3.1 Introduction

This chapter has several goals. Among other things, we introduce terminology and facts about the Upper Paleolithic (UP); explain why conventional theories of economic growth do not adequately capture the dynamics of population and technology during this period; develop our own formal model of these matters; suggest that climate change was the driving force behind population growth and technological innovation in the Upper Paleolithic; and argue that our model can account for some empirical patterns identified by archaeologists. The ideas in this chapter provide foundations for our analysis of the transition to sedentism (Chapter 4) and the transition to agriculture (Chapter 5).

Following an overview in this section, we offer empirical information on climate, population, and technology in Sections 3.2–3.4. Section 3.4 argues that standard models of economic growth conflict with crucial facts about the Upper Paleolithic. In particular, any model predicting continuous expansion at an exponential or faster rate contradicts the observation that regional populations were commonly flat, cyclic, or subject to collapse.

This motivates our effort in the second part of the chapter to construct a formal model better suited to the archaeological evidence about the Upper Paleolithic. In our framework the equilibrium growth rate for population and technology is zero almost all of the time, with transitory periods of growth triggered by climate change. While some readers will not want to delve deeply into the mathematics in this part of the chapter, we urge all readers to look at the verbal portions, which often discuss important conceptual points. Proofs of all formal propositions are available at cambridge.org/economicprehistory.

Section 3.5 presents a model of technological innovation based on learning by doing. In our approach, technology is more likely to improve for food resources that are more intensively exploited, and it remains constant for any resources not currently in use. Section 3.6 studies the short-run allocation of labor time across individual food resources. Section 3.7 adopts a long-run framework in which population is endogenous and evolves according to Malthusian dynamics. Section 3.8 uses a very-long-run framework where technology is also endogenous and evolves according to the model of learning by doing in Section 3.5. There can be many equilibria in the very long run, and foraging societies can get stuck in what we call a “stagnation trap.” In such situations, some resources are not used because they are unattractive with existing technical knowledge, but knowledge does not improve because the resources are not used. Section 3.9 shows that changes in climate can sometimes enable foraging societies to escape from stagnation traps, leading to episodic technological innovation and population growth.

In the rest of the chapter we resume a verbal presentation. Section 3.10 suggests that our model can explain two empirical patterns in the archaeological record. One of these is the broadening of the diet in the Upper Paleolithic (the so-called broad spectrum revolution). The other is the increasing ability of human groups to colonize harsh natural environments such as Siberia. Section 3.11 is a summary and Section 3.12 is a postscript.

We begin by explaining some terminology for the benefit of non-archaeologists. Newcomers to the literature will frequently encounter references to the Lower, Middle, and Upper Paleolithic, where the term Paleolithic means “old stone age.” In all three of these periods, stone tools were used for hunting and gathering. The Neolithic period, or “new stone age,” refers to the use of stone tools for agriculture (see Chapter 5).

The boundary between the Lower and Middle Paleolithic is traditionally dated to about 250 KYA. Until recently, the accepted date for the advent of anatomically modern humans (AMHs, or Homo sapiens) was around 200 KYA, so the Lower Paleolithic was associated only with archaic humans; that is, members of the genus Homo, but not Homo sapiens. The most famous of these is Homo neanderthalensis. However, new discoveries from Morocco have pushed the appearance of Homo sapiens back to 300 KYA or earlier (see Section 1.1), creating overlap between AMHs and the Lower Paleolithic. Because very little is known about the technology used by AMHs at such early dates, we ignore the Lower Paleolithic here.

The boundary between the Middle and Upper Paleolithic is more relevant for our purposes. We will address the question of dates for this boundary below. The distinction between the two periods is based on differences in the methods used to create stone tools. The Middle Paleolithic is generally associated with a relatively primitive technique called “flake technology” while the Upper Paleolithic exhibits a more advanced technique called “blade technology” (for a description of the latter, see Fagan and Durrani, Reference Fagan and Durrani2019, ch. 5).

In blade technology, the tool producer (or knapper) starts with a prepared core of stone material, often flint. A punch stone and hammer stone are used to split fragments from the core, which are called blades. The blades are then shaped into a variety of tools including knives, scrapers, burins, and so on. The Upper Paleolithic toolkit represents an advance over the Middle Paleolithic in several ways: (a) raw materials such as flint are used more efficiently; (b) specialized tool types are made for specialized uses; (c) each tool type follows a standardized pattern; (d) stone tools are used to make further tools out of bone, antler, or ivory; and (e) composite tools can be created using multiple materials. The ability to carve bone or ivory made possible the creation of new tools like fishhooks and sewing needles. Composite techniques also widened the range of tools; for example, spears could involve a wood handle, a bone tip, and resin to attach the tip to the handle.

Later improvements in hunting methods included spear throwers as well as bows and arrows. Other markers of the Upper Paleolithic are grinding and pounding tools for processing wild plant foods; long-distance trade in raw materials; food storage facilities; structured hearths; specialized habitation areas for butchering, cooking, waste disposal, and sleeping; and artistic items such as ornaments, paintings, and musical instruments.

The traditional date for the boundary between the Middle and Upper Paleolithic is around 40–45 KYA, the period when AMHs were arriving in Europe. This amounts to a distinction between the technology used by Neanderthals already living in Europe and the technology brought to Europe by the first AMH immigrants. However, the technology of Upper Paleolithic Europe closely resembles technology used by AMHs in South Africa as early as 60–80 KYA (Mellars, Reference Mellars, Mellars, Boyle, Bar-Yosef and Stringer2007; Mourre et al., Reference Mourre, Villa and Henshilwood2010; Nowell, Reference Nowell2010). Mellars argues that the pioneering blade technology of South Africa fueled the spread of AMHs to Asia and Europe, a view that remains debatable. In any event, 40–45 KYA may represent not so much a qualitative boundary in the technical progress of AMHs as a milestone in their geographic dispersal (see our discussion in Section 2.3 on what we mean and don’t mean by the word “progress” in the context of technical change).

Archaeologists often refer to the Upper Paleolithic as a revolution in comparison with the Middle Paleolithic (Bar-Yosef, Reference Bar-Yosef2002a). The dramatic contrast in technological capabilities between the two periods warrants this label. From an economic perspective, however, the innovation process was remarkably slow. More than 200,000 years elapsed between the emergence of anatomically modern humans and the earliest widely accepted evidence of blade technology around 80 KYA. Tens of millennia more elapsed before the Neolithic transition to agriculture began around 12 KYA in southwest Asia. While spearthrowers and the bow and arrow were breakthroughs for the people who invented them, the average pace of technical progress in the Upper Paleolithic was negligible by comparison with early agricultural societies, let alone industrial societies.

Blade technology is a classic example of what economists would call a general-purpose technology (Lipsey et al., Reference Lipsey, Carlaw and Bekar2005). In this chapter we treat the existence of blade technology as given and do not attempt to explain the transition from the Middle to the Upper Paleolithic (for an intriguing demographic hypothesis about the rise and spread of UP culture see Powell et al., Reference Powell, Shennan and Thomas2009, Reference Powell, Shennan, Thomas, O’Brien and Shennan2010). Our focus is on the development of techniques specific to individual food resources. For example, nets can be used to catch birds, snares can be used to catch hares, weirs can be used to catch fish, and sickles can be used to harvest wild grasses. Our model of learning by doing in Section 3.5 addresses resource-specific technologies of this kind.

There is no uniform end date for the Upper Paleolithic. In the European context, archaeologists sometimes treat the Upper Paleolithic as terminating with the onset of the Holocene climate regime around 11.6 KYA (see Figure 1.1 in Chapter 1). For Europe, the period between the start of the Holocene and the arrival of agriculture is often called the Mesolithic. Because agriculture arrived at different times in different subregions of Europe, the timing of the shift from the Mesolithic to the Neolithic varies with location.

In the context of the Levant or other parts of southwest Asia, archaeologists often refer to the period between the Last Glacial Maximum and the Holocene (about 21–11.6 KYA) as the “Epi-Paleolithic,” meaning “Final Old Stone Age.” This terminology works well for southwest Asia because the Neolithic transition to agriculture roughly coincided with the arrival of the Holocene around 11.6 KYA (see Chapter 5). The distinguishing technological features of the Epi-Paleolithic in southwest Asia include microblades and, by about 14.5–13.0 KYA, increasing sedentism (see Chapter 4).

These terminological schemes are not necessarily a good fit for other regions of the world such as Africa and China. In the New World, the period known as “Archaic” is sometimes regarded as paralleling the Upper Paleolithic of the Old World. However, in many parts of the world the Upper Paleolithic never led to indigenous agriculture or to indigenous metal tools. In much of Africa, Asia, and the Americas, and all of Australia and the Arctic, an Upper Paleolithic toolkit remained in use until contact with the global economy in the last few centuries.

Direct evidence on Upper Paleolithic social institutions is scarce, so ethnographic analogies are helpful in painting a picture. There is broad agreement that prior to the rise of sedentary communities, the basic social unit was the mobile foraging band. Bands observed by anthropologists generally have about 25 members, with 7–8 healthy adults who can search for food on a full-time basis (Kelly, Reference Kelly2013a, ch. 7; French, Reference French2016, 176–179). These figures vary across societies, but they likely reflect a rough compromise between the insurance benefits from sharing food within the group and the fact that larger groups would deplete local resources too quickly, requiring more frequent residential moves (Kelly, Reference Kelly2013a, ch. 7). Such bands tend to make regular seasonal rounds within a familiar territory (see the description of traditional Shoshone society in Johnson and Earle, Reference Johnson and Earle2000, ch. 3). Several bands may come together annually at times and places of resource abundance or when stored food is available, offering opportunities to find mates, share information, trade exotic items, and engage in ritual activities.

Beyond the day-to-day benefits from food sharing within a band, the key strategy for coping with negative environmental shocks that are local and temporary is reciprocity among neighboring bands. This may involve sharing of food or access to territory (Kelly, Reference Kelly2013a, ch. 6). Storage technology is usually limited and tends to play a minor role in risk management. For negative shocks that are widespread and prolonged, the only practical response is a search for better resources in more distant territories.

Of course, small egalitarian bands of mobile foragers were not the only form of social organization in the Upper Paleolithic. Graeber and Wengrow (Reference Graeber and Wengrow2021, 86–92) cite a few examples from the period of settlements that include stone temples, elaborate burials, and massive monuments. But these findings are rare and are not the focus of this chapter.

3.2 Climate

Climate reconstruction (paleoclimatology) involves observations on polar ice, alpine glaciers, ocean and lake sediments, tree rings, calcite precipitated on the walls of caves, growth bands in corals, and the chemistry of marine shells. Polar ice cores have more than 40 distinct climate-related properties and provide data on temperature, rainfall, ice volume, and atmospheric CO₂ concentrations (EPICA, Reference Bentley, Alexander and Shennan2004; McManus, Reference McManus2004). For an introductory review of Ice Age climate history, see Woodward (Reference Woodward2014, chs. 8–9).

Figure 3.1 summarizes climate history for the last 420,000 years using data from the Vostok ice core in Antarctica. Notice that from a climatic standpoint this covers the Late Pleistocene and the Holocene. These terms should not be confused with the terms Paleolithic and Neolithic, which are defined by technology rather than climate. The data in Figure 3.1 were published by Petit et al. (Reference Petit1999) and the graph was taken from Creative Commons (2019). The top panel shows temperature variation in degrees Celsius relative to the Holocene, the middle panel shows atmospheric carbon dioxide in parts per million by volume, and the bottom panel shows atmospheric dust concentration. The latter is a proxy for aridity, so the bottom panel can be interpreted as the inverse of precipitation.

Figure 3.1. Glacial cycles from Antarctic ice core

From other Antarctic ice cores it is known that the pattern of Ice Ages depicted in Figure 3.1 extends back at least 800,000 years (EPICA, Reference Bentley, Alexander and Shennan2004). The underlying cause is a complex set of cycles involving the Earth’s orbit. Each cycle lasts about 100,000 years, with glacial periods lasting about 90,000 years separated by interglacial periods of about 10,000 years. During glacial periods, large volumes of water are locked up in ice sheets on the continents and sea levels are consequently low. In the past this has created bridges between land masses that are currently separated by ocean. The best known is Beringia, a former land bridge between Siberia and Alaska.

The three time series are not perfectly correlated, but glacial periods (or stadials) tend to have low temperatures, low CO₂, and low precipitation, while interglacial periods (interstadials) tend to have the opposite conditions. It is also evident from Figure 3.1 that there are major temperature fluctuations within each glacial period. By comparison with the Holocene, the last Ice Age involved a decrease in global mean annual temperature of 5–10 degrees Celsius, a 30% reduction in atmospheric CO₂ concentration, and a drop in sea level of 125 meters (Cronin, Reference Cronin1999).

Several climate events that will be significant for later discussions are visible in Figure 3.1. The Eemian period of 126–116 KYA was the most recent interglacial before the Holocene and was slightly warmer at its peak. The temperature fell rapidly starting around 116 KYA, recovered during 100–80 KYA, and fell rapidly again until 65 KYA. This was followed by a moderate phase during 65–30 KYA. The Last Glacial Maximum (LGM) brought a low point around 21 KYA. Temperature recovered quickly beginning around 19 KYA. A temperature drop called the Younger Dryas (YD) can be seen around 13.0–11.6 KYA. Although this reversal appears brief on the scale of Figure 3.1, it lasted over 1000 years and had major demographic repercussions (see Section 3.3). The Holocene brought warmer, wetter, and more stable conditions beginning at 11.6 KYA. During the Holocene there have been various global and regional climate fluctuations, which will be discussed as they become relevant in later chapters.

3.3 Population

There is a large archaeological literature dedicated to Pleistocene and Holocene demography. French (Reference French2016) reviews the many proxies archaeologists have used to infer population change. The approach in this section involves a method called “dates as data.” First, we provide a quick introduction to radiocarbon dating techniques.

Archaeologists frequently estimate the age of organic materials by measuring the concentration of a naturally occurring radioactive isotope of carbon called ¹⁴C relative to the more common stable isotopes ¹²C and ¹³C. Plants and animals absorb ¹⁴C from their environments while alive, and this isotope decays at a known rate after an organism dies. Older samples therefore have lower concentrations of ¹⁴C. In practice the true date for a sample is estimated with error. The extent of this uncertainty is sometimes expressed by a confidence interval, and sometimes by providing an entire probability distribution.

Radiocarbon dating has three limitations. First, it does not work for non-organic material such as stone. Second, dates are unreliable further back than about 40–50 KYA because earlier samples have too little ¹⁴C. Third, the atmospheric concentration of ¹⁴C varies over time, creating the need for calibration techniques that translate “radiocarbon years” into calendar years. All radiocarbon dates in this book are calibrated and can be treated as calendar years unless the contrary is explicitly stated. Additional information about radiocarbon dating methods is available in introductory archaeology textbooks.

The idea behind “dates as data” runs as follows. If the population in a geographic region in a given time period is large, it is likely to generate extensive material remains that can be discovered by archaeologists. Thus if a relatively large number of observed radiocarbon dates fall in a given time interval, one can infer the presence of a relatively large population during that interval, and conversely if there is a relatively small number of radiocarbon dates in a different time interval. This yields information about how the population changed over time in a region. Archaeologists sometimes use supplemental data to estimate absolute population levels.

Early research using “dates as data” placed individual estimated dates in temporal bins, and used the number of dates in each of the bins to generate a histogram indicating how population changed over time. The size of the bins was usually adjusted to reflect the degree of uncertainty in the dating process. For example, if a typical date had a 95% confidence interval of 1,000 years, the researcher might choose 1,000 years as the size of a bin. Since about 2010, many archaeologists have begun to exploit the entire probability distribution for each date, where the sum of the radiocarbon probabilities is treated as a proxy for population. Various sources of bias in this approach are discussed by French (Reference French2016), who also provides extensive references.

In this chapter we limit attention to regions for which data extend far back into the Pleistocene (Europe and Siberia), for which Upper Paleolithic technology continued into the Holocene (North and South America), or both (Australia). Other regions (including southwest Asia, Japan, and the Sahara) will be discussed when appropriate in Chapters 4 and 5. For brevity we focus on recent studies and often omit citations to earlier research on the same geographic region.

We begin with Europe during 30–13 KYA, a period that covers the growth and retreat of ice sheets associated with the Last Glacial Maximum. Tallavaara et al. (Reference Tallavaara, Luoto, Korhonen, Järvinen and Seppä2015) derive population estimates in two ways. First, they use a “dates as data” approach of the kind described above, where 3,718 calibrated median dates from 895 archaeological sites are placed in bins with 1,000-year durations. They compare these results with population estimates obtained through climate modeling in conjunction with population densities for recent hunter-gatherer societies. Both techniques indicate a large contraction in human population size and range due to the LGM beginning around 27 KYA, with a minimum around 23 KYA and a rapid recovery during 19–13 KYA. The simulations indicate an initial European population of 330,000 at 30 KYA, a minimum of 130,000 at 23 KYA, and 410,000 at 13 KYA. The simulations also show that even in the depths of the LGM, human populations extended north as far as central France, southern Germany, and the southern parts of Ukraine and European Russia, with around 36% of Europe remaining habitable. Tallavaara et al. (Reference Tallavaara, Luoto, Korhonen, Järvinen and Seppä2015, 8235) remark that “climatic conditions were crucial drivers of last glacial human population dynamics.”

Fernández-López et al. (Reference Fernández-López de Pablo, Gutiérrez-Roig, Gómez-Puche, McLaughlin, Silva and Lozano2019) have studied population dynamics in the Iberian peninsula for 16.6–8 KYA using summed probability distributions with 907 radiocarbon dates. The later part of this interval extends into the Holocene, but the Iberian population were still foragers at this time, so the results are relevant here. The authors found several statistically significant upward and downward deviations from a null exponential model. Three qualitative phases were apparent: (a) exponential growth in a period of improving climate called the Last Glacial Interstadial; (b) collapse followed by prolonged stagnation during the Younger Dryas and the first half of the Early Holocene; and (c) logistic growth in the second half of the Early Holocene. The population during phase (a) was positively correlated with temperature and precipitation. The population in phase (c) became stable as temperature and precipitation stabilized during the second half of the Early Holocene.

Population trends in Siberia have been studied by Kuzmin and Keates (Reference Kuzmin and Keates2005) for 46–12 KYA using 437 dates (these authors do not discuss calibration issues, and their dates appear to be uncalibrated). They group radiocarbon dates at individual sites into “occupation episodes” of 1,000 years each. On this basis, they conclude that population density was low until 36 KYA, increased gradually during 36–16 KYA, and increased more rapidly from 16–12 KYA. They find that Siberia was not depopulated at the Last Glacial Maximum. Their conclusions for the LGM are controversial (see Section 3.10).

Australian population trends from 50 KYA to European contact are examined by Williams (Reference Ashraf and Galor2013), who uses 4,575 calibrated radiocarbon dates from 1,750 archaeological sites. The median dates are assigned to 200-year bins. Williams finds low populations through the late Pleistocene, with a drop of about 60% associated with the Last Glacial Maximum during 21–18 KYA. A slow stepwise increase begins around the time of the Holocene, near 12 KYA, with population recovering to pre-LGM levels by about 9.6 KYA. Williams suggests that high-amplitude environmental shocks may have kept the population low during the Late Pleistocene, with climate stabilization in the Holocene allowing population growth. The Holocene shows several population fluctuations that are probably driven by climatic variation, with a peak population in the Late Holocene. Williams suggests that a founding group of 1,000–3,000 people around 50 KYA and a maximum population of 1.2 million around 500 years ago would be consistent with the data. The average annual growth rate for the entire 50,000–year interval is about 0.01% with a range between 0.07 and −0.03.

Several recent articles have focused on North American paleodemography. Two points should be kept in mind in this regard. First, recent evidence indicates that human settlement occurred in North America by 23–21 KYA, but radiocarbon data usable for paleodemography are not available until around 15 KYA. Second, although much of the western portion of the continent was occupied by mobile foraging populations until European contact, sedentary foraging was important in the Pacific Northwest and farming became important east of the Mississippi River as well as in the southwestern US. We focus on the findings for mobile foragers.

Peros et al. (Reference Peros, Munoz, Gajewski and Viau2010) use 25,198 calibrated dates for all of North America, where the median probability value for each date is assigned to a 200-year bin. The dates run from about 15 KYA until European contact. The most rapid growth rate was shortly before 14 KYA. Population was low and relatively constant until roughly 6 KYA, when it began to increase. Because plant domestication is known to have occurred in the eastern half of the continent starting about 4 KYA (see Chapter 5), results after that date are not relevant here. The authors do not report any population decline associated with the Younger Dryas.

Anderson et al. (Reference Anderson, Goodyear, Kennett and West2011) use summed radiocarbon probabilities to investigate the possible effects in North America from the cold and arid period of the Younger Dryas, which occurred during 12.9–11.6 KYA. Their 682 calibrated radiocarbon dates run from 14–11 KYA. They find a large population increase before the YD followed by a rapid drop (perhaps up to 30–50%) in the first few centuries of this period. There is a gradual rebound over the next 900 years in the later stages of the YD, suggesting adaptation to the adverse climatic conditions. This pattern holds for all regions of the continent and is supported by evidence on projectile point frequencies as well as quarry use. Anderson et al. describe similar results using radiocarbon data for Europe, Asia (excluding the Middle East), and Africa. Interestingly, the Middle East does not display a significant population drop with the onset of the Younger Dryas, a point to which we will return in Chapter 5.

Kelly et al. (Reference Kelly, Surovell, Shuman and Smith2013) explored the relationship between population and climate for foragers in the Bighorn Basin of Wyoming over the last 13,000 years, using the summed probabilities derived from several hundred radiocarbon dates. Fluctuations in population were complex, with several peaks and troughs where intervals of population growth were associated with cooler and wetter conditions. An increase in temperature of one degree Celsius led to a 45% decline in population, and a decrease in moisture of 50 mm led to a 26% decline in population after a lag of 300 years. However, the authors were unable to distinguish migratory effects from natural population growth or decline.

Zahid et al. (Reference Zahid, Robinson and Kelly2016) studied forager populations in Wyoming and Colorado during the last 15,000 years using summed probabilities for 7,900 radiocarbon samples. Because this is a larger geographic area than that studied by Kelly et al. (Reference Kelly, Surovell, Shuman and Smith2013), migration effects may be less of a concern. Following an early jump in population, Zahid et al. found an exponential trend for 13–6 KYA with an average annual growth rate of 0.04%, but with several large upward and downward deviations from this trend that lasted for centuries. Population leveled off and declined during 6–3 KYA. The authors attribute the earlier exponential trend to climate change or endogenous biological processes rather than technical innovation. They do not propose a specific explanation for the population decline following 6 KYA.

Our last stop is South America. Goldberg et al. (Reference Goldberg, Mychajliw and Hadly2016) use 5,464 radiocarbon dates from 1,147 sites to compute summed probabilities for 15–2 KYA. Dispersal by the original colonizers is not visible because the earliest dates are scattered throughout the continent. The authors find a rapid increase both in radiocarbon dates and occupied sites during 13–9 KYA, followed by oscillations around a constant mean during 9–5.5 KYA. They suggest that a peak around 11 KYA could reflect an overshoot associated with a megafaunal extinction. Growth resumed in the interval 5.5–2 KYA. The best fit was provided by a two-phase model with logistic growth until 5.5 KYA followed by an exponential growth trend thereafter. The authors note that the timing of exponential growth coincided with a shift toward agriculture across much of South America.

The main lessons we take away from this review of demographic research on five continents are that mobile foragers with Upper Paleolithic technology tended to converge on equilibrium populations for the environments they inhabited, and that there were often very strong climatic effects on population size. Some periods of slow population growth appear to be consistent with slow technological progress in a natural environment where resources were fluctuating around a constant mean. But steady exponential growth was certainly not the norm in the Pleistocene, and probably not in the Holocene either. Even when an exponential trend was sustained for a substantial period, large and statistically significant deviations were usually superimposed on the trend, and regional population contractions were not unusual.

3.4 Technology

If one wanted to argue that the Upper Paleolithic was a technically progressive period, one could. Direct archaeological evidence reveals considerable evolution in the design of stone tools, including the development of microblades. New hunting methods such as the spear thrower and bow and arrow were also developed. Two other empirical patterns are evident. First, people began to rely less on hunting large mammals and more on small prey like hares and birds, along with a wide array of plant foods. This process is known as the broad spectrum revolution. Second, people colonized harsh environments such as the Arctic, which required new hunting technologies as well as innovations with respect to clothing, shelter, and transport.

On the other hand, one could also argue that the Upper Paleolithic was technically stagnant. Tens of thousands of years elapsed between the beginnings of blade technology around 80 KYA and the beginnings of agriculture around 12 KYA. Precursors of modern society like metallurgy and writing are even more recent. Strikingly, much of the world continued to rely on stone-age foraging techniques until European contact in the last 500 years. This suggests that Upper Paleolithic technology was a stable way of life, and that unusual circumstances were needed to push societies toward other economic systems.

In our view, a theoretical framework that can reconcile these two perspectives will need to have the following features:

1. (a) Technology should be endogenous, with innovation being slow, sporadic, and frequently absent for long periods of time.

2. (b) Population should be endogenous, with Malthusian dynamics.

3. (c) Nature should be exogenous, with changes in climate, geography, and ecosystems serving as triggers for changes in technology and population.

4. (d) The theory should be consistent with the climate history in Section 3.2.

5. (e) The theory should be consistent with the population history in Section 3.3.

6. (f) The theory should account for the broad spectrum revolution and the expansion of human populations into harsh environments.

We will construct a model along these lines in Sections 3.5–3.9.

In this section, we review several previous efforts to endogenize technology. We begin with a model of economic growth by Kremer (Reference Kremer1993). Although this is about three decades old, it is worth discussing because it is a pioneering application of growth theory to prehistory and because it crystallizes key issues in a relatively simple way. In fairness we should say that Kremer’s central concern is the transition to modern economic growth, not economic prehistory, but he does claim that his model is relevant for pre-agricultural societies. Other efforts to construct comprehensive theories of economic growth having relevance for prehistory, such as Galor (Reference Galor, Aghion and Durlauf2005, Reference Bar-Yosef2011), will be discussed in Chapter 12.

Kremer’s starting point is a sequence of estimates for world population beginning a million years ago and ending with the twentieth century. The early figures in the series are crude and should not be taken seriously. Much more sophisticated estimates are available today (see Goldewijk et al., Reference Goldewijk, Beusen and Janssen2010), and we need not linger over measurement issues here. The more interesting questions are theoretical in nature.

Kremer’s model rests on two causal relationships. First, he argues that the size of the world population is determined by technological productivity in a Malthusian manner. His model of population is a simplified version of the one in Chapter 2, where short-run adjustment processes are ignored and population is assumed always to equal its long-run equilibrium level given the current productivity level. Equivalently, the average product of labor is always equal to the fixed long-run standard of living y* from Chapter 2.

Second, he argues that the rate of productivity growth is proportional to the level of world population. Note that this claim involves a linkage between a growth rate and a level, in contrast to the Malthusian part of the model that provides a linkage between the level of population and level of productivity. Kremer’s argument for this second causal relationship is that a larger population implies a larger number of independent inventors, and therefore more useful inventions per unit of time. Although Kremer studies several ways of specifying the determinants of productivity growth, we focus on his basic linear model. The idea that a higher population level yields a higher productivity growth rate is common among economists (see Becker et al., Reference Becker, Glaeser and Murphy1999; Galor and Weil, Reference Galor and Weil2000; Jones, Reference Jones2001; Galor, Reference Galor, Aghion and Durlauf2005; Olsson and Hibbs, Reference Olsson and Hibbs2005; Aiyar et al., Reference Aiyar, Dalgaard and Moav2008).

Putting together the causal relationships from the two preceding paragraphs gives a simple prediction: the current population growth rate will be proportional to the current population level. Kremer argues (plausibly) that this provides a good fit to the data over the last several thousand years, except in the twentieth century when the Industrial Revolution and the associated demographic transition became important. For our purposes, the main points are that the model predicts continuous growth in both population and productivity, and that the growth process will be faster than exponential.

This is clearly inconsistent with our criteria for a satisfactory theory of Upper Paleolithic technology. First, it is incompatible with the fact that technical progress was usually sporadic or absent. Second, the prediction about population growth conflicts with findings that regional populations often exhibited stasis, cycles, or collapse. And finally, the model ignores the powerful demographic effects of the natural environment.

As Richerson et al. (Reference Richerson, Boyd and Bettinger2001) have pointed out, when tiny rates of technical change are compounded for tens of millennia, they have enormous consequences for population, the standard of living, or both. Even a small founding population could bring all of Asia to modern hunter-gatherer population densities within a few thousand years. Over the 50 millennia in which humans have occupied Australia, a tiny rate of population growth can take a founding population from the low thousands to the low billions (Williams, Reference Ashraf and Galor2013). This is true even for a small constant rate of exponential growth. If the growth rate is an increasing function of the population level, as in Kremer (Reference Kremer1993), the results are still more explosive. Because there is no sign of sustained population growth (or improving living standards, or greater social complexity) before the first known sedentary communities at about 15 KYA, the equilibrium growth rate must have been zero almost all of the time.

Another problem with the Kremer framework requires comment. In a Malthusian setting it is reasonable to hypothesize that observed population growth could be explained by unobserved technological progress. But the Kremer model ignores the possibility that population growth could have been stimulated by climate improvements with technology held constant (e.g., the recovery from the Last Glacial Maximum, or the shift from the Pleistocene to the Holocene). It also ignores the fact that the colonization of new land masses can yield world population growth with a constant technology. For inferences of unobserved technological progress to be persuasive, one must argue that population grew within a fixed geographic region with fixed natural resources, and that this increase was not driven by migration from external sources.

Archaeology and anthropology also have a literature linking population size to innovation, based on social learning and cultural transmission (Shennan, Reference Shennan2001; Bentley et al., Reference Bentley, Alexander and Shennan2004; Henrich, Reference Henrich2004; Powell et al., Reference Powell, Shennan and Thomas2009, Reference Powell, Shennan, Thomas, O’Brien and Shennan2010; Boyd et al., Reference Boyd, Richerson and Henrich2011; Richerson, Reference Richerson2013; Crema and Lake, Reference Crema and Lake2015). The usual (although not invariable) conclusion is that a higher population level leads to more rapid social learning and technical change. Eventually this yields greater cultural complexity, better adaptation to the environment, and/or enhanced biological fitness. These ideas have been applied in discussions of the Upper Paleolithic (Shennan, Reference Shennan2001; Powell et al., Reference Powell, Shennan and Thomas2009, Reference Powell, Shennan, Thomas, O’Brien and Shennan2010).

The archaeological literature on social learning is consistent with the part of the Kremer model where the innovation rate is an increasing function of the population level. However, it omits the other part of the Kremer model where population is endogenized through Malthusian dynamics. If we were to add a Malthusian population equation, we would be back to the same positive feedback loop where technology and population can both grow explosively. We are not aware of any formal modeling in archaeology that tackles this issue, although the potential for positive feedback has been noted (Shennan, Reference Shennan2001, 14; Powell et al., Reference Powell, Shennan and Thomas2009, 1301; Reference Powell, Shennan, Thomas, O’Brien and Shennan2010, 145). This problem could be mitigated by a concave relationship between the population level and the innovation rate, as is usually found in social learning models, rather than the linear relationship used in the simplest version of the Kremer model.

Anthropological data have been used to test the claim that technical innovation is an increasing function of population size. Some studies support this hypothesized link while others do not (Collard et al., Reference Collard, Buchanan and O’Brien2013). We are not surprised by these mixed results. Virtually all such studies use (small) samples of recently observed foraging or farming societies. The hypothesis being tested treats population as exogenous and technology as endogenous. But on time scales where technology is endogenous (and perhaps at or near an equilibrium level), population will almost certainly also be endogenous for Malthusian reasons (and also at or near an equilibrium level). Regressing technology on population thus involves regressing one endogeneous variable on another, resulting in simultaneity bias. In a model where both technology and population are endogenous, the exogenous variables are those determined by nature (climate, geography, and ecosystems).

Our own story about technical change in the Upper Paleolithic is closely related to other theories about social learning that have been influential among archaeologists. For analytic convenience, our formal model adopts a very simple approach to social learning where learning is initiated through mistakes in copying, which can be regarded as a form of accidental experimentation. Directional learning occurs through comparisons among the accidentally generated techniques for harvesting a particular resource. However, our modeling framework could be extended to accommodate deliberate experimentation.

It is useful to compare our analysis with that in Henrich’s (Reference Henrich2016) highly influential book The Secret of Our Success. Henrich’s treatment of collective learning is much more detailed than ours. It stresses the innate ability of humans to copy from each other and the importance of social norms for the transfer of new knowledge within a society and to future generations. We accept Henrich’s richer view of collective learning and see it as complementary to our simplified account. However, we differ from Henrich in that our analysis is motivated by the observation that technological progress was dramatically uneven during the Upper Paleolithic. Why were there periods of technical improvement separated by long periods of apparent stagnation? Henrich does not address this question but elements of his analysis suggest a pathway to an answer.

Let social norms be the primary mechanism for knowledge transfer, as Henrich suggests. Strict adherence to social norms related to technology could theoretically shut down both experimentation and knowledge transfer. Under what conditions will norms be flexible enough to allow for technological change? Giuliano and Nunn (2021) find that tradition and cultural persistence are weaker in less stable environments. We can link this idea with Henrich’s (Reference Henrich2016, 301) view that climate fluctuations are a source of intensified pressure for social learning. Perhaps climate fluctuations simultaneously loosened the restrictive grip of social norms on new technologies. Our simpler approach yields a relationship between climate and technical innovation that is more easily tested, but it could potentially be expanded to include social norms as a mediating factor.

A brief comment on our relationship to the framework of Boserup (Reference Boserup1965) is also in order. Boserup famously argued that population growth puts downward pressure on living standards, which stimulates technological innovation (often called “intensification” by archaeologists and anthropologists). If population growth is viewed as an exogenous variable, we would reject this approach. However, Wood (Reference Wood1998) provides a synthesis of Boserupian and Malthusian ideas where population is treated as endogenous.

Even so, our theory diverges from Boserup. We do not treat falling standards of living (whether caused by a movement along a declining average product curve, or by a downward shift of the curve) as a causal stimulus for innovation. We do sometimes treat a negative climate shock as the trigger for technical change (see our models of agriculture and manufacturing in Chapters 5 and 10, respectively), but our causal mechanisms differ from those of Boserup. Perhaps more importantly here, we also believe positive climate shocks, which raise living standards in the short run, were a key trigger for innovation in the Upper Paleolithic. Details will be given in Sections 3.9 and 3.10. We make similar arguments regarding positive climate change as a stimulus for sedentism in Chapter 4.

We suggest the following way of thinking about Upper Paleolithic technology. Nature provides many potential food resources that could be exploited if a society had suitable technical knowledge. In a static environment, foragers become very competent at exploiting some subset of these resources, but face long-run stagnation because (a) there is an upper bound on productivity for each resource, (b) latent resources remain unexploited due to the limitations of prevailing knowledge, and (c) knowledge cannot improve for resources that are never used.

To escape from such a trap, a foraging society must be exposed to shocks from nature. For example, an improved climate tends to increase population in the long run. If this Malthusian effect is large enough it may become attractive to exploit latent resources. Once this occurs, learning by doing will generate improvements in the techniques used to harvest, process, or store these new resources. If such knowledge gains are irreversible, a series of positive and negative climate shocks can yield a ratchet effect in technological capabilities, with occasional episodes of population growth and technical innovation.

We develop these ideas formally in Sections 3.5–3.9. Our theory can explain the prolonged periods of economic stagnation found in many pre-agricultural societies, both in the distant past and among recent hunter-gatherers. In Section 3.10 we suggest that our theory can also account for some macro-level patterns in the Upper Paleolithic, including the broad spectrum revolution and the colonization of extreme environments.

3.5 Learning by Doing

The next several sections develop a formal model of Upper Paleolithic economic growth that conforms to the requirements (a)–(f) identified in Section 3.4. We begin with a model of technical innovation along lines similar to those of the social learning models used in archaeology and anthropology, where children learn techniques from adults. Our approach makes a number of simplifications. We assume children can imitate any adult, without special preferences for biological parents or prestigious role models; we assume children can compare techniques among themselves and adopt the best ones available, so there is no variation among the members of any given generation in the techniques used; and we study a limiting case where the number of imitation events goes to infinity while the mutation rate goes to zero. This yields directional learning over multiple generations, where techniques may improve or remain static but never deteriorate.

Consider a list of natural resources $\mathrm{r}=1..\mathrm{R}$. These could include blueberries, rabbits, and so forth. Each resource can be converted into food through the use of labor. We treat food as a homogeneous output, ignoring distinctions among calories, protein, vitamins, and so on. The production function for converting resource r into food is

${\mathrm{F}}_{\mathrm{r}}\left({\mathrm{a}}_{\mathrm{r}},{\mathrm{k}}_{\mathrm{r}},{\mathrm{n}}_{\mathrm{r}}\right)={\mathrm{a}}_{\mathrm{r}}{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right){\mathrm{f}}_{\mathrm{r}}\left({\mathrm{n}}_{\mathrm{r}}\right)$(3.1)

where a_r is the abundance of resource r (regarded as a flow provided by nature in a given time period); k_r is the technique used to harvest it; and n_r is the labor used for harvesting. The abundance of each resource is determined by climate, geography, and ecosystems. However, climate is the only determinant of resource abundance that will be varied in this chapter, so for convenience we often refer to $\mathrm{a}=\left({\mathrm{a}}_1..{\mathrm{a}}_{\mathrm{R}}\right)>0$ as “climate.”

For fixed a_r and k_r, each production function in (3.1) has the same general shape as the production function from Chapter 2. We impose the following technical conditions.

Each function has a positive and finite marginal product ${\mathrm{f}}_{\mathrm{r}}^{\prime }\left(0\right)$ when labor input is zero. We will say that resource r is active if ${\mathrm{n}}_{\mathrm{r}}>0$ and latent if ${\mathrm{n}}_{\mathrm{r}}=0$. The finiteness of the marginal products at zero will be important when we discuss conditions under which a latent resource becomes active. As in Chapter 2, the function ${\mathrm{f}}_{\mathrm{r}}$ exhibits diminishing marginal productivity (the second derivative is negative).

Dropping the r subscript momentarily, one example of a function satisfying the requirements in A3.1 is $\mathrm{f}\left(\mathrm{n}\right)=1–{\mathrm{e}}^{-\mathrm{mn}}$, where the marginal product at zero is ${\mathrm{f}}^{\prime }\left(0\right)=\mathrm{m}>0$. This functional form is consistent with an interpretation where ${\mathrm{a}}_{\mathrm{r}}$ is the resource flow in nature, ${\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right)$ is the fraction of ${\mathrm{a}}_{\mathrm{r}}$ that is potentially accessible with the current technique ${\mathrm{k}}_{\mathrm{r}}$, and ${\mathrm{f}}_{\mathrm{r}}\left({\mathrm{n}}_{\mathrm{r}}\right)$ is the fraction of ${\mathrm{a}}_{\mathrm{r}}{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right)$ that is actually harvested.

Techniques are modeled as binary strings of uniform length Q so ${\mathrm{k}}_{\mathrm{r}}=\left({\mathrm{k}}_{\mathrm{r}1}..{\mathrm{k}}_{\mathrm{rQ}}\right)\in {\left\{0,1\right\}}^{\mathrm{Q}}$. Let ${{\mathrm{k}}_{\mathrm{r}}}^{\ast }$ be the best method for transforming resource r into food: $0<{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right)<{\mathrm{g}}_{\mathrm{r}}\left({{\mathrm{k}}_{\mathrm{r}}}^{\ast }\right)$ for all ${\mathrm{k}}_{\mathrm{r}}\ne {{\mathrm{k}}_{\mathrm{r}}}^{\ast }$. The function ${\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right)$ is increasing in the number of digits of ${\mathrm{k}}_{\mathrm{r}}$ that match ${{\mathrm{k}}_{\mathrm{r}}}^{\ast }$. It does not matter which digits are matched. The number of matching digits indexes the level of technical knowledge regarding resource r.

Q is assumed to be large enough that an exhaustive search for the ideal strings is impractical. Instead, each generation inherits a repertoire of techniques from its parental generation. The repertoire available to the adults of generation t is ${\mathrm{K}}^{\mathrm{t}}=\left\{{\mathrm{k}}_1^{\mathrm{t}}..{\mathrm{k}}_{\mathrm{R}}^{\mathrm{t}}\right\}$. This array summarizes the current state of technological knowledge for all natural resources, where this knowledge is freely available to all adults of generation t.

The repertoire ${\mathrm{K}}^{\mathrm{t}+1}$ for the next generation is derived as follows. Let there be ${\mathrm{N}}^{\mathrm{t}}$ adults in period t. Each adult is endowed with one unit of labor time. The society’s labor allocation is ${\mathrm{n}}^{\mathrm{t}}=\left({\mathrm{n}}_1^{\mathrm{t}}..{\mathrm{n}}_{\mathrm{R}}^{\mathrm{t}}\right)$ where ${\sum }_{\mathrm{r}}{\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}={\mathrm{N}}^{\mathrm{t}}$. All adults use the same time allocation, so each person devotes ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}/{\mathrm{N}}^{\mathrm{t}}$ time units to resource r.

A typical child in the period t has X opportunities to watch parents or other adults harvest resources, where X is a large number. ${\mathrm{Xn}}_{\mathrm{r}}^{\mathrm{t}}/{\mathrm{N}}^{\mathrm{t}}$ of these observations will involve resource r (there are no observations for latent resources with ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}=0$). Every time a child i sees the exploitation of resource r, the string ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}}=\left({\mathrm{k}}_{\mathrm{r}1}^{\mathrm{t}}..{\mathrm{k}}_{\mathrm{rQ}}^{\mathrm{t}}\right)$ is copied. For each of the Q positions on the string, with probability p an error is made in copying the current digit, and with probability 1-p the digit is copied accurately. Mutations are independent across loci, observations, and agents. A copy is denoted by ${\mathrm{k}}_{\mathrm{rix}}^{\mathrm{t}}$ where x indexes observations.

Whenever a copy of ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}}$ is made, child i uses a negligible amount of labor ε $>0$ to determine the marginal product ${\mathrm{a}}_{\mathrm{r}}^{\mathrm{t}}{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{rix}}^{\mathrm{t}}\right){\mathrm{f}}_{\mathrm{r}}^{\prime }\left(0\right)$. If the new copy achieves a higher value than the best previous copy, the child retains the new copy and discards any earlier copy. Otherwise, the best previous copy is retained and the new copy is discarded. At the end of this learning process, child i has a best string for resource r, which we denote by ${\mathrm{k}}_{\mathrm{ri}}^{\mathrm{t}+1}$.

The number of children who survive to adulthood is ${\mathrm{N}}^{\mathrm{t}+1}$. When the parents from period t die and their children become adults, these new adults compare their strings for resource r and coordinate on (one of) the best available. The result for resource r is ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}+1}=\text{argmax}\left\{{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{ri}}^{\mathrm{t}+1}\right)\text{for}\mathrm{i}=1..{\mathrm{N}}^{\mathrm{t}+1}\right\}$.

This process gives updated strings for the resources r having ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}>0$. In most of this chapter, we will assume passive updating for latent resources: that is, ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}=0$ implies ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}+1}={\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}}$. We discuss alternative productivity assumptions for latent resources at the end of Section 3.9. The entire updated repertoire ${\mathrm{K}}^{\mathrm{t}+1}=\left\{{\mathrm{k}}_1^{\mathrm{t}+1}..{\mathrm{k}}_{\mathrm{R}}^{\mathrm{t}+1}\right\}$ is freely available to all adults of generation $\mathrm{t}+1$.

Let ${\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}}$ be the number of correct digits in the string ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}}\in {\mathrm{K}}^{\mathrm{t}}$. To model technical progress for active resources, we need to know the probability distribution over ${\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}+1}$.

Proposition 3.1

(learning by doing).

Assume ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}>0$ so resource r is active in period t. Define ${\mathrm{\rho }}^{\mathrm{t}}\equiv {\mathrm{N}}^{\mathrm{t}+1}/{\mathrm{N}}^{\mathrm{t}}$ to be the number of children who survive to adulthood in period $\mathrm{t}+1$ per adult in period t. Let the number of observations per child (X) approach infinity and the mutation probability (p) approach zero with $\mathrm{\lambda }\equiv \mathrm{Xp}>0$ held constant. In the limit,

$\begin{array}{lcll}\text{Prob}\left({\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}+1}={\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}}\right)& =& exp\left[-{\mathrm{\lambda n}}_{\mathrm{r}}^{\mathrm{t}}{\mathrm{\rho }}^{\mathrm{t}}\left(\mathrm{Q}-{\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}}\right)\right]& \text{and}\\ \text{Prob}\left({\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}+1}={\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}}+1\right)& =& 1-exp\left[-{\mathrm{\lambda n}}_{\mathrm{r}}^{\mathrm{t}}{\mathrm{\rho }}^{\mathrm{t}}\left(\mathrm{Q}-{\mathrm{q}}_{\mathrm{r}}^{\mathrm{t}}\right)\right]& \end{array}$

All other transition probabilities for q_r^t+1 go to zero in the limit.

Proposition 3.1 shows that technical evolution for an active resource is directional. Either the number of correct digits in the string is unchanged and so is productivity, or there is a one-step improvement in the string for resource r with an associated productivity gain.

The intuition behind this result is not hard to grasp. In order to obtain technical regress for the string for resource r, it would be necessary for every observation of ${\mathrm{k}}_{\mathrm{r}}^{\mathrm{t}}$ to be worse than the status quo. In the limit as the number of observations goes to infinity, the probability of this outcome goes to zero. But technical progress only requires that at least one observation be better than the status quo. The probability of this outcome does not go to zero (though the probability of multiple productivity steps does). Proposition 3.1 does not require a large population. Each child only needs to have many opportunities to observe adult behavior, which could be true even in a small foraging band.

The probability of technical progress is an increasing function of the mutation rate, the labor input for the resource, the population growth rate, and the current distance from the ideal string. Other things equal, progress is more likely for resources involving larger labor inputs because children make more observations for these resources. It is also more likely when the population growth rate is higher because then there are more children who can compare notes on the copies they have made. Finally, progress is more likely when the distance from the ideal string is greater (the current technique is worse), because then there are more opportunities to learn something useful. One corollary of Proposition 3.1 is that if the transition probabilities are continuous at ${\mathrm{n}}_{\mathrm{r}}^{\mathrm{t}}=0$, productivity for a latent resource is constant with certainty. This provides a theoretical rationale for the idea that strings for such resources are passively updated.

Notice that if we fix the fraction of total time devoted to each resource, a higher population N implies a higher labor input n_r for every active resource r, and therefore an increased probability of technical progress for every resource (as long as its current string remains imperfect). This is consistent with the idea that a larger population leads to more innovation. Moreover, the relationship between population and innovation is concave. A less standard result is that a higher rate of population growth (ρ) also stimulates technical progress. However, when a society is close to a Malthusian equilibrium this parameter is close to unity, so the population growth effect may not play a significant role in practice. We often refer to processes of technical change like those modeled in this section as learning by doing. The everyday sense of this phrase is that an individual person can become better at a task by doing it repeatedly, perhaps refining methods through trial and error or improving skills through practice. Our use of the phrase instead refers to social learning in which one generation’s techniques are an improvement on the techniques of the preceding generation. The fundamental idea is that one generation must actively use a resource or technique in order for the next generation to attain higher productivity for that particular resource or technique.

3.6 Short-Run Equilibrium

The next three sections develop increasingly inclusive equilibrium concepts. We start in the present section with the short run, in which time allocation is endogenous and population, technology, and climate are all exogenous. Section 3.7 goes to the long run, where we treat population as endogenous using Malthusian dynamics. Section 3.8 goes to the very long run, where we also treat technology as endogenous through the model of learning by doing developed in Section 3.5. At this point, the only remaining exogenous variable will be climate. Section 3.9 will show how climate change can induce changes in all of the other variables. The relationships among the short, long, and very long runs are summarized in Table 3.1.

Table 3.1. Endogenous and exogenous variables

At the moment we are only concerned with the allocation of time across various resources, and for this purpose it is useful to adopt more compact notation. Let ${\mathrm{A}}_{\mathrm{r}}\left({\mathrm{a}}_{\mathrm{r}},{\mathrm{k}}_{\mathrm{r}}\right)\equiv {\mathrm{a}}_{\mathrm{r}}{\mathrm{g}}_{\mathrm{r}}\left({\mathrm{k}}_{\mathrm{r}}\right)$ be the productivity of resource r when its natural abundance is ${\mathrm{a}}_{\mathrm{r}}$ and technique ${\mathrm{k}}_{\mathrm{r}}$ is used. As mentioned earlier, ${\mathrm{A}}_{\mathrm{r}}\left({\mathrm{a}}_{\mathrm{r}},{\mathrm{k}}_{\mathrm{r}}\right)$ can be interpreted as the amount of resource r that is potentially accessible for harvesting under current technology. The corresponding vector of productivities will be written A(a, K). When a and K are being held constant, as in this section, we drop these arguments and write this vector as $\mathrm{A}=\left({\mathrm{A}}_1..{\mathrm{A}}_{\mathrm{R}}\right)$.

In keeping with our general premise that a small group with frequently interacting members can resolve coordination, distribution, and free rider problems (see Chapter 1), we assume a foraging band allocates the total adult labor supply N across the individual resources to maximize total food, which is shared equally. The resulting time allocation will be called a short-run equilibrium (SRE).

Definition 3.1 A short-run equilibrium for the productivities $\mathrm{A}>0$ and population $\mathrm{N}\ge 0$ is a labor allocation $\mathrm{n}\left(\mathrm{A},\mathrm{N}\right)=\left[{\mathrm{n}}_1\left(\mathrm{A},\mathrm{N}\right)..{\mathrm{n}}_{\mathrm{R}}\left(\mathrm{A},\mathrm{N}\right)\right]$ that achieves

$\mathrm{H}\left(\mathrm{A},,,\mathrm{N}\right)\equiv max{\Sigma }_{\mathrm{r}}{\mathrm{A}}_{\mathrm{r}}{\mathrm{f}}_{\mathrm{r}}\left({\mathrm{n}}_{\mathrm{r}}\right)\text{subject}\text{to}{\mathrm{n}}_{\mathrm{r}}\ge 0\text{for}\text{all}\mathrm{r}\text{and}{\Sigma }_{\mathrm{r}}{\mathrm{n}}_{\mathrm{r}}=\mathrm{N}.$(3.2)

The next proposition characterizes the optimal time allocation.

The scale effect in part (a) says that if population increases, with all productivities held constant, more labor time will be devoted to every active resource. The substitution effect in part (b) says that if productivity increases for a particular active resource r, with all other productivities and population held constant, more labor time will be devoted to r and less time will be devoted to all other active resources. The remaining parts say that the food per person y(A, N) falls when population rises, that food per person goes to zero as population goes to infinity, and that food per person has a finite upper bound equal to the largest marginal product among all of the resources, evaluated at zero labor time.

The model of this section shows why diet breadth might expand when population grows. Consider Figure 3.2, where the group population N is on the horizontal axis and the marginal products for two resources $\left(\mathrm{r}=1,2\right)$ are on the vertical axis. The graph is drawn such that the marginal product for resource 1 always exceeds the marginal product for resource 2 when the same amount of labor time is devoted to each. In the language of diet breadth models (Bettinger, Reference Bettinger2009), resource 1 is unambiguously more highly ranked.

Figure 3.2. Population and marginal products for two resources

If $\mathrm{N}<{\mathrm{N}}^{\mathrm{a}}$ in Figure 3.2, it is optimal to allocate all labor to resource 1 because then ${\mathrm{A}}_1{\mathrm{f}}_1^{\prime }\left(\mathrm{N}\right)>{\mathrm{A}}_2{\mathrm{f}}_2^{\prime }\left(0\right)$ and any attempt to transfer labor from resource 1 to resource 2 would result in less total food. This is what an economist would call a corner solution (or a boundary solution). However, when $\mathrm{N}>{\mathrm{N}}^{\mathrm{a}}$ this is no longer the case. If all labor were to be used for resource 1, we would have ${\mathrm{A}}_1{\mathrm{f}}_1^{\prime }\left(\mathrm{N}\right)<{\mathrm{A}}_2{\mathrm{f}}_2^{\prime }\left(0\right)$ and the marginal product for resource 1 would be lower than the marginal product for resource 2. This implies that if a small amount of labor were transferred from 1 to 2, the gain in food output from resource 2 would outweigh the loss in output from resource 1. In this case optimality requires that positive labor time be devoted to each resource (what an economist would call an interior solution). Furthermore, optimal time allocation requires equal marginal products; that is, ${\mathrm{A}}_1{\mathrm{f}}_1^{\prime }\left({\mathrm{n}}_1\right)={\mathrm{A}}_2{\mathrm{f}}_2^{\prime }\left({\mathrm{n}}_2\right)$ where ${\mathrm{n}}_1>0$, ${\mathrm{n}}_2>0$, and ${\mathrm{n}}_1+{\mathrm{n}}_2=\mathrm{N}$.

It may seem counterintuitive that even though one resource clearly dominates in Figure 3.2, a group might want to use both. But because resource 1 has a diminishing marginal product, if enough labor is used to harvest this resource, its marginal product will eventually become so low that it is worthwhile to exploit resource 2 as well. This provides an elementary theory about the linkage between population and diet breadth: as population rises, a group eventually shifts from exploiting one resource to exploiting two. It should be clear that the same logic extends to three or more resources. We will return to this point in Section 3.10 in connection with the broad spectrum revolution.

3.7 Long-Run Equilibrium

This section extends the theory of Chapter 2 to determine the population N. We assume that the conditions for short-run equilibrium from Section 3.6 are satisfied in each time period, and we continue to treat technology and climate as exogenous. Accordingly, we continue to use the productivity notation $\mathrm{A}=\left({\mathrm{A}}_1..{\mathrm{A}}_{\mathrm{R}}\right).$ For simplicity we consider a closed population where migration issues do not arise.

Let time periods be indexed by $\mathrm{t}=0,1..$, where a time period is the length of a human generation (about 20 years). We assume adults live for one period. During this time, they produce food, have children, and then die. When the adults from period t die, they are replaced by their surviving children, who become the adults for period $\mathrm{t}+1$.

Suppose a typical adult has $\mathrm{\rho }\left(\mathrm{y}\right)$ children who survive to adulthood, where y is the adult’s food income. We have already assumed that food is shared equally among adults, so y is the same for everyone. In a simple interpretation, each adult has the same number of surviving children. However, if there is random variation across adults in numbers of offspring, we can also think of $\mathrm{\rho }\left(\mathrm{y}\right)$ as the average number of children as long as there are enough adults that the random variation across individuals washes out. Integer problems involving the number of children are ignored. As in Chapter 2, $\mathrm{\rho }\left(\mathrm{y}\right)$ is a continuous and increasing function with a unique food income ${\mathrm{y}}^{\ast }>0$ such that $\mathrm{\rho }\left({\mathrm{y}}^{\ast }\right)=1$.

The population evolves according to

${\mathrm{N}}^{\mathrm{t}+1}=\mathrm{\rho }\left[\mathrm{y}\left({\mathrm{A}}^{\mathrm{t}},{\mathrm{N}}^{\mathrm{t}}\right)\right]{\mathrm{N}}^{\mathrm{t}}$(3.3)

where $\mathrm{y}\left(\mathrm{A},\mathrm{N}\right)$ is food per adult as defined in Proposition 3.2(c). We impose the following technical restriction to simplify population dynamics.

This says that if the initial population ${\mathrm{N}}^0$ yields an income above the equilibrium level ${\mathrm{y}}^{\ast }$, so population tends to rise, it will rise steadily along the path leading to the equilibrium at ${\mathrm{N}}^{\ast }$. By the same token, if ${\mathrm{N}}^0$ yields an income below ${\mathrm{y}}^{\ast }$, so that population tends to fall, it will fall steadily along the path leading to ${\mathrm{N}}^{\ast }$. Thus MPA rules out oscillations around ${\mathrm{N}}^{\ast }$ along the adjustment path. It holds when the direct positive effect of ${\mathrm{N}}^{\mathrm{t}}$ on ${\mathrm{N}}^{\mathrm{t}+1}$ in (3.3) outweighs the indirect negative effect of ${\mathrm{N}}^{\mathrm{t}}$ through $\mathrm{y}\left({\mathrm{A}}^{\mathrm{t}},{\mathrm{N}}^{\mathrm{t}}\right)$. Note that the latter indirect effect is negative because as N rises, y falls, and this dampens the rate of growth $\mathrm{\rho }\left(\mathrm{y}\right)$.

We can now give a formal definition of long-run equilibrium. The central idea is that we need a constant population $\mathrm{N}\left(\mathrm{A}\right)$ that solves (3.3) for a given productivity vector A. We also require that the time allocation for this population satisfy the requirements for a short-run equilibrium.

If the population is zero in period t, it must also be zero in period $\mathrm{t}+1$, and this satisfies D3.2. Hence $\mathrm{N}=0$ is always an LRE. The more interesting question is whether there is a non-null LRE having $\mathrm{N}\left(\mathrm{A}\right)>0$. If such an LRE exists, it must have $\mathrm{y}\left[\mathrm{A},\mathrm{N}\left(\mathrm{A}\right)\right]\equiv {\mathrm{y}}^{\ast }$ in order to have $\mathrm{\rho }\left({\mathrm{y}}^{\ast }\right)=1$. Proposition 3.3 determines when a non-null LRE can occur.

Part (a) shows that if at least one resource is productive enough, it is possible to support a positive population in the long run. Part (b) shows that it is impossible to have a positive population in the long run if no resource is productive enough to achieve food income ${\mathrm{y}}^{\ast }$.

Remark:

For brevity we omit a formal definition of stability for LRE. When Proposition 3.3(a) is relevant, the fact that y(A, N) is decreasing in N, plus monotone population adjustment from A3.2, guarantees that the non-null LRE with $\mathrm{N}\left(\mathrm{A}\right)>0$ is globally asymptotically stable and the null LRE with $\mathrm{N}=0$ is unstable. When Proposition 3.3(b) is relevant, the null LRE with $\mathrm{N}=0$ is globally asymptotically stable.

In every non-null LRE, per capita food consumption is ${\mathrm{y}}^{\ast }$. As the vector A varies with climate or technology, the long-run population $\mathrm{N}\left(\mathrm{A}\right)$ will vary but the long-run living standard will not. This is the Malthusian feature of the model: an improvement in climate or technology increases food consumption per person in the short run but leads to a larger population with unchanged food per person in the long run.

Figure 3.3 illustrates the results from Proposition 3.3. When some natural resources are sufficiently abundant and/or technology is sufficiently productive, the curve $\mathrm{y}\left(\mathrm{A},\mathrm{N}\right)$ has a vertical intercept above ${\mathrm{y}}^{\ast }$ and a positive LRE population $\mathrm{N}\left(\mathrm{A}\right)$ exists. The arrows show directions of population adjustment and indicate the stability of $\mathrm{N}\left(\mathrm{A}\right)$. However, if resources are too scarce and/or technology is too unproductive, the vertical intercept for $\mathrm{y}\left(\mathrm{A},\mathrm{N}\right)$ is at or below ${\mathrm{y}}^{\ast }$ and the only LRE is the one with zero population.

Figure 3.3. Population levels in long-run equilibrium

A concept that will be useful later is the habitation boundary. Suppose there is just one food resource and just one technique for obtaining it. The productivity A is then a scalar rather than a vector. In general, there is a boundary productivity level ${\mathrm{A}}^{\mathrm{b}}$ with the feature that $\mathrm{A}\le {\mathrm{A}}^{\mathrm{b}}$ implies a zero long run population and ${\mathrm{A}}^{\mathrm{b}}<\mathrm{A}$ implies a positive long run population. For a given technology, ${\mathrm{A}}^{\mathrm{b}}$ determines a boundary between a geographic region where natural resources are sufficient to support a positive human population and an adjacent region where they are not. For example, there could be a latitude in northern Asia beyond which foragers cannot survive because natural resources are too scarce. The habitable geographic region will expand when technology improves, a point to which we return in Section 3.10 when discussing the colonization of harsh environments.

3.8 Very-Long-Run Equilibrium

The models in Sections 3.6 and 3.7 showed how time allocation and population could be treated as endogenous variables. Here we continue down this path by making technology endogenous as well, leaving only climate as a source of exogenous shocks.

We begin with an informal interpretation of the model developed thus far. Let ${\mathrm{N}}^{\mathrm{t}}$ be the number of adults at the start of period t and let ${\mathrm{K}}^{\mathrm{t}}$ be their technology. The adults observe the climate ${\mathrm{a}}^{\mathrm{t}}=\left({\mathrm{a}}_1^{\mathrm{t}}..{\mathrm{a}}_{\mathrm{R}}^{\mathrm{t}}\right)$. Together, ${\mathrm{K}}^{\mathrm{t}}$ and ${\mathrm{a}}^{\mathrm{t}}$ determine the productivities ${\mathrm{A}}^{\mathrm{t}}$. The adults then collectively choose a time allocation ${\mathrm{n}}^{\mathrm{t}}$ that maximizes total food output. This also maximizes the per capita food income ${\mathrm{y}}^{\mathrm{t}}$. Early in the period, the adults form couples and the flow of food ${\mathrm{y}}^{\mathrm{t}}$ determines the number of live births per couple. Later in the period, these children begin to observe the production activities of the adults. At the same time the flow of food ${\mathrm{y}}^{\mathrm{t}}$ determines how many children survive to adulthood. When children become adults, their parents die and the period ends. The ${\mathrm{N}}^{\mathrm{t}+1}$ surviving children become the adults of period $\mathrm{t}+1$. They compare techniques for harvesting each resource and choose the best possible repertoire ${\mathrm{K}}^{\mathrm{t}+1}$. The process then repeats.

We use the term “very-long-run equilibrium” (VLRE) for a situation in which the LRE requirements are satisfied and in addition, the prevailing technological repertoire K is transmitted to the next generation with probability one. Proposition 3.1 showed that for an active resource $\left({\mathrm{n}}_{\mathrm{r}}>0\right)$, there is a positive probability of technical progress whenever ${\mathrm{k}}_{\mathrm{r}}\ne {\mathrm{k}}_{\mathrm{r}}^{\ast }$. This implies that in a VLRE, the equilibrium repertoire K must include the ideal string ${\mathrm{k}}_{\mathrm{r}}^{\ast }$ for every active resource, so that productivity for each of these resources is at its upper bound. Moreover, given prevailing technical knowledge, it must be optimal to assign a zero labor input to each latent resource (n_r = 0).

This leads to the following formal definition.

One interpretive remark is important here. The term “very long run” might suggest that technology evolves on a longer time scale than population does. This is not the case. In reality, technology and population evolve on the same time scale (one human generation at a time), according to the dynamic rules described in Sections 3.5 and 3.7. However, for analytic purposes it is often useful to hold technology constant while allowing population to evolve. This is the setting of the long-run model in Section 3.7. In the very long run, by contrast, both variables are allowed to evolve simultaneously (both are endogenous).

To characterize the set of VLREs, we need additional terminology and notation. Let S ⊆ {1 . . R} be a non-empty set of resources. A VLRE is said to be of type S if k_r = k_r* for r ∈ S and k_r ≠ k_r* for r ∉ S. All opportunities for technical progress have been exhausted for the resources in the set S but such opportunities remain available for the resources not in this set. If the ideal string k_r* is available for resource r, we use the notation A_r* for the associated productivity a_rg_r(k_r*).

Next we construct a specific repertoire of type S. Let k_r^min be the string having minimal productivity for resource r. Define the repertoire K^S by setting k_r^S = k_r* for r ∈ S and k_r^S = k_r^min for r ∉ S. Thus, we assign the best possible technique to each resource in the set S and the worst possible technique to each resource not in S. The next proposition describes the conditions under which there is a non-trivial VLRE of type S with the specific repertoire K^S. It also shows that all of the other VLREs of type S are essentially identical, in the sense that they have the same population and time allocation.

Part (a) says that there must be at least one resource for which the ideal string k_r* is available, and the resource can support a positive population at the food income y*. If the latter condition is not satisfied, the equilibrium population must be zero. Part (b) says that the marginal product of labor for the society as a whole (H_N) must be at or above the marginal product for every latent resource. If this were not true, one could increase food output by using some latent resource, so time allocation would not be optimal. The only differences among non-null VLREs of type S involve techniques for the latent resources r ∉ S. These must be sufficiently unproductive but are otherwise indeterminate.

A resource that satisfies A_r*f_r′(0) > y* as in condition (a) of Proposition 3.4 will be called a staple. Such a resource can support a positive population even in the absence of any other resource. Every non-null VLRE of type S must have at least one staple in the set S. It can be shown that every staple in S must be active. Any resources with A_r*f_r′(0) ≤ y* will be called supplements. The set S could include one or more supplements, and these resources may be either active or latent.

Every resource r ∉ S is latent, regardless of whether it is a staple or a supplement. Condition (b) from Proposition 3.4 requires that for each of these resources, there must be a harvesting method so unproductive that it would not be optimal to exploit the resource. It seems plausible that for each resource there is some highly unproductive technique of this kind. As a result, many different VLREs could exist, with different resources being active and latent (or more precisely, different specifications of the set S).

The following corollary to Proposition 3.4 provides a general test for the existence of a VLRE with positive population.

The corollary says that whenever at least one staple exists, there is a non-null VLRE of the form (K*, N*, n*). We call this the maximal VLRE because although other VLREs could exist, no VLRE can support a larger population, and any VLRE with fewer active resources will have a smaller population. A VLRE with population below the maximum associated with (K*, N*, n*) will be called a stagnation trap, because a larger population could be supported if suitable technological knowledge were available. When no staple exists, even ideal technical knowledge is inadequate to maintain a viable population.

3.9 The Effects of Climate Change

We are now ready to address a central theoretical question: What environmental conditions are most conducive to technical progress? In particular, can climate change help a society escape from a stagnation trap?

The first task is to show that the system converges to a VLRE from any initial state. This is done in Proposition 3.5. The second task is to study the impact of climate shocks on technology, population, and labor allocation. Proposition 3.6 provides such an analysis for neutral shocks that affect all resources in the same proportion. We show that in this case, positive shocks can stimulate progress while negative shocks cannot.

We then discuss biased shocks in a setting with two resources. Our analysis shows that negative shocks biased toward latent resources can generate progress, but this depends on the outcome of a race between rising productivity and declining population. Finally, we discuss the possibility of regress when active resources shut down.

Before defining convergence to a VLRE, we need to spell out how techniques and population are updated over time for a fixed array of resource abundances a = (a₁ . . a_R) > 0 called “climate.” Let (K^t, N^t) be the state in period t. We obtain (K^t+1, N^t+1) as follows.

1. (a) K^t determines the productivities A^t = [a₁g₁(k₁^t) . . a_Rg_R(k_R^t)].

2. (b) A^t and N^t determine SRE labor allocation n^t and food output H(A^t, N^t) as in (3.2).

3. (c) N^t and H(A^t, N^t)/N^t ≡ y(A^t, N^t) determine N^t+1 as in (3.3).

4. (d) K^t, n^t, and N^t+1 determine the probability distribution over K^t+1 as in Proposition 3.1. This yields a new state (K^t+1, N^t+1).

Proposition 3.1 is expressed using q_r^t (the number of digits in k_r^t that match the ideal string k_r*), but it generates a probability distribution for k_r^t+1 conditional on k_r^t because there is an equal probability of mutation at every locus of k_r^t that does not already match k_r*. We assume existing strings for latent resources are retained with probability one. The laws of motion in (a)–(d) above yield the following results.

Part (a) shows that the system always reaches a stationary technology in finite time. Once this occurs, population converges to the corresponding Malthusian level and the allocation of labor time converges to the corresponding optimum for this technology and population. Because regress is impossible for active resources and strings remain constant for latent resources, the productivity vector {A^t} must be non-decreasing.

We know from part (b) that the system converges to some VLRE. However, it is difficult to say much about the probability of converging to any specific VLRE because this depends in a complex way on how mutations affect the productivities, which affect population dynamics, which feed back to the mutation probabilities as in Proposition 3.1.

Part (c) does provide some information about the population dynamics. There are three possibilities. If population is initially below the LRE level, it always rises, because Malthusian forces push population in this direction and could be reinforced by technical progress. If population is initially at the LRE level, it cannot decrease and may remain at a constant level (with no technical progress) or rise (if progress occurs). If population is initially above the LRE level, it may fall forever (for Malthusian reasons), or it may fall for some finite number of periods and then never fall again (because technical progress eventually outweighs Malthusian tendencies).

We next consider responses to climate shocks and associated changes in resource abundances. Recall that technical progress enables a society to support a larger long-run population with a given set of natural resources. This definition requires us to distinguish population growth due to technical change (holding climate constant) from population growth due to climate change (holding technology constant).

Our approach is based on the following thought experiment. Let the system be in a VLRE corresponding to some initial climate state a⁰. Suppose the climate jumps from the initial state a⁰ to a new state a′ and stays in the new state for a long time. The system will eventually converge to a VLRE for the new climate state. Finally, suppose climate jumps back to the original state a⁰ and stays there for a long time. The system will now converge to a VLRE for the original climate. If the population in the final equilibrium is higher than in the initial equilibrium, this can only be due to technical progress along the adjustment path.

The next proposition carries out this thought experiment for a case where climate change influences all productivities in the same proportion. Hence the abundances of all plant and animal species move up or down together, so nature as a whole becomes either more generous (positive shock) or less generous (negative shock). In either situation, the shock is neutral because we are ruling out substitution effects among the resources.

Proposition 3.6(a) shows that a negative neutral shock cannot stimulate technical progress, because it cannot change the repertoire K. Population falls and some of the previously active resources may become latent (the diet may narrow). Reversing the shock returns the system to the original VLRE and restores the previous population, as long as the society has not gone extinct in the meantime. Restoration of the status quo population reflects the absence of technical progress in response to the climate change.

Proposition 3.6(b) shows that a positive neutral shock can stimulate permanent progress. A necessary and sufficient condition for this outcome is that the shock must lead to exploitation of a latent resource whose technique can be improved. This cannot occur in the short run through substitution effects because relative resource abundances are held constant. Instead, the key causal channel is a scale effect involving population.

Without technical change, the improved climate would lead to a larger population $\mathrm{N}\left({\mathrm{\theta A}}^0\right)$ in the long run. This population growth could make one or more latent resources active, as explained at the end of Section 3.6. If so, the technical repertoire K improves through learning by doing, and population expands beyond the level $\mathrm{N}\left({\mathrm{\theta A}}^0\right)$ induced by the climate change alone. But if the scale effect is too small to activate a latent resource, or productivity is already at its upper bound for each newly active resource, technology remains static and population grows only to the extent that climate permits.

If the climate returns to its original state ${\mathrm{a}}^0$ and technology has not improved in the meantime $\left({\mathrm{K}}^{\prime }={\mathrm{K}}^0\right)$, clearly population must return to its original level ${\mathrm{N}}^0$. The same is true if technology improves as a result of the climate shock, but not by enough to alter the set of active resources used in the original climate ${\mathrm{a}}^0$. For progress to become visible in the population level $\left({\mathrm{N}}^{″}>{\mathrm{N}}^0\right)$ after climate reverts back to ${\mathrm{a}}^0$, the string for at least one previously latent resource must improve to a point where the old labor allocation ${\mathrm{n}}^0$ is no longer optimal at the old population level ${\mathrm{N}}^0$. In this case, at least one new resource will be used after the climate returns to ${\mathrm{a}}^0$. Simultaneously, some of the resources that were previously active at the climate ${\mathrm{a}}^0$ may be abandoned due to substitution effects.

Provided that strings for latent resources are conserved, the new technical plateau will be permanent. Proposition 3.6(a) shows that subsequent negative shocks cannot force a technological retreat. The result is a ratchet effect in which technological knowledge gradually improves. But unlike conventional growth models, our framework predicts a punctuated process where intermittent productivity gains stimulated by positive climate shocks can be separated by long periods of stagnation during which technology does not change and the average population growth rate is zero. During these intervals population will fluctuate in response to climate, and individual resources may go in and out of use as a result, but there is no lasting improvement in technological capabilities.

Thus far we have made two key assumptions: that climate shocks are neutral and that strings for latent resources are conserved. In the rest of this section we consider the consequences of relaxing each assumption.

Shocks biased toward or against particular resources create short-run substitution effects that can activate latent resources even before population has time to adjust. It will be convenient to discuss these effects in the context of two resources (R = 2). In all cases we start from an initial VLRE (K⁰, N⁰, n⁰) associated with climate a⁰ in which resource 1 is active and resource 2 is latent, and we assume the new climate is permanent. Because a negative shock to a latent resource cannot affect the system, we ignore this case.

3.10 Archaeological Applications

The theoretical framework constructed in this chapter can shed light on instances of technological innovation in the archaeological record. We will consider two examples here: the broad spectrum revolution, and colonization of the Arctic.

3.11 Conclusion

We have developed a theory to explain why technological progress was slow and sporadic during the Upper Paleolithic. The theory presented in Sections 3.5–3.9 satisfies the following criteria from Section 3.4.

1. (a) Technology is endogenous and the rate of productivity growth in very-long-run equilibrium is zero. However, occasional technical progress can occur.

2. (b) Population is endogenous and described by Malthusian dynamics.

3. (c) Climate change is the main driver of changes in technology and population.

4. (d) The model is consistent with the climate history in Section 3.2.

5. (e) The model is consistent with the population dynamics in Section 3.3.

6. (f) The model suggests explanations for the broad spectrum revolution and the gradual colonization of extreme environments.

In our framework, foragers exploit a subset of the resources provided by nature. Although they often reach high productivity levels for these resources, further advances require the use of latent resources, which is not attractive in light of existing knowledge. We call this a stagnation trap. Climate shocks can cause foragers to experiment with latent resources. This generates punctuated equilibria where foragers acquire enhanced technical capabilities and higher populations at successive plateaus. If a foraging society can retain enough knowledge of inactive techniques, it may benefit from a ratchet effect through which technology progresses over archaeological time. We will build on these ideas to explain the origins of sedentism and agriculture in Chapters 4 and 5.

3.12 Postscript

This chapter is based on the article “Stagnation and innovation before agriculture” by Dow and Reed (Reference Dow and Reed2011) in the Journal of Economic Behavior and Organization. We received advice on drafts of the original article from Matthew Baker, Cliff Bekar, Mark Collard, Brian Hayden, Stephen Shennan, Lawrence Straus, an anonymous referee, and colleagues at Simon Fraser University, the University of Copenhagen, and the Canadian Network for Economic History. Scott Skjei assisted with the research, and the Social Sciences and Humanities Research Council of Canada provided financial support. As always, the opinions are those of the authors.

Sections 3.1–3.4 are new for this volume, as are the graphs. References to the archaeological literature have been updated and Section 3.10 has been entirely rewritten. The formal model is identical to the one in Dow and Reed (Reference Dow and Reed2011). When we constructed the original model, we were aware of the climate history in Section 3.2, some of the early findings from paleodemography (although none of the specific studies in Section 3.3), and the evidence on the broad spectrum revolution in Section 3.10. We were not aware of the information about Siberian climate, population, or technology.

4.1 Introduction

From the origins of anatomically modern humans until after the Last Glacial Maximum around 21,000 years ago, almost everyone lived in small, mobile foraging bands. Starting around 15,000 years ago, foragers in some regions such as southwest Asia and Japan began to develop permanent settlements. Sedentary foraging predated agriculture by several millennia and accelerated with the onset of the Holocene 11,600 years ago, which brought a warmer, wetter and more stable climate. The best evidence for early sedentism is from temperate zones. Among modern hunter-gatherer societies, those in tropical rainforests and the Arctic have remained the most mobile (Kelly, Reference Kelly2013a).

Sedentism can be defined in various ways and is a matter of degree, so we need to clarify our use of this term. First, it is important to recognize that mobile foraging groups do not just move at random across the landscape. A common pattern involves the use of seasonally shifting base camps over an annual cycle, with hunting and gathering on trips away from each base camp. When anthropologists and archaeologists refer to sedentism, they often mean settlements that are at least partially occupied year-round.

Evidence used by archaeologists to infer sedentism at a site includes the presence of plants and animals from all four seasons; the presence of species that flourish when in frequent contact with humans (e.g., mice, rats, and sparrows); substantial investments in dwellings, earthworks, ceremonial structures, or monuments; greater use of cemeteries; and site-specific investments in food processing and storage facilities. Marshall (Reference Marshall2006) cites the durability of dwellings and the size of settlements as the most crucial markers.

While we accept these indicators of sedentism, we require more than simply the year-round occupation of a site. A time period in our model is defined to be one human generation. We are therefore interested in the conditions under which adult children stay in the same location as their parents, even when facing multi-year environmental shocks that reduce the abundance of local food resources. We call a community sedentary if it is robust to such negative shocks over periods lasting decades or centuries. We sometimes refer to this phenomenon as sustained sedentism to differentiate it from more temporary or fragile forms of sedentism. A time interval lasting generations or centuries makes it possible for major structures to be built, for burial practices to develop, for the regional population to grow, and for new technologies and institutions to evolve.

Why was sustained sedentism an important transition, one that (as in the words of our subtitle) shaped the world? The development of large permanent communities was a massive social upheaval relative to the ancestral lifestyle of small mobile bands. We will unpack some of the implications in what follows.

Ethnographic data indicate that group sizes among sedentary foragers are much larger than for mobile foragers (Kelly, Reference Kelly2013a, 171–172). Sedentism is also correlated with other important variables. Using original data from Keeley (Reference Keeley1988, Reference Keeley, Barton, Roberts and Roe1991), Rowley-Conwy (Reference Rowley-Conwy, Panter-Brick, Layton and Rowley-Conwy2001, 40–44) shows that ethnographically known foragers fall into two distinct clusters. Those with low sedentism have low population relative to natural productivity, low use of food storage, and low stratification. Those with high sedentism have high levels on these other dimensions. A classic example of the latter kind is provided by the societies on the northwest coast of North America (Ames and Maschner, Reference Ames and Maschner1999). Such societies illustrate Kelly’s point that sedentary foraging is correlated with “social hierarchies and hereditary leadership, political dominance, gender inequality, and unequal access to resources” (Reference Ferguson and Fry2013a, 104). Anthropological evidence strongly suggests that even if sedentary foraging had never led to agriculture, it would still have led to economic inequality, and it would probably have led to greater warfare over land. We pursue these ideas in Chapters 6–8.

The other (and larger) reason why sustained sedentary foraging shaped the world is that at least in some cases, it was a precursor to agriculture. We want to be careful in our comments on this matter because there is a long and unhelpful tradition of conflating sedentary foraging with agriculture. Sloppy language such as “the transition to sedentary agriculture” should therefore be avoided. It is essential to recognize that the transition to sedentary foraging was distinct from the transition to agriculture, and requires a distinct theoretical explanation (Marshall, Reference Marshall2006). Indeed, we will propose in this chapter that the transition to sedentism was largely driven by positive climate trends, while we argue in Chapter 5 that transitions to agriculture were often driven by negative climate shocks.

Sedentary foraging is clearly not a sufficient condition for a subsequent transition to agriculture. For example, millennia of sedentary foraging never led to indigenous agriculture in Japan or along the northwest coast of North America. When domesticates finally arrived, they were imported from elsewhere. We discuss the case of Japan further in Section 4.3. Nor is year-round sedentism a necessary precondition for agriculture. In the American Southwest, for example, imported domesticates like maize appear to have been integrated into a foraging system involving seasonal mobility, with truly sedentary villages arising a millennium or more later (Wills, Reference Wills1988).

Nevertheless, we believe that sustained sedentism contributed to several pristine agricultural transitions. One leading example is southwest Asia (see Section 4.3). In this region, sedentism supported a large population and led to technological innovations that were pre-adapted for cultivation. For reasons to be explained in Chapter 5, both factors made a subsequent transition to agriculture more likely. Sedentism may also have led to institutional innovations with respect to property rights that increased the probability of an agricultural trajectory, as some economists have asserted (see Section 4.2). However, robust foraging communities arose 3,000 years before agriculture and were clearly not the proximate cause of this trajectory in southwest Asia. Although sedentism helped set the stage, we will argue that climate and geography took the leading roles. Our hypothesis about the agricultural transition in southwest Asia will be presented in Chapter 5.

This chapter assumes that people in a region can freely migrate from one location to another. Thus, as discussed in Section 1.2, we are studying a transition along the main sequence from cell A to cell B in Figure 1.2, where institutions allow open access for both mobile and sedentary foragers. To be sure, the sedentary foragers described by anthropologists often exercise group property rights over valuable locations, exhibit stratification, and engage in warfare. We take up these subjects in Chapters 6–8, where we explore sedentary foraging with closed access (cell B′ in Figure 1.2) or stratification (cell B″ in Figure 1.2). But when mobile foragers began their journey toward sedentism, their institutions probably did not yet involve rigid property rights or social stratification. These are more likely to be effects of sedentism rather than causes. Therefore, we would like to know whether an initial impetus toward sedentism can develop in a setting of free mobility. We will answer in the affirmative.

What caused mobile foragers to become sedentary foragers? A common answer involves direct effects of nature. The idea is that people are mobile when the location of food resources is constantly shifting, or when important resources are not available all in one place, or when local resources are rapidly depleted. People become sedentary when nature provides a rich, diverse, and reliable assortment of resources at a single location, and these resources are not easily depleted.

We don’t doubt that this is part of the story, but it is far from the whole story. As we will see in Section 4.3, sedentism was not just a matter of people settling down in one place. It also involved new food sources, larger settlements, larger regional populations, and technological innovation. These changes often included more use of plant or aquatic foods as compared with prey animals; fixed investments in mortars, ovens, and storage facilities; and more durable housing (Kelly, Reference Kelly2013a, 122–128). How can we explain the recurrent pattern of greater dietary breadth, population growth, and technical change?

We develop a formal model that addresses these questions. We consider a region with many individual production sites, where the weather at each site can be good or bad. Good weather is associated with abundant food resources and bad weather is associated with scarce food resources. In our model, a climate regime is defined by the probability distribution over weather conditions (temperature, precipitation, etc.) at each site. For a given climate these random draws are independent across sites and time periods, so a site that is good in one period can be bad in the next. A change in climate refers to a change in the probability distribution for weather events. The rate of sedentism is the fraction of the local population that remains in place when a site switches from good to bad weather. This is our measure of how robust communities are to natural shocks.

The shift from the Last Glacial Maximum to the Holocene involved both better mean weather and decreased variance in weather (see Section 3.2; Richerson et al., Reference Richerson, Boyd and Bettinger2001; Woodward, Reference Woodward2014, chs. 8–9). These trends were sometimes associated with sedentism well before the Holocene (see the discussion of southwest Asia in Section 4.3), although more often sedentism developed in the early Holocene. Climate shifts of this kind could have led to a positive rate of sedentism through three causal mechanisms.

First, let total regional population be constant in the short run. The lower variance shrinks the productivity gap between sites with good weather and sites with bad weather. To take the simplest case, imagine that the productivity of good sites does not change but bad sites become better. If the bad sites improve enough, diminishing returns at the good sites could make it attractive for some agents to start using bad sites. It follows that when a site switches from good weather to bad weather it will no longer be entirely abandoned.

Second, in the long run the better mean weather conditions result in population growth through Malthusian dynamics. Even if the decreased variance in weather is not sufficient to cause sedentism by itself, a higher regional population could lead to some use of sites with bad weather due to diminishing returns at sites with good weather. It can thus yield a positive rate of sedentism. Neither of these two mechanisms requires technical change, and each is reversible in the sense that if climate reverts to its earlier mean and variance, the rate of sedentism will eventually return to its original level.

The third causal mechanism involves technological innovation. We assume that agents can use two methods of food collection. We will refer to these metaphorically as “hunting” (a shorthand term for mobile methods) and “gathering” (shorthand for stationary methods). Suppose that a society initially uses only hunting, and suppose climate change leads to Malthusian population growth. This will reduce the marginal product of labor in hunting. At some point gathering may begin, with new food resources being targeted. If so, learning by doing could increase the productivity of gathering over time. The rising productivity of gathering will then reinforce the population growth generated by climate change. This trajectory can lead to sedentism even when climate change and population growth alone would not. Assuming that new knowledge about food acquisition methods is retained over time, this process creates a ratchet effect such that even if climate reverts to its original state, greater dietary breadth, increased population, and new techniques of gathering can persist. The sedentism rate can thus remain above its original level.

The rest of the chapter is organized as follows. Section 4.2 reviews a number of existing theories about the origins of sedentism from both anthropology and economics. Section 4.3 reviews archaeological evidence on sedentary foraging, including the leading cases of southwest Asia and Japan.

Sections 4.4–4.10 develop our formal model. We consider three different time spans: the short run, the long run, and the very long run. In the short run, the allocation of labor across sites and food procurement methods is endogenous, while the aggregate regional population, technology, and climate are exogenous. In the long run, the regional population becomes endogenous while technology and climate remain exogenous. In the very long run, technology becomes endogenous and only climate is still exogenous. This parallels the structure of our modeling efforts in Chapter 3.

Section 4.4 describes food acquisition at an individual site. Section 4.5 considers the regional economy, which has a continuum of production sites. The sites are exposed to idiosyncratic variations in food availability due to weather shocks. Property rights are characterized by open access. As explained above, this is a logical starting point for the development of sedentism, and we see it as a reasonable approximation when population density is low and the agents can move relatively easily from one site to another through kinship ties. A short-run equilibrium (SRE) is an allocation of regional population across production sites such that no individual agent can gain by changing sites.

Section 4.6 adopts the Malthusian assumption that the population growth rate is an increasing function of food per capita, as was discussed in Chapter 2 and Section 3.7. A long-run equilibrium (LRE) is an SRE such that the regional population stays constant over time. Section 4.7 adds technological innovation where the gathering technology can improve through learning by doing, along lines discussed in Section 3.5. A very-long-run equilibrium (VLRE) is an LRE such that technological capabilities remain constant over time, as was discussed in Section 3.8.

Section 4.8 constructs a baseline VLRE that captures central features of the Last Glacial Maximum. These include a climate where weather conditions have a low mean and high variance, a low population, the use of mobile food acquisition techniques, and complete abandonment of sites whenever natural resources become locally scarce. We then consider a permanent shift to a new climate regime with a higher mean and a lower variance. Section 4.9 studies three causal mechanisms by which climate amelioration can lead to sedentism. The first operates in the short run through the reduction of variance in weather conditions across sites. The second operates in the long run through Malthusian population growth, and the third combines population growth with technical innovation.

Section 4.10 examines the consequences of sedentism for demography and living standards. Assuming sedentism made children less costly, as is often said to be true, our theory predicts a shift toward increased fertility, increased child mortality, decreased food per capita, and increased regional population. Section 4.11 follows with some comments on how our model relates to theories based on resource depletion.

Section 4.12 reviews further archaeological evidence. Section 4.13 summarizes our conclusions and discusses a few extensions of the theory. Section 4.14 is a postscript. Proofs of the formal propositions are available at cambridge.org/economicprehistory.

4.2 Theories of Sedentism

There are five main alternatives to our explanation for the origins of sedentism, three from anthropology and two from economics. Price and Brown (Reference Price and Brown1985) describe two traditional theories from anthropology, which they label as “pull” and “push.” According to the pull hypothesis sedentism arose due to resource abundance, which encouraged people to reduce their mobility. According to the push hypothesis foragers were forced to adopt sedentism by resource scarcity, which caused a shortage of food relative to population. This led to greater dietary breadth and more time devoted to harvesting and processing wild food resources, which in turn led to lower mobility. These theories have various empirical inadequacies (Kelly, Reference Kelly1992).

A third anthropological explanation is summarized by Kelly (Reference Kelly2013a), who argues that sedentism was driven by regional population growth, which led to group packing and made it harder for groups to relocate when local resources were depleted. Sedentism by one group raised the cost of moving for other groups, and therefore sedentism tended to spread. Sedentism typically emerged in places where the local resource base was hard to deplete (e.g., on coasts or rivers where aquatic resources were available). Sedentism led to further population growth through increased fertility, so once the process was set in motion it became self-reinforcing. Technology evolved toward the use of lower-value but less-easily-depleted food resources.

We agree with Kelly that population growth and technical change were mutually reinforcing in the transition to sedentism. Our model does not include resource depletion because we want to show that the qualitative evidence on the transition to sedentism can be explained without appealing to this variable. Indeed, we will argue that sedentism was triggered by the greater resource abundance brought about by a better climate. We will, however, discuss resource depletion issues further in Section 4.11. Another contrast with Kelly is that we do not emphasize the group ownership of production sites in this chapter, although we discuss and model it extensively in Chapters 6–8. We also depart from Kelly by endogenizing population through Malthusian dynamics, and by emphasizing technical ratchet effects along the lines of Chapter 3.

Acemoglu and Robinson (Reference Acemoglu and Robinson2012, 137–143) offer an economic and political story about the origins of sedentism. Their account centers on Natufian society in southwest Asia, which we will discuss in Section 4.3. Acemoglu and Robinson note that the Natufians were sedentary foragers well before they began to domesticate plants and animals. They observe that mobility was costly for the young and old, made food storage difficult, and prevented use of productive but heavy implements such as grinding stones. However, “while it might be collectively desirable to become sedentary, this doesn’t mean that it will necessarily happen. A mobile group of hunter-gatherers would have to agree to do this, or someone would have to force them” (139). “In order for sedentary life to emerge, it therefore seems plausible that hunter-gatherers would have had to be forced to settle down, and this would have to have been preceded by an institutional innovation concentrating power in the hands of a group that would become the political elite, enforce property rights, maintain order, and also benefit from their status by extracting resources from the rest of the society” (139). “The emergence of political elites most likely created the transition first to sedentary life and then to farming” (140). “Institutional changes occurred in societies quite a while before they made the transition to farming and were probably the cause both of the move to sedentarism [sic], which reinforced the institutional changes, and subsequently of the Neolithic Revolution” (141).

One can argue that there was mild inequality among the Natufians after sedentism and before agriculture, although some archaeologists call them egalitarian (Grosman and Munro, Reference Grosman, Munro, Enzel and Bar-Yosef2017, 704). There is also evidence for mild inequality in southwest Asia after the spread of agriculture during the Holocene. We will return to both points in Chapter 6. However, there is no evidence at all that the mobile foragers who preceded the sedentary Natufians had either political elites or stratification. In fact there is no evidence that any prehistoric or more recent society of mobile foragers has had a political elite or economic stratification. Acemoglu and Robinson appear to believe that an elite could have emerged prior to sedentism through random institutional drift, but they offer no further explanation as to how or why this could have happened. They also do not explain how an elite in a mobile band could have forced other band members to settle down, or what their motivation for this would have been. Chapter 6 will develop a theory about the origins of inequality that is more consistent with archaeological evidence. In this chapter, we ignore the alleged role of coercion as a causal factor in the transition to sedentary foraging.

Bowles and Choi (Reference Bowles and Choi2013, Reference Feder2019) view sedentism in the context of the transition to cultivation, which eventually led to farming communities dependent upon domesticated crops. They think sedentism initially arose at atypically rich production sites where wild resources were sufficiently concentrated to allow sedentary living, and to make defending the resources worthwhile. They also emphasize the reduction of climate volatility, which lessened the pressure for mobility.

Their causal sequence runs as follows. Initially production technology was based on mobile hunting and gathering. Climate change led to warmer and more stable weather conditions favorable to plant growth, which resulted in a few particularly well-endowed production sites. The existence of these sites motivated a transition to greater sedentism. Finally, sedentism created an environment where the coevolution of property rights and agriculture could occur. We will return to the ideas of Bowles and Choi when we discuss the origins of agriculture in Chapter 5.

4.3 Some Archaeological Evidence

This section provides background information about the rarity of sedentism at the Last Glacial Maximum and its emergence as climate conditions improved. We then take a closer look at two regions where early sedentism is well documented: southwest Asia and Japan. These cases offer guidance for our formal model. We conclude this section with some remarks on the relationship between sedentism and agriculture.

In the first three chapters, we used the notation “KYA” to abbreviate dates going back tens of thousands of years. In this chapter and those to come, it will be convenient to switch to the notation “BP,” meaning years before the present. This naturally raises the question of what is meant by the present. In archaeological publications from the second half of the twentieth century, the present was usually defined by convention to be 1950 AD (or CE, meaning “current era”). However, in the twenty-first century some scholarly work has begun to treat the year 2000 CE as the definition of the present. For the time scales of this book a difference of 50 years is negligible, because 95% confidence intervals for radiocarbon dates are often ±100 years or more. Readers can therefore enjoy the simplicity of round numbers and treat 10,000 BP as 10,000 years before 2000. As always we use calibrated (calendar) years rather than uncalibrated (radiocarbon) years unless otherwise stated.

The Last Glacial Maximum (LGM) occurred around 21,000 BP in calendar years, or around 18,000 BP in uncalibrated radiocarbon years. The contributors to two volumes edited by Gamble and Soffer (Reference Gamble and Soffer1990) and Soffer and Gamble (Reference Gamble and Soffer1990) survey archaeological evidence from this period for 30 regions of the world. Many regions had settlement sizes ranging from small (probably temporary hunting camps or tool-making workshops) to medium (probably seasonal base camps) and large (probably residential camps with occasional population aggregation). Some foraging groups in areas of high ecological diversity may have collected food within relatively compact territories, and such groups probably had higher population densities than elsewhere.

Nevertheless, in the 600+ pages of the two volumes edited by Soffer and Gamble, only one author claims evidence for a sedentary lifestyle (involving permanent structures in Siberia; see Madeyska, Reference Madeyska, Gamble and Soffer1990, 32). Archaeological evidence for sedentary communities at the time of the LGM may not have survived or may have been overlooked. But if there were such communities during this period, they were apparently rare and ephemeral.

Aside from southwest Asia and Japan, which will be discussed in detail below, sedentism arose early in the Holocene in Portugal, the Baltic region, and parts of Africa. For Portugal in the first 5,000 years of the Holocene, Straus (Reference Aikens, Akazawa, Straus, Ericksen, Erlandson and Yesner1996) reports the appearance of cemeteries, dugout structures, and large shell middens, with a highly diversified diet from hunting, fishing, other aquatic resources, and probably abundant nuts, berries, and tubers. Straus describes the settlements as “semi-sedentary.” For Eastern Europe and the Baltic region, Dolukhanov (Reference Dolukhanov, Straus, Ericksen, Erlandson and Yesner1996) doubts that year-round settlements existed in the late Pleistocene, but by the early Holocene climate change had led to “comparatively large settlements of permanent and semi-permanent character in Baltic lagoons and inland lacustrine depressions, with an economy based on the exploitation of a wide spectrum of wildlife resources” (168). Sedentary foraging arose in North Africa and East Africa around the same time (Barham and Mitchell, Reference Barham and Mitchell2008, 351–355).

Such evidence for early sedentism is important, but it is necessary to emphasize that in other places there was no such transition. For example, Hiscock (Reference Hiscock2008, 252–253) doubts that sedentary villages existed in Australia prior to European contact. The same was probably true for most tropical rain forests, deserts, and the Arctic.

4.4 The Production Site

We consider a site with L agents. Each agent is endowed with one unit of labor time, and each agent is of negligible size relative to the population at the site. The food produced at the site is shared equally among the agents, who allocate their labor time to maximize total food output. Our use of the term “site” will be flexible and may involve a relatively large geographic area within which mobile foragers search for food.

Food can be obtained by hunting or gathering. We use “hunting” as a generic label for mobile food procurement activities involving a portable toolkit, minimal processing, and quick consumption. Aside from literal hunting, this could include the collection of berries or other easily harvested wild plant foods. We use “gathering” as a generic label for stationary activities involving fixed assets, extensive processing, or storage. Fixed assets could include grinding stones, ovens, pottery, or storage pits. Further examples outside the literal definition of gathering include animal traps and fishing weirs.

Total food output (in calories) is given by

$\mathrm{Y}=\mathrm{\theta }\left[\mathrm{f}\left({\mathrm{L}}_{\mathrm{f}}\right)+\text{kg}\left({\mathrm{L}}_{\mathrm{g}}\right)\right]$(4.1)

where f and g are the production functions for hunting and gathering respectively; $\mathrm{\theta }$ is the weather; ${\mathrm{L}}_{\mathrm{f}}\ge 0$ is the labor input for hunting; ${\mathrm{L}}_{\mathrm{g}}\ge 0$ is the labor input for gathering; and $\mathrm{k}>0$ is the productivity of gathering. The parameter $\mathrm{\theta }$ is neutral between the two activities and reflects the general abundance of food resources at the site.

The production functions have the following properties:

One example of such a production function is $\mathrm{f}\left(\mathrm{L}\right)=\mathrm{A}\left(1-{\mathrm{e}}^{-\mathrm{mL}}\right)$ where $\mathrm{m}>0$ and A can be interpreted as the natural supply of some wild food resource available for harvesting.

At a benchmark site where $\mathrm{\theta }=1$, the maximum food output is

$\mathrm{H}\left(\mathrm{L},\mathrm{k}\right)\equiv max\left\{\mathrm{f}\left({\mathrm{L}}_{\mathrm{f}}\right)+\text{kg}\left({\mathrm{L}}_{\mathrm{g}}\right)\text{subject}\text{to}{\mathrm{L}}_{\mathrm{f}}\ge 0,{\mathrm{L}}_{\mathrm{g}}\ge 0,\text{and}{\mathrm{L}}_{\mathrm{f}}+{\mathrm{L}}_{\mathrm{g}}=\mathrm{L}\right\}$(4.2)

The solution in (4.2) is unique due to the strict concavity of the objective function. The optimal labor allocation and the function H have the following properties.

Part (a) of Proposition 4.1 says that if gathering productivity (k) is small enough, only hunting is used at low levels of L and both activities are used at higher levels of L. Conversely, part (b) says that if k is large enough, only gathering is used at low levels of L and both activities are used at higher levels of L. Part (c) is a boundary case. Part (d) shows that the marginal product of labor is positive and decreasing. Part (e) shows that food per capita h(L, k) decreases as labor input rises. Part (f) shows that food per capita is finite at $\mathrm{L}=0$ and goes to zero as L goes to infinity. For an unoccupied site, food per capita is defined to be $\mathrm{h}\left(0,\mathrm{k}\right)\equiv max\left\{{\mathrm{f}}^{\prime }\left(0\right),\mathrm{k}{\mathrm{g}}^{\prime }\left(0\right)\right\}$.

As in Chapters 2 and 3, for the production function f(L_f) the marginal product of labor and the average product of labor are equal to the same finite constant at zero input. The production function g(L_g) has the same feature. This extends to the function H(L, k), which has ${\mathrm{H}}_{\mathrm{L}}\left(0,\mathrm{k}\right)=\mathrm{h}\left(0,\mathrm{k}\right)$ where H_L(0, k) is the marginal product of labor at zero and h(0, k) is the average product of labor at zero, as in Proposition 4.1(f)(i).

4.5 Short-Run Equilibrium

This is the first section of the book in which we need to distinguish between sites and regions. We think of a region as a contiguous geographic area for which there is no in- or out-migration. This may be true because the area is circumscribed by mountains, deserts, or oceans, or because heavily populated zones are far apart and it makes sense to ignore migration between such zones. In empirical discussions we often refer to regions like southwest Asia, sub-Saharan Africa, or Mesoamerica that correspond to areas studied by archaeological specialists. In such contexts we assume that events in separate regions evolved independently.

Within a region there are generally multiple production sites. In Chapters 4–5 we assume that individuals can move freely among these sites. For simplicity we ignore the costs associated with travel over intra-regional distances. We also assume there are no social barriers to movement among sites, either because population densities are too low to make exclusion feasible, or because kinship links enable population to flow relatively freely between sites when this is economically advantageous to the individuals involved. Property rights therefore involve open access. In Chapters 6–8 we consider physical and social barriers to mobility across sites.

The number of sites in a region depends on the model. In Chapters 4–6 it will be convenient to assume a continuum of sites. In Chapters 7–8 we will study models where a region has only two distinct sites or territories. Finally, in Chapter 10 we will consider a region with one large site and many small sites.

Now consider a continuum of production sites uniformly distributed on the unit interval and indexed by $\mathrm{s}\in \left[0,1\right]$. The weather at an individual site s is binary:

We think of the regional economy as having a fixed total supply of food resources under a given climate regime. The weather in the current period determines how this supply is distributed across sites. There is no aggregate uncertainty.

We define the short run to be a time period over which regional population $\mathrm{N}>0$ does not change (one human generation, or about twenty years). Due to open access, it must be true in equilibrium that no agent can obtain a higher food income by changing sites. This motivates the following definition.

1. (a) ${\mathrm{\theta }}_{\mathrm{A}}\mathrm{h}\left({\mathrm{L}}_{\mathrm{A}},\mathrm{k}\right)=\mathrm{y}$ with ${\mathrm{L}}_{\mathrm{A}}>0$

2. (b) ${\mathrm{\theta }}_{\mathrm{B}}\mathrm{h}\left({\mathrm{L}}_{\mathrm{B}},\mathrm{k}\right)\le \mathrm{y}$ with equality if ${\mathrm{L}}_{\mathrm{B}}>0$

3. (c) ${\mathrm{\lambda L}}_{\mathrm{A}}+\left(1–\mathrm{\lambda }\right){\mathrm{L}}_{\mathrm{B}}=\mathrm{N}.$

The logic behind this definition runs as follows. Due to $\mathrm{N}>0$, if we had ${\mathrm{L}}_{\mathrm{A}}=0$ we would need to have ${\mathrm{L}}_{\mathrm{B}}>0$. The resulting food per agent at a site of type B would be ${\mathrm{\theta }}_{\mathrm{B}}\mathrm{h}\left({\mathrm{L}}_{\mathrm{B}},\mathrm{k}\right)$, which is less than ${\mathrm{\theta }}_{\mathrm{A}}\mathrm{h}\left(0,\mathrm{k}\right)$, the food an agent could obtain by moving to an unoccupied site of type A. To prevent such incentives for movement, we must have ${\mathrm{L}}_{\mathrm{A}}>0$ as in condition (a) of D4.1. The associated food income per agent is denoted by y.

If ${\mathrm{L}}_{\mathrm{B}}=0$ then we must have ${\mathrm{\theta }}_{\mathrm{B}}\mathrm{h}\left(0,\mathrm{k}\right)\le \mathrm{y}$ to avoid the movement of agents from sites of type A to unoccupied sites of type B. If ${\mathrm{L}}_{\mathrm{B}}>0$ then sites of type B must provide the same income as sites of type A, because otherwise agents would move from sites of one type to sites of the other. This gives condition (b) in D4.1. The local populations at the individual sites must add up to the regional population N, so we require ${{\int }_0}^1{\mathrm{L}}_{\mathrm{s}}\mathrm{ds}={\mathrm{\lambda L}}_{\mathrm{A}}+\left(1‒\mathrm{\lambda }\right){\mathrm{L}}_{\mathrm{B}}=\mathrm{N}$ as in condition (c).

There is a unique SRE for each $\mathrm{N}>0$. To see why, let η be the inverse function h^‒1, where we temporarily suppress the parameter k to simplify notation. This gives

${\mathrm{L}}_{\mathrm{A}}=\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{A}}\right)\text{where}\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{A}}\right)\equiv 0\text{for}\mathrm{y}/{\mathrm{\theta }}_{\mathrm{A}}\ge \mathrm{h}\left(0\right)$(4.3a)

${\mathrm{L}}_{\mathrm{B}}=\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{B}}\right)\text{where}\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{B}}\right)\equiv 0\text{for}\mathrm{y}/{\mathrm{\theta }}_{\mathrm{B}}\ge \mathrm{h}\left(0\right)$(4.3b)

The regional “demand” for labor is $\mathrm{D}\left(\mathrm{y}\right)\equiv \mathrm{\lambda \eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{A}}\right)+\left(1‐\mathrm{\lambda }\right)\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{B}}\right)$. The equilibrium food income y must satisfy

$\mathrm{D}\left(\mathrm{y}\right)\equiv \mathrm{\lambda \eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{A}}\right)+\left(1‐\mathrm{\lambda }\right)\mathrm{\eta }\left(\mathrm{y}/{\mathrm{\theta }}_{\mathrm{B}}\right)=\mathrm{N}$(4.3c)

The demand D(y) is zero for $\mathrm{y}\ge {\mathrm{\theta }}_{\mathrm{A}}\mathrm{h}\left(0\right)$. It is continuous and increases as y decreases for $\mathrm{y}\le {\mathrm{\theta }}_{\mathrm{A}}\mathrm{h}\left(0\right)$, with $\mathrm{D}\left(\mathrm{y}\right)\to \infty$ as $\mathrm{y}\to 0$. This implies the existence of a unique food income y(N) such that (4.3c) holds for the given $\mathrm{N}>0$. More formally:

In the rest of this section we show how the qualitative nature of SRE depends on the parameters N and k. It is easy to see from the definition of SRE that either $0={\mathrm{L}}_{\mathrm{B}}<{\mathrm{L}}_{\mathrm{A}}$ or alternatively $0<{\mathrm{L}}_{\mathrm{B}}<{\mathrm{L}}_{\mathrm{A}}$. We assume throughout this section that ${\mathrm{f}}^{\prime }\left(0\right)>\mathrm{k}{\mathrm{g}}^{\prime }\left(0\right)$, because we want to focus on situations where an initially low population density implies that only the hunting technique is employed. It follows from Proposition 4.1(a) that either ${\mathrm{L}}_{\mathrm{Af}}>0$ and ${\mathrm{L}}_{\text{Ag}}=0$ (only hunting is used at sites of type A) or ${\mathrm{L}}_{\mathrm{Af}}>0$ and ${\mathrm{L}}_{\text{Ag}}>0$ (both hunting and gathering are used at sites of type A). When sites of type B are active, ${\mathrm{L}}_{\mathrm{Bg}}>0$ implies ${\mathrm{L}}_{\text{Ag}}>0$ due to the greater population at sites of type A. Thus if gathering is used anywhere, it must be used at sites of type A.

These considerations yield four possibilities for the structure of SRE:

1. (I) Only type A sites are active $\left({\mathrm{L}}_{\mathrm{B}}=0\right)$ and only hunting is used $\left({\mathrm{L}}_{\text{Ag}}=0\right)$.

2. (II) All sites are active $\left({\mathrm{L}}_{\mathrm{B}}>0\right)$ and only hunting is used $\left({\mathrm{L}}_{\text{Ag}}=0\right)$.

3. (III) Only type A sites are active $\left({\mathrm{L}}_{\mathrm{B}}=0\right)$ and both techniques are used $\left({\mathrm{L}}_{\text{Ag}}>0\right)$.

4. (IV) All sites are active $\left({\mathrm{L}}_{\mathrm{B}}>0\right)$ and both techniques are used $\left({\mathrm{L}}_{\text{Ag}}>0\right)$.

Figure 4.1 illustrates these four regions in (N, k) space for a fixed climate regime $\left({\mathrm{\theta }}_{\mathrm{A}},{\mathrm{\theta }}_{\mathrm{B}},\mathrm{\lambda }\right)$. Regions I and II where only hunting is used are associated with low values of the gathering productivity k, while regions III and IV where both hunting and gathering are used are associated with higher values of k. Regions I and III where sites of type B are inactive correspond to low values for the regional population N, while regions II and IV where all sites are active correspond to higher values of N.

Figure 4.1. Short-run equilibrium as a function of regional population and gathering productivity for a fixed climate regime

The locations of the regions in Figure 4.1 depend only on the productivity ratio ${\mathrm{\theta }}_{\mathrm{B}}/{\mathrm{\theta }}_{\mathrm{A}}$ and not the absolute productivity levels $\left({\mathrm{\theta }}_{\mathrm{A}},{\mathrm{\theta }}_{\mathrm{B}}\right)$. The boundary for the use of the gathering technique (labeled ${\mathrm{L}}_{\text{Ag}}=0$ in Figure 4.1) is determined using Proposition 4.1(a). The relevant equation is

${\mathrm{f}}^{\prime }\left({\mathrm{L}}_{\mathrm{A}}\right)=\mathrm{k}{\mathrm{g}}^{\prime }\left(0\right)$(4.4)

When only sites of type A are active (regions I and III), the local population density ${\mathrm{L}}_{\mathrm{A}}=\mathrm{N}/\mathrm{\lambda }$ is obtained directly from condition (c) in the definition of SRE. Substitution into (4.4) gives the desired relationship between N and k. When all sites are active but gathering is not used, (a) and (b) in the definition of SRE imply ${\mathrm{\theta }}_{\mathrm{A}}\mathrm{f}\left({\mathrm{L}}_{\mathrm{A}}\right)/{\mathrm{L}}_{\mathrm{A}}={\mathrm{\theta }}_{\mathrm{B}}\mathrm{f}\left({\mathrm{L}}_{\mathrm{B}}\right)/{\mathrm{L}}_{\mathrm{B}}$. This and (c) determine local populations ${\mathrm{L}}_{\mathrm{A}}\left(\mathrm{N}\right)$ and ${\mathrm{L}}_{\mathrm{B}}\left(\mathrm{N}\right)$. Substituting ${\mathrm{L}}_{\mathrm{A}}\left(\mathrm{N}\right)$ into (4.4) gives the relationship between N and k corresponding to the boundary ${\mathrm{L}}_{\text{Ag}}=0$ in Figure 4.1.

The boundary for use of sites of type B (labeled ${\mathrm{L}}_{\mathrm{B}}=0$ in Figure 4.1) is derived as follows. We use the notation h(0, k) and η(⋅, k) to indicate that these functions depend on the gathering productivity k. Consider the population threshold

$\mathrm{N}\left(\mathrm{k}\right)\equiv \mathrm{\lambda \eta }\left[{\mathrm{\theta }}_{\mathrm{B}}\mathrm{h}\left(0,\mathrm{k}\right)/{\mathrm{\theta }}_{\mathrm{A}},\mathrm{k}\right]>0$(4.5)

When $\mathrm{N}\le \mathrm{N}\left(\mathrm{k}\right)$ only sites of type A are active $\left({\mathrm{L}}_{\mathrm{A}}>0,{\mathrm{L}}_{\mathrm{B}}=0\right)$. When $\mathrm{N}>\mathrm{N}\left(\mathrm{k}\right)$ all sites are active $\left({\mathrm{L}}_{\mathrm{A}}>0,{\mathrm{L}}_{\mathrm{B}}>0\right)$. This boundary is vertical at $\underset{¯}{\mathrm{N}}$ in Figure 4.1 when gathering is not used, because then the parameter k is irrelevant and $\mathrm{N}\left(\mathrm{k}\right)=\underset{¯}{\mathrm{N}}$ is a constant. When gathering is used at sites of type A, the boundary ${\mathrm{L}}_{\mathrm{B}}=0$ slopes up as indicated in Figure 4.1. This occurs because higher gathering productivity drives a wedge between the A and B sites in a way that favors the type-A sites, which are already gathering. As we explain later, our concept of sedentism focuses on the question of whether bad sites are active, so a key task will be to determine how $\mathrm{N}>\mathrm{N}\left(\mathrm{k}\right)$ can be achieved if it does not hold initially.

4.6 Long-Run Equilibrium

We turn next to population dynamics. Time is discrete with each period equal to one human generation. Agents come to adulthood at the beginning of a period, observe the weather conditions at each site, choose a site at which to work, produce food, have children, and then die at the end of the period. The regional (adult) population in period t will be denoted ${\mathrm{N}}^{\mathrm{t}}$. All adults have the same food income ${\mathrm{y}}^{\mathrm{t}}$ in period t regardless of the site at which they work. This income is obtained from ${\mathrm{y}}^{\mathrm{t}}=\mathrm{y}\left({\mathrm{N}}^{\mathrm{t}},\mathrm{k}\right)$ using Proposition 4.2, where we now include k as an explicit argument in all relevant functions.

The relationship between food and surviving children arises because more parental food leads to higher fertility, lower child mortality, or both. This is the Malthusian aspect of the modeling framework used in this chapter.

As in Chapter 3, population evolves according to

${\mathrm{N}}^{\mathrm{t}+1}=\mathrm{\rho }\left[\mathrm{y}\left({\mathrm{N}}^{\mathrm{t}},\mathrm{k}\right)\right]{\mathrm{N}}^{\mathrm{t}}\mathrm{t}=0,1,.$(4.6)

This equation implies that a non-null stationary population $\mathrm{N}>0$ must yield $\mathrm{\rho }\left[\mathrm{y}\left(\mathrm{N},\mathrm{k}\right)\right]=1$ or equivalently $\mathrm{y}\left(\mathrm{N},\mathrm{k}\right)={\mathrm{y}}^{\ast }$. We formalize this idea in the following definition.

In order to support a positive population in the long run, food income must achieve the level y* needed for demographic replacement. This gives an existence result.

The following monotone convergence assumption simplifies later analysis.