I. INTRODUCTION

A traditional view of historical economic growth is that prior to the Industrial Revolution there was very little change in per capita consumption. (1) This characterization is based largely on data from many studies summarized and augmented in Maddison (1982), which shows that median food consumption has generally been flat. The typical explanation is that humanity was stuck in a Malthusian trap. Roughly, this means that for per capita income above some subsistence level, population increased and below that level, it declined. Consequently, the population converged to a steady state at subsistence. In contrast, since the Industrial Revolution, countries with low per capita income have had high rates of population growth and countries with high per capita income have had much lower (or even declining) rates of population growth. At the same time, per capita consumption has soared.

This "hockey stick" characterization of historical consumption is seen as a puzzle and has led to the development of theories that can accommodate long-run stagnation followed by explosive and then sustained economic growth. The vast majority of these models assume a single composite good subject to a Malthusian constraint prior to the Industrial Revolution, which implies that consumption is kept at subsistence and growth is flat. Models vary however in how they explain explosive growth thereafter. After the Industrial Revolution, researchers generally assert, something fundamental must have changed. Hundreds of papers (we will review many of them in Section II) have been written on this subject, each pointing to a different "something." Perhaps the best-known example is unified growth theory, which explains the transition from long-run stagnation to explosive growth within a single framework and as part of the equilibrium path (Galor and Moav 2002; Galor and Weil 1999; Hansen and Prescott 2002; Jones 2001). For example, Galor and Moav (2002) posit that a single composite good consumed at subsistence due to a Malthusian trap leads to evolutionary pressures where survival favors individuals with abilities that accelerate technological progress. This leads to a state of technology requiring high human capital, which drives down fertility rates.

Over the past 10-15 years, new data coupled with new analyses of existing data have presented a more complicated picture of economic growth prior to the Industrial Revolution (see, for instance, Fouquet and Broadberry 2015 for a recent contribution). Perhaps most striking have been revisions to data from Maddison (1982), which rely less on median food consumption to draw conclusions about historical growth and consumption patterns. We will elaborate on the historical record below, but the new data on per capita gross domestic product (GDP) in the earlier era show a rich and interesting history rather than the history of long-term stagnation suggested by data on median food consumption. This calls into question whether it is appropriate to build theories on the assumption that humanity was subject to a one-good Malthusian constraint prior to the Industrial Revolution.

In this paper, we make the following contribution to growth theory in general and to the modeling of the Malthusian constraint in particular. We argue that it is not the assumption of a Malthusian constraint that is problematic. Rather, it is the assumption of a single composite good coupled with the Malthusian constraint that should be revisited. We start with the basic idea that humanity needs food to survive, but argue that this is a constraint that is as binding today as it was prior to the Industrial Revolution. On the other hand, the story of the Industrial Revolution is not a story about rising food consumption, but of explosive growth in the output and consumption of other goods that do not necessarily enhance survival, but that do enhance the quality of life.

Hence, we are led to study a two-sector model with two outputs: bread, by which we mean basic nutrients, and circuses, by which we mean all those things that make life worth living. We therefore add to a set of two-sector Malthusian models (see, e.g., Strulik and Weisdorf 2008), which we discuss in Section II. In our model, we begin with the most restrictive version of the Malthusian mechanism, where food determines population and in the long run, the steady state is fixed at subsistence. Next, we add a second, nonfood good (circuses), which has no impact on population dynamics, evolutionary fitness, or procreation. In our model, circuses capture goods that are enjoyed, but not necessary for survival. This certainly includes manufactured goods, but also includes entertainment (thus the name) and food items, such as wine or beef, that are not required for survival. (2) We allow technological change to enhance either the bread sector or the circus sector. In this manner, we create a theoretical framework where food production, which governs population, is divorced from the production of other goods, which affect living standards. (3)

The two-sector model generates conclusions about the impact of sector-specific shocks. A distinctive conclusion from our model is that technological change that primarily enhances the bread sector--the Neolithic Revolution being the most dramatic instance--will largely increase population, but can actually lead to a decline in living standards measured by per-capita circus production. In other words, our model accommodates the idea that a positive technology shock can leave everyone worse off. The first implication can be derived from a standard, single-good Malthusian model. However, long-run declines in living standards would require additional assumptions since in a single-good Malthusian model, living standards simply return to subsistence following a shock, which is equal to their preshock level.

Our model also implies that technological change that primarily enhances the circus sector--Ancient Rome being a key example-will largely increase long-run per capita GDR but leave population the same or even lower. In the standard Malthusian model, a technological change can increase per capita GDP only in the short run and must increase population.

We validate the theory against a variety of historical incidents for which data are available. Our procedure is to first determine which sector the technological shock impacted and then evaluate whether subsequent demographic and economic changes match our model predictions. Besides the Neolithic and Industrial Revolutions, we examine the Song dynasty in China, the Irish Potato Famines, the Black Death, and the Great Divergence. These historical episodes are quite different, but are all consistent with our model. Note, the aim is not to deny the possibility that some increases in historical living standards were effectively consumed by commensurate increases in population. That story is fully accommodated by our model and would occur if increases in bread productivity are not accompanied by sufficient increases in circus productivity. While episodes of this sort are well documented, we also provide evidence that there were sustained increases in living standards prior to the Industrial Revolution that were not immediately followed by increases in population. We argue from that data that this is because circus productivity rose, but bread productivity did not. This latter situation appears common in modern times. Our study has a very simple message: in light of recent revisions to historical data, it appears that the latter situation, though perhaps uncommon, also arose in ancient times. Models of historical growth should therefore be designed to accommodate this possibility.

Of particular interest is our analysis of the Great Divergence because our model suggests a novel account. The Great Divergence refers to the worldwide growth in the late Medieval period after Columbus and preceding the Industrial Revolution. Growth rates of total GDP in Europe and China were similar during this period, but in Europe per capita incomes went up while in China they declined. Our explanation of this is relatively straightforward: the import of New World crops represented a significant improvement in bread technology in those places suitable to their cultivation. Maize was suitable for cultivation in China and had a big impact on Chinese agriculture. Not so in Europe. We know that circus technology improved during this period. Even if we assume that circus technology improved as much in China as in Europe, the fact that bread technology improved in China but not Europe leads to the conclusion that per capita income should have risen in Europe relative to China. Therefore, our theory of the Great Divergence is that China was dragged down by agricultural improvement. This conclusion is buttressed if we observe that China was not the only place in which New World crops could be easily cultivated. The potato turns out to be particularly well suited to Ireland. Hence, if the theory is correct, Ireland--which surely had access to the same or similar circus technologies as (nearby) England--should nevertheless have seen a much greater population increase together with a per capita income fall relative to England--dragged down as they were by the potato. This is exactly what happened.

A second point to emphasize is the prediction that our model makes about the negative technological shock in the bread sector in Ireland due to the potato blight that led to a famine in the mid-1800s (sometimes known as the Great Famine). Our model predicts a big decline in population, which accords with well-known facts. The model also predicts a large increase in per capita GDP--measuring from before the blight to the new steady state--which is a less obvious consequence of a negative shock to the bread sector. As we document, it is indeed the case that during this period per capita GDP in Ireland increased much more rapidly than in England.

In the next section, we review the literature on historical economic growth. We highlight some papers that are similar to ours so that our key departures are clear. Next, we present the theoretical model. This leads to the basic hypotheses that improvements in bread technology increase population and lower income while increases in circus technology leave population unchanged and increase income (and conversely for technological regression). We then examine the key historical epochs for consistency with these hypotheses: the Neolithic Revolution, the Roman Empire, the Black Death plague, the Great Divergence, and the Industrial Revolution.

II. PREVIOUS LITERATURE ON HISTORICAL GROWTH

In this section, we discuss the relevant literature in economics on historical economic growth. To facilitate comparison across papers, we summarize them in Table S1 found in Appendix S1, Supporting Information. For each paper, the table includes: assumptions on preferences and technology, historical evidence on the abrupt shift in growth patterns to which the paper appeals and main findings.

We organize this review of literature around our key departures from earlier work. A first set of papers model early growth using a single-good subject to a Malthusian constraint. This assumption seems to be at odds with (1) new evidence of sustained high living standards in history and (2) the fact that food consumption has not exploded after the Industrial Revolution. A second set of papers considers two sectors. Most allow nonfood goods to affect population growth. Moreover, many two-good frameworks limit substitutability between the sectors, for example, only allowing nonfood goods to affect utility once a minimum amount of food is consumed. In contrast, we study a two-good setting where only one good, bread, drives population growth. We do not impose that goods be nonsubstitutable below some level of consumption. As we will demonstrate, our model can generate sustained rises and declines in living standards (depending on the sequence of shocks) despite a binding Malthusian constraint.

A. Malthusian Traps in Single Composite Good Economies

Appendix Table SI highlights how nearly all of the papers we review are motivated by the assumption that growth prior to 1760 was negligible. (4) To explain this approximation, many researchers assume there is a single composite good that is subject to a Malthusian constraint and that economic growth is driven by endogenous technological advancements. The biggest problem with the single composite good is that living standards and food consumption are inextricably linked. As a result, explaining how Malthusian stagnation could be followed by explosive economic growth requires an abrupt shift in economic fundamentals (preferences or production functions) after some kind of living standards threshold is reached. For example, Arifovic, Bullard, and Duffy (1997) present a model where, through an adaptive learning process, and beyond a certain level of accumulated human capital, an economy moves from a low-growth to high-growth equilibrium. Other papers (e.g., Becker and Barro 1988; Lucas 2002; Razin and Ben-Zion 1975) employ dynastic utility functions to examine how endogenous fertility patterns interact with changes in technology. In these latter papers, there is a fundamental shift in how children are valued that is used to explain how stagnation gave way to growth.

A related approach, sometimes known as unified growth theory, develops models that explain within a single framework Malthusian stagnation as well as subsequent explosive and then sustained growth. The idea is that the Malthusian trap itself affected population dynamics in ways that would eventually lead to an escape from the Malthusian trap. For example, Galor and Moav (2002) argue that Malthusian pressures led to the evolutionary selection of traits that would lead to higher economic growth. Like other authors, however, they make the assumption that prior to the Industrial Revolution, growth in standards of living (and differences in living standards across countries) was negligible. In our view, this assumption is at odds with ongoing revisions to historical data. This means that different theories should be developed that accommodate recent empirical work. Herein lies our contribution.

In a second type of research on historical growth, authors posit models in which growth prior to 1760 is not necessarily flat. A key example is in Acemoglu and Zilibotti (1997), who develop a model emphasizing the high variability of output during early stages of development. In their setup, output growth is slow, but also subject to randomness. (5) They explain this with a model where capital projects are few and subject to indivisibilities, which limits risk-spreading and encourages investment in safer projects that are less productive.

Authors positing models of pre-Industrial Revolution growth have recognized the inconsistency of some historical data (indeed, the data they use to motivate their own models) with Malthusian assumptions, and so they are left with the task of reconciling their models with the widespread view that growth was negligible. One illustrative example is Temin (2013), who provides a detailed and convincing account of high living standards in Ancient Rome. Nevertheless, rather than argue that his evidence contradicts the standard view of Malthusian bleakness until the Industrial Revolution, he instead argues that Ancient Roman living standards constituted a temporary aberration, that is, an exception to what was otherwise a widespread Malthusian trap. We disagree. Instead, the model we posit in Section III develops conditions under which a binding Malthusian constraint can coexist with increases in living standards. (6)

Another example is Baumol (1990). Like most other authors, he states his general acceptance that pre-1760 growth is flat. In other passages, however, he expresses some doubts regarding the common practice of relying on food production to infer growth patterns. This leads to a noticeable tension in the paper. The model in Baumol (1990) focuses on the allocation of entrepreneurial resources to explain growth differences, the aim being to explain great leaps in economic growth. However, the historical evidence he offers points to opportunities in eras and places--that are neither Great Britain nor post-Industrial Revolution--where great wealth could be gained. An example is the High Middle Ages, when a smaller and shorter Industrial Revolution occurred. After listing a number of technological improvements, including, for example, better woven woolen goods which surely raised utility, Baumol (1990) states, "In a period in which agriculture probably occupied some 90 percent of the population, the expansion of industry in the twelfth and thirteenth centuries could not by itself have created a major upheaval in living standards." However, in an accompanying footnote, he writes, "But then, much the same was true of the first half century of 'our' Industrial Revolution," by which he refers to the one beginning in the eighteenth century. Nonetheless, Baumol (1990) goes on to say: "[I]t has been deduced from what little we know of European gross domestic product per capita at the beginning of the eighteenth century that its average growth in the preceding six or seven centuries must have been modest, since if the poverty of the later time had represented substantial growth from eleventh-century living standards, much of the earlier population would surely have been condemned to starvation." This of course is not true in a two-sector model where living standards rise due to greater circus consumption and can be low without starvation.

B. Two-Sector Economies

Several papers are similar to ours in studying a two-sector model. One example is Hansen and Prescott (2002), who assume two types of production: a Malthus technology that uses a fixed factor (land) and a Solow technology that does not. Their motivation for the Malthusian technology is that its output has a considerable share of food in it and therefore requires land while the Solow production function results in mostly nonfarm output (or "factory" output) and therefore requires very little land. The Solow technology, which produces nonfarm output, appears similar to our circus sector technology. The biggest difference here is that while they divide the sectors production wise (land using or not), we divide the sectors consumption wise (bread and circus have different demographic effects). Essentially, there is only one consumption good in their economy. Their model, however, implicitly assumes that individuals consider food and nonfood items to be substitutable reproductively on a one-to-one basis. Given that their population growth function depends on consumption in general and not just on consumption of food, the model violates the basic Malthusian premise of population growth being kept in check by the availability of food. (7)

In a related paper, Yang and Zhu (2013) allow the Solow technology to govern both nonagricultural production, which never uses land, and farm production, which does initially use land. After industry develops, capital replaces land in farm production. In Yang and Zhu (2013), however, it is total consumption, agricultural and nonagricultural, that drives population growth. Furthermore, defining consumer preferences by requiring food consumption to constitute a specific amount of total income with the rest being spent on nonfood items essentially makes nonfood production the true driver of population growth.

Another set of papers that are similar to ours considers two consumption sectors. In Voigtlander and Voth (2013b), high growth is achieved by a major shock to population, the Black Death, which raised wages and also induced demand for goods produced in cities. As a result, wages in cities rose, which attracted new citizens. As death rates in urban centers were higher, population growth was kept in check. In a different paper offering a compelling explanation of economic responses to the Black Death, Voigtlander and Voth (2013a) argue that a new Malthusian equilibrium arose prior to the Industrial Revolution where population is lower and wealth is higher. This occurs since land abundance induced by the plague led to a shift toward land-intensive pastoral agricultural production in which women had a comparative advantage. Higher female employment raised the opportunity cost of marriage and childbearing, which lowered population and increased wealth. In Voigtlander and Voth (2013b), a form of lexicographic preferences means that nonfood goods are enjoyed only after subsistence is reached. However, population dynamics are driven both by food and nonfood consumption, with increases in the latter raising mortality rates. (8) In Voigtlander and Voth (2013a), minimum consumption of a composite good is effectively assumed (the marginal utility of consumption below a "basic needs" level is large and positive) and the only other good in the economy over which women have preferences is their offspring, which is also tied to survival. In other words, in both papers, all goods affect survival and substitution between goods is limited. (9)

Another similar paper is Davies (1994) who models beef as a commodity that brings higher utility per calorie than potatoes. (10) Here, the idea is to add a subsistence constraint to classical demand analysis to show the Giffen effect. In particular, if humans consume meat and potatoes and are above but near subsistence (an assumption that violates Malthusian precepts), an increase in potato prices could lead individuals to fall below subsistence unless they replaced beef with more potatoes. Here, the subsistence constraint is at the individual level versus the population level, which means that like other papers substitutability between goods is limited. In another paper, Taylor and Brander (1998) model a food sector and an "other goods" sector, which includes moai, the mysterious monumental statues on Easter Island. Their focus is on steady state dynamics to explain how the population of Easter Island may have been wiped out by an "overcorrection" of population growth. In their setup, only labor is used in the nonfood sector. Also, the food good is not subject to a strict Malthusian constraint. Instead, higher per-capita food consumption is assumed to increase the rate of population growth.

More similar to us, Strulik and Weisdorf (2008, 2009) and Weisdorf (2008) develop sophisticated theories on how industrial productivity growth could restrain fertility--and thus raise living standards under a Malthusian constraint--by making children relatively more expensive than manufactured goods. Strulik and Weisdorf (2008) is probably the closest paper to ours. They model two sectors where population growth depends solely on prices in the food sector. There are three key differences. First, the model in Strulik and Weisdorf (2008) uses learning by doing and studies balanced growth equilibria. In contrast, we focus on technology shocks. Therefore, the model in Strulik and Weisdorf (2008) has difficulty explaining why certain areas of the world experienced technology shocks that raised living standards considerably and then collapsed for reasons that have nothing to do with demography. Still, Strulik and Weisdorf (2008) do give evidence that population growth is responsive to food shocks. A second difference is that our Malthusian model is more strict--we use level of food rather than price of food to affect population. Third, there is no demand for children in our model. In contrast, Strulik and Weisdorf (2008) assume that utility is linear in children and manufactured goods and that children determine the growth rate of population in a linear way. This would be analogous to our assuming that bread and circuses are perfect substitutes, which we do not. Therefore, their conclusions are rather different. In particular, we show that introducing a nonfood sector that enhances utility, but that does not affect population growth, is sufficient to deliver the patterns of economic growth observed both before and after the Industrial Revolution in the presence of (and despite) the binding constraint that humans need food to survive.

C. Maddison Data and Recent Revisions

Nearly all of the papers we have discussed (and that are listed with more detail in Appendix Table S1) rely at least to some extent on data from Maddison (1982) to motivate their models explaining historical growth patterns. Recent historians have found a richer picture. We will consider some specific instances of historical growth subsequently.

It is worth mentioning that the original dataset was meant to be a work-in-progress. The following excerpt from Bolt and van Zanden (2013) indicates this point.

   In view of the new research that has been done,
   many of the pre-1820 estimates (and all the pre-1600
   figures) had to be modified. Maddison was of course
   aware of this: his strategy was to produce numbers
   even if a solid basis for them did not always exist,
   expecting that scholars might disagree and do new
   work to show that he was wrong. In this way he
   induced many scholars to work on these themes and to
   try to quantify long-term economic development. This
   was a highly successful strategy, but not always understood
   and appreciated by his colleagues: thanks to his
   pioneering work and the many, many reactions to it,
   we can now present a much more detailed overview of
   long-term economic growth than when he started his
   project in the 1960s. (11)

Educated guesstimates constitute a valuable approach, but they are vulnerable to the contamination of the Malthusian presumption. As pointed out in a footnote in Galor and Weil (2000), parts of the Maddison (1982) data dealing with the pre-Industrial Revolution era may have been imputed with a Malthusian framework in mind. In other words, data imputed by presupposing Malthusian stagnation is used to motivate and test models of Malthusian stagnation. (12)

Turning to the evidence, a number of authors have begun to revise data on historical consumption. an undertaking that, according to the quotation above, was originally intended. Many results from this undertaking have been collected in Fouquet and Broadberry (2015) and Bolt and van Zanden (2013). We list some key contributors in Table 1 along with a note on their evidence and their main findings. In fact, some are not even that recent. Finley (1965) noticed that there was early evidence of technological progress and improvements to available goods. More recently, Broadberry et al. (2015) and van Zanden and van Leeuwen (2012) find evidence of pre-industrial persistent economic growth in England and Holland, respectively. This is reflected in several recent revisions to the pre-1820 estimates of per capita GDR The revised data show an economic landscape far from stagnation. We collect these revisions (and compare them to Maddison's original numbers) in Table 2. For instance, Broadberry et al. (2015) show British per capita GDP doubling between 1270 and 1700 and doubling again between 1700 and 1870. Furthermore, van Zanden and van Leeuwen (2012) find that Dutch per capita GDP more than tripled between 1000 and 1500. Malanima (2011) finds Italian per capita GDP to be declining between 1400 and 1820 from a level significantly higher than was suggested in the initial Maddison dataset. Beyond suggesting a revision of our understanding of economic performance of these regions prior to 1820, these new estimates also justifiably cast doubt on the other flat parts of the pre-1820 Maddison dataset, which are currently under scrutiny.

III. MODEL

We now develop a model that incorporates the following ideas. Humans need food to survive and population is therefore bound by a biological constraint. This was as true prior to 1760 as it is today. Thus, imposing a Malthusian constraint relating food supply and population growth is a reasonable approximation, but the constraint applies to modern economies, too. On the other hand, humans neither need nor want to consume much more food than is biologically necessary. Therefore, explosive growth in food consumption has never been and should never be an expected byproduct of economic growth. Instead, growth in living standards should be measured in light of consumption of other, nonsubsistence goods. Finally, imposing the same type of "Malthusian" constraint with regard to the consumption of all other goods does not make sense as most other goods are not necessary for survival, but do affect the quality of life.

Therefore, our approximation of how consumption and demographic change interact throughout history is a model that captures our need for food along with our desire for everything else. The model we present is simple, based on standard neoclassical economic theory and capable of delivering the result that technology change can drive economic growth. The key innovation is to impose a Malthusian constraint that links the supply of basic nutrients to population growth, and to also permit a second good that is enjoyable, but that does not affect demographics, either through a Malthusian constraint or through some other device like evolutionary fitness. Notice that, absent the assumption of Malthusian bleakness for most of human history, we are no longer compelled to assume something else that would produce an abrupt escape from it. Rather, we show that in spite of a Malthusian constraint that binds food to population growth, changes in technology are sufficient for explaining growth patterns prior to 1760 along with growth patterns thereafter.

In particular, we examine a simple two-sector model. There are two goods, which we call bread X and circuses Y. These are produced using two inputs, land L and people N. The endowment of land is fixed at L, while at a particular moment of time the endowment of people (labor) is denoted by N. The number of people, however, will be endogenously determined by a Malthusian dynamic. We focus on the most standard formulation for technology: the Cobb-Douglas.

The technologies for bread and circus production are of the following form.

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Here, B, C represent overall productivity in the bread and circus sector. Our assumption about the production technology is that output elasticity in the bread sector with respect to land is greater than that in the circus sector: that is, that [[gamma].sub.B] [less than or equal to] [[gamma].sub.C] [less than or equal to] 1. The economy is subject to standard resource constraints,

[L.sub.B] + [L.sub.C] [less than or equal to] L, [N.sub.B] + [N.sub.C] [less than or equal to] N.

There is a single representative consumer whose per capita consumption of the two goods is denoted by

X = X/N, y = Y/N

and has utility function,

U(x,y) = (([x.sup.1-[rho]] - 1)/(1 - [rho])) + (([y.sup.1-[sigma]] - 1)/(1 - [sigma]))

with [rho], [sigma] > 0. In other words, man does not live by bread alone. This reduces to the constant elasticity of substitution (CES) utility function if [rho] = [sigma] but in order to allow bread to be an inferior good, we assume that [sigma] [less than or equal to] [rho]. Our other primary assumption is that the benefit of circuses does not decline too much as consumption of circuses increases: we assume that [sigma] < 1 which in the CES case means that the utility function exhibits greater substitutability than the Cobb Douglas. (13) Note, however, that we do not assume that bread and circuses are perfect substitutes. Moreover, we discuss how most of our results carry through under alternative specifications of the utility function in Section III.C, where we discuss robustness of key modeling predictions to alternative assumptions.

Turning to population dynamics, we begin with the idea that economists do not see individuals as maximizing survival and the Malthusian model does not suppose that population grows above the level of subsistence and that everybody instantly dies the moment food consumption falls below subsistence. We consider a population dynamic in which lower food consumption leads to lower birth rates and higher death rates. Furthermore, we assume that the trade-off between bread and circuses for an individual is the usual type of trade-off studied by economists. Specifically, for any given population level N, we model the economy as a competitive market for bread and circuses. Equivalently, according to the first welfare theorem, we assume that labor is allocated between the two sectors to maximize the utility of the representative consumer.

Population is assumed to move according to the usual Malthusian dynamic with respect to food consumption. We use [bar.x] to denote the subsistence level of bread--circuses being pleasant but unnecessary for survival. Recalling that x is the per capita output of bread, it follows that population increases if x > [bar.x]and declines if x < [bar.x]. Hence, we will be interested in the Malthusian long-run steady state equilibrium population level where N = X/[bar.x]. (14) Given that long-run steady state consumption of bread is subsistence, our measure of living standards will be given by per capita consumption of circuses y, which is a measure of per capita GDP above and beyond that which is needed to survive. Moreover, to make sensible comparisons across time, we compute real per capita GDP by assigning fixed weights to bread and circuses. Since in any Malthusian equilibrium bread consumption is fixed, increases or decreases in circus production or consumption are reflected in corresponding changes in real per capita GDP.

A. Discussion of the Model

Before discussing comparative statics, we examine the key assumptions in our model. First, we discuss the measurement of bread, circuses, land, and labor. Then, we look at evidence in support of the assumption that food is bound by a Malthusian constraint. Finally, we show evidence on the degree of substitutability of bread and circuses even among groups who are arguably at or near subsistence.

Inputs and Outputs. In our model, we refer to two goods: bread and circuses. We use this terminology instead of agriculture and manufacturing to emphasize what each good in the model represents. Bread, in our model, represents nutrients needed for subsistence as required by a strict interpretation of the Malthusian constraint. (15) Circuses include all goods that can enhance utility, but which are not required for subsistence. Understood this way, circuses can include food items that are enjoyable to eat, but which are not need for survival (e.g., meat or sweets rather than cereal), entertainment, and nice-looking clothing (consider chemical dyes, which played a large role in the second German industrial revolution). Likewise, circuses contain many goods that are not manufactured at all, entertainment being one example. Our division of goods in this manner seems to be consistent with the idea that the Industrial Revolution brought about a decrease in the cost of luxury items. For example, chemical dyes did not lower the cost of making clothes but it lowered the cost of making nice-looking clothes.

Given this interpretation of the two goods in the economy we model, it seems straightforward to include land in the production of circuses, that is, to assume [[gamma].sub.C] < 1. (16) Indeed, if circuses include types of food that are not needed for subsistence, such as meat, it is natural to assume that land is an input. This assumption becomes even more compelling if "land" is understood to include everything in fixed supply independent of population that might be used either for bread or circuses, including chemicals (for fertilizer), energy, metals, and other mineral deposits (plows vs. trains), forests (for wood and other building materials for farms or factories), farmland (for food vs. chemical products), and water.

To buttress the argument that our interpretation of what land entails is appropriate and, moreover, that the inclusion of land in the production of circuses is sensible, we consider urbanization. Urbanization is a proxy measure for the amount of land that is not used for bread production, but is instead used in the circus sector. Obviously, urban land is used for housing, though land used in the bread sector would also require housing. The fact is that land used to build cities is not being used, generally, in agriculture. Our estimates of arable land used in cultivation in Appendix S3 that show the amount of urbanized land is considerable (Foley et al. 2005). To measure urbanization, we assess how much arable land is used for something other than agriculture across countries. In Appendix S3, we report actual land use for several countries. Urbanization rates vary considerably. There are highly agricultural countries, such as Australia and Canada (5%-7%); intermediate ones, such as Spain and the United States (12%-20%); and highly urbanized countries, such as France, Germany, Italy, and Japan (25% to over 50%). (17) These numbers suggest that much land is used in something other than agriculture, which we believe supports the assumption that land is dedicated to the production of nonsubsistence items, which in our model are captured as "circuses."

Food Consumption Near Subsistence. Our model assumes that per capita food consumption remains at (or near) subsistence most of the time. Prior to the Industrial Revolution, there is strong evidence this is true. Indeed, for a long time, economic historians only had data on food consumption and found it was quite flat (the handle of the hockey stick). However, food consumption remained flat also after the Industrial Revolution. Below we report U.S. food consumption in the twentieth century: it did not increase much until about 1980 (Figure 1). (18) The implication is that GDP includes many goods that do very little (in our model: nothing) to encourage survival, but do constitute economic growth. Economists do not see individuals as maximizing survival. Rather, agents trade off longevity with other consumables that improve the quality of life.

Food and Circuses Are Partial Substitutes. In our framework, we assume that bread and circuses are at least partially substitutable. This is different from authors who assume, for example, that agents derive utility from nonfood items only after a certain subsistence level has been reached. It is precisely this degree of substitutability that makes the economy substitute circuses for bread in the face of circus sector improvements making larger populations unsustainable in equilibrium, leading to smaller economies with higher per capita GDP.

It turns out that such substitutability has considerable empirical support. Banerjee and Duflo (2011) extensively study consumption by very poor people in the modern era--those who live near subsistence. The authors document how poor people in the developing world spend a significant fraction of their income on weddings and funerals--and afterwards lament their lack of food. "In Udaipur, India, where almost no one has a television, the extremely poor spend 14 percent of their budget on festivals." When television sets are available they are widely owned--and the poor people that own them maintain that television is more important than food. When people who are hungry are given a choice between more calories or better flavor, they often choose better flavor. These are scarcely new or controversial observations: much the same point is made by George Orwell in The Road to Wigan Pier (Orwell 1937). Relatedly, in modern health economics, the value of a medical innovation is sometimes misleadingly computed focusing solely on its effect on survival. Recent work has shown that this value should take into account that dynamically optimizing agents are willing to forfeit years of life when doing so can increase comfort or consumption. (19)

B. The Malthusian Equilibrium and Comparative Statics

We characterize equilibrium and describe comparative statics via a set of propositions, stated in this section. Detailed derivations needed for solution of the model are found in Appendix S2. Proofs for propositions stated in this section are also in Appendix S2. In our model, a short-run equilibrium is the allocation of-land and labor between the two sectors that maximizes utility for a given population N. A Malthusian (or long-run steady state) equilibrium is a short-run equilibrium, where N is chosen so that consumption is at subsistence: x = [bar.x]. We now give the basic properties of equilibrium, proven in Appendix S2.

PROPOSITION 1. For any [bar.x], there is a unique Malthusian equilibrium.

The following result is immediate and shows that our utility function captures the key fact that circus consumption and circus expenditure move in the same direction:

PROPOSITION 2. In a Malthusian equilibrium, changes in technology that lead to higher per capita circus consumption y result in an increase in the share of expenditure on circuses y[U.sub.y]([bar.x], y)/([bar.x][U.sub.x]([bar.x], y))= [y.sup.1-[sigma]]/[[bar.x].sup.1-[rho]].

Define relative circus productivity

[mathematical expression not reproducible]

to be a measure of land productivity in the circus sector relative to the bread sector. It is convenient to parameterize the economy by B, [zeta] In this case, a technological improvement means a higher B, a higher C, or some combination of the two. Such an improvement is bread-favoring if [zeta] declines and circus-favoring if [zeta] increases. Our main result is

PROPOSITION 3. If [[gamma].sub.C] < 1, then long-run real per capita GDP y and the ratio of labor used in circuses [N.sub.C]/[N.sub.B] is a strictly increasing function of relative circus productivity [zeta]. If [[gamma].sub.C]= 1, long-run per capita GDP and the ratio of labor used in circuses [N.sub.C]/[N.sub.B] depends only on absolute circus productivity C and is an increasing function.

We do not view the implication of [[gamma].sub.C] = 1 (labor ratio is independent of bread productivity) as an attractive one. It is true that the land ratio moves in the same direction as the labor ratio. However, if there is more land in bread than in circuses, then a greater percentage of the population (compared to percentage of land) moves between sectors. Consider a concrete example. If 20% of the land is in circuses and this drops to 10% and if the coefficient of land in circus production is one-sixth and in bread it is one-half, then originally 56% of the population is in circuses, but this drops to 36%. Thus, 20% of the population shifts against only 10% of the land. Regardless, we show below that if we assume complete capital immobility but that some fixed factor is used in the production of circuses then the [[gamma].sub.C] < 1 result is robust.

PROPOSITION 4. If [[gamma].sub.C] < 1, then population is strictly increasing without bound in B. So, given an initial B and [zeta] and a technology shock [zeta]', there exists [bar.B] such that N' > N iff B' > [bar.B]. Moreover, [bar.B] [less than or equal to] B iff [zeta]' < [zeta]. If [[gamma].sub.C] = 1 population is strictly increasing without bound in B and decreasing in C.

In words, bread-favoring technological improvement lowers per capita income and raises population. There are three mechanisms to explain this. First, higher bread productivity raises population, which makes land more scarce, thereby lowering circus productivity. Second, improvements in bread productivity shift labor into the circus sector. Third, increased population, holding fixed the relative amount of labor in the circus sector, increases the absolute amount of labor in that sector, and if that sector has decreasing returns, this lowers productivity in that sector. In contrast, circus-favoring technological improvement raises per capita income and raises population if the improvement in bread technology is sufficiently great and lowers population if the improvement in bread technology is too small. With circus-favoring technological regression (for instance B declines with [xi] fixed due to climate change or crop disease), per capita income goes up and population goes down.

A remark on our choice of technology parameterization is due. Relying on B and [zeta] instead of B and C allows for the clean characterization of the relationship between technological changes and growth as captured by the propositions above. More importantly, we are interested in what happens when technological change improves both B and C. We have the intuition that if C goes up fast enough relative to B, then we should see an increase in per capita income. One way to look at this is to assess what happens if they go up in proportion. But there is nothing special about this particular benchmark and in fact we discovered that it is not a terribly helpful one. Instead, we use as our benchmark the ratio [zeta] which allows us to obtain an exact characterization of what it means for C to increase enough relative to B that per capita income increases. Of course we want to know that when both C and B increase, we can get population to increase as well, but that is well identified by Proposition 4 which says that any technological change leads to increased population if it is big enough.

We also give one short-run result concerning exogenous decreases in the population:

PROPOSITION 5. For any N, there is a unique short-run equilibrium. If [rho] > 1, then a decrease in N results in a short-run equilibrium with a higher wage (in bread units) and higher per capita GDP.

This shows that this model exhibits the same basic characteristics as a variety of other models used to analyze the consequences of shocks such as the Black Death plague.

We can summarize the relevant comparative statics with the following hypotheses about the long run:

* A technological improvement that lowers relative circus productivity will increase population, decrease per capita GDP, and lower expenditure share on circuses.

* A technological improvement that increases relative circus productivity will increase per capita GDP, increase expenditure share on circuses, increase population if it is strong, and decrease population if it is weak.

* A technological shock that lowers productivity in the bread sector will decrease population, raise per capita GDP, and increase expenditure share on circuses.

* A shock that decreases population in the short run results in a higher wage and higher per capita GDP.

C. Robustness of Model Predictions to Alternative Assumptions

We consider the robustness of key model predictions to alternative assumptions. Calculations are in Appendix S2. We provide a brief summary here. We focus on the assumption that land is used in circus production. In fact, allowing land to affect circus production embodies two assumptions: there is a single fixed factor and it is used in both sectors. In contrast, many models assume that only labor is used in the circus sector. We agree that labor is relatively more important in the circus sector, an assumption we maintain throughout the paper. However, the assumption that only labor is used in the circus sector likewise has broader implications: it assumes that land is not used in circuses and, more broadly, that no other fixed factor is used in circuses. We evaluate each in turn. As we show in Appendix S2, in a model where there is no fixed factor used in circuses, our results on C and y remain intact, but B has no effect on y. Next, we focus on a model where land is used in both sectors, but cannot be switched between sectors. However, we require decreasing returns to scale (with respect to labor) due to the use of a fixed factor (that does not need to be land). We show that it is still the case that y falls if B increases.

These alternative models demonstrate that independence between B and y obtains only if we are willing to assume constant returns to scale to labor in the circus sector. Empirically, we argued above that fixed factors are used in both sectors and that in addition some of the same fixed factors are used in both sectors. Even if we think that bread and circuses do not compete for the same natural resources, and even if we believe that there is no real scarcity of land and other natural resources, land and natural resources vary in their quality. Even if there is essentially unlimited coal, there is by no means an unlimited amount of coal that can be easily and cheaply mined; one important tale in the industrial revolution is the introduction of steam power to remove water from ever deeper and more costly coal mines. Hence, we think our model is the right one. However, our results are robust to there being a lack of substitutability between fixed factors used in the two sectors. Moreover, as we will detail below when we discuss historical evidence in line with our modeling predictions, the prediction that increases in B lead to lower j along with lower [L.sub.C]/[L.sub.B] seems consistent with the difference between Europe and China. It also helps to explain Ireland, which other theories such as those in Voigtlander and Voth (2013b) do not.

In Appendix S2, we provide additional evidence that our main results, in particular, that increases in B lead to lower y, are robust to additional assumptions. This includes versions of the model where o is set to 1 or 0. We also show that main results hold if we do not specify a particular functional form for utility over x, but just require the function to be differentiable at [bar.x]. Finally, we consider an assumption that is similar to one in Voigtlander and Voth (2013b) that increases in circuses lower population due to the higher rate of mortality in cities. The qualitative features of our model do not change.

IV. HISTORICAL EVIDENCE FOR MODEL IMPLICATIONS: POSITIVE TECHNOLOGY SHOCKS

The model presented in the previous section generates predictions about the impact of shocks on demographics and consumption. We now discuss whether historical evidence supports these model implications. We study historical epochs characterized by technological shifts and then examine subsequent demographic and economic transitions. The key to inspecting historical evidence in light of model predictions lies in our ability to determine the sector in which technology change occurred. The more precise predictions of the model map sector-specific shocks (or combinations of shocks) to shifts in population and living standards.

We discuss the following historical epochs: the Neolithic Revolution, the Classical Period (including Ancient Rome and Song China), the Late

Medieval period (in particular, the Great Divergence), and the Industrial Revolution. In the following section, we also discuss "negative" shocks, including the Black Death and two potato famines in Ireland, episodes which we argue are also accommodated by our model. The historical examples we refer to are chosen because they are periods characterized by large technology shocks that led to substantial demographic and economic transition, which means it is possible to observe changes many centuries later. (20) To organize our discussion, for each epoch, we first discuss evidence that identifies the type of shock (or combination of shocks). Second, we refer to model hypotheses about population and consumption given the type of shocks. Third, we discuss whether data support model implications.

A. Neolithic Revolution

Perhaps the best historical example of an improvement to bread technology without a commensurate rise in circus technology is the Neolithic Revolution, which is widely understood as an agricultural transition, moving humanity from hunting and gathering to farming. The Neolithic Revolution likely began in the Fertile Crescent around 10000-8000 BC and in other places between 10000 and 5000 BC. It brought the domestication of plants and animals and led largely nomadic groups to create permanent agrarian settlements.

Technology Change. The Neolithic Revolution is generally characterized as a revolution in the production of food and the dissemination of new farming techniques. As Bocquet-Appel (2011) writes, "The major change that arose from this 'revolution' was, in evolutionary time, the number of potential mouths it was possible to feed per [km.sup.2]" (p.560). When distinguishing the Neolithic from other eras, scholars have used the concept of the "Neolithic Package," which is the collection of elements common across space and particular to the Neolithic period (Cilingiroglu 2005). There is some debate as to what, precisely, the "package" contains. (21) However, the list of contents is generally seen as including pottery, cultigens, and domesticates. In contrast, there is little evidence that circus technology advanced as much as bread technology during the Neolithic Revolution.

Model Implications. According to our theory, a bread technology improvement where there is no improvement in circus technology should lead to growth in population density and a fall in living standards. That population density would increase after an improvement in bread technology is also accommodated by a standard single-good Malthusian model. In that model living standards would first rise then fall back to their original level--the "Malthusian trap."

In the single-good model, however, there is no possibility of a long-term decline in living standards. Living standards can fall to preshock subsistence and no further. Our two-good model differs in that it predicts that living standards can fall below preshock levels since there are fewer circuses to go around, but more mouths to feed.

Evidence. A key prediction of our model is that an increase in bread production is followed by an increase in population density. It is well known that population levels and density rose during the Neolithic Revolution as individuals formed permanent settlements around farms (Bocquet-Appel 2011). In terms of the magnitude, Shennan et al. (2013) discuss evidence of a sixfold rise in population density as a result of improvements in agricultural technology over the course of the Neolithic Revolution. Kremer (1993) argues that population could have risen much more.

A number of authors have claimed that living standards fell after the Neolithic Revolution. In light of post-Revolution falls in living standards, Diamond (1987) provocatively calls the adoption of agriculture "the worst mistake in the history of the human race." We find this view somewhat shortsighted since it is difficult to argue that today's living standards are worse than those of hunter gatherers. However, in the aftermath of the Neolithic Revolution there is considerable evidence of falling living standards. (22) Guzman and Weisdorf (2011) point to evidence of worse health and nutrition and longer working days among early adopters of agriculture in comparison to hunter-gatherers and Rowthorn and Seabright (2010) claim that scarce resources needed to be diverted to defending agricultural land. Shennan et al. (2013) use cemetery data to establish increased mortality, which they attribute to lack of clean drinking water and infectious disease as villages were formed and people were bound to farms and were therefore compelled to live in close proximity to one another. Relatedly, Bandy and Fox (2010) claim that the advent of villages led to higher levels of social conflict. Higher mortality rates did not outweigh rising population. However, they do provide some evidence of lower living standards. (23)

It is important to point out that later technology shocks would alleviate some of the pressures associated with permanent settlements based around agriculture. Innovations in sanitation technology would prevent disease, developments in architecture would improve dwellings and social innovations would lead to better tools to resolve conflicts. Moreover, per capita income would eventually explode during the Industrial Revolution, which may never have occurred absent agriculture and permanent settlements. Therefore, later technology shocks cast doubt on the claim that agriculture was a "mistake." Still, there is good evidence of falling living standards following a positive shock to food production. This pattern is difficult to reconcile with a standard single-good Malthusian model, which would instead predict that living standards would fall back to subsistence following population growth after a technology shock. In contrast, our model with two sectors accommodates the possibility that living standards (in our model: circuses per capita) could fall if bread technology supports a larger population left with fewer nonfood goods per capita.

B. The Classical Period: Ancient Rome

We now focus our attention on Ancient Rome, in particular, the first 100 years AD. The reason is that there is good evidence of progress in technologies related to the circus sector, but also evidence that bread technology remained relatively unchanged.

Technology Change. Finley (1999) provides accounts of continuous progress in grape and olive-pressing equipment during this period in Rome. Moreover, Greene (2000) reports that fieldwork projects around the Mediterranean have shown that these technologies were applied throughout the region during Roman times. Here, it is important to note that grapes and wine are not goods that would fit into our bread sector since they are not necessary for survival and should not therefore be seen as governing population dynamics. Instead, these are goods that improve the quality of life and are therefore in the circus sector. (24) Other evidence of nonagricultural innovation includes water power, use of pumps in mining, mass production of pottery, bricks, glass, and paper in factories, aqueducts, fountains, and baths.

At the same time, Greene (2000) states that "Finley ... saw no selective breeding and no changes in tools or techniques for plowing, harvesting, or irrigation, although he did acknowledge modifications of land use." In other words, there is evidence of large improvements in technologies related to goods in the circus sector and also evidence that technology in the bread sector remained about the same, relying on old techniques.

Model Implications. According to our theory, a circus technology improvement where there is little improvement in bread technology should lead to no growth in population density and an increase in living standards. Note that a standard single-good model with a Malthusian constraint only permits short-run improvements to quality of life in the form of higher per capita income until population growth catches up, consuming output in excess of what is needed for subsistence. In contrast, our model accommodates the idea that a circus technology shock can improve living standards if there is no commensurate rise in bread technology. The reason is that the economy cannot sustain additional mouths to feed, but can produce more utility-enhancing goods per person.

Evidence. Some economic historians argued that the baths, monuments, colosseums, and palaces that Roman civilization was famous for were only available to a select few while ordinary individuals were just as poor as their counterparts in any other ancient society. (25) There are two issues with this argument. First, from a theoretical point of view, per capita income includes the income of the very rich--the huge increase in world per capita income during the Industrial Revolution (discussed in more detail below) is a huge increase in mean income and median income changed very little until the 1980s. Second, more recent research has shown that the scale of Roman production of consumer goods was beyond what a tiny fraction of elites could use. For example, near the River Tiber, there is a mound of pottery waste, known as Monte Testaccio. The mound is composed of the fragments of an estimated 53 million amphorae, which once contained 1.6 billion gallons of olive oil that was imported to a nearby port in ancient times. No elite class could have consumed so much oil. Although anecdotal, this piece of evidence is in line with what scholars of Ancient Rome have carefully established: that the average Roman lived a far better life than the average European in the post-Roman period. As Ward-Perkins (2005) points out, for example, the poorest Roman rural houses had tiled roofs, something unavailable even to the elites of the post-Roman era. What's more, pottery and tiles are heavy and difficult to move around. If they can be produced in such a large scale, and traded across thousands of miles as archaeological findings suggest, it stands to reason that other consumer goods were traded, too, but are simply found less frequently because they are more perishable.

Further evidence on the Roman economy comes from the demand for copper currency. The mining of copper emitted pollution that was captured in ice in Greenland. In fact, ice data reveal the Roman period as one of the three peaks of world-wide copper production (Hong et al. (1996)). The contrast between the peaks and the time in between is striking. Immediately after Rome collapsed, world-wide copper production fell to less than a seventh of its previous level. A summary account of the Roman economy and the prosperity it brought to common Romans can be found in Temin (2013), who argues that Roman Italy was comparable to the Netherlands in 1600 and estimates per capita GDP over the entire Roman Empire at $1,000 dollars in year 1900.

If, in Roman times, there was an improvement to circus technology and no commensurate improvement to bread technology, the model predicts that population density would remain relatively flat. Indeed, when we look at evidence from Italy, we find little evidence of explosive population growth. In particular, Frier (2000), using data from a variety of sources, including McEvedy and Jones (1978) shows that population in Italy in AD 14 was 7 million (with a land mass of 2,500,000 [km.sup.2]). In AD 164, population had risen to 7.6 million. Population density (per [km.sup.2]) is calculated at 28 in AD 14 and 30.4 in 164. That Rome developed technologically and population did not rise casts doubt on the assumption of a single composite good subject to a Malthusian constraint as a valid way to model historical growth. The lack of population growth in Italy during the Roman era is important for a second reason: Rome did eventually fall, so it is possible to argue that traditional Malthusian forces were at work and high per capita income was simply a transitory phenomenon until Malthusian forces caught up. The problem with this argument is that to be true it would have to be the case that population density was rising so that eventually population outstripped food supply. There are many reasons for the fall of Rome--but increased population density in Italy is not one of them. (26)

C. The Classical Period: Song China

We have argued that Ancient Rome provides a counterexample to the view that sustained increases in living standards occurred first during the Industrial Revolution. Here, we argue that the Song dynasty in China offers another. The period we consider began in 960 and ended in 1127. The period saw a flowering of culture, science, and commerce--along with an increase in population.

Technology Change. China during the Song dynasty is generally characterized by a number of technological advancements, both in agriculture and in other sectors. For agriculture, there were increases in plowable land and improvements to irrigation. In other sectors, there were a number of developments, including gunpowder, the use of coal as fuel, improvements in iron and steel production, the introduction of banknotes, and the development of joint stock trading companies and improved international commerce.

Model Implications. In our model, a sufficiently large improvement in technology should lead to increases in population, but has an ambiguous impact on living standards. If the shock to B dominates, living standards would be expected to fall (as is the case with the Neolithic Revolution). If C dominates, living standards would rise.

Evidence. There is evidence that population grew in Song China, roughly doubling from about 55 million to 100 million (Maddison 2003). However, in Maddison's data, which relies largely on food intake, Song China's GDP per capita is estimated to be $450, compared to $600 in Ming China (the next dynasty). This modest rise may not tell the whole story. Like Ancient Rome, Song China enjoyed great prosperity in industry and commerce. Moreover, Song left reliable administrative records with which we can reconstruct economic life with some confidence. According to Liu (2015), only a third of the Song government's tax revenue came from agriculture. The other two thirds were from commerce and manufacturing. In comparison, agriculture contributed as much as 84% to the Ming government's tax receipts. The difference is not because Song levied lower taxes on agriculture. Just the opposite, Song's agricultural tax was even higher than Ming's. Its total tax was three times as large because Song's industry and commerce were highly developed. The relative importance of the Chinese dynasties and their abundant records tell us that Maddison's estimates may not be correct. Following Pomeranz (2009), historians such as Morris (2011) have argued that Song's standard of living was similar to that in Europe as late as the eighteenth century: substantially higher than Maddison. This may explain why Marco Polo marveled at Chinese civilization (he arrived when the Mongols were about to conquer the Southern Song), whereas later European visitors to Ming and Qing China were hardly as taken by what they saw. This is not surprising. In the intervening years, Europe developed and China declined. The rise and fall were not in the absolute size of the economy--population grew in both ends of Eurasia during the time in between--but in the ratio of industry to agriculture, which determined average living standards.

D. Late Medieval Period: The Great Divergence

The Great Divergence describes shifts in relative per capita income in China versus Europe (specifically England) during the late Medieval period. It is known as a "divergence" since per capita income soared in Europe and not in Asia (Allen 2001) even though China and Europe looked fairly similar economically circa 1750 (see, e.g., Pomeranz 2009). The period includes the Industrial Revolution, which we will discuss below. It also coincides with the introduction and adoption of maize in China between the 1500s and the 1900s, which we argue played an important role in the divergence.

Generally, the Great Divergence is explained as soaring circus technology in Europe contrasted with stagnant technology in China. Here, we offer a novel perspective based on data from Ireland. To understand, first note that our model accommodates the idea that a positive shock to technology can lead to a fall in living standards. This can occur in spite of a positive circus technology shock if this is accompanied by a large bread technology shock. We argue that this dynamic occurred not only in China, but in Ireland when the potato was introduced, which greatly increased the number of people who could be fed. Therefore, despite the likelihood that Ireland had access to the same circus technology as nearby England, Ireland grew poorer than England.

In fact, the divergence in Ireland looks exactly like that of China, there was a huge increase in total GDP similar to England, but with declining per capita income and greatly increased population. Apparent similarities between Ireland and China lead us to a reinterpretation of the Great Divergence between China and England. We discuss evidence showing that China imported New World crops, specifically maize, and that this led to higher population growth and lower living standards. This evidence provides a rather different perspective on the Great Divergence, which is usually interpreted as China falling behind; it suggests that the problem was that China pulled ahead--in agriculture.

Technology Change. Between 1500 and 1900, roughly corresponding to the Qing dynasty, both China and Europe incorporated New World crops. In the case of China, maize became a staple crop. However, there is little evidence of growth in other technologies. In fact, in his very detailed study of the Great Divergence, Pomeranz (2009) argues against what he calls the Weberian view--that China grew differently than Europe because of some cultural differences that favored growth in one place over the other. Rather, the argument is that the location of coal deposits and the discovery of the New World (exogenous shocks to production technology, especially manufacturing) meant that Europe was able to produce more goods, but China was not. Thus, the period in China is an example of an increase in bread productivity (via the incorporation of maize) without a commensurate rise in circus productivity.

Similarly, in Ireland, a more agrarian economy, the potato seemed to play an outsized and early role in the economy. Not only was it widely adopted Ireland, but it also led to relatively large changes in food output. The potato more than doubled caloric output--Mokyr (1981) claims that it quadrupled output in Ireland. Moreover, Mokyr (1981) points out that introduction of the potato may have led to further improvements in bread technology, encouraging the development of practices such as crop rotation, which would further exacerbate the discrepancy between bread and circus technology.

In summary, there was a large positive shock to bread technology in China and also in Ireland. In England, there is a large shock to circus technology but not bread technology. This circus technology was also available in Ireland, but perhaps not in China.

Model Implications. In Ireland and China, population should go up but per capita income should go down or up less quickly. The model predicts a greater increase in population, but lower living standards in Ireland and China compared to England.

Evidence. The adoption of maize in China led to growth in population density. Causal estimates are provided in Chen and Kung (2011), who find using IV estimates that a decade of maize planting resulted in a 5.6% annual increase in population density between the period 1500 and 1900. Chen and Kung (2011) follow Acemoglu, Johnson, and Robinson (2001, 2005) in using urbanization as a way to measure income. They find some evidence of declines in urbanization following maize adoption in China. Recall, an implication of the standard Malthusian model is that population growth following a technology shock would drive living standards back down to subsistence. However, if urbanization reflects income, it would appear that maize adoption in China caused living standards to decline relative to their preshock levels, which is a prediction of our model and which is not accommodated by the standard model.

Turning to Ireland, Nunn and Qian (2011) show that potato cultivation led to explosive population growth in parts of the Old World, leading some regions, such as Ireland, to a Malthusian-like poverty trap. After the introduction of the potato, there is evidence that in comparison to England, population grew rapidly, but that per capita consumption did not. Between the early 1600s (about when the potato was introduced, see, e.g., Nunn and Qian 2011) until about 1800, Irish population grew from roughly between 500,000 and 1,000,000 people (depending on the source) to about 6,000,000 at the start of the 1800s. It should be noted that most sources put the starting point at the lower number, which would suggest a 12-fold increase in population. In England, over the same period population rose 2-3 times, from 4.2 million to 10.5 million (see, e.g., McEvedy and Jones 1978) Over the same time period. English per capita income roughly doubled from 1,082 to 2,108 in year 2000 U.S. dollars (Bolt and van Zanden 2013). (27) Unfortunately, we do not have estimates in per capita consumption in Ireland over the same period. The numbers are not found in the updates to the Maddison dataset, including Bolt and Zanden (2014). Nevertheless, contemporary accounts indicate that Ireland (compared to England) remained mired in poverty and insofar as there was per capita growth it was not on the English scale, nor did it accrue to the Irish as most of Ireland's wealth (in the form of land) was controlled by and benefitted the English (Moody, Martin, and Byrne 1976).

E. The Industrial Revolution

We now turn our attention to the Industrial Revolution. We argue that the Industrial Revolution was the result of technology shocks to both circuses and bread, where the former dominated. In single-good models, it is necessary to introduce non-Malthusian demographic forces to explain why the Industrial Revolution raised per capita income. Our model is consistent with the same demographic forces being at work before and after the Industrial Revolution.

The Industrial Revolution lasted from about 1760 to 1820 and was the period when a largely agrarian economy was replaced by one where industry and manufacturing dominated. The Industrial Revolution is generally seen as the first period in history when living standards rose sustainably--though other work sees rises in living standards for the median individuals occurring much later. What is less controversial is the nature of changes that occurred in the economy. Whereas the shift in technology during the Neolithic Revolution mostly improved bread production, in the Industrial Revolution, it was the productivity of sectors for goods that were not essential to survival that rose.

Technology Change. The argument is not that bread technology did not improve, and it did, as agriculture and transportation improved. The argument is that circus technology improved more. Several papers report factor productivity for the economy as a whole and for agriculture, in particular. For example, Feinstein (1981) reports that total factor productivity (TFP) growth in all sectors was roughly 1.3% per year after 1800. In contrast, Kogel and Prskawetz (2001) and Crafts (1980) report TFP growth in agriculture to be 0.9% per year between 1820 and 1840. Given the relatively low share of manufacturing in the economy, generating TFP growth of this magnitude for the entire economy would require large relative growth in TFP in manufacturing versus agriculture.

Increases in living standards do not generally (and should not be expected to) accompany increases in food consumption. The key is to divorce food consumption with the consumption of other goods. To illustrate this point, we turn once more to modern data. We consider growth in the dollar value of musical instruments per person, again from 1929 to 2000, in comparison to food (Figure 2, normalized to $ 1.00 and 1 calorie per person, respectively, in 1929). Calorie growth is flat. However, the dollar value of the stock of musical instruments per person is anything but flat, a fact that likely did nothing to improve survival, but certainly led to an improvement in the quality of life for many people.

Model Implications. The simple two-sector model leads to a straightforward explanation of what took place: large and positive shocks to circus technology absent commensurate rises in bread technology lead to widespread and explosive growth in living standards.

Evidence. The Industrial Revolution led to massive and sustained rises in living standards that were unprecedented in human history. Population rose as well, but not so much as to push humanity back to subsistence. Finally, there is little evidence that food consumption per capita changed very much (though Fouquet and Broadberry 2015 show evidence that the quality of calories may have improved).

Much research is dedicated to the project of making sense of the Industrial Revolution in light of centuries of slow and sporadic growth. We believe that our model offers a unified explanation that accommodates various episodes throughout history that other models have trouble explaining. Technology growth in either the bread or the circus sectors occurred throughout history. Many shocks were such that additional output would be consumed by larger populations. In other cases, living standards declined since circus technology did not improve as quickly as bread technology (e.g., due to the introduction of new crops well suited to the climate). Finally, in other cases, living standards improved since circus technology improved more rapidly than bread technology. This occurred on a small but significant scale in Ancient Rome and on a massive scale during the Industrial Revolution.

V. HISTORICAL EVIDENCE FOR MODEL IMPLICATIONS: NEGATIVE SHOCKS

To test model implications, we also study two negative shocks: the Irish Potato Famines and the Black Death.

A. The Irish Potato Famine

Here, we return to Ireland and potato cultivation, focusing on the Irish Potato Famine that occurred between 1845 and 1852. In the popular view of Malthus, population outstrips the food supply leading to famine that reduces the population. The Irish famine is sometimes used to illustrate this point. As should be clear, however, the potato famine was not caused by overpopulation, but by an exogenous shock--a crop disease (potato blight) that occurred worldwide (Donnelly 2012; Mokyr 1981). The same observation can be made about the earlier famine in Ireland in 1740-1741 and no doubt an event that influenced Malthus's writing in 1798. Like the better known potato famine, the earlier Irish famine has no causal link to population: it was caused by an exogenous change in climactic conditions--unusually cold winters; it was also a negative technological shock.

The idea is not that Ireland lacked access to circus technology from England, which is difficult to assume given proximity. Rather, Ireland was well suited for potato cultivation which means that it experienced a larger relative shock to bread technology and, as our two-sector model implies, it is the relative size of these shocks that matters. Therefore, by the time of the blight (the negative shock to bread technology), Ireland had a large population sustained by a single crop characterized by low living standards.

Technology Change. In our model, the potato blight is a negative technology shock to B: it lowered the amount of food that could be produced with a given amount of labor and land. It was not due to overpopulation, but was likely due to reliance on a single strain of potato, which was an inexpensive way to sustain a large population. In other words, a third factor (reliance on a single strain) made sustaining a large population possible and also likely led to the blight (Fraser 2003). It is important to note that the shock to B was a long-term shock. It was caused by the introduction of an endemic fungal disease Phytophthora Infestans to which there were no resistant potato strains until the late twentieth century. (28) An effective fungicide was developed only in about 1900. (29) After the blight potato cultivation became difficult in Ireland except in a few locations where the potatoes were protected by the local microclimate.

Model Implications. A drop in B with C fixed also raises [zeta]. This implies that in the long-run population must fall but per capita income should rise. The model also implies that a high B economy with most land and labor dedicated to the agricultural sector is very vulnerable to B shocks since the shock cannot be offset by shifting land or labor from the circus sector. Hence, population has to adjust very quickly.

Evidence. The evidence supports both the short-and long-term effects of a drop in bread technology. The theory predicts that a consequence of the famine is not only that population should decline, but per capita GDP should go up. During the famine about 1 million people starved and another million emigrated so that Ireland lost about 25% of its population (Woodham-Smith and Davidson 1991). Maddison (2003) contains per capita GDP in Ireland and England before and after the famine. The per capital GDP in year 2000 U.S. dollars for Ireland for 1820 and 1870 are 877 and 1,775. Comparable numbers for England are 2,074 and 3,190, respectively. Together, population decline and rapid growth in per capita GDP in Ireland versus England seem to support the predictions of the model. (30) This point deserves some emphasis: our model predicts that a consequence of a negative technological shock to the bread sector not only lowers population but also increases per capita income. This is not such an obvious conclusion--yet in Ireland a little noted consequence of the potato famine is an enormous increase in per capita income.

B. The Black Death

A canonical example often used to support the one-good Malthusian model is the Black Death plague which led to a massive decline in population. The reason is that there is evidence that the surviving population had higher living standards after the plague had subsided. The plague reached its peak in the mid 1300s and resulting deaths are estimated at 75-200 million (Gottfried 2010).

Technology Change. The shift is not a technology change, but instead is a negative shock to the population coupled with no change to technology.

Model Implications. Our model implies that a decline in population should be followed by population growth. This is what the single-good model would imply. Given prevailing technology and the fact that land is fixed and that marginal returns decline, more individuals could be sustained. In other words, the standard model would predict more food production until the Malthusian constraint is met. Until then, people consume more, but living standards return to subsistence. In contrast, our model accommodates the idea that fewer individuals were needed to produce food to sustain the smaller population. These individuals shifted, therefore, to circus production, which raised living standards.

Evidence. There is ample evidence of population recovery following the Black Death, which is in line with our model predictions. Model implications also accommodate the flourishing of art, culture, and science, that is, rises in living standards that occurred after the Black Death (Clark 2016; Herlihy and Cohn 1997; Pamuk 2007). Interestingly, Clark (2016) sees little evidence of greater agricultural productivity until the mid 1500s, which would further suggest movement of labor to the circus sector.

VI. CONCLUSION

The traditional Malthusian point of departure is the following: living standards were roughly the same everywhere in the world until about 1760. As measured by per capita GDP, living standards, as recent research has shown and we have documented, have varied enormously over time and space. We do not argue with the well-established observation that growth after the Industrial Revolution has been rapid and sustained, dwarfing earlier growth, which was comparatively sporadic and slow. What we do argue against is the subsequent jump to approximating earlier growth as no growth, which we view as an oversimplification leading to theories that must explain the Industrial Revolution as arising from a shift in fundamentals rather than changes in technology. In other words, we argue that early growth should be taken at face value. Our model provides one attempt at explaining them while still assuming a Malthusian constraint.

We have proposed a simple two-sector model according to which the Industrial Revolution is not seen as some sort of aberration in human behavior. Rather our theory says that improvements in bread technology will largely result in population growth while improvements in circus technology will largely result in growth in per capita GDP. We hope that this theory will lead to more serious work exploring ancient economic growth such as the recent work of Temin (2013). Indeed, as Temin (2013) says, the distinguishing feature of modern economic growth is not that it takes place, but rather that it takes place so quickly. In particular, our model suggests a key role is played by [zeta] as a measure of relative technological change in the bread and circus sector and it would be useful if economic historians made a systematic effort to measure this.

Perhaps it is useful to conclude by indicating what we hoped to accomplish by writing this paper and what we learned in the course of writing it. We started with the idea that in a two-sector model it makes a difference whether technological shocks hit the bread or circus sector, with the former favoring population growth and the latter growth in per capita GDP. We constructed a simple two-sector Malthusian model and verified that it indeed has this property. This accomplished our goal of accommodating the basic observation that prior to the Industrial Revolution big changes in technology--generally favoring agriculture--had little impact on living standards while the opposite was the case in the Industrial Revolution. In those historical cases where technology did favor circuses, we find that indeed per capita income did increase. Other data support the basic validity of the model; for example, we have the fact, perhaps not so well-known, that the population density of Italy changed little during the Roman period. Without the model we would not have thought to consider the impact of the Irish Potato Famine on per capita GDP; we simply assumed that it had a bad effect. In fact, as the model predicts, it went up quite a lot.

We certainly did not set out to explain the Great Divergence--rather we were puzzled by the implication of the model that a big increase in circus technology, if offset by an even larger increase in bread technology, would not lead to increased per capita income, but would rather be reflected in increased population. In discussions of the Great Divergence, the great increase in Chinese population is noted, along with the fact that this was fueled in large part by improvements in agricultural technology brought about by the import of New World crops. By contrast, most of Europe showed little impact from New World crops due to unsuitable growing conditions. This dovetails with the theory. Moreover, when we went to look at places in Europe where New World crops did have a big impact--Ireland--we discovered the same story as China. In Ireland, population grew much faster than in England and per capita GDP rose much more slowly. Hence, in light of our model, we argue that the Great Divergence is less due to the unavailability of new circus technology in China, but more to the perverse effect of improved bread technology.

ABBREVIATIONS

CES: Constant Elasticity of Substitution

GDP: Gross Domestic Product

HIV: Human Immunodeficiency Virus

TFP: Total Factor Productivity

REFERENCES

Acemoglu, D., and F. Zilibotti. "Was Prometheus Unbound by Chance? Risk, Diversification, and Growth." Journal of Political Economy, 105(4), 1997, 709-51.

Acemoglu, D., S. Johnson, and J. A. Robinson. "Reversal of Fortune: Geography and Institutions in the Making of the Modern World Income Distribution." Quarterly Journal of Economics, 117(4), 2001, 1231-94.

--. "The Rise of Europe: Atlantic Trade, Institutional Change, and Economic Growth." American Economic Review, 38(4), 2005, 546-79.

Allen, R. C. "The Great Divergence in European Wages and Prices from the Middle Ages to the First World War." Explorations in Economic History, 38(4), 2001, 411-47.

Alvarez-Nogal, C., and L. Prados de la Escosura. "The Rise and Fall of Spain (1270-1850)." Economic History Review, 66(1), 2013, 1-37.

Arifovic, J., J. Bullard, and J. Duffy. "The Transition from Stagnation to Growth: An Adaptive Learning Approach." Journal of Economic Growth, 2(2), 1997, 185-209.

Bandy, M. S., and J. R. Fox, ed. Becoming Villagers: Comparing Early Village Societies. Tucson: University of Arizona Press, 2010.

Banerjee. A. V., and E. Duflo. Poor Economics: A Radical Rethinking of the Way to Fight Global Poverty. New York: Public Affairs, 2011.

Baumol, W. J. "Entrepreneurship: Productive, Unproductive, and Destructive." Journal of Political Economy, 98, 1990, 893-921.

Becker, G. S., and R. J. Barro. "A Reformulation of the Economic Theory of Fertility." Quarterly Journal of Economics, 103, 1988, 1-25.

Bocquet-Appel, J.-P. "When the World's Population Took Off: The Springboard of the Neolithic Demographic Transition." Science, 333(6042), 2011, 560-61."

Bolt, J., and J. L. van Zanden. "The First Update of the Maddison Project: Re-Estimating Growth before 1820." Mimeo, University of Groningen, 2013.

--. "The Maddison Project: Collaborative Research on Historical National Accounts." Economic History Review, 67(3), 2014, 627-51.

Broadberry, S., M. S. Bruce, A. K. Campbell, M. Overton, and B. Van Leeuwen. British Economic Growth, 1270-1870. Cambridge: Cambridge University Press, 2015.

Chaney, E., and R. Hornbeck. "Economic Dynamics in the Malthusian Era: Evidence from the 1609 Spanish Expulsion of the Moriscos." The Economic Journal, 126(594), 2016, 1404-40.

Chen, S., and J. K.-s. Kung. "A Malthusian Quagmire? Maize, Population Growth, and Economic Development in China." Mimeo, Hong Kong University of Science and Technology, 2011.

Cilingiroglu, C. "The Concept of 'Neolithic Package' Considering Its Meaning and Applicability." Documenta Praehistorica, 32, 2005, 1-13.

Clark, G. "The Condition of the Working Class in England, 1209-2004." Journal of Political Economy, 113(6), 2005, 1307-40.

--. "Microbes and Markets: Was the Black Death an Economic Revolution?" Journal of Demographic Economics, 82(02), 2016, 139-65.

Crafts, N. F. R. "Income Elasticities of Demand and the Release of Labour by Agriculture during the British Industrial Revolution." Journal of European Economic History, 9(1), 1980, 153-68.

Davies, J. E. "Giffen Goods, the Survival Imperative, and the Irish Potato Culture." Journal of Political Economy, 102, 1994, 547-65.

Diamond, J. "The Worst Mistake in the History of the Human Race." Discover, 66, 1987, 64-66.

Donnelly, J. S. Jr. The Great Irish Potato Famine. Stroud, UK: The History Press, 2012.

Feinstein, C. H. "Capital Accumulation and the Industrial Revolution," in The Economic History of Britain since 1700, edited by R. Floud and D. McCloskey. Cambridge: Cambridge University Press, 1981, 128-42.

Finley, M.I. "Technical Innovation and Economic Progress in the Ancient World." Economic History Review, 18(1), 1965, 29-45.

--. The Ancient Economy. Berkeley: University of California Press, 1999.

Foley. J. A., R. DeFries, G. P. Asner, C. Barford, G. Bonan, S. R. Carpenter, F. S. Chapin, M. T. Coe, G. C. Daily, H. K. Gibbs, J. H. Helkowski. T. Holloway, E. A. Howard, C. J. Kucharik, C. Monfreda, J. A. Patz, I. C. Prentice, N. Ramankutty, and P. K. Snyder. "Global Consequences of Land Use." Science, 309(5734), 2005, 570-74.

Fouquet, R., and S. Broadberry. "Seven Centuries of European Economic Growth and Decline." Journal of Economic Perspectives, 29(4), 2015, 227-44.

Fraser, E. D. G. "Social Vulnerability and Ecological Fragility: Building Bridges between Social and Natural Sciences Using the Irish Potato Famine as a Case Study." Conservation Ecology, 7(2), 2003, 9.

Frier, B. W. "Demography," in The Cambridge Ancient History, Chapter 27, edited by A. K. Bowman, P. Garnsey, and D. Rathbone. Cambridge: Cambridge University Press, 2000, 787-811.

Galor, O., and O. Moav. "Natural Selection and the Origin of Economic Growth." Quarterly Journal of Economics, 117, 2002, 1133-91.

Galor, O., and D. N. Weil. "From Malthusian Stagnation to Modern Growth." American Economic Review, 89(2), 1999, 150-54.

--. "Population, Technology, and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond." American Economic Review, 90, 2000, 806-28.

Goodfriend, M., and J. McDermott. "Early Development." American Economic Review, 85(1), 1995, 116-33.

Gottfried, R. S. Black Death. New York: Simon and Schuster, 2010.

Greene, K. "Technological Innovation and Economic Progress in the Ancient World: M.I. Finley Reconsidered." Economic History Review, 53(1), 2000, 29-59.

Guzman, R. A., and J. Weisdorf. "The Neolithic Revolution from a Price-Theoretic Perspective." Journal of Development Economics, 96(2), 2011, 209-19.

Hansen, G. D., and E. C. Prescott. "Malthus to Solow." American Economic Review, 92(4), 2002, 1205-17.

Herlihy. D., and S. K. Cohn. The Black Death and the Transformation of the West. Cambridge, MA: Harvard University Press, 1997.

Hersh, J., and J. Voth. "Sweet Diversity: Colonial Goods and the Rise of European Living Standards after 1492." Mimeo, Wharton School, University of Pennsylvania, 2009.

Hong, S., J.-P. Candelone, C. C. Patterson, and C. F. Boutron. "History of Ancient Copper Smelting Pollution during Roman and Medieval Times Recorded in Greenland Ice." Science, 272(5259), 1996, 246-49.

Jones, C. I. "Was an Industrial Revolution Inevitable? Economic Growth over the Very Long Run." Advances in Economic Growth, 1(2), 2001, 1-43.

Kogel. T., and A. Prskawetz. "Agricultural Productivity Growth and Escape from the Malthusian Trap." Journal of Economic Growth, 6(4), 2001, 337-57.

Kremer. M. "Population Growth and Technological Change: One Million BC to 1990." Quarterly Journal of Economics, 108(3), 1993, 681-716.

Lipsey, R. G., K. I. Carlaw, and C. T. Bekar. Economic Transformations: General Purpose Technologies and Long-Term Economic Growth. Oxford: Oxford University Press, 2005.

Liu, W. G.-L. "The Making of a Fiscal State in Song China, 960-1279." Economic History Review. 1(68), 2015, 48-78.

Lucas, R. E. The Industrial Revolution: Past and Future. Cambridge, MA: Harvard University Press, 2002.

Maddison, A. Phases of Capitalist Development. Oxford: Oxford University Press, 1982.

--. Monitoring the World Economy, 1820-1992. Paris: OECD Publishing, 1995.

--. The World Economy Historical Statistics. Paris: OECD Publishing, 2003.

--. Contours of the World Economy I-2030AD: Essays in Macro-Economic History. Oxford: Oxford University Press, 2007.

Malanima, P. "The Long Decline of a Leading Economy: GDP in Central and Northern Italy, 1300-1913." European Review of Economic History. 15(2), 2011, 169-219.

McEvedy, C., and R. Jones. Atlas of World Population History. Harmondsworth, Middlesex: Penguin Books Ltd, 1978.

Miech, R. A., S. K. Kumanyika, N. Stettler, B. G. Link, J. C. Phelan, and V. W. Chang. "Trends in the Association of Poverty with Overweight among US Adolescents, 1971-2004." Journal of the American Medical Association, 295(20), 2006, 2385-93.

Moav, O., and Z. Neeman. "Saving Rates and Poverty: The Role of Conspicuous Consumption and Human Capital." The Economic Journal, 122(563), 2012, 933-56.

Mokyr, J. "Irish History with the Potato." Irish Economic and Social History, 8, 1981, 8-29.

Moody, T. W.. F. X. Martin, and F. Byrne. A New History of Ireland: Early Modern Ireland, 1534-1691. Vol. 3. Oxford: Clarendon Press, 1976.

Morris, I. Why the West Rules--For Now: The Patterns of History, and What They Reveal about the Future. New York: Random House LLC, 2011.

Nunn. N.. and N. Qian. "The Potato's Contribution to Population and Urbanization: Evidence from a Historical Experiment." Quarterly Journal of Economics, 126(2), 2011, 593-650.

Orwell, G. The Road to Wigan Pier. New York: Penguin Books, 1937.

Pamuk, S. "The Black Death and the Origins of the 'Great Divergence' across Europe, 1300-1600." European Review of Economic History, 11(3), 2007, 289-317.

Papageorge, N. W. "Why Medical Innovation Is Valuable: Health, Human Capital, and the Labor Market." Quantitative Economics, 7(3), 2016, 671-725.

Parente, S. L., and E. C. Prescott. "Changes in the Wealth of Nations." Federal Reserve Bank of Minneapolis Quarterly Review, 17(2), 1993, 3-16.

Pomeranz, K. The Great Divergence: China, Europe, and the Making of the Modern World Economy. Princeton, NJ: Princeton University Press, 2009.

Razin, A., and U. Ben-Zion. "An Intergenerational Model of Population Growth." American Economic Review, 65(5). 1975, 923-33.

Rowthorn R., and P. Seabright. "Property Rights, Warfare and the Neolithic Transition." Toulouse School of Economics Working Paper No. 10-205, 2010.

Shard, D. "The Neolithic Revolution: An Analogical Overview." Journal of Social History, 7(2), 1974, 165-70.

Shennan, S., S. S. Downey, A. Timpson, K. Edinborough, S. Colledge, T. Kerig, K. Manning, and M. G. Thomas. "Regional Population Collapse Followed Initial Agriculture Booms in Mid-Holocene Europe." Nature Communications, 4, 2013, 1-8.

Strulik, H., and J. Weisdorf. "Population, Food, and Knowledge: A Simple Unified Growth Theory." Journal of Economic Growth, 13(3), 2008, 195-216.

--. "The Determinants of Subsistence Income in a Malthusian World." Discussion Paper, School of Economics and Management of the Hanover Leibniz University, 2009.

Taylor, M. S., and J. A. Brander. "The Simple Economics of Easter Island: A Ricardo-Malthus Model of Renewable Resource Use." American Economic Review, 88(1), 1998, 119-38.

Temin, P. The Roman Market Economy. Princeton, NJ: Princeton University Press, 2013.

Voigtlander, N., and H.-J. Voth. "Why England? Demographic Factors, Structural Change and Physical Capital Accumulation during the Industrial Revolution." Journal of Economic Growth, 11(4), 2006, 319-61.

--. "How the West 'Invented' Fertility Restriction." American Economic Review, 103(6), 2013a, 2227-64.

--. "The Three Horsemen of Riches: Plague, War, and Urbanization in Early Modern Europe." Review of Economic Studies, 80(2), 2013b, 774-811.

Ward-Perkins, B. The Fall of Rome and the End of Civilization. Oxford: Oxford University Press, 2005.

Weisdorf. J. L. "Malthus Revisited: Fertility Decision Making Based on Quasi-Linear Preferences." Economics Letters, 99(1), 2008, 127-30.

--. "Why Did the First Farmers Toil? Human Metabolism and the Origins of Agriculture." European Review of Economic History, 13(2), 2009, 157-72.

Woodham-Smith, C., and F. Davidson. The Great Hunger: Ireland 1845-1849. London: Penguin Books, 199L

Wu, L. "Does Malthus Really Explain the Malthusian Trap?" Mimeo, Department of Economics, University of California Berkeley, 2013.

Yang, D. T., and X. Zhu. "Modernization of Agriculture and Long-Term Growth." Journal of Monetary Economics, 2013.

van Zanden, J. L., and B. van Leeuwen. "Persistent but Not Consistent: The Growth of National Income in Holland 1347-1807." Explorations in Economic History, 49(2), 2012, 119-30.

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Appendix S1. Previous Research

Appendix S2. Model Solutions and Proofs for Propositions

Appendix S3. Land Use

Table S1. Economic Literature on Historical Growth

(1.) For example, Jones (2001) writes, "For thousands of years, the average standard of living seems to have risen very little, despite increases in the level of technology and large increases in the level of the population." Or, for example, Hansen and Prescott (2002) who begin with the assertion that "Prior to 1800 living standards in world economies were roughly constant over the very long run ..." Or Galor and Weil (1999, 2000) who assert: "This Malthusian framework accurately characterized the evolution of population and output per capita for most of human history. For thousands of years, the standard of living was roughly constant, and it did not differ greatly across countries."

(2.) Given this definition of circuses, we assume that both land and labor are used in the production of circuses. So that it is clear which results are affected by this assumption, we also study an alternative model, where land is not used in the production of circuses.

(3.) In a Malthusian framework, bread consumption is at subsistence. Thus, when we discuss "living standards," we mean "circuses per capita."

(4.) For example, Arifovic, Bullard, and Duffy (1997) cite Maddison (1982) to conclude "Prior to industrialization, all of today's highly developed economies experienced very long periods, epochs, of relatively low and stagnant growth in per capita income ... While these data are highly aggregated and necessarily involve some guesswork, few economists would question the picture they paint." Goodfriend and McDermott (1995) make a similar assertion: "Maddison presents population and per capita growth rates since 500 AD. Although the numbers are highly aggregated both across countries and over time and are obviously imprecise, they tell a dramatic story. For the thousand years following the fall of Rome, there was little net progress in population and none in per capita product." Other sources that are often used as evidence of stagnant growth include McEvedy and Jones (1978), Parente and Prescott (1993), and Clark (2005, 2016), among others.

(5.) Relatedly, Voigtlander and Voth (2006) emphasize randomness in explaining why the Industrial Revolution began in Great Britain versus, say, France.

(6.) Similarly, Chaney and Hornbeck (2016), studying the expulsion of Moriscos from Spain in 1609, find evidence that a negative population shock led to 150 years of high living standards. They point out that their results suggest nuance in the Malthusian era that could be "reconciled with an augmented Malthusian framework." We provide one possible augmentation that is consistent with sustained living standards above subsistence.

(7.) It should be noted that Hansen and Prescott (2002) is not the only paper we have discussed in which a model is proposed that is not really Malthusian. In Acemoglu and Zilibotti (1997) and Arifovic, Bullard, and Duffy (1997), land is not a factor of production, which strictly speaking may be seen as violating a key Malthusian assumption.

(8.) In Voigtlander and Voth (2013b), increased circus consumption could lead to lower population (due to higher mortality rates in cities). Similarly, in our model, circus consumption does not raise population.

(9.) Somewhat similarly, in Moav and Neeman (2012), the circus good is replaced with one that serves a function as conspicuous consumption and therefore improves evolutionary fitness at the individual level. In our setting, circuses play no such evolutionary role.

(10.) Relatedly, Lipsey, Carlaw, and Bekar (2005) study a highly involved three-sector model that includes a public good sector funded through taxation.

(11.) Broadberry et al. (2015) echo the point that Maddison data were in part based on "guesstimates" meant to induce further research.

(12.) Maddison (1995), it should be noted, argues that data imputation in Maddison (1982) does not presuppose Malthusian stagnation.

(13.) Unlike Voigtlander and Voth (2013b), we do not use the Stone-Geary linear expenditure formulation with a subsistence level for food. While this does allow bread to be an inferior good it implies for fixed bread consumption that the expenditure share of circuses remains constant as circus consumption increases. As we discuss below, the evidence suggests this is not the case. However, in Appendix S2, where we study alternative modeling assumptions, we show our results are robust to [sigma] = 1, which includes the Stone-Geary formulation used by Voigtlander and Voth (2013b).

(14.) Allowing technology to grow continuously would pull the average bread consumption slightly away from the steady state equilibrium. We abstract from this complication because introducing it would not qualitatively affect results.

(15.) In our model, the meaning of "bread" could be extended to include nonfood goods, such as the most basic clothing needed to generate enough warmth to survive. To minimize confusion, however, throughout the paper, we say that bread embodies the basic nutrients needed for survival.

(16.) In Appendix S2, we include results from a version of the model where we assume that land has no role in circus production.

(17.) Cultivation excludes grazing, but grazing is used to produce meat, which would belong in the circus category.

(18.) Using per capita food supply as the sole measure of societal wealth and well-being is also problematic since in the United States and other developed (and even developing) countries, obesity has been associated with poverty rather than affluence. See, for example, Miech et al. (2006).

(19.) In particular, Papageorge (2016) shows that men infected with human immunodeficiency virus (HIV) often favor drugs with fewer side effects even if their choices put their survival at risk. Our model of bread and circuses captures a similar trade-off between the quality and the quantity of life.

(20.) It is important to acknowledge that we are selecting data based on the magnitude of shocks.

(21.) There is also some debate about the use of the terms since it may obscure variation across space in how the Neolithic Revolution played out.

(22.) For an early example, Shard (1974) discusses the adoption of agriculture as an "agonizing transition."

(23.) Other researchers have asked why individuals would adopt the new technology in the first place given the costs it imposed on early adopters (Weisdorf 2009).

(24.) Sweets would be another example. Hersh and Voth (2009) show that sugar, tea, and coffee--luxuries by almost any measure, even if they are technically food--added the equivalent of over 16% of household income to British welfare by the eighteenth century.

(25.) A similar point is made in Temin (2013).

(26.) In relatively underpopulated areas, particularly France and Spain, the adoption of Roman agricultural technology did lead to a substantial increase in population density.

(27.) See also The Maddison Project, http://www.ggdc.net/ maddison/maddison-project/home.htm, 2013 version.

(28.) See https://en.wikipedia.org/wiki/Phytophthora_infestans.

(29.) See http://www.apsnet.org/publications/apsnetfeatures/Pages/Fungicides.aspx.

(30.) It is worth pointing out that dynamic models in which the population effect works through the choice of number of children are not going to get this right. The decline in population in Ireland occurred through two mechanisms: starvation and emigration. Likewise, for historical episodes characterized by improvements in technology, immigration alongside fertility is an important factor.

ROHAN DUTTA, DAVID K. LEVINE, NICHOLAS W. PAPAGEORGE and LEMIN WU *

* First Version: June 20, 2013. We are grateful to NSF Grant SES-08-51315 for financial support. For helpful comments, we thank George Akerlof, Francisco Alvarez-Cuadrado, Pranab Bardhan, Gregory Clark, Bradford DeLong, Jan De Vries. Barry Eichengreen, Hulya Eraslan, Mukesh Eswaran, Ali Khan, Ronald Lee, Peter Lindert, Salvatore Modica, Martha Olney, Gerard Roland, and Yingyi Qian along with seminar participants at UC Berkeley, UC Davis, McGill University, Peking University, and Tsinghua University and members of the Monday Reading Group at Washington University in St. Louis. The authors would also like to acknowledge that a similar paper by one of the authors (Wu 2013) was begun independently and in parallel to the current paper. In that paper, as in this one, a two-sector model is studied where welfare can be considerably higher than subsistence despite a Malthusian constraint.

Dutta: Assistant Professor, Department of Economics, McGill University, Montreal, Quebec H3A2T7, Canada. Phone +1 514 398 5611, Fax +1 514 398 5611, E-mail rohan.dutta@mcgill.ca

Levine: Joint Chair Economics and RSCAS, Department of Economics, European University Institute, San Domenico, Fiesole 50014, Italy; Department of Economics, European University Institute and Washington University in St. Louis, St. Louis, MO 63130. Phone +39 055 468 5954, Fax +39 055 468 5954, E-mail david @ dklevine.com

Papageorge: Broadus Mitchell Assistant Professor, Department of Economics, Johns Hopkins University, Baltimore, MD 21218. Phone 410 516 4938, Fax 410 516 7600, E-mail papageorge@jhu.edu

Wu: E-mail leminwu@pku.edu.cn

doi: 10.1111/ecin.12479

Online Early publication July 25, 2017

Caption: FIGURE 1 Calories Per Capita in the U.S. from 1929 to 2000

Caption: FIGURE 2 Calories Per Capita and Value of Musical Instruments Per Capita in the United States from 1929 to 2000

TABLE 1
Literature from History on Pre-Modern Growth

Paper            Evidence               Key Findings

Finley (1965)    Various                Some evidence
                                        of technological progress
                                        in the ancient world.

Broadberry       Own dataset            Evidence of sustained and
et al. (2015)                           persistent (if slow)
                                        growth in England,
                                        including growth in
                                        consumption goods.

Malanima         Data on Northern       Evidence that pre-
(2011)           Italian agricultural   Industrial Revolution
                 production and         growth in Northern and
                 urbanization           Central Italy was
                                        stagnant.

Alvarez-Nogal    Own dataset            Evidence that pre-
and Prados                              Industrial Revolution
de la Escosura                          growth in Spain was
(2013)                                  stagnant.

van Zanden and   Own dataset            Evidence of pre-Industrial
van Leeuwen                             Revolution persistent
(2012)                                  growth in Holland.

Fouquet and      Several datasets       Reject the "received
Broadberry                              wisdom" that growth in
(2015)                                  Europe prior to the
                                        Industrial Revolution
                                        was stagnant.

TABLE 2
Original Maddison Data on Historical Consumption
Compared with Recent Revisions (in Bold)

Year   Italy   N. Italy    Holland    Holland

1       809        *         425         *
1000    450        *         425         *
1348     *     1,486#        *       876#
1400     *     1,716#        *      1,195#
1500   1,100   1,503#       761     1,454#
1600   1,100   1,336#      1,381    2,662#
1700   1.100   1,447#      2,130    2,105#
1800     *     1,336#        *      2,609#
1820   1,117       *        1,838        *

Year   England    England    Spain    Spain

1        400         *        498       *
1000     400         *        450       *
1348      *       919#         *     907#
1400      *      1,205#        *     819#
1500     714     1,134#       661    846#
1600     974     1,167#       853    892#
1700    1,252    1,540#       853    814#
1800      *      2,200#        *     916#
1820    1,706        *       1.008      *

Notes: This table presents historical data originally
from Maddison (2007) alongside more recently revised data.
Following Maddison (2007), units are per capita GDP levels
in 1990 Geary-Khamis dollars. Data from other papers were
converted to the same units in Bolt and van Zanden (2013).
For each country, the Maddison data (in regular type) are
listed adjacent to the more recently revised data
(in bold and italicized type). Note that data are not always
available for exactly the same years, in which case the cell
has a star. Revised data sources are as follows: Malanima (2011)
for North Italy, van Zanden and van Leeuwen (2012) for Holland,
Broadberry et al. (2015) for England, and Nogal and de la Escosura
(2013) for Spain.

Note: Original Maddison Data on Historical Consumption
Compared with Recent Revisions are indicated with #.