_ _ _ _ _ _ ___ W PS od 61 J
POLICY RESEARCH WORKING PAPER 2692
Introduction to Property This paper inaugurates the
mathematical treatment of
Theory property theory, proving the
two fundamental theorems
The Fundamental Theorems for the property system that
correspond to the two
fundamental theorems for the
David Ellerman competitive price system.
The World Bank
Development Economics
Office of the Senior Vice President U
October 2001
I POLICY RESEARCH WORKING PAPER 2692
Summary findings
The market system consists of a price mechanism built In this paper Ellerman inaugurates the mathematical
on the foundation of a system of property and contract. treatment of property theory. In contrast with earlier
In many developing and transition economies the market work in "law and economics" and the "new institutional
system functions poorly. In many cases, if not most, the economics," this approach uses principles drawn from
malfunctioning is not simply in the price system (for jurisprudence and does not attempt to reduce "law" to
example, anti-competitive activities) but in the 'economics` in the sense of efficiency considerations
underlying property system (such as contracts being such as the minimization of transactions costs. The main
breached, and externalities in the sense of transfers not results are the two fundamental theorems of property
covered by contracts). theory that are analogous to the two fundamental
Economic theory tends to take the functioning of the theorems of price theory that, in essence, state that:
system of property and contract for granted and focuses e A competitive equilibrium is Pareto optimal.
on the operation of the price mechanism. Property * Given a Pareto optimal state, there exists a set of
theory focuses on that underlying system of property prices such that a competitive equilibrium at those prices
and contract. would realize that Pareto optimal state.
This paper-a product of the Office of the Senior Vice President, Development Economics-is part of a larger effort in the
Bank to understand the institutional basis for a private property market economy. Copies of the paper are available free
from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Beza Mekuria, room MC4-358,
telephone 202-458-2756, fax 202-522-1158, email address bmekuria@cKworldbank.org. Policy Research Working Papers
are also posted on the Web at http://econ.worldbank.org. The author may be contacted at dellerman@worldbank.org.
October 2001. (26 pages)
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about
development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The
papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this
paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the
countries they represent.
Produced by the Policy Research Dissemination Center
Introduction to Property Theory:
The Fundamental Theorems
David Ellerman
World Bank*
We have now run over the three fundamental laws of nature, that of the stability of possession, of
its transference by consent, and of the performance of promises. 'Tis on the strict observance of
those three laws, that the peace and security of human society entirely depend; nor is there any
possibility of establishing a good correspondence among men, where these are neglected.
David Hume 1739
Table of Contents
Introduction
Graph Theoretical Preliminaries
Directed Graphs
Assignments to Arcs and Nodes
The Net Inflow or Boundary Operator
Basic Mathematical Results
Chain Groups
The Augmentation of any Boundary is Zero
The Associated Market Graph
Any Node Assignment with Zero Augmentation is a Boundary
Introductory Property Theory
Preliminaries
Normative Concepts of Property Theory: Initial Treatment
The Contractual Mechanism of the Property System
The Fundamental Theorems of Property Theory: First Formulation
Intellectual Background
Introducing Property Rights Explicitly
Stock-Flow Considerations
The Invisible Judge: The Market Mechanism of Legal Appropriation
The Fundamental Theorems with Property Rights
The Responsibility Principle
The Fundamental Theorems and the Responsibility Principle
Conclusion
Appendix 1: Proofs of Theorems 1 and 2
Appendix 2: Analysis of Externalities and Breaches
References
*The findings, interpretations, and conclusions expressed in this paper are entirely those of the author and
should not be attributed in any manner to the World Bank, to its affiliated organizations, or to the members
of its Board of Directors or the countries they represent.
Introduction
The market system consists of the price mechanism built upon the foundation of the system of
private property and voluntary contracts. In the economic analysis of the price mechanism
("price theory"), the appropriate operation of the underlying property system is assumed. Here
that assumption is relaxed to focus the analysis on the operation of the system of property and
contracts ("property theory"). In many transition and developing countries, the market system as
a whole does not function well. In many-if not most-cases, the failure is not simply in the
price mechanism but in the underlying system of property and contract.
The main attributes of the (competitive) price system are demonstrated in two fundamental
theorems that relate competitive equilibria to Pareto optimal (or allocatively efficient)
allocations. The theorems show that (1) under certain assumptions, a competitive equilibrium is
Pareto optimal, and that (2) under certain assumptions, given a Pareto optimal state, there exists a
set of prices such that a competitive equilibrium at those prices will realize the Pareto optimal
state. ' My purpose here is to develop the mathematical framework of property theory in a simple
(one-period flow) model and to prove the two theorems for the property system that correspond
to the two fundamental theorems about the competitive price system. These results inaugurate
the field of mathematical jurisprudence in its own right-where, unlike the work in the field of
law and economics or in most of the new institutional economics, there is no attempted reduction
of the jurisprudential principles to efficiency considerations such as the minimization of
transactions costs.
It is sometimes noted that human freedom is an end in itself, not just a means to achieve
efficiency. A decentralized private property system has virtues that are much more robust than
the highly special conditions of a competitive equilibriUM2_such as the realization of people's
intentional decisions with social coordination and mutual adjustment carried out by voluntary
contracts and such as the principle of reaping the fruits of one's labor. These broader virtues are
attributes of the system of private property and voluntary contracts and have been emphasized by
economist-philosophers such as John Locke, David Hume, Adam Smith, Frank Knight, Michael
Polanyi, and Friedrich Hayek. They are part of the subject matter of property theory.
Graph Theoretical Preliminaries
Directed Graphs
Intuitively, a "directed graph" has a finite set of points or nodes with arrows, or directed arcs
between them.
' See Arrow 1951, Debreu 1951, Gale 1955, or Quirk and Saposnik 1968.
2 The work of Stiglitz [see 1994 for a summary] and colleagues in information economics shows that these
assumptions are even more special and less robust than previously thought.
A Directed Graph
In more technical terms, directed graph G =(Go,Gl,t,h) is given by a set Go of nodes (numbered i
= 1,...,I), a set GI of arcs (numbered j = 1,2,...,J), and head and tailfinctions h,t:G,1-Go which
indicate that arc ej is directed from its tail, the t(ej) node, to its head, the h(ej) node.
Arc ej
Each node represents a group or party interpreted always as a natural person or group of persons
acting together as a unit, and the arcs between nodes represent the "channels" for transfers
between the parties.
Assignments to Arcs and Nodes
Real n-dimensional vectors3 (each component representing a type of commodity including
money as a component) will be assigned to the arcs to represent transfers between parties. A
function T:G1-R+ is a arc assignment indicating that, given an arc (or "edge") ej, the vector
T(ej) is transferred from the t(ej) party (on the tail of the arrow) to the h(ej) party (on the head of
the arrow) with negative components representing transfers in the opposite direction. For
instance, if T(ej) = (Q,O,-K,O), then Q units of the I" commodity are transferred from the t(ej)
party to the h(ej) party with K units of the third commodity transferred in the opposite direction.
3 There is an alternative to using n-dimensional real vectors with positive and negative components. For over five
centuries, double-entry bookkeeping has been used to describe the stocks and flows of scalar notions of value within
and between parties. In earlier work two decades ago to develop mathematical machinery for property theory,
double-entry bookkeeping was mathematically formulated for the first time (!) and then generalized to n dimensions
to describe the stocks and flows of physical quantities that underlie conventional scalar accounting [see Ellerman
1982, 1986, or 1995]. The key idea was that the "T-accounts" of double-entry accounting were the ordered pairs in
the group of differences construction which extends the monoid of non-negative numbers or even non-negative n-
vectors, which have no additive inverses, to an additive group with additive inverses. The same ordered-pairs trick
is used multiplicatively to extend the multiplicative monoid of integers to the rationals which include multiplicative
inverses (division) and where the ordered pairs are written vertically as "fractions." Double-entry bookkeeping is
mathematically formulated and extended to n dimensions by using (horizontally written) ordered pairs or "T-
accounts" where the debit and credit entries are non-negative n-vectors. Property theory could be developed using
that machinery of n-dimensional double-entry accounting. However, the additive group of T-accounts of non-
negative n-vectors is isomorphic to the additive group of R' so property theory will be developed simply using RW
with vectors of both positive and negative components.
2
T(ej) =(Q, 0, -K, 0)
(t) ~~Arc ej (
In due course, various types of transfers will be considered (e.g., voluntary or not, intended or
actual) but for now the focus is on the underlying mathematical machinery.
A node assignment S: Go->R assigns real n-dimensional vectors to the nodes to represent net
flows within the parties represented by the nodes. Note well that since a node is not "directed,"
there is no standard interpretation of the signs in the components of the vectors assigned to
nodes. Components of one sign represent commodities consumed, used up, or added to stock,
and the components of the other sign represent commodities produced or subtracted from stocks
during the time period.
There is no need to segregate the parties, represented by nodes, into "producers" and
"consumers"; all parties are capable of both activities. If the value of S at node ni is S(ni) =
(Q,0,-K,0) , then it might mean that the ith party produced Q (or removed it from previous
stocks) and used up or consumed K (or added it to stocks) during the time period. If S
represented production activities as positive components with consumption as the negative
components, then -S would represent the consumption activities as positive with production as
the negative components.
The Net Inflow or Boundary Operator
Given an arc assignment of transfers T, each node will have certain property vectors transferred
into it and out from it. The net in-transfer at each node can then be computed, and it would be a
node assignment AT: Go0->R determined by the given arc assignment T. If (3, 2, -3, 0) was
transferred into a node and (2, -6,0, 1) was transferred out of the node, then the net in-transfer or
net inflow is:
(3,2,-3,0)- (2,-6,0, 1)=(1,8,-3,-1).
In-transfer Out-transfer
(3, 2,-3, 0) ( ,-6, ,1
Net Inflow
=(1, 8, -3, - 1)
Given the arc assignment T, the node assignment AT is defined at a node ni by adding all the
T(ej) where the arc ej has node ni at the head of the directed arc, and subtracting T(ej) where arc
ej has node ni at the tail of the arc. In technical terms, the node assignment AT is defined as:
aT(ni)= ET(ej)- ET(ej)
h(e3)=ni t(ej)=ni
3
for nodes ni = nj,...,n1. The node assignment AT is called the net inflow or, in mathematics,
boundary of the arc assignment T [e.g., Aigner 1979, 358; Giblin 1977, 27; or "divergence" with
the opposite sign convention in Rockafellar 1984, 1 1]. Thus a is the boundary operator that
carries arc assignments T into node assignments oAT. If a commodity is both transferred to and
away from a party during the time period, then it will cancel out in the calculation of the net
inflow at that node.
Basic Mathematical Results
Chain Groups
Let C(G1, RW) be the set of arc assignments and let C(Go, RW) be the set of node assignments.
Each will be taken as an additive group by taking (T + T')(e) = T(e) + T'(e), and similarly for
nodes. It is sometimes convenient to write the members of these groups as formal sums called
"chains" and then the groups are "chain groups." For instance if the arcs are ej for j = 1,... ,J then
the arc assignment T could be written as the formal sum:
T(ej)ei + T(e2)e2 + ... T(ej)ej
called a 1-chain [e.g., Giblin 1977, 26]. The formal sums in C(Go, RW) are called 0-chains. The
boundary or net inflow function defined by
aT(ni)= ET(ej)- ZT(ej)
h(ej)=ni t(ej)=ni
is a group homomorphism:
a: C(GA, R') -> C(Go, Re).
Given a node assignment S, its augmentation Aug(S) is the sum in R' of values assigned to all
the nodes ni for i = 1,...,I:
Aug(S)= ES(ni)
i=l
which gives a group homomorphism: Aug: C(Go, RT) - RW.
The Augmentation of any Boundary is Zero
In the boundary AT of any arc assigmnent T, each value T(e) on an arc e occurs exactly once
positively at the node on the head of the arc and exactly once negatively at the node on the tail of
the arc. Thus the augmentation of any boundary is zero (see Appendix 1 for the proof).
Theorem 1: For any T in C(G1, RW), Aug(8T) = 0.
4
The image Im(a) of the boundary homomorphism is a subgroup of C(Go, R ) consisting of all the
node assignments that are the boundary of some arc assignment. The kernel Ker(Aug) of the
augmentation is the subgroup of all the node assignments mapped to 0. Theorem 1 then says that
the image of the boundary is contained in the kernel of the augmentation, Im(a) c Ker(Aug).
The Associated Market Graph
The economic motivation of some of the concepts will become clearer if one associates with
each graph G, the associated market graph Gm with the "market" added as an extra (fictitious)
node m, with an arc added directed from the market node m to each node n of G, and with the
original arcs deleted (so each party relates only to the market).
0
S (n)
Associated Market Graph Gm
With each node assignment S on G, there is an associated arc assignment Ts on Gm where S(n) is
assigned to the arc from the market node m to the node n and clearly any arc assignment on Gm
could be associated with a unique node assignment on the original graph. Indeed, the chain
group of arc assignments on Gm is isomorphic to the chain group of node assigmnents on G.
Then the net inflow or boundary of Ts at m is minus the augmentation of the node assignment S,
aTs(m) = -Aug(S),
and the sum of the boundaries of Ts over all the original nodes of G is the augmentation of S:
aTs (n ) = Aug(S).
i=l
The augmentation of any boundary is zero (Theorem 1), and, in the case of Ts, it has the special
fonn: Aug(aTs) = Aug(S) - Aug(S) = 0. Taking the augmentation of a node assignment on the
original graph is equivalent to computing the "quantity balance" or net outflow at the market
node in the augmented graph. Taking the positive components in an arc assignment from the
market node m to any other node n as the quantity demanded by the party n from the market and
negative components as the quantity supplied by n to the market, then the overall excess supply
would be the net inflow aTs(m) = -Aug(S) at m.
5
Now the idea is to use the market-style reasoning of the associated market graph to approach
Theorem 2: given a node assignment S satisfying the quantity-balance condition Aug(S) = 0,
there exists an arc assignment T such that AT = S. This proposed Theorem 2 states that
Ker(Aug) : Im(8) so it is the converse to Theorem 1. The strategy is to show how the theorem
is trivial on the associated market graph and then to get the general proof by carrying over that
market-style reasoning to the original graph G.
Given an arbitrary node assignment S on G, it is trivially extended to a node assignment S* on
the node set of Gm by assigning S*(m) = -Aug(S) and otherwise S*(n) = S(n). The extra node m
serves as the "market-maker" by absorbing the excess supply from S [i.e., S*(m) = -Aug(S)], so
(by construction) Aug(S*) = 0 and Ts is an arc assignment on the associated market graph such
that aTs = S*.
Any Node Assignment with Zero Augmentation is a Boundary
By reproducing this market-maker reasoning within the original graph G, I arrive at the proof of
Theorem 2. The proof is the customary one in homological algebra [e.g., Giblin 1977, pp. 31-2]
which seems to be arrived at without any knowledge of the underlying market interpretation.
Instead of the extra market node m, I select arbitrarily any node nm in G to play the role of
"market-maker" to absorb whatever is the excess supply from the other nodes. Instead of
constructing a node assignment S* with zero augmentation, I start with an assignment S with
Aug(S) = 0. Instead of the arcs constructed to any node n from m, I need the notion of a "path"
to any other node n from the market-maker node nm. That requires some definitions.
Given nodes n and n*, a path from n to n* is a sequence of arcs ee'e". ..e* such that the head or
tail of e (respectively e*) is the node n (resp. n*), and where each two consecutive arcs in the
path share a conmmon node. An arc occurs positively (resp. negatively) along the path if the
direction of the arc (as an "arrow") is along (resp. against) the direction of the path from n to n*.
Clearly each path can be identified with a 1-chain ħeħe'ħe". ..ħe* in C(GI, Rn) where the
coefficient on each arc is determined by whether the directed arc lies positively or negatively
along the path. The negative of a path from n to n* is a path from n* to n.
A graph is connected is there is a path between any two nodes. G is henceforth assumed to be
connected.
Theorem 2: Let S be a node assignment on a connected graph G such that Aug(S) = 0. There
exists an arc assignment T on G such that AT = S, i.e., Ker(Aug) c Im(8) [see Appendix 1 for a
proof].
Theorems 1 and 2 together imply that Ker(Aug) = Im(a). In the language of homological
algebra, the sequence
C(G,,R)-a->C(GO,R) Aug 4R
is exact. Theorem 2 says that given a set of production-consumption intentions for each party
such that the quantities balance or "market clears" overall, there exists a set of bilateral
6
exchanges along the given channels connecting the parties that would support the production-
consumption decisions.
Theorem 2 can also be given a more indirect proof (without the market-maker motivation) using
linear algebra.4 The theorem can be generalized by adding upper and lower capacities to the arcs
and then accordingly strengthening the existence conditions. The generalized existence results
were proven by David Gale [1957] and A. J. Hoffman [1960], and were stated in a clear general
form by Rockafellar as the Gale-Hoffman "Feasible Distribution Theorem" [1984, p. 71].
Introductory Property Theory
Preliminaries
Both the property system and the price system (which operates on top of the property system) are
concerned with the general problem of socially coordinating the individual decisions of the
parties. Competitive price theory is a theory about how a decentralized system of competitive
markets can carry out this function in a certain optimal or efficient manner. The decentralized
system of property and contract is concerned not with optimization but with the more
rudimentary matter of voluntary and intentional actions; it uses the system of voluntary contracts
as the mutual-adjusting and socially coordinative mechanism. The theory has a "quantity" flavor
since prices (equilibrium or disequilibrium, competitive or not) are only relevant in determining
the quantities of money or other commodities that might be traded in a voluntary exchange.
Price theory models the parties as making decisions or, at least, determining intentions by
optimizing some objective function within feasibility constraints. But for the purposes of the
property theory theorems, we can take intentions and consents as primary data. The optimizing
(or satisficing) of objectives and the satisfaction of the feasibility constraints is all in the
background and need not be explicitly considered for property-theoretic purposes.
A property system is concerned with inter-party interactions and ignores the unintended effects
on one party not caused by other parties. It is a concern for the property system if a farmer's crop
is stolen, but not if the crop is lost in a storm. The difference is that another party was involved
in a theft, not in storm damage. Theft, fraud, and coercion are involuntary or unintentional
transfers of the possession of commodities between parties. The concept of factual "possession"
is constantly being redefined by the laws and cases in a legal system. In order to emphasize the
inter-party aspect, once a commodity comes into the possession of a party, it stays that way until
it comes into the possession of another party (even though the possessor may not be physically
using the commodity).
Given the intended actions of the various parties, one way to connect and coordinate them would
be to have a bureaucracy operating a central clearing house-the "socialist" solution. Indeed,
that centralized approach is used within most large organizations.5 A system of legal contracts
4See Slepian 1968, Theorem 3, p. 92, or Strang 1986, Box IQ, p. 74 for a clear treatment.
5 Some large organizations are experimenting with market-like contracting mechanisms between subunits where the
decentralized decisions are otherwise assured or assumed to be in line with organizational goals, e.g., Pinchot and
Pinchot 1993, Ackoff 1994, or Halal 1996. The property theoretic theorems could also be applied, mutatis
mutandis, to this situation within a large organization.
7
and property rights is a decentralized alternative (which still requires the operation of centralized
legal authorities) that emphasizes the decision-making autonomy of the parties. Property theory
shows how that system addresses the basic problem of realizing the compossible intentions of the
various parties.
Price theory tends to ignore the actual bilateral or multilateral structure of exchanges by treating
each party as dealing with the market (after the fashion of the associated market graphs
considered above). This treatment of market transactions as if they all went through a central
clearing house has inspired various socialist attempts to "mimic the market." Property theory
restores the focus on decentralized transactions directly between the parties. Even a purchase
from a vending machine (the standard example of an impersonal transaction with "the market")
is actually a bilateral exchange between the customer and the operator of the vending machine.
Hence the theory uses general connected graphs to model transactions (instead of the particular
type of graph seen in the associated market graphs). Instead ofjust considering the node vectors
of the parties (which was sufficient on an associated market graph), the transfers between parties
must also be included. The transfers in an arc assignment T are said to support the actions given
by a node assignment S if aT = S.
In the fashion of introductory treatments of consumption and production in price theory, a one-
period flow model is used. Assignments to arcs and nodes will represent flows of commodities
during the time period. "Commodities" refers to separable private goods that can be transferred
between parties, not goods with public or jointness attributes where decentralized mechanisms
need to be supplemented by other social choice mechanisms.6 When a vector is assigned to a
directed arc, the positive components represent transfers of those commodities along the arc (as a
"trade channel" between the parties represented by the nodes) while the negative components
represent transfers against the direction of the arc. Flows at the nodes result from either
production-consumption activities or from changes in stocks or inventories made by the parties
during the time period. Nodes are undirected so the interpretation of the positive and negative
components is determined by the context.
Normative Concepts of Property Theory: Initial Treatment
The basic normative and institution-free (e.g., price-free) notion in price theory is that of Pareto
optimality (or allocative efficiency). The basic institutional notion in price theory is that of a
competitive equilibrium in the price system. The "Fundamental Theorems of Welfare
Economics" show in what sense the price system is efficient by relating the notions of a
competitive equilibrium and Pareto optimality. The first theorem is that (1) under certain
assumptions, a competitive equilibrium is Pareto optimal; and the second theorem is that (2)
given a Pareto optimal social state, there exists, under certain conditions, a set of prices such that
a competitive equilibrium at those prices yields or supports that Pareto optimal state.
In order to construe Pareto optimality in our simple model, a utility or welfare function would be
associated with each party (node) and it would be a function of the node assignment. A Pareto
optimal node assignment would be a feasible one such that no party's welfare function could be
6 See, for example, Frank Knight's treatment of government by discussion [1956, 266-7].
8
increased by another feasible assignment without making another party worse off. Each party
would be at a maximum that is compossible with the maximum achieved by the other parties.
Underlying optimizing behavior is the broader notion of intentional action (which is taken here
as a primitive notion). If one ignores the optimization aspect in favor of simply considering a set
of intentional actions by the parties, then the compossibility or social consistency requirement
reduces down to the quantity balance condition, i.e., that the augmentation of the node
assignment representing intended actions is zero. Thus to find the property-theoretic analogue of
the notion of "Pareto optimality," intentional action replaces optimizing action and the quantity
balance condition replaces the compossibility condition on the optimizing of the different parties.
A node assignment is said to be a social intentional action if it represents intentional decisions of
the parties that satisfy the "social" or compossibility condition of a quantity balance (i.e., has
zero augmentation). "Intentional" is stronger than and entails "voluntariness." In realizing a
social intentional action, each party is not just passively consenting to how it is affected but is
carrying out an intentional or deliberate plan of action and is defacto responsible for the results.
Let SIA, as in "social intentional action," stand for any node assignment representing intentional
production-consumption and inventory-change decisions or plans of the parties that satisfy the
quantity balance condition, i.e., Aug(SIA) = 0. The sign conventions are chosen so that the
positive components of SIA at a node represent the commodities the party intends to consume or
add to stocks while the negative components represent the commodities the party intends to
produce or subtract from stocks. The negative -SIA describes the same socially consistent plans
but with the opposite sign convention (i.e., with production as positive and consumption as
negative).
The Scottish Enlightenment (e.g., Hume and Smith) and the later Austrian literature in
economics have emphasized the role of markets to coordinate free activity as an end in itself, not
simply as a means to achieve an efficient allocation of resources. The property-theoretic
theorems given below (initial formulation) could be seen as an attempt to formally render that
role of a private property market economy. More recently, Sen [1999] has emphasized that
while freedom may play an instrumental role in reaching a Pareto optimal state, it also has a
constitutive role. The notion of a social intentional action and the property theorems show in a
simple way how the property system realizes this constitutive role of freedom, the realization of
people's socially consistent intentional and responsible actions.
The Contractual Mechanism of the Property System
Property theory is concerned with how a decentralized system of voluntary exchange will realize
social intentional actions that are compossible in the sense of the quantity balance condition just
as price theory is concerned with how a decentralized system of competitive markets will realize
the compossible optimizations of a Pareto optimal (or "allocatively efficient") state.
The contractual mechanism is an institutional means to deal with the transfers of physical
quantities of commodities between the parties. Formally the idea is quite simple; enforce certain
types of transfers (arc assignments) whose boundary is intentional. Intuitively, these transfers
are the mutually voluntary contracts. The possibility of fraud renders the notion of a
9
"voluntary" transfer as more complicated than simply a transfer accompanied by "consent." A
party might consent to pay $10 for a certain commodity but unknowingly receive a worthless
substitute. The transaction has the appearance of voluntariness but the consent was to a rather
different transaction. Thus "mutual voluntariness" is always intended in the strong sense of a
knowing "meeting of the minds" about the actual commodities exchanged.
Legal transfers or contracts are defined as being mutually voluntary transfers. Let VC, as in
"voluntary contracts" (in spite of the redundancy since all contracts are voluntary), stand for any
arc assignment representing mutually voluntary transfers of commodities between the parties. In
view of the interpretation, it is reasonable to assume that the boundary of voluntary contracts is a
set of intentional decisions by the parties. If the party at a node voluntarily agrees to each of the
transfers in VC in which the party is involved, then surely the net inflow aVC at that node
represents the party's intentional decisions SIA. In any case, the assumption aVC = SIA is made.
From the strictly formal viewpoint, it only matters that the contractual mechanism enforces the
arc assignments which have a certain attribute ("voluntariness") whose boundary has the attribute
desired of node assignments ("intentionality").
The transfers actually carried out might diverge from the expectations based on mutually
voluntary contracts. A farmer might intentionally use inputs to produce some outputs but part of
the outputs might be taken by another party involuntarily or the farmer might have stolen or
"converted" some of the inputs from another party. Let FT, as in "factual transfers," be the arc
assignment representing the transfers in the possession and control of commodities that actually
takes place (involuntarily or voluntarily) between the parties during the period. This means that
AFT represents the actual or realized net inflows at the nodes. How will the contractual
mechanism lead to the realized node assignment aFT being a social intentional action?
The "equilibrium" in the contractual mechanism, FT = VC, is when all factual transfers of
commodities are mutually voluntarily and all mutually voluntary agreements are fulfilled by the
corresponding factual transfers. It would be misleading to take VC as "given" data to be
compared to the "given" actual transfers FT. The contractual mechanism is a negative feedback
system that operates to eliminate any difference between VC and FT. There are two ways this
matching is brought about by the contractual mechanism.
Given a factual transfer, the contractual mechanism works to insure that it is covered by or can
be construed as a voluntary contract (i.e., works to eliminate property externalities) by allowing
legal recourse for damages to any aggrieved party (e.g., an alleged victim of a theft or
conversion). The factual transfers not covered by voluntary contracts are called externalities.
And given an intended voluntary transfer representing a meeting of the minds, i.e., a contract, the
contractual mechanism will work to insure that the contract is fulfilled by the indicated transfer
in the possession and control of the commodities between the parties. This is typically done by
allowing legal recourse for damages to one party if that party has fulfilled its part of the contract
but the other party has allegedly breached its part. The legal contracts unfulfilled by factual
transfers are called breaches [see Appendix 2 for more analysis of externalities and breaches].
10
This negative feedback contractual mechanism aims at the "equilibrium" FT = VC which will be
called a contractual matching since it means the actual or realized transfers match the intended
mutually voluntary transfers of commodities. It should be noted that the price theory theorems
already assume that this contractual mechanism has done its job to achieve a matching. That is,
the price-theoretic fundamental theorems assume that there are no property externalities (no
extra-contractual transfers) and that the contracts determined by optimizing decisions are
fulfilled. Our concerns are precisely those prior questions of insuring that actual transfers in
possession are covered by contracts (no externalities) and that contracts are fulfilled by actual
transfers.
For the moment, I have avoided any mention of "property rights" so they can be introduced later.
The initial formulation of the two fundamental property theorems deal with the case where the
actual transfers match the contractual transfers, and the contractual mechanism can be described
in terms of insuring that actual transfers of commodities are covered by contracts and that
contracts are fulfilled by actual transfers. For the next two theorems, it is not necessary to
introduce "property rights" nor to interpret the contracts as transferring property rights. The
theorems will be later restated in those terms.
The Fundamental Theorems of Property Theory: First Formulation
What is the property theoretic analogue to the price theoretic theorem that a competitive
equilibrium is Pareto optimal? It is the rather straightforward result that if there is a contractual
matching, then a social intentional action is achieved.
Theorem 3: First Fundamental Theorem (Initial Formulation): If there is a contractual matching,
then the actions realized by the parties constitute a social intentional action.
Proof. With VC = FT, the realized actions AFT are intentional since aFT = aVC (= SIA) and
satisfy the quantity balance condition by Theorem 1. QED
Thus if the contractual mechanism does its job to insure that all actual transfers are mutually
voluntary and that all intended mutually voluntary contracts are fulfilled by the indicated
transfers, then the decisions realized by the parties are intentional and socially consistent. By the
contractual mechanism, the parties are led, as if by an invisible hand, to realize their intentions in
a socially compossible way.
The "converse" theorem will investigate when a given social intentional action of the parties can
be supported and realized by the contractual system of voluntary transfers.
Theorem 4: Second Fundamental Theorem (Initial Formulation): Given a set of intended actions
SIA which satisfy the quantity balance condition Aug(SIA) = 0, there exists a supporting set of
transfers T between the parties such that if T is accepted by the parties as voluntary contracts
VC, then the contractual matching would realize the social intentional action SIA.
Proof: The existence of T follows directly from Theorem 2 and the rest is immediate from the
definitions. By the existence of T accepted as VC, we have OVC = SIA, and the matching VC
FT implies AFT = aVC = SIA. QED
11
Although formulated at a high level of abstraction, Theorems 3 and 4 show the simple
effectiveness of the decentralized system of private property and contracts to realize the
intentions of the various parties so long as those intentions are socially consistent. The
contractual mechanism works if prices are in equilibrium or not, if markets are competitive or
not, and, indeed, if the exchanges involve money or only barter.
Intellectual Background
The ideas of the contractual mechanism in the property system and the price mechanism in the
competitive market system as constituting spontaneous orders or invisible-hand mechanisms
were developed during the Scottish Enlightenment principally by David Hume and Adam Smith.
Indeed it would seem appropriate to attribute the intellectual description of the contractual
mechanism to Hume just as the description of the competitive market system is attributed to
Smith. In Hume's Treatise of Human Nature (first published in 1739 well before Smith's Moral
Sentiments in 1759 or Wealth of Nations in 1776), he focuses on the operation of the system of
property and contract and develops the principles involved in the legal machinery enforcing a
contractual matching.
We have now run over the three fundamental laws of nature, that of the stability
of possession, of its transference by consent, and of the performance of promises.
'Tis on the strict observance of those three laws, that the peace and security of
human society entirely depend; nor is there any possibility of establishing a good
correspondence among men, where these are neglected. [Hume 1978 (1739),
Book III, Part II, Section VI, p. 526]
The "transference by consent" means that factual transfers are covered by voluntary contracts (no
externalities), and "performance of promises" means that voluntary contracts are fulfilled by the
corresponding factual transfers (no breaches) so, together (by Theorem 7 in Appendix 2), Hume
describes a contractual matching FT = VC.
In Michael Polanyi's treatment of the system of property and contract as a spontaneous order, he
cites similar rules that Leon Duguit extracted from French jurisprudence.
In the Code Civil of France (leaving out of account the law of the family) Duguit
finds only three fundamental rules and no more-freedom of contract, the
inviolability of property, and the duty to compensate another for damages due to
one's own fault. Thus it transpires that the main function of the existing
spontaneous order of jurisdiction is to govern the spontaneous order of economic
life.... No marketing system can function without a legal framework which
guarantees adequate proprietary powers and enforces contracts. [Polanyi 1951,
185]
Frank Knight emphasizes the point that the exchange system works with separable private goods
so the participants do not have to come to an agreement about a common end (unlike members of
a unitary organization who would require government by discussion or some other social
decision-making procedures), and he reduces the rules of justice to the essentials.
12
The supreme and inestimable merit of the exchange mechanism is that it enables a
vast number of people to co-operate in the use of means to achieve ends as far as
their interests are mutual, without arguing or in any way agreeing about either the
ends or the methods of achieving them. It is the 'obvious and simple system of
natural liberty.'... The only agreement called for in market relations is acceptance
of the one essentially negative ethical principle, that the units are not to prey upon
one another through coercion or fraud. [Knight 1956, 267]
Friedrich Hayek cites both Hume's rules of justice as well as the modem contribution of Duguit
[Hayek 1976, 40]. Hayek used the term "catallaxy" to refer to "the special kind of spontaneous
order produced by the market through people acting within the rules of the law of property, tort
and contract." [Hayek 1976, 109]
The decisive step which made such peaceful collaboration possible in the absence
of concrete common purposes was the adoption of barter or exchange.... All that
was required to bring this about was that rules be recognized which determined
what belonged to each, and how such property could be transferred by consent.
[Hayek 1976, 109]
Hayek goes on to note [p. 185] that it "was these rules to which David Hume and Adam Smith
emphatically referred as 'rules of justice'..." In Smith's The Theory of Moral Sentiments, Smith
argued that society need not depend on "mutual love and affection" or on beneficence.
Society may subsist among different men, as among different merchants, from a
sense of utility, without any mutual love or affection; and though no man in it
should owe any obligation, or be bound in gratitude to any other, it may still be
upheld by a mercenary exchange of good offices according to an agreed
valuation. [Smith 1969 (1759), Part II, Section II, Chapter III, 124]
Hayek argues that even without common purposes or mutual bonds, it is the observance of the
rules of disinterested exchange that Smith sees as the "justice" which is
the main pillar that upholds the whole edifice. If it is removed, the great, the
immense fabric of human society, that fabric which, to raise and support, seems,
in this world, if I may say so, to have been the peculiar and darling care of nature,
must in a moment crumble into atoms. [Smith 1969 (1759), Part II, Section II,
Chapter III, 125]
Recently, Charles Lindblom [2001] has explored the essentials of the market system. This
system of social coordination and peacekeeping is based on the personal liberty to act and on
each party having a protected sphere of property.
13
These two rights, liberty and property, seem like perverse foundations for a
system of broad social coordination, for they guarantee that, except for specific
prohibitions, people can do as they wish. Yet coordination would seem to require
that people do not simply do as they wish but instead bend to the requirements of
cooperation and peacekeeping. All the more interesting, then, are the ways in
which the two rights support cooperation and peacekeeping. [Lindblom 2001, 53]
The third requirement is the quid pro quo that yields voluntary contracts between the parties.
The three customs and rules [(DE) that is, liberty, property, and quid pro quo
voluntary exchanges] create a widespread process of mutual adjustment in which
each participant explores innumerable possibilities of benefit for both self and
other, thus innumerable opportunities for cooperating and reducing conflict.
[Lindblom 2001, 54]
Introducing Property Rights Explicitly
Stock-Flow Considerations
In the factual transfers FT, it is the possession and control of physical quantities of commodities
that is transferred. "Property rights" are constructed so that they are what are transferred by the
voluntary contracts VC enforced by the legal system. The definition of "property rights" needs
to cover the whole "life cycle": how property rights are initiated, transferred, and terminated.
First I must go "inside the black box" at each node to analyze the stocks and flows within a time
period. The basic identity is:
Ending Stock = Beginning Stock + Inflows - Outflows, or
Net Increase in Stock = Net Inflows.
For the factual flows of physical quantities of commodities, the net inflows from transfers with
the other parties are given at each node by the boundary AFT of the factual transfers. But there
are also increases and decreases due to the production (creation) or consumption (destruction) of
commodities. The production or creation of a commodity will be called thefactual
appropriation of the commodity, and the consumption, using up, or otherwise destruction of the
commodity would simply be treated as the negative of factual appropriation. Hence let FA', as in
"factual appropriation," be the node assignment giving factual creation and destruction of
commodities by each party during the time period. The sign conventions are taken so that the
positive components of FA' represent commodities produced or otherwise created, while the
negative components represent the commodities consumed, used up, or otherwise destroyed
(always in net terms) during the time period.7
Finally let AFH, as in "change in factual holdings," be the node assignment representing the
changes in the stocks, inventories, or holdings in the factual possession and control of the parties
during the time period (positive components represent increases in holdings and negative
7 The sign convention to represent appropriation is in accord with the convention for production sets in economics
where produced outputs are positive and used-up inputs are negative, e.g., Quirk and Saposnik 1968, 27.
14
components represent reductions in holdings). Since FA' and AFT are two forms of net inflows
(from production/consumption within the party and from other parties), the stock-flow identity
for the possession and control of physical commodities is: AFH = FA' + aFT.
The flows in property rights are rendered in the same way with aVC being the net transfers in
property rights to each party from other parties. The creation of a property right is the "legal
appropriation" of the right, and the termination of a legal property right will be treated as the
negative of appropriation.8 Let LA', as in "legal appropriation," be the node assignment with the
positive (negative) components representing the initiation (termination) of property rights by the
parties during the time period. Finally let ALH, as in "change in legal holdings," be the node
assignment representing the changes in the existing property rights to commodities already held
by the parties during the time period. Then the stock-flow identity for legal property rights is:
ALH = LA'+ aVC.
There are two ways to treat the effects of the passage of time on stocks. One way is to treat the
"same" stock as carrying over into the next time period and only becoming one period "older."
The other way is to treat the carrying over of stocks as the using up of the stocks in the first
period and the creation in the next period of one-period-older stocks. It will simplify our
formalism and not sacrifice any interpretation if I choose the second method.
Thus the factual equation might be viewed as: 0 = [-AFH + FA'] + aFT, where the term in square
brackets is the broadened "production-consumption" term including the "production" of the
stocks carried over from the prior period and the "consumption" of the stocks carried over to the
next period. Now I broaden the interpretation of factual appropriation to include these stock
changes, i.e., FA = [-AFH + FA'], so that the factual equation can be simplified to: 0 = FA +
aFT. Since the boundary of the factual transfers is the net inflow, the equation could be
expressed as the:
Factual Boundary Conditions:
FA = -0FT
"Factual Appropriation" = "Net Factual Outflows."
The same reinterpretation will be applied to the legal flow identity so the legal appropriation LA
= [-ALH + LA'] includes the negative of the changes in legal holdings. Then the equation would
be the:
Legal Boundary Conditions:
LA = -aVC
"Legal Appropriation" = "Negative Boundary of Voluntary Contracts."
8 Indeed the termination of rights was an original meaning of "expropriation." "This word [expropriation] primarily
denotes a voluntary surrender of rights or claims; the act of divesting oneself of that which was previously claimed
as one's own, or renouncing it. In this sense, it is the opposite of 'appropriation'. A meaning has been attached to the
term, imported from foreign jurisprudence, which makes it synonymous with the exercise of the power of eminent
domain, .... " (Black 1968, 692, entry under "Expropriation") Since "expropriation" now has this acquired meaning,
I will treat the "expropriation (termination) of rights to the assets +X" as the "appropriation of the liabilities -X."
15
The Invisible Judge: The Market Mechanism of Legal Appropriation
I have already considered the interpretation of the voluntary contract term VC and the
contractual mechanism implemented by the legal authorities. It remains to interpret what I have
called the "legal appropriation" term LA which equals -aVC by the formal stock-flow identities.
What is the legal mechanism, in effect, used by the legal authorities to initiate and terminate
property rights, and does it equal -aVC? This turns out to be an "invisible hand" mechanism of
some interest.
Consider, by contrast, an example of the "visible hand" of the legal authorities intervening to
legally assign initial or terminal property rights to a party. Suppose goods owned by one party
have been destroyed allegedly by another party. The owner takes the other party to court and
sues for damages. If no trial or judgment was made, then the owner would "swallow" the costs
of the destroyed property and would have, in effect, terminated the legal property right-which
could be viewed formally as the "appropriation of the liability" for the property. But when the
trial is held, the legal authorities try to ascertain the facts and to assign the legal liability or legal
responsibility for destroying the property to the charged party if that party is found to indeed be
defacto responsible for destroying the property (i.e., Duguit's duty to "compensate another for
damages due to one's own fault"). The payment of material damages to the owner, in effect,
makes one last transfer of the property right to the guilty party where that property right is
terminated.9 This is expressed in simple terms in the china shop sign: "If you break it, you
bought it."
This operation of the visible hand of the legal authorities allows us to conceptualize the "laissez
faire" mechanism of appropriation that is, in effect, used when the legal authorities do not
intervene to make any assignments. The last owner of a commodity that was consumed, used up,
or otherwise destroyed is, in effect, the terminal owner (when that party has no legal recourse to
get the legal authorities to enforce one last transfer of ownership to some other party). Since the
"destruction of a plus" can be expressed mathematically as the "creation of a minus," I will say
that the terminal owner of a consumed, used up, or otherwise destroyed commodity laissezfaire
appropriates the liability for that property. This yields a laissezfaire or market mechanism for
the legal termination of property rights, i.e., being the terminal owner of the property rights.
Who is the terminal owner of a property right during the time period? The voluntary contracts
transfer ownership, and many parties might buy and sell a commodity during the period. The
boundary operator computes the net inflow at the parties, namely the inflows not cancelled by
any corresponding outflows during the time period. Hence the positive components of aVC at a
node gives the property for which that party is the terminal owner. The negative boundary -8VC
would at each node represent those terminally owned commodities as the negative components.
By the legal flow equation, LA = -aVC, those are precisely the negative components or
"liabilities" in the LA node assignment. Thus we see the interpretation of the negative
9 There may also be punitive damages or fines which function not as a price to make a transaction but as a penalty to
enforce the constraint to stay within the market mechanism rather than provoke the intervention of the "visible
judge" (e.g., see the different roles of prices and penalties in the "penalty method" in linear programming).
16
components in LA as representing a laissez faire or market mechanism of appropriation of
liabilities.
Turn the signs around and we have the market mechanism of appropriation for positive property
(or "assets"). Property might be bought and sold many times during the time period but the party
who is the initial seller, in effect, appropriates that property right if the legal authorities do not
intervene to reassign that property right. This is illustrated in ordinary production. For example,
a family farm would purchase or already own the inputs (seed, fertilizer, use of land, and so
forth) and these inputs would be used up in production. With no legal action, the farm is the
terminal owner of those inputs and thus, by the market mechanism, legally appropriates those
liabilities. Then the farm would be the initial seller of the produced outputs. Since the farm had
borne the costs of all the inputs ("appropriated the input-liabilities"), no other party would have
legal recourse to challenge the farrn's initial sale of the produced outputs. In that manner, one
could say that the farm laissez faire appropriated the liabilities for the used-up inputs and the
ownership rights to the produced outputs. 0 Of course, many commodities are both produced and
used up within a party but they are of no regard to the property system which is concerned with
interparty relationships.
How would those initial sales or legal transfers show up in the formal machinery? Since the
boundary of VC computes the net inflows, the negative of that boundary would be the legal
outflows not cancelled by any corresponding inflows. Thus the positive components of -aVC at
a node represent the "first sale" or initial legal transfers of the property rights that are laissez
faire appropriated by the party at that node during the time period. And by the legal flow
equation LA = -aVC, those are precisely the positive components in the value of LA at the node.
Thus we have arrived at an interpretation of the term LA as expressing the laissez faire or market
mechanism of appropriation-so the label "legal appropriation" is fitting. The market
represented by the voluntary contracts VC acts, in effect, as an "invisible judge" to impute the
legal responsibility for the appropriated "assets and liabilities" according to the equation: LA =
-aVC. The appropriation of assets is the beginning of contract, and the appropriation of
liabilities follows the end of contract. Hence the laissez faire appropriation principle is:
"Appropriation is the (negative) Boundary of Contract"
("negative" since the boundary computes the net purchases or inflows rather than net sales or
outflows). This completes the description of the life cycle of legal property rights: how they are
initiated, transferred, and terminated by the market-based system of property and contract.
The laissez faire mechanism shows the effects on property rights when the legal authorities do
not intervene, i.e., when the law "lets it be." This could include the appropriation of assets from
'1 The closest Hume came to recognizing the legal appropriation of assets in production was his rather inadequate
notion of "accession": "We acquire the property of objects by accession, when they are connected in an intimate
manner with objects that are already our property, and at the same time are inferior to them. Thus the fruits of our
garden, the offspring of our cattle, and the work of our slaves, are all of them esteem'd our property, even before
possession." [Hume 1978 (1739), Book III, Part II, Section III, p. 509] He did not explicitly recognize the
algebraically symmetric appropriation of liabilities.
17
and the termination of property rights to the commons, e.g., the harvesting of fish from the ocean
or nitrogen from the air, or the emission of various by-products into the ocean or air. To limit
these activities is create the basis for a legal action brought by some public or private party that
would lead the legal authorities to intervene to set aside the let-it-be judgment of the "invisible
judge." The theorems developed below apply to the laissez faire mechanism. The theorems do
not apply to other explicit acts of the legal authorities which nevertheless affect the property
system, e.g., enclosures, homestead acts, state-sponsored taking or destruction of property, or
various legalizations of conquest.
One might wonder why the description of the laissez faire mechanism of appropriation seems to
be new in spite of the great simplicity of the mechanism. There seems to be a combination of
reasons. One reason is that discussions of appropriation tend to be restricted to a rather mythical
state of nature [e.g., Locke 1960 (1690)] or original position in the philosophical literature.
Secondly, the role of appropriation in ordinary production tends to be overlooked due to the
common but ill-formed idea that the ownership of the product is part of the ownership of some
already-owned capital assets. Hume's notion of "accession" might be included as an example. In
any case, this idea is easily defeated by considering the case when the capital assets are rented
out rather than the other factors being hired in. In general, it is the party who acquires the
contractual position of being the terminal owner of the used-up inputs (e.g., when capital assets
are rented, their services are bought and then used up) who has the legally defensible claim on
the produced outputs which can then be sold without legal challenge. Thus that party, by virtue
of its contractual position (not its asset ownership), legally appropriates the liabilities for the
used-up inputs as well as the rights to the produced assets. The net value of those assets and
liabilities is the "residual" so it might be said that residual claimancy is a contractual position,
not an attribute of owning land or other capital assets.
Tlhe idea that "being the firm" is a property right that is owned rather than a contractual position
is nevertheless quite common in the economics and legal literature, not to mention in lay beliefs.
Entrepreneurs are "bidding for ownership of the firms" [Hirshleifer 1970, 124] and become the
"owners of the productive opportunity" [Ibid., 125]. A proprietor may sell "the rights to the
transformation function" or "his rights to the venture" [Fama and Jensen 1996, 341] to another
proprietor. The entrepreneur is the "owner of a production function" [Haavelmo 1960, 210] and
even Robinson Crusoe "owns the firm" [Varian 1984, 225]. But the most common fallacy
occurs when the ownership of a corporation (which owns various economic resources) is
confused with the contractual role of the corporation. It is only the contractual role (e.g., hiring
in labor or hiring out capital), that determines whether a corporation or any other resource owner
ends up being the firm (legally appropriating the assets and liabilities created in production) or
only a resource supplier to that firm. If one continually thinks that "being the firm" is an owned
property right (like "ownership of a production function"), then the market mechanism of
appropriation will be entirely overlooked-as indeed seems to have happened.
Property theory also differs markedly from the "law and economics" literature since the latter
attempts to reduce legal rules and juridical mechanisms to efficiency considerations such as the
minimization of transactions costs. Similarly "the 'new institutional economics,' [consists] in
large part of transactions cost analysis of property rights, contracts and organizations."
18
[Rutherford 2001, 187] In contrast, property theory uses jurisprudential norms central to the
subject matter such as the responsibility principle introduced in the next section. The
fundamental theorems relate the responsibility principle to the legal mechanism for the laissez
faire appropriation of the assets and liabilities created in production-consumption activities. By
attempting to reduce "law" to "economics," the law-and-economics literature and the new
institutional economics neither formulates the responsibility principle nor the market mechanism
for legal appropriation.'I
The Fundamental Theorems with Property Rights
The Responsibility Principle
The development of the concept of "property rights" (previously only implicit in the framework)
together with the market mechanism of appropriation allows a richer restatement of the
fundamental theorems. The laissez faire mechanism of appropriation, like the price mechanism,
is an institutional arrangement that can be judged according to certain normative principles.
Previously I considered the notion of a socially-possible realization of intentional actions as the
normative principle used to evaluate the contractual mechanism.
How can one evaluate the market mechanism of appropriation? The relevant juridical principle
is indicated by the example given of the legal authorities intervening to assign or impute the legal
liability for destroyed property. The principle was to assign or impute the legal or dejure
responsibility (i.e., the legal liability) for the destroyed property to the party defacto responsible
for destroying the property (again, this is Duguit's legal responsibility to "compensate another for
damnages due to one's own fault"). Applied to both the creation and termination of property
rights, the juridical responsibility principle is: impute dejure (or legal) responsibility in
accordance with defacto responsibility. 2 Under what circumstances does the market
mechanism of appropriation "automatically" (i.e., with no legal intervention) satisfy the
responsibility principle?
To relate the responsibility principle to our previous treatment, it suffices to consider the
connection between intentional actions and de facto responsibility. When people carry out
intentional actions, then they are de facto responsible for the results of those actions. Previously
SIA stood for a socially consistent set of intentional decisions by the parties about production
and consumption (which includes intended changes in stocks under our present interpretation).
When realized, SIA would stand for defacto responsibility. The sign conventions for SIA were
chosen so that the positive components represented commodities consumed or used up while the
negative components represented commodities produced. Hence -SIA would be a quantity-
balanced node assignment with the positive (resp. negative) components at each node
representing the assets (resp. liabilities) for which the party represented by the node is de facto
responsible when the intentional actions are realized.
" For example in the Putterman and Kroszner anthology [1996] of papers on the "economic" nature of the firm,
none of the papers pose the question of appropriation in their treatment of the firm. The question of appropriation in
the firm is similarly ignored in the "economics of property rights" [e.g., Furubotn and Pejovich 1974] and in the so-
called "property rights approach" to the firm [e.g., Hart and Moore 1990; Hart 1995].
12 This principle can be taken as a modem and symmetrical treatment of the old Lockean [Locke, 1960 (1690)] or
natural rights principle [see Schlatter 1951] of people getting the "fruits of their labor." See Ellerman 1992 for more
analysis of the responsibility principle.
19
To summarize the connections between the underlying interpretations: the boundary of voluntary
contracts are intentional decisions, and when intentional decisions are realized, then the parties
are defacto responsible for the results.
The Fundamental Theorems and the Responsibility Principle
Theorem 5 is Theorem 3 restated in terms of responsibility.
Theorem 5: First Fundamental Theorem of Property Theory: If there is a contractual matching,
then the market mechanism of appropriation satisfies the responsibility principle.
Proof. Since VC = FT, the realized actions AFT are the social intended actions SIA = aVC and
that common node assignment represents what the parties are defacto responsible for consuming
(with production listed as negative). The negative of that node assignment gives that for which
the parties are defacto responsible for producing (with consumption as negative). Thus "assets
and liabilities for which the parties defacto responsible" = -aFT = -SIA = -OVC = LA = "assets
and liabilities for which the parties are dejure responsible," i.e., the responsibility principle is
satisfied. QED
Thus the "natural system of property and contract" operates, as if guided by an "invisible hand,"
to satisfy the basic juridical principle of imputing legal responsibility in accordance with defacto
responsibility. When there is a contractual matching, the system operates as an "invisible judge"
to correctly impute legal responsibility. The contrapositive of the theorem states that if the
responsibility principle was violated within the production-consumption activities of the parties
represented by the nodes then it would show up as a mismatch in the transactions represented on
the arcs. This is a property-theoretic refutation of Marx's charge that there could be exploitation
in the "hidden abode of production" while the sphere of exchange "is in fact a very Eden of the
innate rights of man" [Marx 1967, 176].
Theorem 6: Second Fundamental Theorem of Property Theory: Given the proposed de facto
responsible actions -SIA that satisfy the quantity balance condition Aug(SIA) = 0, there exists a
supporting arc assignment T such that if T is accepted by the parties as voluntary contracts VC
transferring property rights, then the contractual matching and market mechanism of
appropriation would assign legal responsibility (LA) in accordance with the realized defacto
responsibility (-SIA).
Proof: The existence of T follows directly from Theorem 2 and the rest is immediate from the
definitions. By the existence of T accepted as VC, we have aVC = SIA, and the matching VC =
FT implies that the net inflow realized is AFT = iVC = SIA. The realized net outflow is -SIA =
-aVC = LA. QED
Conclusion
The first fundamental theorem shows the underlying logic of the natural system of property and
contract. The contractual mechanism enforces a contractual matching that keeps the legal
ownership correlated with the defacto possession and control of commodities. The laissez faire
20
mechanism defines the legal appropriation of assets and liabilities in terms of the contracts.
Since people will have the first (respectively, last) possession of what they produce (consume),
the market-based mechanism will ensure that they legally appropriate the rights to the assets they
produce (the liabilities for the assets they consume or use up). Thus Hume's contractual
mechanism reaches back to enforce the natural rights principle associated with John Locke.
Appendix 1: Proofs of Theorems 1 and 2
Theorem 1: For any T in C(GI, Rn), Aug(aT) = 0.
Proof:
Aug(MT =' ZT(ej)- ZT(ej)
u =l h(ej)=ni t(ej)=n J
= E[T(ej)-T(ej)]=O
j=1
where, in the last term, the positive T(ej) is from the node at the head of ej and the negative T(ej)
is from the node at the tail of the arc. QED
Theorem 2: Let S be a node assignment on a connected graph G such that Aug(S) = 0. There
exists an arc assignment T on G such that AT = S, i.e., Ker(Aug) S Im(a).
Proof: Let nm be any node of G which will be called the "market-maker" node. Since the sum of
all the S(n) over all the nodes of G is 0,
- ,S(n) = -S(nm).
nfnm
Hence the O-chain S can be written as: ZS(n)[n -nm ]=S. Since G is a connected graph, for
nfnm
any node n different from the market-maker, there is a path ħeħe'ħe". . .ħe*from nn to n where
the coefficients ħ1 are determined according to whether the arc is oriented along (+1) or against
(-1) the path. Let Tn = S(n)[ħeħe'ħe". . .ħe*]. I show that aTj = S(n)[n - nm] so that the sum of
the Tn is the desired T such that AT = S. I consider the endpoints of the path separately from the
interior points. Consider a point n' on the path where, say, e_>(node n'), e" and where
the arc e' is in the direction of the path. Then the corresponding term S(n)[+e' - e"] in Tn
contributes nothing to the boundary since aTn (n') = {+ S(n) - (+S(n))}n' = 0. In a similar
manner, the other three possible cases for S(n)[ħe'ħe"] also have zero boundary. Hence it
remains to consider the four cases that arise at the end points e and e* of the path from nm to n.
21
8{S(n)[+ e + * + e - =S(n)nm + S(n)n = S(n)[n - nm]
aIS(n)[+ e + - e *= -S(n)n m - (- S(n)n) = S(n)[n - nm]
JS(n)[- e ħ .+ e *= +(- S(n)nm )+ S(n)n = S(n)[n - nm]
a(S(n)[- e ħi - e *D = +(-S(n)nm ) -(- S(n)n) = S(n)[n - nm I
In each case, aTn = S(n)[n - nm] so T = E Tn yields an arc assignment T such that
n*nm
aT= ZaTn = ES(n)[n-nm]=S. QED
n#nm n#n,
By the proof, we see any party can be selected as the market-maker and the desired bilateral
exchanges can be constructed as exchanges of each other party with the market-maker so that the
resulting net inflow at the market-maker is equal to the market-maker's intended production-
consumption decision-so all the intentions can be realized by the bilateral exchanges.
Appendix 2: Analysis of Externalities and Breaches
A real vector is non-negative if each component is non-negative (zero or positive). A real vector
can always be decomposed into the difference of two non-negative vectors. Given two vectors
X = (xl,.. .,x,) and Y = (y1,. . .,y,n) in R", let max(X,Y) be the vector with the maximum of xi and
y; as its ih component, and let min(X,Y) be the vector with the minimum of xi and yj as its ith
component. The positive part of X is:
Xt = max(X,O),
the maximum of X and the zero vector [the notation 0 is used to denote the zero scalar, zero
vector, and the zero arc and node assignments]. The negative part of X is:
X = max(-X,O) = (-X)
where also X = -min(X,O). Both the positive and negative parts of X are non-negative (note the
slight abuse of language in calling the non-negative vector X the "negative part" of X). Every
vector X can be represented as the difference of the positive and negative parts, which is called
the "Jordan decomposition" in Analysis:
x=X+-x-
Jordan decomposition.
Two non-negative vectors X and Y are said to be disjoint if min(X,Y) = 0, i.e., if each
component is zero in one or the other (or both) of the vectors. The positive and negative parts of
any vector are disjoint, i.e., min(X+,X-) = 0, and their difference is X. All these definitions and
results extend immediately to arc and node assignments by applying them to each vector
assigned to the arcs or nodes.
22
There are two types of mismatches between the factual transfers FT and the voluntary contracts
(legal transfers) VC. The factual transfers not covered by contracts will be called externalities,
symbolized EXT, (or "conversions" using more juridical terminology) and the contracts not
fulfilled by the factual transfer or delivery of the legally transferred commodities will be called
breaches, symbolized BRE (i.e., contracts that were breached). The Jordan decomposition of
vectors allows precise definitions:
EXT = [FT+ - VC+]+ - [FT- - VC-]+
BRE = [VC+ - FT+]+ - [VCU- FT]+.
For instance, let us consider commodity vectors that are three-dimensional and, on arc ej oriented
from party A to B, we have FT(ej) = (5, -7, 2) and VC(ej) = (3, -9, -3).
FT(ei) = (5, -7, 2)
VC(ej) = (3, -9, -3)
Then EXT(ej) = [(5, 0, 2) - (3, 0, 0)]+ - [(O, 7, 0) - (0, 9, 3)]+ = (2, 0, 2)+ - (0, -2, -3)+ = (2, 0, 2)
and similarly, BRE(ej) = (0, -2, -3).
~ EXT(ej) = (2, 0, 2)
(NA)~ oDB
BRE(ej) = (0, -2, -3)
Thus party B converted 2 units of the 1st commodity and 2 units of the 3rd commodity from party
A, and party B also breached contracts to transfer 2 units of the 2nd commodity (note that 7 units
were fulfilled) and 3 units of the third commodity to party A. The interpretation of the mixed
signs in the third component of FT(ej) = (5, -7, 2) and VC(eQ ! = (3, -9, -3) is interesting. Not
only did B not fulfill the contract to transfer 3 units of the 3r commodity to A but "added insult
to injury" by converting an additional 2 units.
It might be noted that the difference between the factual and legal transfers (FT - VC) is the
same as the difference between the externalities and breaches (EXT - BRE), i.e.,
EXT - BRE
= [FT+ - VC+]+ - [FT - VC-] - {[VC - FT+]+ - [VC- FT]+ }
= [FT+ - VC+]+ - [VC+ - FT+]+ + [VC - FT-] - [FT - VCJ1
= [FT+ - VC+]+ - {-[FT+ - VC+++ [VC- FT+ - {-[VC- FT1}+
= [FT+ - VC+]+ - [FT+ - VC+]F + [VC7- FT]+ - [VC-- FT]-
= [FT+ - VC+] + [VC_- FT-]
= [FT+ - FT] - [VC+ - VC-]
=FT-VC.
23
In the example, EXT(ej) - BRE(e1) = (2, 0, 2) - (0, -2, -3) = (2, 2, 5) = (5, -7, 2) - (3, -9, -3) =
FT(ej) - VC(ej).
This also means that there is a common overlap of FT and VC that could be thought of either as
the fulfilled contracts or as the transfers covered by contracts. Moreover, that overlap is the
factual transfers except for the externalities and it is also the voluntary contracts except for the
breaches, i.e., "Overlap" = FT - EXT = VC - BRE. That overlap is the natural definition of the
intersection of FT and VC:
VCr)FT = VC - BRE = FT - EXT.
To illustrate graphically with two-dimensional vectors, let FT(ej) = (-2, 5) and let VC(ej) = (-7,
3). Then EXT(ej) = (0, 2) and BRE(ej) = (-5, 0) and the overlap or intersection is VCnFT(ej) =
(-2, 3). Dropping the reference to the arc ej, this is illustrated as follows.
FT=(-2, 5)=EXT+VCnFT
VCnFT = (-2,3)
VC = (-7,3)
=BRE + VCnFT EXT = (0, 2)
BRE =(-5, 0)
Graph 1: Graphical Illustration of FT EXT + VCnFT and VC BRE + VCrFT
Since FT - EXT = VC - BRE, when EXT = 0 and BRE = 0, then there is a contractual matching,
FT = VC. Conversely when there is a contractual matching, FT = VC, then clearly FT+ = VC+
and FT- = VC- so EXT = 0 and BRE = 0. These results are summarized as:
Theorem 7: There is a contractual matching (FT = VC) if and only if there are no externalities
(EXT = 0) and no breaches of contracts (BRE = 0).
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