Origins Inequality

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The Origins of Inequality:

A Theoretical Framework from Economics

Gregory K. Dow ([email protected])

Clyde G. Reed ([email protected])

Department of Economics

Simon Fraser University

January 2009

Highly preliminary; please do not cite without permission from the authors.

These notes were prepared for the Workshop on the Emergence of Hierarchy and

Inequality organized by Sam Bowles at the Santa Fe Institute, Feb 13-15, 2009.

The authors are grateful to the Social Sciences and Humanities Research Council of

Canada for financial support. SSHRCC is not responsible for the opinions expressed.


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1. Introduction

These notes sketch a projected paper on the origins of inequality. We develop an

economic model that attempts to explain the associations among (a) rising technological

sophistication; (b) increasing population density; and (c) increasing inequality based upon

control over land. The model should provide some insights into both the cross-sectional

comparisons of societies carried out by anthropologists, and the dynamics of prehistoric

inequality inferred from archaeological data. The theoretical argument applies equally to

hunter-gatherer and agricultural societies. This presentation is preliminary and intended

only for discussion purposes.

Our understanding of the facts from anthropology and archaeology is roughly as

follows. Nomadic foraging societies have little inequality. They typically have divisions

of labor by age and sex but there are few differences in consumption or wealth, and none

that are hereditary. Food sharing is common within bands. There may be territoriality at

the band level, but individual households do not have property rights to land. Sedentary

foraging societies are more diverse. Some seem relatively egalitarian while others have

substantial inequality across and within sites.

With the spread of agriculture, inequality becomes more pronounced and is based

largely on control over land. This is true both for chiefdoms (e.g. the northwest coast of

North America and Hawaii) and early states (e.g. ancient Egypt and the Incan empire).

Archaeological evidence about inequality includes walls or fortifications, which suggest

that the insiders had a standard of living better than (and hence attractive to) outsiders.

Stratification can sometimes be inferred from teeth and bones, which provide information

about health and nutrition, and from the presence of prestige goods in burial sites.


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[Note: For a polished paper we need to discuss several important cases in detail.

Ideally we are looking for cases that bridge the entire transition from egalitarianism to

stratification, and have been studied intensively by archaeologists or anthropologists.

Candidates include (a) the Fertile Crescent; (b) the Jomon; (c) the northwest coast of

North America; and (d) Hawaii. There is a literature in economics on the emergence of

property rights and/or territoriality, which we will discuss in a more complete draft.]

How can we explain a trajectory toward increasing inequality over time? We

think a key fact is the correlation of rising population density with rising inequality, both

across sites and within sites. Although this correlation is not perfect, it is too obvious to

ignore. It seems relevant both for cross-sectional comparisons of the kind made possible

by the Standard Cross-Cultural Sample, and also for the longer run dynamics studied by

archaeologists. At the same time, one cannot simply say that higher population density

causes inequality, because (at least in economics) no one believes that population is an

exogenous variable in the very long run. A model that is persuasive to economists must

treat population as endogenous, and explain why population and inequality tend to rise

simultaneously. It should also trace this process back to changes in variables that can

plausibly be treated as exogenous, such as climate or technology.

Our general framework runs as follows. We consider a region with many sites

that vary in their quality (due to water availability, soil characteristics, elevation, or other

features). There are two technologies: one for producing food, and one for excluding

outsiders from a site. Exclusion requires a sufficiently high density of insiders. We will

model this idea in a fairly general way. One possibility is that a sufficiently high density

of food producers may be enough to exclude people (for example, if intruders are easily


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detected and expelled or killed by cooperating neighbors). Another possibility is that a

subset of insiders could specialize in exclusion activities such as building fences or

patrolling a perimeter. However, these specialists must be fed and do not add to the

group's food supply, so it is not obvious that a strategy of this sort will be attractive.

The other important feature of the model is Malthusian population dynamics. We

assume that fertility is a linear function of income and that the mortality rate is constant.

In long run equilibrium, this yields constant food per capita for the region as a whole.

Thus if climate or production technology improves, the productivity gains are absorbed

by population growth. When this Malthusian assumption is combined with endogenous

property rights, it can be shown that higher region-wide productivity (whether caused by

better climate or better food production technology) leads to higher population density,

which leads to more enclosure of high-quality sites.

This mechanism yields growing inequality across sites: as technology improves,

insiders at closed sites become better off while people at open sites become worse off. At

least in the early stages, this process implies increasing poverty among non-insiders (this

must be true because per capita food consumption for the region as a whole is constant).

A key goal of the paper is to derive theoretical conditions under which poverty continues

to deepen as climate or technology improves.

A straightforward extension gives stratification within closed sites. This occurs if

insiders can admit outsiders without giving them any property rights over land. Instead,

the outsiders are given a 'wage' equal to food consumption per person at the open sites.

As region-wide productivity rises, we should initially see growing inequality across sites

(with the best sites being closed first) and then growing inequality within sites (again, this


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We hypothesize that conditions of this kind gave rise to the earliest instances of

systematic inequality across and within production sites. In any case, they are sufficient

to do so. However, actual historical cases of inequality may be over-determined, in the

sense that more than one set of sufficient conditions applies simultaneously.

The rest of the paper is organized as follows. Section 2 develops a formal model

of an individual production site where we take the site quality, the outside options of the

agents, and region-wide productivity as exogenously given. We are especially interested

in how the nature of the equilibrium at the site (open or closed, hired labor or not) varies

with site quality. This section introduces the production functions for food and exclusion

and shows how to derive the size of the elite and commoner populations as functions of

site quality and the outside option. We also characterize the per capita income of the elite

and the break points in the site quality distribution between (i) open access; (ii) closed but

no commoners; (iii) closed with commoners and internal stratification.

Section 3 combines endogenous property rights with Malthusian population at the

level of the region as a whole. At this stage the outside option of the agents, which was

exogenously given in section 2, becomes an endogenous variable. Here we will model

fertility, mortality, and migration among sites. We rely on simple assumptions that yield

sharp analytic conclusions. Section 4 discusses the relationship between productivity and

inequality. Throughout sections 2-4, we omit formal proofs in order to clarify the general

structure of the model.

Section 5 concludes by discussing empirical implications of the model that could

be tested with archaeological or anthropological data, and also theoretical extensions.

Readers not interested in the formal analysis may want to skip directly to this section.


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2. The Production Site

The production function for food at an individual site is

(1) H(L,M) = !sf(L)g(M)

where !is a productivity parameter reflecting the state of climate and technology; s is site

quality; L is labor devoted to food production; and M is land. Food output is measured in

calories. The model can be applied equally well to foraging or agricultural economies, so

we refer generically to L as 'food labor'. We normalize the land at a site to be M = 1 with

g(1) = 1. The function f(L) has standard properties: f(0) = 0, f"(L) > 0 for all L !0, f""(L)

< 0 for all L !0, f"(0) = +#, and f"(+#) = 0.

There is a continuum of agents so each individual agent is negligible relative to

the mass of agents at the site as a whole (L thus refers to a mass of labor, not the labor of

an individual agent). Each agent is endowed with one unit of time. These endowments

are used for food production, exclusion of outsiders, or some combination of the two (we

ignore leisure). The outside option of each individual agent is w, which is the food per

person that is available by migration to another site. There is an infinitely elastic supply

of outsiders who will enter the site if they can obtain more than w by doing so, and are

not excluded by existing insiders.

Suppose for the moment that a group of insiders can prevent entry by outsiders

(the technology by which this is accomplished will be discussed later in this section). We

refer to such a group as an elite. An elite of size e divides its time between food labor (ef)

and guard labor (eg) subject to the constraint ef+ eg= e. Available food is shared equally

among the members of the elite. Each member receives


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(2) y(ef, eg) = {max c !0!sf(ef+ c) - wc}/(ef+ eg)

where c is the number of commonersadmitted to the site by the elite. Commoners cannot

appropriate any land, but the elite may allow them to supply food labor in exchange for a

wage w equal to the outside option. This may or may not be a profitable strategy for the

elite (the optimal number of commoners could be zero).

The optimal choice of commoners in (2) gives the first order condition

(3) !sf"(ef+ c) "w with !sf"(ef+ c) = w if c > 0.

Let efbe the boundary value of efsuch that c > 0 for e f< efand c = 0 for ef!ef. This

boundary satisfies zf"(ef) $1 where z $!s/w. We have ef%(0, #) for all z > 0. Due to

the concavity of the production function, efis an increasing function of z.

Now define the surplus (above outside opportunities) for a member of the elite as

(4) v(ef, eg) = y(ef, eg) - w

When ef!efwe have c = 0 in (2) and surplus is v(ef, eg) = !sf(ef)/(ef+ eg) - w. When ef 0 in (2) and surplus is v(ef, eg) = !sf(L)/(ef+ eg) - w where L = ef+ c is

food labor from the elite and commoners together, and zf"(L) $1 from (3). In the latter

case, total food labor L depends only on z $!s/w and remains constant for all ef"efas

long as z remains fixed. As elite food labor declines over this range, commoner labor is

substituted on a one-for-one basis.

Next consider the set of (ef, eg) pairs such that v(ef, eg) = 0. This set is graphed in

Figure 1 for a fixed value of z. All points below the boundary in Figure 1 give positive


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surplus to members of the elite. Elites with time allocations on or below the boundary

are therefore viable. We denote this set by V(z). At points above the boundary, surplus

is negative and members of the elite are better off taking the outside option w.

To the left of the vertical line at ef(z), any decrease in food labor by the elite is

offset by an equal increase in commoner labor so food output stays constant (although

more is spent on wages for commoners). To the right of this line no commoners are used

and all food is produced by the elite. The horizontal intercept e0(z) shows the amount of

food labor by the elite that yields per capita consumption equal to the outside option w

(and thus zero surplus) when no guard labor is used.

The lines below the zero-surplus boundary in Figure 1 are indifference curves that

correspond to fixed (positive) levels of surplus per member of the elite. Lower curves

correspond to higher levels of surplus. It can be shown that each indifference curve is

downward sloping and has a slope flatter than -1 everywhere. Each curve is linear in the

region to the left of the boundary at ef(z) and strictly concave to the right. Surplus per

capita is a continuous function of (ef, eg). The indifference curves are continuous and

their slopes are also continuous, including at the boundary ef(z).

We next consider the exclusion technology used by the elite. Let

(5) E ${(ef, eg) !0 such that outsiders can be excluded from the site}

A higher density of food producers should make exclusion easier, ceteris paribus, and

thus reduces the need for specialized guard labor. The boundary of E should therefore

have a negative slope. We assume E is closed and strictly convex. It is also monotonic:

if (ef, eg) %E then (ef", eg") %E for all (ef", eg") !(ef, eg). This says that if a site can be


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enclosed by an elite with a given time allocation, then some other elite with at least as

much food labor and guard labor can also enclose the site.

There are two possibilities. First, there may be no intersection between the non-

negative surplus set V(z) and the feasible set E, as shown in Figure 2(a). In this case, no

elite is viable and the site is said to be open. Second, there may be some points (ef, eg)

that are in both sets, as shown in Figure 2(b). An elite is then viable, and the site is said

to be closed. If in addition the elite chooses c > 0, we say that the site is stratified.

When a site is closed, we assume the size of the elite is given by the value of e

that maximizes surplus per member. If elite size is smaller than this, the members can

always raise per capita consumption by recruiting commoners into the elite. If elite size

is larger, attrition through death or other causes tends to shrink the elite in the long run.

These previous members are not replaced because it is not profitable for the remaining

members to do so.

Due to the shape of the indifference curves there is a unique point (e f*, eg*) that

achieves the maximum surplus when a site is closed. For interior solutions, this point is

located at a tangency between the boundary of the feasible set E and the lowest possible

indifference curve from Figure 1. Corner solutions can also occur, as will be explained

below. In either case, we obtain some elite size e* with surplus per member v*. After ef*

is determined, the optimal number of commoners c* follows from (3).

The following results will be used in section 3.

(a) There is a unique A > 0 such that for z < A no elite is viable and the site is open.

For z = A the site is closed but the elite obtains zero surplus. For A < z the site is

closed and the elite obtains positive surplus.


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(b) There is a unique B > A such that for A "z "B no commoners are used (the site

is closed but not stratified). For B < z some commoners are used (the site is both

closed and stratified).

(c) The surplus v*(z) for the elite is an increasing function of z for A "z.

Several important special cases are discussed in the remainder of this section.

First, suppose the constraint set E touches the horizontal axis at some density of

food producers Ehas shown in Figures 3(a) and 3(b). At low enough values of z, the site

is open and there is no elite. The number of 'commoners' is given by the requirement that

all of the open access sites provide food per agent equal to the outside option w (the label

'commoners' is appropriate for such agents because open sites are in the 'commons'). For

such sites we have !sf(L)/L = w or zf(L)/L = 1. Now hold productivity (!) and the

outside option (w) fixed and consider better quality sites so that s and z $!s/w both rise.

As this occurs, the number of food producers L rises due to the strict concavity of the

production function. Since there is no guard labor at open sites, the result is a rightward

movement along the horizontal axis in Figure 3(a) or 3(b).

At z = A an elite becomes viable (with zero surplus). This can occur in two ways.

One case is a corner solution on the horizontal axis, where the zero-surplus indifference

curve at z = A is flatter or equal in slope to the boundary of the constraint set at Ehas in

Figure 3(a). In this situation, L = Ehwhen z = A. An elite then emerges and all members

of the elite produce food (there is no guard labor). Further increases in z raise surplus for

members of the elite but leave the corner solution at (Eh, 0) unchanged until the slope of

the indifference curve through (Eh, 0) is the same as the slope of the constraint boundary.

After this, increases in z cause the solution to become interior and (e f*, eg*) moves up and


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to the left along the boundary of the feasible set E. Elite food labor declines, guard labor

rises, total elite size falls (because the relevant slopes are flatter than -1), and eventually

commoners are used as food producers. In this scenario, elite food labor and guard labor

are continuous functions of the parameter z as we go from open access (z < A) to closed

access (z > A).

The other possibility is an immediate jump to an interior solution when z = A as

in Figure 3(b). In this case, the zero-surplus indifference curve for z = A is tangent to the

boundary of the set E at an interior point (ef*, eg*) > 0. For z < A, we have open access

and an increase in z moves the system along the horizontal axis as before. Once z = A is

reached, an elite becomes viable. This elite is indifferent between the corner solution

[e0(A), 0] and the interior solution (ef*, eg*) but only the latter is feasible. As z increases

further, the system moves up and to the left along the boundary of the feasible set E as

before, but starting from the point (ef*, eg*) shown in Figure 3(b). Accordingly, there is a

discontinuity at z = A where food labor drops abruptly from e0(A) to ef* and guard labor

jumps abruptly from eg= 0 to eg* > 0 when the site becomes closed.

The preceding discussion assumed that the constraint boundary has a finite

horizontal intercept as in Figures 3(a) and 3(b). This is true if a dense enough population

of food producers can directly exclude outsiders without using specialized labor to guard

the site. Under the alternative assumption that some positive guard labor is always

needed for exclusion to be feasible (the set E never touches the horizontal axis), we must

have a discontinuity at z = A because there must be a jump from the horizontal axis to an

interior solution when an elite first becomes feasible.


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Another possibility is that the constraint boundary could have a finite vertical

intercept Evas shown in Figure 4. This does not affect the analysis for z values in the

vicinity of z = A. If Figure 3(a) is relevant, there is still a smooth transition from open to

closed sites, and otherwise there is a discontinuous jump to an interior solution at z = A.

However, a finite vertical intercept is relevant for the behavior of the system when z

becomes large. With such an intercept, eventually the elite goes to a corner solution on

the vertical axis with zero food labor (ef= 0) and maximum guard labor (eg= Ev). Thus,

when z is high enough the elite specializes in guarding the site and relies entirely on

commoners for food production. If the constraint set never touches the vertical axis,

elites always devote positive labor to food production, although this labor input may

approach zero asymptotically as z goes to infinity.

3. Regional Equilibrium

Consider a region with a continuum of production sites. Sites have qualities

indexed by s %[0,1] where s is distributed according to a continuous density function

q(s) > 0 that reflects the number of sites of type s. People can move freely among sites

within the region, subject to possible exclusion by insiders (elites) at some sites. Let n(s)

be the population density at a typical site of type s. This consists entirely of commoners

for open sites, and includes both elites and commoners at closed sites. We use ef(s), eg(s),

e(s), and c(s) to denote elite food labor, elite guard labor, total elite size, and commoner

size at a site of type s. Thus ef(s) + eg(s) = e(s) and e(s) + c(s) = n(s).

Time is discrete. The adults alive in period t have children who become adults in

period t+1. For a given adult in period t, the number of children surviving to adulthood is

&ytwhere &> 0 is a constant and y tis the adult's food income. This demand for children


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defines the function L(!,w,s) for open sites. Since labor is supplied only by commoners

we write this simply as c(x,s). The absence of any elite at the open sites implies c(x,s) $

n(x,s) for all s %[0, a(x)).

On the quality interval [a(x), b(s)], sites are closed but there are no commoners.

The size of the elite is determined by maximizing surplus per elite agent v subject to the

feasible set E, as explained in section 2. Setting c = 0 in (2), the resulting elite size e* =

ef* + eg* is the one that maximizes f(ef)/(ef+eg) subject to (ef, eg) %E. This is a constant

that does not depend upon productivity (!), the outside option (w), or the site quality (s).

The resulting food labor is L(x,s) = ef* and the resulting population density is n(x,s) = e f*

+ eg* = e*.

Finally, consider the interval (b(x),1] on which sites are both closed and stratified.

Because c > 0, total food labor from (3) satisfies sxf"(L) $1. This identity gives the food

labor input L(x,s). The total population also includes guard labor eg. The latter depends

only on the ratio x and the site quality s, so we write n(x,s) = L(x,s) + eg(x,s).

Substituting these results into (8) yields

(9) )0a(x)!sf[c(x,s)]q(s)ds +)a(x)

b(x)!sf(ef*)]q(s)ds +)b(x)1!sf[L(x,s)]q(s)ds

= ({)0a(x)c(x,s)q(s)ds + )a(x)

b(x)e*q(s)ds +)b(x)1n(x,s)q(s)ds}

The only unknown in this equation is the ratio x = !/w. But because !is an exogenous

productivity parameter determined by the natural environment and food technology, the

only truly endogenous variable is the food per capita w obtained at the open access sites.

Once this is determined, we can compute the break points a(x) and b(x) among open,

closed, and stratified sites; the sizes of the commoner and elite populations at each site;


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the division of the elite between food producers and guards; the total food produced at

each site; and the distribution of food between elites and commoners. Total regional

population N falls out as a by-product.

It is not obvious how to solve for w (or x) directly from (9), at least not without

imposing further restrictions on the production and exclusion technology. However, a

few observations can be made. First, food output for sites in the open access interval

[0,a(x)) is !sf[c(x,s)] $wc(x,s). Second, food output for unstratified sites on the interval

[a(x), b(x)] is !sf(ef*) = e*v* + e*w, where e* and v* are the size of the elite and surplus

per member of the elite, respectively. And third, food output for the stratified sites on the

interval [a(x), 1] is !sf[L(x,s)] = e(x,s)v(x,s) + wn(x,s) where e(x,s) is the size of the elite

and v(x,s) is surplus per member of the elite.

Substituting these results into (9) and rearranging gives

(10) (w - ())01n(x,s)q(s)ds +)a(x)

1v(x,s)e(x,s)q(s)ds = 0

When a(x) !1, the second integral is irrelevant and we have w = (. This is intuitively

clear: if all sites on [0,1] are open, then free mobility ensures that every agent receives the

same payoff w in equilibrium (there is no inequality). The Malthusian assumption earlier

in this section guarantees that this payoff is (, which is the per capita food consumption

consistent with a stationary population for the region as a whole. On the other hand,

when a(x) < 1 the best sites are closed. Surplus for the elite is positive at every closed

site, whether or not that site is stratified. Therefore v(x,s) > 0 for every s > a(x). This

implies that w < (if a(x) < 1. Thus if there are any closed sites, commoners must get a

'wage' below the replacement level of food consumption (. This is balanced by food


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consumption per capita y(ef,eg) > (above the replacement level for the elite, such that

population is stationary at an aggregate (regional) level.

Because elite agents have more food per capita than the commoners, the elite have

more surviving children. As a result, there must be a mechanism ensuring that some elite

children become commoners in each generation. Otherwise, the size of the elite at closed

sites would increase beyond the level that maximized food per member. Several such

mechanisms are empirically important, and we will return to this topic in section 5.

4. Productivity and Inequality

Our main concern is with the relationship between productivity and inequality

defined by the equilibrium condition (10). More specifically, we would like to know how

a gain in productivity (!) from a better climate or food production technology affects the

per capita food consumption of commoners (w). We also want to know whether x = !/w

rises or falls, because this indicates how much of any improvement in productivity goes

to commoners (for example, if x remains constant then an increase in productivity makes

commoners better off in the same proportion). Moreover, the breaks a(x) and b(x) in the

site quality distribution depend on this ratio. If a rise in productivity leads to a higher

value of x, more sites will be enclosed and more of the closed sites will be stratified.

It is straightforward from (10) that for any given value of x, there is a unique

value of w consistent with long run regional equilibrium. It is less obvious whether a

given value of w corresponds to a unique value of x (and therefore !). It is also unclear

whether a given productivity !corresponds to a unique wage w, or whether there could

be multiple values of w that satisfy (10) for a particular !.


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(11) N =)0a(x)c(x,s)q(s)ds + )a(x)

b(x)e*q(s)ds +)b(x)1n(x,s)q(s)ds

In the first integral, c(x,s) is an increasing function of x due to the concavity of the food

production function. In the second integral, e* is a constant. In the third integral, n(x,s)

= e(x,s) + c(x,s) is the sum of the elite population (both food producers and guards) plus

commoners. Although the elite population falls at a given site quality as x increases, the

sum n(x,s) rises unambiguously because the increase in commoners exceeds the decrease

in elite food producers, and the number of elite guards rises.

Therefore, the only way for N to decrease as x increases is to have a discontinuity

in population density at a(x) or b(x). But there is no such discontinuity at b(x): as new

sites become stratified, the elite and commoner population densities change continuously.

The remaining possibility is a discontinuity in population density at a(x), where sites go

from being open to closed. As was discussed in section 2 in connection with Figure 3(b),

there is such a discontinuity if the zero surplus indifference curve for z = A is tangent to

the boundary of the feasible set E at an interior point (e f*, eg*). In this case, the marginal

closed site with s = a(x) has an abruptly lower population density e* compared to slightly

worse open sites.

In order to obtain a backward-bending locus in Figure 5, this effect on population

density from changing property rights would have to outweigh the effect of a higher x on

population density at sites where property rights remain unchanged, which always goes in

the opposite direction. If the latter effect is relatively large, or the discontinuity at a(x) is

small, or the site density q(s) is small in the vicinity of a(x) so that the discontinuity has


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little impact on the regional population, a backward-bending curve will not arise, and

there is a unique wage w for each productivity value !.

Furthermore, we showed in section 2 that there are conditions under which no

discontinuity arises. This was true in Figure 3(a), where the constraint boundary had a

finite horizontal intercept and the zero-surplus indifference curve at z = A was flatter than

the constraint boundary at this intercept. The resulting corner solution gives a continuous

population density n(x,s) at the boundary a(x) between open and closed sites. Hence, this

is a sufficient condition for a unique wage at each productivity level.

Can higher productivity make commoners better off? When the first (highest

quality) sites are enclosed, the wage w must fall. But more generally, one can show that

a necessary condition for w to increase as !increases along the locus in Figure 5 (e.g.

from points T to U) is that higher x = !/w must reduce total rent )a(x)1v(x,s)e(x,s)q(s)ds

over the relevant range (see equation 10). Because an increase in x raises surplus per

elite agent v(x,s) at every site quality, the only way for total rent to decrease is for the

size of the elite e(x,s) to decrease in greater proportion, at least for some sites.

We have already shown that on the interval [a(x), b(x)] the size of the elite is a

constant that does not depend on x. Hence if a(x) < 1 "b(x) so that some sites are closed

but none are stratified, w cannot rise with !. In this situation, higher productivity implies

rising total rent at closed sites and falling consumption per capita at open sites. The fact

that !is rising while w is falling means that x is rising. This implies that the dividing line

a(x) between open and closed sites is shifting to the left on the interval s %[0, 1]: more

sites are enclosed and the remaining open sites have lower average quality than before.


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The only other possibility is b(x) < 1 so that some sites (those highest in quality)

are both closed and stratified. In this situation, an increase in !could lead to an increase

in w as long as elite size e(x,s) at the stratified sites drops rapidly enough to offset the

rising surplus per member v(x,s). But even in this case, w can still fall indefinitely. A

sufficient condition for this result is that the feasible set E touches the horizontal axis as

in Figure 3(a), and the boundary of the feasible set is everywhere steeper than -1. This

implies that for any closed site, the optimal elite time allocation (ef*, eg*) remains at the

corner solution where ef* = Ehand eg* = 0 regardless of z (therefore for any productivity

level or site quality). Even if a site is stratified, elite size stays constant and the wages of

commoners must fall as productivity rises. This outcome is most likely when elite food

labor automatically helps to exclude outsiders and specialized guard labor does not lead

to major reductions in the amount of elite food labor required for this purpose. As an

extreme case, this would be true if the constraint boundary in Figure 3(a) were vertical.

So what happens to inequality as productivity improves? The overall message of

this section can be summarized as follows. At low productivity values !, all sites are

open and everyone has equal food consumption w = (. Once !becomes sufficiently

large, the best sites are enclosed and over some range the elite becomes better off while

commoners become worse off. However, at first the 'commoners' are the people located

at open sites, while the 'elite' are the insiders who occupy the closed sites. There is no

stratification between elite and commoners within the same site.

As productivity continues to rise, in general there could be a backward bending

part of the locus in Figure 5 (although we have given conditions that are sufficient to rule

this out). If so, the most reasonable prediction is that the wage level continues to change


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smoothly until !reaches a point like Q in Figure 5. Any further increase in !should

trigger an abrupt drop in the wage w from the point Q to the point S, followed by a return

to smooth changes in the wage to the right of S. If there are multiple backward-bending

segments of the curve in Figure 5, there will be multiple discontinuous drops in the food

consumption of commoners as productivity rises over time.

Whenever a discontinuous drop in w occurs, it implies a discontinuous increase in

the ratio x = !/w. This causes abrupt drops in the break points a(x) and b(x) for the site

quality distribution. The result is an abrupt increase in the number of closed sites and in

the number of stratified sites (if b(x) < 1 is relevant). Indeed, one may see the sudden

appearance of many internally stratified sites where none previously existed.

Backward-bending parts of the curve in Figure 5 do not alter our expectation that

commoners become worse off over time, as long as productivity always grows over time.

If productivity ever declined, however, we would need to consider abrupt upward jumps

in the wage, accompanied by a sudden reduction in the number of enclosed sites. Due to

the existence of multiple equilibria, the time sequence of productivity changes would be

important, introducing an element of path dependency into the story.

As long as stratification has not yet emerged, commoners must become worse off

as productivity improves. Thus there cannot be any range like the one between points T

and U in Figure 5 where !and w both rise as x increases and the slope of the rays from

the origin diminishes. However, after there is stratification, productivity increases could

improve the lot of the commoners. As we discussed earlier, this requires that total rent to

the elite must fall, which can occur only if rising productivity leads to a rapid reduction in


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elite size sufficient to outweigh the rise in elite consumption per capita. If the size of the

elite falls too slowly or not at all, the poor continue to get poorer as the rich get richer.

5. Predictions and Extensions

A number of empirical predictions can be derived from our model, some of which

should be amenable to testing by archaeologists or anthropologists.

(a) Starting from a world with no or minimal exclusion of outsiders from production

sites, an increase in the productivity of food technology (whether due to climate

or technical innovation) should eventually cause some sites to become closed to

outsiders. This enclosure process should begin at the highest-quality sites in a

region, and gradually extend to lower-quality sites as productivity increases.

(b) Closed sites should initially not have any internal stratification. Agents in higher

quality closed sites should have higher per capita food consumption, and agents at

any closed site should be better off than agents at the remaining open sites. The

population density at sites that are closed but unstratified should be no larger than

the density at the best open sites (perhaps with an abrupt downward drop as new

sites are enclosed). Sites that are closed but unstratified should have an internal

division of labor between food production and land protection that does not vary

with site quality or productivity.

(c) If productivity continues to increase, eventually some of the closed sites should

become internally stratified. This process should begin with the highest-quality

sites and gradually extend to lower quality sites. Commoners at the stratified sites

should have the same level of food per capita as the people at open sites. Internal

inequality should be greatest at the highest-quality sites. Food consumption for


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elite agents should rise continuously with productivity. The size of the elite at

stratified sites should either stay constant or fall gradually as productivity rises. If

it falls, over time more elite labor should be devoted to guarding land while less is

devoted to direct food production (eventually all of the elite may guard land and

food may be produced only by commoners; from this point on, further increases

in productivity will not change the size of the elite).

(d) As productivity increases, if there are discontinuous drops in local population

density as successive sites go from being open to closed, there may likewise be

discontinuous drops at the regional level in the food consumption per capita of

commoners. Such abrupt drops would be accompanied by sudden increases in the

number of sites that are enclosed, and also in the number that are stratified (if any

stratified sites exist). But if there is no discontinuity in local population density

associated with changing property rights, the welfare of commoners should be a

continuous function of region-wide productivity.

(e) As productivity rises, commoners should become worse off at least until the

emergence of internally stratified sites. After this point, commoners should

continue to become worse off as long as the total surplus of the elite as a whole

continues to expand. But if the size of the elite declines proportionately more

rapidly than food per member rises, the commoners may see improvements in

their standard of living.

Having summarized the empirical content of our model, we turn to possible extensions.

Kinship. A key observation is that people who share access to territory and keep

outsiders from using it often have close kinship ties, whether genetically or by marriage.


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Moreover, highly stratified pre-state societies generally have hereditary elites who claim

rights to territory based on descent from a common ancestor, and keep careful records of

genealogical relations within the elite. Proximity to a chiefly or kingly line is generally a

source of enhanced status, while the ancestry of commoners is of little interest to anyone.

We suspect that such patterns arise because kinship groups have advantages in

producing local public goods, one of which is exclusion of outsiders and/or maintaining

control over commoners. This follows from familiar arguments involving evolutionary

biology about altruism among genetic relatives. In our model, groups with close kinship

ties should have exclusion technologies with a larger feasible set E as discussed in section

2. As productivity rises, these groups will be the first to find it profitable enclose a site of

any given quality level. This should carry over to stratified sites as kinship-based elites

begin to use commoner labor.

Downward mobility. Our Malthusian framework implies that regional population

increases as productivity increases, and that per capita food consumption for the region as

a whole (averaging over elites and commoners) remains constant. As we mentioned at

the end of section 3, the inequality between elite and commoner implies that these two

groups have unequal numbers of surviving adult children per capita. In order to have a

stable elite size at each site, there must be a mechanism for moving some elite children

into the commoner class. There must also be more likelihood of downward mobility in

higher-quality sites, because the associated elites are relatively richer and therefore have

relatively more children per adult.

Three mechanisms for implementing downward mobility that may be important in

practice are rules of bequest that give an inheritance only to a subset of children (such as


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believe that a common causal path runs from productivity (determined jointly by nature

and technology) to population density, and then to sources of inequality that involve

increasing returns to scale and only become important once population density is high.

The details at the last step are likely to vary with the specific features of the society.

Our present model is about economic inequality, not political hierarchy. We have

attempted to explain how differences in access to land can emerge, and what implications

they have for differences in food consumption. This is quite different from the question

of how authority relationships emerge. However, it may be possible to link the two ideas

by focusing on supervision of subordinates (commoners), who must somehow be induced

to work on behalf of their superiors (elites). We have not modeled how an elite secures

compliance from the commoners who toil on the land, but these methods surely overlap

with the ways in which political leaders secure compliance from their followers.


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Origins Inequality

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