1 – Introduction
1We contrast two views of the price system, as the barometer of valuation and as the means of appropriation. The marginal revolution was understandably valuecentric, perhaps inadvertently masking the importance of appropriation. But the emphasis on valuation grew to become the dominant feature of neoclassical economics. Its summary statement, definitively presented in Debreu [1959], leads either to the interpretation that (a) valuation includes appropriation in the nominal sense that the prices of goods and services determine the distribution of rewards among individuals? or—adopting an alternative definition, below—to the conclusion that (b) valuation can be separated from appropriation, providing tacit support for the continued absence of attention to it.
2The neoclassical connection between these two views of the price system, emphasized by Pigou [1932], is: appropriation is the standard by which the efficiency of valuation is measured. An efficient allocation of resources requires private marginal valuations perceived by individuals to coincide with their social marginal benefits/costs. Failure to achieve that standard is an indicator of inefficiency. We describe a related theme, but one that is based on a different meaning of appropriation.
3Rather than a property of valuation, “appropriation” is a generic description of an individual’s efforts to extract gains from others. The term full appropriation will refer to the upper bound on what an appropriator can hope to accomplish. Evidently, the upper bound is constrained by property rights, which protects each individual from efforts to extract gains by others, and by competition, which protects others from efforts to extract gains by any individual. An early illustration of full appropriation is the determination of land rents by Anderson [1775].
In every country there are various soils, which are endued with different degrees of fertility; and hence it must happen that the farmer who cultivates the most fertile of these can afford to bring his corn to market at a much lower price than others who cultivate poorer fields. But if the corn that grows on these fertile spots is not sufficient to fully supply the market alone, the price will naturally be raised in that market to such a height as to indemnify others for the expense of cultivating poorer soils. The farmer, however, who cultivates the rich spots will be able to sell his corn at the same rate in the market with those who occupy poorer fields; he will, therefore, receive much more than the intrinsic value for the corn he rears. Many persons will, therefore, be desirous of obtaining possession of these fertile fields, and will be content to give a certain premium for an exclusive priviledge to cultivate them? which will be greater or smaller according to the more or less fertility of the soil. It is this premium which constitutes what we now call rent, a medium by means of which the expense of cultivating soils of very different degrees of fertility may be reduced to perfect equality.
5Because the price/acre of two land parcels differ exactly by their fertility (and the value of marginal land is zero), each landlord fully appropriates the value of his entire parcel. Prior to the marginalist revolution, this analysis was invoked to demonstrate that rent is a residual payment: the price of land (having no alternative use) is determined by the price of corn. It was replaced by the neoclassical principles of valuation which determines the prices of corn, land with alternative uses, and other commodities. Connections between the classical theory of land rents and neoclassical marginal analysis did not go unnoticed. On his own contribution, Jevons [1879] said: “The general correctness of the views put forth in preceding chapters derives great probability from their close resemblence to the Theory of Rent, as it has been accepted by English writers for nearly a century,” after which he cites Anderson. This connection will be even stronger in the analysis, below. The landlord’s situation will be the model for competitive appropriation.
6Juxtaposition of valuation and appropriation is intertwined with contrasting views of the meaning and significance of perfect competition. In the neoclassical tradition, perfect competition is the servant of valuation. It is in its handmaiden form—known as the pricetaking assumption—that valuation can be separated from appropriation. The principle that valuation is driven by appropriation is not an explicitly articulated theme of neoclassical economics. We argue that it should, and show that it can, be.
7An oftencited critique of neoclassical theory is its failure to acknowledge the importance of economic institutions, particularly the role of property rights. Coase’s [1960] influential contribution on this subject should therefore be an occasion to rehabilitate the status of appropriation. His rejoinder to the portrayal of the inefficiency of markets highlighted the ability of individuals to tailor their rights to economic circumstances, in pointed contrast to the Pigovian pricetaking/markettaking description of valuation. Nevertheless, the setting he chose to portray the ideal functioning of property rights attenuates the importance of appropriation because his formulation minimizes the need for competition as a market enforcement mechanism which, along with the legal enforcement of property rights, underpins the benefits of appropriation.
8Further questions about the primacy of valuation versus appropriation extend to the function of the price system, itself. To Hayek [1945], the key scarcity underlying the function of the price system is divided knowledge. This observation was made in the context of a debate about the possibility of valuation under central planning. Hayek stressed the practical difficulties central planners would face in aggregating and coordinating the myriad changes in local conditions required to implement changes in prices. Those on the other side of the debate [Lange, 1939? Lerner, 1944] did not find his argument dispositive. Appeal to Walras’ tâtonnement as the neoclassical method of achieving equilibrium prices in a market economy—which relies on the restriction that individuals behave as pricetakers—supports the view that: “The logic of general competitive equilibrium is closely related to that of economic planning.” [Arrow, 1984: v.] Perhaps implicit in Hayek’s critique of the prospects for valuation under central planning were concerns about appropriation. But his analysis did not address those issues explicitly.
9Recognition of appropriation as the modus operandi of economic behavior (alternative to the pricetaking maximization of neoclassical valuation) is now pervasive. This is a direct result of the increased attention economics has more recently paid to incentives. Incentive compatibility, or “what you can get away with,” is synonymous with “what you can appropriate.”
10An unfortunate consequence of the neoclassical legacy separating valuation from appropriation is that a tacit fence now divides the older theme of valuation from the newer theme of incentives. As a specific instance, the efficiency and incentive compatibility properties found in Vickrey [1961], Clarke [1971], and Groves [1973] are widely regarded as the sui generis product of mechanism design. We view their provenance differently: the efficiency/incentive compatibility conclusions of mechanism design follow from the full appropriation properties of perfect competition. More generally, Stiglitz [2002] describes a change in paradigm that occurred when economics began to focus on problems attributable to asymmetric information. The transition is less abrupt when the newer class of problems is seen as originating from the underlying principle that valuation is driven by appropriation.
11Section 2 calls attention to the valuecentric focus of neoclassical theory as the decisive accomplishment of the marginal revolution. Section 3 describes how the pricetaking assumption employed in the model of general equilibrium leads to the conclusion that valuation can be separated from appropriation. Section 4 views Coase’s formulation of the consequences of welldefined property rights in a frictionless environment as following from an overemphasis on valuation and underemphasis on appropriation. Section 5 considers the divided knowledge rationale for the price system emphasized by Hayek as a point of departure for the appropriation role of the price system. Vickrey’s contribution to mechanism design is restated and extended to illustrate the importance of appropriation. Finally, some of the differences between the valuation and appropriation approaches to perfect competition are summarized in Section 6.
2 – The origin of value
12The marginal revolution focused on the principles of valuation. Describing Jevons’ contribution, Wicksteed [1894] wrote:
The law of value, resting as it does on the law of indifference and the phenomena of marginal utility, amounts to nothing in the world but the purchaser will not give more for it than he thinks it is worth to him. This was of course well known to everyone, and is instantly assumed in every economic treatise of whatsoever date? but nevertheless its exact mathematical expression in mathematical language has made an epoch, and is making a revolution in economic science. For it is one thing to be practically familiar with a principle and to assume it in simple cases as a matter of course, and it is another thing to grasp it so consciously and so firmly as never to lose hold of it or admit anything inconsistent with it, however remote from familiar experience and however complicated and abstract may be the regions of enquiry in which we need it as our clue.
14Menger’s version [1871] of the principles of valuation did not rely on infinitesimal calculus, but it was nonetheless a forceful and comprehensive statement overturning the cost of production?labor theory of value. It was, however, Walras [1874] who is credited with the capstone presentation of the principles of valuation. “I might have written Walrasian general equilibrium but there is only one general equilibrium whatever its name. …For there is but one system of the world and Newton was the one who found it. Similarly, there is but one grand concept of general equilibrium and it was Walras who had the insight (and luck) to find it.” [Samuelson 1962]
15There is some ambiguity in this claim. “A collectivist society of any type would necessarily confront the same economic problems, in the formal sense, as an individualistic one… For the principles of marginalism are the logical, mathematical, and hence universal, principles of economy…” [Knight 1936: 255–6]. Is Walrasian general equilibrium a description of price determination in a competitive market economy or is it a description of the general principles of valuation with universal applicability? It cannot be both? otherwise, the features that distinguish collectivist from noncollectivist economies are not accounted for.
3 – Separating valuation from appropriation: I
16The purpose of this section is to demonstrate that the role of the price system in modern general equilibrium allows valuation to be separated from appropriation. Because the appropriation role is not highlighted, the distinction between pricing and efficiency and competition with private property and efficiency is ambiguous.
17Following Debreu’s (1959) presentation, the “data” of the economy is
19where represents the tastes of consumer i, Y_{j} is the set of inputoutput combinations available to producer j, and ? is the aggregate resource vector available to the economy as a whole. The data is defined with respect to a common space, R^{l}. The consumption of consumer i, denoted x_{i}, will be assumed to be nonnegative. Hence, is a (binary) preference relation on R^{l}_{+}×R^{l}_{+}, where means that x_{i} is at least as good as x^{?}_{i} to i. [3]
20A state of E is an [( x_{i}),( y_{j})]. Define x = ?x_{i}, y = ?y_{i}, and
22A state is attainable for E if x_{i} ?R^{l}_{+}, y_{j} ?Y_{j}, and z = 0. Another way of writing z = 0 is (x – w) ? ?_{j}Y_{j}, i.e., the difference between aggregate consumption and aggregate endowment belongs to the set of aggregate production possibilities.
23The data of E is the basis for (Pareto) optimality. A state [( x_{i} ),( y_{j} )] is optimal if it is attainable in E and for any other state [( x^{?}_{i} ),( y^{?}_{j} )] such that and at least one ordering is strict, [( x^{?}_{i} ),( y^{?}_{j} )] is not attainable.
24The crucial hypothesis about individual behavior, referred to as the servant of valuation, is pricetaking maximization.
25The behavior of each consumer is described with the aid of his budget set
27where p · x_{i} is the market value of purchases, the inner product of p and x_{i}. An important feature of the formalities is that the source of is wealth, ?_{i}, is left undetermined. Maximization by the consumer over this set i’s with respect to
28. The consumer’s behavior at prices p and wealth ?_{i} is given by his demand _{?}correspondence
30For the producer, p · y_{j} = ?^{l}_{h = 1} p_{h} y_{h} gives the evaluation of the production vector y_{j} at the prices p. Given the sign convention on inputs and outputs, this is the total revenue from sales minus the total cost of purchases, or profit. The goal of the (pricetaking) producer facing prices p is to find a y_{j} ?Y_{j} to achieve
32The behavior of the producer is summarized by the supply correspondence
34The data in E defines what is possible under any method of economic organization. A distinction is made between E and the description of the institutional arrangement defining the private ownership economy ?. The transition form E to ? adds two elements to determine a consumer’s wealth given prices. They are
 ?_{i}, the consumer’s initial endowment
 ?_{ij} ? 0, the consumer’s share of the profit of firm j.
36subject to the restrictions that ??_{i} = ? and ?_{i}?_{ij} = 1 for all firms j.
37In a private ownership economy, the wealth function of a consumer is
39The essential role of profit shares is to ensure that all the wealth in a private ownership economy is transferred to consumers, where
41The preceding is preparation for the following:
42Definition: An equilibrium of ? (synonymously, a Walrasian equilibrium or a pricetaking equilibrium) is a [( x^{*}_{i} ),( y^{*}_{j} ), p^{*} ] such that
 x^{*}_{i} ? ?_{i} ( p^{*}, ?_{i} ( p^{*} )), (each consumer’s action is utilitymaximizing at the prices p^{*})
 y^{*}_{j} ? h_{j} ( p^{*} ) (each producers action is profitmaximizing at p^{*} )
 x^{*} – y^{*} = ? (market clear).
43First Welfare Theorem. If [(x^{*}_{i}),(y^{*}_{j} ), p^{*} ] is an equilibrium for , then [(x^{*}_{i}),(y^{*}_{j} )] is optimal. [4]
3.1 – A gap between contribution and reward
44The ownership of resources, ?_{i}, and ownership shares of producers profits, ?_{ij}, are two apparently similar pieces of data in the description of the private ownership economy. In particular, with the exception of the labor services component of ?_{i}, the nonhuman components of ?_{i}, such as acres of land, would appear to be on a par with shares of firms in the sense that there is no justification within the model for each of these ownership claims. For example, both could be rationalized as different forms of inherited wealth. Nevertheless, the ways the consumer appropriates the value of these two claims are very different. Once ownership is given and i controls ?_{i}, he can exclude others. Thus, without i’s participation, total resources would be ?_{k?i}?_{k}. When market prices are p, the market value p · ?_{i} is a measure of the amount others would be willing to pay for i to part with ?_{i}. [5]
45No comparable claim can be made for ?_{j}?_{ij}?_{j}(p) because i’s share of the profits of any producer cannot be identified with any contribution by i. For example, it cannot be said that without consumer i’s participation, Y_{j} would be ( ?_{k?i}?_{ij} ) Y_{j}. Appeal to the empirical fact that dividends are paid to equity holders is not a valid interpretation of ?_{j}?_{ij}?_{j}(p). Equities in the actual economy are issued for various reasons including financing constraints and risk aversion? hence equity holders are supplying scare resources. In this model, producers do not face financial constraints and there need not be any risk.
46Thus a remarkable property of the Walrasian general equilibrium model is that it permits a gap between the value of a consumer’s expenditures and the contribution he makes to others. This is not to say that a consumer violates his wealth constraint. The wealth constraint on expenditure is p · x_{i} = ?_{i} ( p ). A gap occurs because the wealth constraint ?_{i}(p) = p · ?_{i} + ?j?_{ij}?_{j}(p) does not require that there be any justification of _{?j}?_{ij}?_{j}(p).
3.2 – Widening the gap
47The exogenous profit shares in the private ownership economy ? are related to the data of the economy E in a way that is reminiscent of what Lange [1938], Lerner [1944], and others called market socialism. To illustrate the idea, begin by making a distinction between resources that are alienable and those that are not. The inalienable resources are the part of ?_{i} such as labor services that cannot be dissociated from individual i. Alienable resources such as land are those that might, in an alternative specification of ?, have been owned by someone else. The distinction is roughly that between human and nonhuman resources. Let ?_{i} = ?_{i} + r_{i}, where is the inalienable and r_{i} is the alienable component of consumer i’s resources.
48In the actual economy, consumers finance the purchase of capital goods owned by producers. In the model of a private ownership economy, realworld financial intermediation is ignored and consumers are regarded as the owners of the economy’s capital stock that they rent to producers. To describe market socialism, transfer the alienable resources from the consumer to the producer sector of the model as follows: suppose that r_{j} are the resources available to producer j, where ?jr_{j} = ?_{i}r_{i} represents the transfer of alienable resources to producers. Producer j’s objective is now
50The value of profits at prices p changes because j does not have to pay for the resources r_{j}, but j’s profit maximizing choice remains the same, i.e.,
52In particular, if resources that j holds can be more profitably sold to others rather than be put to use by j, they will.
53Taking the next step, define the market socialist economy
55where ?_{i} ? 0, ?_{i}?_{i} = 1 replaces profit shares of producers. At prices p, total alienable wealth in ?^{M} is given by
57A consumer’s wealth function for ?^{M} is , the sum of his inalienable wealth and his share of the economy’s alienable wealth. Notice total wealth in ? and ?^{M} is the same since
59Proceeding along the same lines, the method of assigning profit shares in E can be generalized to all wealth, as in Gale and MasColell [1975], by constructing the economy ?^{?} defined by the individual wealth functions ?_{i}(p) = ?_{i}?(p), where ?i > 0 and ?_{i}?_{i} = 1 so that, again, ?_{i}?_{i}(p) = ? (p).
60Whatever the wealth function, each producer’s behavior is the same; and each consumer’s behavior is defined by his preferences and his budget set, which in turn depends only on prices and the wealth function ?_{i}(p). Hence the definition of equilibrium for ? could easily be replaced by equilibrium for ?^{M} or ?^{?}. Further, the mathematical arguments needed to establish existence of equilibrium are the same.
61These observations above are succinctly summarized by Debreu in the following.
62Definition: The state [(x^{*}_{i}), (y^{*}_{j})] is an equilibrium relative to a price system p for the economy E (not ?) if
 x^{*}_{i} ? ?_{i}(p, p x^{*}_{i})
 y^{*}_{j} ? ?_{j}(p)
 x^{*} – y^{*} = ?.
63The concept of an equilibrium relative to a price system allows for an immediate generalization of the First Welfare Theorem.
64Generalized First Welfare Theorem. An equilibrium relative to a price system is an optimum.
65The converse to this result is what is usually meant by the Second Theorem of Welfare Economics. It is given by the following.
66Second Welfare Theorem. For any optimal allocation [(x^{*}_{i}),(y^{*}_{j})] there is a p such that [(x^{*}_{i}),(y^{*}_{j} ), p] is an equilibrium relative to p. [6]
67I.e., by redistributing total wealth, any distribution of welfare can be supported as an equilibrium relative to some price system.
68A possible objection to the above is that in moving to ?^{M} or ?^{?} from ?, the meaning of private ownership is pushed beyond its intended interpretation as a model of private enterprise. To this, Debreu provides an important rejoinder: “Allegiance to rigor dictates the axiomatic form of the analysis where the theory, in the strict sense, is logically entirely disconnected from its interpretations.” The two welfare theorems demonstrate a comprehensive synthesis, which Debreu summarizes by saying “these two essential theorems of the theory of value thus explain the role of prices in an economy.” In accord with his injunction, the axioms of the Walrasian general equilibrium model provide theorems that explain the relation between pricing and efficiency, not appropriation and efficiency. By allowing valuation to be separated from appropriation, the axioms are fundamentally incomplete.
4 – Separating valuation from appropriation: II
69The previous section describes how an overemphasis on valuation and consequent underemphasis on appropriation leads to an incomplete formulation of the importance of both property rights and competition as sources of efficiency in a market economy. This section describes how the same bias can lead to an overestimate of the importance of property rights when insufficient attention is paid to the appropriation role of competition. In keeping with the presentation style of its subject, the previous section followed a formal model. Similarly, here the presentation will be informal.
4.1 – Transactions costs
70Coase [1937] reasoned that except for realworld frictions he called “transactions costs,” there were no limits to the valuation role of the price system. Without those costs, the price system could coordinate the economic activity normally performed within firms. Since the need for firms is selfevident, this was a counterfactual thought experiment to explain the conscious control and direction of resources within a firm as the “supercession of the price system” that must be due to the existence of transactions costs. This view of the price system is seconded by Arrow [1974]: “…[O]rganizations are a means of achieving the benefits of collective action in situations in which the price system fails.”
71It is generally agreed that transactions costs refer to the costs of running the price system. Among the most of significant are the costs of operating the means of payment and finance. Nevertheless, the theory of price determination is framed in an environment ignoring such costs. But the expenditure of resources on money and banking need not be regarded as evidence of the supercession of the price system. Rather, these costs can be seen as enabling the price system to function. A related perspective may be adopted with respect to the existence of firms.
72To illustrate, the question can be posed: How does an individual appropriate the gains from technical knowledge when there are advantages to team production? In certain situations, individuals do sell their knowledge directly to others. More generally, an agreement could be drawn up in which that knowledge—to be applied in all agreedupon conceivable contingencies as they unfold—can be communicated via transactions with others using the notion of complete contracts. When that is not practicable, the appropriation of knowledge requires the possessor to “boss” the other members of the team, which includes the possibility that the others may also be wellsuited to appropriate the gains from their being capable of following their boss. Like the institutions of money and banking, firms can be viewed as contributing to the appropriation function of the price system.
73Our objective is not to question why firms exist. The issue is raised because the zero transactions cost condition under which Coase saw the price system as obviating their existence figures prominently in his discussion of property rights. Later contributors found Coase’s appeal to transactions costs to be a seminal observation allowing economics to move beyond the boundaries imposed by its failure to take them into account. We draw a different lesson. Rather than viewing neoclassical theory as limited by its failure to recognize transactions costs, we see the modifications required to address contemporary concerns as evidence of a gap that was always there—the need to link valuation with appropriation. The transactions costs and appropriation perspectives overlap, but they are not the same.
74Transactions costs can be divided into those that are (i) technological, such as the unavoidable efforts required to write a contract (possibly involving many individuals), and those that are (ii) behavioral, such as efforts individuals employ to improve the contract terms in their favor. Incorporating technological transactions costs may, or may not, be relevant to the problem at issue. For example, the rationalization of firms, given above, focuses on a particular instance of (i). From the appropriation perspective, however, (ii) cannot be eliminated. It is essential to the theme that valuation is driven by appropriation, which leads to an abiding concern for conditions in which appropriation is compatible with efficiency.
4.2 – Coase’s critique and the Coase Theorem
75The context of Coase’s contribution [1960] is a critique of the Pigovian characterization of “market failure”. That a market outcome results in an inefficient allocation of resources necessarily means that there are unexploited opportunities for gain. Why should government intervention? regulation be in a better position to capture those gains than private enterprise? In contrast to the Pigovian presumption that participants take prices and markets as given, Coase adopted a more entrepreneurial perspective. To realize its full potential, he emphasized that participants must be able to exchange whatever the relevant sources of value are—property rights should be welldefined and complete. In addition, opportunities to make such exchanges should be unrestricted— transactions costs should be zero. Under these conditions, Coase argued that markets will not fail. This conclusion is called the Coase Theorem.
76A remarkable feature is that competition is not part of the hypothesis, although the first restatement, by Stigler [1966], did contain perfect competition as requirement for the conclusion. But later commentary by Coase [1988] objected to the need for it.
Of course, the competition of substitutes normally very much narrows the range within which the agreed price must fall, but it must be very rare indeed for both the buyer and the seller to be indifferent as to whether a transactions goes through. …We do not usually seem to let the problem of the division of the gain stand in the way of making an agreement. Nor is this surprising. Those who find it impossible to conclude agreements will find that they neither buy nor sell and consequently will usually have no income. Traits which lead to such an outcome have little survival value, and we may assume (certainly I do) that normally human beings do not possess them and are willing to “split the difference.” (Italics added.)
78In support of his position, Coase appealed to Edgeworth’s [1881] contracting and recontracting among groups of individuals as a costless method of bargaining. Edgeworth’s procedure is now known as the core, a solution concept from the cooperative theory of games. To paraphrase Coase’s interpretation, every solution in the core is necessarily efficient when there are no impediments to bargaining and property rights are welldefined.
79In cooperative game theory, property rights place a lower bound on what a group of individuals can guarantee themselves. However, since gains from trade implies the existence of a surplus, there must also be an added willingness to refrain from actions that, while reducing the overall surplus, lead to a gain for the ones undertaking those actions. For example, suppose a situation in which two individuals can each achieve a utility of from utilizing what they own, but through exchange they can attain a maximum of , which they agree to split as , with ? ? (0,1) and . This presumes both will eschew socially wasteful expenditures that might privately improve their own share. Alternatively, suppose property rights were poorly defined so that each could secure only on their own. The two games would differ with respect to their “threat points.” But the cooperation that is presumed to follow from the absence of transactions costs allows the participants in both games to achieve (possibly different) splits of the same thereby diminishing the importance of property rights as a source of efficiency. A similar conclusion is drawn by Cheung [1982] who argues that with zero transactions costs, but without property rights, individuals can agree to establish them. I.e., starting from u^{0}, they could agree to u, and then to a split of
80An important issue, emphasized below, is that knowledge of one’s own characteristics, e.g., tastes and endowments, is itself private property. Consequently, this information is not costlessly available to others. In the example, above, while each individual may know the sources of his , common knowledge of the determinants of is problematic. The exercise of one’s property rights includes disguising what the benefits and costs might be, an obvious dimension of conflict standing in the way of efficiency.
81Coase’s recipe for achieving efficiency is reliance on property rights, along with removal of impediments to their exchange. In contrast, an appropriation view of the price system implies that imperfections in competition have consequences similar to poorly defined property rights. In each case, appropriation of the surplus becomes a common property resource. It is through the discipline of competition imposed by others that individual property rights can be transmuted into efficient appropriation. Evidently, the appropriation theme is furthered by the language and tools of noncooperative games. [7]
5 – Appropriation role of the price system
82The purpose of this Section is to highlight the connections among information, appropriation and competition.
5.1 – Different kinds of asymmetric information
5.1.1 – Asymmetric information intrinsic to property rights
83Hayek [1945] emphasized
the knowledge of particular circumstances of time and place. It is with respect to this that practically every individual has some advantage over all others because he possesses unique information of which beneficial use might be made, but of which use can be made only if the decisions depending on it are left to him or are made with his active cooperation … The economic problem of society is thus not merely a problem of how to allocate ‘given’ resources—if given is taken to be given to a single mind, which deliberately solves the problem set by these ‘data’. It is rather a problem of how to secure the best use of resources known to any of the members of the society, for ends whose relative importance only those individuals know.
85Local conditions of time and place is divided knowledge, a form of “property rights” which must, in effect, be exchanged. With respect to the model of Section 3, an individual’s tastes, resources, and productive capacities are sources of local knowledge. Hayek pointed to the computational complexity of ever changing local conditions as both the flaw in the central planning approach to valuation and the rationale for a market price system allowing individuals to employ their knowledge in response to prices that succinctly summarize information about relative scarcities.
86Read today, what is striking about Hayek’s essay is the absence of explicit emphasis on incentives. Even if the words are not there, perhaps the incentive issue should be ‘understood.’ To indicate that Hayek’s argument does require bolstering, consider an extract from a more recent commentary by Friedman [1981]. Speaking of the functions that prices serve in a market economy, the author says:
Fundamentally prices serve three functions in such a society. First, they transmit information. …The crucial importance of this function tended to be neglected until Friedrich Hayek published his great article…This function is essential, however, for enabling economic activity to be coordinated. …. A second function that prices perform is to provide an incentive for people to adopt the least costly methods of production and to use available resources for the most highly valued uses. They perform that function because of their third function, which is to determine who gets what and how much—the distribution of income.
88The latter two functions may be readily joined and the author’s position can be summarized as: besides the decentralization role, prices also serve an appropriation function. We go a step further to subordinate the former to the latter. [8]
89We adopt the term privacy to denote the divided knowledge, or asymmetric information, each individual has about himself and his resources. Privacy is an essential feature of property rights.
5.1.2 – Asymmetric information that subverts property rights
90Property rights prevent involuntary exchange, e.g., theft. The phenomenon of pollution is also an instance of an uninvited change in utility. There is another kind of involuntary exchange that occurs in the course of a voluntary transaction that we call delivery problems because there can be a discrepancy between what is agreed upon by one side and what is actually delivered by the other. Adverse selection and moral hazard are wellknown examples. The existence of such phenomena is attributed to asymmetric information. It will be important to distinguish between privacy and delivery problems.
91To illustrate a possible confusion, adverse selection is commonly referred to as “hidden characteristics.” But that term also applies to privacy, i.e., the knowledge each individual has about his own characteristics. The following are formally similar: (1) a buyer of a new car would like to know the costs of his supplier? and, (2) a buyer of a used car would like to know whether he is getting a ‘lemon’ or a ‘peach.’ In either case, the information could be used to improve the buyer’s utility: in (1), it could reduce the cost he pays; and, in (2), it would allow the buyer to know what he is getting. Evidently, (2) is a delivery problem and (1) is not. The difference between what is promised and what is delivered can be regarded as a property rights deficiency, but one that can be avoided by refusing to trade. Similarly, the moral hazard problem of shirking between principal and agent recognizes the possible gap between promise and delivery that results from initiating the relationship.
92It is obviously a more ambitious task to pursue the appropriation theme when property rights can be subverted by delivery problems. Before that is undertaken, the issue of appropriation should be examined in the simpler setting where property rights are not deficient, i.e., there is privacy but no delivery problems. This is the subject of the remainder of the paper.
5.2 – Appropriation in Mechanism Design with Privacy
93The starting point for Vickrey [1961]:
In his Economics of Control, A. P. Lerner threw out an interesting suggestion that where markets are imperfectly competitive, a state agency, through counterspeculation, might be able to create the conditions whereby the marginal conditions for efficient resource allocation could be maintained. Unfortunately, it was not made clear just how this counterspeculation was to be carried out, and to many this term denotes just one more of the empty boxes that rattle around in the economist’s cupboard of ideas. And there appears to have been, in the years since Economics of Control first appeared, no attempt to examine critically just what this intriguingly labeled box might contain.
95Vickrey formulated the problem as: how to give participants the incentive to reveal their willingness to buy and sell so that an efficient allocation of purchases and sales could be determined. In what is now called a direct mechanism, buyers and sellers announce their entire schedules of willingness to buy and sell to a Public Marketing Agency, Vickrey’s term for the mechanism designer or central planner. [9]
96Vickrey’s solution begins with the readily established result that a single seller (buyer) would have no reason to misrepresent if he were rewarded as a perfectly discriminating monopolist (monopsonist). Similarly, in a market with several buyers and sellers, no one would gain from misrepresentation when each extracts all the possible gains subject to the “property rights” of others. Each seller would fully appropriate the gains from his technology, as the landlord fully appropriates the fertility of his land. An analogous condition holds for each buyer when he pays the social opportunity cost of the consumption foregone by others, defined as the payment leaving them no better, nor no worse, off than they would be without the buyers purchase.
97Vickrey demonstrated his conclusion in a model with a divisible commodity where buyers exhibit diminishing marginal utility and sellers exhibit increasing marginal costs. Pricetaking behavior is consistent with the existence of a price at which demand is equal to supply. But his argument established that providing the incentives to reveal information goes beyond the neoclassical marginal conditions that suffice for Walrasian equilibrium. He made the point that:
98§1.The existence of Walrasian equilibrium is not sufficient for incentive compatibility.
99To ensure that individuals would reveal their willingness to buy or sell, buyers should pay less for the inframarginal units than the Walrasian price they pay for the marginal unit, analogously, sellers should receive more for their inframarginal sales than the Walrasian price for the marginal unit they supply. Since this leads to the market running a deficit, Vickrey regarded such a scheme as unsatisfactory.
100While Lerner recognized the need to deal with the monopoly power arising from small numbers, Vickrey’s finding that it could only be resolved at a cost confirmed the traditional conclusion that monopoly was problematic. But he did not interpret his incentive approach as an invitation to challenge the viability of the LangeLerner program. He pursued a practical application, instead.
5.2.1 – Second?price Auction and Quasilinear Utility
101To obtain a solution to the incentive problem, Vickrey chose a simpler setting: only the buyer side of the market is addressed and there is only one indivisible object for sale. Therefore, concerns about valuation of inframarginal units are precluded. Assuming the seller’s reservation value is zero, marginal conditions for the efficient allocation of a divisible commodity are replaced by the requirement that the object should be given to the buyer with the highest valuation. It is readily seen that Walrasian equilibria exist in such a model: the prices at which demand equals supply are anywhere between the value of the highest and the secondhighest buyer’s valuations. These observations imply the following:
102§2. The secondprice auction for a single object mimics the marginal conditions for a divisible commodity in Walrasian equilibrium. The cost of the last infinitesimal unit purchased of a divisible commodity is the same as other buyers’ willingness to purchase the commodity. Others are indifferent between whether or not the buyer purchases the last infinitesimal unit. The same condition holds for the second price auction of a single object.
103§3. The secondprice auction for a single object mimics perfect price discrimination for the winning buyer. A single object requires only a single payment. Nevertheless, replace the means of perfect price discrimination— different prices with respect to quantities and individuals—with its end—a perfect price discriminator captures all the surplus he contributes to others. Like the perfectly discriminating monopsonist, the winning buyer appropriates all the surplus that his valuation contributes to the economy’s gains from trade.
104In the general equilibrium model in Section 3, preferences can be represented by ordinal utility functions. However, starting with Vickrey, many of the contributions in auction theory and mechanism design are formulated assuming the more restrictive hypothesis of quasilinear utility. For example, a buyer’s utility is u(z,m) = v(z) + m, where z is the “nonmoney” commodity and m is the “money” commodity, which may be positive or negative; and, a seller’s production possibilities Y = {(z,m) : m = c (z)} are similarly defined, where c(z) is the quantity of the money commodity required as input to produce the output z. The simplifying characterization of efficiency that follows from quasilinear utility is:
105§4. With quasilinearity, an allocation is efficient if and only if it maximizes total utility among all feasible allocations. Quasilinearity “exaggerates” the marginalist lesson that utility is the basis of cost, value, and welfare because utility can be treated as if it were interpersonally fungible, via transfers of the commonly valued money commodity.
106Exploiting quasilinearity,
107Definition: The marginal product of an individual is the difference between the maximum total gains from trade as a result of the individual’s characteristics being what they are compared to the (maximum) gains if the individual were not present.
108Implicit in the definition is that individuals own their characteristics, e.g., preferences and/or production possibilities. Therefore, in obtaining one’s marginal product, an individual cannot reduce the welfare of others below the upper bound of what they could have achieved on their own.
109The definition of an individual’s marginal product yields another way to view monopoly:
111The only way a monopolist can get his marginal product is if he can achieve the ultimate status of a perfect price discriminator? otherwise, the existence of consumers’ surplus implies that the monopolist is receiving less. When there is more than one seller, the definition of each one as a perfect price discriminator, given the existence of the others, means that each receives its marginal product. Analogous descriptions apply to monopsony in a market with more than one buyer.
112A restatement of Vickrey’s finding for the divisible commodity model is:
113§5. If an individual can appropriate his marginal product he has the incentive to reveal truthfully.
114“Proof:” An individual’s marginal product is the maximum addition that his characteristics contribute to the total gains beyond what they would be otherwise. To appropriate one’s marginal product means the individual receives all of the extra gains. Therefore, the actual added gains from whatever (feasible) allocation is chosen—based on the individual’s reported characteristics—cannot be greater than one’s marginal product. Hence, it pays to tell the truth.
115Applying these definitions to a singleobject auction in which the seller, s, has zero reservation value and the buyer’s reservation values are listed in descending order as v_{1} ? v_{2} ? … ? v_{b} ? … ? v_{n} ? 0,
117When the buyer with valuation v_{1} pays the amount v_{2} he receives his marginal product, as do the other buyers who pay nothing. The marginal product of the seller equals the total gains since without his object the gains are zero.
118Vickrey’s point of departure is perfect competition.
120Walrasian equilibrium is incentive compatible for an individual who is a perfect competitor since there is no prospect of gain from price manipulation. In Vickrey’s smallnumbers example of a market for a divisible commodity where buyers exhibit diminishing marginal utility and sellers increasing marginal costs, the absence of any one individual would change the Walrasian price. Therefore, everyone is an imperfect competitor. Nevertheless, in certain situations, even with small numbers an individual can be a perfect competitor. In the singleobject auction, when the price equals the secondhighest valuation, that would also be a Walrasian price without the highestvalued buyer. Therefore,
121§6. The secondprice auction for a single object makes the winning buyer a perfect competitor. Rewards are distributed as,
123The economy’s reservation price for the object is the lowest price at which demand would be zero, which is evidently v_{1}. If the price of the object is v_{1}, it is the same as the price without the seller. Therefore,
124§7. The firstprice auction for a single object makes the seller a perfect competitor.
126The sum of the marginal products of the buyers and the seller is
128This recalls the inequality Vickrey found for the divisible commodity model where, if all participants were perfect price discriminators, i.e., were to receive their marginal products, payments by buyers would not cover payments to sellers. That qualification is suppressed in the auction version of the market for a singleobject because the seller does not behave strategically or, equivalently, his reservation value is commonly known to be zero.
129The seller’s strategic possibilities can be activated by allowing his costs to be c ? 0 (known only to the seller). Then, assuming (v_{1} – c) ? 0 so that total gains are positive, the above inequality continues to hold for the singleobject market:
131Note that if the seller is promised the payment (v_{1} – c) when it is positive and zero otherwise, he is effectively rewarded as a perfectly discriminating monopolist since he receives his marginal product, and would therefore have the incentive to reveal c.
132Extending the definition of a perfect competitor, above:
133Definition: The singleobject market is perfectly competitive when the price would be the same with or without any participant.
134§8. Assuming v_{1} – c > 0, the single object market is perfectly competitive if and only if v_{1} = v_{2}. In that case, all participants can feasibly receive their marginal products because the sum of the rewards based on individuals appropriating their marginal products can be “financed” by the total gains:
136The seller of the object receives his marginal product, appropriating all the gains through competition on the buyers’ side reducing their gains to zero.
137“Monopoly” denotes a single seller and “monopoly power” refers to the seller’s ability to set the price rather than take it as given. When there is a single object, the seller is necessarily a monopolist. Nevertheless, because perfect competition among buyers forces them to pay v_{1}, the seller’s reward is a maximum when the singleobject market is perfectly competitive market.
138Walrasian equilibrium always exists in a singleobject market, but not necessarily perfect competition. Let V^{n} be the set of those ( v_{1},…, v_{n} ) arranged in descending order with the stipulation that v_{b} ? [0,1], b = 1,…,n; and let V^{n}_{e} (PC) be the subset of V^{n} with the added restriction that v_{1} – v_{2} ? ?, the set of single object markets that are within e of being perfectly competitive. For any n and any ? < 1, (1, 0, 0,…, 0) ? V^{n}V^{n}_{?}(PC) illustrates that even with large numbers of buyers the market may be far from perfectly competitive. But such examples can be regarded as exceptional. For example, if each buyer’s valuation is independently drawn from the same probability distribution, as the number of draws increases the vanishing difference between highest and the secondhighest order statistic implies:
139§9. Existence of perfect competition in a single object market. For any ? > 0,
5.2.2 – Competition in the assignment model
141The previous section illustrated the equivalence among a perfect competitor, receiving one’s marginal product, and fully appropriating the gains one contributes in the elementary setting of a market for a single object. In this section, the setting is extended to the assignment model to show that the same equivalence applies.
142As above, each buyer wishes to purchase at most one object and a seller has only one object for sale. The difference is: (a) there is more than one seller and (b) each seller may supply a commodity unlike any other. As a variant of Anderson’s example, the buyers can be regarded as farmers, each able to cultivate at most one parcel of land, and the sellers are landowners, each owning one parcel. The value v_{bs} ? 0 denotes the number of bushels of corn that farmer b can obtain by using the parcel owned by s, which allows farmers to have different abilities in farming the same parcel. (In a utility–based interpretation, v_{bs} is the utility b obtains from the object initially owned bys.) Denote by c_{s} ? 0, the reservation value, or cost, to landlord s of supplying his parcel. [As a further simplification, sellers’ reservation values can be assumed to be zero, i.e., land has no alternative uses to its owners.]
143By construction, there are as many parcels as there are sellers. The net gains from assigning b to s is
145Inequality between the numbers of buyers and sellers can be eliminated as follows: if there are k fewer sellers than buyers, the set of sellers, denoted by S, can be augmented to where, for all b, v_{bs} = 0 for s ? S?S; if there are k fewer buyers, the set of buyers, denoted by B can be augmented to , where v_{bs} = 0 for all b ? B?B and all s ? S. Assume throughout the following that the number of buyers and sellers is n. The problem is: how to assign farmers to parcels, which is an order of magnitude more difficult than the special case of a singleobject auction.
146Denote the data of the assignment model by A = (V,c), where V = ( v_{bs} ) and c = ( c_{1},…, c_{s},…, c_{n} ). This is a special case of a private ownership economy ? in Section 3. Walrasian equilibrium can be used to describe the valuation problem associated with an optimal assignment. Since sellers are only concerned with the amount they receive, not which buyer they supply, there need only be one price per parcel. Prices are denoted by p = ( p_{1},…, p_{n} ). A buyer may choose , where e_{s} is the vector whose s^{th} component is 1 and zeroes otherwise. Seller s can choose to supply or not to supply, x_{s} ? {e_{s}, 0}. At p, pricetaking maximization yields b and s the net gains
148Following the notation in Section 3, denote by the element(s) in achieving v^{*}_{s}(p) and h_{s}(p) the element(s) in {e_{s}, 0} achieving v_{s}^{*}( p ). Regarding A as a special case of a private ownership economy in Section 3,
149Definition: [Standard Description] {(x_{b}),(x_{s}), p} is a Walrasian equilibrium of A if
 (each buyer’s action is utilitymaximizing at the prices p)
 (each seller’s action is profitmaximizing at p)
 (markets clear).
151This inequality allows for a more succinct description of equilibrium: Definition: [Revised Description] [?, p] is a Walrasian equilibrium for A if
153Each b gains at least as much from the assigned parcel ?(b) = s as from any other, subject to the restriction that he is not forced to farm and the seller receives p_{s} for his parcel subject to the restriction that he is not forced to sell. The “marketclearing” condition is contained in the definition of p and the fact that if ?(b) = s, then b pays p_{s} to s if g_{bs} > 0. If a dummy buyer is assigned to parcel s, pricetaking maximization means he produces nothing and does not pay; and similarly for a buyer assigned to dummy parcel ( = seller).
154The assignment ? yields total gains (bushels of corn) in A = (V,c),
156The maximum total gains in is A is
158Therefore, an assignment p is an optimum if G_{A}(?) = G(A).
159It follows from the Revised Description of Walrasian Equilibrium that:
160§10. If [?;p] is a Walrasian equilibrium for A,
162Moreover, for every optimal ?, there is a p such that [?;p] is a Walrasian equilibrium.
163Let A_{–b} = ( V_{–b}, c ) denote the model without b. This is effectively accomplished by replacing row b in V with zeroes, denoted V_{–b}. Similarly, A_{–s} = ( V_{–s}, c ) removes s, which is accomplished by replacing column s in V by zeroes, V_{–s}. With this convention, the set of ?’s in A are also available A_{–i},i = b, s, although the total gains from ? may differ. Thus,
164Marginal products are
166Comparison between the gain an individual receives in Walrasian equilibrium and his marginal product is similar to the singleobject auction:
167§11. If [?;p] is a Walrasian equilibrium for A,
169Applying §10. and §11.,
170§12. In A,
172Definition: A exhibits full appropriation if
174In the divisible commodity market with a small number participants, the presence or absence of a single participant changes Walrasian prices. In the single object market, however, the range of Walrasian prices is such that the highest price yields the seller his marginal product and the lowest allows the winning buyer to appropriate his marginal product. A related result holds for the assignment model.
175§13. If ? is an optimal assignment, there exist prices p^{b} = ( ^{b}p_{1},… p^{b}_{s},…, p^{b}_{n} ) such that [?;p^{b} ] is a Walrasian equilibrium for A at which buyers receive their marginal products, i.e., for all b,
177and there exist prices p^{s} = ( p^{s}_{1},… p^{s}_{s},…, p ) such that [?;p^{s} ] is a Walrasian equilibrium for A at which sellers receive their marginal products, i.e., for all s, [10]
179It follows from 10. and 13. that if [?, p] is a Walrasian equilibrium, ?(b) = s, and trade occurs, then
181The amount (p^{s}_{s} – c_{s}) is the best the seller can achieve. If the price of s were higher, the fact that ( p_{s}^{s} – c_{s} ) = MP_{s} ( A ) means that the buyer could find another seller with whom he could achieve a net gain of ( v_{bs} – p^{s}_{s} ); moreover, if other buyers and sellers were displaced, they too could find equally satisfactory outcomes. Similarly, the amount ( v_{bs} – p^{b}_{s} ) is the maximum gain to the buyer: if the price were any lower, the seller could find another buyer, …etc…. I.e., p^{b}_{s}is the outside option for the seller and p^{s}_{s}is the outside option for the buyer. Within the bounds p^{s}_{s} > p^{b}_{s}, the division of the surplus between buyer and seller is indeterminate. Unlike Coase, we view those gains as “common property,” with the potentially wasteful efforts by each to appropriate more of the gains that the term suggests.
182In preparation for a resolution of that problem:
183Definition: A is perfectly competitive if there exists a p such that [?, p] is a Walrasian equilibrium for A and for each i = b,s, there exists ?_{–i} such that [?_{–i}; p ] is a Walrasian equilibrium for A_{–i}.
184When A is perfectly competitive, there exists [?, p] such that if ?(b) = s and v_{bs} – c_{s} > 0, then
186I.e., perfect competition implies that efforts to appropriate more of the gains are pointless because if the price were different, the buyer or the seller would have a perfectly substitutable alternative.
187The relation between perfect competition and full appropriation in the assignment model is:
188§14. A is perfectly competitive if and only if it exhibits full appropriation.
5.3 – Existence of perfectly competitive equilibria
189As a historical conceit, imagine that Jevons had used the assignment model to establish the utility origins of the theory of value. Instead of deriving his Law of One Price from the physical identity of commodities, the assignment model provides a utility basis for such a conclusion. While every seller’s commodity is physically distinct, if each buyer regards them as perfect substitutes, i.e., for each buyer ^{b},
191then p_{s} will be the same for all s. This restriction yields an extension of 9 for a market with many buyers and sellers of homogeneous objects, called a doubleauction.
192§15.(Perfect Competition in a Double Auction). If v_{bs} = v_{b}, ?s ? S, and
193(I) there are large numbers of buyers and sellers,
194(II) valuations of buyers are such that for any v_{b} there is v_{b?} such that v_{b} – v_{b?} is small and/or for any c_{s} there is a c_{s?} such that c_{s} – c_{s?} is small, then the assignment model will be nearly perfectly competitive, i.e., the price of commodity will be nearly the same in A as it is in A_{–i}. [11]
195Similar conclusions can be established if the preferences of buyers recognize the differences only among a fixed, finite set of objects. For example, suppose the sellers S = {1,…,n} can always be partitioned into , with m < n and m independent of n. Assume that as n increases, for every b,
197Again, substitution possibilities among buyers and sellers are likely to increase as the number of buyers and/or sellers increases, leading to perfect competition. This is predicated on the assumption that the utility heterogeneity buyers display with respect to sellers products increases sufficiently slowly so that substitution possibilities eventually overwhelm complementarities as the size of the economy increases. Without this qualification, the link between competition and large numbers need not hold. To illustrate the contrary in the assignment model:
199For example, buyers and sellers may have locations, along with prohibitive transport costs, such that the only possible trading partner is the one person on the other side of the market where the individual is located. If, however, as the number of individuals increases, population density is also increasing so that the distance to others is effectively shrinking, then unless buyers are extraordinarily discriminating as well as heterogeneous with respect to sellers, substitution possibilities would predominate, as in Gretsky et al.
5.4 – Divisible Commodities
200The existence of complementarities are limited in the assignment model because all the gains an individual receives in the economy as a whole can be achieved by exchanging with one other person. A similar phenomenon occurs with divisible commodities when increases in the size of the population take place through replication. The private ownership economy ? of Section 3 consisting of m consumers and producers can be extended to a model ?^{r} with rm participants consisting of r copies of the types in ?.
201A necessary condition for Walrasian equilibrium to exist in is the absence of complementarities in ?^{r} compared to ?. I.e., the per capita gains consumers and producers receive in ?^{r} should also be achievable in ? because an equilibrium for ?, when replicated at the same prices, is necessarily an efficient allocation for ?^{r}. With quasilinearity, this necessary condition can be stated as:
202§16. (Complementarities under Replication Precludes Existence of Walrasian Equilibrium). If there is an integer r such that G(?^{r} ) > rG(?) , then Walrasian Equilibrium does not exist in ?.
203The primary source of counterexamples to the existence of Walrasian equilibrium is nonconvexity, a source of complementarity. When nonconvexity precludes the existence of Walrasian equilibrium in ?, it follows that for some r > 1, there are feasible allocations (states of the economy) in ?^{r} such that percapita gains to each type are at least as large, and for some type(s) larger, than are achievable in ?. Nevertheless, if nonconvexities are “bounded,” the complementarities associated with nonconvexity disappear with sufficient replication: for any ? > 0, there exists an such that if ,
205i.e., per capita complementarities are small when the number of replicas is large.
206When large numbers is achieved via replication, inherent substitution possibilities typically, but not always, lead to existence of perfectly competitive equilibrium. We illustrate in a simple, but representative, case. Suppose ? = { v_{b}, c_{s} } consists of a single buyer and a single seller exchanging a divisible commodity. Let [z;p] be a Walrasian equilibrium for ?, i.e.,
208Efficiency of Walrasian equilibrium implies
210i.e., a Walrasian equilibrium represents a particular division of the maximum total gains.
211Replica invariance of Walrasian equilibrium implies
213A feasible allocation for ?^{r}_{–b}, the rreplica without b, is to reduce the quantity supplied by each s to z – z/r, enabling each of the (r – 1) remaining buyers to consume z. Therefore,
215The last inequality follows from
217Since v^{*}_{b}(p) = v_{b}(z) – pz, MP_{b}(?^{r}) ? v^{*}_{b}(p) extends 11 to the case of divisible commodities.
218Because
220the extra revenue each seller receives from supplying the last of r equalsized increments of z/r yields an return that is at least as large as the extra cost. If the inequality is strict, the seller adds to his profit by selling z rather than z – z/r. Therefore, b’s purchases at price p are contributing a positive surplus to each of the sellers. If pz/r = c_{s} ( z ) – c_{s}(z – z/r ), then the profit each seller makes from supplying z or z – z/r is the same—hence, the seller is indifferent. In that case, getting z while paying pz, buyer b is paying the social opportunity cost of his purchase—the amount such that the members of ?^{r}_{–b} are just indifferent between supplying b with z or nothing. When there is equality, p is a Walrasian equilibrium price for ?^{r}_{–b}, i.e., b is a perfect competitor, i.e.,
222A feasible allocation for ?^{r}_{–r}, the rreplica without s, is to increase the quantity supplied by each of the remaining r – 1 sellers to z + z/(r – 1), enabling each of the r buyers to consume z. Therefore,
224The last inequality follows from:
226Similarly, because
228the extra cost of supplying z/(r – 1) is at least as large as the extra payment at price p. When the inequality is strict, the fact that s receives pz for contributing z in ?^{r} means that he is contributing a positive surplus to the members of ?^{r}_{–b} since they could not do as well on their own. If, however, c_{s}(z + z/(r – 1)) – c_{s}(z) = pz/(r – 1), sellers are again indifferent between supplying z or z + z/(r – 1) at the price p and
230Therefore,
231§17. If [z;p] is a Walrasian equilibrium for ? and the inequalities in ( i ) and ( ii ) are satisfied as equalities for some r, then Walrasian equilibrium [z;p] in ?^{r} exhibits full appropriation, i.e., MP_{i}(?^{r}) = v^{*}_{i}(p), i = b, s, and
233The directional derivative of c_{s} at z in the direction –z is
235Since the above inequalities (i) and (ii) hold for all r,
237If c_{s} is differentiable at z, then pz = – Dc_{s}(z; – z) = Dc_{s}(z; z).
238Analogous arguments focusing on adjustments in the allocations to buyers rather than sellers yields,
239§18. If [z;p] is a Walrasian equilibrium for ? = {v_{b}, c_{s}} and either v_{b} or c_{s} is differentiable at z, then , and
241Provided that either v_{b} or c_{s} is differentiable at the allocation z, the discrepancy between an individual’s marginal product and the reward received under Walrasian equilibrium vanishes as the number of replicas increases. To illustrate the need for differentiability, consider the following:
Let v_{b}(z) = min{z, (z + 1) / 2} and c_{s}(z) = max{0,2(z – 1)}, z ? 0; hence, v_{b} is a polyhedral concave function and c_{s} is a polyhedral convex function, both of which fail to be differentiable at z = 1. Moreover, Dv_{b} (1; – 1) = – 1 and Dc_{s} (1; – 1) = 0. It is readily verified for this ? = { v_{b}, c_{s} } that [1;p] is a Walrasian equilibrium for any p ? [½,1]; therefore, replicating [1;p] yields a Walrasian equilibrium for ?^{r}. Evidently, there is no p such that – Dv_{b}(1; – 1) = Dc_{s}(1; – 1).
242The indeterminacy of prices caused by the absence of both functions to be differentiable at z = 1 means that there is indeterminacy in the distribution of the gains from trade, v^{*}_{b}(p) + v^{*}_{s}(p) = v_{b}(1) – c_{s}(1) = 1, between buyers and sellers. This provides an opportunity for manipulation. For example, suppose p = ½. If one seller were to “change” his costs by reporting , the economy consisting of r buyers, r – 1 sellers of the original type and one seller would yield the unique Walrasian equilibrium price p = 1, increasing the seller’s profits from pz = (½) 1 to (1).9. This conclusion holds for all r. The failure of differentiability among all buyers and sellers at the Walrasian allocation allows for the possibility that even with large numbers, small changes can have nonnegligible implications for prices.
6 – Perfectly competitive versus Walrasian Equilibrium
6.1 – Walras shortcut
243Counterexamples such as Examples 1 and 2 notwithstanding, the results in 9, 15. and 18. are representative of a more general conclusion:
245This would seem to confirm the validity of Walras’ pricetaking shortcut as abstracting the essence of the competitive theory of value: even though Walrasian equilibrium for a model with a finite number of individuals, call it ?—such as in Section 3 or the finite numbers models in the previous Section—does not fulfill the conditions of perfect competition, nevertheless if that model were replicated as ?^{r} a sufficient number of times, competition would typically be nearly perfect? and since a Walrasian equilibrium of ?^{r} repeats the equilibrium of ? on which it is based, little is lost by confining attention to ?.
246We disagree. Because Walras’ characterization can be formulated without the necessary competitive accompaniments, it puts too much emphasis on valuation and not enough on appropriation. We call attention to these contrasting perspectives on the meaning of competitive equilibrium with the aid of a familiar diagram.
Walrasian and Perfectly Competitive Equilibrium
Walrasian and Perfectly Competitive Equilibrium
247The intersection of aggregate market demand and supply is highlighted by the large dot. Figure 1 also includes a magnification of the dot to illustrate the situation from the perspective of an individual participant when the market is perfectly competitive. In the magnification, aggregate demand and supply lies beyond the individual’s field of vision: the horizontal line through the dot depicts the individual as facing perfectly elastic demand or supply, characterized by the equality?MP_{i}(?) =G(?) describing full appropriation.
248Walras’ pricetaking shortcut makes the magnification an optional, hence unnecessary, component of the analysis. It is therefore a weakening, or “generalization,” of the conditions required for perfectly competitive equilibrium. As emphasized above, the perfunctory treatment is implicitly rationalized by the neoclassical focus on valuation. The following remarks explore differences between the Walrasian depiction of competitive equilibrium versus the more restrictive conditions underlying the magnification.
249I. Equilibrium: coordination versus substitution
250Focus on valuation emphasizes that equilibrium is primarily a problem of coordination. In the Walrasian tradition as presented in Section 3, the intersection is found by first constructing the quantities buyers and sellers wish to trade at various prices and then finding a price at which these plans would be consistent. Equilibrium is conceived as the problem of finding prices such that:
252The pricetaking formulation of perfect competition is more than a convenient simplification. It is taken literally. “Each individual participant in the economy is supposed to take prices as given and determine his choices as to purchases and sales accordingly? there is no one whose job it is to make a decision on price.” [Arrow, 1959: 43] While exogeneous price adjustment can be regarded as a heuristic that finesses the difficult problems of how equilibrium is attained, it also reinforces the view that competitors can be relieved of responsibility for taking initiative beyond reacting to prices.
253From the appropriation perspective, some of the spotlight on the coordination role of prices should be redirected to the reasons why those prices are competitive? namely, the existence of substitution possibilities among individuals, a feature that is missing in the pricetaking description, above. For example, when those substitution possibilities are introduced, equilibrium under perfect competition can be regarded as the elimination of arbitrage opportunities rather than the elimination of excess demands. An arbitrage approach leads directly to description of equilibrium exhibited in the magnified portion of Figure 1 that bypasses pricetaking demand and supply. (See Makowski and Ostroy [1998].) Note that when commodities are indivisible and heterogeneous, e.g., houses, aggregate demand and supply for a standardized commodity such as housing is a fiction. Competition is based on substitutability among individual (owners of) houses.
254II. Value versus distribution
255Focus on valuation emphasizes that the prices of all goods, inputs and outputs, determine individual rewards. Schematically, pricetaking highlights the fact that
257This leads to the ambiguities between market pricing and market socialism described in Sections 3.1 and 3.2. The appropriation view of the price system argues that it should be the other way around: competition among individuals over rewards determines the prices of goods and services. The summary statement as to why competition leads to efficient price determination is that a perfectly competitive environment allows individuals to fully appropriate what they can contribute:
259The reason imperfect competition leads to inefficiency is that these goals are mutually incompatible:
261Thus, instead of characterizing the inefficiency of monopoly compared to the efficiency of perfect competition in marginalist terms as the difference between choices equating marginal revenue to marginal cost compared to choices equating price to marginal cost, the lesson emphasized here is that full appropriation leads to efficiency because it removes the wedge between the reward an individual receives and his social contribution—not only on the marginal unit supplied, but on all of the inframarginal units as well. Conversely, when the seller cannot fully appropriate his contribution he will undersupply. Failure of the Walrasian tradition to formally incorporate the connections between competition and appropriation is the reason mechanism design has been credited as the origin of incentive compatibility.
262III. Marginal analysis: intensive versus extensive
263Commenting on the theory of rent, Jevons [1879] remarked: “It has not been usual to state this theory in mathematical symbols, and clumsy arithmetical illustrations have been employed instead? but it is easy to show that the fluxional calculus is the branch of mathematics which most correctly applies to the subject.” In more recent terminology, Jevons recognized that there was a connection between the extensive margin applied to (indivisible) land parcels and the intensive margin that is the mathematical locus of the marginal revolution, i.e., the characterization of pricetaking maximization in terms of firstorder conditions with respect to the commodity margin which determines individual demands and supply responses, hence the determinants of Walrasian equilibrium. [12]
264The indivisible individual is the margin of analysis for appropriation. Hence, in addition to prices for units of a divisible labor commodity, i.e., the intensive margin, contracts for individuals fits this framework seamlessly. Reasons why the relevant margin is extensive rather than intensive are attributed by Makowski [1979] to “personalized nonconvexity.” Whether intensive or extensive, the condition describing perfect competition is full appropriation with respect to individuals. Full appropriation is a differentiability condition with respect to individuals. When the number of individuals is finite, perfect competition implies “discrete differentiability,” a condition that is difficult, but not impossible (see 8., 14., and 17.), to achieve. With large numbers, differentiability with respect to individuals is more readily, but not universally (see Example 2), attained via the predominance of substitution possibilities among commodities. The following Section illustrates how the presence of complementarities systemically precludes differentiability with respect to individuals.
6.2 – Economy?wide complementarities
265The demonstrations in §9., §15., and §18. that large numbers ? perfect competition owe their validity to the fact that substitutability predominates as the population grows. But large numbers is compatible with environments where complementarities among individuals are not overwhelmed by substitution possibilities. And there are good reasons to believe that this is a description of our economic environment.
266To illustrate, consider the following example with heterogeneous divisible commodities allowing each buyer to gain from each seller.
267The utility of i in ?^{n} is
269where
271The resource endowment of i in ?^{n} is
273The number of individuals determines the number of goods available for consumption. Interpret the fact that endowments of personalized commodities are increasing with population size as the result of unmodelled specialization allowing each person to become more productive at supplying “his” commodity. Equally important is the feature that consumers’ tastes for increasing variety is a source of neverending gains. Otherwise, if there were only a finite number of commodities, increases in productivity would result in limited gains since utility is satiated when individual consumption of any commodity reaches 3/2. [13]
274Symmetry implies that total gains are maximized when each individual has an equal share of existing commodities. In ?^{n}, therefore, z_{ic} = n^{–1} for all i,c = 1,…,n. The parameters are chosen so that p^{n} = (1,…,1;3,…, 3,…) are the Walrasian prices for ?^{n}, where (1,…,1) ? ?^{n} are the prices of those commodities available in ?^{n}; and, the prices of commodities not available are set equal to their reservation values, 3, the minimum price at which demand would be zero. At prices p^{n}, utilitymaximizing demand by each consumer for each available commodity in ?^{n} is 1, establishing that demand equals supply. Therefore, since v_{c}(1) = 3 × 1 – 1^{2} = 2 and v^{*}_{i}(p^{n}) = n × [2 – 1] = n,
276Hence, per capita gains G(?^{n})/n = n ( = per capita wealth) are increasing with the size of the population. Nevertheless, each individual is becoming small relative to the size of the economy in the sense that per capita gains/wealth in ?^{n} are the fraction 1/n of total gains/wealth. Unlike §16., however, increasing per capita gains are not due to replication, allowing this version of complementarity to be compatible with Walrasian equilibrium.
277Denote the absence of i in ?^{n} by ?^{n}_{–i}. Walrasian prices for ?^{n}_{–i}, denoted p^{n}_{–i}, are obtained from p^{n} by replacing the price of the i^{th} commodity by its reservation price, 3. Hence,
279In contrast to the substitution possibilities built into replication, here the predominance of complementarities among individuals precludes full appropriation as the population increases. Specifically the per capita discrepancy
281is the consumers’ surplus that each individual contributes to the (n – 1) others when each is able to buy 1 unit at a cost 1, which is increasing with n. The absence of full appropriation is reflected in the unexploited monopoly power of individuals in Walrasian equilibrium however large the population may be. If an individual were to withhold approximately 1/4 of his total supply so that is the supply available to others for c = i after i retains 3/2 units for himself, the Walrasian equilibrium price for the remainder would yield a price of 3/2. Therefore, i could obtain a total revenue of
283which exceeds the revenue obtained from selling n – 1 units at a price of 1.
284This example juxtaposes two themes of Adam Smith:
285(A) specialization and division of labor are the source of gains in welfare, limited by the extent of the market;
286(B) competitive private enterprise is the way to achieve those gains.
287We interpret (A) to mean (a) that economywide complementarities with respect to gains from trade persist as the size of the market (population) grows. The example challenges the claim that (B) can be represented as (b) full appropriation. In our view, the tension between (A) as (a) and the idealization of (B) as (b) does not negate the validity of either. Rather, it shows that large numbers does not necessarily resolve appropriation problems.
6.3 – Ordinal utility
288“Marginal product” measures the difference between comparable magnitudes. Because the analysis, above, relies on quasilinearity, individual utilities can be added and subtracted and an individual’s marginal product can be defined. From the onset of the marginal revolution, however, it has been a wellaccepted tenet that utility is not interpersonally comparable, hence differences in utility are not comparable. Moreover, modern statements of the theory of value, such as in Section 3, replace individual utility by a preference ordering. Since these conditions imply that total utility is both meaningless and unnecessary, the above conclusions might seem to be of limited application.
289In fact, the key idea behind appropriability is an ordinal extension of the notion of marginal product that is based on indifference. And the above concepts were originally formulated without reference to utility functions [Ostroy 1980? Makowski, 1980]. To illustrate, recall that the notion of an individual’s marginal product was introduced by appeal to a perfect price discriminator. Define a perfect discriminator in a nonquasilinear, i.e., ordinal utility, setting as an individual who extracts all the surplus he contributes, leaving others indifferent between what they can achieve on their own and what they can achieve when the individual participates— called an allocation in which each individual contributes no surplus to others. Just as Vickrey found for the divisible commodity model with quasilinearity and small numbers of buyers and sellers that Walrasian equilibrium yields each participant a share of the total gains that is less than their marginal product, an analogous result holds in the ordinal utility setting: the reward individual i receives in an ordinal utility model of Walrasian equilibrium does not fully capture the surplus that i contributes to others because if the others did not trade with i, they (typically) would not be able to obtain those (ordinal) levels of satisfaction on their own. Similarly, the relation between full appropriation and perfectly competitive equilibrium when there is quasilinearity is replaced by the analogous condition in the ordinal model that i contributes neither a positive nor a negative surplus to others: at perfectly competitive prices, i extracts all the surplus he contributes, leaving others indifferent. Although it is a special case, key results concerning appropriation obtained under quasilinearity— such its relation to incentive compatibility and efficiency—can be extended to ordinal utility [Makowski, Ostroy and Segal, 1999]. There is a strong family resemblance between perfect competition with and without quasilinearity.
Notes

[1]
Department of Economics, UC Davis (1makowski@ucdavis.edu)

[2]
Department of Economics, UCLA (ostroy@ucla.edu)

[3]
In Debreu’s treatment, consumption takes place in some subset of R^{l} denoted by X_{i}.

[4]
This conclusion only requires a local nonsatiation assumption on preferences.

[5]
In the absence of active competition, it is only a rough measure.

[6]
This conclusion requires convexity assumptions on preferences and production possibilities.

[7]
Nevertheless, the cooperative theory of games, and particularly the core are relevant. Edgeworth did not introduce his notion of bargaining as an end in itself. Rather, he employed it to demonstrate the conditions under which it would yield determinacy. In contemporary terminology, determinacy means that (1) the core coincides with Walrasian equilibrium and (2) Walrasian equilibrium is (locally) unique. [His masterservant illustration is an example of Walrasian equilibria that fails to satisfy (2).] The conditions leading to determinacy and those leading to full appropriation nearly coincide.

[8]
See Makowski and Ostroy [1993].

[9]
In contradistinction to Hayek’s emphasis on informational economy, a direct mechanism is profligate since it requires individuals to communicate the data defining their characteristics. This does not preclude the possible existence of a more informationally economical indirect mechanism, which might implement the direct mechanism.

[10]
See Gretsky, Ostroy and Zame [1999] and Shapley and Shubik [1972].

[11]
Conditions (I) and (II) are sufficient; they are not necessary. Edgeworth pointed out—in his MasterServant example—that condition (I), by itself, is not sufficient: consider v_{b} = 1 all b, cs = 0 all s, and the number of buyers equals the number of sellers, n. Then, for every n, any p ? [0, 1] is Walrasian equilibrium price.

[12]
Reflecting advances in the mathematics of optimization, modern presentations of the theory of value such as in Section 3 dispense with the differential calculus. These advances show that the equalities defining the linearity of directional derivatives, i.e., differentiability, can be replaced by inequalities, as in Example 2.

[13]
It is important that there are not only an unbounded number of commodities, but they are truly heterogeneous, i.e., the utility gains cannot be closely approximated by any finite number of commodities, as they could be if the commodities were highly substitutable.