The author wishes to thank Keijiro Otsuka and Nancy McCarthy for helpful comments on an earlier draft. The results reported in the paper are part of a broader research project on the evaluation of natural resources financed by the World Bank. For this, the usual disclaimer applies.
Common property and the social standard
The fundamental sontingent contract
The option value of appropriation
According to Schlager and Ostrom (1992), five basic rights are most relevant for the use of common resources. These are defined as follows:
While these claims may characterize the stakeholders’ structure of resource use, they are unlikely to resolve the problem of ownership and control. In fact, a basic confusion of languages appears to affect most literature on the argument of property. For at least one influential school of thought, “a property right is an enforceable authority to undertake particular actions in a specific domain” (Commons 1968). On the other hand, for most economists who have recently looked into the question (Williamson 1994; Hart 1997), property is characterized by residual rights, that is, by the claims to what survives after all other claims have been satisfied.
The economic point of view appears to include the alternative that considers property as the process of appropriating “bundles of rights” in the sense that any rights not specifically given to one particular class of stakeholders will coalesce into the “bundle” secured by ownership. This also implies, however, that the specific assignment of rights to specific subjects cannot be considered ownership because it lacks the encompassing characteristic of residuality. The case of common-resource use is particularly relevant in this respect, since the multiplicity of rights that can be given out for alternative uses (for example, access and withdrawal) makes residual rights crucial for social efficiency. For example, if the members of the local community secure access and withdrawal rights, residual claimants are vested with management and alienation rights. These rights were indeed at the origin of enclosure and appropriation, as they arose to limit nonresidual claims to their ex ante, well-specified nature.
More generally, following Hart (1995), we can assert that the imperfect nature of contractual relations makes ex ante arrangements differ from ex post outcomes in unpredictable ways. This renders most contracts contingent on the state of the world precarious and risky, especially in the case of natural resources. Because of the inherent uncertainty associated with the vesting of customary rights and the instability in the power relations among competing groups, rent seeking and opportunism are likely to be especially strong in the case where access and withdrawal to a given resource are not bundled together in strong property rights. In this context, ex post arrangements are likely to involve continuous and substantial renegotiations of ex ante agreed rights. The role of residual rights is thus likely to encompass management and exclusion and, as an extreme measure to resolve conflict, alienation. In a very general sense, therefore, contracts can be conceived as a way of assigning contingent rights and corresponding responsibilities under uncertainty and incomplete information. In other words, contracts are inherently stipulations on risk sharing between two basic parties: a primary risk-holder and a residual owner.
Norway offers an interesting example of common property as a residual claim. In this country, different types of commons, mainly differentiated on the basis of ownership of the grounds, are a prominent feature of natural-resource management. Today Norwegian commons can be classified in three broad categories: state commons, bygd (community) commons, and private commons. The characterizing difference among state, bygd, and private commons is the ownership of ground. While in a state common, the state (central government) is the owner of the ground; in the bygd and the private commons, the commoners own the ground. What distinguishes bygd and private commons from co-ownership is that in the bygd commons more than 50 percent of the commoners are owners of the ground and in the private commons less than 50 percent of the commoners own the ground.
Ownership of the ground covers an important role as a container of what is called the remainder. This is defined as a bundle of residual rights encompassing all rights not explicitly assigned to the common. Hydroelectric power, for example, is one of these remainder rights, which emerged only recently (after being ignored for more than 100 years) as a consequence of a new technology. On one hand, thus, the remainder can be seen as a nucleus of rent seeking and appropriation that provides the holder of residual rights with risks and opportunities. In turn these constitute the incentive to oversee the resource and make sure that the owner reaps the benefits that pertain to his or her rights. On the other hand, the residual rights vested onto the remainder suggest specific responsibilities for maintenance and monitoring of the resource and offer a tax basis for the government to enforce conservation policies.
The contrast between common and remainder rights brings to the fore the point that rights have a dual nature—“the opportunity set enhancement of those who have rights and the opportunity set restriction of those who are exposed to them” (Samuels 1974, 122). Every definition of claims imposes benefits and costs, the enhancement of some opportunity sets, and the simultaneous restriction of others. Externalities are thus ubiquitous and reciprocal—any definition or redefinition, assignment or reassignment, or change in the degree of enforcement of rights benefits some interests and harms others (Medema and Samuels 1996). The externality remains, in different form; it is merely shifted, as was made clear by Ronald Coase in The Problem of Social Cost (1960). The contingent nature of benefits and costs are the consequence both of the inherently incomplete nature of all contracts, and of the random nature of asset yield. This sets the stage for sharing the predictable rights and obligations, and prominently, the risk arising from the unpredictable.
More precisely, because the assignment of rights concerns possible actions under alternative contractual arrangements, limited information creates a context where uncertainty matters. Two types of uncertainty appear to be relevant in this respect: the unknown outcome of the random variables, for which assignment of rights enables appropriation or use; and the behavior of the contractual parties under alternative circumstances. In both cases the rights tend to circumscribe the faculty of undertaking an action that would not be feasible under alternative assignments. Thus, the option value of stakeholders’ rights—that is, the options open as a consequence of the assignment—constitutes a characterizing feature of any contractual distribution of rights.
This chapter shows that the assignment of rights over the random yield of a natural resource can be modeled as a problem of partitioning the sample space into mutually exclusive subsets, one of which, the remainder, has a residual nature. Moreover, the random nature of the underlying variable creates a risk, which becomes the main differentiating factor in all alternative assignments of rights. This risk, which may be measured as the value of the put option1 corresponding to the parties’ default rights, is de facto the main object of contention among the stakeholders involved and can be demonstrated to be the ultimate determinant of the extent of residual rights under alternative regimes.
1 The right to sell a fixed amount of the yield of the resource at a predetermined price within a given time
This chapter
Delimiting the extent of what is privately and what is publicly owned can be interpreted as the result of creating social institutions to regulate the distribution of contingent rights and responsibilities. While these may consist of complex arrangements, whose meaning may be largely contextual, their ultimate functioning will depend on a relatively simple operation: the establishment of a social standard (Scandizzo and Knudsen 1980, 1996). This can be seen as a key feature of a contract that redistributes contingent rights by partitioning a given distribution into two parts: the part above the standard and the part below the standard. Depending on whether the standard is a minimum threshold (such as a poverty line) or a maximum limit (such as a pollution quota), the contract provides for an appropriate compensation being extracted from one part of the distribution to improve the other part. Thus the social standard can be seen as a way of specifying a socially desirable distribution (over individuals or states of nature) under the constraint that the only operation feasible is truncation of one of the tails, or possibly both.
Because the standard allows a separation of the outcomes of an underlying random variable into two nonoverlapping subsets, its application can also be seen as risk redistributive. The redistribution consists in attributing to one of the components of the distribution (and the corresponding contingent rights) the responsibility of covering any shortfall between actual outcomes and the standard itself. As such, it can be self-sustaining, as in a self-financing, negative income-tax program or in a self-liquidating buffer stock.
In the case of natural-resource management, at least three different applications of a social standard come to mind. First, a maximum limit to the amount of natural resource used by individuals (firms and, possibly, consumers) can be set as a share of an aggregate preservation target to be applied to each potential user or only to some users. Second, a safety-first criterion can be used, by requiring that the use of the natural resource in question be not above a maximum limit defined as the one desirable in the least favorable state of nature. Third, the possibility of irreversible loss can be captured by a social standard that reflects the option value of the resource under uncertainty, that is, the risk of using a resource whose future value may turn out to be higher than expected.
A social standard can also be interpreted as a yardstick for determining the distribution of a given type of risk between two subgroups of a population, one of which is defined as a residual claimant. In the case of a common resource, the standard can be interpreted as a line dividing the rights of the “commoners” from those of the public or the private owners. In turn, the latter are defined as residual claimants of the resource after the “commoners” are compensated and brought up to the social standard by a sufficient transfer of usage rights. The degree of compensation to which the commoners are entitled is the social standard, while the right to default on such compensation is a complement of residual rights of the individual proprietors. The individual proprietors have the responsibility of ensuring that all nonresidual claimant rights are satisfied, and can appropriate the residual surplus of any asset only after these responsibilities have been met. If this is not possible, however, they have the right to default. In a riskier environment, the standard may be expected to be more generous, and the rights to default on the part of the individual owners correspondingly higher. Thus, the higher the risk, the more likely the arrangements relying on common property, and vice versa.
In the case of enclosures of eighteenth-century England, for example, private individuals reduced production risk by fencing and farming village land intensively. The ensuing fall in the willingness to share risks was instrumental in creating a class of landless poor, who had to migrate to the cities in search of employment and income. Appropriation was thus possible because society accepted a new, lower poverty line, whereby the people excluded from the traditional use of common land experienced a drastic fall in income. Compensation in general was not paid because a new social standard evolved that tolerated, to a much larger extent, individual poverty; and because the new owners eluded payment (Zaretsky 1976).
Another example comes from pastoralism, which in the semi-arid African regions is characterized by high variability of rainfall, low population density, and high transaction costs (McIntire 1993). These conditions prevent the existence of conventional-factor (land, labor, and capital) markets so that contracts are generally complex, provide for risk sharing and common-resource management, and rely on normative notions of the rights to exploit pastures (Thompson and Wilson 1994). Poorer households are explicitly taken into account in these institutional arrangements, and the generosity with which they are treated appears directly related to the high risks of the environment (Sakurai 1995).
In the case of natural resources, the possibility of depletion or irreversible damage gives rise to yet another partition between primary and residual claimants. The present generation, in fact, may be imagined as vested of residual rights in the sense that it can claim the whole lot of natural resources, once an appropriate reserve is made to avoid the future generation’s finding itself with an amount short of the social standard. The value of the resources that the present generation can claim is thus equal to the total amount that is expected to be available to both generations minus the amount that it deems necessary to ensure that the future generation is able to enjoy the standard. A key residual right is the value of the option of not making provisions beyond what is presently considered a reasonable complement to individual efforts of the members of future generations.
For example, the inhabitants of the Sahel address land conservation by enhancing mobility and by organizing collective access-control over their own resources. In the more densely populated areas, on the other hand, attempts to develop active conservation policies are more evident and they appear to go hand in hand with land-use intensification, exclusive resource control, and technological change (Vedeld 1997). The concern for conservation thus appears to be positively related to the reduction of production risk, while the social standard for food security is much higher in the riskier areas.
How a social standard is established is a question that can be viewed from the point of view of both positive and normative economics. The rise of a social standard is part of a complex process of developing structures for social action through common goal-setting, institution-building and norm-designing. As such, it may be related to the notion of social justice as fairness developed by Rawls (1974, 1996) and his school of thought. That the residual claimants of the social product may be defined as those who are willing to take the burden to provide for the needy may be seen as an implicit stipulation of a “fair” social contract. The thought experiment behind the veil of uncertainty, whereby one chooses the society that maximizes the well being of the least well off, can also be seen as a way of explaining the emergence of a social standard and the willingness of those above it to forego part of their income to improve the condition of the people below it.
From a normative point of view, on the other hand, a social standard can be established by answering a very general question: What partition of a statistical population (of persons or events) is consistent with a program that transfers resources from one group to another for a given unit cost of the transfer? For example, how do we define a mutually consistent poverty line and poverty eradication program if the cost of transferring resources from the rich to the poor is c dollars for each dollar transferred? If Rawls’ criterion of maximizing the income of the poor is adopted, the solution to this problem can be obtained easily by imposing the condition that the post-transfer income of the people charged with the transfer (that is, their income minus the total cost of the transfer) be not less than the ex post income of the people who benefit from the transfer. If the transfer cost, c, is zero, the answer is that the class of poverty lines with the desired characteristics is bounded from above by the average income of the population. For c greater than zero, on the other hand, it can be shown that such a Rawlsian poverty line is given by average income minus c times the option value of becoming poor.
As residual claimants of national income, in other words, the “rich” should be prepared to transfer to the “poor” a maximum income equal to what they would expect behind the veil of uncertainty (that is, expected income) minus an allowance to reflect the uncertainty of their future condition. This allowance in turn is calculated as the resource loss that would occur should poverty be eradicated totally at the expense of the rich. Thus, under such an interpretation of the Rawlsian rule, the minimum socially tolerable difference between the rich and the poor is the gap that could not be closed by a transfer program.
Because the setting of a social standard immediately recalls the notion of social justice, it is important to see the problem of ownership from a point of view entirely alternative to Rawls’. One such point of view is given by Nozik (1977, 1993), who rejects the idea that rights should be judged on the basis of a “consequentialist” principle, that is, on the effects of their distribution on the well-being of any members of the society. Building upon Locke’s conception of property rights, Nozik proposes two procedural principles to determine social justice: legitimate acquisition, and efficiency of original appropriation compared with nonappropriation. The first principle states that “A person who acquires a holding…from someone else entitled to that holding, is entitled to that holding” (Nozick 1977, 151). Thus, given a distribution of holdings at time t, its “just evolution” at time t + 1 requires that all transfers of rights are voluntary and lawfully contractual in nature. The second principle defines the criterion to judge original appropriation or justice in acquisition. This principle states that “a person who acquires a holding in accordance with the principle of justice in acquisition is entitled to that holding” (Vedeld 1997, 151). In this respect, Nozick argues that the initial acquisition of holdings from the previously unowned natural world is acceptable so long as the appropriation “leaves no one worse off than she would have been had that part remained un-owned” (Roemer, 1996, 3).
Nozick’s view is opposite to Rawls’ in that he openly rejects the idea that societies may develop their own standards of justice on the basis of what is deemed desirable as a consequence of the application of the allocation of rights. His second principle, however, cannot escape some consideration of the consequences of appropriation. This consideration is somewhat weaker than Rawls’ but is vulnerable to two objections. First, in many cases (most notably for natural resources), determining whether appropriation “leaves no one worse off” may be impossible. Future generations, for example, may be considerably worse off as a consequence of the appropriation of natural resources and their succeeding depletion by lawful owners. Second, no compelling reason exists to consider nonappropriation as the sole yardstick by which to judge efficiency. Joint ownership (including, as a subclass, all types of public ownership) may very well be a superior alternative.
In spite of these objections, Nozick’s proposal is interesting in that it may provide an additional (rather, an alternative) useful criterion for evaluating the vesting of rights as the partition of risk, through the application of a social standard. Nozick’s first principle, in fact, outlines a procedure by which the vesting of rights may gain social approval. Any transfer of rights is likely to be validated if it is at the end of a sufficiently long sequence of past contracts. What is required for such a validation is a certain stability of the same partition of claims and responsibilities invoked by the contract. In turn, this implies that the dividing line between the amount of risk that each of the two parties agrees to bear also must be correspondingly stable. In other words, the social standard is itself legitimate only to the extent that it is validated by an unchallenged (or not successfully challenged) series of contracts upholding it. Nozick’s second principle can be interpreted as a criterion to evaluate the establishment of a social standard from an original state, where property is unowned. The standard can be used in this case to justify a different distribution of contingent rights and responsibilities from the original one. This redistribution, in turn, is validated by its strong Pareto-efficient consequences.
While for some contracts unforeseeable contingencies may conceivably be of minor concern, in fact virtually no transfer of rights can occur outside the realm of basic uncertainty. Any contract can be described as a logical sequence of uncertain claims and counterclaims in response to an original act of appropriation. The sequence ends when it finds an acceptable balance among the rights and responsibilities of the contracting parties. The balance should be acceptable not only to the parties engaged in negotiating the contract, but also to the broader community, whose laws and regulations may validate the clauses of the contract, provide a forum for complaints, and allow any conflicts to be litigated, negotiated, and ultimately resolved by consensus or enforcement.
In their barest form, contracts can be studied in a world with one time period, where asset returns are uncertain in the sense that expectations entertained ex ante are not necessarily realized ex post. Under these conditions, the basic contract is stipulated ex ante and implemented ex post. It can be described as arising from a negotiating relation between two parties, which for the moment is simply indicated as party A and party B, and an asset producing a random yield y. The random yield is characterized by a probability distribution function F(y) and a density f(y), which are commonly known.
Formally, the logical sequence describing the contract can be expressed as its value Vi (i = A,B) for the contracting parties:
VA = Ey – P + R. [1]
VB = P – R. [2]
In its simplest form, the contract originates from the appropriation of a commonly held resource by party A. As a consequence, the contract provides a value to party A that is the algebraic sum of three terms:
In turn, the value of the contract for B is given by A’s payment minus A’s default rights. Party A thus holds the “residual rights” in the contingent sense that she can appropriate what is left once B’s rights are satisfied. If this is not the case, either because A fails to honor her promise to B or because she can no longer enforce her ownership through third parties, “residual” rights are transferred to B. In the absence of stipulated restrictions by the two parties or by the law, VA and VB are also the liquidation values of the contract for the two parties, that is, the minimum amount of money that each of them would accept to alienate the rights conferred by the contracts.
Equations (1) and (2) express the simplest string of values characterizing a contract. Two main ways in which this basic structure can be complicated are the specification of additional payments to cover predictable contingencies, and the provision of clauses contingent upon acts of one or both agents. While both of these complications have been examined in the literature (see, for example, Grossman and Hart 1983; Hart and Moore 1990), they are not essential to understanding the fundamental nature of the contract, which rests only on the three basic elements indicated: expected asset value, compensation to the party excluded (or enforcement costs to exclude), and default rights.
To investigate further the shaping of rights arising from this formulation in equations (1) and (2), consider residual rights. The contingent nature of these rights arises as a consequence of the imperfection of the contracts and is independent of the will of the negotiating parties. In the basic structure designed in this chapter, they arise from an external principle of law: limited liability, that is, the fact that satisfaction of any obligation cannot exceed the repayment capacity of the subject involved. In this case, limited liability implies that party A may default on her payment to party B if such a payment exceeds the income-generating capacity of the asset. This information enables equations (1) and (2) to be rewritten as follows, respectively:
dF(y). [3]
. [4]
The default value R equals the difference between the payment and asset yield in the “unfavorable” states of nature, that is, in those states where the payment could not be made because it would exceed the yield. This value, R(P), is thus an expected gain for party A (the contingent holder of residual rights) and an expected loss for party B. It can also be considered a risk—the risk of default, which is charged to the payee (party B) whether the contract provides for it or not.
By developing R by parts, the following is obtained:
.
This mathematical form describes more explicitly the residual value as a measure of risk.2 An alternative way of formulating equations (1) and (2) in light of 3 and 4 is as follows:
.
.
This formulation shows that appropriation engenders a form of “vertical” sharing of the yield of the asset involved. This means that property rights—defined as a bundle of access, withdrawal, and alienation rights, even when they are not decomposed into their constituent rights—are always shared to some degree by the contractual parties involved, by virtue of the principle of contingent residuality.
2 This can be seen by appealing to the concept of stochastic dominance of second degree. If two
assets, 1 and 2, with different yield distributions F1(y) and F2(y), are compared, asset 1 dominates, stochastically, second-degree asset 2 if the following is true for all P:
Thus, equation (5), as a building block of stochastic dominance, can be considered a local measure of risk.
Another point is that party B, who holds the residual under the unfavorable states of nature, need not be a private party but may instead represent the community at large or a local community having a specific, original claim to the resource appropriated by A. In many contracts, on the other hand, society—or any additional stakeholder holding a recognizable claim as a consequence of the contract—might be included as a third, implicit agent.
The provision of a given payment P is a crucial determinant of the structure of the contract for three main reasons:
For example, P may be the amount of rent in a lease contract, the price of the asset in a sale contract, the bidding price at an auction, the principal and the interest on a loan, the enforcement cost of privatization, or any other ex ante promise for a payment whose actual disbursement is contingent upon ex post outcomes.
Because of its dual characteristic of a payment and a dividing line among “superior” and “inferior” states of the world, P may also be given a broader interpretation. In a social contract where the stochastic variable is income distribution, productive resources may be seen as appropriated by the “well off,” defined as the people whose income is above the “poverty line,” P. In this case, the social contract stipulates the promise of the “well off” to provide for the needy. This is accomplished by a transfer sufficient to ensure that all people below the minimum level of income (or basic commodities) are given enough to meet the basic need requirement P.
In this Rawlsian interpretation of the social contract, social justice would be served by the society where P was the highest possible. The height of P, however, would depend both on the level of society’s total wealth, Ey, and on the value of default rights, R. In this case, in fact, such a value depends on the wealth of the people below the poverty line, P, and measures their capacity to approximate the social standard without the help of the well off.
However, returning to the main frame of reference in this discussion, how is P established in general? Focusing on this question requires noting that, once the two parties have agreed on the terms of the contract, the interest of A is to minimize P, while the interest of B is of course the opposite.3
3 For A to credibly commit herself to the contract, she has to stipulate the payment and the default rights in a way that ensures her incentive to default only if y < P. this will occur if residual rights are determined as follows:
[1.1]
where D is the default line .
In other words, R stipulates that if y < D, A will default on the payment but will turn the asset back to B. In this case, maximizing VA with respect to D results in the following:
[1.2]
This is < 0 for P = D. Thus, under the definition of the residual rights as expressed in equation 1.1, A can commit herself credibly to the contract if she may default only when she cannot sustain the payment (y < P), because it is in her interest to push the default line up to P.
The conditions of the contract can be investigated by using the structure of a two-by-two, noncooperative game. A basic version of such a game is shown in Figure 8.1, where it is assumed that the two parties can adopt two alternative strategies: attempt to appropriate and accept not to appropriate. The pay-off matrix in the figure shows an ex ante sharing of the asset plagued by an external cost, C, which only one of the two parties (say party A) would be able to eliminate under appropriation. This may be the case, for example, if A is an individual, while B is a collective party, so that any free-rider problem (such as overstocking) would continue to occur under ownership by B.
FIGURE 8.1 Payoff matrix for the single-period game
Party B | |||
Appropriate |
Not appropriate | ||
Party A |
Appropriate |
|
|
Not appropriate |
|
| |
Under what conditions can party A be expected to assume ownership of the asset, with the agreement of the other party, provided a compensation 
is paid or even without her agreement, if appropriate costs to enforce appropriation are incurred? Clearly these conditions are realized if the outcome in the northeast corner of Figure 8.1 is a Nash equilibrium, that is, if it is the locus of the best possible response for both parties.
A necessary and sufficient condition for appropriation to be a best possible response for A is as follows:
Ey – 
> α(Ey – C) and
αEy >
;
[8]
and for B,
> (1 – α)(Ey – C) and (1 –α
)(Ey – C) > Ey ––
C, [9]
where the compensation, 
and 
, is defined as the net risk of default, that is,
= PA – R(PA) and 
= PB – R(PB).
When the two inequalities are combined, the following is obtained:
.
Now the conditions to obtain a stable Nash solution must be considered. These can be derived by maximizing the Nash product (Harsanyi and Selten 1992) between the gains of each party with respect to the ex ante situation (the southeast corner of the payoff matrix):
where 
= the Nash value.
Performing the maximization with respect to
results in the following:
If this scheme is enforced, the parties share an equal improvement from A’s appropriation equal to one-half of the external costs C. Equation (13) can be written as follows:
This expression can be solved explicitly if the form of F(y) is known. For example, if F(y) is a uniform distribution in the interval [0,1], the only feasible solution is as follows:
.
More generally, if the geometric average is maximized, the following results:
[16]
where 0 < θ < 1.
The following value leads the two parties to share the gain C from privatizing the asset with the proportions aθ for party A and (1 – a) θ for party B:
(17)
Consider now the condition expressed in equation (11). This condition refers to the case where party B would appropriate the asset and party A would oblige accepting the compensation 
. The Nash solution would require the following:
However, this would imply a compensation compatible with equation (11), so that the alternative solution, in which A appropriates and B is compensated, would be accepted. In fact, it is easy to check that any mutually acceptable level of compensation 
would lead to payoffs lower than the payoff that could be realized by both parties by the level of compensation 
in the interval indicated in equation (10).
The framework developed can be generalized by considering the problem of appropriation in a multitemporal contract. At a time t, a given asset offers a yield y that changes stochastically over time according to a Brownian process of the form:
where
µ and σ = constants; and
dz = a random variable with expected values Ez and Ez2 = dt.
The ex ante appropriation value of the asset can be considered a call option,4 whose value is as follows:
where
E = expectation;
T = the time at which appropriation is made;
r = the discount rate; and
V(yt) = the value of the asset at time T.
It is assumed that µ < r; otherwise the expectation in equation (21) could be made indefinitely large by choosing a larger T.
4 The right to buy a fixed amount of the yield of the asset at a predetermined price within a given period of time.
By solving the maximization problem, equation (21) (Dixit and Pindyck 1994), the payoff matrix of Figure 8.1 can be reformulated as shown in Figure 8.2.
FIGURE 8.2 Payoff matrix for the multi-period game
Party B | |||
Appropriate |
Not appropriate | ||
Party A |
Appropriate |
|
|
Not appropriate |
|
| |
In Figure 8.2, δ = r – µ and y/δ represents the expected present value of the yield stream yt, when its initial level is y. In fact, because of the Brownian process assumption, E(yt) = yert, and discounting at the appropriate rate r results in the following:
.
The expressions L(PB)yB1 and L(PA)yB1 represent the value of the option of appropriating the whole resource from each of the two parties. It is possible to show (Dixit and Pindyck 1994) that L(PB)yB1 and L(PA) are constants that depend on the compensations PB and PA respectively paid by A and B. b1 is a parameter that depends only on the parameters of the process underlying y and on the discount rate as follows:
.
From equation (23), it follows (Dixit and Pindyck 1994) that β1 is the positive root of equation (23) and that β1 and ∂β1 / ∂σ < 0.
For appropriation to be profitable for A, the following conditions can be derived, from Figure 8.2:
and
.
Similarly, for party B,
, [25]
and
.
Consider first the inequality in equation (24). For a given PB, appropriation becomes profitable for party A when the increase in payoff from the prior situation, that is, the difference of the two sides of equation (24), equals zero:
.
The point at which appropriation is jointly profitable is also the optimum value of y. (Waiting does not pay any longer.) Thus, the first derivative of VA in equation (26) with respect to y should also be zero. (This is called the “smooth pasting condition,” see Dixit and Pindyck 1994) that is,
.
Substituting this expression in equation (26) results in the following:
,
where y* is the value of y, at the optimum, for a given PB. Thus, for any given y, it will be profitable to appropriate for party A if and only if the following is true:
.
In other words, for a given y, party A will be able to enjoy an appropriation rent, if she can negotiate a deal where PB is lower than the level at which it just becomes profitable to appropriate. This is compatible with the improvement of B’s payoff unless PB falls below the limit set by the first inequality in equation (25).
Consider now the prospect of appropriation from B’s point of view. Her gain under this hypothesis would be the difference between the right-hand and the left-hand sides of the second expression in equation (25).
A procedure similar to the one used for A yields the following:
.
If a = 1 – a and PA = PB, clearly y** > y* because, under the same privatization cost, A’s appropriation is more beneficial, since it eliminates the externality C.
This is even more so if, as it can be expected, a < 1 – a. In fact, the smaller party should generally be expected to appropriate before the larger one, for two reasons: it will be easier for her to reduce the externality (because she has to respond to a smaller number of co-owners and possibly only to herself), and she will able to offer a better deal to the other party.
From equation (30), it can be argued that, for a given y, for appropriation to be profitable to party B, it is necessary that the following be true:
.
Comparison of equations (29) and (31) readily shows that for α < 1 – α, then 
; where
PA and PB are the maximum acceptable levels of compensation that each appropriating party may be willing to pay.
The contingent value of the contract, under the dynamic model presented in the section The Option Value of Appropriation, above, appears to depend on a different feature than in the one-period model. This was in fact characterized by default risk, which functioned as a put option for the holder of residual rights (the appropriating party) on the asset. For the multiperiod contract, on the other hand, this model has assumed that the payment to the expropriated party is made at the beginning so that all risk is borne by the new owner. The appropriating party, however, does hold an option in the form of the right to wait before she makes the move to appropriate. Rather than holding a put option as in the one-period case, the appropriating party kills a call option, which she holds before moving to the new contract from the previous position.
The appropriation contract may be framed, however, in a context of limited liabilities and default rights, by assuming that compensation to the expropriated party is not paid immediately but, at least partly, after a certain number of periods. In this case, rather than waiting, the appropriating party may appropriate and then consider the probability of default.
To explore this possibility, assume that at time t party B agrees to forego her rights on the asset in exchange of a compensation PB to be paid in equal installments 
. In this case the appropriation condition for A will be as follows:
[32]
where = the price at which appropriation will occur.
Once appropriation has been accomplished, however, at time t + t, party A faces a different prospect. If Ey(t + τ) = yeu(t+τ) is sufficiently high; in fact, she will able to pay the contract price and keep the asset.
If the expected value of y at any time is not sufficiently high, however, A may decide to default. Because of limited liability, default may not cause any loss in party A’s personal wealth, if any, but will presumably prevent her return among the beneficiaries of common property. Party A’s prospective gain from default at time t + t may thus be indicated as follows:
. 
[33]
where
R(PB)yβ2 = the value of the put option held by B as her right to default;
R(PB) = a constant that depends on the value of the promise to pay PB; and
β2 = the negative root of the quadratic equation in equation (23).
Using the smooth pasting condition ∂VA/∂y and substituting into equation (33) results in the following, to avoid default:
[34]
where PeB denotes the price at which appropriation will cease.
From equations (32) and (34), it may be concluded that, for any level of expected yield, appropriation will not occur and, if it has occurred, will cease if the contract price exceeds a multiple of the present value of the asset. Such a multiple will be a function of the variance:
.
[35]
An increase in risk will cause the entrance fee for appropriation to decrease but the default level to increase.
The difference between the entry and the exit price, in fact, is much more important than what it may appear from the algebraic expressions. If plausible values of the parameters are used to study the effect of risk increases, the values of the entry and exit price and
PeB diverge dramatically. This is shown in Table 8.1, which reports the values of
β1, β2, /y, and
PeB/y both under the hypothesis that the discount rate is not adjusted and then when it is adjusted for risk. The adjustment is made using the capital-asset-pricing formula
ρ = r + Фρym ×σ,
where
r = the riskless rate (the market price for risk),
ρ ym = the correlation coefficient between the asset (yield) and the market price, and
σ = the standard deviation of y.
Table 8.1 and Figures 8.3 and 8.4 show that, as variance increases, unadjusted entry compensation levels quickly stabilize at about 90 percent of expected yield. Exit levels instead increase rapidly so much that even moderately high values of the variance seem to imply no respect of abandoning the asset.
Adjustment for risk appears to act mostly on the entry level, which is substantially reduced. Exit levels, even though they are lower than in the unadjusted case, are so high that they still seem to rule out any possibility of default.
The dramatic divergence in the values of the two thresholds is mainly due to the large differences in absolute value between the values of the risk parameters b1 and b2, since the latter approaches zero very quickly as the variance increases. Thus, for even moderately high values of the variance, a new owner will be willing to pay very little compensation to the excluded party. However, once she holds her new possession, it will take a very large price to make her default.
TABLE 8.1 Entry and exit threshold levels for increasing risk
S.D., |
Risk-adjusted discount rate, |
Compensation thresholds | ||||||
Risk parameters |
Unadjusted for risk |
Adjusted for risk | ||||||
Beta1 |
Beta2 |
Entry |
Exit |
Entry |
Exit | |||
0.01 |
1.902498 |
–2.1025 |
0.024 |
0.970289 |
2.951249 |
0.821115 |
2.529642 | |
0.11 |
1.818182 |
–0.2 |
0.064 |
0.92106 |
12 |
0.536602 |
7.111111 | |
0.21 |
1.809998 |
–0.10524 |
0.104 |
0.916037 |
21.00497 |
0.493925 |
11.61977 | |
0.31 |
1.806895 |
–0.07141 |
0.144 |
0.914121 |
30.00687 |
0.472252 |
16.12309 | |
0.41 |
1.805262 |
–0.05404 |
0.184 |
0.91311 |
39.00787 |
0.455485 |
20.62485 | |
0.51 |
1.804255 |
–0.04347 |
0.224 |
0.912486 |
48.00849 |
0.439621 |
25.12594 | |
0.61 |
1.803571 |
–0.03636 |
0.264 |
0.912061 |
57.00891 |
0.423185 |
29.62668 | |
0.71 |
1.803076 |
–0.03125 |
0.304 |
0.911754 |
66.00922 |
0.405511 |
34.12721 | |
0.81 |
1.802702 |
–0.02739 |
0.344 |
0.911522 |
75.00945 |
0.386251 |
38.62762 | |
0.91 |
1.802409 |
–0.02439 |
0.384 |
0.91134 |
84.00963 |
0.365205 |
43.12794 | |
1.01 |
1.802174 |
–0.02198 |
0.424 |
0.911193 |
93.00977 |
0.342254 |
47.62819 | |
1.11 |
1.80198 |
–0.02 |
0.464 |
0.911073 |
102.0099 |
0.317317 |
52.1284 | |
1.21 |
1.801818 |
–0.01835 |
0.504 |
0.910972 |
111.01 |
0.290343 |
56.62858 | |
1.31 |
1.801681 |
–0.01695 |
0.544 |
0.910887 |
120.0101 |
0.261293 |
61.12873 | |
1.41 |
1.801562 |
–0.01575 |
0.584 |
0.910813 |
129.0101 |
0.230142 |
65.62886 | |
1.51 |
1.80146 |
–0.0147 |
0.624 |
0.910749 |
138.0102 |
0.196868 |
70.12897 | |
1.61 |
1.80137 |
–0.01379 |
0.664 |
0.910693 |
147.0103 |
0.161458 |
74.62906 | |
1.71 |
1.80129 |
–0.01299 |
0.704 |
0.910644 |
156.0103 |
0.123899 |
79.12915 | |
1.81 |
1.801219 |
–0.01227 |
0.744 |
0.9106 |
165.0104 |
0.084184 |
83.62923 | |
Note: S.D. indicates standard deviation.
FIGURE 8.3 Entry unadjusted and adjusted for risk
Now, consider the probability of appropriation and default. It is reasonable to hypothesize that the larger the number of people seeking ownership rights is, the smaller the ratio between the compensation that they have to pay and the asset yield is. Thus, in a riskier society, by virtue of equation (34), the number of owners should be fewer, since only a few will find attractive a very low ratio between the price and yield of the asset. However, a comparatively larger share of owners will hold on to the asset, once it is appropriated, and will not default. A higher variability, in fact, implies, other things being equal—that “it pays to wait” before exercising the default option and forego the rights to the asset.
These results can also be interpreted in a context where A’s “payment” denotes the expenditure for privatizing, and where default rights may be renegotiated. Thus, for example, some access to A’s land may be granted to B in the event of a poor realization. Alternatively, A’s access to the remaining commons may be reduced, so that A receives less in bad years (because of this lack of access).
FIGURE 8.4 Exit unadjusted and adjusted for risk
Thus, the interesting feature of a risky situation is not that the appropriator may default, but that, ex post, either she must expend even more resources to keep others out (if she realizes a relatively good yield) or that she wants to access what remains of the commons but may not be able to (if she realizes a relatively bad yield). This also implies that, in a riskier society, owners will tend to be more generous toward the people who were excluded from the asset and to whom compensations are due. If the contract between the owner and the excluded party takes the form of a loose arrangement that can be periodically negotiated, for example, a higher risk may be expected to result in higher level of payment. Alternatively, as risk decreases, the established owners’ willingness to pay will be reduced and, at the same time, a higher level of appropriation will occur.
Because technical progress typically involves both an increase in average yield and a decrease in its variability, its effect on appropriation is ambiguous. On one hand, it will tend to increase the price that would-be owners are willing to pay to appropriate a commonly held asset. On the other hand, it may increase or decrease the degree to which established owners will be willing to pay compensations to the commoners. If the effect of risk prevails over the increase in average yield, in fact, a higher rate of default, as well as lower propensity to increase the level of compensation to the commoners, should be expected to accompany technical progress.
In conclusion, in a higher-risk society, the following should be expected:
Consider now the efficiency of private property from the point of view of resource allocation. Assume that the asset appropriated may be combined with one or more inputs x, at a given market price w, according to a neoclassical production function:
q = q(x)u, [36]
where u is a random variable in [0,1].
For example, x may represent land, livestock or other inputs, while u is random shocks. The appropriation contract can be reformulated as follows:
[37]
where
In the E,P,R formulation of the contract, the following can easily be checked:
[38
The following can be obtained through using the E,P,R formulation and developing R in equation (40) by parts:
. [39]
Differentiating with respect to x, equating to zero, and solving for qxEu yields the following, after some simplification:
[40]
From equation (42), it can be argued that the marginal productivity will be greater or less than factor remuneration according to whether the following is true:
[41]
Solving for P, from equation (39), can further yield the following: qxEy is greater or less than i according to whether P is greater or less than qEy – wx + R/F(up). However, it can be easily checked from equation (37) that VA > 0 will require P < qEy – wx + R. Thus, the range of the possible values of P for which underutilization of resource occurs is much larger than the alternative on the privately appropriated land. Contrary to the “tragedy of the commons,” which derives from a tendency to overexploit a common resource, appropriation may thus cause a “tragedy of the private.” In any case, for up ≠ 0, allocation will necessarily be inefficient, according to whether party A has a higher or lower incentive to combine productive resources with the asset appropriated. It can also be shown that, for the Nash solution VA = VB, marginal productivity will be greater than the wage rate, so that underexploitation of resources will ensue.5
5 For the Nash solution VA = VB, the value of up is and equation 41 will imply that qxEy > w, since requires only that , which is always true for .
In the appropriation–nonappropriation game, it can thus be argued that the condition to benefit both parties becomes
, [42]
where r and l represent, respectively, the external costs under common access and the efficiency losses due to limited liability—all measured in units of the efficient solution 
.
Thus far, it has been assumed that party B is somewhat passive, in the sense that P and w are established by the market or by a bilateral bargaining process independent of factor allocation. Alternatively, however, it can be assumed that P is determined on the basis of a participation constraint for B (Mitra 1983).
In this case, it might be argued that, since there are no transaction costs, Coase theorem would apply. Even in the absence of transaction costs, however, efficiency may not be reached if part of the burden of default risk has to be borne out by the owners of the resource. More specifically, consider the following contract formulation:
[43]
and
where
[44]
where ux= wx
q(x)
Equations (43) and (44) indicate that party A treats the payment to the input owner (possibly herself) as having seniority rights with respect to party B. Thus, the compensation P is paid only after the market value of the input has been paid. Furthermore, it is assumed that limited liability holds both for A and for B and that the market rate w is given and nonnegotiable.
Given these assumptions, consider A’s problem as the maximization of VA in equation
(43) given the requirement that the expected income VB be equal to a given amount 
.
The following is obtained through substitution of equation (44) into equation (43):
[45]
which can be written as
[46]
The first-order conditions for the maximization of equation (48) can be written as follows:
[47]
where
Solving for w yields the following:
[48]
with decreasing returns θ < 1 and w > (∂q/∂x)Eu, denoting overuse of the resource, with respect to its private optimal use. This overuse is the result of the fact that limited liability allows both A and B to default when u <wx/q.
This chapter has explored the significance of appropriation and the transfer of rights in the context of imperfect contracts. It has identified the basic form of appropriation in an E,P,R contract, whereby a party acquires the right to enjoy a random yield y and in exchange, agrees to pay to (at least) another party a price P, whose payment is subject to a right of default R. This chapter has proposed to consider default rights as the fundamental consequence following appropriation of residual rights and the basic risk burden faced by nonappropriating stakeholders.
While risk aversion in one or both parties has bearing on the negotiation of the transfer price, default risk asserts itself independently of subjective preferences. It always corresponds to a put option that the vesting of the rights create and that neither party can avoid. Ultimately, however, both the price paid and the particular distributional solution must be found through negotiation. The question of a “fair price,” in particular, seems to be a legitimate problem in this kind of exchange and precedes logically and, perhaps, has preceded chronologically, all questions of efficiency. The transfer price’s being considered as a social standard emphasizes the fact that the transfer of rights generally involves problems of fairness and provision for the needy.
In an intertemporal context, the problem of acquiring a property asset can be distinguished from that of maintaining the asset. In both cases, risk increases the value of waiting, that is, the value of not modifying the status quo. Higher risk will result in a lower acceptable cost to appropriate the asset and thus, presumably, in a lower number of people taking the action to appropriate. For those who already own assets, on which compensations are due to other parties by virtue of explicit or implicit contracts, the situation is similar. Increases in risk will imply, in fact, that these owners will default under comparatively higher dues. Fewer appropriators, therefore, will choose to exercise the limited liability option, under higher risk. The quantitative impact of risk, however, is very different in the two cases: moderate on the threshold compensation acceptable to seek ownership, and extremely strong on the default threshold.
The default threshold can be interpreted as the compensation paid by the owners of wealth (the rich) to the people who do not own substantial resources (the poor). Interpreting the average value of the payment made by the rich to the poor as a social standard emphasizes the fact that the transfer of rights generally involves problems of fairness and prevision for the needy. As such, the price paid by the holders of residual rights to the other parties is likely to reflect the social consensus on the sharing of the risks among the parties involved.
The riskier the environment in which the deal itself is consummated, the more likely the social standard is to be more generous toward the commoners that are deprived of their rights. The reason for this is that, in a riskier environment, a higher compensation is likely to be more acceptable to the rich and more necessary to the poor. Higher risk and a correspondingly higher social standard for food security, for example, can be interpreted as a form of communal property arrangement where access to common resources is enhanced through mobility, reciprocity. and other arrangements (see, for example, Vedeld 1997).
Nevertheless, the distribution may be separated from the efficiency problem through the assumption that the original condition is resource sharing and joint management. Against this standard, it can be shown that any act of appropriation may generate both benefits and costs through the elimination of the externalities associated with collective action on one hand (the “tragedy of the commons”) and through the interjection of inefficiencies due to default risk on the other.
Fuller appropriation of rights to a simple asset thus appears to be a more desirable strategy than partial appropriation. In fact, privatization appears justified at least on the grounds that the ownership of a nexus of rights from the part of single owner reduces her incentive to default and provides, ex ante, a more credible basis for the transfer of rights.
On the other hand, neither joint ownership nor full appropriation provide clear-cut rules to choose a superior solution from the point of view of social justice and efficiency. Joint ownership, as a solution that maximizes total output, in fact, is an extreme case where distribution is totally undetermined, while efficiency is at a maximum. However, this can be so only if it is accepted that the assumption that the ensuing distribution will display a pattern of incentives validating the joint maximization assumption. Similarly, appropriation may solve the problem of distribution in a clear-cut way, but leaves efficiency to be determined by the arrangements between the appropriating and the expropriated parties. Even if transaction costs are disregarded, compensation arrangements are likely to result in inefficiencies due to limited liability and the opportunity to shift some of the default costs onto other parties.
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