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 non-human 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 non-human 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, real-world 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 = ?_ir_i represents the transfer of alienable resources to producers. Producer j’s objective is now

49

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.,

51

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

54

55where ?_i ? 0, ?_i?_i = 1 replaces profit shares of producers. At prices p, total alienable wealth in ?^M is given by

56

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

58

59Proceeding along the same lines, the method of assigning profit shares in E can be generalized to all wealth, as in Gale and Mas-Colell [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.

4.1 – Transactions costs

70Coase [1937] reasoned that except for real-world 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 self-evident, 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 agreed-upon 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 well-suited 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 neo-classical 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 well-defined 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.

77

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 well-defined.

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 u⁰, 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]

5.1 – Different kinds of asymmetric information

5.2 – Appropriation in Mechanism Design with Privacy

93The starting point for Vickrey [1961]:

94

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. Price-taking 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 neo-classical 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 infra-marginal 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 Lange-Lerner 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 infra-marginal 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 second-highest buyer’s valuations. These observations imply the following:

102§2. The second-price 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 second-price 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 “non-money” 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:

110

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 single-object auction in which the seller, s, has zero reservation value and the buyer’s reservation values are listed in descending order as v₁ ? v₂ ? … ? v_b ? … ? v_n ? 0,

116

117When the buyer with valuation v₁ pays the amount v₂ 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.

119

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 small-numbers 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 single-object auction, when the price equals the second-highest valuation, that would also be a Walrasian price without the highest-valued buyer. Therefore,

121§6. The second-price auction for a single object makes the winning buyer a perfect competitor. Rewards are distributed as,

122

123The economy’s reservation price for the object is the lowest price at which demand would be zero, which is evidently v₁. If the price of the object is v₁, it is the same as the price without the seller. Therefore,

124§7. The first-price auction for a single object makes the seller a perfect competitor.

125

126The sum of the marginal products of the buyers and the seller is

127

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 single-object 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 (v₁ – c) ? 0 so that total gains are positive, the above inequality continues to hold for the single-object market:

130

131Note that if the seller is promised the payment (v₁ – 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 single-object market is perfectly competitive when the price would be the same with or without any participant.

134§8. Assuming v₁ – c > 0, the single object market is perfectly competitive if and only if v₁ = v₂. 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:

135

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 v₁, the seller’s reward is a maximum when the single-object market is perfectly competitive market.

138Walrasian equilibrium always exists in a single-object market, but not necessarily perfect competition. Let Vⁿ be the set of those ( v₁,…, v_n ) arranged in descending order with the stipulation that v_b ? [0,1], b = 1,…,n; and let Vⁿ_e (PC) be the subset of Vⁿ with the added restriction that v₁ – v₂ ? ?, the set of single object markets that are within e of being perfectly competitive. For any n and any ? < 1, (1, 0, 0,…, 0) ? VⁿVⁿ_?(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 second-highest order statistic implies:

139§9. Existence of perfect competition in a single object market. For any ? > 0,

140

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

144

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 single-object auction.

146Denote the data of the assignment model by A = (V,c), where V = ( v_bs ) and c = ( c₁,…, 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 = ( p₁,…, 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, price-taking maximization yields b and s the net gains

147

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 utility-maximizing at the prices p)
• (each seller’s action is profit-maximizing at p)
• (markets clear).

Taking advantage of the special properties of the assignment model, we adopt a more convenient description. Because there are as many buyers as sellers, a feasible allocation of resources, or state of the economy, can be described by a permutation , where ?(b) = s indicates that farmer b is assigned to the parcel owned by seller s, with the option of whether or not to cultivate it. For any p and ?,

150

151This inequality allows for a more succinct description of equilibrium: Definition: [Revised Description] [?, p] is a Walrasian equilibrium for A if

152

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 “market-clearing” 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, price-taking 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),

155

156The maximum total gains in is A is

157

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,

161

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

165

166Comparison between the gain an individual receives in Walrasian equilibrium and his marginal product is similar to the single-object auction:

167§11. If [?;p] is a Walrasian equilibrium for A,

168

169Applying §10. and §11.,

170§12. In A,

171

172Definition: A exhibits full appropriation if

173

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 = ( ^bp₁,… 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,

176

177and there exist prices p^s = ( p^s₁,… 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]

178

179It follows from 10. and 13. that if [?, p] is a Walrasian equilibrium, ?(b) = s, and trade occurs, then

180

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_sis the outside option for the seller and p^s_sis 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

185

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,

190

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 double-auction.

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,

196

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:

198

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 non-convexity, 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 per-capita gains to each type are at least as large, and for some type(s) larger, than are achievable in ?. Nevertheless, if non-convexities are “bounded,” the complementarities associated with non-convexity disappear with sufficient replication: for any ? > 0, there exists an such that if ,

204

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.,

207

208Efficiency of Walrasian equilibrium implies

209

210i.e., a Walrasian equilibrium represents a particular division of the maximum total gains.

211Replica invariance of Walrasian equilibrium implies

212

213A feasible allocation for ?^r_–b, the r-replica 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,

214

215The last inequality follows from

216

217Since v^*_b(p) = v_b(z) – pz, MP_b(?^r) ? v^*_b(p) extends 11 to the case of divisible commodities.

218Because

219

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.,

221

222A feasible allocation for ?^r_–r, the r-replica 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,

223

224The last inequality follows from:

225

226Similarly, because

227

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

229

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

232

233The directional derivative of c_s at z in the direction –z is

234

235Since the above inequalities (i) and (ii) hold for all r,

236

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

240

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:

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 non-negligible implications for prices.

6.1 – Walras short-cut

243Counterexamples such as Examples 1 and 2 notwithstanding, the results in 9, 15. and 18. are representative of a more general conclusion:

244

245This would seem to confirm the validity of Walras’ price-taking short-cut 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.

Figure 1

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’ price-taking short-cut 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:

251

252The price-taking 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 price-taking 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 price-taking 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, price-taking highlights the fact that

256

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:

258

259The reason imperfect competition leads to inefficiency is that these goals are mutually incompatible:

260

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 infra-marginal 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 price-taking maximization in terms of first-order 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

268

269where

270

271The resource endowment of i in ?ⁿ is

272

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 never-ending 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 pⁿ = (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 pⁿ, utility-maximizing demand by each consumer for each available commodity in ?ⁿ is 1, establishing that demand equals supply. Therefore, since v_c(1) = 3 × 1 – 1² = 2 and v^*_i(pⁿ) = n × [2 – 1] = n,

275

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 pⁿ_–i, are obtained from pⁿ by replacing the price of the i^th commodity by its reservation price, 3. Hence,

278

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

280

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

282

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 economy-wide 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 well-accepted 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 non-quasilinear, 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.