Good Fences Make Good Neighbors:
Endogenous Property Rights in a Game
of Conflict∗
John R. Boyce† and David M. Bruner‡
August 26, 2008
Abstract: This paper derives the conditions under which property rights
can arise in an anarchy equilibrium. The creation of property rights requires
that players devote part of their endowment to the public good. In the Nash
equilibrium, no player contributes to the provision of property rights protection.
Therefore, players are left with two alternatives. A king who provides property
rights protection paid for by a tax on endowments completely eliminates conflict.
However, a king who can eliminate conflict can also take the surplus for himself.
As a despotic king inefficiently taxes endowments, players have an incentive to
find a solution that keeps power in their own hands. Thus in a social contract,
players first credibly commit part of their endowments to providing property
rights and then allocate the balance of their endowments between production
and conflict. A social contract can also drive conflict to zero. However, as the
number of players rises, the private provision of property rights through a social
contract results in positive levels of conflict and lower levels of aggregate welfare
than under a benevolent king.
1
Introduction
Demsetz (1967) was one of the first to offer an economic theory of the origin
of property rights. Demsetz observed that the Montagnais bands of the Algonquians of Labrador created property rights for beaver ponds once the fur trade
made the establishment of property rights sufficiently valuable. He postulated
that exogenous increases in the value of a resource will lead to the establish∗ We have benefited from comments by Diane Bischak, Christopher Bruce, Herb Emery,
Joanne Roberts and Scott Taylor. Scott Odland provided excellent research assistance. All
remaining errors are our own. The title is from Robert Frost, “Mending Wall” (1914).
† Professor of Economics, University of Calgary, 2500 University Drive, N.W., Calgary,
Alberta, T2N 1N4, Canada. email: [email protected] telephone: 403-220-5860.
‡ Assistant Professor of Economics, Appalachian State University, Boone, North Carolina,
28608 U.S.A. email: [email protected].
1
ment of property rights for that resource.1 While this may be the case, it is
also possible that an increase in the value or productivity of a resource may
instead result in an increase in conflict. This appears to be the case in Nigeria,
where the discovery of oil raised the level of conflict rather than resulting in
the formation of well-defined property rights (Sala-i-Martin and Subramanian,
2003). Similarly, the current conflict in Darfur has been attributed to a drought,
which raised the marginal value of water. However, instead of property rights
for water organically arising, deadly conflict has enveloped the region, with some
estimates suggesting over 400,000 dead.2
In the absence of property rights, there exists what Hirshleifer (1995) calls
an anarchy equilibrium.3 If subsistence consumption is not subject to theft
(Murphy, Shleifer and Vishny, 1993), when resources are of insufficient value
in production, players may simply consume their endowments in a subsistence
economy. Thus there is no need for property rights to arise, and as a consequence, players need not spend resources on conflict. However, as the resource
endowment becomes more valuable in the sense of Demsetz, players will wish to
reallocate some of their endowment from subsistence consumption to production. But if produced goods are subject to thievery, in the absence of property
rights, players also have to devote some resources to protecting their production.
Therefore, as the value of production rises, the level of conflict also rises. But
this means that the social value of creating property rights rises as well, since
conflict, which only redistributes existing production, is socially costly because
it reduces the resources available for production.
Therefore, players have an incentive to solve the public goods problem of
providing property rights protection for all players. The question is, under
what conditions can property rights arise in an anarchy conflict equilibrium?
Our first finding is quite negative about the possibility that the property rights
can arise organically. If players simultaneously allocate their endowments between production, conflict, and a contribution to the public good of property
rights protection, then in the Nash equilibrium, zero protection for property is
provided. Thus property rights suffer from an extreme version of the problem
of private provision of public goods.
We then consider two alternatives to private provision of public property
rights. In the first, property rights are provided by an external player, a chief,
a lord, or a king, who offers protection of property in exchange for the right
to tax his ‘citizens’. Only if the value of creating property rights is sufficiently
high, is a king able to create and enforce property rights. However, when that
condition is satisfied, the king is able to enforce perfectly defined property rights,
in the sense that the economy under a king has no conflict. But allowing a king
1 See
Anderson and Hill (1975), Umbeck (1977), and the June 2002 special issue of the
Journal of Legal Studies.
2 “Hundreds Killed in Attacks in Eastern Chad,” Washington Post, April 11, 2007, p. A.10.
3 On conflict models more generally, see Bush and Mayer (1974), Dixit (1987), Hirshleifer
(1978, 1991a, 1991b, 1995, 2000), Garfinkle (1990), De Meza and Gould (1992), Skaperdas
(1992, 1996, 2003), Grossman (1991, 1994, 2001 2002), Grossman and Kim (1995), Neary
(1997a, 1997b), Hotte (2001), Baker (2003) Gonzalez (2005, 2007), and Hafer (2006).
2
to create property rights is a risky strategy, since a king who has the power
to protect property may use that power to take property. A despotic king will
exploit this power (e.g., Winetrobe (1990), Olson (1993), Grossman (2002)).
While this has distributional consequences, since a despotic king can take all of
the surplus he creates, it can also have efficiency consequences. Under the form
of contract between king and subjects common in the middle ages, that of a
tax on endowments,4 we show that a despotic king is unable to solve the puzzle
of how to create incentives to generate surplus while simultaneously exploiting
that surplus.
Therefore, given both the distributional and efficiency risk to devolving
power, it is in the interest of players to find an alternative in which property
rights can be established – but without relinquishing their say in how those
rights will be protected. In what we call a social contract game, players make
the establishment of property rights an antecedent to their allocation between
production and conflict. In the social contract stage of the game, a public goods
provision game occurs in which players simultaneously allocate part of their endowment towards the protection of property rights. Then in the conflict stage
of the game, players allocate the remainder of their endowment between conflict
and production. In this game an organic and credible form of property rights
arises that does not require externally provided force. This occurs because the
strategic effect of property rights protection is able to overcome the incentive
to free-ride on others’ provision of property rights protection that so dominates
the Nash equilibrium.
This sort of Lockean social contract ideal, is, of course, not costless. Players
still must commit resources away from productive uses towards the establishment of property rights. In a rational expectations equilibrium, they do this
by correctly anticipating how property rights affect their behavior in the subsequent conflict game. The subgame perfect Nash equilibrium level of property
rights protection is Pareto improving, provided that the Demsetz condition is
satisfied that establishment of property rights is sufficiently valuable. However,
the minimum level of productivity of the endowment when used in production
under which property rights can be established in the social contract game is at
least as great as, and as the size of the population rises, is strictly greater than,
the minimum level of the Demsetz parameter under which a king (benevolent
or despotic) can create property rights. This occurs because unlike a king, a
social contract cannot fully eliminate the incentive to free-ride in the provision
of the public good of property rights protection. Thus the Demsetz condition
that the value of establishing property rights be sufficiently high is a necessary
condition in order that well-defined property rights be established, but it is not
a sufficient condition to establish these property rights.
4 William
the Conquerer, the Norman who became king of England after defeating the
Anglo-Saxon king Harold in the Battle of Hastings in 1066, provides the starkest example of
the way in which a king funded his government by a tax on the endowment. William declared
all land “terra regis,” or the king’s land. This lead Bloch (1960, at p. 188) to conclude, “All
land was held of a lord and this chain, which was nowhere broken, lead link by link to the
king.”
3
This paper is closest in spirit to the conflict models by Grossman (2002)
and Hafer (2006). Each of those authors also considered an economy in which
property rights are endogenously determined. In Grossman (2002), a king provides property rights. Like Grossman, we find that a king creates property
rights that drive conflict to zero. However, Grossman does not consider either
of the organic alternatives to a king that we consider. Hafer (2006), like us,
considers a model in which property rights arise in an organic fashion. In her
model, there are two productive processes, and one of these requires a resource
that is subject to thievery. The security of property rights evolves over time
as players learn about the “type” of the other player through repeated conflict
interactions. Players for whom ownership of the resource yields higher marginal
productivity than their outside alternative are more inclined to defend it. As
other players learn this over time, conflict diminishes and a form of property
rights based on “might-makes-right” occurs. Like Hafer, the model we consider
has an outside alternative for which there is no conflict, although unlike Hafer,
the outside option of subsistence consumption only plays a role in determining the minimum level of the Demsetz parameter under which property rights
can be developed. Like Hafer, our model also requires an explicit expenditure
on property rights creation. However, property rights in our model arise as a
result of private or public contributions to the public good of property rights
protection, rather than through repeated conflict. Thus in our model, not only
do property rights arise, but so does a primitive state, one with a purpose and
the means to accomplish that purpose.5
The conflict models of Hirshleifer (1995) and Grossman and Kim (1995) are
based on the rent-seeking model of Tullock (1980). Because we are interested
in analytical solutions, we simplify the conflict technology in two ways. First,
in the phrase of Hirshleifer (1995), the model we consider has a ‘decisiveness’
parameter equal to one.6 This means that in a symmetric conflict game with
no property rights, each player’s proportion of their own production that they
appropriate is N1 . Second, as we wish to emphasize the role of property rights,
we allow the contest success function to be asymmetric in the sense that in a
symmetric game property rights make one’s appropriation of one’s own output
greater than N1 and one’s expropriation of the other players output less than
1
N . We obtain this by a simplification of the conflict success function used by
Grossman and Kim (1995) in which the property rights parameter enters the
5 Semi-democratic forms of governance first appeared in Mesopotamia in the Sumerian citystates. In The Epic of Gilgamesh, King Gilgamesh (c.2700bc) wished to go to war. He first
sought permission of the elders of the city to do so, and when that was not forthcoming,
he appealed to an “assembly of all the men of the city of fighting age” (Saggs, 1989, at pp.
35-36). Thus men had some say in their public affairs in some of the earliest examples of citystates. However, the Egyptians and Persians were autocracies throughout their histories. The
Athenians in sixth to fourth centuries bc were the first to be ruled directly by an assembly
of its own citizens. Thus in the Greek city-states we see the first formations of the social
contracts of the form we study.
6 In Tullock, the probability player i investing x wins a prize worth R when N playi
PN
m . In Hirshleifer, the proportion of one’s own
ers compete for the prize is pi = xm
j=1 xj P
i /
N
m
endowment, that one appropriates is pii = xm
j=1 xj , where m is the decisiveness parami /
eter.
4
conflict success function additively rather than multiplicatively.7
2
Examples of Property Rights Creation
We begin by discussing seven examples in which property rights have been created out of anarchy equilibria. The first four examples are cases where property
rights arose organically. In the fifth example, property rights are created by a
benevolent king, and in the sixth example, property rights are created organically in response to a despotic king. The final example contrasts how the Allies
settled with Germany after World War I and then after World War II.
2.1
The California Gold Rush 1848-1850
When gold was discovered in California in January 1848, California had just
become a territory of the United States following a war with Mexico. As it was
not until September 1850 that California became a state, there was no legal
foundation for property rights during the gold rush (Umbeck, 1977). By the
end of 1848, between 5,000 and 10,000 miners were in the area, but the the
area was large enough that new arrivals simply moved elsewhere on the rivers.
However, in 1849, an additional 40,000 people arrived, and by the end of 1852
the population of California had increased by 150,000. During the period 18481850, property rights were established and protected by miners using informal
organizations. The size of claims would be decided at a miner’s meeting, usually
by majority rule, and the miners would stake their claims by marking their
territory and then working it. As long as the owner did this, the other miners
would help keep “claim jumpers” off of each other’s property. Most, but not
all, of these claims were later recognized by California law (Clay and Wright,
2005).
2.2
Water Rights in 15th Century Valencia
Ostrom (1990, at pp. 73-79) discusses the formation of property rights to water
in Medieval Spain. The Spanish had created irrigation canals to carry water to
fields. Farmers were assigned property rights to the water in which water could
be used sequentially by the farmers. These property rights were secured by a
two-fold system. First, each successive user had an incentive to monitor the
previous user since as soon as the previous farmer’s fields were irrigated, the the
farmer could begin irrigating. Second, a farmer who took more than his share
7 Grossman and Kim distinguish between appropriation (z ) and expropriation (x ) effort
i
i
and allow for differential relative effects of appropriative and expropriation technologies. In
their model, the proportion of one’s own production that one appropriates is pii = θzi /(θzi +
xj ). We differ from Grossman and Kim and by allowing θ to enter pii additively rather than
multiplicatively, and (like Hirshleifer (1995)) by not distinguishing between zi and xi . See
Skaperdas (1992, 1996), Hirshleifer (1991b, 2000), and Grossman and Kim (1995, n. 6 at
p. 1279) for detailed discussions of alternative specifications for the technology of conflict.
Mueller (2003, at p. 379) provides a rent-seeking analog to the contest function we use.
5
was brought before a council of all farmers. Thus the threat of sanctions by
other farmers provided security of property rights. Similar systems of property
rights evolved in Swiss grazing fields, Japanese forests, and Nova Scotia fisheries.
2.3
The American West 1870-1900
Anderson and Hill (1975) examined how property rights to grazing lands were
established in the American West. The stock-growers banded together into associations which attempted to restrict entry into grazing lands. These groups
were involved in extra-legal schemes to prevent entry, including violence, but
they also were active in lobbying state, territory, and federal governments to enact laws to protect their interests. Following the harsh winter of 1886-87, which
killed off large numbers of cattle, memberships in the stock-growers associations
dropped from 416 to 183 in Wyoming by 1889 and by two thirds in Montana
over the same time period as the benefit of providing the public good declined.
2.4
Texas Oil Fields
Libecap and Wiggins (1985) consider the case of the establishment of property
rights to oil fields. While surface rights existed, the migratory nature of oil
meant that a field was common property. The Slaughter field in west Texas,
discovered in 1936, is 71,000 acres in surface size. An attempt to unitize production on the whole field failed, but twenty-eight sub-units were created. Libecap
and Wiggins (1985, at p. 694) estimated that by 1975, 427 wells had been drilled
along the boundaries of the sub-units to re-inject water that has been recovered
along with the oil and gas. The purpose of these wells was to prevent migration
of oil across subunits; they were drilled at a total cost of 156 million dollars. As
of 2002, there were approximately 2400 total wells drilled on this field.8 Data
from the Oil and Gas Journal and the Texas Railroad Commission indicates
that 1.3 billion barrels of oil have been extracted from this field since 1936.9
Using the average production rate of 18.2 million barrels of oil per year, and
using estimates of the real oil prices from 1936 to present,10 the present value
(at 4 percent real interest rate) of revenues from oil production on the field in
2007 dollars is roughly 9 billion dollars. Using Libecap and Wiggins estimate of
a cost of 360,000 dollars per well in 1975, and assuming that the cost of the 2400
total wells drilled was spent in 1936, then the total drilling costs were on the
order of 3.3 billion 2007 dollars. Thus ignoring natural gas production, which
was substantial, the field appears to have generated over five billion dollars in
net of drilling cost revenues for its owners. Therefore, the dissipation of rents
by the 427 injection wells on the subunit boundaries was small relative to rents
8 Source: International Oil and Gas Development Yearbook (Review of 2002), Vol. 73, p.
705, International Oil Scouts, Mason Map Service, Austin, Texas.
9 See “U.S. Fields with Ultimate Oil Recovery Exceeding 100 Million BBL”, Oil and
Gas Journal, January 26, 1998, p.
82, and TRRC’s “General Production Query”,
http://webapps.rrc.state.tx.us/PDQ/home.do
10 BP Statistical Review of World Energy 2008, www.bp.com/statisticalreview
6
received. By prohibiting migration of oil across sub-unit boundaries, these wells
could be viewed as a public good which prohibited even worse dissipation of
rents had the oil been allowed to migrate.
2.5
Athenian Democracy 594-322BC
At the beginning of the sixth century bc Athens was ruled by an oligarchy
formed from the families the Pentakosiomedimnoi, who owned the land. While
property rights existed in the oligarchy era, it was their insecurity due to the
lack of a modern financial system that brought forth the enlargement in the
franchise in 594bc. In order to borrow, the lower classes offered themselves
as collateral. If they defaulted, they were sold into slavery. A wave of crop
failures around 600bc created much unrest. Concern that the oligarchy would
be replaced by a tyranny supported by the lower classes (as had happened several
times previously) led the Areopagus, the highest council of Athens, to ask a man
named Solon to write a new constitution. Solon agreed to do so on the condition
that it would not be altered for ten years.11 Solon’s most important change was
to allow the lowest of four classes of citizens, the Thetes, membership in the
assembly, the Ekklesia, which voted on all important matters and served as
juries, the Heliaia, in legal disputes. However, membership in the Areopagus,
which controlled agendas, was still restricted only to Pentakosiomedimnoi (see
Stanton, 1990, at p. 66-67). The importance of Solon’s constitution was that
juries in disputes over transactions would now be made up of all of the classes,
and the Thetes, being largest in number, would be able to prevent enslavement
as a remedy to future property disputes. The Pentakosiomedimnoi gave up
some control over future decisions, but gained security of their property.
2.6
The Magna Charta
The Magna Charta, which the English King John I accepted in 1215, limited the
rights of the King over his barons. Prior to the Norman invasion, Saxon King
Cuthred declared in 745 that “all gifts of former kings. . . in country houses, and
in villages and lands, and farms and mansions. . . shall remain firm and inviolate,
as long as the revolution of the pole shall curry the lands and seas with regular
movement around the starry heavens [sic]” (Barrington, 1900, at p. 35). However, upon assuming the throne in 1066, King William the Conquerer declared
all land in England “terra regis”, repudiating all previous claims.12 When in
need of money, William would take lands that he had previously granted and
11 Solon understood subgame perfection. Upon issuing the new constitution, he took a ten
year trip abroad, reasoning that while the Pentakosiomedimnoi had agreed not to change the
constitution themselves, they could still persuade him to change the constitution!
12 The Domesday Book was the assessment of the value of all the king’s holdings. The
estimated value of all land in England outside of the towns and cities in 1087 was £73,000.
“Of this sum the king and his family received £17,650; his servants and officials, the king’s
sergeants, £1,800; the church £19,200; and some few entrusted Englishmen £4,000. The
remainder, amounting to a sum of £30,350, was apportioned out to some 170 baronies as
rewards for the Normans who had shared in the enterprise of conquest” (Poole, 1955, at p. 2)
7
resell them to the highest bidder, sometimes “taking them away from the purchaser, and again selling to one who would bid higher” (Barrington, 1900, at
p. 55). William and his successors also imposed various new taxes on their
citizens.13 In 1209, in a dispute with Pope Innocent over who should become
Archbishop of Canterbury, King John confiscated all church lands. The Pope
excommunicated John and absolved all nobles of their oaths of fealty to John.
In 1215 some 2000 earls, barons and knights marched upon London to force
John to accept the Magna Charta. Section 12 stated that no taxes could be
imposed by the king without the consent of a council of nobles; section 9 allowed
property to be passed on to heirs of the noble’s choosing; and section 28-31 and
39 forbid the taking of property by the king without due process (Barrington
(1900, at pp. 228-250), Poole (1955, at pp. 474-76)). Thus the Magna Charta
was both an economic document (allowing for transfer of land ownership) as
well as a political document (requiring that the nobles approve new taxes).
2.7
War Reparations and the Marshall Plan
At the end of World War I, the Allies imposed war reparations on Germany
in the amount of 269 billion gold marks (about $400 billion in current dollars).
Keynes (1925) presciently argued that “The existence of the great war debts
is a menace to financial stability everywhere” (p. 25). With the rise of Hitler
being in part attributed to the onerous war reparations, at the end of World
War II the Allies chose a much different path, using the Marshall Plan to rebuild
Germany.
3
Model Assumptions
We now present a formal model of conflict in which security of property is
a public good. Consider a game in which there are N ≥ 2 players, indexed
i = 1, . . . , N . Each player has an endowment of ω units of a resource. There
are four different goods that the endowment may be used to produce. First,
the endowment may be consumed directly. This how a hunter-gatherer society
treats its endowment. Each unit of the endowment consumed in this fashion
yields one unit of utility, which we shall call “subsistence” consumption, as it
corresponds to the minimum level of possible equilibrium utility. On the other
hand, the endowment can be invested to produce a consumable good, which
we call “corn”. An investment of ki units of endowment into corn production
produces Aki units of utility. The parameter A is the Demsetz parameter that
tells how valuable corn is relative to consuming the endowment directly through
13 In 1084, William imposed a tax of six shillings on every hide (60-120 acres). His successor,
Rufus, in 1096 imposed taxes on the Church of 10,000 marks, which the church paid by melting
chalices and robbing their crucifixes (Barrington, 1900, at p. 69). In 1109, Henry II, imposed
another tax of six shillings per hide. King John I, in 1203, after losing Normandy in a war
caused by John’s taking the wife of one of his barons as his own, imposed a tax equal to 1/7th
of the value of the barons’ holdings to pay for his war debts.
8
subsistence consumption. We assume that stealing the endowment is not worthwhile, since a unit of the endowment must be combined with a unit of labor to
produce consumption. However, corn can be stolen. Therefore, if A ≤ 1, planting corn produces less than could be obtained through subsistence consumption,
hence, insecure property rights for corn are irrelevant. Thus A > 1 is necessary
for either conflict or property rights to arise. As we shall see, the lower bound
on A under which property rights may arise is larger than than this.
The other two goods in which an player may invest his endowment are
“guns”, xi , and the provision of property rights protection, which we call “policing”, yi , although we shall include in policing all aspects of protection of property rights including prevention, enforcement, dispute resolution, and sanctions.14 Guns serve two purposes. They are a tool for protecting one’s own
property and for stealing the property of others. An increases in xi increases
the share of i’s own corn production that i appropriates and it increases the
share of the other players’ corn production that i expropriates. Thus guns are
the mechanism of conflict. In contrast, an increase in policing increases the
share of player i’s own corn that i appropriates but it reduces the share of other
players’ corn that i expropriates. While an increase in i’s guns increases his
appropriation and his expropriation, an increase in policing has an asymmetric
effect on appropriation and expropriation. Guns make the owner better off and
others worse off; policing is a public good that makes the provider both better
off and worse off.
Policing can be P
provided either privately or by the state, if one exists. PriN
vate policing, Y ≡ i=1 yi , is paid for by a contribution from the endowment
of each individual. Public policing, Ψ, is paid for through a tax, τ , imposed
by the state. In addition, a player has an underlying natural advantage over
thieves in protecting his own property, denoted by the parameter θ. This could
be due to a barrier such as a mountain or a river that divides one’s property
from others, or due to the psychology of the endowment effect. Y , Ψ, and θ
are perfect substitutes. The sum θ + Ψ + Y measures the advantage a player
has in appropriating his own corn production relative to expropriating the corn
production of others. Therefore, the conflict technology has the following properties. The proportion of player i’s corn production that player i appropriates
is given by
θ + Ψ + Y + xi
,
i = 1, . . . , N,
(1)
pii =
θ+Ψ+Y +X
PN
where X ≡
i=1 xi , and the proportion of player j’s corn production that
14 In ancient Greece, the policing investment was that its subjects participate in the assembly, to serve in the military when called, and to act as magistrates. In medieval England,
William the Conquerer required of his subjects to build some 1200 castles. In the settling
of the American west, stock-growers joined associations which made rules about when cattle
could be rounded up and branded, and to influence legislation on the use of brands, barbed
wire, and other legislation of interest to them. The miners of the California gold rush acted
both as makers and enforcers of rules regarding mining claims. In the Texas oil field, the investment in injection wells prevented migration of oil across subunit boundaries. All of these
are subsumed into our use of the word ‘policing.’
9
player i gets to expropriate is
pij =
xi
,
θ+Ψ+Y +X
i 6= j, i = 1, . . . , N.
(2)
The proportion of i’s corn production that i appropriates is increasing in xi , θ,
Ψ and Y , and decreasing in X−i ≡ X − xi . The proportion i expropriates from
others is increasing in xi and decreasing in X−i , θ, Ψ and Y . Since pji is the
amount of i’s corn production that player j expropriates from i, corn production
is either appropriated or expropriated. Hence,
pii +
N
X
pji = 1, for all i = 1, . . . , N.
(3)
j6=i
It is natural to think of the quality of property rights in terms of pii . When
pii = 1, (3) implies that property rights are perfectly protected. When pii < 1,
property rights are insecure. It is clear from (1) that property rights are perfectly
secure if and only if x1 = x2 = · · · = xN = 0. This specification differs from
Grossman and Kim (1995) in that the sum θ + Ψ + Y enters additively in (1)
and (2) rather than multiplicatively. Our specification has the limitation that
absent property rights, the natural advantage does go to the expropriator as
can happen in Grossman and Kim (1995) and in Hirshleifer (1995), but it has
the advantage that it easily yields closed form solutions.
Absent state or privately provided protection of property rights, (1) and (2)
implies that the parameter θ creates an asymmetry that favors the holder of
the endowment. We assume that the natural advantage to protecting one’s own
property is limited:
Assumption 1. ω > θ ≥ 0.
Assumption 1 limits how much of his endowment each player devotes to guns
in the conflict equilibrium. It also plays a role in determining whether or not
property rights can be created.
Each player’s utility is the sum of what he appropriates from his own corn
production and what he expropriates from the corn production of the other
players, plus his subsistence consumption:
ui = pii Aki +
N
X
pij Akj + ci ,
i = 1, . . . , N.
(4)
j6=i
Each player simultaneously maximizes his utility by choosing how he allocates
his after-tax endowment, ω−τ , across the four possible choices: corn production,
private provision of policing, subsistence consumption, and guns:
ki = ω − τ − xi − yi − ci ,
i = 1, . . . , N.
(5)
Thus ki is the residual from the choices of ci , xi , yi and the rate of taxation, τ .
10
As we have assumed that each player’s endowment is identical, we restrict
our attention to symmetric equilibria in which x1 = x2 = · · · = xN ≡ x ≥ 0,
y1 = y2 · · · = yN ≡ y ≥ 0, and c1 = c2 · · · = cN ≡ c ≥ 0. However, all of the
results presented can be derived with few alterations if we were to allow the
endowments to differ.15
4
The Nash Equilibrium
Let us suppose that no state exists. Therefore, Ψ = τ = 0. Our objectives are
to see how well property rights are protected absent a state, and to characterize
the Nash equilibrium in terms of the Demsetz productivity parameter, A.
In the Nash equilibrium, each player simultaneously chooses xi , ci , and yi
to maximize ui , taking the actions of the other player as given. The first-ordernecessary-conditions for player i are


N
X
∂p
∂p
∂ui
ii
ij
= A
ω − xi − yi − ci +
ω − xj − yj − cj − pii  ≤ 0,
∂yi
∂Y
∂Y
j6=i
i = 1, . . . , N, (6)
∂ui
= 1 − pii A ≤ 0,
∂ci
i = 1, . . . , N,
(7)


N
X
∂p
∂p
∂ui
ij
ii
ω − xi − yi − ci +
ω − xj − yj − cj − pii  ≤ 0,
= A
∂xi
∂xi
∂xi
j6=i
i = 1, . . . , N. (8)
From (1) and (2), the rates at which the appropriation and expropriation parameters change as xi , yi , and ci increase are
∂pii
∂pii
X−i
=
=
2 ,
∂Y
∂xi
θ+Y +X
∂pij
−xi
=
and
2 ,
∂Y
θ+Y +X
∂pii
∂pij
=
= 0,
∂ci
∂ci
∂pij
θ + Y + X−i
=
2 .
∂xi
θ+Y +X
(9)
Each unit of the endowment allocated to any of subsistence consumption,
guns or policing has an opportunity cost in foregone appropriated corn production of pii A. From (7), the marginal benefit from an increase in subsistence
consumption is simply the direct increase in the utility from the subsistence
15 When endowments differ, players with lower endowments devote a larger portion of their
endowment to conflict in the Nash equilibrium (Hirshleifer, 1991a). A king of either type
taxes those with a larger endowment at a higher rate, and under a social contract, those with
a larger endowments devote more resources towards policing.
11
consumption, as an increase in ci has no effect upon the proportion of one’s
own corn production appropriated nor upon the proportion of others corn production expropriated. From (8), the marginal benefit from an increase in guns
is the increase the amount player i’s own corn production that player i gets
to appropriate, plus the increase in the amount of the other players corn production that player i gets to expropriate. From (6), the net marginal benefit
from an increase the size of policing is the increase in the share, pii , that player
i appropriates from his own investment in corn production, less the reduction
in the share, pij , of the other players corn production that player i gets to expropriate. Therefore, an increase in expenditures on guns by i increases both
i’s appropriation and expropriation shares, and an increase in expenditures on
policing by i increases the i’s appropriation share but decreases i’s expropriation
share. This asymmetry implies that players will spend more on guns than on
policing in the Nash equilibrium. The extent of this asymmetry is given in the
following proposition:
Proposition 4.1. In the symmetric Nash equilibrium to the conflict game, each
individual contributes zero to the public good of property rights protection.
Proof. See the Appendix.
While private provision of a public good is well known to result in under
provision of the public good relative to the social optimum (Samuelson, 1954),
here the problem is particularly acute. No player wishes to contribute a positive
quantity to policing in the symmetric Nash equilibrium. This occurs because
(9) implies that ∂pii /∂Y = (N − 1)∂pij /∂Y , thus the gain to appropriation
is just offset by the loss in expropriation. Therefore, Proposition 4.1 implies
that property rights are remain insecure.16 This result is unaffected by the size
of A, which suggests that the Demsetz hypothesis does not hold in the Nash
equilibrium.
Given that y N E = 0, absent state-provided property rights, the symmetric
Nash equilibrium condition for the choice of guns, x, and subsistence consumption, c, given by (8) and (7), respectively, can be written as
∂ui
A(θ + x) =
(ω − c − x)(N − 1) − (θ + x) ≤ 0,
∂xi
(θ + X)2
1 ∂ui
=
(θ + X) − A(θ + x) ≤ 0,
∂ci
θ+X
i = 1, . . . , N,
i = 1, . . . , N.
(8’)
(7’)
The next result shows that k < ω:
Proposition 4.2. Under Assumption 1, no Nash equilibrium exists in which
players devote their entire endowment to production.
Proof. See the Appendix.
16 It
can be shown that this result also holds with the Grossman and Kim (1995) conflict
technology, and that this result also holds in our model with asymmetric endowments.
12
This result occurs because setting c = x = 0 results in positive marginal
utility to xi and maybe even to ci by (8’) and (7’), respectively.
Given Propositions (4.1) and (4.2), there are therefore three types of equilibria that may arise. We shall refer to equilibria where xN E > 0, k N E > 0 and
cN E = 0 as the Hobbesian conflict Nash equilibrium (HCNE), since conflict and
production are both positive in this equilibrium.17 Similarly, we shall refer to
equilibria of type where xN E = k N E = 0 and cN E = ω as the Rousseauian subsistence Nash equilibrium (RSNE), as there is neither conflict nor production in
this equilibrium.18 Finally, we shall call equilibria in which k N E > 0, xN E ≥ 0
and cN E > 0 the Lockean subsistence-conflict Nash Equilibrium (LSCNE), as
there is simultaneously conflict, production and subsistence consumption in
these equilibria.19
Let us consider the Hobbesian conflict Nash equilibrium. In this equilibrium,
cN E = 0, but xN E > 0 and k N E > 0. Then (8’), (5), (3) and (4) imply that
y N E = 0, xN E =
ω+θ
A(ω + θ)
ω(N − 1) − θ N E
,k
=
, and uN E =
.
N
N
N
(10)
By Assumption 1, the level of conflict in the HCNE is positive for all N ≥ 2.
Hence, property is less than perfectly secure in the HCNE:
E
pN
=
ii
ω+θ
(N − 1)ω − θ
E
and pN
.
ij =
Nω
N (N − 1)ω
A necessary condition to be in the HCNE is that cN E = 0. From (7’), this
implies that the Demsetz parameter must satisfy the following condition:
cN E = 0, xN E > 0, and k N E > 0 for all A such that A ≥ Ā ≡
Nω
.
ω+θ
(11)
Call the state of nature in which A ≥ Ā the Hobbesian state of nature since it
yields equilibria in which players do not try to reduce conflict by allocating some
of their endowment to subsistence consumption. By inspection, the minimum
value of the Demsetz parameter, Ā, such that the HCNE occurs is increasing in
N.
Next, consider the Rousseauian subsistence Nash equilibrium. In this equiE
librium, cN E = ω and xN E = k N E = 0. Since xN E = 0 implies that pN
= 1,
ii
17 Thomas Hobbes (1651, Chapt. 13 at p. 185) wrote that “during the time men live without
a common power to keep them all in awe, they are in that condition which is called war; and
such a war as is of every man against every man” from which he deduced that “the life of
man [is] solitary, poor, nasty, brutish, and short” (Hobbes, 1651, Chapt. 13 at p. 186).
18 Jean-Jacques Rousseau (1762), states “Men are not natural enemies, for the simple reason that men living in their original state of independence do not have sufficiently constant
relationships among themselves to bring either a state of peace or a state of war” (Rousseau,
1762, Book 1, Chapt. 4 at p. 145).
19 John Locke (1690), fits between Rousseau and Hobbes in his views of the state of nature.
In his view, as in Hobbes, the state of nature involved conflict, but, like Rousseau, he believed
that conflict could be overcome by means other than autocracy: “The state of nature has a
law of nature to govern it, which obliges every one: and reason, which is that law, teaches all
mankind, who will but consult it” (Locke, 1690, Sect. 6 at p. 5).
13
w
cNE
w- q
w +q
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å
2
kNE
w -q
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å
2
q
xNE
êêê
A
`
A
A
Figure 1: The Conflict Nash Equilibrium with N = 2 Players.
(7’) implies that cN E > 0 only if A ≤ 1. When A < 1, players wish to devote
their entire endowment to subsistence consumption. Therefore, utility in the
RSNE is equal to uN E = ω. Define  ≡ 1 to be the upper bound on the Demsetz parameter A such that the RSNE occurs. We call the region where A < Â
the Rousseauian state of nature as it is characterized by an absence of conflict.
For values of A in this region,
y N E = xN E = k N E = 0 and cN E = ω, for A < Â.
(12)
Property rights are perfectly defined in the RSNE since subsistence consumption
cannot be stolen and no one devotes any of their endowment to anything but
subsistence consumption.
The intermediate case is the Lockean state of nature, where  ≤ A < Ā.
The LSCNE is characterized by cN E > 0, xN E ≥ 0 and k N E > 0. These are
the simultaneous solutions to (8’), (7’) and (5):
y N E = 0, xN E =
θ(A − 1) N E
θA
θ
,c
=ω−
, and k N E =
,
N −A
N −A
N −A
for  ≤ A < Ā. (13)
Let us characterize the LSCNE. First, as A increases at Â, k N E jumps from
zero to a positive value and cN E drops by the same amount. (See Figure 1, which
displays the equilibrium choices for a range of values of the Demsetz parameter,
A, for the case where N = 2 and θ = 14 ω.) However, no such jump occurs
at Ā. The upper boundary to the Lockean state of nature, Ā, is increasing in
14
N , but the lower boundary, Â. is invariant to N . Assumption 1 implies that
Ā > Â, so that the Lockean state of nature always exists. The final interesting
property of the LSCNE is that uN E = ω, which is the same as the utility in
the RSNE. This occurs because the marginal utility of subsistence consumption
is one, and in the LSCNE, subsistence consumption coexists with production
and conflict. The marginal product from corn production appropriated is one
E
E
because at xN E , pN
= 1/A which means that pN
ii
ii A = 1. Similarly, using (9),
∂ui
NE
= ω for all A ≤ Ā.
it can be shown that ∂xi = 1 as well. Thus u
Thus, we have characterized the Nash equilibrium. These results are summarized in the following proposition:
Proposition 4.3. The symmetric Nash equilibrium is characterized as follows:
(i)
For A < Â, the RSNE satisfies (12),
(ii)
for  ≤ A < Ā, the LSCNE satisfies (13),
(iii)
for Ā ≤ A, the HCNE satisfies (11).
To summarize, by Proposition 4.1, in no Nash equilibrium is private provision
of the public good positive. By Proposition 4.2 there is no Nash equilibrium in
which players devote all of their endowment to corn production. By Proposition
4.3, all else constant, as the Demsetz parameter, A, increases the proportion
of the endowment devoted to corn production and to conflict increases in the
LSCNE. However, corn production and conflict are unaffected by changes in A
in either the RSNE or the HCNE. By Proposition 4.3, the LSCNE exists only
because ω(N −1) > θ by Assumption 1. Finally, holding the Demsetz parameter
constant and increasing N increases the likelihood of the equilibrium being in
the Lockean state of nature.
Perhaps the starkest example of how the Demsetz parameter determines the
nature of the Nash equilibrium is provided by Diamond (1996). Around 1000ad,
Polynesians settled both New Zealand and the Chatham Islands, some 500 miles
southeast of New Zealand. The rich environment of New Zealand allowed the
Maori population to prosper. In contrast, the Moriori who settled the Chatham
Islands found a cold climate unsuited to the Polynesian agriculture. While the
Maori grew to a rich and warlike society of over 100,000 people, the Moriori
society reverted to an unstructured hunter-gatherer society of approximately
2000 people. In 1835, upon learning of the existence of the existence of the
Chatham Islands, 900 Maori sailed there. Upon arriving, the Maori declared
the Moriori their slaves. The Moriori, who “had a tradition of resolving disputes
peacefully,” intended to share their resources with the Maori, but before an offer
could be made the Maori attacked. A survivor described the ensuing slaughter:
“[The Maori] commenced to kill us like sheep. . . . [We] were terrified, fled to the
bush, concealed ourselves in holes underground and in any place to escape our
enemies. It was of no avail; we were discovered and killed—men, women, and
children indiscriminately” (p. 53).
15
5
Property Rights by Social Contract
Now we consider a variation in game in which property rights may arise even
absent a king. We continue to assume that there is no state. Hence Ψ = τ = 0.
Suppose that players break the game into two stages. In the social contract
stage, each player first simultaneously voluntarily contributes an allocation of yi
from their endowment for the provision of property rights. In the conflict stage,
after the size of the policing, Y , has been realized, each player simultaneously
allocates his remaining endowment between subsistence consumption, guns, and
corn production. We call this a social contract, since players make the allocation
of yi prior to making each of the other economic decisions. As the equilibrium
is subgame perfect, the allocations to Y are credible.
Subgame perfection requires that we solve the game using backwards induction. Suppose that A is sufficiently large such that conflict occurs in the
second stage of the game. We know from the analysis in Section 4 that the
Nash equilibrium to the second stage game involves zero subsistence consumption when A ≥ Â and that Y N E = 0. However, here we allow y1 , y2 , . . . , yN to
take arbitrary values when we consider the choices in stage two of the game,
although we continue to assume that c1 = c2 = · · · = cN = 0. (We later check
to see if these assumptions are valid.) The choices of xi and ki depend upon
the values of y1 , y2 , . . . , yN from the first stage decisions. As the intermediate
steps to finding the subgame perfect Nash equilibrium are quite tedious, they
are relegated to the proof of the proposition:
(N +1)ω
Proposition 5.1. For all A ≥ ASC ≡ N2(N
ω+θ) , the social contract subgame
Nash perfect equilibrium (SCSPNE) satisfies
(N ω − θ)(N − 2) SP
ω − θ SP
Nω + θ
,y
=
,k
=
,
(N + 1)N
N +1
N (N + 1)/2
(14)
A(N ω + θ)
2
N −2
SP
pSP
,
and
u
=
.
= ,
=
ij
N
N
N (N + 1)/2
cSP = 0, xSP =
pSP
ii
For A < ASC , the SCSPNE is equal to the LSCNE or the RSNE.
Proof. See the Appendix.
Thus unlike the Nash equilibrium in which zero policing is chosen, in the
subgame perfect Nash equilibrium, where the allocation to policing occurs prior
to the allocation of the remainder of the endowment, there is a positive amount
of provision to the public good. This occurs because there is a strategic effect of
reduced conflict by other players when the amount of policing increases. This
can be seen by recalling that θ and Y are prefect substitutes, and (10) implies
that investment in guns in the second stage is decreasing in θ and hence also in
Y . Thus, policing and guns are strategic complements. It is this effect that is
absent in the Nash equilibrium.
When N = 2, conflict is completely eliminated. However, the SCSPNE does
not fully eliminate conflict. In the limit as N → ∞, k SP = y SP = uSP = 0,
16
and xSP = ω. This occurs because as N grows, the effect any player can have
upon influencing the behavior of the balance of the population diminishes. Thus
for N sufficiently large, there is little strategic effect from investing in policing.
(ω−θ)
From (14), the size of the aggregate provision to policing is Y SP = NN
+1 .
SP
This is increasing in N , but is bounded from above by Y
≤ ω − θ. In
contrast, total guns expenditure is increasing roughly linearly in N . The result
is that the proportion of one’s own corn production that one appropriates is
pSP
ii = 2/N , which is diminishing towards zero as N increases. In contrast, the
share of the other N − 1 competitors corn production that i gets to expropriate
is pSP
ij = (N − 2)/N , which tends towards one as N grows large. Thus players
have ever increasing success in stealing from the dwindling production of corn
of others.
Hence, in the Slaughter oil field in west Texas, with over a hundred separate
operators, a portion of the rents were dissipated by the use of boundary injection
wells. Clay and Wright (2005) argue that gold mining claims were much less
secure than had been claimed by Umbeck (1977), noting that with the influx of
around 150,000 people into California during the gold rush, “every two or three
claims supported at least one lawyer” (at p. 170). The Greek cities of Athens
and Sparta, with roughly a 30,000 citizens each (Hansen, 1991, at pp. 90-94),
though able to cooperate in the war against Persia in 480bc, were never able
to completely eliminate conflict between themselves. Thus in large societies,
conflict dominates the SCSPNE.
The condition under which cSP = 0 in a social contract is
AN
∂ui
< 0.
=1−
∂ci
2
Given that pSP
ii = N/2, this inequality holds only for A < N/2. Whether or not
this condition ever binds depends upon what the minimum level of A is such
that a player does better by contributing zero to the public good and reverting
to the Nash behavior. For A < Ā, that condition requires that uSP ≥ ω. Thus
from (14),
N (N + 1)ω
cSP = 0 if, and only if, A > ASC ≡
.
(15)
2(N ω + θ)
Thus ASC > N/2 since ω > θ by Assumption 1. Therefore cSP = 0 for all A
such that a constitutional subgame perfect Nash equilibrium exists.
5.1
Reneging
In order for the social contract to be effective, it must be that players cannot
renege on their investment in the public good protection of property rights. In
the California mining example, the investment in property rights protection was
an agreement by miners to jointly protect one another’s claim. Clay and Wright
(2005) give an example where a group of miners who were approached by a larger
group of claim jumpers decided to renege upon their earlier agreement and to
let the claim jumpers have a share of their claims. Reneging forces players back
17
into the Nash equilibrium. If a player can use his investment in a gun to protect
everyone’s mines or he can use it to jump another miner’s claim, then in the
Nash equilibrium, he will always allocate the use of the gun to conflict rather
than to the public good of property rights protection.
There are several ways in which this outcome can be avoided. In the Texas oil
field example, the commitment of using the boundary injection wells is enforced
by the cost of removing that well from its present purpose of water injection
and using it to extract oil. Thus, commitment is solved by the putty-clay
nature of the investment in the public good. Even when investment is of a
putty-putty nature, commitment can be achieved in some instances. In the
Valencia irrigation example, monitoring other player’s actions was rational in
the subgame because each player had an incentive to ensure that he got his
turn at using the water. In revolutionary settings, a similar commitment is
enforced by the knowledge that the leaders will be punished if the revolution
is unsuccessful. Furthermore, in the Valencia example part of the investment
in the public good went to paying an veedors, who acted as the law enforcer,
both ensuring that farmers helped with canal repairs and gathering information
used in dispute resolutions. Thus, if the contribution to the public good can be
thought of as an acceptance of a social contract in which one subjects himself to a
state that can enforce property rights, then that state provides the commitment.
6
Autocratically Provided Property Rights
In this section we derive two benchmarks by which we can compare the social
contract equilibrium. In both, we suppose that a third party, whom we call a
king, offers to create property rights by providing a state-sponsored policing of
size Ψ to supplement the existing natural property rights of size θ between the
players. He does this in exchange for the right to impose a lump sum tax of τ on
the endowment of each player. The two cases we consider are (i ) a benevolent
king, the Aristotelian ideal, who devotes all of the tax revenues to supplement
policing and chooses the size of state-sponsored policing to maximize social
welfare; and (ii ) a despotic king, who keeps any surplus tax revenues above the
costs of supplying the policing for himself, and chooses the size of the police
force to maximize the surplus he is able to grab from his citizens. While these
would be equivalent in total surplus if the kings could tax output, they are not
equal when the kings are forced to tax the endowment.
A tax on the endowment was common in the contracts between a king and his
subjects in the feudal system in medieval Europe. This tax was probably used
because it does not subject the king to the moral hazard problem that occurs
when output is taxed. Bloch describes the method of taxation as follows:
The powerful individual who forced his weaker neighbor to submit
to him was apt to require the surrender of his property as well as
his person. The lesser men, therefore, in offering themselves to the
chief, also offered their lands. The lord, once the bond of personal
subordination had been sealed, restored to his new dependent the
18
property thus temporarily surrendered, but subject now to his superior right, expressed by the various obligations imposed upon it.
This great movement of land surrender went on at every social level
during the Frankish period and the first feudal age (1960, at p. 171).
The system of vassal homage arose response to the anarchy of the era following
the collapse of the Roman Empire. Again, quoting Bloch:
Neither the State nor the family any longer provided adequate protection. The village community was barely strong enough to maintain order within its own boundaries; the urban community scarcely
existed. Everywhere, the weak man felt the need to be sheltered by
someone more powerful. The powerful man, in his turn, could not
maintain his prestige or his fortune or even ensure his own safety except by securing for himself, by persuasion or coercion, the support
of subordinates bound to his service (1960, at p. 148).
In the Nash equilibrium, players take the tax rate τ and the king’s choice of
Ψ as given when choosing how to allocate their after-tax endowment between
corn production, guns, subsistence consumption and private provision of the
public good of property rights protection. Therefore, in the symmetric Nash
equilibrium, the first-order-necessary conditions for the choices of y, c, and x,
respectively, satisfy
θ+Ψ+Y +x
∂ui
= −A
≤ 0,
i = 1, . . . , N,
(16)
∂yi
θ+Ψ+Y +X
θ+Ψ+Y +x
∂ui
=1−A
≤ 0,
i = 1, . . . , N,
(17)
∂ci
θ+Ψ+Y +X
∂ui
A [(ω − x − c − y − τ )(N − 1) − (θ + Ψ + Y + x)]
≤ 0, i = 1, . . . , N.
=
∂xi
θ+Ψ+Y +X
(18)
As in the case where no king exists to create property rights, the first-order
condition (16) for yi is negative for all feasible values of y, x, and c. Hence:
Proposition 6.1. In the symmetric Nash equilibrium under a king, each player
sets y ∗ = 0.
Therefore, when a benevolent king exists, only the king provides property
rights protection. But as Proposition 4.1 implied there was no private provision
of property rights protection in the anarchy equilibrium, there is no crowding
out of private provision.
We saw above that if any of the endowment were consumed directly as subsistence consumption that the utility is equal to ω. This result occurs with a king
as well, which implies that a king cannot improve welfare if his citizens have an
incentive to consume part of their endowment as subsistence consumption. Thus
the interesting outcomes are occur in the Hobbesian state of nature. Absent a
king, this occurs when A ≥ Ā. From (18), (4) and (5), for any feasible values of
19
Ψ and τ , when A ≥ Ā the level of investment in guns and corn production and
the corresponding symmetric equilibrium utility satisfy the following:
x∗ (Ψ, τ ) =
6.1
(N − 1)(ω − τ ) − θ − Ψ ∗
ω+θ+Ψ−τ
, k (Ψ, τ ) =
N
N
A(ω + θ + Ψ − τ )
∗
and u (τ, Ψ) =
.
N
(19)
A Benevolent King
A benevolent king taxes in proportion to the opportunity cost of providing
property rights. Since a state-sponsored police force of size Ψ costs a total of Ψ
units of endowment, to balance the budget the benevolent king chooses the tax
rate such that
Ψ = N τ.
(20)
For a given tax rate τ , the budget balancing size of a police force provided by
the king is given by (20). Therefore, from (19), the equilibrium values of x∗B (τ ),
∗
kB
(τ ) and u∗B (τ ) depend upon the tax rate τ :
x∗B (τ ) =
ω + θ + (N − 1)τ
ω(N − 1) − θ − τ (2N − 1) ∗
, kB (τ ) =
,
N
N
A ω + θ + (N − 1)τ
.
and u∗B (τ ) =
N
(21)
∗
It is clear that u∗B (τ ) and kB
(τ ) are increasing in τ and that x∗B (τ ) is decreasing
in τ . Therefore, a benevolent king maximizes welfare by setting τ just large
enough to drive x∗B (τ ) to zero. Thus from (21) and from (20),
(N − 1)ω − θ ∗
(N − 1)N ω − N θ
, ΨB =
,
2N − 1
2N − 1
Nω + θ
A(N ω + θ)
∗
∗
x∗B = c∗B = yB
= 0, kB
=
, and u∗B =
.
2N − 1
2N − 1
τB∗ =
(22)
∗
Each citizen devotes kB
of his endowment to corn production and the remainder
to paying taxes for the provision of the policing. This means that property
rights are perfectly enforced, since x∗B = 0 implies that p∗ii = 1 and p∗ij = 0.
i
Furthermore, evaluated at x∗B = 0, we see from (17) that ∂u
∂ci = 1 − A, which is
negative for all A > Â. Thus at the optimal tax rate, τB∗ , no subject wishes to
switch to subsistence consumption to avoid the tax for any A > Â.
The net gain to society under a benevolent king is the difference in aggregate
∗
utility, ∆SB
≡ N (u∗B − uN E ), under the benevolent king relative to the Nash
equilibrium:

N

for  ≤ A < Ā
 2N −1 ω[N (A − 2) + 1] + θA
∗
(23)
∆SB
=

 A(N −1) ω(N − 1) − θ,
for
A
≥
Ā.
2N −1
20
The next proposition gives the minimum level of the Demsetz parameter under
which a king can create property rights:
Proposition 6.2. Under Assumption 1, a benevolent king improves welfare for
all values of A > AK , where
AK ≡
ω(2N − 1)
⊂ (Â, Ā),
Nω + θ
(24)
Proof. See the Appendix.
Thus the minimum Demsetz parameter under which a benevolent king can
create property rights is AK > 1. Since AK > Â, a benevolent king is able
to create and perfectly enforce property rights only if there is conflict, since
xN E > 0 for all A > Â. However, the presence of conflict is not sufficient to
ensure that a benevolent king is able to create property rights, since a king
cannot exist when  < A < AK , even though conflict occurs in the Nash
equilibrium. Second, the minimum value of the Demsetz parameter, AK , such
that a king can arise is increasing at a decreasing rate in N . Differentiating AK
ω(ω+2θ)
K
with respect to N yields dA
dN = (N ω+θ)2 . This is positive, but decreasing in N .
The limit of AK as N → ∞is AK = 2.
6.2
A Despotic King
A despot uses his power of taxation to expropriate wealth for his own consumption.20 The potential for despotism by the king is especially dangerous since
even a benevolent king chooses Ψ∗B at just high enough level so that citizens give
up their guns. Here, we show that the problem is more serious than a simple
distributional problem—the surplus under a despotic king, under some conditions, is less than the surplus generated by a benevolent king. The problem is
caused because the despotic king takes too much of the endowment relative to
a benevolent king.
The surplus the despotic king earns is the difference between his tax revenues
and his costs of providing state-sponsored policing:
RD = N τ − Ψ.
(25)
∗
He sets the tax rate, τD
such that his citizens are indifferent between the Nash
equilibrium outcome without a king and the Nash equilibrium outcome in which
20 This corresponds the the ”tinpot” form of dictatorship in Winetrobe (1990). Mueller
(2003, at p. 409) gives the examples of the Roman emperor Nero, who composed and sang in
public, bribed his way to winning in Olympic games, and was alleged to have played his lyre
while Rome burned. French King Louis XIV and English King Henry VIII are other examples
of kings whose consumption (houses and wives, respectively) was deemed extravagant. In
modern times, Imelda Marcos, wife of a Philippines dictator, became famous for her 3,000
pairs of shoes. In contrast, totalitarian dictators wish to control the lives of their subjects.
Hitler, for example, lived quite simply.
21
the king taxes them at rate τ and provides public good equal to Ψ.21 There are
two cases to consider, depending upon whether A ≥ Ā or A < Ā.
When A ≥ Ā, the equilibrium payoffs absent a king are the payoffs in the
A
(ω + θ). When each citizen takes Ψ
HCNE. From, (10), that utility is uN E = N
and τ as given when choosing xi and ki , the utility with a king is given by (19).
Thus the constraint faced by a despot is
A
A(ω + θ + Ψ − τ )
≥ (ω + θ) ≡ uN E when A ≥ Ā.
(26)
N
N
Taking the tax rate as given and solving for the size of the police force that
just makes each player indifferent between having a king who provides property
rights protection of Ψ and doing without a king and experiencing the HCNE
utility yields Ψ∗D (τ ) = τ . Substituting this into the surplus function (25) yields
R∗ (τ ) = (N −1)τ . Thus the surplus the despotic king earns is strictly increasing
∗
∗
∗
in τ . Feasibility requires that x∗D (τD
) ≥ 0 and that kD
(τD
) ≥ 0. From (19),
∗
this means that the τD that maximizes the despot’s surplus, given the behavior
∗
∗
) = 0. Therefore, under a
such that x∗ (τD
of his citizens, is the value of τD
despotic king, when A > Ā, the equilibrium is given by
u∗D (τ, Ψ) =
(N − 1)ω − θ
ω+θ ∗
, τD = Ψ∗D =
,
N
N
(N − 1)[(N − 1)ω − θ]
∗
, when A ≥ Ā. (27)
and RD
=
N
∗
∗
The values in (27) should be familiar. The yD
, c∗D , and kD
are identical to the
HCNE allocations to subsistence consumption, private provision of policing,
and corn production, respectively, given in (10). Furthermore, the tax charged
∗
is identical to the HCNE allocation to guns, and is
by the despotic king, τD
independent of the Demsetz parameter value A. Like a benevolent king, a
despotic king eliminates conflict in his regime by setting the level the policing
sufficiently high so as to prevent any investment in guns by his citizens. By
∗
∗
Assumption 1, he is able to improve welfare since ∆SD
= RD
> 0 for all A ≥ Ā
and for all N ≥ 2. But he is unable to induce his citizens to increase their
investment in corn relative to the HCNE, as he simply replaces the conflict
between individual citizens with exploitation by the king. As a result, while the
despotic king eliminates conflict, he is not able to induce efficiency in production.
Next, consider the case where  ≤ A < Ā. The utility each citizen earns in
the LSCNE absent a king is uN E = ω. Solving for the size of state-sponsored
policing, Ψ∗D (τ ), that equates utility yields
∗
∗
yD
= c∗D = x∗D = 0, kD
=
Nω
− ω − θ + τ.
A
Therefore, the despotic king’s surplus is
Nω
∗
RD
(τ ) = N τ − Ψ(τ ) = (N − 1)τ −
+ ω + θ.
A
Ψ∗D (τ ) =
21 Bloch (1960, at p. 146) notes that the vassal was often referred in medieval times as the
‘man of mouth and hands,’ to his lord. The reference to ‘mouth’ is taken to imply that the
lord is responsible for providing the vassal with the means to provide for himself.
22
∗
∗
Again, RD
is strictly increasing in τ , which means that (19) implies that τD
∗
is chosen to set xD = 0. Since we saw above that this condition is sufficient
to yield c∗D = 0 for all A > Â, we conclude that this is a Nash equilibrium.
Therefore, the despotic king chooses
∗
τD
=
(A − 1)(ω) ∗
(N − 1)ω
, ΨD =
− θ, when AK ≤ A < Ā.
A
A
(28)
Hence,
∗
∗
yD
= c∗D = x∗D = 0, kD
=
ω
ω[N (A − 2) + 1] + θA
∗
, and RD
,
=
A
A
when AK ≤ A < Ā. (29)
∗
The first thing to observe about (29) is that the value of A such that RD
=
0 is A = AK . Thus a despotic king can arise at the same minimum level
of the Demsetz parameter that a benevolent king can arise. Second, the tax
rate and the investment in corn production under the despotic king does not
equal the corresponding LSCNE levels. This is because there is no subsistence
consumption in the Nash equilibrium with a despotic king when AK ≤ A < Ā.
Both a benevolent and despotic king drive conflict to zero. Describing the
order which William the Conquerer brought to England during his reign of 10661087, it was remarked that “It was such than any man, who was himself aught,
might travel over the kingdom with bosom full of gold unmolested; and no man
durst kill another however great the injury he might have received from him
[sic]” (Barrington, 1900, at p. 57).
7
The Social Contract vs. Autocracy
Now we can compare the equilibrium under social contract with the equilibrium
in which property rights are provided by a king.
We begin by showing that the minimum value of the Demsetz parameter
under which a social contract can arise is at least as large as that under a king.
Proposition 7.1. The minimum Demsetz parameter under which a social contract can create property rights is at least as great as than the minimum Demsetz
parameter under which a king can create property rights:
ASC = AK
N =2
if
.
ASC > AK
N >2
Proof. See the Appendix.
It should not be surprising that as N gets large that a social contract has
difficulty creating perfect property rights, for we saw that as N grows, the freeriding problem in social contract overwhelms the strategic incentive to provide
property rights protection.
23
Next, let us compare the utility of citizens under a social contract with
that under a benevolent king. The surplus created under a social contract is
the difference between utility under the Nash equilibrium and under the social
contract:

ωN [2A−(N +1)]+2Aθ

if ASC ≤ A < Ā

N +1
∗
∆SSC =
(30)

(N −1)A(ω−θ)

if A ≥ Ā
N +1
Proposition 7.2. A benevolent king produces at least as great of surplus as a
social contract for all N ≥ 2 and for all A ≥ AK :
∗
∆SSC = ∆SB
N =2
if
.
∗
∆SSC < ∆SB
N >2
Proof. See the Appendix.
A social contract can do as well as a benevolent king only when N = 2.
For all N > 2, in the social contract, the incentive to free-ride in providing
protection of property rights exceeds the strategic incentive to over invest in
property rights protection.
Next, let us compare the surplus created under a social contract with that
under a despotic king. The surplus created under a despotic king is given by
(29) when AK ≤ A < Ā and by (27) when A ≥ Ā. Let us compare the surplus
created for the case where A ≥ Ā.
Proposition 7.3. When A ≥ Ā, the surplus created under a social contract
exceeds the surplus created under a despotic king if A ≥ A∗ ≥ Ā, where
A∗ =
[(N − 1)ω − θ](N + 1)
.
N (ω − θ)
(31)
A sufficient condition for this to occur is that ω > θ > ω/3.
Proof. See the Appendix.
What Proposition 7.3 implies is that a social contract is able to do better
than a despotic king so long as the Demsetz parameter is sufficiently large.
However, as A∗ is increasing in N , an implication of this proposition is that the
minimum level of the Demsetz parameter under which this is true is increasing
in N . Indeed, in the limit as N → ∞, A∗ → ∞, so that for very large societies,
even a despotic king does better than a social contract. This occurs for the same
reason that a benevolent king was always able to produce higher surplus than a
social contract: either type of king is able to overcome the free-riding problem
that tends to dominate the social contract equilibrium as N grows large.
Figure 2 illustrates the equilibrium payoffs in the symmetric equilibria for
values of N ≥ 2. This is drawn holding the values of N , ω, and θ fixed at
N = 3, ω = 1, and θ = ω/3. Below the value AK , no king nor social contract
can arise, so utility in the Nash equilibrium is U N E = ω. Above the value AK , a
24
DSB , DSCD , RD
DSB
RD
DS CD
`
AKA ACD
êêê
A
A*
A
Figure 2: Net Surplus under Kings and under social contract.
king, benevolent or despotic, can arise. The despotic king creates more surplus
than a benevolent king for values of AK < A ≤ Â.22 For values of A > Â, a
benevolent king creates more surplus than a despotic king. For values of A ≥ Ā,
the surplus created by a despotic king is constant in A. A social contract can
only arise for A ≥ ASC , which, because N > 2, is greater than AK . A social
contract always does worse than a benevolent king, but for values of A ≥ A∗ , a
social contract provides higher surplus than does a despotic king.
8
Conclusions
This paper has examined whether or not the Demsetz hypothesis that property
rights arise when the value of creating them is sufficiently large holds in a gametheoretic conflict equilibrium. Our main finding is that the answer is a qualified
yes. While in the Nash equilibrium the public goods problem of property rights
protection overwhelms society, we have shown that there are two ways that this
problem can be overcome, and that each of these ways requires that a version
of the Demsetz hypothesis hold.
Those two mechanisms are an external mechanism, whereby a king offers to
create and enforce property rights, and an internal mechanism, whereby players
agree in a social contract stage to give up some of their endowment for the
purpose of creating property rights. In each case, we find that property rights
can be developed only if the value of the good that is subject to conflict is sufficiently high. We also find that the minimum level of the Demsetz parameter
22 This
result is shown in the Appendix.
25
under which property rights can be defined also rises as the number of players
rises. We believe that this explains not only why the Greek city-states were unable to develop democracy at a wider level, but also why some modern conflicts,
such as the one in Darfur and the instability of Nigeria, can be explained by the
large numbers problem. In addition, while we find that while property rights
are perfectly defined when the number of players is only two, that under a social
contract, as the number of players rises, the security of property rights declines.
This is in contrast to what happens under the Aristotelian ideal of a benevolent
‘philosopher’ king. However, it is likely that the Aristotelian ideal king is as
distant today as he was two millennium ago. Finally, we find that property
rights defined by a social contract can only dominate a despotic king when the
Demsetz parameter is quite large. This is consistent with evidence presented in
Acemoglu et al. (2008) that a number of rich countries remain autocratic.
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A
A.1
Mathematical Appendix
Proof of Proposition 4.1
Proof. In the symmetric Nash equilibrium, (9) implies ∂pii /∂Y = −(N −1)∂pij /∂Y .
Therefore, (6) can be written as
∂ui
= −pii A < 0,
∂yi
i = 1, . . . , N.
Thus in the symmetric Nash equilibrium, each player sets yi = 0.
A.2
Proof of Proposition 4.2
Proof. Suppose not. Suppose that players devote their entire endowment to
production, then xN E = cN E = 0. However, xN E = cN E = 0 implies that (8’)
can be written as
∂ui
A
=
(N − 1)ω − θ ,
∂xi
θ
which is positive by Assumption 1. This contradicts xN E = 0.
29
A.3
Proof of Proposition 5.1
Taking y1 , y2 , . . . , yN as given, but holding c1 = c2 = · · · = cN = 0, we may
write the utility of the ith player as
ui =
h
i
X
A
(θ + Y + xi )(ω − xi − yi ) + xi
(ω − xj − yj ) ,
θ+Y +X
j6=i
i = 1, . . . , N. (32)
A.3.1
The Conflict Stage
The first-order-necessary conditions for the choice of x1 , x2 , . . . , xN satisfy
h
X
∂ui
A
(θ + Y + X−i )(ω − xj − yj )
=
X
(ω
−
x
−
y
)
+
−i
i
i
∂xi
(θ + Y + X)2
j6=i
i
− (θ + Y + xi )(θ + Y + X) = 0,
i = 1, . . . , N. (33)
Solving the joint system of (33) for xi (Y, yi ) yields23
xi (Y, yi ) =
(N − 1)(N ω − Y )2 − N 2 (θ + Y )(ω − yi )
,
N 2 (N ω − Y )
i = 1, . . . , N. (34)
Therefore, substituting (34) into (1) and (2) yields
pii (Y, yi ) =
(N − 1)(N ω − Y )2 + N 2 (Y + θ)[(N − 1)ω − Y + yi ]
,
N 2 (N ω − Y )2
i = 1, . . . , N, (35)
and
pij (Y, yi ) =
(N − 1)(N ω − Y )2 − N 2 (Y + θ)(ω − yi )
,
N (N − 1)(N ω − Y )2
i = 1, . . . , N. (36)
The amount invested in corn production, ki (Y, yi ), is found from the resource
constraint (5):
ki (Y, yi ) = ω − yi − xi (yi , Y ) =
N 2 (N ω + θ)(ω − yi ) − (N − 1)(N ω − Y )2
,
N 2 (N ω − Y )
i = 1, . . . , N. (37)
Summing of over the j 6= i of the kj yields:
N
X
j6=i
kj (Y, yi ) =
[(N − 1)ω − Y + yi ](N ω + θ) − (N − 1)2 (N ω − Y )2
,
N 2 (N ω − Y )
i = 1, . . . , N. (38)
23 Our method of solving (33) was to solve it for the N = 2, N = 3, and N = 4 cases, and
then use induction to find the form given in (34).
30
Substituting (37),(35)-(38) into the utility function (32) yields, after some simplification, the value function in terms of yi and Y :
A N 2 ω(ω + θ) + N (N − 2)ωY + Y 2 − N 2 yi (Y + θ)
ui (Y, yi ) = 2
,
N
Nω − Y
i = 1, . . . , N. (39)
A.3.2
The Social Contract Stage
Given the utility functions (39), each player in the public goods provision stage
chooses yi , taking the y−i as given. Hence, the first order condition in the choice
of yi is
n
A
∂ui
= 2
(N 2 − 1)Y 2 + N Y [N θ − (N − 2)ω]
2
∂yi
N (N ω − Y )
o
− N 2 [yi (N ω + θ) − (N − 1)(ω − θ)ω] = 0,
i = 1, . . . , N. (40)
Imposing symmetry on (40), so that y1 = y2 = · · · = yN ≡ y, yields
∂ui
A(N − 1)[ω − θ − (N + 1)y]
= 0.
=
∂yi
N 2 (ω − y)
Solving this for y yields the subgame perfect level of private provision to policing,
and substituting these results back into (34)-(39) yields the results in (14).
A.4
Proof of Proposition 6.2
Proof. Let us first show that this is true when A ≥ Ā. Relative to the HCNE
in which there is no king, the gain to aggregate welfare is given by the lower
expression in (23). By Assumption 1, this is positive for all N ≥ 2.
Second, when  ≤ A < Ā, the gain in welfare relative to the LSCNE utility
of uN E = ω when there is no king is given by the upper lower expression in
−1)−θ
∗
(23).Evaluated at A = Â, this expression equals ∆SB
= − ω(N
2N −1 , which is
Nω
negative by Assumption 1. Evaluated at A = Ā = ω+θ , this expression equals
−1)[ω(N −1)−θ]
∗
∗
∆SB
= ω(N(2N
, which is positive by Assumption 1. Given that ∆SB
−1)(ω+θ)
∗
∗
is increasing in A, and that AK solves ∆SB
= 0, ∆SB
> 0 for all A > AK .
Next, we show that  < AK < Ā. By Assumption 1, (N − 1)ω > θ. Thus
(2N − 1)ω > N ω + θ, so that AK > 1 ≡ Â To show that AK < Ā, note that by
Assumption 1,
(N − 1)ω > θ
(N − 1)2 ω > (N − 1)θ
(N 2 − 2N + 1)ω > (N − 1)θ
N 2 ω + N θ > (ω + θ)(2N − 1)
Ā ≡
Nω
ω(2N − 1)
>
≡ AK .
ω+θ
Nω + θ
31
This completes the proof.
A.5
Proof of Proposition 7.1
Proof. We saw above that a king (benevolent or despotic) is able to create
−1)
property rights only if A ≥ AK = ω(2N
N ω+θ . For all N ≥ 2,
0 ≤ (N − 2)(N − 1)
0 ≤ N 2 − 3N + 2
2(2N − 1) ≤ N (N + 1)
AK =
(2N − 1)ω
N (N + 1)ω
≤
= ASC .
Nω + θ
2(N ω + θ)
When N = 2, the minimum value of the Demsetz parameter under which a
social contract can exist is the same as when a king can exist. However, for
N > 2, the inequalities hold strictly.
A.6
Proof of Proposition 7.2
Proof. Since a social contract cannot arise for values of AK ≤ ASC ≤ A, we
can restrict our attention to the case where A > ASC . From (22), the utility
ω+θ)
each citizen earns under a benevolent king is u∗B = A(N
2N −1 . From (14), the
ω+θ)
utility each citizen earns under a social contract is uSP = NA(N
(N +1)/2 . Since the
reference utility level is the same no matter what the value of A, the difference
in equilibrium utilities is proportional to the difference in surplus created. This
difference is
u∗B − uSP =
A(N − 2)(N − 1)(N ω − θ)
≥ 0 for all A ≥ ASC .
2N 2 + N − 1
This difference is zero when N = 2, but is strictly positive for all N > 2.
A.7
Proof of Proposition 7.3
Proof. When A ≥ Ā, the difference in surplus created is
∗
RD
− ∆S ∗ SP = (N − 1)
h (N − 1)ω − θ A(ω − θ i
−
.
N
N +1
This expression is decreasing in A, and it is equal to zero when A = A∗ . Thus
so long as A∗ ≥ Ā, surplus is higher under a social contract for A ≥ A∗ than
under a despotic king.
That θ > ω/3 is a sufficient condition for A∗ > Ā can be seen by noting
2ω
. Thus A∗ > Ā for θ > ω/3. That
that when N = 2, A∗ = 3/2 and Ā = ω+θ
this is true for all N can be seen by noting that ∂A∗ /∂N =
32
(N 2 +1)ω+θ
N 2 (ω−θ)
and
∂ Ā/∂N =
ω
ω+θ .
As
ω 2 + (2N 2 + 1)ωθ + θ2 > 0
(N 2 + 1)ω 2 + (N 2 + 1)ωθ + θ2 > N 2 ω 2 − N 2 ωθ
[(N 2 + 1)ω + θ](ω + θ) > N 2 (ω − θ)ω
∂A∗ /∂N ≡
ω
[(N 2 + 1)ω + θ]
>
≡ ∂ Ā/∂N.
2
N (ω − θ)
ω+θ
Thus θ > ω/3 is sufficient to ensure that A∗ > Ā for all N .
A.8
Comparing a Despotic King with a Benevolent King
∗
, given by (27) when A ≥ Ā, and
The surplus created by a despotic king is RD
by (29) when AK ≤ A < Ā. The surplus created by a benevolent king are given
∗
∗
yields
from ∆SB
by (23). Subtracting RD

(N −1)[(N −1)ω−θ][N (A−2)+1]

if A ≥ Ā

N (2N −1)
∗
∗
∆SB − RD =
(41)

 [A(N ω+θ)−ω(2N −1)][N (A−2)+1] if A ≤ A < Ā
K
N (2N −1)
The first expression in brackets of the condition for A ≥ Ā is positive by Assumption 1. The first expression in brackets in the numerator of the case where
AK ≤ A < Ā is positive for all A > AK . The denominators are each positive
in sign. This leaves the term N (A − 2) + 1 to be signed. Let à ≡ 2NN−1 denote
∗
∗
is increasing in A,
− RD
the value of A such that this term is zero. Since ∆SB
∗
∗
∗
∗
for values of A ≥ Ã, ∆SB ≥ RD , but for values of A < Ã, ∆SB
< RD
. For all
θ > 0, it follows that à > AK . Thus for values of AK < A < à a despotic king
creates more surplus than a benevolent king, even though both are attempting
to maximize the surplus.
The reason for this result can be seen in the constraints the two kings face. A
benevolent king chooses {τ, Ψ} to maximize utility of the representative player
subject to the constraints that the budget balances, Ψ = N τ , and that the
level of conflict can at most be driven to zero: x(τ, Ψ) ≥ 0. A despotic king
chooses {τ, Ψ} to maximize his rents, R = N τ − Ψ, subject to the constraints
that x(τ, Ψ) ≥ 0 and that u(τ, Ψ) ≥ uN E = ω. Given that utility in (19) is
increasing in Ψ and decreasing in τ , the iso-welfare curves thus have a slope
of +1 in Ψ − τ space (see Figure 3), and are increasing for movements in the
northwesterly direction.24 The benevolent king faces the inequality constraint
x(τ, Ψ) ≥ 0, and the equality constraint Ψ = N τ . Thus the constraint set of
the benevolent king is denoted by the thick line along the budget constraint
24 As A increases, the constraint u(τ, Ψ) ≥ uN E = ω shifts downward. For A greater than
Ā, the level of utility in Nash equilibrium becomes uN E > ω, so the constraint becomes the
Ψ = τ , which does not shift. This constraint is shown as the gray colored long-dashed line
that is parallel to the u(τ, Ψ) ≥ ω constraint. The corresponding iso-rents curve for the despot
passes through the intersection of that constraint and the x(τ, Ψ) ≥ 0 constraint in Figure 3.
33
G
Nt
êêê
R*D» A<A
êêê
R*D »A¥A
u=u*B
x¥0
B
GB*
GD*
u¥uNE
D
u¥w
0
-R*D
t
tB* tD*
Figure 3: The Constraint Sets for Benevolent and Despotic Kings.
out to the x(τ, Ψ) ≥ 0 boundary, which is a curve with a positive intercept and
a slope of −(N − 1). As the budget constraint has a steeper slope than the
iso-welfare curve (N > 1), the benevolent king chooses {τB∗ , Ψ∗B } at the corner
solution labeled “B”. The despotic king, on the other hand, wishes to maximize
R, Thus his iso-rents curve is increasing in τ , decreasing in Ψ, and has a slope
of +N in Figure 3. The constraint that u ≥ uN E yields constraint curves with
slopes of +1. For A ≥ Ā, the constraint that binds is given by (26), and for
AK ≤ A < Ā, the constraint that binds is u(τ, Ψ) ≥ ω. In both cases the
∗
constraint x(τ, Ψ) ≥ 0 binds. Thus the despotic king chooses {τD
, Ψ∗D } at the
corner solution labeled “D”. Thus the benevolent king’s constraint set is tighter
than that of the despotic king.
34