanc and Register2003; Gat, Reference Gat2006; Pinker, Reference Pinker2011) argue that humans are a violent and aggressive species, and from the beginning have lived in a state of frequent warfare. This has led to a literature that views war as a biological selection mechanism for other human traits (see Ferguson, Reference Ferguson and Fry2013a, for a discussion and references). For example, Bowles and colleagues argue that warfare provides an evolutionary explanation for the co-existence of altruism within human groups and hostility across groups (Bowles, Reference Bowles2006, Reference Bowles2009, Reference Bowles2012; Choi and Bowles, Reference Choi and Bowles2007; Bowles and Gintis, Reference Bowles2011).

Claims of pervasive warfare in prehistory have been vigorously contested (Fry, Reference Fry2006, Reference Fry2012, Reference Fry2013; Fry and Soderberg, Reference Fry and Soderberg2013; Ferguson, Reference Ferguson2018). While the “war” school argues that 15 to 25 percent of adult males died in warfare in prehistory (LeBlanc, Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020, 42), the “peace” school argues that the cases advanced as examples of early warfare are often not convincing and, in any case, are not representative of a reality in which warfare was rare. LeBlanc (Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020, 50) argues that the “peace” school ignores bias in the evidence allegedly favoring a lack of warfare in contemporary ethnographic studies, and that this bias renders claims for the prevalence of peaceful societies “virtually worthless.”

We contribute to this debate by studying the theoretical conditions under which early warfare could have occurred. As noted in Section 7.1, our focus is specifically on group conflict over land rather than other forms of lethal group violence such as raiding. These other types of violence may have predated warfare over land. The preponderance of the evidence for early warfare among foragers is associated with raiding (Gat, Reference Gat2006, 184; LeBlanc, Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020, 42).

We are not concerned here with the question of whether a propensity toward war is part of human nature. If one believes that the relevant part of the human genome has been stable for the last 15,000 years or so, and if one wishes to explain variations in the incidence of warfare across societies during that period, then an explanation rooted in biology would be an explanation of a variable by a constant. We agree with Gat (Reference Gat2006, chs. 3–5) that war may or may not promote the deeper biological goals of survival and reproduction, depending on the circumstances. We will consider the biological subgoal of food acquisition, and study the roles of natural resources, technology, and population as causal factors in the prevalence of warfare.

Boix (Reference Boix2015) argues that cooperation within small foraging bands was destabilized by the inequality resulting from technical advances. In this scenario individuals specialized into roles as producers or predators, depending on their comparative advantages. Two trajectories tended to arise: Either the producers banded together to defend themselves against the predators, or the predators banded together to exploit the producers. Boix (Reference Boix2015, ch. 4) describes innovations in military technology and resulting changes in political structure. These innovations include metal weapons, horse domestication, and gunpowder, which are generally not relevant for the societies of interest in this chapter. However, we will return to ideas from Boix in Chapters 8, 11, and 12.

There is a small economic literature on conflict in prehistory. Baker (Reference Baker2003) constructs a model where groups compete for access to land. In equilibrium there is no actual warfare. North et al. (Reference North, Wallis and Weingast2009) and Rowthorn and Seabright (Reference Rowthorn and Seabright2010) study the effects of early warfare on the development of institutions and agriculture respectively. Bowles and co-authors, as mentioned above, study the effects of early warfare on the evolution of cooperation within social groups.

More generally, a number of economists and political scientists have developed theoretical models of conflict over resources. In economics it is common to have a first stage where the distribution of power is determined by the choices made by leaders, and a second stage determining whether war or peace occurs (Garfinkel and Skaperdas, Reference Garfinkel, Skaperdas, Hartley and Sandler2007; Acemoglu et al., Reference Acemoglu, Golosov, Tsyvinski and Yared2012; Bhattacharya et al., Reference Bhattacharya, Deb and Kundu2015). In political science the distribution of power is usually exogenous and the focus is on the issue of whether bargaining can avert costly warfare (Fearon, Reference Fearon1995; Powell, Reference Powell1996, Reference Powell2006). Theorists from both disciplines agree that if complete contingent contracts were feasible, peace would prevail. Fearon (Reference Fearon1995) argues that such contracts can be impossible to negotiate or enforce due to asymmetric information, indivisibilities, or a lack of commitment devices.

Relationships among pre-state groups of foragers or farmers are “anarchic” in much the same way that relationships among states in the modern world are anarchic. Thus similar commitment problems arise. As Fearon (Reference Fearon1995, 402) points out, repeated game effects cannot deter defections that will result in the death of one player. When a concession does not alter the prizes at stake or the probabilities of winning, groups will be tempted to renege after the concession is sunk. At best a territorial concession by a weak group might reduce the benefit of an attack to a strong group and thus forestall it. However, compact sites with highly valued resources create indivisibility problems that are difficult to resolve in this way.

We will show that even in this unpromising setting, powerful forces favor peace. In particular, we endogenize the probability of winning a conflict by having it depend on group sizes. The group sizes are determined in the short run by the migration decisions of individual agents, and in the long run by the effect of food income on fertility. Both mechanisms promote peace without any need for bargaining.

We also show that when individual mobility is costly enough, exogenous shocks can lead to warfare. Other authors have made related points. Powell (Reference Powell2006) finds that a rapid change in the distribution of power can result in warfare. In contrast to Powell, we highlight shocks to the value of the prizes at stake rather than to the power of each side to secure a prize. Chassang and Miguel (Reference Chassang and Padró i Miquel2009) present a model of civil war and show that a transient negative shock to labor productivity can lead to warfare by reducing the current opportunity cost of fighting relative to the future value of the assets seized in war. Roche et al. (Reference Roche, Müller–Itten, Dralle, Bolster and Müller2020) generalize and extend this model. We explore a different causal channel that involves shocks to site-specific productivities, rather than aggregate shocks and resulting inter-temporal tradeoffs.

7.3 Evidence on Early Warfare

The model in this chapter assumes that groups are egalitarian with respect to consumption and decision-making. Small-scale foraging and farming societies often fit this description. As we discussed in Chapter 6, mobile foragers are normally egalitarian apart from gender and age distinctions. Sedentary foragers are sometimes stratified but not inevitably so (Kelly, Reference Kelly2013a, ch. 9). Johnson and Earle (Reference Johnson and Earle2000) provide ethnographic examples of foraging and farming groups with up to a few hundred members that lack stratification. Some but not all engage in warfare, including warfare over land.

Two archaeological markers are widely used to infer early warfare: skeletons showing signs of deadly force, and defensive structures or settlements in easily defended locations. Other markers include specialized weapons and artistic depictions. The latter are more relevant for chiefdoms or states than for small egalitarian groups (see Haas and Piscitelli, Reference Haas, Piscitelli and Fry2013, 178–181, for a discussion of the severe problems in identifying warfare images in early rock art). In an archaeological context it can be very difficult to separate warfare over land from warfare having other motivations, and readers should bear this in mind throughout the rest of this section.

Even when skeletons show obvious evidence of violence, it can be difficult to distinguish warfare from individual homicide, execution, religious sacrifices, and non-warfare-related cannibalism. Uncertainties in the dating of skeletons can also make it difficult to distinguish cases where many deaths occurred simultaneously (a massacre) from cases where deaths occurred over a span of decades or centuries (a cemetery).

Defensive structures and defensively located settlements, while evidence of the threat of warfare, are not evidence of actual warfare. They tend to deter groups that may be considering an attack. Such investments could have been undertaken in the belief that they would prevent war or reduce its likelihood. They could also have been motivated by a belief that in the event of an attack, the defenders would be more likely to prevail.

The best-documented example of early large-scale violence is from the cemetery of Jebel Sahaba, which is located in the Nile valley in southern Egypt and has been dated between 18,600–13,400 BP (Crevecoeur et al., Reference Crevecoeur, Dias-Meirinho, Zazzo, Antoine and Bon2021). Of the 61 individuals examined, 41 had at least one healed or unhealed lesion, 38 had indications of trauma, and 25 had clear projectile impact marks. There is no patterning of trauma or projectile impacts by age or sex. Some individuals had experienced multiple episodes of violence in their lives. An assortment of weapons was used, including light arrows, heavier arrows, and spears.

Crevecoeur et al. reject the notion of a single catastrophic event at Jebel Sahaba and do not believe the evidence supports face-to-face battles. They favor the view that the community was a target for small recurrent attacks such as raids or ambushes over a short time scale. Large cemetery areas suggest some degree of sedentism and variation in lithic industries indicates differing cultural traditions. The authors conclude that climate change between the Last Glacial Maximum and the African Humid Period was probably responsible for triggering violent competition among culturally distinct groups of semi-sedentary hunter-fisher-gatherers.

Another case of inter-group violence has been documented for the early Holocene (10,500–9500 BP) at Nataruk, near what would then have been a lagoon on Lake Turkana in Kenya (Lahr et al., Reference Lahr2016). Details are available for twelve skeletons found in situ. Ten had lethal lesions, including five with sharp-force trauma probably involving arrows and five with blunt-force trauma to the head. The area was a fertile lakeshore that supported a substantial population of hunter-gatherers. The use of pottery may indicate storage and a reduction in mobility. The authors suggest that the massacre could have been part of a raid for resources (territory, women, children, or stored food).

Ferguson (Reference Ferguson and Fry2013b) reviews archaeological data on prehistoric warfare for Europe and the Near East. For Europe, the Upper Paleolithic shows negligible evidence of war, and barely any evidence of interpersonal violence. In the Mesolithic, from the onset of the Holocene around 11,600 BP until the arrival of agriculture, warfare is “scattered and episodic.” This period is associated with increasing sedentism, more food storage, more distinctive group identities, and greater inequality. Approximately 500–1,000 years after the Neolithic transition to agriculture, warfare becomes widespread, and in the Copper, Bronze, and Iron Ages, warfare is the norm. Crevecoeur et al. (Reference Crevecoeur, Dias-Meirinho, Zazzo, Antoine and Bon2021) likewise believe that fatal trauma became more frequent in Europe during the Mesolithic.

For the Near East, Ferguson (Reference Ferguson and Fry2013b) begins with early Natufian society around 15,000 BP, which was pre-agricultural and similar to the European Mesolithic. There is no skeletal evidence of war in the southern Levant for the next 10,000 years. Likewise there is no evidence of fortifications in this period. Peace came to an end in the southern Levant in the early Bronze Age around 5200 BP, a time that coincided with the formation of the Egyptian state. The apparent absence of war over this long span is remarkable in light of the intensive archaeological study the region has received; its history of climate shocks, technological innovation, and population fluctuation (see Sections 4.3 and 5.3); and the possibility of social stratification during some periods (see Section 6.11).

Ferguson reviews a substantial body of evidence indicating warfare in other parts of the Near East, including Anatolia and the northern Tigris area, at least as early as the Pottery Neolithic beginning around 8400 BP. At some sites warfare probably goes back to the Pre-Pottery Neolithic A during 11,600–10,500 BP when agriculture was beginning to spread in southwest Asia. Warfare continued through the Copper and Bronze Ages.

Using data on 16,820 hunter-gatherer burials at 329 archaeological sites in central California for 1530–230 BP, Allen et al. (Reference Allen, Bettinger, Codding, Jones and Schwitalla2016) find that sharp-force trauma is correlated with resource scarcity measured by net primary productivity (NPP). These traumas were probably contributing factors for death in 95% of the cases where they occurred. There was no similar correlation for blunt-force traumas, which were much less likely to be the cause of death. The authors note that if populations were distributed over the landscape to equalize resources per person, differences in violence would not be caused directly by differences in scarcity. However, groups in areas with lower NPP (net primary productivity) might have had larger and more poorly defined territories, leading to greater conflict with neighboring groups. The NPP data are modern and do not provide information about past climate shocks.

The ethnographic literature provides direct evidence on warfare among hunter-gatherers. Modern hunter-gatherers may differ in important ways from those who lived thousands of years ago. Contemporary hunter-gatherers live in extreme environments, and behavior in these societies has been influenced significantly by the outside world (Lee and Daly, Reference Lee and Daly1999). Thus caution must be exercised when extrapolating back from modern examples to the frequency of warfare (whether over land or not) in prehistory.

The Standard Cross Cultural Sample or SCCS is a representative sample of 186 well-documented and culturally independent pre-modern societies (Murdock and White, Reference Murdock and White1969, Reference Murdock and White2006). Two results from this data set stand out (Kelly, Reference Kelly2013a, Reference Kelly and Fry2013b). The first is that while there is evidence for warfare in all types of societies, the incidence of warfare is lower for egalitarian nomadic foragers than it is for non-egalitarian sedentary foragers. The second (first reported in Keeley, Reference Keeley1996) is that warfare is correlated with population pressure but not with population density, where population pressure refers to the ratio of population to total food, and population density refers to the ratio of population to total land. Keeley (Reference Keeley1996) argues that the losers in early warfare are frequently driven off or killed, a finding that is consistent with the formal model we will develop later.

Fry and Soderberg (Reference Fry2013) use the SCCS to study 148 episodes of lethal violence in 21 mobile forager bands. They conclude that “most incidents of lethal aggression may be classified as homicides, a few others as feuds, and a minority as war” (270). Kelly (Reference Kelly2000, ch. 2) observes that in a sample of 25 foraging societies drawn from the SCCS, war is strongly associated with segmented social structures that generate group identities. Ember and Ember (Reference Ember and Ember1992) use the SCCS to study the effect of ecological variables on war. They find that by far the best predictor of warfare is a history of natural disasters and that chronic scarcity has no independent effect on warfare.

Kelly (Reference Kelly2013a) makes some additional observations about foraging societies that are particularly relevant to warfare over land. All such societies have ways of assigning individuals to specific tracts of land and allowing them to secure access to others (154–161). In some instances individuals can move flexibly across social groups. There is a widespread but not universal pattern among foragers in which “connections to land are social and permeable rather than geographic and rigid” (156). Perimeter defense is rare, but the need to ask permission to use another group’s territory is common (it is normally granted). When foragers become sedentary, however, population is usually so high that residential movement is not possible without displacing another group; “war appears when mobility is not an option” (205).

A growing historical literature supports the idea that environmental shocks can trigger war. Zhang et al. (2006, Reference Zhang, Zhang, Lee and He2007) find a positive association between abnormally cold periods and domestic rebellions in China during 1000–1911 AD. They argue that the causal mechanism involves Malthusian population growth during warm periods followed by reduced food output in cold periods. Tol and Wagner (Reference Tol and Wagner2010) find a similar pattern for most of Europe during 1000–1990 AD, where unusually cold periods are associated with more violent conflict. Bai and Kung (Reference Bai and Kung2011) find that droughts are correlated with nomadic invasions of China between 220 BC and 1839 AD. They argue that drought had larger effects on food supply for pastoralists than for farmers in central China. In a meta-analysis of econometric research involving modern (post-1950) data, Burke et al. (Reference Burke, Hsiang and Miguel2015) estimate that a one-standard deviation increase in temperature raises intergroup conflict by 11.3%, with smaller but still substantial effects for drought or extreme rainfall.

7.4 Production and Warfare

The formal model has only two sites or territories, in contrast to the model from Chapter 6 that had a continuum of sites. We ignore the technology of exclusion used by insiders to prevent entry by individual outsiders and focus instead on the technology of combat between two organized groups. Connections to the formal model in Chapter 6 will be addressed in Section 7.9.

This section treats site qualities and group sizes as exogenous and examines the conditions under which war or peace will occur. The main results are as follows. First, war and peace are determined only by the ratio of the site productivities and the ratio of the group sizes. The absolute levels of these variables do not matter. Second, for a given productivity ratio peace occurs when the regional population is distributed across sites in a way that makes food per capita similar across sites. Extreme population distributions lead to large differences in living standards across the sites and result in war, where the attacker is the group with the larger size and lower food per capita.

Consider two sites $\mathrm{i}=\mathrm{A},\mathrm{B}$. Each agent is risk neutral and maximizes expected food consumption. Agents regard death as equivalent to being alive without food, where both yield zero utility (but see Section 7.8). Each agent is endowed with one unit of labor time. Individual agents are of negligible size relative to the total population at a site.

The production function for food is

${\mathrm{Y}}_{\mathrm{i}}={\mathrm{s}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{i}}^{\mathrm{\alpha }}\text{with}0<\mathrm{\alpha }<1\mathrm{i}=\mathrm{A},\mathrm{B}$(7.1)

where ${\mathrm{s}}_{\mathrm{i}}>0$ is the quality or productivity of site i and n_i is the population of site i at the time of production. Site qualities are determined by permanent geographic features such as lakes, rivers, good soil, and diverse local ecosystems, as well as transient factors such as weather. Equation (7.1) can be obtained from a Cobb–Douglas production function with constant returns to scale in labor and land. Here the land area for each site is normalized at unity. The full production function will be used in Chapter 8.

The agents at a site share food equally, so in the absence of warfare each agent at site i receives the average product

${\mathrm{y}}_{\mathrm{i}}={\mathrm{s}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{i}}^{\mathrm{\alpha }-1}\mathrm{i}=\mathrm{A},\mathrm{B}$(7.2)

The two sites together form a region with total population ${\mathrm{n}}_{\mathrm{A}}+{\mathrm{n}}_{\mathrm{B}}=\mathrm{N}>0$.

We will develop a model with the following steps in each period: (i) Malthusian dynamics determine an initial population for each site, (ii) individual agents can migrate between sites, and (iii) the resulting groups decide whether or not to attack. We proceed by backward induction. This section gives the results at step (iii), treating the group sizes $\left({\mathrm{n}}_{\mathrm{A}},{\mathrm{n}}_{\mathrm{B}}\right)$ and site qualities $\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)$ as exogenous. Section 7.5 studies the location decisions at step (ii), and Section 7.6 describes the population dynamics at step (i).

At step (iii) a war occurs if at least one group chooses to attack. Given that a war takes place, the probability that group i wins is

${\mathrm{p}}_{\mathrm{i}}={\mathrm{n}}_{\mathrm{i}}/\left({\mathrm{n}}_{\mathrm{A}}+{\mathrm{n}}_{\mathrm{B}}\right)\text{or}{\mathrm{p}}_{\mathrm{i}}={\mathrm{n}}_{\mathrm{i}}/\mathrm{N}\mathrm{i}=\mathrm{A},\mathrm{B}$(7.3)

The probability of victory for a group is equal to its share in total population. This seems plausible for small-scale societies without specialized warriors or weapons. The military technology in (7.3) assigns no advantage to the attackers or defenders. Our functional form is the simplest of those considered by Garfinkel and Skaperdas (Reference Garfinkel, Skaperdas, Hartley and Sandler2007).

We make several other simplifying assumptions (see Section 7.8 for the effects of relaxing these assumptions). Payoffs are “winner take all” in the sense that the winning group keeps its own site and gains the site of the opposing group. The winners suffer no casualties and spread their population across the two sites to maximize total food output. The losing group is killed or driven away and receives a zero payoff.

When group i with population n_i wins the contest, its total food output is

$\mathrm{H}\left({\mathrm{n}}_{\mathrm{i}}\right)=max\left\{{\mathrm{s}}_{\mathrm{A}}{\mathrm{L}}_{\mathrm{A}}^{\mathrm{\alpha }}+{\mathrm{s}}_{\mathrm{B}}{\mathrm{L}}_{\mathrm{B}}^{\mathrm{\alpha }}\text{subject}\mathrm{to}{\mathrm{L}}_{\mathrm{A}}\ge 0,{\mathrm{L}}_{\mathrm{B}}\ge 0,{\mathrm{L}}_{\mathrm{A}}+{\mathrm{L}}_{\mathrm{B}}={\mathrm{n}}_{\mathrm{i}}\right\}$(7.4)

This requires a few comments. Given that only the victorious group survives, this group gains control over both sites. Thus the winners can distribute their population across the two sites in order to maximize the food output of the group as a whole, which is shared in an egalitarian way. The simplifying assumption that winners suffer no casualties implies that the total labor supply available for allocation between the two sites is n_i.

Due to the nature of the production function in (7.1), it is never optimal to abandon the old site and locate the entire group at the new one. This follows from the fact that the marginal product of labor at a site approaches infinity whenever the labor input at the site approaches zero (this is true regardless of the productivity s_i for the site). Therefore, the winning group always chooses an interior solution with a positive population at each site, and equalizes the marginal products of labor across sites. A different production function might yield a boundary solution where only one site is used (presumably the new one).

Another feature of the production function in (7.1) is that whenever the marginal products are equalized across sites, the average products are automatically equalized as well. This implies that a member of the winning group is indifferent between site A and site B in the aftermath of a war. A different production function might not result in the equalization of the average products, implying a need for side payments across sites to equalize food consumption among the victors.

We can express the function H using the following notation. Let σ be the ratio of the site productivities as in (7.5a) below and let ϕ be the function of the site productivities defined in (7.5b) below. Then the total food output H(n_i) can be written as in (7.5c) and the optimal labor allocation is described by (7.5d):

$\mathrm{\sigma }\equiv {\mathrm{s}}_{\mathrm{A}}/{\mathrm{s}}_{\mathrm{B}}\in \left(0,\infty \right)$(7.5a)

$\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)\equiv {\mathrm{s}}_{\mathrm{A}}/{\left[1+{\mathrm{\sigma }}^{-1/\left(1-\mathrm{\alpha }\right)}\right]}^{\mathrm{\alpha }}+{\mathrm{s}}_{\mathrm{B}}/{\left[1+{\mathrm{\sigma }}^{1/\left(1-\mathrm{\alpha }\right)}\right]}^{\mathrm{\alpha }}$(7.5b)

$\mathrm{H}\left({\mathrm{n}}_{\mathrm{i}}\right)=\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right){{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\alpha }}$(7.5c)

${\mathrm{L}}_{\mathrm{A}}\left({\mathrm{n}}_{\mathrm{i}};\mathrm{\sigma }\right)={\mathrm{n}}_{\mathrm{i}}/\left[1+{\mathrm{\sigma }}^{-1/\left(1-\mathrm{\alpha }\right)}\right]\mathrm{and}{\mathrm{L}}_{\mathrm{B}}\left({\mathrm{n}}_{\mathrm{i}};\mathrm{\sigma }\right)={\mathrm{n}}_{\mathrm{i}}/\left[1+{\mathrm{\sigma }}^{1/\left(1-\mathrm{\alpha }\right)}\right]$(7.5d)

The aggregate food output in (7.5c) has the same structure as the production function in (7.1) but the individual site productivity is replaced by the aggregate productivity ϕ(s_A, s_B). In (7.5d), the optimal labor allocation for the winning group has properties one would expect: the higher the productivity of site A relative to site B, and thus the larger the ratio σ, the more labor the winning group allocates to site A, and conversely. Equation (7.5c) gives an expression for per capita food consumption among the winning group, which we write as

$\mathrm{h}\left({\mathrm{n}}_{\mathrm{i}}\right)=\mathrm{H}\left({\mathrm{n}}_{\mathrm{i}}\right)/{\mathrm{n}}_{\mathrm{i}}=\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right){{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\alpha }-1}$(7.6)

War occurs when group A attacks, when group B attacks, or when both attack. If group j ≠ i attacks, group i is indifferent between attacking or not because a war occurs in either case. If group j does not attack, it is optimal for group i to attack whenever ${\text{p}}_{\text{i}}{\text{h(n}}_{\text{i}}{\text{) > s}}_{\text{i}}{\text{n}}_{\text{i}}^{\mathrm{\alpha }-1}$ where the left side is the expected food per person from a war and the right side is the food per person from peace. Thus whenever this inequality holds it is a dominant strategy for i to attack. Assuming a group does not attack in situations of indifference, we obtain the following result:

$\text{peace occurs if}\mathrm{and}\text{only if}{\mathrm{p}}_{\mathrm{i}}\mathrm{h}\left({\mathrm{n}}_{\mathrm{i}}\right)\le {\mathrm{s}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{i}}^{\mathrm{\alpha }-1}\mathrm{for}\mathrm{i}=\mathrm{A},\mathrm{B}$(7.7)

We next consider how war and peace are determined by the group sizes (n_A, n_B) and site qualities (s_A, s_B). Using (7.6) and (7.7), the functions in (7.8) provide lower bounds on the probability of victory p_A or p_B that would motivate an attack by A or B respectively.

$\begin{array}{l}{\mathrm{x}}_{\mathrm{A}}\left(\mathrm{\sigma }\right)\equiv {\mathrm{s}}_{\mathrm{A}}/\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)\in \left(0,1\right)\\ {\mathrm{x}}_{\mathrm{B}}\left(\mathrm{\sigma }\right)\equiv {\mathrm{s}}_{\mathrm{B}}/\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)\in \left(0,1\right)\end{array}$(7.8)

These functions depend only on the ratio σ ≡ s_A/s_B.

Lemma 7.1

The functions x_A and x_B have the following properties.

${\mathrm{x}}_{\mathrm{A}}\left(\mathrm{\sigma }\right)+{\mathrm{x}}_{\mathrm{B}}\left(\mathrm{\sigma }\right)>1\mathrm{for}\mathrm{all}\mathrm{\sigma }>0$(7.9a)

${{\mathrm{x}}_{\mathrm{A}}}^{\prime }\left(\mathrm{\sigma }\right)>0\mathrm{and}{{\mathrm{x}}_{\mathrm{B}}}^{\prime }\left(\mathrm{\sigma }\right)<0\mathrm{for}\mathrm{all}\mathrm{\sigma }>0$(7.9b)

${\mathrm{x}}_{\mathrm{A}}\left(0\right)=0\mathrm{and}{\mathrm{x}}_{\mathrm{A}}\left(\infty \right)=1$(7.9c)

${\mathrm{x}}_{\mathrm{B}}\left(0\right)=1\mathrm{and}{\mathrm{x}}_{\mathrm{B}}\left(\infty \right)=0$(7.9d)

Our first proposition uses Lemma 7.1 to show that the occurrence of war or peace depends only on the ratio of the site qualities and the ratio of the group sizes.

These results are illustrated in Figure 7.1. The productivity ratio σ is on the horizontal axis. Group A’s population share n_A/N, which gives the probability p_A that A wins a war if one occurs, is on the vertical axis. For a fixed productivity ratio σ, a war occurs if either A or B has a large enough population share. The boundary cases n_A/N = 0 and n_A/N = 1 lead to war only in the trivial sense that the grand coalition consisting of all agents in the region attacks an empty site and wins with certainty.

Figure 7.1. War and peace determined by the ratio of site productivities and the ratio of group sizes

We adopt the term trivial war in referring to boundary cases of this sort, because graphically such points are in the warfare region of Figure 7.1. However, in trivial wars no land is transferred between groups and no one dies. The only result is an allocation of labor across sites to maximize total food as in (7.4). We refer to non-trivial war when we are discussing conflicts between two groups of positive size. In these cases one group dies and the other group takes its land. Only non-trivial wars are empirically relevant.

Peace occurs for intermediate population allocations. The dashed curve indicates the locus where average products are equal across sites, which is always in the interior of the peace region. Any population allocation that comes sufficiently close to equalizing the average products, and therefore the standards of living across sites in terms of food per capita, will ensure that peace prevails.

The utility functions of the individual agents are

$\begin{array}{l}\mathrm{war}:\\ \text{peace}:\end{array}\begin{array}{lll}{\mathrm{u}}_{\mathrm{A}}\left({\mathrm{n}}_{\mathrm{A}},{\mathrm{n}}_{\mathrm{B}}\right)& =& \left[\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)/\mathrm{N}\right]{\mathrm{n}}_{\mathrm{A}}^{\mathrm{\alpha }}\\ {\mathrm{u}}_{\mathrm{A}}\left({\mathrm{n}}_{\mathrm{A}},{\mathrm{n}}_{\mathrm{B}}\right)& =& {\mathrm{s}}_{\mathrm{A}}{\mathrm{n}}_{\mathrm{A}}^{\mathrm{\alpha }-1}\end{array}\begin{array}{c}\mathrm{and}\\ \mathrm{and}\end{array}\begin{array}{lll}{\mathrm{u}}_{\mathrm{B}}\left({\mathrm{n}}_{\mathrm{A}},{\mathrm{n}}_{\mathrm{B}}\right)& =& \left[\mathrm{\varphi }\left({\mathrm{s}}_{\mathrm{A}},{\mathrm{s}}_{\mathrm{B}}\right)/\mathrm{N}\right]{\mathrm{n}}_{\mathrm{B}}^{\mathrm{\alpha }}\\ {\mathrm{u}}_{\mathrm{B}}\left({\mathrm{n}}_{\mathrm{A}},{\mathrm{n}}_{\mathrm{B}}\right)& =& {\mathrm{s}}_{\mathrm{B}}{\mathrm{n}}_{\mathrm{B}}^{\mathrm{\alpha }-1}\end{array}$(7.10)

In wartime an agent prefers to be in a larger group because this increases the probability of victory. In peacetime an agent prefers to be in a smaller group because this increases the average product of labor and therefore raises food consumption per capita.

7.5 Individual Mobility

We now move back to the migration stage and examine the location decisions of the individual agents. This section is complex but the main results can be summarized as follows. Given the productivities of the sites and the distribution of regional population across them, an individual agent can form expectations about whether there will be war or peace based upon the results from Section 7.4. An agent must then choose a location. We will be interested in situations where no individual agent wants to switch sites given their (correct) expectations with regard to war or peace.

Suppose that an initial distribution of population across sites is inherited from the past (in Section 7.6 this will be determined by Malthusian dynamics). We will show that individual migration, if it occurs, will either push the system toward a boundary solution with trivial war (agents move to the site of the group that is expected to win) or toward a peaceful outcome (agents move to the site with higher food per capita). This shows that non-trivial wars cannot occur if migration occurs. Thus, non-trivial wars require mobility costs high enough to block migration, so agents do not flee from an impending conflict.

We assume an initial population allocation m = (m_A, m_B) ≥ 0 with m_A + m_B = N. We want to know how individual migration determines a final allocation n = (n_A, n_B) ≥ 0, which then leads to group decisions about war or peace as in Section 7.4. The individual agents observe the site qualities (s_A, s_B) and ignore their own influence on the sizes of the groups at each site. Each agent takes the locations of all other agents as given.

An agent initially located at site i who moves to site j ≠ i only receives a fraction η ∈ (0, 1] of the utility u_j available to non-movers at j ≠ i. The penalty 1-η may reflect a preference to live near family and friends, or to remain in a group with familiar customs and beliefs. It may also reflect a stigma imposed on outsiders by members of the other group. We use a multiplicative migration cost because under this assumption, only the relative group sizes will be important in our Malthusian model in Section 7.6.

Equilibrium is defined so that no agent wants to change sites.

Definition 7.1 Fix σ > 0 and N > 0. Consider allocations n = (n_A, n_B) ≥ 0 with n_A + n_B = N. We call n a locational equilibrium (LE) in any of the following cases: n = (N, 0); n = (0, N); or n > 0 with

${\mathrm{u}}_{\mathrm{A}}\left(\mathrm{n}\right)\ge {\mathrm{\eta u}}_{\mathrm{B}}\left(\mathrm{n}\right)\mathrm{and}{\mathrm{u}}_{\mathrm{B}}\left(\mathrm{n}\right)\ge {\mathrm{\eta u}}_{\mathrm{A}}\left(\mathrm{n}\right)$(7.11)

The boundary cases (N, 0) and (0, N) are always equilibria because an agent who moves to an empty site is attacked by all other agents and loses with certainty, resulting in zero utility. In these boundary cases only one of the inequalities in (7.11) holds because utility is positive at one site and zero at the other. In interior cases both inequalities must hold.

The next task is to identify population distributions that lead to particular war or peace outcomes (as in Proposition 7.1) while simultaneously forestalling migration (as in D7.1). For example, suppose we seek an allocation leading to a war in which B attacks A, while ensuring that no individual agents flee their current locations. This is clearly true for n = (0, N), which yields a trivial war and is also a locational equilibrium.

The interesting cases involve interior allocations. Now to have a war in which B attacks, we must satisfy both Proposition 7.1(a) and (7.11). For a given productivity ratio σ Proposition 7.1(a) requires n_A/n_B < (1−x_B)/x_B. We obtain the utility levels for warfare from (7.10) and substitute these into (7.11) to obtain η^1/α ≤ n_A/n_B ≤ (1/η)^1/α. Combining these, we require η^1/α ≤ n_A/n_B ≤ min {(1/η)^1/α, (1−x_B)/x_B}, where n_A/n_B must be strictly less than (1−x_B)/x_B. More generally we obtain the following restrictions on the ratio n_A/n_B.

The sets LE_B and LE_A in Proposition 7.2 may be empty or non-empty. We will discuss the relevant conditions for each case below.

Figure 7.2 shows a case where LE_B exists and LE_A does not (the interpretation of the arrows will be discussed later). Group size ratios n_A/n_B in the interval (0, (1−x_B)/x_B) lead to an attack by B due to Proposition 7.1(a). To keep agents at A from fleeing to B or vice versa, we also require η^1/α ≤ n_A/n_B ≤ (1/η)^1/α. In Figure 7.2 the upper bound on LE_B is given by (1/η)^1/α but in other situations it could be given by (1−x_B)/x_B instead.

Figure 7.2. Migration and locational equilibrium

The peace interval LE_P in Proposition 7.2(b) is always non-empty. When η = 1 so there is no mobility cost, LE_P contains the single point n_A/n_B = σ^1/(1−α) where the average products are equal across sites. As η falls and mobility costs rise, the interval LE_P grows (see Figure 7.2). When η is close to zero, every peaceful allocation in Proposition 7.1(b) is an LE and the interval LE_P in Figure 7.2 becomes [(1−x_B)/x_B, x_A/(1−x_A)].

From Proposition 7.2(a) the interval LE_B is non-empty if and only if η^1/α < (1−x_B)/x_B. Using the properties of x_B(σ) from (7.9) this requires a sufficiently large σ, so site A is a sufficiently valuable prize for group B. When η = 1 so there is no mobility cost, this reduces to x_B(σ) < 1/2. Similarly Proposition 7.2(c) implies that LE_A is non-empty if and only if x_A/(1−x_A) < (1/η)^1/α. This requires a sufficiently small σ, so site B is a sufficiently valuable prize for group A. When η = 1 this reduces to x_A(σ) < 1/2.

Given a fixed value of σ and assuming the relevant intervals exist, LE_B and LE_A both expand as mobility costs rise. This occurs simply because when migration is more expensive, agents are less inclined to move despite an impending war. When the warfare intervals exist they may either be adjacent to the peace interval or separated from it by an interval of non-LE allocations. However, the ratios n_A/n_B in the warfare interval LE_B are always less than those in the peace interval LE_P because group B must be strong enough to make an attack worthwhile. Likewise the ratios in LE_A always exceed those in LE_P.

Now consider a given initial (pre-migration) allocation m ≥ 0. We want to know how the decisions of individual agents lead to a final (post-migration) allocation n ≥ 0. When m = (N, 0) or (0, N) we already have a locational equilibrium by D7.1 and none of the agents wants to change sites. The same is true for any interior allocation m > 0 such that m_A/m_B is in one of the three LE intervals described in Proposition 7.2.

We therefore focus on cases where m is interior but not an LE. This implies that one of the conditions in (7.11) must be violated. Because the other condition in (7.11) must be satisfied, agents can only move in one direction. For example, if u_B(m) < ηu_A(m) then agents migrate from B to A and no agents migrate from A to B. In this situation we study allocations n = (n_A, n_B) ≥ 0 with n_A + n_B = N such that n_A ≥ m_A. Migration stops when the system reaches an allocation where u_B(n) = ηu_A(n). This implies that u_A(n) ≥ ηu_B(n) also holds, so we have an LE. If there is no interior n_A ≥ m_A at which this occurs then all agents go to site A, and the final allocation is n_A = N and n_B = 0. If the initial allocation has u_A(m) < ηu_B(m) the same logic applies but with the roles of A and B reversed. This reasoning leads to the following definition.

The intuition behind D7.2 can be clarified by considering the directions in which the agents flow. Suppose the system is not yet in locational equilibrium. When war is expected the utility functions are those for war in (7.10), and agents flow toward the site with the larger current population. This process is self-reinforcing and can only end with a boundary equilibrium or a transition to the peace interval. When peace is expected the utility functions are those for the peace case in (7.10), and agents flow toward the site with the higher average product. This reduces the difference in average products between the sites, and migration stops when this gap is small enough relative to the mobility cost. If the average product is initially higher at site A, migration stops at the lower bound of the peace interval LE_P in Proposition 7.2, and if the average product is initially higher at site B, migration stops at the upper bound of the peace interval LE_P.

For completeness we need to consider the cases where LE_B is empty or not, and where LE_A is empty or not. The formal results are as follows.

Figure 7.2 shows the migration process for the case where LE_B exists and LE_A does not, where the arrows indicate the direction of migration. If m_A/m_B is below LE_B, group B will attack, we have u_A(m) < ηu_B(m), and agents flow from site A to site B. From the war utilities in (7.10) this migration process raises u_B and lowers u_A, reinforcing the initial inequality. Population continues to flow in the same direction until all agents are at site B and we have $\mathrm{n}=\left(0,\mathrm{N}\right)$.

Next suppose ${\mathrm{m}}_{\mathrm{A}}/{\mathrm{m}}_{\mathrm{B}}$ is between LE_B and LE_P. If ${\mathrm{m}}_{\mathrm{A}}/{\mathrm{m}}_{\mathrm{B}}<\left(1-{\mathrm{x}}_{\mathrm{B}}\right)/{\mathrm{x}}_{\mathrm{B}}$ so group B will attack, we use the war utilities from (7.10). The fact that m_A/m_B is above LE_B implies ${\mathrm{u}}_{\mathrm{B}}\left(\mathrm{m}\right)<{\mathrm{\eta u}}_{\mathrm{A}}\left(\mathrm{m}\right)$, so agents flow from B to A. As long as B will attack, migration raises u_A and lowers u_B so the preceding inequality continues to hold. Once the system reaches ${\mathrm{n}}_{\mathrm{A}}/{\mathrm{n}}_{\mathrm{B}}=\left(1-{\mathrm{x}}_{\mathrm{B}}\right)/{\mathrm{x}}_{\mathrm{B}}$ we use the peace utilities from (7.10). Now migration from site B to site A lowers u_A and raises u_B. This continues until ${\mathrm{u}}_{\mathrm{B}}\left(\mathrm{n}\right)={\mathrm{\eta u}}_{\mathrm{A}}\left(\mathrm{n}\right)$ at the lower bound of LE_P. Migration then stops and we have a peaceful LE.

Finally suppose ${\mathrm{m}}_{\mathrm{A}}/{\mathrm{m}}_{\mathrm{B}}$ is above LE_P. When peace is expected, population flows toward B and migration ends at the upper bound of LE_P. When A is expected to attack, population flows toward A until all agents are at site A and we have $\mathrm{n}=\left(\mathrm{N},0\right)$.

Several implications follow from Proposition 7.3. First, if the initial allocation m is not an LE, the system must move to one of four final allocations: (N, 0), (0, N), the lower bound of LE_P, or the upper bound of LE_P. None is consistent with non-trivial war. Thus, non-trivial warfare can only arise if the system starts from an initial population allocation m that is already in LE_B or LE_A, and mobility costs keep all agents at their initial sites. If the initial allocation m yields peace but is not an LE, the final allocation also yields peace and migration reduces the gap in average products across the sites.

The interval LE_P is stable in the sense that a small deviation to a point that is not an LE causes migration flows that bring the system back to LE_P. The same is true for the boundary cases (N, 0) and (0, N). But if the warfare intervals LE_A and LE_B exist they are unstable, in the sense that a small deviation to a point that is not an LE will lead either to a boundary case or an allocation in the peace interval LE_P. Points in the interior of LE_B or LE_A are neutrally stable because a small deviation leaves the system in LE_B or LE_A.

When mobility cost is zero $\left(\mathrm{\eta }=1\right)$ three locational equilibria always exist: the boundary cases with trivial war, and the peaceful allocation with equal average products. If ${\mathrm{x}}_{\mathrm{B}}\left(\mathrm{\sigma }\right)<1/2$ then LE_B contains the single point ${\mathrm{n}}_{\mathrm{A}}={\mathrm{n}}_{\mathrm{B}}=\mathrm{N}/2$ and it is empty otherwise. A symmetric result holds for LE_A. At most one of these two interior warfare equilibria can exist. If either one does, it is unstable. Thus a positive mobility cost is a necessary condition for a stable equilibrium with non-trivial warfare.

We now pause to summarize the results so far. For an exogenously fixed ratio of site productivities $\mathrm{\sigma }={\mathrm{s}}_{\mathrm{A}}/{\mathrm{s}}_{\mathrm{B}}$, Proposition 7.1 derived the group size ratios n_A/n_B that lead to an attack by B, peaceful coexistence, or an attack by A. We then added the requirement that individual agents who foresee these outcomes not change sites. The group size ratios n_A/n_B consistent with this individual rationality requirement were given by the intervals LE_B, LE_P, and LE_A in Proposition 7.2.

Next we examined migration when the initial population allocation is not in one of these intervals. Proposition 7.3 showed that if the system is not already at a locational equilibrium with non-trivial warfare (LE_B or LE_A), then individual migration cannot take the system to such an equilibrium. Instead it either pushes the system to a boundary case with trivial warfare, or to the boundary of the peace interval LE_P. In Section 7.6 we will study how migration costs and productivity shocks can interact to yield non-trivial wars.

7.6 Malthusian Dynamics

This section embeds the model of Sections 7.4 and 7.5 in a Malthusian framework where population growth depends on food income. Chapter 6 used a similar framework to study inequality in small-scale societies.

The role of Malthusian mechanisms in the present model is to use the population levels from the two sites at the end of one period to generate new population levels at the beginning of the next. The latter populations provide the starting point for the migration process in Section 7.5, which then leads to the combat decisions in Section 7.4. Under the assumptions of this section, the group size ratio in one period along with the sequence of productivity ratios for the current period and all future periods suffices to determine all of the future group size ratios. This information also suffices to determine whether there is war or peace in each period.

We show that there are two necessary conditions for warfare over land. First it is necessary that there be no individual migration in response to the population allocation determined by Malthusian processes. This condition is required because we have shown in Section 7.5 that when migration occurs, it leads either to a boundary outcome where no actual fighting happens (trivial war) or to peace. The second condition is that there must be a sufficiently large exogenous shock to the ratio of the site productivities, either due to a change in nature or a change in food technology. Without this, Malthusian mechanisms will maintain the group size ratio within the interval needed for peace. A corollary is that a series of wars requires a series of shocks because in the aftermath of a war, population is reallocated across sites in a way that restores peace. The two necessary conditions are jointly sufficient; if both are satisfied then the model predicts that a war will occur.

Denote periods by t = 0, 1, . . where a period is one human generation. For an individual agent let ρy be the number of that agent’s children who survive to be adults, where y is the agent’s food income and ρ > 0 is a constant. This captures the idea that adults with more food income are more fertile, have healthier children, or both. Adults in period t produce food, have children, and then die at the end of the period. The children become adults in period t+1 at the site where their parents produced food.

Let there be some sequence of productivity ratios ${\mathrm{\sigma }}^{\mathrm{t}}={\mathrm{s}}_{\mathrm{A}}^{\mathrm{t}}/{\mathrm{s}}_{\mathrm{B}}^{\mathrm{t}}$ for t = 0, 1, . . These are given exogenously by technological and environmental factors. Each period starts with an initial allocation of agents ${\mathrm{m}}^{\mathrm{t}}=\left({\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}},,,{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}}\right)$ inherited from the past. The agents can then move from their birth site to the other site as in Section 7.5. These moves (if any) lead to the final allocation ${\mathrm{n}}^{\mathrm{t}}=\left({\mathrm{n}}_{\mathrm{A}}^{\mathrm{t}},,,{\mathrm{n}}_{\mathrm{B}}^{\mathrm{t}}\right)$. Once the final allocation has been determined, the two groups decide whether to attack and payoffs are obtained as in Section 7.4.

Suppose first that peace occurs in period t. This can happen only when the final allocation n^t > 0 is interior. The food income of an agent at site A is ${\mathrm{y}}_{\mathrm{A}}^{\mathrm{t}}={\mathrm{s}}_{\mathrm{A}}^{\mathrm{t}}{\left({\mathrm{n}}_{\mathrm{A}}^{\mathrm{t}}\right)}^{\mathrm{\alpha }-1}$ and the food income of an agent at site B is ${\mathrm{y}}_{\mathrm{B}}^{\mathrm{t}}={\mathrm{s}}_{\mathrm{B}}^{\mathrm{t}}{\left({\mathrm{n}}_{\mathrm{B}}^{\mathrm{t}}\right)}^{\mathrm{\alpha }-1}$. The Malthusian linkage between food and surviving children gives ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}={\mathrm{\rho y}}_{\mathrm{A}}^{\mathrm{t}}{\mathrm{n}}_{\mathrm{A}}^{\mathrm{t}}$ and ${\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}={\mathrm{\rho y}}_{\mathrm{B}}^{\mathrm{t}}{\mathrm{n}}_{\mathrm{B}}^{\mathrm{t}}$. Thus

$\text{peace in period}\mathrm{t}\text{yields}{\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}={\mathrm{\sigma }}^{\mathrm{t}}{\left({\mathrm{n}}_{\mathrm{A}}^{\mathrm{t}}/{\mathrm{n}}_{\mathrm{B}}^{\mathrm{t}}\right)}^{\mathrm{\alpha }}$(7.12)

This result shows that in peacetime the ratio of the group sizes at the start of period t+1 is a strictly concave function of the ratio prevailing at the end of period t.

Now suppose instead that a war occurs in period t and the winner is W ∈ {A, B}. Group W allocates its population ${\mathrm{n}}_{\mathrm{W}}^{\mathrm{t}}$ across sites to maximize total food output as in (7.4) while the opposing group disappears. Let the resulting number of agents at each site be ${\mathrm{L}}_{\mathrm{A}}^{\mathrm{t}}$ and ${\mathrm{L}}_{\mathrm{B}}^{\mathrm{t}}$ where ${\mathrm{L}}_{\mathrm{A}}^{\mathrm{t}}+{\mathrm{L}}_{\mathrm{B}}^{\mathrm{t}}={\mathrm{n}}_{\mathrm{W}}^{\mathrm{t}}$. Due to the fact that the production function from (7.1) has constant output elasticity, the equalization of marginal products across sites implies equalization of average products. Thus every member of W has the same food income ${\mathrm{y}}^{\mathrm{t}}=\mathrm{H}\left({\mathrm{n}}_{\mathrm{w}}^{\mathrm{t}};{\mathrm{s}}_{\mathrm{A}}^{\mathrm{t}},{\mathrm{s}}_{\mathrm{B}}^{\mathrm{t}}\right)/{\mathrm{n}}_{\mathrm{W}}^{\mathrm{t}}$. Using ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}={\mathrm{\rho y}}^{\mathrm{t}}{\mathrm{L}}_{\mathrm{A}}^{\mathrm{t}}$ and ${\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}={\mathrm{\rho y}}^{\mathrm{t}}{\mathrm{L}}_{\mathrm{B}}^{\mathrm{t}}$ with (7.5d) implies that

$\mathrm{war}\text{in period}\mathrm{t}\text{yields}{\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}={\left({\mathrm{\sigma }}^{\mathrm{t}}\right)}^{1/\left(1-\mathrm{\alpha }\right)}$(7.13)

This result shows that in wartime the ratio of the group sizes at the start of period t+1 is a constant determined solely by the relative productivities of the sites. Because the winners distribute their period-t population across sites to maximize total food, the effect of group size ratios from earlier periods is erased and population adjustments restart from scratch. Notice that (7.13) applies both when n^t > 0 so the war is non-trivial, and also when the war is trivial because n^t = (N^t, 0) or (0, N^t).

Lemma 7.2 shows that the initial group size ratio ${\mathrm{m}}_{\mathrm{A}}^0/{\mathrm{m}}_{\mathrm{B}}^0$ and the sequence of productivity ratios {σ^t} determine the subsequent group size ratios ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}}$ for all t ≥ 1. This information also suffices to determine whether war or peace occurs in each period. Accordingly, our results do not depend on the absolute productivity levels (${\mathrm{s}}_{\mathrm{A}}^{\mathrm{t}},{\mathrm{s}}_{\mathrm{B}}^{\mathrm{t}}$), the absolute population levels N^t, or the identities of the winners in particular conflicts.

Now suppose we are given σ^t and ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}}$ for period t. As in Lemma 7.2, we can compute ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}$. We want to know the conditions under which a non-trivial war will occur in period t+1. This question is readily answered using Proposition 7.3. Such a war occurs if and only if the ratio ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}$ is in the set LE_B or LE_A. If neither is true then either the ratio is in LE_P or m^t+1 is not an LE. In the former case we have peace. In the latter case migration leads to peace or to formation of the grand coalition followed by a trivial war. The following proposition formalizes these ideas.

The necessity of (a) is straightforward. We have already shown that any time individual agents migrate from one site to another, a non-trivial war is impossible. Thus we are limited to cases in which no migration occurs in period t+1. If a war is expected the utility functions are obtained from (7.10). The absence of migration implies that these utilities must satisfy (7.11). Putting these conditions together, the initial population ratio in period t+1 (which is also the final population ratio) must satisfy Proposition 7.4(a). In this situation the mobility costs incorporated in the parameter η are large enough to prevent migration despite an impending war.

The necessity of (b) has the following intuition. If we use Malthusian dynamics to compute ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{m}}_{\mathrm{B}}^{\mathrm{t}+1}$ and take this ratio as given, we know the initial population share ${\mathrm{m}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{N}}^{\mathrm{t}+1}$ for group A. Because we have ruled out migration as explained in connection with Proposition 7.4(a), this must also be the final population share ${\mathrm{n}}_{\mathrm{A}}^{\mathrm{t}+1}/{\mathrm{N}}_{\mathrm{A}}$. As shown in Figure 7.3, this fixes a horizontal line that determines an interval of productivity ratios σ^t+1 consistent with peace. The latter is the interval $\left[{\mathrm{\sigma }}_{\mathrm{A}}^{\mathrm{t}+1},{\mathrm{\sigma }}_{\mathrm{B}}^{\mathrm{t}+1}\right]$ in Proposition 7.4(b). To have a war the productivity ratio σ^t+1 must be outside this interval so ${\mathrm{\sigma }}^{\mathrm{t}+1}\notin \left[{\mathrm{\sigma }}_{\mathrm{A}}^{\mathrm{t}+1},{\mathrm{\sigma }}_{\mathrm{B}}^{\mathrm{t}+1}\right]$.

Figure 7.3. Productivity ratios for war and peace in period t+1 given initial group sizes

The key ideas from Proposition 7.4 can now be summarized. Malthusian dynamics generate a population allocation at the beginning of period t+1. If warfare over land is to occur, this allocation cannot be altered by migration. Moreover, given the probabilities of winning determined by the population shares, the productivity ratio between sites must be high or low enough that an attack is attractive for one group or the other. The conditions in Proposition 7.4 are sufficient because for given initial group sizes, a war will occur when migration is costly enough and the productivity ratio is extreme enough.

Proposition 7.4 has the following important implication.

This follows from the fact that the productivity ratio σ^t is an interior point of the interval $\left[{\mathrm{\sigma }}_{\mathrm{A}}^{\mathrm{t}+1},{\mathrm{\sigma }}_{\mathrm{B}}^{\mathrm{t}+1}\right]$ as indicated in Figure 7.3. In order for war to occur σ^t+1 must involve a large enough shock relative to σ^t. When σ^t+1 = σ^t Proposition 7.4 already rules out a non-trivial war. The corollary also excludes trivial wars in the sense defined in Section 7.4.

The corollary shows that when the productivity ratio is constant over some time interval, there can be at most one war in this interval and it must occur at the start of the interval. Peace prevails thereafter. The reason is that war will equalize average products across sites and hence will equalize population growth rates across sites. This maintains the existing group size ratio, which maintains the equality of the average products. Thus peace continues. A series of wars therefore requires a series of shocks.

A useful thought experiment is to shut down individual migration by having η close to zero. This guarantees that (a) in Proposition 7.4 is satisfied. However, (b) is also necessary for war. In a stationary productivity environment Malthusian fertility effects ensure that (b) never holds. Even without migration this is enough to preserve peace.

Any exogenous variable that can alter the relative productivities of the sites can serve as a trigger for warfare. One such variable is production technology. Another is climate. Such shocks can easily change the relative productivities of sites or territories. For example, reduced rainfall may lower productivity substantially at sites far from lakes or rivers but only slightly at sites with access to surface water (see Chapter 5). Historical correlations between climate shocks and warfare are well documented (see Section 7.2).

We close this section with some remarks on the relationship between our theory and that of Malthus (Reference Malthus and Appleman1798 [2004]). Our framework is “Malthusian” only in the sense that we rely on a positive linkage between food per capita and fertility (or a negative linkage with child mortality). This is the operative check on population growth. We do not rely on mortality from warfare as a check on population. In fact our model predicts that in a stationary productivity environment, regional population will converge to an equilibrium level without warfare.

Some anthropologists argue that warfare will occur in a demographic steady state. For example, Harris (Reference Harris1977, 60) suggests that reproductive pressure leads to both warfare and female infanticide, and that infanticide is “a savage but uniquely effective solution to the Malthusian dilemma.” In contrast, we believe fertility effects can restrain population growth even without any mortality effects derived from warfare or infanticide.

7.7 Empirical Implications

Three mechanisms tend to promote peace in our model: mobility between groups, Malthusian fertility effects, and reallocation of population after a war. All three assign larger populations to better sites. The resulting positive association between group size and site quality tends to deter attacks. A productivity shock can trigger a war by creating a temporary imbalance between group size and site quality, provided that the imbalance is not undone through individual migration. We remind the reader again that whenever we use the term “war” in this section, we are referring specifically to warfare over land.

Our theory helps explain Ferguson’s (Reference Ferguson and Fry2013b) finding that in Europe warfare was rare or non-existent in the Upper Paleolithic, sporadic in the Mesolithic, and common in the Neolithic and later. The small foraging bands of the Upper Paleolithic probably had fluid social boundaries, with easy individual mobility across groups through exogamous marriage and kinship networks. At low population densities it would also have been hard to exclude outsiders from a territory, yielding open access and rough equality of average products of labor (see Chapter 6). The bands of the Upper Paleolithic almost surely had Malthusian fertility dynamics. Together these factors make warfare over land unlikely.

The shift to the Mesolithic and the Neolithic was associated with rising sedentism (see Chapter 4) and social segmentation (Kelly, Reference Kelly2000, ch. 4). This would make it harder for individuals to circulate among groups. In addition, community sizes were growing. Research on recent small-scale societies suggests that this probably made endogamous marriage more common (Dow et al., Reference Dow, Reed and Woodcock2016), reducing kinship linkages across groups and increasing mobility costs for individuals. In combination with the shocks resulting from technological and environmental change, this would have made warfare more common. A possible source of such shocks was the diffusion of agriculture across Europe, which was accompanied by increased warfare (see Section 7.3).

Our results are also consistent with patterns detected in anthropology. The model accounts for the empirical finding that warfare is a function not of population density but rather of pressure on food resources. Population density can be high at good sites due to individual migration or Malthusian dynamics without leading to warfare. However, war may well occur when shifts in productivity mean that a group has many mouths to feed and few resources with which to feed them. The model correctly predicts that warfare will be more common when shocks from natural disasters or climate change are larger and more frequent. And finally, our prediction that costlier individual mobility makes warfare more likely is consistent with ethnographic evidence that warfare occurs more often in societies with strictly enforced group identities.

We conclude this section with an ethnographic case based on oral histories from the Enga people of highland New Guinea (Wiessner and Tumu, Reference Wiessner and Tumu1998, ch. 5). The Enga are divided into many tribes, clans, and subclans. All such groups are egalitarian with no one in a position of authority other than “big men.” These groups engage in foraging and farming in varying proportions. The Enga have a history of warfare in recent centuries, with the winning groups frequently pushing the losers off their land. The losers then had to move elsewhere. Although the proximate causes of conflict were often trivial, warfare resulted in major land redistributions among groups.

Two related questions are why warfare occurred at all, and whether it had been going on forever or had a recent origin. One place to look for an explanation would be climate shocks, and indeed Wiessner and Tumu note that there were occasional shocks due to frost or drought. These could have served as triggers for war. But perhaps more importantly, the system was far from equilibrium because the arrival of the sweet potato led to substantial regional population growth (see Weissner and Tumu, 1998, ch. 4, and Section 2.6 of this book). This caused considerable conflict. In particular, some people were pushed out of land at higher elevations that had previously been good for hunting and gathering but now became useful for farming. We suspect that this was a case where a large technological shock altered the relative productivities of sites and that our model of egalitarian warfare may apply. Unfortunately, however, we lack archaeological data on the frequency of warfare before and after the arrival of the new food technology.

7.8 Extensions of the Model

We have made a number of assumptions about warfare that could be questioned. In many cases, altering our assumptions would make war less likely. For example, one could assume that winning groups suffer injuries or deaths, that winning groups may not gain full control over the losing group’s land, that attackers are risk averse rather than risk neutral, that groups preparing an attack incur an opportunity cost in the form of lost food output, that defenders have advantages due to their knowledge of local geography, that defenders can construct fortifications, that potential attackers have kinship connections with defenders, and so on. Because the assumptions used in our formal modeling tend to bias the analysis toward warfare, modifying these assumptions would tend to strengthen our conclusion that war was generally rare among small egalitarian groups in prehistory.

Another assumption that could be modified involves the utility function. We treat death as equivalent to being alive without food. But having no food may be better than death (perhaps more food will arrive tomorrow and one has a chance of surviving until then). This “happy to be alive” utility function makes the prospect of death seem worse in relative terms and would tend to discourage warfare, much as risk aversion would.

On the other hand, one can argue that a minimum amount of food is necessary for life, so a positive amount of food below this subsistence threshold would be useless. This “nothing to lose” utility function would make a group more inclined to attack a rival group when a food shortage looms, much as risk seeking would.

The question is whether these ideas have empirical implications. For example the “happy to be alive” case may apply when agents can smooth consumption through storage or pooling mechanisms while the “nothing to lose” case may apply when mean food output is low, variance is high, and smoothing is infeasible. The influence of such variables on the incidence of early warfare should be open to empirical testing.

In principle, our framework applies both to foragers and farmers. But farmers make greater site-specific investments in land clearance, irrigation systems, terracing, tree planting, and so on, and therefore are more inclined to defend their sites rather than flee from an attack. At the same time improved sites become more attractive to potential attackers, especially those whose own sites suffer from declining productivity due to soil erosion or exhaustion. The combination of endogenous investment and local resource depletion may tend to generate warfare cycles in egalitarian farming societies, especially if there are also barriers to individual movement among groups (see the examples of the Yanomamo, the Tsembaga Maring, and the Central Enga in Johnson and Earle, Reference Johnson and Earle2000).

We conclude this section with a remark on institutions. As discussed in Section 7.3, the southern Levant enjoyed about 10,000 years during which evidence for warfare is almost non-existent. From the perspective of our theory this may be surprising, given the numerous environmental and technological shocks affecting the region during this period, including an early transition to agriculture. War was clearly not restrained through state suppression of violence because no state existed at the time.

Our theory suggests an alternative: perhaps institutions of a more decentralized kind helped to transfer population across communities in response to shocks. Examples could include exogamous marriage, shared language, shared religion, or shared norms. Regional networking among local elites could also have facilitated individual migration when relative productivities changed. This is a possibility even for an agrarian society where warfare might otherwise have been expected. More generally, we believe good institutions can help to promote peace even when third party enforcement is absent.

7.9 Inequality and Warfare

The model of this chapter may appear complicated in some ways but it is simple in one way: It only involves two sites located within an isolated region. Chapter 6 had a much richer setting with a continuum of sites and a range of property rights institutions, including sites that were open, sites that were closed but unstratified, and sites that were closed and stratified. The latter two cases generated insider–outsider inequality and elite–commoner inequality respectively. In this section we build various bridges between the results of Chapters 6 and 7.

The food technologies in the two models are identical (compare Sections 6.5 and 7.4). In both cases we use a Cobb–Douglas production function with inputs of labor and land, and we normalize the land input at unity. Chapter 6 decomposes site productivity into a regional effect from climate and/or technology and a local effect from geography, while Chapter 7 collapses these effects into a single parameter, but this is unimportant. The two models also assume identical Malthusian dynamics where an agent’s surviving offspring are proportional to the agent’s food income (compare Sections 6.7 and 7.6).

In Chapter 6 we assumed free mobility among sites within the commons, which equalized food per capita across all open sites. This equalization of average products is precisely the condition that ensures peace in Section 7.4. Therefore groups at open sites in Chapter 6 will not engage in warfare. This conclusion might need to be modified, of course, if there are costs of individual mobility across the sites in the commons, which could be physical, social, or informational in nature.

For sites that are closed but unstratified the issues are more subtle. In the model of Chapter 6 these sites were internally egalitarian but insiders blocked further entry by individual outsiders. This required that insider groups have a minimum size, which we called “d” in Chapter 6. Because closed sites had a range of productivities but the same group size, their food per capita varied from relatively low to relatively high.

A tempting approach in thinking about such sites runs as follows. Because all of the closed but unstratified sites have the same group size, they have equal military power, so in a conflict between two such groups each has an equal probability of winning. Then the question is whether differences in site qualities are large enough to induce the poorer group to attack the richer group. The technology of exclusion used by insiders prevents migration across groups, so individual mobility will not limit the incentives for war.

The problem with this reasoning is that in the model of Chapter 7 we assumed the victorious group would control both sites. In a world with just two sites and no outsiders this makes sense. But in the context of Chapter 6 there are many individual agents who might like to enter a site, and a group that hopes to gain control of a new site will need to exclude these outsiders after it wins the war. If the group intends to use both the old and new sites, as we assumed in Section 7.4, then it does not merely need a critical mass of “d” at its old site; it also needs “d” at its new site. This raises the question of how the winners find the necessary personnel to control and exploit both sites without suffering any entry by outsiders at one place or the other.

One possible answer is that this personnel constraint makes it impossible for the winners to control both sites, so they move all of their members to the new and better site while abandoning the old one. The downside for the winners is that by putting all of their labor input into one site, they obtain a lower food per capita than what they would get by spreading the same labor input across the two sites. This is true not just for the particular production function adopted in Section 7.4 but for the entire class of production functions where the marginal product of labor approaches infinity as labor input approaches zero, so interior solutions are optimal. Therefore if the personnel constraint binds, the gain to the winners of a war is smaller than it would otherwise be and warfare is less likely than the model of Chapter 7 suggests.

Both for this reason and in order to gain additional military strength in a world where warfare is common, an insider group may prefer to have more members than the minimum (d) needed to exclude unorganized outsiders. Ethnographic evidence suggests that in egalitarian societies with frequent warfare, local groups often recruit outsiders to enhance their military prospects. If an insider group chooses to have a membership size of at least 2d rather than d, the personnel constraint disappears, because if the group wins a war it will be able to control both sites. But there are still tradeoffs: a larger group may be beneficial in wartime but it lowers food per capita in peacetime. In principle, it would be possible to solve the latter problem by requiring new group members to pay entry fees to the old group members, but in practice this may not be feasible.

A further wrinkle involves Malthusian population dynamics. In Chapter 6 we showed that within the set of closed but unstratified sites, differences in food per capita would lead to population growth at higher quality sites and population decline at lower quality sites. We suggested in Chapter 6 that in the former case this would cause some children of the insiders to face downward mobility into the commons, while in the latter case it would cause insiders to recruit new commoners until group property rights were restored. An institutional environment with warfare opens up further possibilities.

Rather than departing for the commons, the surplus children of the insiders at rich sites may remain available to assist with military activity and may provide the personnel needed to control sites gained through warfare. The implications for insiders at poor sites who suffer population losses depend upon the sequence of events. If decisions about war take place before such groups can be replenished through recruitment from the commons, these groups will be militarily weak and vulnerable to attack (although their sites are not very valuable prizes). Under such conditions the Malthusian model of Section 7.6 may apply and individual mobility costs combined with productivity shocks may lead to war.

Another direction in which the model could be extended is to give up the idea of sites being indivisible units. We can instead think of each site as having some total land area (which was normalized to unity in Section 7.4). Instead of thinking of the parameter “d” from Chapter 6 as a minimum number of agents needed to close an entire site, we can think of it as the minimum density of organized insiders per unit of land that would have to be present in order to exclude unorganized outsiders. With these modifications we can treat the sites as divisible territories rather than indivisible units, and model warfare over fractional slices of territory. This eliminates the discrete nature of the “d” versus “2d” issue discussed above, and (depending on other details of the model) might lead to wars where groups with modest military advantages try to shift territorial boundaries in their favor.

In addition to sites that are open, or closed but unstratified, Chapter 6 developed a theory to explain the emergence of stratified sites with elite–commoner inequality. The next item on the agenda is to consider warfare in societies with stratification. Chapter 8 creates a formal model to describe wars of conquest among rival elites. There we argue that elite warfare tends to be chronic, even without mobility costs or productivity shocks.

7.10 Postscript

This chapter is based on “The economics of early warfare over land” published in the Journal of Development Economics (Dow et al., Reference Dow, Mitchell and Reed2017). Leanna Mitchell played an equal role in writing the version of this paper that was originally submitted to JDE. Dow and Reed rewrote the paper at the request of the JDE editor and are responsible for this chapter. Leanna received her PhD from the Department of Economics at Simon Fraser University in 2021.

Comments on the original article were provided by Cliff Bekar, Sam Bowles, David Burley, Timothy Earle, Curt Eaton, Mukesh Eswaran, Brian Ferguson, Lawrence Keeley, Patrick Kirch, Ian Kuijt, James Kai-sing Kung, Patricia Lambert, Gerard Padró i Miquel, Omer Moav, Peter Richerson, Stephen Shennan, Bruce Winterhalder, and two anonymous referees. We also thank participants at the 2014 meeting of the Canadian Economics Association and the 2015 symposium “Warfare in Interdisciplinary Perspective” organized by the Human Evolutionary Studies Program at Simon Fraser University. HESP, the Department of Economics at SFU, and the Social Sciences and Humanities Research Council of Canada provided financial support, and Ideen Riahi and Haiyun Kevin Chen supplied outstanding research assistance. We received valuable feedback on an earlier draft of this chapter from Samuel Bowles, Richard Lipsey, John Chant, Jon Kesselman, Craig Riddell, Herbert Grubel, and Richard Harris. Carles Boix gave us insightful comments on our analysis of warfare in both Chapters 7 and 8. All opinions are those of the authors.

We have expanded Section 7.1 substantially in order to create linkages with the other chapters in Part III of the book. Sections 7.2 and 7.3 are largely unchanged relative to the original journal article, although the discussion of the literature has been updated. The formal model in Sections 7.4–7.6 is unchanged but we have added intuition for the benefit of non-economists. Sections 7.7 and 7.8 have been lightly edited except for the discussion of the Enga at the end of Section 7.7, which is new. We were unaware of this case when the model was constructed. All of Section 7.9 on the connections with Chapter 6 is new.

8.1 Introduction

The preceding chapter studied warfare over land among egalitarian groups. Here we extend our study of warfare to elite groups in stratified societies. The formal model in this chapter, although similar to the one in Chapter 7 with respect to the production and military technologies, gives very different results. We will argue that elites are likely to engage in chronic warfare over land and have a strong bias toward territorial expansion. Our investigation of the origins of the state in Part IV (especially Chapter 11) will make use of these ideas, but for now we are only concerned with warfare in stratified non-state or pre-state societies.

In Chapter 7 we distinguished three kinds of warfare in early societies: raiding, displacement, and conquest. Raiding involves theft of moveable property or the capture of humans. It usually puts a premium on surprise, speed, and rapid escape. This is quite different from the military technology used to transfer land ownership permanently from one group to another. We want to reiterate that we are not offering a theory of raiding.

Displacement involves the seizure of land from a rival group while driving off or killing the previous occupants. Our model in Chapter 7 examined wars of displacement among internally egalitarian groups. In this chapter we will consider conquest, which we define to be the seizure of land from an opposing elite, where the previous elite is driven off or killed but commoners are left in place to pay land rent to the new elite.

We base our approach on the model of stratification developed in Chapter 6. In that model an elite begins as an organized insider group that achieves the threshold size needed to exclude unorganized outsiders from a specific site. Stratification occurs when these insider groups find it profitable to hire commoner labor to work their land. An elite enjoys land rent due to its collective ownership of the site, where the rent is equal to food output at the site minus elite expenditures on labor inputs. The wage paid to a commoner is equal to the food income that agent could get at an open site located in the commons.

We remind the reader that as in Chapter 6, when we use the terms “wage” or “labor market,” we mean that elites provide food in exchange for labor services, not that there is a monetary payment in the modern sense. Our theoretical results would be identical with an alternative set of institutions where commoners pay “rent” to elites in a “land market,” by which we mean that workers make food payments to elites in exchange for access to land. In this chapter we extend the idea of a “labor market” to include warriors who receive food payments from the elite in exchange for military services.

Here we ask whether elites will find it profitable to engage in organized warfare with neighboring elites in order to capture additional land rent. The short answer is yes. The incentives for warfare among egalitarian groups are limited by two main factors: the presence of diminishing returns to land for a fixed labor input, and the fact that a group’s military power is limited by its size (for detailed discussions see Section 7.9 in Chapter 7 and Section 8.8 below). Both of these constraints are absent in the case of elite warfare.

The first factor restraining warfare in egalitarian societies is that groups have limited labor inputs, and as a result face diminishing returns to the accumulation of land. In contrast, elite groups can hire unlimited commoner labor at a parametric wage, which means they can scale up labor and land simultaneously if they acquire more land through warfare. Assuming the old and new land have equal quality, an elite’s total land rent is proportional to the land area it controls. The fact that land has diminishing returns when the labor input is fixed becomes irrelevant because labor is not fixed. Variations in land quality do not affect the substance of this argument, because it is still true that an elite’s total land rent is simply the sum of the land rents obtained from its individual territories. There is no tendency for the value of incremental land acquisitions to decline as an elite accumulates more land.

The second factor restraining warfare in egalitarian societies is that groups have limited military power because they have fixed sizes. Elite warfare differs because an elite can recruit an army of potentially unlimited size by hiring warriors at a parametric wage. As we will explain later, the warriors must be offered booty if they win a war to compensate them for the prospect of death if they lose. In some cases, the land rents that would be obtained by a victorious elite are insufficient to finance the required booty. But generally speaking, when the sizes of the rival combatant groups are determined by profit maximization rather than individual migration or Malthusian dynamics, the guardrail of limited military power is removed.

For these reasons our formal model leads to a stark conclusion: a pair of elites will not pursue policies of peaceful coexistence. Instead the elites either attack or flee. If both elites attack, there is a war. If one attacks while the other flees, the winner grabs the territory of the loser without a fight. We call this “winning through intimidation.” What we do not obtain is a stable equilibrium in which each elite chooses an army size that successfully deters an attack by the opposing elite.

When the degree of inequality between elites and commoners is low, so the total land rent at stake in a conflict between rival elites is small relative to the cost of an army, open war may be avoided because intimidation causes one side to flee with certainty. But there are also equilibria in which open warfare and intimidation each occur with positive probability. When the degree of inequality between elites and commoners is sufficiently high, these become the only possible equilibria, so open warfare must sometimes occur. Army sizes for such equilibria are larger than when intimidation succeeds with certainty, and military expenditures dissipate the entire land rent of the elites, so war is maximally wasteful. Open war and intimidation both put territories under unified control, enabling the successful elite to engage in geographic expansion.

These rather extreme conclusions can be qualified in a number of ways. Two constraints on elite warfare are provided by (a) limitations on elite personnel needed to administer new territories and (b) limitations on the fiscal capacity of an elite to recruit warriors. We also discuss restraints on elite conflict involving preferences, geography, defensive technology, institutions, and culture (see Section 8.9). Although these factors may sometimes offer paths to peaceful coexistence, the key message from the model is that elites have strong incentives to conquer more territory, either through overt war or covert intimidation, so any restraints on aggressive behavior need to be equally strong.

Section 8.2 reviews theories of elite warfare and Section 8.3 surveys empirical evidence. We already discussed the literature on early warfare in Chapter 7 so we will keep these sections short and emphasize issues specific to stratified societies.

The next four sections develop the formal model. Section 8.4 introduces basic ideas. The main results are in Sections 8.5 and 8.6. Events unfold in two stages. First rival elites choose army sizes (what we call the recruitment stage). Once the army sizes have been determined and observed by each elite, there is a second stage at which each elite decides whether to attack, defend, or flee (what we call the combat stage). It is analytically convenient to start by treating army sizes as exogenous and examining the decisions made at the combat stage, which we do in Section 8.5. We then move back to the recruitment stage in Section 8.6 and study how army sizes are chosen, assuming that each elite understands how army sizes will influence outcomes at the combat stage.

Section 8.5 shows that there are no peaceful equilibria at the combat stage where both elites choose to defend. For most army sizes, either one elite or the other prefers to attack. There is a particular ratio of army sizes where each elite is indifferent between attacking and defending. But even in this situation there is an advantage to attacking, because the opposing elite might flee and it would then become possible to seize the other elite’s land without a fight. This breaks the tie in favor of aggression.

Section 8.6 shows that open warfare of a deterministic kind is not an equilibrium at the recruitment stage, because each elite has an incentive to shift to a larger army size in order to win by intimidation. However, open warfare does arise probabilistically when armies are large enough. Section 8.7 addresses the role of fiscal constraints on warfare. Such constraints can limit warfare if elite–commoner inequality is very high, but do not prevent probabilistic warfare at low to moderate levels of inequality. One might expect that richer elites would have a greater ability to finance wars but in our model this is not true, because booty is paid out of the total rent appropriated by the victorious elite rather than being paid up front using an elite’s current (smaller) land rent. Readers who want to skip most of the math should at least scan Section 8.4 to understand the general structure of the model, but can omit Sections 8.5–8.7 if they like.

Section 8.8 compares the warfare models in Chapters 7 and 8, and comments on applications of the formal model from Chapter 8 to archaeological evidence. Section 8.9 discusses factors omitted from the formal model that could limit elite warfare in practice. We will return to elite warfare in the context of state formation, so we will defer further discussion of regional cases until Chapter 11. Section 8.10 concludes with an overview of the theory of pre-state institutional development we have constructed in Chapters 6–8. Proofs of all formal propositions can be found at cambridge.org/economicprehistory.

8.2 Theories of Elite Warfare

We are aware of relatively few theoretical contributions from either economists or archaeologists that focus specifically on warfare among societies having stratification but lacking states. The literature on warfare among state-level societies is huge, but does not speak to the issues we are addressing here.

As we discussed in Section 7.2, there has been intense debate between writers (sometimes called “hawks”) who regard warfare as a constant feature of prehistory and writers (sometimes called “doves”) who believe it was rare or non-existent among early egalitarian societies, and only became commonplace with the proliferation of stratified societies in the early Neolithic. The hawks often rely on genetic arguments to explain constant warfare and have trouble explaining variations in the incidence of war across time and space in the last 10,000–15,000 years. The doves often rely on ideas about non-observed institutions that are said to have maintained peace through most of prehistory.

While there is strong disagreement between hawks and doves over the incidence of early warfare and its causes, there are also several areas of agreement reflected in two recent publications by major figures in the debate (Ferguson, Reference Ferguson2018; LeBlanc, Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020). The areas of agreement include the following:

1. (a) Warfare is defined as lethal organized conflict between groups.

2. (b) The earliest examples of warfare involve raiding, not set battles over territory.

3. (c) Archaeological evidence for very early warfare is weak and difficult to interpret (i.e., evidence for warfare and evidence for peace are both hard to find in the pre-Neolithic).

4. (d) There is strong evidence that stratified societies are prone to constant warfare.

5. (e) There are dangers of sample selection biases in using ethnographic data to infer warfare in prehistory.

6. (f) Climate change played an important role as a trigger for warfare.

7. (g) Raiding and warfare over land can be included in the same analysis.

We accept points (a)–(f) but believe it is important to provide separate theories of raiding and warfare over land, because the relevant military technologies differ substantially.

LeBlanc (Reference LeBlanc, Fagan, Fibiger, Hudson and Trundle2020) believes that the causal channel runs from a human genetic predisposition to violence, to constant warfare, to stratification. However, he treats the genetic predisposition as a hypothesis, not a fact (2020, 47–48). He explains the lack of warfare data in early egalitarian societies by the fact that populations were smaller and the time period was earlier compared to stratified societies. He does not see this lack of evidence for warfare as a reason to think there was actually less warfare. Rather, it is evidence that the markers of warfare are less visible to us today.

Ferguson (Reference Ferguson2018) believes that constant warfare resulted from the transition to complexity and stratification. He lists several conditions that promote war, including sedentism, regional population growth, livestock or other valuable resources, social complexity and hierarchy, trade, group boundaries and collective identities, and severe environmental changes. He also lists several conditions that promote peace, including cross-group linkages of kinship and marriage, cooperation in food production and food sharing, flexible social arrangements, norms that value peace, and recognized means for conflict resolution. We view these correlations as suggestive of causal connections, but explicit theory is required to make the case convincing.

In contrast to these authors we separate raiding from warfare over land. We offer a theory of stratification (see Chapter 6) that does not depend on warfare. We show that there is little economic incentive for warfare in egalitarian societies except during periods of sudden environmental or technological change (see Chapter 7). And we show in this chapter that in stratified societies there is an economic incentive for constant warfare that is independent of genetic predispositions for violence, population change, environmental shocks, and the other items on Ferguson’s list of warfare-inducing conditions. While we do not rule out the possibility that good institutions (or other factors) might restrain such warfare, it seems clear from the data that potential solutions of this kind often fail.

Rowthorn et al. (Reference Rowthorn, Guzmán and Rodríguez–Sickert2014) note that most theories of pre-modern stratified societies share three common elements: a class distinction between warriors and peasants, a food surplus generated by peasants that supports the warriors, and hereditary social positions. They use the microeconomic tool of rational individual choice to model such societies.

At the regional level the authors assume exogenous numbers of egalitarian and stratified societies. In egalitarian societies producers defend themselves using readily available weapons. In stratified societies a class of warriors defends the society using specialized weapons that others do not possess. Malthusian dynamics determine the sizes of the warrior and peasant classes in the long run, and the greater cost of warrior children leads to economic inequality in favor of the warriors. The relative military powers of the individual societies determine the land available for food production in each society.

Our model differs in several ways. First, we distinguish between an elite class of landowners and the warriors they employ. The elite guard their land against intrusion by individual outsiders and use the resulting land rent to recruit warriors for combat against the warriors employed by rival elites. Second, in our theory the size of the warrior class is determined by profit maximization on the part of the landowning elite, not directly by Malthusian mechanisms. Third, in the model of Rowthorn et al. warriors give peasants land to farm. By contrast in our model the elite recruits warriors to enhance its land rent and warriors provide no benefit to commoners. Fourth, the direction of causality differs. Rowthorn et al. take warfare for granted and explain the degree of inequality, while we take inequality for granted and explain the prevalence of warfare.

Rowthorn et al. argue that their theory can explain instances where changes in military technology led to changes in the relative numbers of warriors and peasants, and the degree of economic inequality between them. Most of their examples involve state-level (although pre-modern) societies, while our focus is on pre-state institutions.

Boix (Reference Boix2015, ch. 4) similarly argues for a causal channel running from warfare to inequality. He asserts that changes in military technology have led to dramatic changes in political and social institutions, including the degree of inequality between economic classes. Important innovations in military technology have included developments with respect to metallurgy (copper, bronze, carburized iron); horses (chariots, stirrups); and gunpowder (firearms, cannons). We agree that military technology has had far-reaching effects on political, social, and economic institutions. However, we make two points.

First, most of the military technologies discussed by Boix are not relevant for the societies studied here. The societies we consider did not have bronze weapons, let alone iron or steel. They certainly did not have domesticated horses or gunpowder. If we take prehistory to be defined by the absence of written records, our overlap with Boix mainly involves the use of copper weapons in southern Mesopotamia, a case to be discussed in Chapters 9 and 10. The emergence of the Mesopotamian city-states, accompanied by the first written records, substantially predates their eventual political unification by Sargon, which Boix attributes in part to the development of bronze weapons (2015, 132–133).

The second point is more theoretical. We start from the stratification model in Chapter 6 and consider the incentives for warfare between two neighboring elites. Our theory about the origins of stratification does not involve warfare. It simply requires a technology of exclusion through which organized insiders can establish property rights over valuable sites by repelling unorganized outsiders. In our approach the existence of elites leads to warfare, not the other way around.

We do not want to dismiss the argument by Boix that causality could also flow in the opposite direction, where warfare leads to inequality or stratification. Indeed we will see in Chapter 11 that many archaeologists believe warfare created a trajectory leading to pristine state formation. But in this chapter we hold military technology constant, change the institutional structure from egalitarian to stratified, and look at the effects on warfare, rather than changing military technology and looking at the effects on institutions. We will return to Boix (Reference Boix2015) in Chapter 12 when we discuss the transition from prehistory to the modern world.

Pandit et al. (Reference Pandit, Pradhan and van Schaik2017) develop a theory of warfare in small-scale societies based on biological fitness. Individuals join attacking coalitions voluntarily if this enhances their fitness. The probability that a coalition wins a war depends on the difference in summed fighting abilities of the attacking and defending sides. After a war, surviving individuals have ranks in the fused society that reflect their fighting abilities.

A key question involves the impact of inequality with respect to fighting ability within each group, which determines access to economic resources and is correlated with reproductive success. The authors conclude that other things being equal, egalitarian groups have a military advantage relative to despotic ones. The reason is that when inequality is high, lower-ranking individuals are less likely to benefit from a successful war than those of higher rank, so elites have trouble recruiting coalition members. The authors point out that skew in intrinsic fighting ability is unlikely to vary much across societies, but actual fighting ability may depend upon training, weapons, or wealth. They also note that large complex societies typically overwhelm their small and less complex neighbors, so in such cases aggregate military power clearly outweighs the effects of skew within groups.

The theory of Pandit et al. resembles ours in some ways. For example, it assumes the use of simple weapons, and rules out state-level phenomena such as the imposition of coercive taxes on conquered societies. On the other hand, Pandit et al. also rule out the employment of mercenary warriors by elites, which we permit, and highlight differences in fighting ability across individuals, which we ignore. We assume individual agents are motivated by food rather than directly by fitness, although of course one is a means to the other. More generally, we differ from Pandit et al. in stressing the economic implications of stratification and warfare rather than direct biological implications.

8.3 Evidence on Elite Warfare

By contrast with warfare between egalitarian groups, where there is a serious dispute over the frequency and dating of warfare (see Section 7.3), there is a consensus that warfare between stratified groups was widespread in prehistory.

Kyriacou (Reference Kyriacou2020, 44–45) studies the association of warfare and stratification using ethnographic data for the societies in the Standard Cross Cultural Sample. He estimates four regressions using ordinary least squares, with two alternative dependent variables (social stratification and class stratification) and two alternative independent variables (frequency of warfare or fighting, and casualty rate among the combatants). Population density is controlled in all four regressions, which have either 79 or 133 observations according to data availability. All of the coefficients display the expected positive sign, with the frequency of warfare significant at the 10% level (social stratification) or the 5% level (class stratification), and the casualty rate significant at the 1% level for both of the stratification measures. As discussed in Section 8.2, the direction of the causal arrow can be debated, but a positive correlation clearly exists in the ethnographic data.

Gat (Reference Gat2006) reports on the features of warfare in pre-state, pre-urban agricultural societies where stratification is common. He calls these groupings “tribal societies.” They are much larger (typically about 2,000 people) than simple hunting and gathering or early agricultural groups (typically about 500 people; 2006, 176). Both, however, were based on kinship circles. They also shared similar armaments that were privately owned and generally of poor quality: spears, axes, clubs, knives, bows and arrows, shields, and occasional leather armor (185–186). Early stratification involved the selection of a chief who had limited authority, was often elected, and was often the war leader. Another form of early leadership was the “big man.” In these stratified societies, food producers did not play an important role in warfare (298–299). The maximum size of the warrior group was around 200 (228). Open battles became more common over time as the element of surprise became more difficult to achieve in raiding expeditions. These open battles were increasingly fought over land (186). Our model incorporates a number of these features, although we do not address the leadership structure within the elite class.

Before presenting the formal model we briefly mention some regional cases that have both stratification and warfare in order to give a flavor of their varying relationship. Most were discussed in earlier chapters so we forego detailed descriptions here. Further cases will be discussed in Chapter 11 in the context of state formation.

8.4 The Formal Model

Consider two stratified sites $\mathrm{i}=\mathrm{A},\mathrm{B}$. We continue to use our standard term “sites” to distinguish geographically separate food production locations, although in the present context they can equally well be called “territories.” As in the model of Chapter 6, at each site an organized elite prevents unorganized outsiders from entering. The elite also hires commoners to work the land. In contrast to the model of Chapter 6, we simplify here by assuming that the elite agents supply no labor to food production. In peacetime an elite merely guards its territory, employs commoners, and enjoys land rent. All agents (elite and commoner) are risk neutral and maximize expected food consumption.

Food is produced from inputs of labor and land using a Cobb–Douglas function with constant returns to scale:

${\mathrm{Y}}_{\mathrm{i}}={\mathrm{\theta }}_{\mathrm{i}}{{\mathrm{L}}_{\mathrm{i}}}^{\mathrm{\alpha }}{{\mathrm{Z}}_{\mathrm{i}}}^{1-\mathrm{\alpha }}\mathrm{i}=\mathrm{A},\mathrm{B}$(8.1)

where ${\mathrm{Y}}_{\mathrm{i}}$ is food, ${\mathrm{\theta }}_{\mathrm{i}}$ is the productivity of site i, ${\mathrm{L}}_{\mathrm{i}}$ is commoner labor, and ${\mathrm{Z}}_{\mathrm{i}}>0$ is land. The productivity ${\mathrm{\theta }}_{\mathrm{i}}>0$ is determined by nature and technology. As usual $0<\mathrm{\alpha }<1$.

The minimum density of elite agents per unit of land needed to exclude outsiders and maintain property rights over a territory is $\mathrm{e}>0$. The corresponding parameter value in Chapter 6 was d. We adopt the new notation in order to highlight our new assumption that elites do not produce food and instead only guard land.

In order for elite property rights to be secure we require ${\mathrm{e}}_{\mathrm{i}}/{\mathrm{Z}}_{\mathrm{i}}\ge \mathrm{e}$ where e_i is the size of elite i. When each elite is of the minimum necessary size, ${\mathrm{e}}_{\mathrm{i}}={\mathrm{eZ}}_{\mathrm{i}}$ for $\mathrm{i}=\mathrm{A},\mathrm{B}$. But for reasons to be explained later in this section, it will be useful to consider situations where the elites have excess capacity in the sense that ${\mathrm{e}}_{\mathrm{i}}>{\mathrm{eZ}}_{\mathrm{i}}$. The sizes of the elites are exogenous throughout.

The stratified sites are embedded within a regional economy that includes a large commons with open access as described in Chapter 6. The food income of agents in the commons is $\mathrm{w}>0$. The elites at sites A and B treat w as parametric and hire commoners at this wage rate to work on their land. The resulting land rents are

${\mathrm{r}}_{\mathrm{i}}=max\left\{{\mathrm{\theta }}_{\mathrm{i}}{{\mathrm{L}}_{\mathrm{i}}}^{\mathrm{\alpha }}{{\mathrm{Z}}_{\mathrm{i}}}^{1-\mathrm{\alpha }}-{\mathrm{wL}}_{\mathrm{i}}\text{subject}\mathrm{to}{\mathrm{L}}_{\mathrm{i}}\ge 0\right\}\mathrm{i}=\mathrm{A},\mathrm{B}$(8.2)

Due to the Cobb–Douglas technology these land rents are strictly positive. It can also be shown that the rent from site i is proportional to the land area associated with the site, so r_i = β_iZ_i where β_i > 0 is a constant determined by the exogenous parameters θ_i, α, and w. The sites (or territories) may have differing rents due to differences in their productivity or land area. We denote aggregate land rent by R = r_A + r_B.

For a given land area Z_i and elite size e_i we say that elite i is economically viable if its land rent r_i exceeds its opportunity cost we_i. The latter is what the elite agents could obtain by abandoning their site, moving to the commons, and producing food there. We assume the productivity θ_i of each site is high enough to guarantee we_i < r_i for i = A, B so there is an economic benefit from being in the elite, at least in peacetime. The members of the elite divide their land rent equally so each individual member receives r_i/e_i.

The degree of stratification in the society will be indexed by R/w, which is total rent normalized by the commoner wage. The lower bound for normalized rent is

${\mathrm{e}}_{\mathrm{A}}+{\mathrm{e}}_{\mathrm{B}}<\mathrm{R}/\mathrm{w}$(8.3)

This must be true if both elites are economically viable in the sense defined above.

Each elite can recruit an army, which can be used to attack the rival elite. Elite i hires a number of warriors m_i ≥ 0 from the commoner class where each warrior is paid an expected food income equal to w. Combat involves generally available weapons and the members of an army produce no food.

In Sections 8.5 and 8.6 we will construct a two-stage model. At the first stage the elites recruit their armies simultaneously. At the second stage the army sizes (m_A, m_B) ≥ 0 are fixed and observed by both elites. The issue at the second stage is whether either of the elites wishes to attack the other. If neither elite wants to attack, there is peace. If one or both elites chooses to attack, there is war. We also allow a third option: elite i can flee to the commons and accept the reservation utility we_i available at open sites.

We proceed by backward induction. In Section 8.5 we examine the combat stage where (m_A, m_B) has already been determined and the elites make decisions about whether or not to attack. In Section 8.6 we move back to the recruitment stage where each elite hires its army (correctly anticipating the decisions to be made at the combat stage), and study Nash equilibria with respect to the choice of army sizes. This yields results about whether war or peace will prevail, and how these outcomes are related to the degree of stratification summarized by R/w.

In wartime the probability that elite i = A, B wins against j ≠ i is

${\mathrm{p}}_{\mathrm{i}}={\mathrm{m}}_{\mathrm{i}}/\left({\mathrm{m}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{j}}\right)={\mathrm{m}}_{\mathrm{i}}/\mathrm{M}\text{where}\mathrm{M}\equiv {\mathrm{m}}_{\mathrm{A}}+{\mathrm{m}}_{\mathrm{B}}$(8.4)

The probability that elite i wins is independent of who attacked whom. This combat technology is identical to the one used in Chapter 7. As in that chapter, the probability of winning is determined by the relative sizes of the combatant groups. The only difference is that here the size of each combatant group is chosen by its sponsoring elite rather than being determined by individual migration or Malthusian dynamics.

If elite i wins, it retains the rent r_i from its existing site, appropriates the rent r_j from the site j ≠ i, and compensates its warriors in a manner to be described below. If elite i loses, both the elite and its army die, resulting in zero payoffs (see the remarks on utility functions in Section 7.8). The commoners who engage in food production do not participate in combat, face no risk of death, and are indifferent toward the identity of the winning elite.

At this point we flag an issue about property rights over the conquered territory. We will assume that if elite i wins a war it gains the prize R = r_i + r_j = r_A + r_B where this aggregate land rent is a constant that is independent of the sizes of the armies involved. The armies only affect the probability of victory, not the prize at stake. But how does elite i extend its property rights to the newly conquered territory at site j? If we have e_i = eZ_i so elite i has the minimum density needed to control its home territory, any transfer of elite agents from site i to site j makes it impossible for elite i to fully guard its original territory, leading to entry by unauthorized outsiders and erosion of the home rent r_i.

The simplest way to address this problem is to assume that each elite has excess capacity so e_i ≥ e(Z_A + Z_B) for i = A, B. In this scenario the victorious elite can always spare enough personnel to occupy the other territory and establish property rights over it. For example, the agents placed in charge of the new territory may be offspring of the elite who would otherwise have moved down into the commoner class (see Section 6.10). For this excess capacity solution to work we need e(Z_A + Z_B) < r_i/w for i = A, B so each elite can still cover its own opportunity cost we_i as discussed earlier. This will be true as long as the productivities θ_i in (8.1) are high enough relative to the wage w.

Next consider the payments to warriors from a sponsoring elite. If there is peace, warriors are paid their opportunity cost w, which is what they can obtain in the commons. We assume that warriors are “on call” and do not produce food because they must be ready to defend against a possible attack (or launch one). We ignore leisure, training costs, or the social status associated with the role of being a warrior. This gives elite i the payoff

$\text{peace}:{\mathrm{r}}_{\mathrm{i}}-{\mathrm{wm}}_{\mathrm{i}}\mathrm{i}=\mathrm{A},\mathrm{B}$(8.5)

If a war occurs, an individual warrior faces a positive probability of being killed and receiving zero. We assume that warriors are not forced to fight; joining the army is a voluntary transaction. To compensate for the prospect of death a warrior must be offered a food income in the event of victory that exceeds the peacetime wage w. We refer to the wartime wage b_i as “booty.” The booty must satisfy the participation constraint ${\mathrm{p}}_{\mathrm{i}}{\mathrm{b}}_{\mathrm{i}}=\mathrm{w}$ so expected food income per warrior is equal to what warriors could obtain in the commons. The payoff to elite i from war is therefore

$\mathrm{war}:{\mathrm{p}}_{\mathrm{i}}\left(\mathrm{R}-{\mathrm{b}}_{\mathrm{i}}{\mathrm{m}}_{\mathrm{i}}\right)={\mathrm{p}}_{\mathrm{i}}\mathrm{R}-{\mathrm{wm}}_{\mathrm{i}}=\left[{\mathrm{m}}_{\mathrm{i}}/\left({\mathrm{m}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{j}}\right)\right]\mathrm{R}-{\mathrm{wm}}_{\mathrm{i}}\mathrm{i}=\mathrm{A},\mathrm{B}$(8.6)

There is an infinitely elastic supply of commoners willing to become warriors if they are offered an expected income of w, and all of the candidate warriors are perfect substitutes for military purposes. The elite offers the minimum booty needed to attract an army and captures all the ex ante surplus from warfare.

Under our “excess capacity” assumption about the elite’s ability to take over any newly conquered territory, when elite i is victorious it gains the aggregate rent R. This can be used to finance the booty owed to its army. The warrior participation constraint gives ${\mathrm{b}}_{\mathrm{i}}=\mathrm{w}/{\mathrm{p}}_{\mathrm{i}}=\mathrm{wM}/{\mathrm{m}}_{\mathrm{i}}$ so the total payment owed to a victorious army is ${\mathrm{b}}_{\mathrm{i}}{\mathrm{m}}_{\mathrm{i}}=\mathrm{wM}$. Because the winning elite i appropriates the total rent R, it can pay its army as long as

$\mathrm{w}\left({\mathrm{m}}_{\mathrm{A}}+{\mathrm{m}}_{\mathrm{B}}\right)\le {\mathrm{r}}_{\mathrm{A}}+{\mathrm{r}}_{\mathrm{B}}\text{or}\mathrm{M}\le \mathrm{R}/\mathrm{w}$(8.7)

This fiscal constraint involves both m_A and m_B because a warrior’s willingness to fight depends on the probability of death, which depends on the sizes of both armies. There is no subscript $\mathrm{i}=\mathrm{A},\mathrm{B}$ in the condition $\mathrm{M}\le \mathrm{R}/\mathrm{w}$. When (8.7) holds, the winning elite will be able to pay its own troops their promised booty, whichever elite that may be. When (8.7) does not hold, neither elite will be able to do so even if it proves to be the winner. As a result neither will be able to recruit an army of the desired size, because all prospective warriors will correctly foresee a failure to deliver adequate booty in the event of victory. We will show in Section 8.7 that this fiscal constraint can sometimes limit the ability of elites to engage in open warfare.

Even if financial solvency is not an issue, one might question why the victorious elite would actually pay its warriors their promised wages ex post. Clearly in prehistory there were no legally binding contracts about such matters. One possible answer is that such games were played repeatedly and the elite wanted to build a reputation for paying its warriors. Another possible answer is that unhappy warriors, having been organized as an army, could take control of the elite’s home territory and/or newly conquered territory. We will not delve into such details here and simply assume that elites honor promises to their warriors as long as their budget constraints allow them to do so.

8.5 The Combat Stage

In this section we treat army sizes $\left({\mathrm{m}}_{\mathrm{A}},{\mathrm{m}}_{\mathrm{B}}\right)\ge 0$ as exogenous and investigate elite decisions regarding warfare. The only restriction placed on army size is that when we are interested in decisions by elite i, we assume ${\mathrm{m}}_{\mathrm{j}}>0$ so elite i’s probability of victory in (8.4) is well defined even when ${\mathrm{m}}_{\mathrm{i}}=0$. The case ${\mathrm{m}}_{\mathrm{A}}={\mathrm{m}}_{\mathrm{B}}=0$ will be addressed in Section 8.6.

We assume $0<{\mathrm{r}}_{\mathrm{i}}/\mathrm{w}-{\mathrm{e}}_{\mathrm{i}}$ for $\mathrm{i}=\mathrm{A},\mathrm{B}$ so in peacetime each elite is economically viable and enjoys a standard of living above that of commoners. This section ignores the fiscal constraint $\mathrm{M}\le \mathrm{R}/\mathrm{w}$ from (8.7) but we will return to it in Section 8.7.